JAN S 191 i GIFT LIBRARY OF THE UNIVERSITY OF CALIFORNIA. GIFT OF g 4 g 6 A TEXT-BOOK OF PHYSICS " BY S. E. COLEMAN, S.B., A.M. HEAD OF THE SCIENCE DEPARTMENT AND TEACHER OF PHYSICS IN THE OAKLAND, CALIFORNIA, HIGH SCHOOL; AUTHOR OF "A PHYSICAL LABORATORY MANUAL," "NEW LABORATORY MANUAL OF PHYSICS," AND "THE ELEMENTS OF PHYSICS" D. C. HEATH & COMPANY BOSTON NEW YORK CHICAGO COPYRIGHT, 1911, BY D. C. HEATH & Co. PREFACE THE present tendency in physics teaching is to attach less importance to the formal and academic features of the subject, and to lay greater stress on the applications of physics in daily life. This change of front is in accord with the general movement in education which seeks to give the subjects of instruction a more useful content, drawn from the social and industrial life of the time. Judged by this standard, the kind of physics which views the util- ities from afar or ignores them altogether is discredited. But a word of caution is clearly in order. A new truth is never the whole truth. The older physics had much to commend it, despite the caustic criticisms of zealous reformers; for "the most practical thing in the world is the foundation of pure science upon which applied sci- ence rests." To eliminate or minimize the fundamentals while attempting to teach their applications is not to pro- vide a "royal road to learning," but a fool's highway to pretentious ignorance. Pure and applied science are equally essential to a well rounded course. The one alone is barren; the other, when not well founded in the first, is superficial, disconnected, and trivial. A first course in physics should compass results which are in themselves worth while. It need not on that account be any the less valuable as a preparation for college; but a course which recognizes only the latter goal is not likely to come within hailing distance of any other. The subject matter should be drawn largely from the common surround- iii 227432 iv PREFACE ings and affairs of life, about which the pupil already knows something and about which it concerns and interests him to know more. Scientific education should begin with and develop out; of the science of common things. Genuine knowledge is not gained by the contemplation of laws and principles in the abstract. A general truth can be seen only through the medium of particular instances. It is given the immature student to see the great generali- zations of science only in part, at the best; but the partial view may be clear and vivid, if adequately grounded in experience. The premature introduction of Newton's laws of motion, presented as physical axioms (!) and buttressed mainly by formulas and problems thrice removed from the pupil's experience, has only served to bring the subject of dynamics into disrepute. The principle of the conserva- tion of energy, as traditionally presented, is a further ex- ample of a great generalization reduced to empty verbiage a sort of word puzzle, which claims the attention for a moment and is then dismissed for good and all. Hence it is that elementary physics has been only a loose aggrega- tion of subjects, having little apparent relation to one another; for the essential unity of the science is to be found only in the idea of energy and its conservation in all physi- cal phenomena. "The doctrine of energy plays in physical science the same role as does the doctrine of evolution in biological science, since it furnishes concepts and a termi- nology in which all forms of physical phenomena may be expressed. This terminology and these concepts are partic- ularly useful, because they are derived from the idea of mechanical work, which is one of the most immediate and familiar of the concepts drawn from daily experience." Mechanical principles in general run through the whole of physics, and serve as the necessary basis for its rational presentation. While it does not follow from this that all PREFACE V of mechanics must come first, the author is strongly of the opinion that this is the best plan. It is true that many topics in mechanics and heat are so closely related that they may, with some show of reason, be thrown together. But such an arrangement, if it aims at correlation (and it appar- ently has no other warrant), defeats its own purpose; for the several parts of mechanics stand in a more intimate and vital relation to one another than they do to the topics in heat or any other branch of physics. Owing to the diversity which exists between physics courses planned under different conditions, the scope of a text-book written for general use is necessarily a compro- mise. If it is restricted to the "essentials" which form the common ground of all courses, it must be largely supple- mented as local conditions may require. If its scope is broadened to include a more complete survey of phenomena and principles and a great variety of industrial applications, it will contain more than should be attempted with most classes, and the problem of the teacher will be to select that which best suits his purpose. The latter type of book is to be preferred for several reasons. In the first place it is difficult to make effective use of reference books for sup- plementary work, especially with large classes; and, at best, this is a time-consuming expedient. On the other hand, supplementary material incorporated in the text itself is ready at hand when wanted, and its presence invites atten- tion and stimulates interest. // is expected that the teacher, in using this book, will omit considerable portions at his discretion, and will touch lightly on other portions which, for his purpose, are to be regarded as of minor importance. To cover the entire course in detail would require a year and a half with most classes. The subject matter has been arranged and presented with due regard to its correlation with the laboratory course. vi PREFACE If the laboratory experiments are to have any particular value, they must fit into the general scheme of the text, just as the class-room experiments do. This correlation has been worked out in connection with the author's New Laboratory Manual of Physics, published by the American Book Company. This preface would be unduly lengthened were any attempt made to review the plan of the book in detail or the treatment of the different subjects. On such matters it may be assumed that the book will speak intelligibly for itself. CONTENTS PAGE CHAPTER I. INTRODUCTION . I CHAPTER II. MATTER AND FORCE. PHYSICAL MEASUREMENTS I. The Three States of Matter 7 II. Force and Inertia 9 III. Physical Measurements 15 CHAPTER III. STATICS or LIQUIDS I. Introduction . . ... . . 22 II. Gravity Pressure in Liquids 24 III. Transmission of Applied Pressure by Liquids .... 33 IV. Buoyancy of Liquids 36 V. Specific Gravity 39 CHAPTER IV. STATICS OF GASES I. Atmospheric Pressure 43 II. Laws of Gases 52 III. Applications of the Mechanics of Fluids 61 CHAPTER V. STATICS or SOLIDS I. Concurrent Forces 69 II. Parallel Forces 78 III. Moments of Force 80 IV. Effect of Weight on the Equilibrium of Bodies .... 84 V. Elasticity. Stresses and Strains 92 CHAPTER VI. DYNAMICS I. Motion 104 II. Newton's Laws of Motion 120 III. The Laws of Motion in Special Cases 135 IV. Work and Kinetic Energy 150 V. Machines 159 VI. Energy 176 VII. Dynamics of Fluids 183 vii viii PHYSICS PAGE CHAPTER VII. THE MOLECULAR THEORY OF MATTER I. The Structure of Matter 196 II. Molecular Properties of Gases 202 III. Molecular Properties of Liquids 207 IV. Molecular Properties of Solids . . 214 CHAPTER VIII. HEAT I. Nature of Heat 218 II. Temperature. 220 III. Conduction and Convection 225 IV. Radiation 230 V. Changes in Volume and Pressure 240 VI. Measurement of Heat. Specific Heat 248 VII. Fusion and Solidification 252 VIII. Vaporization and Condensation 260 IX. Heating and Ventilation of Buildings 283 X. Heat and Other Forms of Energy 287 XI. Heat Engines 292 CHAPTER IX. SOUND I. Origin and Transmission of Sound . 306 II. Properties of Musical Sounds 322 III. Sympathetic and Forced Vibrations. Resonance . . . 342 CHAPTER X. LIGHT I. Nature and Transmission of Light 359 II. Intensity of Illumination. Candle Power 368 III. Reflection of Light . . . . . . . ... . 372 IV. Refraction of Light . . ... . . . .. *. . 389 V. Lenses 405 VI. The Eye ;........ 416 VII. Optical Instruments 425 VIII. Dispersion of Light. Color 437 CHAPTER XL MAGNETISM I. Properties of Magnets 457 II. The Magnetic Field . . . . . . ... . . 466 III. The Earth's Magnetic Field 472 PHYSICS ix PAGE CHAPTER XII. ELECTROSTATICS 477 CHAPTER XIII. ELECTRODYNAMICS I. Introduction 501 II. Primary Cells : 503 III. The Magnetic Action of a Current 517 IV. Measurement of Electric Currents 531 V. Ohm's Law 537 VI. Laws of Resistence 541 VII. Measurement of Resistance and Electromotive Force . . 545 VIII. Electrical Energy. Heating Effects of Electric Currents . 553 IX. Elctromagnetic Induction 564 X. Chemical Effects of the Electric Current 600 CHAPTER XIV. RADIATIONS. THE ELECTRICAL NATURE OF MATTER I. Spectra and Spectrum Analysis 610 II. Electric Oscillations and Waves. Electromagnetic Theory of-Light 617 III. Electric Conduction through Gases. Cathode and Rontgen Rays 622 IV. Radioactivity. Electrical Theory of Matter .... 630 APPENDIX 638 INDEX 641 A TEXT-BOOK OF PHYSICS A TEXT-BOOK OF PHYSICS CHAPTER I INTRODUCTION 1. Scientific Education. From infancy a child is busy acquiring facts concerning the strange world in which he finds himself. It matters little what may chance to turn up, he wants to scrape an acquaintance with it, whether it is a bug or a circus elephant, a jumping-jack or a steam- engine. So earnest is he about this business that he be- sieges his elders with questions, and fairly earns the title of the human interrogation-point. He absorbs miscel- laneous information as naturally as a sponge absorbs water. In his earlier years the boy's interest is awakened only by the things that appeal directly to his senses. He is intent upon seeing, feeling, hearing, and tasting. He wants to know what this or that thing is called, what it does, and what it is for. In all this the child is taking the first step toward scien- tific knowledge. The first step does not carry him far, to be sure, but it is a necessary step. The boy is gathering the raw materials of science, not with any thought of their possible future value, but just because he wants to know. During these early years the collection of 'facts stored up in his small brain resembles a scrap-heap of odds and ends rather than a well kept museum. By and by he begins 2 "... v A'. HYSICS * to fat ; his' >pellecjual -house -in order. This is the second step in his scientific education, and, like the first, it is taken without conscious purpose. Some day, perhaps, it occurs to him that the flight of his kite is not altogether unlike the flight of a bird, and he begins to compare their behavior. Both can remain in the air indefinitely, although objects in general fall to the ground unless they have some visible support. Light objects are borne about by the wind and remain in the air for some time, and the kite also requires a wind; but a bird can fly in still air, although its body may be large and heavy. A bird evidently keeps from fall- ing and maintains its flight by flapping its wings; but how does the motion of the wings bring about this result? Here is the puzzle, and the boy pokes about meditatively among his scrap-heap of facts for something to match it. Ah, here it is! Flying must be something like swimming. Swimming is a part of the boy's personal experience, and he knows that to drive his body forward he must push back on the water with his hands and feet. Then it must be that the swift downward stroke of the bird's wings against the air gives its body an upward impulse which keeps it from falling. In some way, probably due to the shape of the wings, the impulse is partly forward as well as upward; otherwise there would be no forward motion. Speculating thus about flying, the boy recalls that he has seen a balloon rise in the air without the aid of a wind and without wings to flap. Evidently it is neither like a kite nor a bird. Here, then, is a new problem: To find out what makes a balloon rise. By the time a boy reaches the age of twelve or fourteen years his head is full of such problems. No sooner is one solved to his satisfaction than others come to take its place. He has outgrown the chance world of his earlier INTRODUCTION 3 years, in which the things that happen just happen, and is beginning to realize that there is a connected scheme of things, or plan, in which everything that happens has its proper place. His interest is now centered in the how and the why of things. How does the engine work? How does the phonograph talk, or the telephone? Why do the hills mock him with their echo? How does the lens of his camera make the picture on the film? Why does his gun "kick" when he fires a shot? What keeps the people on the under side of the earth from falling off? What makes the moon change from crescent to quarter and from quar- ter to full moon every month? What makes the winds? And so on without end. By dint of thinking and asking questions the boy arrives at some sort of answer to his problems. Thus in a rather aimless fashion he is engaged in sorting over the facts of his experience, comparing, classifying, and drawing gen- eral conclusions which may be of use to him. As a rule, however, his conclusions are vague and inaccurate, for his acquaintance with facts is limited and he is not a trained thinker. He has made a good start in his scientific educa- tion, and has made it in the natural way; but this is about as far as the haphazard experiences and interests of daily life will carry him. The next step in advance demands systematic, purposeful effort, directed toward a definite end. The opportunity for such effort is afforded by the science courses of school and college. In the pursuit of any branch of science the student's incomplete and frag- mentary acquaintance with the main facts of the science is pieced out by observation and experiment in the labora- tory and the class-room, under the guidance of the teacher. Having this substantial acquaintance with a wide range of facts at first hand the student will be able to make effect- 4 PHYSICS ive use of a text-book in gaining a more comprehensive knowledge of the subject than would be possible through his personal observation and experience alone. The teacher, the laboratory, and the text-book are all essential to the best results. 2. Physics and its Place among the Sciences. The knowledge of the material universe is subdivided, for con- venience, into several branches, called the natural sciences. The biological sciences treat of living things; the physical sciences deal with inanimate matter in all its forms, and with the changes and processes which it undergoes. The physical sciences are physics, chemistry, astronomy, geology, meteorology, and mineralogy. Physics is the broadest of the natural sciences and shares with chemistry the honor of being the foundation of all the others. All changes, occurrences, or processes which take place in the material world are of either a physical or a chemical nature. Any action or process in which matter changes from one kind to another is a chemical process. Combustion or burning is the most familiar example. The substance burned unites with oxygen from the air, forming certain gases which pass off into the air. The rusting of iron and the decay of animal and vegetable matter are further examples. A physical process is one that does not involve a change of matter from one kind to another; e.g. the melting of ice and the evaporation of. water. Water is the same substance whether it exists as ice, liquid water, or water vapor. A change from one to the other is a change of state, or of physical condition. It is this distinction between chemical and physical change which serves as the dividing line between chemistry and physics. INTRODUCTION 5 The five great departments of physics are Mechanics, Heat, Sound, Light, and Electricity, the last including the closely related subject of Magnetism. Mechanics treats of the action of forces in determining the state of rest or motion of bodies. It presents the fundamental principles which are applied in the construction and use of machines. A boy in learning the control of his own body in walking, running, jumping, balancing, alighting from a moving car, riding a bicycle, etc., is gaining experience in mechanical matters. Mechanics is the fundamental branch of physics. Its general principles run as a network through all depart- ments of the science. The phenomena* of heat, sound, light, and electricity with which every one becomes acquainted through the experiences of daily life will serve, in a general way, to indicate the field covered by these branches. 3. Physics as a Study. Physics, rightly studied, is not a burden to the memory but an aid to the understand- ing. The facts of physics are easily remembered when they are understood; if they are not understood, it matters little whether they are remembered or not. Information is valuable as an intellectual possession, but the ability to think accurately is of much greater value. The student of physics has at his disposal one of the best means of developing this ability. A fact clothed in slovenly or ambiguous phrase cuts a sorry figure. Physics is an exact science and finds suitable expression only in exact speech. Training in the use of * A phenomenon, as the term is used in science, is any action or occur- rence perceived by the senses, however familiar and commonplace it may be. The visible happenings in nature are collectively termed natural phe- nomena; and if they involve only physical processes, they are further classified as physical phenomena. 6 PHYSICS the mother tongue is one of the lasting benefits to be derived from the study of the subject. The material results of science and invention are about us on every hand. To be able to use them is practically a necessity; to understand them, in some measure at least, is a necessary part of a liberal education; to add to them is to contribute something toward human progress. A knowledge of physics is a valuable aid toward all these ends. CHAPTER II MATTER AND FORCE. PHYSICAL MEASUREMENTS I. THE THREE STATES OF MATTER 4. Matter exists in three physical states or conditions, called the solid, the liquid, and. the gaseous states. Much of physics depends upon the characteristic properties which distinguish the states of matter from one another. Thus we have the mechanics of solids, the mechanics of liquids, and the mechanics of gases. These properties, therefore, require some attention at the outset. Liquids are distinguished from solids by the fact that they tend to flow, and hence must be contained in vessels. Every solid, on the contrary, has a shape of its own, which it tends to preserve. Some solids, e.g. stone and iron, offer great resistance to a change of shape; others, such as wet clay and putty, can readily be molded into any form. But even the small amount of resistance offered by soft solids distinguishes them from liquids. Many of the physical properties of gases may be learned from a study of the air, which is a mixture of several gases, principally nitrogen and oxygen. Although the air is everywhere about us, we are ordinarily unconscious of its existence unless it is in motion. When it is in motion, we recognize it as a current of air, a breeze, or a wind. We commonly call a vessel "empty" when it is full of air; and seldom stop to think that when the so-called empty ves- sel is being filled with a liquid or a solid, the air in it is being pushed out. 7 8 MATTER AND FORCE It will help toward clear thinking on this point to push an inverted tumbler into a vessel of water. The water does not rise to fill the tumbler, being prevented from doing so by the confined air; but, when the tumbler is slowly inclined, the air escapes in a succession of bubbles, and the water enters at the same time to take its place. This simple experiment shows that a body of air confined in any space tends to keep other bodies out of that space; and the same is true of all gases. But we know that, after a bicycle or an automobile tire is fully inflated, much air must still be pumped in to make it hard. Now air can be forced into the fully inflated tire only by com- pressing the air already in it into a smaller space; and experience teaches that the compression of the confined air can be carried as far as the strength of the tire or of the operator will permit. The great compressibility of air can be shown simply by pushing in the piston of a bicycle pump or other compression pump, while the outlet is closed with the finger. A vigorous push will compress the air per- haps to one half or even to one third of its original volume. When the piston is released, the air expands and drives it back. All gases are highly compressible and expansible, like air. When any quantity of gas, however small, is ad- mitted into an otherwise empty space it instantly expands so as to fill the space completely. If the above experiment is repeated with the com- pression pump filled with water, it will be found that the water is as unyielding as a board, for the piston cannot be pushed in at all. All liquids and most solids are only very slightly com- pressible. Even under very great pressure their change of volume is commonly so slight as to escape notice; and, for all practical purposes, they are regarded as incompres- sible. Hence great compressibility and expansibility are distinguishing properties of the gaseous state. 5. Summary. A solid tends to preserve a definite shape and volume. A liquid tends to preserve a definite volume, but has no shape of its own, since its parts move FORCE AND INERTIA 9 readily over one another. A gas has neither a self-deter- mining shape nor volume; it is highly compressible, and tends to expand indefinitely. Since both liquids and gases flow, they are classed together as fluids. The fluidity of gases can be shown in an inter- esting way by pouring carbonic-acid gas upon a lighted can- dle, from a jar filled with the gas, just as water is poured from a vessel. The gas, being considerably heavier than air, falls in a stream upon the candle and extinguishes it. 6. Changes of State. Vapors. The gaseous form of a substance which exists as a liquid at ordinary tempera- tures is called a vapor. Many substances exist in two or all of the physical states. Water is a familiar example. The metals and some other solids can be liquefied and vaporized by the application of heat. Some solids, such as wood, undergo a chemical change with the application of heat, instead of a change of state. II. FORCE AND INERTIA 7. Force. The word force, as used in physics, is a gen- eral term for any push or pull. The following are familiar examples of forces: the pull exerted by a horse upon a wagon; the push or pull by which a door is opened; the strong push or pressure exerted by a hammer at the in- stant it strikes a nail; the downward pressure of a book against a table upon which it is lying and the upward sus- taining pressure of the table against the under side of the book; the pressure of a liquid against the bottom and sides of the containing vessel. 8. Inertia. We learn from daily experience that a body acquires motion only as the result of an applied force, exerted upon it by some other body. For example, a ball is sent flying through the air by a vigorous push of the hand 10 MATTER AND FORCE in the act of throwing it, or by a blow with a bat; a high velocity is imparted to a rifle-ball by the pressure of the gases from the powder exploded behind it; and a wagon is started by the pull exerted by the horses upon it through the traces. It is also a matter of common observation that moving bodies come to rest more or less slowly after the forces that start them cease to act. A book slides over a table when started with a sudden push, but quickly stops; a ball can be made to roll a long distance over a smooth, level surface, as a sidewalk, but gradually loses speed till it comes to rest; and a wagon goes only a short distance after the horses cease to pull This behavior of moving bodies is not due to any tendency of the bodies themselves to come to rest, but is the effect of opposing forces which are devel- oped by the rubbing of one surface over another. Such a force is called friction. Friction acts as a resistance to motion, and tends to bring moving bodies to rest. The smoother the surfaces are, the less friction becomes; hence a body slides farther on a smooth surface than on a rough one. A skater, for example, can go a long distance with- out effort after getting up speed, the friction between skates and smooth ice being very light. Rolling friction is, in general, much less than sliding friction; hence the use of wheels on vehicles of all sorts. Ball bearings (Fig. i) reduce friction still further by substituting rolling friction for sliding friction at the axle. Another hindrance to motion is the resistance of the air. This resistance is small upon a body moving slowly, FIG. i. Ball Bearings. FORCE AND INERTIA n but rapidly increases with the velocity. For high veloci- ties, such as those of an express-train or a rifle-ball, it is very great. Bodies can, of course, be stopped by other forces than friction. A ball is stopped, when caught, by the pressure of the hands against it. The general truth to be gathered from such facts as the above is that the existing state of rest or motion of a body can be changed only by means of a force of some sort acting upon the body from without; in other words, a body can not of itself alone move, if at rest, or change its motion, if moving. This is true of all matter, solid, liquid, and gaseous, animate and inanimate. Matter of itself tends to continue in whatever state of rest or motion it may chance at that instant to be. This prop- erty of passiveness is called the inertia of matter, or simply inertia. 9. Action of Forces With and Without Contact. Weight. All of the forces previously mentioned are exerted by direct or indirect contact of the body exerting the force and the body upon which the force is exerted. Thus a horse in drawing a wagon pushes on the collar with his shoulders, the collar pulls on the traces, and the traces pull on the wagon. Certain forces, however, act without any material connection between the bodies concerned. The forces exerted by a magnet are of this sort. Pieces of iron move toward a magnet and cling to it. We know from this behavior of the iron that it is acted upon by a force whose direction is toward the magnet, although there is nothing whatever to show how this force is exerted. Similarly, the fact that an unsupported body falls indi- cates that a downward force is acting on it. This force is in some unknown way due to the earth, and we think of the earth as exerting a pull or attraction, by which it tends 12 MATTER AND FORCE to draw all bodies toward its center. The attraction ex- erted by the earth upon any body is called the weight of the body. (As a result of the earth's rotation on its axis, the weight of a body is very slightly less than the earth's attraction for it, except at the poles.) Forces such as those exerted by a magnet or by the earth are none the less real because we do not know how they act. The muscular sensation of effort which we experience in resisting these forces is convincing evidence that they are real. A piece of iron, when held near a strong magnet, pulls the hand toward the magnet with a force that is eas- ily felt. The iron pulls the hand because it is itself pulled. Similarly, when we lift any object it exerts a downward pull upon the hand and arm, which it does only because there is an equal downward force (its weight) acting on it. 10. Balanced and Unbalanced Forces. A force acting alone on a body always sets it in motion or changes its existing motion. A stone flying through the air affords a good example; for its weight is practically the only force acting on it during its flight, the resistance of the air being very small. The weight of the stone causes a continuous decrease of speed if the stone is rising vertically, a contin- uous increase of speed if it is falling vertically, and a con- tinuous change of both speed and direction if it is moving obliquely. Two or more forces acting on a body at the same time may be so opposed to each other that the body behaves exactly as if none of the forces were acting. Such forces are said to balance each other, or to be in equilibrium, and are called balanced forces. For example, when two boys pull equally and in opposite directions on a cart, the two pulls are in equilibrium, and the cart remains at rest. Similarly, the weight of a body, at rest or in motion on a FORCE AND INERTIA 13 level surface, is sustained or balanced by the upward pres- sure of the surface; and the weight of a body suspended by a cord is balanced by the upward pull of the cord. In both cases the sustaining force is equal and opposite to the weight of the body. These examples illustrate the simplest case of balanced forces, namely, that of two equal forces acting in opposite directions along the same line. An example of three forces in equilibrium is shown in Fig. 2. The body is supported by two cords. Each cord pulls obliquely upward on the body, and these two pulls together bal- ance the weight of the body. A single force acting on a body is necessarily an unbal- anced force; two or more forces acting together may be either balanced or unbalanced, depending, in part, upon their relative directions. These matters are to be studied further in the following chapters. 11. Resultant Force. In many cases where two or more forces act at the same time upon a body they affect its behavior (as regards rest or motion) exactly as some one force would . This sm^l(e_^qui valeri t Jorce. is .called the result- ant of the given forces. For example, if a boy pulls on a cart with a force of 1 5 pounds, and another boy pulls with him, exerting a force of 25 pounds, the effect on the cart will be the same as if one boy alone pulled with a force of 40 pounds in the same direction. In general, the resultant of any number of forces which act along the same line and in the same direction is a force equal to their sum, acting along the same line and in the same direction as the given forces. The resultant of two forces which act in opposite direc- tions along the same line is a force equal to their difference, acting along the same line and in the direction of the 14 MATTER AND FORCE greater. Thus if the forces exerted on the cart in the above example are in opposite directions, they will together be equivalent to a single force of 10 pounds acting in the direction of the 25-pound pull., The resultant of two equal forces acting in opposite direc- tions along the same line is zero,, since the two forces ex- actly neutralize each other. The resultant of any set of balanced forces is zero, for the same reason. 12. The Mutual Action of Two Bodies. Force is always a two-sided action. Whenever one body exerts a force on another, the second body exerts at the same time an equal and opposite force on the first. This is often evident from the effects produced. For example, when one marble strikes another, it sets that one in motion, and is itself stopped or retarded by the opposite force which the second marble ex- erts on it. The mutual action of a ball and a bat is a similar case. When a bullet strikes a board the force it exerts makes a hole in the board; the equal and opposite force exerted on trie bullet by the board stops the bullet. FIG. 3. Since these familiar examples fur- nish no direct evidence that the forces exerted by two bodies on each other are equal, it will be instructive to try an experiment especially contrived to show this fact. The apparatus consists of two hardwood or ivory balls, sus- pended as shown in Fig. 3. One of the balls is drawn aside and released. It falls, strikes the other ball, and instantly stops; while the other swings out as far (very nearly) as the first ball would have gone if its motion had not been hindered. Since the balls are exactly alike and the one loses as much motion as the other gains, it follows PHYSICAL MEASUREMENTS 15 that the forces which they exert on each other are equal. (If the balls were of unequal size they would still exert equal forces on each other; but the proof of this involves matters not yet considered.) The forces exerted between two bodies at rest are also equal and opposite. Thus, when the hand is pressed against a wall, the wall presses back on the hand with equal force. A book lying on a table exerts a downward pressure equal to its weight; the resistance offered by the table acts as an equal upward pressure on the book. PROBLEMS 1. Discuss any phenomena with which you are familiar that show the inertia of water; the inertia of air; the inertia of your own body* 2. In what direction is an inexperienced person likely to fall on alighting from a rapidly moving car? Why? 3. Discuss some good example of balanced forces. Of unbalanced forces. 4. What forces are acting on a wagon when drawn at a uniform rate on a level road? Are they balanced or unbalanced? 5. A boy exerts a lifting force of 75 Ib. on a stone weighing 200 Ib. (a) Is this a balanced or an unbalanced force? (6) What balanced forces are act- ing on the stone? 6. Is it the forces exerted by or upon a body that affect its state of rest or motion? 7. Make a list of any phenomena which seem to you to indicate (a) that some bodies are without inertia; (&) that there is matter which has no weight; (c) that any body can exert a force on another without the other exerting at the same time an equal and opposite force on it. If you find any such seeming exceptions to the statements made in the text, save the list for future study. III. PHYSICAL MEASUREMENTS 13. Measurement and Units of Measurement. Experi- mental work in physics consists largely in measuring the different kinds of physical quantities, such as length, sur- face, volume, force, velocity, time, mass, etc. Any kind of quantity is measured by finding^ how many times it con- tains a certain fixed or standard amount of that kind of quantity. This standard amount is called a unit; and there 16 MATTER AND FORCE are various units in common use for measuring each kind of quantity. Thus for measuring length we have the inch, foot, meter, centimeter, etc. On account of the great simplicity of the metric system of measures, it is almost exclusively used in scientific work. It is the only system that we need consider here. 14. Units of Extension. The primary unit of length in the metric system is the meter. It is defined as the dis- tance between two lines on a certain metallic rod preserved in the archives of the International Metric Commission, in Paris, the rod being at the temperature of melting ice. This distance was intended to be one ten-millionth of the distance on the earth's surface from the equator to either pole ; but it is now known to be a trifle less than this frac- tion. The meter is equal to 39.37 inches. Its advantage over the yard lies in the fact that its subdivisions are deci- mal fractions. The tenth part of a meter is called a decimeter (dm.), the hundredth part a centimeter (cm.), and the thou- sandth part a millimeter (mm.). The centimeter is the customary unit of length for scientific purposes, and is the only one that the pupil will ordinarily use in the laboratory. Thus a length of 3 dm. 5 cm. 7.5 mm. is written 35.75 cm. An inch is approximately 2.5 cm. (See Tables I and II of the Appendix.) The square centimeter (sq. cm. or cm. 2 ) and the cubic centimeter (cc., ccm., or cm. 3 ) are the customary units of area and of volume respectively. Since a square decimeter is 10 cm. in length and in width, it contains 100 sq. cm.; and since a cubic decimeter is 10 cm. in each of its three dimensions, it contains 1000 ccm. A cubic decimeter, when used as the unit of liquid measure, is called a liter. It is slightly greater than a quart. PHYSICAL MEASUREMENTS 17 15. Weight. The weight of a body (Art. 9) is constant at any one place on the earth, but decreases slightly with increase of altitude above the general level of the earth, as when a body is carried up a mountain or up in a balloon. A body weighing 500 Ib. at sea-level would lose one pound of its weight when taken to a height of about 4 miles. The weight of a body increases slightly as it is taken from the equator toward either pole. This, as will be explained later, is partly due to the rotation of the earth and partly to the fact that the earth is not a perfect sphere. A body weighing 189 Ib. at either pole would weigh only 188 Ib. at the equator. 16. Mass. The quantity of matter in a body remains constant unless , some portion of it is removed or other matter added to it ; but the volume of a body can be changed in various ways without gain or loss of matter. For ex- ample, a fixed quantity of air or other gas can be compressed to one half, one tenth, or one thousandth of its original volume ; or it -can be allowed to expand to any number of times its original volume. So also 100 cu. cm. of ice-cold water expands to 104 cu. cm. when heated to the boiling point, or to 109 cu. cm. when frozen; but there is no gain of matter with the increase of volume in either case. Evi- dently the volume of a body cannot be taken as the meas- ure of the quantity of matter in it. On the other hand, the weight of a given portion of mat- ter at any one place on the earth's surface remains con- stant under all conditions. The weight of a body may, therefore, be taken as the measure of the quantity of mat- ter in it. Moreover, other facts that we are not prepared to consider here show that equal weights of all substances contain equal quantities of matter. i8 MATTER AND FORCE The quantity of matter in a body is called its mass. It follows from the above that the mass of a body is meas- ured by its weight, and that any two bodies having equal weight (at the same place) have equal mass. Although the weight of a body changes slightly when it is taken to a different latitude or a different altitude, its mass remains absolutely constant, for a change of location does not involve a gain or a loss of substance. 17. Units of Mass and of Force. The principal unit of mass in the metric system is the gram. Like the meter, it is now defined with reference to a standard kept at Paris. It was originally taken as the mass of a cubic centimeter of distilled water at the temperature of its greatest density (nearly ice-cold), and this is the useful definition for the purposes of elementary physics. The mass of a cubic centi- meter of fresh water at any moderate temperature is so nearly equal to one gram that the difference may be dis- regarded. Large masses are generally expressed in kilo- grams, a kilogram being equal to 1000 grams. The pound mass is the principal unit of mass in the English system. The weight of a unit mass is taken as a unit of force. The familiar unit, of course, is the pound weight. Thus if we say that a horse exerts a pull of 150 Ibs. in drawing a load, we mean that the pull is equal to the earth's attrac- tion for a i5o-pound mass. The weight of a gram mass is the principal metric unit of force. A unit of mass and the corresponding unit of force have the same name. Thus we speak of a mass of so many grams, meaning a certain quantity of matter, or a force of so many grams, meaning a certain push or pull. This double use of the terms is unfortunate; but one can always tell from the connec- tion in which they are used whether mass or force is referred to. PHYSICAL MEASUREMENTS 19 From what has been said concerning the variation of weight, it is evident that a unit of weight is not exactly the same at all places on the earth's surface; but the variation is so slight as not to be a matter of practical importance. 18. The Measurement of Mass (Weighing). We make use of the equal attraction of the earth for equal masses in weighing with an equal-arm balance (Fig. 4). The arms are the two halves of the beam, from which the pans are suspended. When equal downward forces are ex- erted on the pans, the beam comes to rest in a horizontal position. Hence if the beam takes this position when there is a certain load in each pan, we know that the loads have equal weight and consequently equal mass. FIG. 4. To find the mass of a body it is placed in one pan and balanced with standard masses in the other. The process is called weighing, and the standard masses are commonly called weights. (Note that in this sense a "weight" is a certain standard piece of matter, not a force.) 19. The Unit of Time. The rotation^ of the earth on its axis is constant. The period of one . complete rota- tion is, therefore, an invariable natural unit of time, and is called a "sidereal day." Even the best chronometers are not perfectly accurate, and they are corrected by com- parison with the earth's rotation, as determined by the apparent motion of the stars. Owing to the earth's annual motion round the sun, the solar day, or the time from "high 20 MATTER AND FORCE noon" to "high noon," is slightly longer than the sidereal day and is also slightly variable. The average length of the solar day for the entire year, or the "mean solar day," is divided into 24 hours, the hour into 60 minutes, and the minute into 60 seconds. These are the time intervals indicated by clocks and watches. The second is the unit of time regularly used in scientific work. 20. Fundamental Units. English and Metric Systems. - We have seen that the units of surface and of volume are derived from the units of length. Velocity is expressed in terms of a unit of length and a unit of time, e.g. in miles per hour, feet per second, etc. Similarly, almost all phys- ical quantities (and there are many) can be expressed in terms of the units of length, mass, and time; hence these are called the fundamental units. The fundamental units of the metric system are the centimeter, the gram mass, and the second; from which it is often termed the centi- meter-gram-second or the C.G.S. system. The foot, the pound mass, and the second are the fundamental units of the English or foot-pound-second (F.P.S.) system. 21. Density. A piece of iron weighs more (has a greater mass) than a piece of wood of the same size. In ordinary language we say that iron is heavier than wood, wood is lighter than water, cork is very light, etc. It is understood that such statements refer to the weights of equal volumes of the substances; but the language is not exact, and hence is not adapted to scientific use. The mass of a unit volume of a substance is called its density. The density of pure cold water is i gram per cu. cm. (by definition of the gram mass), or 62.4 pounds per cu. ft. The density of cast iron is 7.2 g. per cu. cm. or x PHYSICAL MEASUREMENTS 21 449 Ib. per cu. ft. Its density is thus 7.2 times as great as the density of water. The density of a substance is determined by measuring the mass and the volume of any convenient portion of it, and computing from these measurements the mass of one cubic centimeter. PROBLEMS 1. Would the weight of a body appear to differ in different latitudes and at different altitudes; (a) when accurately determined with an equal-arm balance; (b) when accurately determined with a spring balance? Give the reasons for your answers. 2. Is the density of a body affected by taking it to a different latitude or altitude? 3. Is abound of iron heavier than a pound of wood? What is implied in the familiar statement that "iron is heavier than wood"? Show that the statement that "iron is denser than wood" leaves nothing to be implied. Which form of statement is to be preferred? 4. Letting v denote the volume of a body, d its density, and m its mass, write the formula (equation) expressing the relation of these three quantities to one another. Write this equation expressing (a) the value of m in terms of v and d; (b) the value of d in terms of v and m; (c) the value of v in terms of d and m. Note. In physics it is customary to represent the value of a physical quantity, whether known or unknown, by the initial letter of its name. 5. The volume of a stone is 630 cm.; its mass is 1575 g. Find its density. 6. What is the volume of 1000 g. of mercury? Of 1000 g. of brass? Of 1000 g. of aluminum? (See table of densities in the Appendix.) *7. What is the mass of i cu. dm. of lead? Of i cu. dm. of marble? 8. Find the densities of water, quartz, and gold in pounds per cubic foot, from the densities in grams per cubic centimeter given in the table. (See also the table of equivalents in the Appendix.) 9. From the known densities of ice and water, show whether water expands or contracts in freezing. 10. Criticize the statement: i ccm. = i g.; also the statement i ccm. of water = i g. In what different ways may the truth of the matter be correctly expressed? CHAPTER III STATICS OF LIQUIDS I. INTRODUCTION 22. The Problems of Mechanics may be briefly described as problems in equilibrium and problems in motion. On the basis of this classification Mechanics is subdivided into Statics and Dynamics. The present chapter deals with the equilibrium of liquids, or Hydrostatics. Common observation teaches that liquids exert pressures. Pipes, tanks, and dams must have a certain strength to resist the pressure of water, or they will burst. A ship rides safely on the water; yet its enormous weight has no other support than the yielding liquid. The designer of a ship must know before the keel is laid how far it will sink when launched, and how much farther with a full cargo. He must also know that the ship will float upright, and not " turn turtle." Evidently the mechanical behavior of liquids and of bodies floating on them or immersed in them is definite and dependable. To understand this behavior one must know the general facts or principles of liquid pressure. A few preliminary ideas concerning pressure in solids will help us. 23. Transmission of Pressure Through Solids. Applied Pressure and Gravity Pressure. When an object is pushed with a stick held in the hand, the force (pressure) is trans- mitted from the hand to the object through the stick. The stick sustains the pressure throughout its length; and, if 22 INTRODUCTION 23 it is not strong enough to withstand this pressure, it will bend or break at the weakest place. This is an example of an applied pressure acting upon and transmitted through a solid. The weight of any body gives rise to a pressure which is similarly transmitted throughout the body. In a brick wall, for example, each brick transmits to those be- neath it the pressure exerted upon it by all the overlying bricks, and adds to that a pressure equal, to its own weight. The pressure therefore increases from top to bottom of the wall, and at any level its amount is determined by the weight of the overlying bricks. Such pressures are called gravity or weight pressures, to distinguish them from exter- nal or applied pressures. The pressure at the bottom of a fac- tory chimney is a gravity pressure when considered with respect to the chimney in which the pressure originates; it is an applied pressure when considered as an external force acting on the foundation which supports the chimney. 24. Lateral Pressure Due to Weight. The gravity pres- sure in a wall of masonry is vertical. Each brick or stone presses up on its neighbors above and down on its neigh- bors below, but not laterally on its neighbors at the same level. In a pile of sand or shot each individual crowds in between its neighbors, causing a pressure sideways as well as upward and downward, as is shown by the tendency of the pile to spread at the bottom. To make the sides of the pile vertical, supporting surfaces must be provided to sustain the lateral pressure. Similarly, the weight of a liquid causes lateral and oblique as well as vertical pressures within it. These pressures are more fully developed in liquids than in a pile of shot, for the particles of a liquid are free to move over one an- other, while in shot there is considerable friction. Hence shot remains in a sloping pile and a liquid does not. STATICS OF LIQUIDS \ u \ FIG. 5. II. GRAVITY PRESSURE IN LIQUIDS 25. Relation Between Pressure and Depth. Pressure in Different Directions. The pressure at different depths and in different directions in water can be observed with the aid of glass tubes of equal length (60 cm. or more), closed at the top and shaped at the lower end as shown in Fig. 5. When such a tube is lowered into a tall glass jar filled with water, the water enters its lower end to a greater or less dis- tance according to the pressure ; for the water can enter only as the confined air is com- pressed, and the compression increases with the pressure. The water pushes farther in as the tube is lowered, showing that jthe_pressuie, increases with-the depth. When the differ- ent tubes are inserted to the same depth, the water enters an equal distance in all, showing that the pressure at a given depth is the same in the various directions tested. By other methods which permit exact measurement it is found that the gravity pressure at any poinl in a liquid at rest is proportional_Jo_Jhe depth of the point below the free surface of the liquid, and at any point the pres- sure is the same in all directions. This behavior is explained as follows. The pressure at any level is due to the weight of the over- lying liquid; 'and, since the liquid FlG . 6 . is of the same density at all depths GRAVITY PRESSURE IN LIQUIDS 25 (liquids being practically incompressible), the weight of liquid in a vertical column is proportional to the depth of the column. Thus at a depth of 2 cm. the pressure is twice as great as at a depth of i cm. ; at 3 cm. it is three times as great, etc. (Fig. 6). Further, since each particle of the liquid is free to move, it would not remain at rest if the pressures upon it in different directions were unequal. 26. Direction of Fluid Pressure Against a Surface. The pressure of a liquid at rest is perpendicular to any surface against which it is exerted, e.g. the walls of the containing vessel or the surface of an immersed body. This also is due to the fact that the particles of a liquid are free to move. If the pressure were oblique to any surface, the liquid would flow along it. 27. Pressure in Vessels of Different Sizes and Shape. Let a funnel and a glass tube be connected by a rubber tube and partly filled with water (Fig. 7). The water stands at the same level in the funnel and the tube, whether they are vertical or inclined at any angle. This beha- vior Suggests One Of the FlG - 7- A liquid "seeks its own level." most important general truths in the mechanics of liquids. Imagine a plane cutting across the tube at its lowest part m. The pressure 'of the water in the funnel tends .to push water past m and up into the glass tube\, while the pressure of the water in the glass tube tends to push water past m in the opposite directionand up into the fun- nel. Since there is no flow in either direction, these grav- ity pressures must be equal. When either side is lowered or inclined, equilibrium is destroyed, and water immedi- 26 STATICS OF LIQUIDS ately flows to that side until the two surfaces are again at the same level. Evidently the pressures are equal at the bottom of the funnel and the tube only when the depth of the water (measured vertically) is the same in both. The greater mass (or weight) of the water in the funnel does not affect the result in the least. This agrees with the well-known fact that water stands at the same height in the spout of a kettle as in the body of the vessel, or the fact that water will rise in a pipe only to the level of the surface in the tank or reservoir from which it comes. In general The gravity pressure of a liquid at a given depth, either within the body of the liquid or against any surface, is wholly independent of the size and shape of the vessel. This can be further shown with three vessels, a, b, and c (Fig. 8), having bottoms of the same area and filled to the same depth with the same liquid. A disk, A, serves as a bottom for each of the vessels FIG. 8. The factors of gravity pressure are depth and density only. in turn, being held in place by the upward pull of a cord. This cord is attached to an arm of a balance, by means of which the same pull is exerted in each case. With the adjustment shown for vessel a, the liquid is poured in till the pressure becomes great enough to force the disk from the bottom. It will be found that this requires the same depth of liquid in the three vessels. Only with the vessel a, GRAVITY PRESSURE IN LIQUIDS 27 however, is the downward force of the liquid on the disk equal to the weight of the liquid. The vertical sides of this vessel do not help to support the weight of the liquid. In vessel b the liquid is partly supported by the slanting sides, which press obliquely upward against the liquid (the pressure being perpendicular to the surface). In vessel c the slanting part of the side presses obliquely downward against the liquid, thus, in effect, supplying a part of the downward force which is exerted on the bottom. 28. Relation Between the Pressure of a Liquid and its Density. We have seen that the gravity pressure of a liquid against any surface is proportional to the depth of the liquid above it, because the weight of the vertical col- umn of liquid overlying the surface is proportional to the depth. But the weight of such a column is also propor- tional to the density of the liquid; hence we should expect the gravity pressure at equal depths in different liquids to be pro- portional to their densities. Experiment shows that this is the case. If a small quantity of mercury is poured into a tall U-tube, and one of the arms is then nearly filled with water, the liquids will stand as shown in Fig. 9. The pressure at the same level c and d in the two arms must be equal, since below : _ ff that level we have the same liquid on both sides. But the pressure at c is due to the column of water be, and the pressure at d to the mercury column ad. Now it is found by measurement that the water j t ' y pressures at column is 13.6 times as high as the mercury column c and d are ad. Since therefore a column of mercury rs-s as equa ' high as the water column exerts an equal pressure, a mercury col- umn of the same height as the water would exert a pressure 13.6 times as great. But the density of mercury is 13.6 times that of water; which agrees with the general conclusion that the gravity pressures at equal depths in different liquids are proportional to the densities of the liquids. 28 STATICS OF LIQUIDS 29. Summary of the Laws of Gravity Pressure in Liquids. The general facts or laws of liquid pressure which we have been considering are as follows: 1. 'The pressure at any point in a fluid at rest is the same in all directions. 2. The pressure of a fluid at rest is perpendicular to any surface with which it is in contact. 3. The gravity pressure in a liquid at rest is proportional to the depth and to the density of the liquid. The first two laws hold for gases as well as for liquids and for both gravity pressure and applied pressure; hence the more general form in which they are stated. 30. Digression on Natural Laws. Experience teaches that, in nature, wherever and whenever the same conditions are repeated, the same results follow. Natural phenomena, when fully understood, always disclose uniformity, order, system. Without this uniformity in nature, science would be impossible, and the innumerable applications of scien- tific principles which we see on every hand would also be impossible. To mention a single instance, if the behavior of electricity under the same conditions were not invariable and dependable, electrical power could never have been brought under control and made the tremendously useful servant that it is at the present day. The uniform behavior of matter, or the unvarying course of phenomena, under the same conditions is known as a natural law. A natural law is a fact in nature before it is discovered, as well as afterward. The laws of gravity pres- sure in liquids and the law of applied pressure (Art. 34) were discovered by the French scientist Blaise Pascal about the middle of the seventeenth century. Laws of nature within the domain of physics are called physical laws. They are met with in considerable number in all GRAVITY PRESSURE IN LIQUIDS 29 branches of the subject. Thus in certain respects all gases behave alike, and this uniform behavior constitutes the laws of gases. In certain respects all bodies behave alike under the action of force, and this uniform behavior con- stitutes the laws of motion. Light is always reflected in a definite manner from polished surfaces, and this beha- vior is known as the law of reflection of light. When a natural law is discovered, the dicoverer formu- lates a statement of the fact as he sees it. This statement is itself called a law (sometimes a principle). Laws in this sense are brief descriptive statements of the manner of behaving not the behavior itself. It is well to bear in mind these two meanings of the word law, as used in sci- ence. Natural processes are never amended; but state- ments of what men believe to be the fact sometimes require modification in consequence of later and more accurate or more complete information on the subject. 31. Pressure as a Measured Quantity. The term pres- sure, as we have used it thus far, may be taken to mean either the whole force exerted by the liquid upon any given area or the force exerted upon each unit area of the surface. Whenever we have to deal with fluid pressures numerically, it is necessary Jx> restrict the use of the term to one of these two possible meanings, in order that there may be no misunderstanding; and it is agreed, among both scien- tists and engineers, that pressure shall always mean the force per unit area. Thus when an engineer speaks of a boiler pressure of 150 Ib. he means that the steam exerts a force of 150 Ib. against each square inch of the boiler surface. Pressure is measured in pounds per square inch, pounds per square foot, grams per square centimeter, etc. The force exerted by a fluid against the whole of a given area (whether greater or less than a unit area) is called 30 STATICS OF LIQUIDS the total force or the thrust. This distinction of terms is strictly adhered to in all that follows, both in the mechanics of liquids and the mechanics of gases. The term pressure at a point means the force per unit area at the level of the point. The rules for computing gravity pressure are derived from the laws. The thrust of a liquid against the bottom of a vessel having vertical sides is equal to the weight of the liquid. (Why?) The pressure on the bottom is equal to the weight of the vertical column of liquid whose base is any unit area of the bottom, and this is equal to the product of the depth and the density of the liquid. (Why?) Hence the general rule: The gravity pressure at any point in a liquid is equal to the product of the depth of the point below the free surface of the liquid and the density of the liquid. Thus in either of the vessels shown in Figs. 10 and n, the pressure at the level MN is everywhere the same, and is equal to hd g. per sq. cm., where h rep- resents the depth of MN below the free surface of the liquid and d the density of the liquid. 32. Rules for Computing the Thrust. Horizontal Sur- faces. Since the pressure of a liquid is uniform over a horizontal surface, the thrust against such a surface is equal to the product of the pressure and the area. Oblique and Vertical Surfaces. The pressure against a vertical or an oblique plane surface increases uniformly from the upper to the lower side. To find the thrust FlG. 10. FIG. ii. GRAVITY PRESSURE IN LIQUIDS 3 1 against such a surface, the average pressure upon it is multiplied by the area. This average pressure is equal to , the actual pressure at the center of the surface. For example, the thrust against a water gate which is 6 ft. wide and 4 ft. high, and the top of which is 8 ft. below the surface, is (10 X 62.4) X (4 X 6) = 14,976 Ib. 33. Applications of Gravity Pressure. Gravity pres- sure is utilized in the water-supply of cities. Wherever possible, reservoirs are located at a sufficient elevation to provide the necessary pressure. Distributing pipes or mains carry the water to all parts of the city, and these connect with the water-pipes in each house. When much water is flowing in the pipes there is considerable loss of pressure due to friction, and water will not rise in the pipes to the level of its source. The reservoir must therefore be higher than any point to be supplied from it. When the reservoir can not be located at a sufficient height to supply the distributing system directly, the necessary pressure is maintained by pumping the water from the reservoir (or sometimes from a lake) into an elevated tank or a tall stand-pipe (Fig. 12). FIG. 12. This stand- pipe at Erie, Pa., holds the water 235 ft. above the level of Lake Erie, whence the supply is drawn; e, the pump house; w, the top of the col- umn of water in the stand-pipe. FIG. 13. Artesian Wells. The flow of artesian wells is due to the gravity pressure of water. If a porous stratum of sand or gravel (c, Fig. 13), lying between two impervious strata and dipping under a lower flat country, becomes filled with water above the level of the ground where a well is bored, an artesian or flowing well will result (Fig. 14). 3 2 STATICS OF LIQUIDS The use of water-power for running turbine water-wheels. (Fig. 159 6) is an important application of gravity pressure. The wheel is at the bottom of a large pipe, which conducts the water to it from a higher level, under a pres- sure which depends upon the ele- vation or " head " of the water above the wheel. The wheels of the Niag- ara Falls Power Company are sup- plied with water under a head of 136 feet. PROBLEMS 1. What is the gravity pressure (a) at a depth of 20 cm. in water? (6) at a depth of 60 cm. in mercury? (c) at a depth of 50 cm. in alcohol? (See table of densities in the Appendix.) 2. What is the pressure in pounds per square foot (a) at a depth of 20 ft. in water? (6) at a depth of 3 mi. in the ocean? (Take 62.4 Ib. per cu. ft. as the density of pure water in all problems. The density of sea water is 1.026 times this.) 3. A rectangular vessel 50 cm. long, 20 cm. high, and 35 cm. wide is filled liquid whose density is 1.5 g. per ccm. Find the thrust (a) against the bottom; (b) against a long side. 4. Find the thrust in pounds against the side of a cylindrical tank, 15 ft. in diameter and 12 ft. high, when filled with water. 5. Find the thrust against a vertical dam, 100 ft. long, against which water stands to a depth of 12 ft. 6. A round hole 2 in. in diameter, in the side of a water tank, is closed with a plug. What is the thrust against the plug when the water stands 8 ft. deep above the center of the hole ? 7. What is the pressure in pounds per square inch due to a 90 ft. head of water? 8. An outlet through the side of a dam is closed by a gate 4 ft. wide and 3 ft. high; and the top of the gate is 9 ft. below the surface of the water. What thrust does the gate sustain? FIG. 14. Artesian Well at Woon- socket, S. D. When photo- graphed, the jet was 97 ft. high. with FlG - TRANSMISSION OF APPLIED PRESSURE 33 III. TRANSMISSION OF APPLIED PRESSURE BY LIQUIDS 34. Pascal's Law. A liquid transmits pressure in the same way whether the pressure is due to its own weight or to an applied force. For example, suppose a vessel, A (Fig. 15), to be filled with water to the level MN, the depth c being 10 cm. The gravity pres- sure of the water against the bottom will be 10 g. per square centimeter. Now if water is added to the additional depth of 20 cm., or to the level CD, the pressure on the bottom will be 30 g. per square centime- ter, and everywhere throughout the original body of water the pressure will be 20 g. per square centimeter greater than it was before. If this additional pressure were exerted by means of a tight-fitting piston, as shown in B, it would be regarded as an external or applied pressure; but its effect would be the same as in the first case, i.e. the applied pressure of 20 g. per square centimeter would be transmitted throughout the water, adding just that much to the original pressure against every unit area of the bot- tom and sides of the vessel. In general A pressure applied to any part of an inclosed fluid is transmitted throughout the fluid, with unchanged intensity, to all parts of the interior surface of the vessel, and its direction is everywhere perpendicular to the surfqce. This is known as Pascal's law, after the French mathematician and physi- cist, Blaise Pasca^ by whom it was_discovejred. An important consequence of Pascal's law is illustrated in Fig. 1 6. The apparatus consists of two cylinders having unequal diame- ters, connected together and fitted with pistons. The pistons rest 34 STATICS OF LIQUIDS upon the water in the vessel, and a pressure exerted by either is transmitted by the water to the other. Since the pressure (force per unit area) is the same against both pistons, the thrusts against them are proportional to their areas. Thus if the area of the larger piston is 50 times that of the smaller, a weight of i kg. placed on the smaller will bal- ance a weight of 50 kg. on the larger. It will be seen from this that water (or any other fluid) is an effective instrument for transmitting and multiplying force. This mechanical principle is applied in a great variety of hydraulic and pneumatic (compressed-air) machines and devices. 35. Applications of Fluid Pressure. The hydraulic or hydro- static press (Fig. 17, a and b) is an important application of Pascal's principle. C is a very strong metal cylinder. In it there is a cast-iron piston, P', working water tight in the head of the FIG. 1 6. Thrust is Proportional to Area. FIG. 170. Section Diagram of the Hydraulic Press. FIG. 176. One-thousand Ton Press used in Steel Car Construction. The fixed platform is at the bottom of the press; the pressure is applied downward. cylinder. The top of the piston carries an iron plate M, on which is placed the substance to be pressed. The fixed upper plate, N, is ATMOSPHERIC PRESSURE 51 are based on other sources of information as well, includ- ing temperature, direction, and velocity of the wind, the course and progress of storms up to the time when the fore- cast is made, and the existing state of the weather; all of which are reported to the central office at Washington by the different stations distributed over the country. The barometer is not to be held responsible for erroneous fore- casts. Its function is to measure atmospheric pressure, and this it dt>es correctly. Since the atmospheric pressure changes at a known rate with change of altitude, the barometer can be used for meas- uring altitudes. To find the height of a mountain by this method, the barometric pressure at its base and at its sum- mit are taken as nearly at the same time as possible. The height to which a balloon ascends is determined in the same way. For moderate altitudes above sea-level, it is approxi- mately correct to compute the change of altitude at the rate of 900 ft. for a fall of the barometer of one inch. PROBLEMS 1. Explain the process of drinking through a straw. 2. When the mercury barometer stands at a height of 76 cm., what will be the height of a barometer the liquid in which has a specific gravity of 1.6? 3. When the barometer stands at 76 cm., a liter of air at o C. weighs 1.293 g- At the same temperature and pressure, what will be the weight of the air in a room 9 m. by 7 m. and 4 m. high? 4. Compute the weight of i cu. ft. of air at o C. and 76 cm. pressure (sp. gr. of air = .001293). 6. What weight of air at this temperature and pressure is contained in a room 20 by 30 ft., and 12 ft. high? 6. The force required to separate Guericke's hemispheres, which were 1.2 ft. in diameter, is equal to the total force of the atmosphere on a flat circular area of the same diameter. Compute it, assuming a pressure of 14 7 lb. per sq. in. 52 STATICS OF GASES 7. (a) The weight of the atmosphere is equal to the weight of an ocean of mercury covering the entire surface of the earth to what depth? (6) What would be the depth of water covering the entire surface of the earth and having equal weight? II. LAWS OF GASES 50. The Behavior of Liquids and Gases Compared. Liquids and gases behave alike in part, since both are fluids. The first and second laws of Article 29 and Pascal's law (Art. 34) are laws of fluid pressure, including both liquids and gases. But, owing to the small density and great compressibility of gases', their behavior in other respects is very different from that of liquids. Vessels for holding liquids must have strength to withstand their gravity pres- sure, and that only. Vessels for holding gases must have strength to withstand the force developed by compressing them, and practically that only, for the gravity pressure is negligibly small. For example, suppose a cylindrical steel tank for holding oxygen or hydrogen to be 12 in. in diameter and 4 ft. high, and to contain either of these gases under a pressure of 200 Ib. per sq. in. The thrust against the top of the tank would be over 22,000 Ib., while the thrust on the bottom would exceed this only by the weight of the gas, which, if it was oxygen, would be about 3.4 Ib., or, if hydrogen, less than I Ib. The gravity pressure of gases is, in general, negligibly small; with the single exception of the atmosphere; and that only on account of its great height. 61. The Elastic Force of Fluids. All fluids offer resist- ance to compression, liquids a very great resistance to any appreciable compression, gases comparatively little. The resisting pressure developed within a fluid by compres- sion is called its elastic force. This is always equal to the LAWS OF GASES 53 applied pressure; in fact, it is the applied pressure, trans- mitted throughout the fluid in accordance with Pascal's law. All fluids expand to their original volume after com- pression, when the added pressure is removed. In other words, fluids have perfect elasticity of volume. To illustrate: The pressure at a depth of 3 miles in the ocean is, in round numbers, 7000 Ib. per sq. in. The loss of volume under this pressure is approximately 2%. If a bottle, filled and sealed at this depth, were brought to the surface, it would burst, unless it was capable of withstanding the elastic force of 7000 Ib. per sq. in., which the water would exert in consequence of being compressed. Similarly, each cubic centimeter of the atmosphere near sea-level is subjected to a pressure of more than 1000 g. against each of its sides (a force approximately 800,000 times as great as its weight), and it reacts with an equal pressure (elastic force) in resisting further compression. Surprisingly great as the elastic force of gases is in comparison with their weight, the contrast is even greater with liquids. While water is about 800 times as dense as air under ordinary condi- tions, the resistance that it offers to a given com- pression is about 20,000 times as great as thkt of air. It must be remembered, however, that the elastic force in a body of liquid ceases when the liquid has expanded to a definite volume, while in a mass of gas it continues (with diminishing in- tensity) however great the expansion may be. 52. Measurement of Gas Pressure. An in- FlG - 32. Open strument for measuring the elastic force or pres- sure of a gas in a closed space is called a pressure gage or ma- nometer. Manometers are made in a variety of forms, adapted to the amount of pressure which they are intended to measure. The open-tube manometer (Fig. 32) is commonly used for measur- ing pressures only slightly greater or less than that of the atmosphere. It consists of a glass U-tube partly filled with water or mercury, with a rubber tube attached to one arm for making connection with the vessel in which the gas is contained, and a scale for measuring the 54 STATICS OF GASES height of the liquid in the two arms. When such a manometer is connected with the gas pipes of a building and the gas turned on, the liquid is pushed down in the arm in which the gas is admitted. The pressure of the gas upon the surface a is equal to the pressure at the same level, c, in the other arm; and we know that the pressure at c is the sum of the atmospheric pressure on b and the gravity pressure of the column of liquid, be. Hence the pressure of the gas exceeds the pressure of the air by an amount equal to the gravity pressure of the liquid column be. This excess of pressure can be computed in grams per square centimeter, if so desired, from the measured height of the column and the known density of the liquid. To find the whole pressure of the gas, the pressure of the atmosphere (determined by the barometer) must be added. The vacuum gage (Fig. 33) is used for measuring gas pressure in partially exhausted vessels. It is a closed-tube manometer, having no air or other gas in the closed arm. In construction and action it is like the siphon barometer, with the exception that its closed arm is commonly much shorter, and the mercur y completely fills this arm when under atmos- pheric pressure. While the air or other gas is be- ing pumped from a vessel to which a vacuum gage is attached, the mercury continues to fill the closed arm for some time, if the origi- nal pressure was more than sufficient to sustain the full height of the column. It is only after Jhe mercury begins to fall that the difference of level measures the pressure. When the mercury stands at the same level in the two arms, the inclosed space with which the gage is connected is a perfect vacuum. The steam gage (Fig. 34) is used for measuring high pressures, as in steam boilers, and in tanks of compressed air and other gases. A is a cock in a small pipe connecting the gage with the tank or boiler; B is a bent tube of elliptical section, as shown at the bottom of the figure, one end of which is joined to the cock and the other closed and free to move. The free end works a sector D and pinion E, by means of the connecting link C. D is pivoted at its center, and E carries a pointer, F, which revolves with it. When the pres- sure within the bent tube increases, the cross-section of the tube LAWS OF GASES 55 Section of tube B. FIG. 34. Dial Pressure Gage. becomes more nearly circular, and the tube tends to straighten. This causes the sector to move, and the pointer to indicate the pressure on the scale. 53. Units of Fluid Pres- sure. The pressure of a gas is often expressed in terms of the height of -the mercury or water column that it can sup- port in a manometer, e.g. we speak of a pressure of so many centimeters or millimeters of mercury in the receiver of an air pump . Great pressures are generally expressed in atmos- pheres, as a pressure of 500 atmospheres. An atmosphere is equal to the pressure of a column of mercury 76 cm. high. This unit is derived from the average pressure of the atmosphere at sea-level. It is constant, and wholly independent of the existing barometric pressure at any time or place. The principal units of fluid pressure are the gram per square centimeter, the pound per square _inch, the centimeter or the inch of mercury, the centimeter or the inch of water, and the atmosphere. The student should be able to com- pute a pressure in terms of any one of these units, when its value in terms of any other one of them is known. 54. Boyle's Law. For twenty years or more after the barometer was invented (1643), the experimental study of atmospheric pressure and the general behavior of the air under different pressures was diligently pursued by physi- cists in the principal countries of Europe. The air pump, 56 STATICS OF GASES invented by Guericke in 1650, opened up a new field of in- vestigation that excited wonder and curiosity. In England the mechanics of the air was first studied by Robert Boyle (1627-1691). He was especially interested in measuring what he~~termed "the spring of the air" under different degrees of compression, or the relation be- tween the different volumes of the same mass of air and the corresponding pressures exerted by (and upon) it. For this purpose he prepared a |~a a bent glass tube (Fig. 35), having the shorter arm closed and the longer arm open. Having poured in a little mercury, he adjusted the quantity of air in the closed arm so that the mercury stood at the same level in both arms (A, Fig. 35). The air in the closed tube was then under the same pressure as the outside air, which the barometer showed to be equal to 29 inches of mercury. Having measured the length of the air column in the closed tube, he poured in more mercury till the length of the air column was reduced exactly one half. He then observed "not without delight and satisfaction" that the mercury stood 29 inches higher in the open arm than in the other (B, Fig. 35). Since the added pressure of this column of mercury was equal to the pressure of the atmosphere, the pressure upon the confined air had been doubled in reducing its volume one half. Other measurements, taken with different pressures, showed the same relation, i.e. the volume of the confined air decreases at the same rate that the pressure upon it increases, and vice versa. Later and more accurate investigations have shown that this relation holds very approximately (not exactly) for FIG. 35. Boyle's Experiment. LAWS OF GASES 57 all gase.s, until the pressure is so great that the gas is not far from liquefying. This behavior is summed up in Boyle's law: The volume of a given mass of any gas varies inversely as the pressure upon it, provided the temperature remains constant. If Vi denotes the volume of a mass of gas when the pressure upon it is P 1} and V 2 its volume under a differ- ent pressure P 2 , the temperature remaining the same, the algebraic statement of Boyle's law will be Pi:P,::7,:7i. From this proportion it follows that PiVi = P Z V 2 ' } i.e. Sit a constant temperature, the product of the pressure and volume of a given body of gas is constant. The volume of a gas does not vary inversely as the pressure if its temperature changes; for a change of temperature will itself produce either a change of pressure or a change of volume. 55. Effect of Pressure on the Density of a Gas. When a gas is compressed, its density increases in proportion to the decrease of its volume; but the pressure also increases in proportion to the decrease of volume, provided the temperature remains constant (Boyle's law). Hence the density of a gas at a constant temperature is directly proportional to the pressure upon it. 56. Height and Density of the Atmosphere. The density of the air is less at higher altitudes because the pressure is less. The height to which the last scanty remnant of the atmosphere extends is unknown, but it is variously estimated at from 100 to 200 miles. It is known to extend above fifty miles; yet the density decreases so rapidly that the pressure at a height of 3.4 miles is only one half as great as at sea-level (Fig. 36) ; from which it follows that one half of the air lies below the latter elevation. (Why?) Men have ascended to higher altitudes than this upon mountains, and, in a few 58 STATICS OF GASES instances, to a height of 6 or 7 miles in balloons. At high altitudes the mass of air taken into the lungs with each breath is greatly re- duced, and breathing must be more rapid to make up the deficiency. A very little exertion brings on a violent struggle for air, accompanied FIG. 36. Diagram of the Atmosphere. The region of convection currents, clouds, and storms extends to a height of about 10 km. (6.21 mi.). Above this is the isothermal layer, extending to an unknown height. It is exceedingly dry and cold. Above the " limit of twilight " the air is too rarefied to reflect sun- light to the earth; but clouds of fine dust from Krakatoa floated in this region for two years after the eruption in 1883. These were seen at night by re- flected sunlight; hence the term "noctilucent." Higher still, the existence of a scanty atmosphere is demonstrated by the light of falling meteors (due to the friction of the air), and, last of all, by the aurora. by a feeling of suffocation. The insufficient pressure upon the body and the intense cold are further disagreeable and even dangerous features of very high balloon ascensions. If the atmosphere were of the same density throughout as at sea- level, it would extend only to a height of about five miles, LAWS OF GASES 59 57. Buoyancy of the Air. The law of buoyancy holds for bodies in air as well as for bodies immersed in liquids, and for the same reason (Art. 37). A body in air is buoyed up by a force equal to the weight of the air displaced by it. A cork rises to the. surface in water and a balloon rises in air under like conditions. In both cases the weight of the body is less than the buoyant force. The gas in a balloon does not of itself exert a lifting force. On the contrary, it is subject to the force of gravity, as all matter is; but its weight is much less than that of the displaced air, leaving an excess of buoyancy more than sufficient to support the weight of the balloon and the occupants of the car, just as a cork may carry up through water a pebble that is tied to it. The buoyant force of the air upon solids and liquids is always very small in comparison with their weight. Upon water it is about slo of the weight, and upon lead less than Woo; upon the body of a grown person it is about 3 ounces. The ordinary circumstances of life afford no means of detecting this buoyant force upon solids and liquids. (A bird, a kite, or a feather is not sustained in air by buoyancy, but by pressure due to the inertia of jail.) Upon gases, however, the buoyancy is relatively large. It exceeds the true weight of any gas less dense than air, e.g. hydrogen and illuminating gas, and all such gases tend to rise. The weight of a gas is always expressed as its true weight, i.e. its weight in a vacuum, the buoyancy of the air being much too large to be disregarded. 58. Summary of the Laws of Gases. The laws of gases include Pascal's law (Art. 34), Boyle's law (Art. 54), and the law of buoyancy or Archimedes' principle (Art. 37). To these must be added the law of Charles (Art. 206), which ex- presses the effect of temperature upon the volume of a gas. The first two laws of Art. 29 hold for gases; but they are included in Pascal's lam- Gravity pressure in gases is so small as to be practi- cally negligible, eja^pt_in__the_case of the atmosphere, and in this case it is not proportional to the depth, since the density of the atmosphere is not uniform, 60 STATICS OF GASES PROBLEMS 1. In ascending a mountain will the fall of the barometer during each thousand feet of the ascent be greater or less than for the preceding thousand feet? Why? 2. Describe the process by which the air enters the lungs in breathing. Criticize the expression "drawing in a breath." 3. (a) At what depth in fresh water is the gravity pressure equal to one atmosphere? (6) At what depth in salt water? 4. From what depth in freshwater must a bubble of gas start in order that its volume may be doubled by the time it reaches the surface? 6. A bag of feathers and a piece of iron weigh exactly a pound each in air. Which has the greater true weight? Which has the greater mass? 6. Is the buoyant force upon the body of a swimmer affected by the greater or less inflation of his lungs? 7. A cubic decimeter of gas is under a pressure of 100 cm. of mercury. What will be its volume at the same temperature under a pressure of 30 cm. of mercury? 8. A liter of gas is taken under a pressure of one atmosphere. What will be its volume at the same temperature under a pressure of 100 cm. of mercury? 9. Two liters of gas under a pressure of one atmosphere will have what volume when the pressure is reduced to 900 g. per sq. cm.? 10. Compute the height to which the earth's atmosphere would extend if it had the same density at all altitudes as at sea-level, assuming that density to be .0012 g. per can., and taking the pressure at sea-level as 76 cm. of mercury. 11. The length of the air column in the closed arm of a Boyle's law appa- ratus is 25 cm. when the mercury stands 20 cm. higher in the closed arm than in the open arm. What will be the length of the air column when the mercury stands 30 cm. higher in the open arm than in the closed arm, the atmospheric pressure being 75 cm.? 12. An open manometer (Fig. 37) is connected with a vessel containing air at o C. The mercury stands 15 cm. FIG. 37. higher in the outer arm of the gage than in the inner arm, and the barometer reads 75 cm. Compute the density of the air in the vessel. (The density of air at o C. is .001293 g- per ccm. under a pressure of 76 cm. of mercury.) THE MECHANICS OF FLUIDS 61 III. APPLICATIONS OF THE MECHANICS OF FLUIDS 59. The Air Pump. The air pump, designed for remov- ing air or other gas- from a closed vessel, was invented by Otto von Guericke. It has been improved at various times, and is now made in many forms differing greatly from one another in details of con- struction and in effectiveness. FIG. 38. Air Pump with Pressure Gage. Figure 38 represents one of the older forms, involving only the earliest and simplest principles. The pump consists of a metal cyb'nder in which fits an air-tight piston operated by the handle. There are two valves, namely, the piston, vate a and the inlet valve b, the latter covering the end of the tube that leads to the bottom of the cylinder. The valves open upward only, as shown in the figure. The simplest form of valve consists of a piece of thin leather or oiled silk, placed so as to cover the hole and fastened, at one edge. The valve closes the opening air tight whejLpressed against it f and leaves it open when pushed in the opposite direction. The pump is con- nected by the tube to an opening, 0, at the center of a flat metal plate, PQ, upon which stands a receiver, R. The action of the pump is as follows. Suppose the piston to be at rest at the bottom of the cylinder. Both valves will be closed, being held down by their weight. During the up-stroke of the piston, 62 STATICS OF GASES the small amount of air beneath it expands and fills the increased space, and its pressure decreases proportionally. The atmospheric pressure upon the top of valve a being now greater than the pressure from beneath, this valve is firmly closed. When the downward pressure upon b is sufficiently diminished, the pressure of the air in the tube beneath this valve lifts it, permitting some of the air in the receiver to escape into the space below the piston. As soon as the piston stops rising, the lower valve is closed by its own weight. On pushing the piston down, the air beneath it is compressed. This air can not escape through the lower valve, since the increased pres- sure only closes this valve more tightly. When the amount of com- pression is such that the density of the confined air is slightly greater than that of the atmosphere, the upper valve is forced open, permit- ting the air to escape. These processes are repeated with every stroke of the piston, thus gradually removing the air from the receiver. The limit of possible exhaustion is reached when the pressure of the air remaining in the receiver is insufficient to lift the lower valve, or when the quantity of air that enters the cyl- inder with the up-stroke is so small that it can not be compressed enough to raise the upper valve. The newer forms of air pumps do not depend upon the pressure of the air for operating the valves, and are therefore capable of pro- ducing a more nearly perfect vacuum. They are commonly provided with metal valves, which are operated automatically by a simple mechanism attached FIG. 39. Exhaust and Compression J Pump. to the piston or to the .piston-rod. 60. Compression Pumps. If the valves of the pump shown in Fig. 38 were made to open downward, the ac- tion of the pump would be reversed, and it would force air into the receiver. It would thus become a compres- sion pump. THE MECHANICS OF FLUIDS FIG. 40. Hand Bellows. A pump may be constructed so as to serve both purposes, as shown in Fig. 39. The air (or other gas) enters the pump through the valve A, which opens inward, and is driven out through C, which opens outward. Hence if a closed vessel is attached at C, air will be forced into it; if attached at A, the air will be exhausted from it. {Describe the action in detail.) A bicycle pump is a simple form of compression pump. (Ex- amine one and describe its action.) The bellows is a simple device for supplying a large volume of air under moderate pressure. If it has only one compartment, as in the hand bellows (Fig. 40), the flow is intermittent; if it has two compartments, as in the blacksmith's bellows and the organ bellows, the flow is continuous. Powerful compression pumps, operated by engines, are much used for compressing air on a large scale for various industrial purposes, as for operating air brakes, air drills, pneumatic ham- mers, etc., and for keeping diving bells, caissons, and tunnels sup- plied with air while work is being done under water. (Fig. 41 .) |f 61. The Lifting Pump. The common lifting or suction pump (Fig. 42) is one of the oldest mechanical devices, its use dat- ing from the fourth century B.C. It is similar to the air pump in its construction and action. The valves open upward, as shown in the figure. A pipe extends from the cylinder or barrel of the pump to some distance below the surface of the water in the well or cistern. The piston is operated by means of a handle, acting as a lever. The pump at first acts as an air pump in exhausting the air from the pipe. While this is taking place, the pressure of the air remaining FIG. 41. Diving Bell. 6 4 STATICS OF GASES in the pipe decreases, and the greater pressure of the atmosphere upon the water in the well pushes water up into the pipe, just as mercury is forced up and sustained in a barometer tube. After the pump is filled with water in this manner, the closing of the lower valve during the down-stroke of the piston prevents the return of the water into the pipe. At the same time the valve in the piston is forced open, and t the water flows through it into the space above. At the beginning of the up-stroke the valve in the piston falls, and the water above it is lifted out. Since the entire pressure of the at- mosphere at sea-level can sustain a column of water only to a height of about 10.3 meters (34 ft.), the lower valve would have to be within that distance of the water in the well even if the pump were capable of producing a perfect vacuum. The actual limit of distance for a good pump is about 26 feet. FIG. 42. Lift or Suction Pump. 62. The Force Pump. In the force pump the second valve is placed at the entrance to the discharge pipe, B (Fig. 43). There is no valve in the piston. The action of the pump during the up-stroke of the piston is the same as in the lifting pump. (Which valve is open? Which closed?) With the down- stroke of the piston the water in the barrel of the pump is forced into the discharge pipe. The height to which water can be forced in this pipe depends only upon the strength of the pump, being in no way affected by atmospheric pressure. FIG. 43. Force Pump. THE MECHANICS OF FLUIDS A force pump is generally provided with an air chamber, which is connected with -the discharge pipe. During the down- stroke of the piston some of the water is forced into this chamber, and compresses the air. During the up-stroke the expanding air drives the water out of the chamber, thus maintaining a contin- uous flow. FIG. 44. Sectional View of a Steam Pump. Force pumps are used to raise water to a higher level in filling tanks, reservoirs, and stand-pipes, and to deliver it under great pressure, as in hydraulic presses (Art. 35) and fire engines. They are operated by hand, by windmills, and by engines (Fig. 44). 63. The Siphon. A bent tube or pipe for conveying liquids over an elevation from a higher to a lower level is called a siphon (Fig. 45, a and b). Either a rigid or a flexible tube will serve the purpose. To start a small siphon it may be held with the bend down and filled, then, with a fin- ger over each end, inverted FIG. 450. The Siphon. FIG. 456. Aspirating Siphon. and placed in position; or it may first be placed in position, and the air then exhausted by "applying the mouth to the lower end. The liquid will continue to flow as long as one end of the siphon is covered by it and the other 66 STATICS OF GASES end is below the level of its surface (i.e. below ab in the figure); but if the outlet of the siphon is also immersed, the flow will cease as soon as the liquid reaches the same level in the two vessels. To explain the action of the siphon we may suppose it to be stopped by closing the outlet, c, with the finger. The liquid will then be at rest, and the laws of pressure for liquids in equilibrium will hold. At points a and b in the tube, on a level with the surface of the liquid, the pressure is the same as that of the atmosphere. (How do we know?) The pressure at c is equal to this plus the gravity pressure of the liquid column be. When the finger is removed, the only upward pressure at c is that of the atmosphere, which leaves the gravity pressure of the column be unbal- anced. This unbalanced downward pres- sure, acting on the liquid in the siphon, causes- it to flow. The liquid would part at the top and run out at both ends, leav- ing the siphon empty, if it were not for the pressure of the atmosphere, which, acting inward at both ends, holds the liquid in a continuous column and compels it all to flow in the same direction. The siphon is useful in drawing liquids from vessels where pouring is inconvenient, and in removing the upper part of a liquid when the lower part is of a different kind or quality or contains sediment. It is usu- ally provided with a suction tube (Fig. 456) for starting the flow without permitting any of the liquid to enter the mouth. To start such a siphon the lower end is closed while the air is exhausted through the suction tube. 64. The Balloon. Since the buoyant force upon a balloon is equal to the weight of the displaced air, its amount depends only upon the size of the balloon. The carrying capacity of a balloon is determined by the difference between the buoyant force and its own weight, including the true weight of the gas with which it is filled. Hence it is an advantage to use hydrogen, which is the least dense of gases; but illuminating gas is generally used, as it is cheaper and more easily obtained. The first balloons were inflated with hot air. If a balloon is not fully inflated at the start, the gas within it Baling Bags Guide or Trail Rope FIG. 46. Balloon. THE MECHANICS OF FLUIDS 67 expands as the balloon rises, in consequence of the diminishing atmospheric pressure upon it. As long as there is room for this expansion, the buoyant force remains constant, for the increase in the volume of the displaced air offsets the decrease in its density. As a balloon rises after becoming fully distended, the buoyant force decreases until it is no greater than the true weight of the balloon and all it carries. It then ceases to rise, unless lightened by throw- ing out sand, a supply of which is carried for this purpose. When the aeronaut wishes to descend, he opens a valve at the top of the balloon, and some of the gas escapes. PROBLEMS 1. Over how great an elevation can water be siphoned? Why? Over how great an elevation can mercury be siphoned? Would a siphon work in a vacuum? Explain. . . 2. (a) At ordinary temperatures and under a pressure of one atmosphere a cubic meter of air weighs about 1.2 kg., a cubic meter of hydrogen about .083 kg., and a cubic meter of illuminating gas about .74 kg. Assuming these values, what is the buoyant force upon a balloon containing 500 cu. m. of hydrogen?, (b) How great a weight will this buoyant force sustain in addition to the weight of the hydrogen? 3. With what volume of illuminating gas must a balloon be filled to rise, if the empty balloon, the car, and the occupants together weigh 500 kg.? 4. Will the true weight of a body be greater or less than its weight in air when weighed on an equal-arm balance with brass weights (a) if the density of the body is the same as that of brass? (b) if its density is less? (c) if its density is greater? 5. The human heart is a pair of force pumps. Consult any physiology for a description of it and its action. Compare its construction and action with that of a force pump as described above. How is a continuous flow of blood maintained in the arteries; in other words, what corresponds to the air chamber of an ordinary force pump? CHAPTER V STATICS OF SOLIDS 65. Introduction. Every portion of matter on the earth, whether at rest or in motion, is constantly acted upon by a number of forces. A body is always subject to the action of gravity (its weight) ; and, except in the very unusual case of a body falling freely in a vacuum, it is at every instant acted upon by one or more other forces as well. When a body at rest remains at rest, or a body in motion continues with unchanging motion, we conclude without further evidence that the joint effect of all the forces then acting upon it is nil or zero, so far as the body as a whole is concerned; and the forces thus acting are said to be in equilibrium, or to constitute a set of balanced forces (Art. 10). Many examples have been studied in the previous chapters. The weight of a floating body and the buoyant force exerted upon it by the liquid in which it is floating constitute a pair of balanced forces. In this case, as in many others, the pressure of the air may be left out of account, since its net result is a buoyant force which is negli- gibly small. Forces not in equilibrium have also been incidentally considered, e.g. the forces acting on a cork released under water. The buoyant force in this case. is- greater than the weight of the cork; and the excess of buoyancy, being unbalanced, pushes the cork to the surface. It is only under certain special conditions that a set of forces, acting together on a body, will be in equilibrium, and it is with these conditions or relations that we are prin- cipally concerned in the present chapter. This study yields 68 CONCURRENT FORCES 69 many interesting and important facts concerning the ordi- nary behavior of bodies, and affords at least some insight into the mechanical principles involved in the arch, the truss, the suspension cable, and other structural forms employed in building houses, bridges, etc. In passing from the mechanics of fluids to the mechanics of solids, it should be noted that forces acting on solids may be and often are concentrated practically at points; hence, as a rule, we shall not have occasion to consider areas or force per unit area. I. CONCURRENT FORCES 66. Equilibrium of Two Forces. The relations that must exist among two or more forces in order that they may balance each other are referred to as the conditions necessary for equilibrium, or, simply, the conditions of equilibrium. The conditions necessary for the equilibrium of two forces were briefly considered for solids in Art. 10, and they have been exempli- fied repeatedly in the study of fluids. However, a further study of FIG. 47. Two Forces not in Equilibrium. this simple case can hardly fail to add to the pupil's understanding of it. For this purpose we may use two spring balances and a board, the latter resting upon a number of small marbles or bicycle balls placed on a table (Fig. 47). Cords are attached to nails at A and B. Horizontal forces are applied to the board through these cords, and are measured by the balances. If these forces are in equilibrium with each other, the board will remain at rest; if they are not in equilib- rium, it will move, since the friction is inappreciable. The experi- ment yields the following results: (i) When equal forces are applied in opposite directions but not along the same line (Fig. 47), the board 70 STATICS OF SOLIDS will not be in equilibrium, but will turn round until the forces act along the same line (Fig. 48). The board will then be in equilibrium. (2) When the applied forces are opposite and have the same line of FIG. 48. Two Forces in Equilibrium. action, but are unequal, the board will be pulled in the direction of the greater force. (3) When the forces are either equal or unequal, but not opposite in direction, the board will be moved. Hence, in general Two forces acting upon the same body balance each other when and only when they are equal in magnitude, opposite in direction, and have the same line of action. A and B are the points of application of the forces. A force has the same effect upon a solid when it 'is applied at any other point in the same line of action. Thus, if either of the equal and opposite forces in the above experiment were applied at C (Fig. 48) instead of at A or B, the two would still be in equilibrium. 67. The Elements of a Force. The effect of a force depends upon three things, namely, its magnitude, its direc- tion, and its line of action (or its point of application) . These are called the elements of a force. They must all be considered in describing and comparing forces and in discussing their effects, whether the forces are balanced or unbalanced. 68. The Geometrical Representation of Forces. The relative magnitudes and directions and the points of appli- cation of a set of forces can be accurately shown in a dia- gram in which each force is represented by a straight line. The direction of the force is represented by the direction CONCURRENT FORCES 71 of the line, as indicated by an arrow-head placed on it; the magnitude of the force, by the length of the line; and the point of application of the force, by the point from which the line is drawn (the end of the line from whrch the arrow-head points)^ The method is illustrated in Fig. 49, which represents two forces having a common point of application, 0, and differing in direction by a right angle. We see that the force represented by OB is. twice as great as the other, since the line representing it is twice o as long; but the diagram does not give the numerical value of the forces unless the scale adopted in the construction is known. The magnitude of a force can be represented on any scale desired. Thus i cm. may represent 10 g., 100 g., 500 g., etc.; but all forces must be represented on the same scale in- the same figure^ The geometrical representation of forces and their relation to one another is exceedingly useful, and is constantly employed in this and the following chapters. The pupil will have practise in the con- struction and use of such diagrams in connection with the laboratory work and in the solution of problems. 69. Equilibrium of Three Concurrent Forces. Forces whose lines of action meet in a point are called concurrent forces. Concurrent forces may or may not have a common point of application, but, if not, the lines along which they act meet in a point when produced. Experiment shows that three concurrent forces are in equilibrium only when certain definite relations, as regards magnitude and direction, exist among them. A simple form of apparatus for studying these relations is shown in Fig. 50. Three cords are tied to a ring and a spring balance pulls on each. Nails or clamps are provided to hold the STATICS OF SOLIDS FIG. 50. balances in position on a large board or a table. The ring moves to a position in which the three pulls exerted on it are in equilibrium. The lines of action of these forces lie in the same plane and are concur- rent at or near the center of the ring. The forces act out- ward from this point in the directions of the three cords, and their magnitudes are given by the readings of the scales. In order to determine the rela- tions which exist among the forces, they are represented in magnitude and direction by the lines a, b, and c, respectively (Fig. 51), according to the rules given above. If the experimental work and the construction are accurate, it will be found that the diagonal R of the par- allelogram constructed upon any two of these lines as sides is equal to the third line, and is in exactly the opposite direc- tion from O. The conditions necessary for equilibrium may therefore be stated as follows : Three concurrent forces, acting on the same body, are in equi- librium only when their lines oj action lie in the same plane and their magnitudes and direc- tions are such that, if the lines representing any two of them be taken as the sides of a parallelogram, the concurrent diagonal of this parallelogram will be equal and opposite to the line representing the third force. FIG. 51. Three Concur- rent Forces, a, b, and c, in Equilibrium. CONCURRENT FORCES 73 70. Resultant Force. The resultant of two or more forces, acting upon the same body, is the single force that would produce the same effect upon the body as the given forces, if it were substituted for them (Art. n). It is frequently necessary in studying mechanical problems to "find" the resultant of given forces, i.e. to determine the magnitude, direction, and line of action of the single equivalent force. In doing this we are not in the least con- cerned with the actual or possible substitution of such a force for the given forces. Such a substitution may or may not be possible. For example, if a cork weighing 20 g. is released under water and the buoyant force of the water upon it is 100 g., the resultant of these forces is a force of 80 g., acting vertically upward; by which we mean that the cork rises through the water just as rapidly as it would if it were acted upon by a single upward force of 80 g. in place of its weight and buoyancy. The actual impossibility of making the substitution in this case does not enter into the question. The process of finding the resultant of two or more given forces is called the composition of forces. In the case of forces acting in both directions along the same line, the resultant is found by subtracting the sum of all the forces that act in one direction along the line from the sum of all the forces that act in the opposite direction. For example, if two ropes are attached to opposite ends of a log and two boys pull, one with a force of 30 Ib. and the other with a force of 40 Ib., upon one rope in the direction of the length of the log, and a third boy pulls with a force of 50 Ib. in the opposite direction upon the other rope, the resultant of these forces will be a force of 20 Ib. in the direction in which the two boys pull. Whether the log will move or not will depend upon whether a force of 20 Ib. is sufficient to overcome the friction between the log and the ground. Other methods are required for the composition of forces acting at an angle with one another or along different parallel lines, as is shown in the following articles. 74 STATICS OF SOLIDS 71. Resultant of Two Concurrent Forces. Parallelo- gram of Forces. Any two of three forces in equilibrium may be regarded as together balancing the third; hence (by definition) their resultant is the single force which would also balance the third ; hence, further (Art. 66) , this resultant must be equal and opposite to the third and must act along the same line. Referring now to Fig. 51, it will be recalled that the ^diagonal R is 'equal and opposite to. c; hence it correctly represents the resultant of the forces denoted by a and b. The sides a and b and the diagonal R of the parallelogram correctly represent the relations between the two concur- rent forces and their resultant irrespective of any third force ; hence // two concurrent forces are represented by lines drawn from the same point, the concurrent diagonal of the parallelogram constructed upon these lines as sides will represent their result- ant in magnitude and direction. This construction is known as the parallelogram of forces. The numerical value of the resultant is found by accurately measuring the length of the diagonal, and computing the force that this length of line represents according 'to the scale adopted in the construction. The accuracy of the numerical result will depend, of course, upon the care and skill exercised in constructing the parallelogram and measuring the diagonal. The resultanf of two concurrent forces can always be found by the parallelogram construction. It can also be computed by the rules of trigonometry. In a few special cases it can be computed from the relations established in plane geometry. The most important of these cases is that of two forces acting at an angle of 90. In this case the resultant is equal to the square root of the sum of the squares of the given forces (since the square of the hypothenuse of a right triangle is equal to the sum of the squares of the other two sides). CONCURRENT FORCES 75 72. Composition of More than Two Concurrent Forces. The resultant of any number of concurrent forces can be found by combin- ing the resultant of any two of them with a third, their resultant with a fourth, and so on till each force has been included once in the con- struction or computation. The last resultant is the resultant of all the forces. It should be noted that we have the privilege of com- bining the forces in any order; for it sometimes happens that the resultant can easily be computed from geometrical relations when a certain order is followed, while any other order leads to difficulties. (See, for example, the eleventh problem in the following set.) 73. Equilibrant. The single force that would balance one or more given forces is called their equilibrant. The equilibrant of any number of forces is equal and opposite to their resultant. (Why?) Either of two forces in equilib- rium is the equilibrant of the other; and any one of three forces in equilibrium is the equilibrant of the other two (Fig. 51)- 74. Resolution of a Force. Component Forces. It is frequently necessary in studying the effects of a force to consider it as being replaced by two or more concurrent forces which are together equivalent to it. The process of finding the required set of equivalent forces is the reverse of composition, and is known as the resolution of the given force into its components. It is effected by constructing the parallelogram of forces, as will readily be understood by referring to Fig. 51. Since R in this figure denotes a force which is equivalent to the forces denoted by a and b, it follows that the forces denoted by a and b are together equivalent to the force denoted by R. The following example will serve to illustrate. A block of wood weighing 1200 g. is placed on a plane, AB (Fig. 5 2), which is inclined at an angle such that the length of the plane AB is three times its height BC. How great must 76 STATICS OF SOLIDS friction be to keep the block from sliding, and what pressure does the block exert upon the plane? As indicated in the ques- tions, the weight of the block (represented by OW in the figure) gives rise to two effects: (i) a tendency of the block to slide down the plane, and (2) a pressure of the block against the plane. The first effect would be produced by a force FIG. 52. Forces OF and OP are together of a certain magnitude acting equivalent to weight OW. on the block in the direction in which it tends to slide (parallel to the plane) ; . and the second effect would be produced by a force of a certain magnitude acting on the block in the direction in which it presses against the plane (perpendic- ular to the plane). The problem then consists in finding the magni- tudes of these two forces, which together would be equivalent to the weight of the block. We have, therefore, to construct the parallelo- gram of forces, having given the diagonal OW and the directions OM and ON (but not the lengths) of two adjatent sides. The sides OF and OP of this parallelogram represent, therefore, the magnitudes as well as the directions of the forces sought. Triangles OFW and ABC are similar. (Why?) Hence 7^. = -TR = " 5 from which OF 3 3 = 400 g. If a force of 400 g. is sufficient to overcome friction, the block will slide; if not, it will remain at rest. The pressure against the plane, represented by OP, is VI2O0 2 4OO 2 = 1131.2 g. A given set of two or more forces has only one resultant, as is shown by the fact that only one parallelogram can be constructed when two adjacent sides and the included angle are given; but any number of different sets of forces can be found which are equivalent to a given force, since any num- ber of parallelograms can be constructed on a given line as a diagonal. When there are only two components and their directions are given or determined, there is but one solution, as in the above problem. CONCURRENT FORCES 77 PROBLEMS Note. The following problems are all to be solved by computation, based upon known geometrical relations. A figure drawn with a rough approximation to accuracy, as in geometrical demonstrations, will serve to present these relations to trie eye. 1. A weight of 100 kg. is supported by two cords mak- ing equal angles with the horizontal and an angle of 120 with each other. What is the tension on each cord? 2. What would be the tension on each cord supporting FIG. 53- the above weight if one made an angle of 30 with the horizontal, and the other 60 ? 3. What would be the tension on each cord if each made an angle of 45 with the horizontal? 4. A ball is placed on a plane inclined at an angle of 30. What fraction of its weight tends to cause motion down the plane? 5. If the weight of the ball in the previous problem is 2 kg., what pressure does it exert upon the plane? 6. A picture weighing 20 Ib. is hung by a cord passing over a nail, the two parts of the cord mak- ing an angle of 60 with each other. ' What is the tension of the cord? 7. A square space is enclosed by passing a rope around 4 posts at the corners, and the tension of the rope is 50 Ib. What is the magnitude and the direction of the resultant force on each post? 8. The beam AB (Fig. 54) of a derrick is in- clined at an angle of 30 with the vertical center post AC, and a weight of 3 tons hangs from the upper end of the beam. Find the tension of the horizontal cable BC. 9. A hammock is suspended from two hooks at the same height and 12 ft. apart. A person weighing 160 Ib. sits B at the center. What is the tension of the ropes (a) if the center of the hammock is 4 ft. below the level of the hooks? (6) if it is 2.5 ft. below? 10. A street lamp weighing 80 Ib. is supported by a bracket projecting 3 ft. from a wall (Fig. 55) . The brace AC meets the wall 4 ft. below the tie-rod, AB. Is the force sustained by the tie-rod a pull or a push? the force sustained by the brace? Compute the forces sustained by each. FIG. 54. STATICS OF SOLIDS FIG. 56. 11. Three ropes pull horizontally upon a post (Fig. 56). The tension of A is 50 lb., of B i6o.'lb., and of C nolb. A and B pull in the same straight line, and C at right angles to-them. Find the resultant force on the post. 12. As the angle between two forces increases from o to 180, how does their resultant vary? What is the value of the resultant at the beginning? at the end? II. PARALLEL FORCES 75. Equilibrium of Three Parallel Forces. Parallel forces are forces having parallel lines of action. It is found by experiment that, if three parallel forces acting on the same body are in equilib- rium, the following condi- tions are always fulfilled: 1. The three forces, f\, / 2 , and / 3 (Fig. 57), are in one plane. 2. The two outside forces act in the same direction, and the inside force in the oppo- site direction. 3 . The inside force is equal to the sum of the other two. 4. The outside forces are inversely proportional to the distances (i.e. the shortest, or per- pendicular distances) of their lines of action from the line of action of the inside force; i.e. f\ :/2 :: d 2 : d i} or/i^i =f^d^ It should be noted that the inside force is nearer the larger of the outside forces if they are unequal; if they are equal, it is midway between them. The points of application of the forces need not lie in a straight line (Fig. 58). Any one of the three forces may be regarded as the equilibrant of the other two. FIG. 57. Three Parallel Forces in Equilibrium. PARALLEL FORCES 79 76. Resultant of Two Parallel Forces Having the Same Direction. When three parallel forces are in equilibrium, the two outside forces together balance the third force; hence their resultant would also balance it. This result- FIG. 58. General Case of Three Par- FIG. 59. Resultant of Two Parallel allel Forces in Equilibrium. Forces, f\ and /2. ant, / 3 (Fig. 59), must, therefore, have the same line of action as the third force, and must be equal to it in magni- tude and opposite in direction ; hence The resultant of two parallel forces having the same direc- tion is equal to their sum, it acts in the same direction as the component forces, and its line of action divides the distance between them into parts which are inversely proportional to the forces. PROBLEMS 1. ?Two boys, A and B, carry a load between them suspended from a pole 5 ft. long. The load is 2 ft. from ^4's end. What fraction of it does A carry? 2. If the load weighs 161 lb., where must it be hung in order that A may carry Q2lb.-of it? 3. Two horses draw a plow by means of a doubletree, D (Fig. 60), at each end of which is a singletree, S. What are the relative magnitudes and directions of the three forces acting on the doubletree, if its arms are of equal length? If the dcyubletree is 4 ft. long, what must be the length of each arm in order that one of the horses shall draw three fifths of the load? FIG. 60. 8o STATICS OF SOLIDS 4. A man carries a weight of 20 Ib. on the end of a stick 3 ft. long, placed over his shoulder. He holds the stick at the other end. What is the pressure on his shoulder (a) if the distance from his shoulder to his hand is 2 ft.? (6) if this distance is i ft.? III. MOMENTS or FORCE 77. The Rotative Action of a Force. The conditions which determine the greater or less effect of a force in pro- ducing or opposing rotation are well illustrated in the child's game of seesaw. FIG. 61. Moments of Force in Equilibrium, fidi facto. Two boys of equal weight will sit at equal distances from the axis about which the board turns; but if they are of unequal weight, the lighter boy will sit at the greater distance. The boy at the top makes his end descend by leaning backward, thus, in effect, increasing his distance from the axis. The boy at the bottom decreases his distance by leaning forward. Evidently the effectiveness of a force (in this case the weight of the boy) in producing rotation depends in part upon the magnitude of the force and in part upon its distance from the axis. The exact relation is easily derived from an experiment patterned after the seesaw. A slender stick (meter rod) is supported on a horizontal axis (a nail), through a hole so situated that the rod comes to rest in a horizontal position. Two weights, either equal or unequal, hung from the rod on opposite sides of the axis can be so adjusted as to leave the rod in equilibrium (Fig. 61). With this adjustment, the tendency of the one weight to pull its end of the rod down is balanced by the equal and opposite tendency of the other weight to pull its end down. In every such case the weights are MOMENTS OF FORCE 81 inversely proportional to their distances from the axis (f\ :/2" 02 : fli), or the product of one force and its distance from the axis is equal to the product of the other force and its distance. If one of the weights is replaced by a spring balance, a measured pull can be exerted in i . ,'' \ i any direction (Fig. 62). It will then be found j\ that the force fa required to maintain equi- _\/i \ 2 librium increases as the direction of the force FlG 62 changes from the vertical toward the horizon- tal. But, at the same time, the distance 02 from the line of action of the force to the axis decreases, so that the product faa^ remains equal to the product f\a\. The products f\a\ and /2#2, therefore, measure the effectiveness of the forces in producing or opposing rotation. The effectiveness of a force in producing or opposing rotation about a given axis is called the moment of the force with respect to that axis. The moment of a force is meas- ured by the product of the force and the distance (i.e. the perpendicular distance) of its line of action from the axis. This perpendicular distance is called the arm of the force. Thus in the above experiment ai is the arm of the force /i with respect to an axis at 0, and /i#i is the moment of the force; a* is the arm of the force / 2 , and/ 2 a 2 is its moment with respect to the same axis. 78. Equilibrium of Moments. The general condition necessary for the equilibrium of a body with respect to rotation is that the sum of all the moments of force tend- ing to turn the body in one direction round any axis must be equal to the sum of all the moments of force tending to turn the body in the opposite direction round the same axis. Direction round an axis is termed clockwise, if it is as the hands of a clock turn, and counter- clockwise, if in the opposite direction round. A clockwise 82 STATICS OF SOLIDS and a counter-clockwise moment may act on the same side of an axis and be in equilibrium (Fig. 63). In the case shown in Fig. 64 there are two clockwise moments, and /33, and one counter-clockwise moment, f\a\. Hence the condition for equilibrium with respect to, an axis at O is/i#i =/202 + faaz- The force repre- sented by /4 does not tend to cause rotation in either direction, since its line of action passes through the axis. Its arm is zero and its moment is zero. The same is true of the pressure at the axis. The law as stated above holds for any body, whether actually supported on an axis or not. An axis may be assumed in any position, and, if the body is in equilibrium, the sum of the clockwise moments, taken with respect to the assumed axis, will be equal to the sum of the counter-clockwise moments, taken with respect to the same axis. Thus in the case of equilibrium shown in Fig. 58, assuming an axis at the point of application of /s, the moment of this force is zero, the moment of /i is/i^i, and the moment of /2 isfodz. But in the previous study of this case we learned that/idi = fzdz; and since these moments are opposite in direction, they fulfil the conditions necessary for equilibrium. Again, if the axis is assumed at the point of appli- cation of /i, the moment of this force is now zero, the moment of fa isfadi, and the moment of /2 is/2 (^i + d%). But/ 2 ( d = vtj also v =-, and t = -. (i) o { >-< 100. Representation and Composition of FlG - 10 Velocities. A velocity may be represented both in mag- nitude and direction by a straight line, just as a force may be. Thus if OB (Fig. 100) represents a velocity of 2 ft. per second toward the north, then OA represents a velocity of 3 ft. per second toward the east. 106 DYNAMICS A body may have two or more independent motions at the same time. Thus a boat rowed across a stream has a motion imparted by the rowing, and also a motion due to the current and equal to it. Suppose the boat to be constantly headed directly toward the opposite shore, and let O (Fig. 101) represent the starting point. OB would be the path of the boat if there were no current. OC is the dis- tance the stream flows while the boat is crossing. The actual motion of the boat relative to the earth is the resultant of these two motions, and its path is repre- sented by OA. If OB and OC be taken to represent the component velocities (in- FIG. 101. stea d O f tne wno ie distances), then OA, which is the concurrent diagonal of the parallelogram constructed on OB and OC as sides, will represent the actual or resultant velocity upon the same scale. Veloc- ities are, in fact, compounded by the same rules as forces, and the construction is called the parallelogram of velocities. 101. Resolution of a Velocity. A velocity, like a force, can be resolved into components in any chosen directions, and the construction is the same as for the resolution of a force. For example, suppose we wish to know at what rate a vessel is advancing northward and at what rate eastward, when it is sailing 30 north of east at a rate of FIG. 102. 12 mi. per hour. If OA (Fig. 102) represents the velocity of the vessel, ON and OE will represent its northerly and easterly com- ponents, respectively, to the same scale. Now it is proved MOTION 107 in geometry that in a right triangle, having an acute angle of 30, the hypothenuse is twice the shorter side. Hence ON represents a velocity of 6 mi. per hour, and OE a velocity of V i2 2 6 2 , or 10.4 mi. per hour. As a further illustration, let us consider how the boat mentioned in the preceding article must be rowed in order to reach the opposite bank at B instead of A . ' The resultant motion is now represented by OB. The component OC, due to the motion of the stream, is the same as before. Hence OB (Fig. 103) is the diagonal of a parallelogram of which one side is OC. The other component motion is therefore represented by OA'. This means that the boat must be constantly pointed in a direction parallel to OA ', and that it would take as long to reach B as it would to row the distance OA ' in still water. PROBLEMS 1. A velocity of 50 mi. per hour is a velocity of how many feet per second? 2. A train runs with a velocity of 23 m. per second. In what time does it run a kilometer? 3. From a train running at the rate of 9 m. per second, a mail-bag is thrown at right angles to the track with a velocity of 4 m. per second. Com- pute the resultant velocity of the bag at the instant it leaves the hand, and draw a figure to show its direction. 4. From a train running at the rate of 12 m. per second, a mail-bag is thrown so that its resultant velocity is equal to that of the train and at right angles to it. What is the magnitude and direction of the velocity im- parted in throwing the bag? 6. An arrow is shot directly backward from the rear of a train with a velocity (relative to the train) equal to that of the train. What is the motion of the arrow? 6. The rotation of the earth carries its surface eastward at the rate of about \ mi. per second (in temperate latitudes). When a ball is thrown up, why is it not left behind (to the west) by the earth in its rotation? io8 * DYNAMICS 7. Four boys, A, B, C, and D (Fig. 104), on the deck of a moving vessel, pass a ball round in the order of the letters. What allowance for the motion of the vessel, if any, must be made by each of the boys in throwing? Give reasons. 8. A vessel sails due N. E. at the rate of 15 mi. per hour. FIG. 104. Compute the northerly and easterly components of its velocity. 9. A boatman wishes to cross a stream where it is 100 m. wide and its velocity .8 m. per second. The component velocity that he imparts to the boat by rowing is 1.2 m. per second, (a) How long will it take him to cross the stream to a point directly opposite to the starting point (Fig. 103)? (b) How long will it take him to cross if he rows as shown in Fig. 101? 10. A boy is riding north with a velocity of 12 mi. per hour, (a) What is the apparent direction and velocity of the wind if the air is still? (b) What if there is an east wind of 20 mi. per hour? 102. Variable Motion. If the motion of a body is variable, qualifying terms must be used in specifying its velocity, such as its velocity at a certain instant, or at a cer- tain point of its path, or its average velocity during a speci- fied interval of time, etc. The unqualified term "velocity" refers to no one of these velocities in particular, and hence has no definite or intelligible meaning. For example, the language is definite when we say that the velocity of a fall- ing body is 128.6 ft. per second at the end of the fourth second from the start. The velocity of a body at a given instant is measured by the distance it would pass over during the following unit of time, if its velocity continued unchanged from that instant. Thus when we say that a train is running at the rate of 30 mi. per hour, we mean that it would run 30 miles in an hour if it continued at its existing rate for one hour. The average velocity of a body during any inter- val of time is equal to the uniform velocity required to cover the same distance in the same time. Thus if an MOTION 109 automobile goes 108 mi. in 6 hr., its average rate is 18 mi. per hour, since this is the uniform rate required to run the given distance in the given time. The actual velocity may vary from zero (during intervals of stopping) to 40 or more miles per hour. It follows from the definition that aver- age velocity is equal to the distance divided by the time. Representing average velocity by v, its definition is expressed by the formula d = vt ; from which v = (2) 103. Acceleration. A change of velocity may consist in a change (increase or decrease) of speed, or a change in the direction of motion, or in a change both of speed and direction. The term acceleration, in its general sense, includes all of these possible changes in velocity. For motion in a straight line, acceleration consists in a change of speed only, and is uniform or constant if the speed in- creases or decreases uniformly with the time (not with the distance passed over). Uniform acceleration in the line of motion is measured by the change of speed per second. For example, the speed of a body is uniformly accelerated if it increases by 3 m. per second every second. At the end of the first second the speed would be 3 m. per second; at the end of 2 sec. it would be 6 m. per second; at the end of 3 sec., 9 m. per second; etc. The motion in this case is said to be accelerated at the rate of 3 m. per second per second. The repetition of the phrase "per second" is necessary, for time is doubly involved in acceleration. With an acceleration of 3 m. per second per second, a velocity of 180 m. per second is gained in a minute; hence this might be expressed as an acceleration of 180 m. per second per minute. no DYNAMICS The principal metric unit of acceleration is an accelera- tion of i cm. per second per second; the principal English unit is an acceleration of i ft. per second per second. Generally the acceleration of bodies is not uniform but variable. A train gains speed less and less rapidly for some distance after starting, until finally the acceleration be- comes zero and the speed constant. The acceleration of a street car is irregular, increasing abruptly whenever more power is turned on, and decreasing steadily during the intervals between. This occurs several times in getting up speed. The best example of uniformly accelerated motion is a falling body, provided its motion is not sensibly affected by the resistance of the air. The acceleration of a falling body varies slightly in different latitudes. In the temper- ate zones it is close to 980 cm. or 32.16 ft. per second per second. It is important to understand that in uniformly accel- erated motion the acceleration is the same, not only for each second, but for any number of seconds or any frac- tion of a second. In other words, the rate of change of velocity is the same during the first hundredth part of a second as it is during any part of the time or the whole of it. Acceleration is called positive when it consists in increase of speed, negative when it consists in decrease of speed. It is customary in common speech and in elemen- tary physics to use the terms retarded motion and retarda- tion instead of negatively accelerated motion and negative acceleration. 104. Digression on the Average Value of a Uniformly Changing Variable. If a board is 6 in. wide at one end and 1 2 in. wide at the other, it does not follow that its average width is or 9 in. By definition, its average width is the uniform width of a board of MOTION in the same length and of equal area. If the shape of the board is as shown in Fig. 105, its average width is 9 in.; for the piece def would fit in the position aeg, giving a uniform width of 9 in. We see also that the average width of the board is its actual width midway be- FIG. 105. FIG. 106. tween its ends. The width of this board increases uniformly from one end to the other; i.e. starting at the narrow end, there is the same increase of width for each foot of length. None of the above relations hold in the case of a board having the shape abcde (Fig. 106) or the shape abcde'. In the one case the average width is obviously less than the half sum of the widths at the ends, and in the other case it is greater, in neither case is it equal to the actual width midway between the ends. Similarly, the average speed of a body, during any time interval within which the speed changes uniformly with the time, is equal to the half sum of the speed at the beginning and at the end of that time interval, and is also the actual speed at the end of half that interval. To illustrate: Since a falling body acquires a velocity of 32.16 ft. per second during each second of its fall, its velocity at the end of four seconds from the start is 4 X 32.16 ft., or 128.64 ft. per second. Since the increase of velocity is uniform and the velocity at the start is (0 + 128.64) zero, the average velocity during the four seconds is - or 64.32 ft. per second. This is also the actual velocity at the end of half that interval, or at the end of two seconds. The whole distance covered in the fall is, of course, the product of the average velocity and the time, which is 4 X 64.32, or 257.28 ft. 105. An Experiment on Uniformly Accelerated Motion. - The motion of a freely falling body is too rapid for con- venient experimental study. This difficulty is overcome by taking instead the motion of a sphere on an inclined plane. The acceleration may be made as small as we 112 DYNAMICS please by diminishing the inclination of the plane; and, if the sphere and the plane are as nearly perfect as may be, the acceleration is uniform, as in the case of a freely falling FIG. 1070. FIG. io7&. body. This method was adopted by the great Italian math- ematician and scientist, Galileo, who by means of it dis- covered the laws of uniformly accelerated motion in the early part of the seventeenth century. Figures 107 a and 107 b represent convenient devices for determining the distances passed over by the sphere in equal intervals of time. The first consists of a polished inclined groove, the cross-section of which is an arc of a circle. A large steel ball, started at one side of the groove near the top, traces a visible wavy path in lycopodium powder or sulphur, dusted over the groove. This path is the result of two motions, namely, a rocking motion from side to side, which serves to mark equal intervals of time, and the accel- erated downward motion. The motion of the ball is so controlled at the start that its downward motion begins at the instant when it first crosses the middle of the groove. Hence the distances between successive points where the path crosses the middle line of the groove are covered in equal times. These distances will be found to be in the MOTION 113 ratio of the numbers i, 3, 5, 7, 9, n, etc., within a small limit of experimental error. The second form of apparatus provides four parallel grooves for four balls, which are released from the same height at the same instant. When the stops are adjusted so that the balls are stopped in succession at distances in the ratio of i, 3, 5, and 7, the balls are heard to strike at equal intervals of time. 106. Analysis of the Results. Distances Passed Over in Successive Seconds. The distances passed over in equal times in either of the above experiments are in the same ratio whether the time interval is one second or not. For simplicity, we shall assume the interval to be one second. If the first distance is denoted by di the distances passed over in successive seconds are di cm., $di cm., 5^ cm., 7 and v = ^2ad. (6) 2a These formulas are algebraic statements of the laws of uniformly accelerated motion in a straight line. (State these laws in words, naming in full the quantities for which the letters stand, instead of the letters.) If any two of the three quantities in any one of the above formulas are given, the value of the third quantity can be found by substituting the given values in the formula. 108. Falling Bodies. The formulas and laws for uni- formly accelerated speed hold for bodies falling freely n6 DYNAMICS from rest, their motion being uniformly accelerated in a straight (vertical) line; but in this case the acceleration is denoted by g, instead of a, since it is due to gravity. (Write, the formulas for falling bodies, and state their meaning in words, calling g " the acceleration due to gravity.") In numerical work the value of g is to be taken as 980 cm., or 32.16 ft. per second per second. 109. Composition of a Constant and a Uniformly Accelerated Velocity. If a body is already in motion when it begins to acquire a uniformly accelerated motion, its velocity at that instant is called the initial velocity, and its velocity at any later instant is the resultant of the initial and the accelerated velocities. Three cases arise, as follows: First Case: When the initial and the accelerated velocities have the same direction. Example: If a ball is thrown vertically downward, its initial velocity as a freely falling body is the velocity imparted in throwing; i.e. it has this velocity when the accelerated motion due to gravity begins. Since the initial velocity and the velocity acquired in falling have the same direction, their resultant is their sum. Let VQ denote the initial velocity of a body which moves in a straight line with a constant acceleration a, v its velocity at the end of / seconds, and d the distance passed over in that time; then V = VQ + at. (7) Since the velocity increases uniformly from the initial velocity VQ to the final velocity VQ + at, the average velocity v during the / seconds is half their sum, or v = $ [ VQ + (o + at) ] = VQ + \ at. Hence d = vt = v t + * a/ 2 . (8) The part v Q t is the distance the body would go in t seconds with the constant initial velocity VQ, and the part i at 2 is the additional distance due to the accelerated velocity. Second Case: When the initial and the accelerated velocities have op- posite directions. Example: A body thrown vertically upward loses velocity at the rate of 980 cm. per second per second, the acceleration due to gravity being negative, or opposite to the direction of motion. MOTION 117 Thus if the initial velocity were 49 m. per second, the body would con- tinue to rise for 5 sec. (49 -~ 9.8 = 5), and at the end of that time it would come to rest. Since the velocity decreases uniformly to zero, the average velocity is half the initial velocity. The height to which the body rises is equal to the product of the average velocity and the time of rise. A ball rolling on a smooth, level surface is also an example, its speed being uniformly retarded by friction. The formulas are as follows, the letters having the same meaning as above: v = VQ at', (9) d = vot - i at 2 . (10) If / is the whole time to the instant when the body comes to rest, the final velocity v is zero, and /= - (Why?), and^o = at. (n) Hence v = i VQ = \ at, and d = vt = i at 2 . (12) (State the meaning of these formulas, and compare with formulas 3, 4, and 5.) Third Case: When the initial and the accelerated velocities are at any angle with each other. Example: The muzzle velocity of a bullet is the initial velocity of its free flight. The initial velocity may have any direction; the accelerated velocity due to gravity is always vertical. Fig. 109 shows a simple device for an experimental illustration of this case. A slender board carries a shelf at one end, the other end is clamped in a vise. When the free end is drawn aside and released, a small object on the forward side of the shelf is driven before it, while, on the other side, the shelf slips from under a second object, which is thus released at the same instant as the first but without initial velocity. The two bodies reach the floor at the same time, showing that the acceler- ated motion in the vertical FlG - IOQ - direction is the same for both. In whatever direction a body may be thrown or projected into space, the accelerated vertical component of Us motion during its flight n8 DYNAMICS is the same as that of a body falling freely from rest, the other component of its motion being the initial velocity. The path of such a body is represented graphically in Figs, no and in. The direction of projection, OD, is horizontal in the first case, and obliquely upward in the second. In both figures OA repre- sents the initial velocity, and Oa rep- resents i g on the same scale. With the initial velocity only, the path of the body would be represented by OD, A B c D d' FIG. no. Path of Projectile, Initial Velocity Horizontal. FIG. in. Path of Projectile, Initial Velocity Oblique. and its position at the end of successive seconds by A, B, C, and D. With the accelerated velocity only, the path and the distances passed over in successive seconds would be as represented by Oabcd. Hence the points a', b f , c', and d' represent the position of the body at the end of successive seconds when these two motions occur at the same time. A smooth curve drawn through these points represents the actual path. The velocity of the body at any point of its path, as at b ' in the figures, is the resultant of the initial and the vertical velocities, the latter being the same as that of a body starting from rest and falling vertically for the same time. The direction of the resultant velocity is, of course, tangent to the curved path at every point. It should be noted that when the acceleration and the motion of the body are not in the same line, one consequence of the acceleration MOTION 119 is always a chdnge of direction of motion, and the path of the body is a curve. There may or may not be a change of speed at the same time. In the case just considered, the acceleration due to gravity results in a change of both speed and direction. PROBLEMS , 1. A street car runs with a constant acceleration of 1.2 m. per second per second for 8 sec. after starting, (a) What is its velocity at the end of that time? (b) What was its average velocity during the 8 sec.? (c) How far does it run in the 8 sec.? 2. A stone falls with a constant acceleration of 980 cm. per second per second. In what time will it acquire a velocity of 35 m. per second? 3. A body moves with a constant acceleration a. (a) How far does it go in the first second? (b) What is its average velocity during the first second? (c) What is the average velocity during the first 6 sec.? (d) What is the average velocity during the sixth second? 4. A train, running with constant acceleration, goes 560 m. during the first minute after starting. Find the acceleration in meters per second per second. 6. A car runs with a constant acceleration of 80 cm. per second per second for a distance of 300 m. (a) What is, then, its velocity? (b) With what average velocity did it run that distance? (c) How long did it take to run this distance? 6. A ball rolling along the ground is uniformly retarded at the rate of 4 m. per second per second. Its velocity at the start is 20 m. per sec. (a) How long will it roll? (b) How far will it roll? 7. How far does a body fall during the first. second? Account for the fact that this distance is numerically equal to half the acceleration. 8. A stone dropped from a cliff strikes the foot of it in 3.5 sec. What is the height of the cliff? 9. Two stones are thrown to the same height, one vertically, the other obliquely. Is the time of flight the same for both? Explain. 10. A stone thrown to the height of a tree reaches the ground in 5 sec. from the time of starting. How high is the tree? 11. A body is thrown horizontally, with an initial velocity of 100 ft. per second, from the top of a tower 150 ft. high. At what distance from the tower will the body strike the ground? 12. An arrow is shot vertically up with a velocity of 42 m. per sec. (a) How long will it rise? (b) How high will it rise? 120 DYNAMICS 13. A ball is thrown upward at an angle of 30 with the horizontal, with an initial velocity of 35 m. per second, (a) What is the time of its flight? (b) How high does it rise? (c) How far from the starting point does it strike the ground? SUGGESTION. Resolve the initial velocity into horizontal and vertical components. The first component is constant; the second is affected by gravity, just as it would be if the first component did not exist. II. NEWTON'S LAWS OF MOTION 110. The General Laws of Dynamics. In the preceding pages we have considered the arithmetical and geometrical relations involved in certain types of motion. These relations may be termed "laws of motion"; but they are mathematical rather than physical laws. "The laws of motion," preeminently so called, are the three general laws of dynamics, which, taken together, completely express the relations between mass, force, and motion. They were first stated in their present form by Sir Isaac Newton in his "Principia" (1687), and are universally known as Newton's first, second, and third laws of motion. 111. The First Law of Motion, or the Law of Inertia. - Every body continues in its state of rest or of uniform motion in a straight line, except in so far as it is compelled by exter- nal force to change that state. This law has already been considered (Art. 8); but the pupil is now in position to understand it more fully. The law asserts that a body at rest would remain at rest if no force acted upon it. This is an ideal case impossible of realization, since gravity, at least, always acts. Assuming the truth of the law, we conclude that, if a body at rest remains at rest, all the forces acting upon it completely neutralize one another's tendency to move the body; and NEWTON'S LAWS OF MOTION 121 these forces, by definition, constitute a set of balanced forces (Arts. 10 and 65). The law asserts further that a body in motion would continue in motion with constant speed in a straight line, if no force acted upon it. This is also an ideal case, and impossible of realization for the same reason as before. Again assuming the truth of the law, we conclude that, if a body in motion continues with constant speed in a straight line, all the forces acting upon it completely neutralize one another's tendency to change the motion of the body; and these forces, by definition, also constitute a set of bal- anced forces. To illustrate: The force necessary to pull a load at a uniform rate over a level surface is, by the law of inertia, equal to the sum of all the resisting forces of fric- tion; and the resultant of all the forces acting on the load, including friction, is zero. When mud flies from the wheels of a carriage, or water from a rapidly revolving grindstone, or a stone from a sling, or an automobile overturns in round- ing a corner too quickly, the phenomenon is nothing more than an exhibition of this universal tendency of moving matter to continue in its existing direction of motion. The body inevitably "flies off at a tangent" to its curved path at the instant when the unbalanced force that was causing the change of direction ceases or becomes inade- quate to the duty expected of it. The law of inertia, as will presently be shown, follows as a corollary, or special case, from the second law of motion. Hence whatever experimental evidence there may be of the truth of the second law holds equally for the first. 112. Relation Between Force and Acceleration, with a Given Mass. The second law of motion, in its usual form, is better understood from a previous study of two other 122 DYNAMICS laws, which are together equivalent to it. These we shall designate as laws 2 a and 26. 20. The acceleration of a given mass is proportional to the resultant force acting upon it, and is in the direction of this force. The law asserts that the acceleration of a body is due to the unbalanced or resultant force acting upon it, and that the two vary in the same ratio. This is illustrated by the motion of a sphere on a plane, when inclined at different angles. The resultant or accelerating force upon the sphere is the component of its weight whose direction is parallel to the plane (Art. 74). This component is doubled when the height of the plane (BC, Fig. 52) is doubled; and the ball then goes twice as far in the same time, showing that its acceleration has been doubled. With any two adjust- ments of the plane, the heights, the unbalanced forces, and the accelerations are all in the same ratio. In firing a cannon we have an excellent example of an enormous unbalanced force and its effect. A twelve-inch cannon 50 ft. long launches a projectile with a muzzle velocity of 3000 ft. per second. This velocity is imparted in about one hundredth of a second, and the average acceleration is approximately 50 miles per second per second. This necessitates an enormous accelerating force, which, with the standard charge of powder and projectile, amounts to no less than 3,600,000 Ib. (1800 tons) at the breech of the gun. If the resultant force upon a body varies during its motion, the acceleration varies in the same ratio; if the resultant force remains constant, the acceleration is also constant. The motion of a sphere on an inclined plane is uniformly accelerated (Art. 105) because the accelerating component of its weight remains constant for all posi- tions on the plane. The resultant force upon a freely falling body is NEWTON'S LAWS OF MOTION 123 its whole weight, if the resistance of the air is negligible; and this is a constant force, hence the acceleration is constant. If the body is light and has a relatively large surface, as a leaf or a sheet of paper, the resistance of the air is relatively large, and the resultant force upon the body as it falls is the difference between its weight and the resistance of the air. Hence the accele- ration in such cases is less than that of compact, heavy bodies. Moreover, the resistance of the air rapidly increases with the velocity, as every bicycle rider knows from experience; hence as a body falls, especially a light one, the resultant force upon it steadily diminishes, and may even become zero (the resistance of the air having be- come equal to the weight of the body). The rate of fall under such conditions is constant, as in ^, , . , i /r . N FIG. 112. An Ex- the case of a rain-drop or a parachute (r ig. 112) treme Case of Air after it has fallen a few hundred feet. Resistance The law holds for negative as well as for positive accel- eration. The rate at which the speed of a body decreases is in proportion to the resultant force acting opposite to the direction in which the body is moving. It takes a greater retarding force to stop a car more quickly, just as it takes a greater accelerating force to start it more quickly. The law further asserts that the acceleration is in the direction of the force. The motion and the resultant force may be in the same direction, in opposite directions, or at any angle with each other; but the acceleration and the force are always in the same line and in the same direction along that line. This is admirably illustrated by the motion of a heavy body in the air. The resultant force is the weight of the body, and the acceleration is always in the direction of gravity (vertically downward), regardless of the direction in which the body may be moving (Art. 109, third case). In all cases the accelerated component of 124 DYNAMICS the motion of a body is in the direction of the resultant force upon it. In addition to this the body may have a constant component of motion (previously imparted) in any direction. The first law of motion is a corollary to the above; for, since force and acceleration are proportional, when the force is zero the acceleration is also zero, and, with zero acceleration, motion remains unchanged both in speed and direction. 113. Relation between Force and Mass, with a Given Acceleration. The second law referred to at the begin- ning of the preceding article is as follows: 2b. The resultant force necessary to produce a given accel- eration is proportional to the mass of the body upon which the force acts. In lifting and carrying bodies, in dragging or pushing them along, in hauling loads in wagons or in cars, and indeed in most cases of motion with which we are familiar through daily observation and experience, friction and weight are the chief hindrances to be overcome; and the effect of mass, being thus obscured, is commonly not rec- ognized at all. The " resultant force" of the law is the resultant of all the forces acting on the body, including weight and friction (which are thus fully allowed for), and is the force that mass makes necessary whenever accelera- tion takes place. If bodies had mass but not weight, and moved without friction (which is true of the earth and other planets as bodies in space), the laws of motion would still be the same. Bearing these facts in mind, let us see how the above law applies in the following cases. If a horse exerts a force of 150 Ib. in drawing a loaded cart at a uniform rate on a level road, it is because the friction opposing the NEWTON'S LAWS OF MOTION 125 motion amounts to 150 lb.; and the resultant or accelerating force on the load is zero. To start the load an additional force is necessary; and, for a given rate of starting, the amount of this additional force is determined solely by the mass of the load (including the cart). A computation, which the pupil will presently be able to make for himself, shows that to impart a velocity of 3 mi. per hour during the first 10 sec. (while the cart goes the first 22 ft.) would require an accelerating force of a little more than 27 lb. per ton of the load. This, it will be observed, is much less than the force ordinarily required to overcome friction. But to impart a velocity of 3000 ft. per second to a thousand-pound projectile while it is moving from the breech to the muzzle of a twelve-inch gun requires an average accelerating force of 1400 t., or 2800 t. of force per ton of mass, a force in comparison with which weight and friction are utterly insignificant. Further light on the meaning of the law is afforded by retarded motion. While friction acts as the chief obstacle to starting in ordinary cases, and thus diverts the atten- tion from the effect of mass, it directly aids stopping. Yet even when friction is greatly increased by applying brakes, time is required to stop a car or other_jvehicle,and the greater its mass the greater is the difficulty in stopping it. The force of gravity (weight) is in no wise responsible for this difficulty. The pupil should impress his mind with the idea of mass by a few experiments in quickly starting and stopping bodies, under such conditions that weight and friction play an unimportant part, such as setting in motion a grindstone or a massive, well balanced wheel, and then endeavoring to stop it, or rapidly twirling a long pole from side to side in the hand. This may seem to be nothing more than a study of inertfa, and that is pre- cisely what it is. Mass and inertia, as measurable quan- tities, are identical. 126 DYNAMICS The proportionality stated in the law is admirably illustrated by falling bodies. All bodies which are not measurably retarded by the resistance of the air, and all bodies without exception in a vacuum (Fig. 113), are equally accelerated in falling. This is readily tested by dropping a pebble and a large stone from the same height at the same time. They reach the ground at the same instant. (Try it.) In such cases the accelerating force is the whole weight of the body, and weight is proportional to mass. Thus a five-pound mass, in falling, is accelerated by a force five times as great as that upon a falling mass of one pound; hence the equal acceleration. When weight increases without an increase of mass, as when a body is taken to a place farther from the equator (Art. 15), its acceleration in falling is proportionately greater (Law 20). Thus at the equator the accel- eration due to gravity is 978 (cm. per second per second), while at either pole it is 983. 114. Absolute Units of Force. The earth's at- traction for the different English and metric units "Guine' md ^ mass su PP nes corresponding units of force (Art. Feather "Ex- J 7)- These are termed gravitational units of force, periment. since they are due to gravity. They vary slightly with latitude and altitude (Arts. 15 and 17); but for most purposes these variations are unimportant. For scientific purposes, however, an invariable unit of force is necessary; and the one chosen has the further advantage of simplifying the statement of dynamical laws and formulas. The dynamical or absolute unit of force is that force which, acting alone or as a resultant force, produces unit acceleration in unit mass. The absolute unit of force in the metric system is called the dyne. It is variously defined in equivalent terms as follows: (i) The dyne is that force which, acting on a mass of one gram for one second, pro- duces a change of velocity of one centimeter per second; or (2) it is that force which, acting continuously, imparts an acceleration of one centimeter per second per second to a mass of one gram; or (3) it is that force which, acting on any mass, changes its momentum at the rate of one unit per second (mass being expressed in grams and NEWTON'S LAWS OF MOTION 127 velocity in centimeters per second). (See Arts. 115 and 116 for meaning of the term momentum.) Since the weight of a gram mass gives it an acceleration 980 times as great as a force of one dyne would give it, it follows (Law 20) that the gram weight is equal to 980 dynes. Strictly speaking, the weight of a gram at sea-level is equal to 978 dynes at the equator, to 980 dynes in the latitude of New York, and to 983 dynes at the pole (Art. 113). It is the gram weight that varies, not the dyne. The English absolute unit of force is called the poundal, and is denned as that force which imparts to a pound mass an acceleration of one foot per second per second. Taking the acceleration of a falling body as 32.16 ft. per second per second, a poundal is y^T* of a pound weight. (Why?) 115. Momentum. Certain dynamical relations, which are presently to be considered, involve the product of the mass of a body and its velocity. This product is called the momentum of the body. It is generally expressed as the product of grams and centimeters per second, or as the product of pounds and feet per second; but other units may be employed. No name has been given to any unit of momentum. Since the mass of a body is constant, its momentum changes only with a change of velocity. Hence the rate at which momentum changes, is measured by the product of the mass of the body and the rate at which its velocity changes; or, more briefly, rate of change of momentum is equal to the product of mass and acceleration. Let m denote the mass of a body, v its velocity, and a its acceleration; then mv is its momentum, and ma is the rate at which its momentum is changing. 116. Newton's Second Law, or the Law of Accelerated Motion. A force of 12 dynes acting on a mass of i g. gives it an acceleration of 1 2 cm. per second per second (Law 20). To impart an equal accelera- tion to a mass of 8 g. would require 8 times as great a force (Law 26), 128 DYNAMICS or 96 dynes. The number of dynes is thus equal to the product of the mass in grams and the acceleration in centimeters per second per second. This relation is general. Expressed as a formula it is / = wa, (13) the units in which the quantities are measured being the dyne, the gram mass, and the centimeter per second per second; or the poundal, the pound mass, and the foot per second per second. Since a force in dynes is changed to grams by dividing by g ( = 980 cm. per second per second) and a force in poundals to pounds by dividing by g (= 32.16 ft. per second per second), the formula becomes ma f = y' (14) when the force is measured either in grams or pounds and the other quantities in the corresponding units. The product ma in either formula is the rate of change of momen- tum (Art. 115); hence the rate of change of the momentum of a body is equal to the resultant force in dynes or poundals acting upon it, and is proportional to the resultant force in grams or pounds. Adding to this the fact that the change of momentum is in the direction of the acceleration and that the acceleration is in the direction of the force, we have The Second Law of Motion: Rate of change of momentum is propor- tional to the resultant force, and takes place in the direction of the force. 117. ^Impulse. The Second Law of Motion Restated. The change of velocity produced in a given mass by an unbalanced force is proportional to the time during which the force acts, as well as to the magnitude of the force. (Why?) Con- sequently, if a force acts only for an in- stant, the motion produced will be slight, unless the force itself is very great. This is especially true if the body has consider- able mass. The force exerted by a bullet in penetrating a board is a good example. The time is so extremely brief that a bullet shot through a board standing on edge, will not overturn it, although it can easily FIG. 114- be overturned with a finger. NEWTON'S LAWS OF MOTION 129 The importance of time in the action of a force is well shown by a simple experiment with a small coin placed on a small card. The friction between them is sufficient to impart the motion of the card to the coin when the card is moved slowly about; but, when it is started suddenly, the coin is left behind. This is neatly shown by snapping the card from under it (Fig. 114). If the blow is aimed successfully, the coin will be left at rest on the finger, friction being insufficient to impart appreciable motion to it in so short a time. (Try it.) Let / denote the force in dynes or poundals acting on a mass m, a the acceleration due to the force, and v the velocity imparted in t seconds; then / = ma, and v = at. Eliminating a, ft = mv. (15) That is, the momentum imparted, or the change of momentum, is equal to the product of the force and the time during which it acts. The product // is called the impulse of the force. Hence one way of stating the second law of motion is as follows: The change of momentum of a body is equal to the impulse which produces it, and is ~in the direction of the impulse. 118. The Third Law of Motion, or the Law of Mutual Action. Newton's third law of motion is as follows : To every action there is an equal and opposite reaction; or, when- ever one body exerts a force on another, the other exerts an equal and opposite force on the first. This law is stated and discussed at some length in Art. 12. It asserts that every force is one of a pair of equal and opposite forces, exerted by two bodies or by two parts of the same body on each other. The two forces involved in " action and reaction " can not, under any circumstances, balance each other, since they act upon different bodies or portions of matter; but either or both may be balanced by other forces acting at the same time. For example, the pressure exerted by a bat against a ball in striking it 130 DYNAMICS is unbalanced, and imparts motion to the ball. The reaction of the ball against the bat is also unbalanced, and checks the motion of the bat. When a piece of iron, placed on an anvil, is struck with a hammer, the downward blow of the hammer is balanced by the equal upward pressure of the anvil, both acting on the piece of iron; hence the iron remains at rest. When a person jumps from a boat, the reaction on the boat is unbalanced and pushes the boat in the opposite direction from that in which the person jumps; but, in jumping from a rock, the reaction against the rock is balanced by friction between it and the ground, and it remains at rest. The equal forces exerted by two interacting bodies upon each other necessarily act for equal times; hence the impulses of the two forces are equal, and, if the bodies are free to move, equal changes of momen- tum are produced in them. Let m\ and mz denote the masses of the two bodies, and v\ and % their respective velocities imparted by mutual action, the bodies being initially at rest and free to move; then m\vi = m^i and m\ : m% :: ^ i, i-e- the velocities imparted to the bodies are inversely proportional to their masses. Thus if a man jumps from a boat whose mass is three times that of his own body, the boat is pushed back with a velocity one third as great as the forward velocity of the man. When a moving body strikes a body at rest, and their mutual actions are unbalanced, the one loses as much momentum as the other gains. 119. Scope of the Laws. The motion of all matter, animate as well as inanimate, is in full accord with Newton's laws of motion. It is a common error to regard the motions of animals and self- propelling machines as exceptions to the law of inertia, because they "make themselves go." The motion of a train presents no difficulty on this score so long as we consider only the cars, for the external force is easily identified; it is the pull of the engine. But what pulls or pushes the engine? We get a clew to the answer when we see the drive-wheels of an engine slip and spin round, as sometimes happens in starting a heavily loaded train. Sharp sand sprinkled on the rails NEWTON'S LAWS OF MOTION (from the sand box above the boiler) remedies the difficulty by increasing the friction between wheel and rail. Thus while friction between the car-wheels and the rails is a hindrance to the motion of the train, between the drive-wheels of the engine and the rails it is a necessity. The drive-wheels exert a backward thrust on the rails; the rails an equal forward thrust (reaction) on the wheels. This forward reaction on the drive-wheels is the external force which not only enables the engine to " move itself " but also to pull the train. Newton's three laws, as already stated, completely express the relations between mass, force, and motion. Starting with these laws as given data, the mathematical physicist can derive the laws of motion in all special cases, such as the laws of curvilinear motion (Art. 122), the laws of planetary motion (Art. 125), the laws of the pendulum (Art. 128), the laws of machines (Arts. 138-147), the laws of vibration of strings in music (Art. 280), etc. Such work, how- ever, lies almost wholly in the field of advanced physics. 120. Galileo and Newton. The creation of the science of dynam- ics is due to Galileo Galilei (1564-1642), an Italian mathematician and physicist. "The first experiments which Galileo made while he was a young professor at Pisa were deci- dedly dramatic. At that time the doctrine that the rate at which a body falls depends upon its. weight was generally accepted as true, merely on the authority of Aristotle. It was even held that the acceleration varies as the weight. Prior to Galileo it had not occurred to any one actually to try the experi- ment. The young professor's tests went con- trary to the doctrine held for two thousand years. Allowing for the resistance of the air, he found that all bodies fell at the same rate, and that the distance passed over varied as the square of the time. With all the enthu- siasm, courage, and imprudence of youth, the experimenter proclaimed that Aristotle, at that time believed by nearly every one to be verbally inspired, was wrong. Galileo met with opposition, but he decided to give his opponents ocular proof. It seems almost as if nature had resorted FIG. 115. Leaning Tower of Pisa. 132 DYNAMICS to an extraordinary freak to furnish Galileo, at this critical moment in the history of science, with an unusual convenience for his pub- lic demonstration. Yonder tower of Pisa had bent over to facilitate experimentation, from its top, on falling bodies. One morning, before the assembled university, he ascended the leaning tower, and allowed a one-pound shot and a one-hundred-pound shot to drop together. The multitude saw the balls start together, fall together, and heard them strike the ground together. Some were convinced, others returned to their rooms, consulted Aristotle, and, distrusting the evidence of their senses, declared continued allegiance to his doctrine." (Cajori's "History of Physics.") Galileo explained the motion of a projectile, and thus, in effect, discovered the first and second laws of motion. He determined the laws of the pendulum (Art. 128), and suggested its use in measuring time. The name of Sir Isaac Newton (1642-1727), an Englishman, stands preeminent in the history of science. As mathematician, astronomer, and physicist, he made invaluable contributions to the progress of knowledge. He formulated the general laws of dynamics which bear his name, and applied them with unexampled skill to the motions of the heavenly bodies (Art. 125). PROBLEMS 1. Why does a falling body, on striking the earth, exert a pressure in excess of that due to its weight? Would the pressure be the same whether the ground was hard or soft? 2. A bullet fired through a plate glass window will often make a smooth hole without cracking the glass. Explain. 3. A nail can be driven by striking it with a hammer, but not by press- ing the hammer steadily against it. Explain. 4. Gravity upon the moon is one sixth as great as upon the earth. Compute the acceleration of a falling body upon the moon. 5. Gravity upon the sun is 27.6 times as great as upon the earth. Compute the acceleration of a falling body upon the sun. 6. How far would a body fall during the first second (a) upon the moon? (&) upon the sun? 7. (a) Would the mass of a given body be the same upon the sun 01 the moon as upon the earth? (6) Would its inertia be the same? NEWTON'S LAWS OF MOTION 133 8. Would it take less powder to fire a cannon ball with a given veloc- ity upon the moon than it would upon the earth? 9. Is it harder for horses to start a loaded wagon or to keep it in uni- form motion? Give reasons. 10. Why does a ball player move his hands quickly backward in the act of catching a swift ball? 11. Is mass, weight, or friction principally responsible for the difficulty experienced in quickly getting up speed on a bicycle? 12. State the meaning of the formula ft = mv when the body has an initial velocity and / is in the direction of motion; also when / is opposite to the direction of motion. 13. How is the weight of a body in dynes obtained from its weight in grams? How is its weight in poundals obtained from its weight in pounds? 14. Account for the motion of a revolving lawn sprinkler. 15. How does the mutual action between the front wheel of a bicycle and the ground differ from that between the rear wheel and the ground? Explain. 16. How does it follow from the second law of motion that an unbalanced force, however small, acting on any mass, however great, will move it or change its existing motion? 17. If equal forces impart equal accelerations to two bodies, how do the masses of the bodies compare, and why? 18. Does a horse pull harder upon a wagon in drawing it than the wagon does on the horse? Explain. 19. Why does an elevator cable pull more than the weight of the car and occupants while gaining velocity going up, and less than the weight of the car and occupants while gaining velocity going down? Is the pull greater or less than the weight, and why, while losing velocity going up and while losing velocity going down? 20. How is the momentum which is produced by a given impulse affected by the mass of the body acted upon? How is the velocity imparted affected by the mass of the body? 21. If equal impulses impart equal velocities to two bodies, how do the masses of the bodies compare, and why? 22. Is the air necessary for the flight of a bird? Why or why not? Discuss as definitely as you can the mechanics of a bird's flight. 134 DYNAMICS 23. Why does a person slip in trying to start, stop, or turn quickly on ice, with ordinary shoes on the feet? How do skates remedy the difficulty? 24. Discuss and compare the mechanics of the standing broad jump and the running broad jump. 25. Why is a locomotive built so that its drive-wheels sustain as much of its weight as is possible? 26. Why do the drive-wheels of an engine sometimes slip, spinning round and round, while the other wheels and the wheels of the cars never do? 27. A bullet is fired from a rifle with a muzzle velocity of 2000 ft. per second. The bullet weighs oz. and the rifle 10 Ib. What is the velocity of the rifle in the recoil? How is this velocity imparted to the rifle? 28. Two boys, A and B, are pulling upon the ends of a rope. A pulls B along. Is he exerting a greater pull than B? Explain. 29. Why does stamping remove mud from the shoes? Why can not all of the mud be removed in this way? 30. The handle can be tightened in the head of an ax by striking the end of the handle against a log, or by holding the ax at rest and striking the end of the handle with a hammer. Explain. 31. A plane is inclined so that i of the weight of a sphere placed on it is effective in causing acceleration down the plane. How far will the sphere roll in 2.5 seconds? 32. A falling body weighs 100 g. What is its acceleration at the instant when the resistance of the air against it is 25 g.? 33. A block weighing 1000 g. slides down a plane 200 cm. long, inclined to a height of 120 cm. The resistance of friction is 275 g. Find (a) the accelerating force, (b) the acceleration, (c) the rate of gain of momentum. 34. The mass of a wagon and its load is 3 tons, and the resistance of friction is 250 Ib. In what time and in what distance will it come to rest if it is moving at the rate of 3 mi. per hour when the horses stop pulling? 35. A body weighing 5 Ib. is projected vertically upward by a constant force/, acting through a distance of 3 ft.; and it rises 100 ft. higher before coming to rest. Find /. SUGGESTION. Find the velocity necessary to enable a body to rise 100 ft. against gravity; the acceleration necessary to impart this velocity in 3 ft.; the resultant force necessary to produce this acceleration in a mass of 5 Ib.; and, finally, the whole upward force /. CURVILINEAR MOTION 135 FIG. 116. Path of a Projectile. III. THE LAWS OF MOTION IN SPECIAL CASES 121. Curvilinear Motion. Change of speed is due to unbalanced force in the line of motion; change of direc- tion, to unbalanced force at right angles to the line of motion. If the result- ant force upon a body is oblique to the line of motion, it is equiva- lent to two compo- nents, one of which is jv in the line of motion and the other at right angles to it. The weight of a projectile is an excellent example. Let LM N (Fig. 1 16) represent the path of a projectile. At L, in rising, the com- ponent Toi the weight causes decrease of speed, and the com- ponent / causes change of direction. At the highest point M, the whole weight/ causes change of direction. At N, in falling, the component T causes increase of speed, and the component / causes change of direction. The path every- where curves in the direction of the component /, and the curvature is greatest at M, where / is greatest. The line of motion at any point of a curved path is the straight line tangent to the curve at that point, and the compo- nent force in the line of mo- tion is called the tangential force. The component force at right angles to the path is called the centripetal force, 136 DYNAMICS because it acts toward the center of the curved path (from the Latin centrum, center, and peter e, to seek). Since change of direction of motion is due to unbalanced centripetal force, the change of direction ceases at the instant when the centripetal force ceases; and the body, from that instant, continues in a straight line, as the re- sult of its inertia. This is readily shown by means of a ball fastened to the end of a string. When the ball is rolled in a circle on the top of a table or on the floor and suddenly released, as at A (Fig. 117), it continues in the direction in which it was moving at the instant of release. Similarly, when a stone is whirled in a sling and released, it " flies off at a tangent." The tendency of bodies to leave a curved path is commonly called a centrifugal tendency (from the Latin centrum, audfugere, to flee, i.e. fleeing from the center); but it is nothing else than the universal ten- dency of moving bodies to maintain a straight course, as expressed in the first law of motion (Art. in). It should be noted that the centrifugal motion which occurs when the centripetal force ceases is not outward along a radius, but outward along a tangent to the curved path. 122. Laws of Centripetal Force. If a ball is suspended by a string from a fixed support and started in a horizon- tal circle (Fig. 118), it will continue to revolve in a slowly diminishing circle (more accurately a spiral) for several minutes. The decrease in the size of the circle is due to friction, chiefly of the air, and may be disregarded. If all friction could be removed, the motion in a circle would continue indefinitely. Disregarding friction, the ball is acted upon by two forces; namely, its weight, W, and the tension, T, of the cord. The vertical component of the tension, v, is equal to W and balances it; the horizontal CURVILINEAR MOTION 137 FIG. 118. component / is unbalanced, and is directed toward the center of the circle. The centripetal force / holds the ball in a circular path, but has no effect on its speed. If the ball is whirled more and more rapidly, the string being held in the hand, the circle in which it revolves grows larger, and the cord becomes more nearly horizon- tal (Fig. 119). The reason for this behavior is that a greater speed is accompanied by a more rapid change of direction, and this necessi- tates a greater centripetal force. The ball consequently moves out until the centripetal force has increased to the required value. It would be difficult from such an experiment to determine the exact relations among the quantities involved in curvilinear motion, since three of them vary at the same time; namely, the centripe- tal force, the radius of curvature, and the velocity. It can be shown, however, both by experiment and by mathematical analysis, that the centripetal force is always pro- portional to the square of the velocity, fl 2 , when the radius is con- stant, and inversely proportional to the radius, r, when the velocity is constant. If in the above ex- periment a heavier ball were used, v and / would change in the same ratio as the weight, and hence also in the same ratio as the FIG. 119. 138 DYNAMICS mass of the ball. This illustrates the general law that, for a given velocity and radius of curvature, the centripetal force is proportional to the mass of the body. When the centripetal force is expressed in dynes or poundals and the other quantities in the corresponding units, the above laws give the formula / = - (Equation for centripetal force.) (16) It can be shown that, for uniform speed in a circle, is the acceler- ation toward the center which results in change of direction; hence the above formula is a special case of the general formula / = ma. 123. Illustrations and Applications of the Laws. In all cases of curvilinear motion upon the earth's surface, the moving body exerts an outward thrust against the ground or other support, and turning is effected by the inward reac- tion of the supporting surface. The necessity for this out- ward thrust and inward reaction to accomplish turning is plainly shown in cases where it is insufficient to meet the demands made upon it, as when a bicycle rider attempts to turn quickly on a wet pavement and the wheel slips out- ward from under him. Un- der like circumstances, an automobile " skids," sliding sidewise toward the outside of the curve. The outward thrust of cars in turning a curve comes upon the outer rail; and for heavy trains moving at high speed, this thrust is enormous, some- times, indeed, so great as to pull out or shear off the spikes which hold the rail in place, causing disastrous wrecks. To reduce this dangerous thrust as much as possible, the road-bed of a track is always raised on the outside of a curve (Fig. 120). The correct FIG. 1 20. The Resultant Force on the Car is toward the Center of the Curve. CURVILINEAR MOTION 139 inclination would be such as to bring the road-bed at right angles to the resultant force between the car and the rails. But since the outward thrust of the car varies as the square of its velocity, and the velocity is not always the same, the inclination adopted is, at best, a compromise. In the figure W represents the weight of the car, P the entire reaction of the track due to the weight and the outward thrust of the car (taken as acting at the center of gravity), and / the resultant or centripetal force upon the car. Centrifugal motion is usefully applied in separating one sub- stance from another, generally a liquid from a solid. The solid is placed in a perforated cylindrical vessel, which is rapidly whirled on a vertical axis. The force of adhesion with which the liquid is held between the particles of the solid or within its pores is insufficient to drag the liquid round in a circular path. The result is that the liquid recedes farther and farther from the axis until it finally reaches the outer surface, where it flies out through the openings of the vessel. Centrifugal machines, acting on this principle, are used to extract honey from the comb, to separate the sirup from sugar in the process of refining, to dry clothes after washing, to separate cream from new milk, etc. The separation of cream depends upon the fact that it is less dense than milk. The denser milk has a greater centrifugal tendency, and consequently moves away from the axis. The cream is thus crowded toward the center, where it is drawn off. 124. " Centrifugal Force." - When a person who is unacquainted with the laws of dynamics sees a body vio- lently leave a curved path and fly off at a tangent, or over- turn, he naturally assumes that this behavior is due to some force which pulls the body out of its course. Hence the idea that centrifugal motion is due to " centrifugal force." This is a fundamental error, which the pupil who has mastered the preceding work of this chapter will not fail to detect. The error consists in the supposition that a body moving in a curved path tends of itself to con- tinue in that path, and that it will do so unless pulled out of it. The truth, as we know, is precisely the contrary; namely, that a moving body tends of itself to pursue a 140 DYNAMICS straight course, and will do so unless pulled or pushed out of it into a curve. It follows that " centrifugal force/' in the above sense, is pure fiction, and, of course, does not enter into any correct discussion of curvilinear motion. Centrifugal force is brought from the realm of fiction to that of fact when defined as the reaction to the cen- FIG. 121. "Loop the Loop." tripe tal force; and this is its only scientific meaning. The outward thrust of a car on the* outer rail of a curve is the centrifugal force which gives rise to the centripetal reac- tion of the rail on the car. The motion of the car is deter- mined by the centripetal thrust on it. The centrifugal thrust is indeed very real, but its effect is expended on the rail and the road-bed; hence the necessity for a strongly built track. PROBLEMS 1. What precautions are necessary in making a turn on a bicycle on a slippery pavement? Discuss these precautions as illustrations of the laws of curvilinear motion. 2. Must a light and a heavy bicycle rider lean equally or unequally in turning the same curve at the same speed? Give reasons. 3. Are tracks inclined more or less on sharp curves than on long ones? What law is illustrated? UNIVERSAL GRAVITATION 141 4. Discuss in definite and accurate terms the overturning of a car, running at high speed on a curve. 5. What is the percentage of increase of the centrifugal thrust on a rail- road curve when the velocity of a train is increased from 40 mi. per hour to 60 mi. per hour? 6. How would you determine experimentally the proper inclination of a bicycle race track on a curve? 7. Just what does a boy do, and why, to change his direction suddenly while running? 8. Discuss the mechanics of "looping the loop" (Fig. 121). 9. A ball weighing 2 kg. is suspended from a cord 50 cm. long, and made to revolve in a circle whose radius is 30 cm. (Fig. 119). Compute (a) the centripetal force upon the ball, and (6) the tension upon the cord. 125. Universal Gravitation. The history of science tells no more interesting or instructive story than that of Newton's discovery of the law of universal gravitation and the work of his predecessors, which made that discovery possible.* Until the sixteenth century it was universally believed that the earth was fixed in space, and that the sun, moon, planets, and fixed stars revolved in the heavens round it. This view was overthrown by a German monk, named Copernicus (1473-1543), who taught that the sun is the center of a system of planets which revolve round it, and that the earth itself is one of them. He held, however, to the old view that the orbits of the planets were circles. The next advance in astronomical science was due to Tycho Brahe (1546-1601), a Danish astronomer, whose observa- tions of the planetary motions, extending over a period of twenty years, were much more extensive and accurate than any that had been made before. These observations were subjected to a searching analysis by Johannes Kep- ler (1571-1630), "a born speculator and thinker," who " after more than four years of assiduous computation, * This story is admirably told in Sir Oliver Lodge's ' Pioneers of Science." 142 DYNAMICS and after trying more than nineteen imaginary paths and rejecting each because it was more or less inconsistent with observation," at last discovered that the orbit of a planet is an ellipse, with the sun at one of its foci (Fig. 122). This is Kepler's first law of planetary motion. Further years of labor brought as their reward the discovery of his sec- ond and third laws of planetary motion. At last it was known what the motion of a planet is; but why planets move thus, rather than in any other fashion, was a question that required the genius of Newton to answer. Previous attempts at an explanation had been based upon the wrong idea that force (unbalanced force) is necessary to maintain a body in motion. It was therefore supposed that a force of some sort must act on a planet in its line of motion to push or pull it along. Newton knew FIG. 122. Orbital Motion of a that such a force . is unnec- essary; but Kepler had shown that the speed of a planet is slightly variable, steadily increasing as the planet moves from the farthest point of its orbit, B (Fig. 122), to the nearest point, A, and stead- ily decreasing from the nearest point to the farthest again, as expressed in his second law of planetary motion. New- ton's problem, therefore, was to account for the elliptical form of orbits and the law of speed in them, assuming that the general laws of dynamics hold in the universe at large. He proved that a central force, directed constantly toward the sun and varying inversely as the square of the distance from it, would accomplish these results, and that a force of any other description would not. He proved further UNIVERSAL GRAVITATION 143 that the moon is held in its orbit by a like force, directed toward the earth, and that this force is identical with the well known force of gravity, which makes bodies fall and gives them their weight. The solution of these problems and others of a similar character showed the existence of a universal attraction or gravitation, as expressed in the following law: Every particle of matter in the universe at- tracts every other particle with a force whose direction is that of the line joining them, and whose magnitude is directly pro- portional to the product of their masses, and inversely pro- portional to the square of the distance between them. According to the law, there is a gravitational attraction between all bodies, large or small. Yet even between masses of several hundred pounds it is exceedingly small small beyond all ordinary means of detecting it. The greatest ingenuity and experimental skill have been exer- cised by various scientists during the past century in accu- rately measuring the attraction between known masses, varying from a fraction of an ounce to several hundred pounds. From the results of these experiments it is known that two spheres of cast iron, each 1.8 m. in diameter, would attract each other with a force of i g. when placed close together. (Actual contact is unnecessary.) Such spheres would weigh 22,000 kg. or 22 metric tons each. Concerning the cause of gravitation, science is still in the dark. It acts without visible or material connection between the attract- ing bodies; yet we must suppose that there is something pervading all space, by means of which and through which it is exerted. It is inconceivable that two bodies, not in contact, should be able to act upon each other with absolutely nothing between them. Since gravitation acts undiminished in a vacuum, and beyond the limits of the atmosphere, it is clear that the means, or medium, for the transmission of gravitational force is not a solid, a liquid, or a gas, and hence is not matter in any of its ordinary forms. 144 DYNAMICS 126. Applications of the Law. Center of Gravitation. Newton proved that the attraction between a sphere and any other body is the same as it would be if the entire mass of the sphere were con- centrated at its center. The attraction between two spheres, as the earth and the moon, is, therefore, inversely proportional to the square of the distance between their centers. In considering the earth's attraction for any body upon its surface, the distance is to be taken as the earth's radius, which, in round numbers, is 4000 mi. Gravitation a Mutual Action. Gravitation between any two bodies is a mutual action, in agreement with the third law of motion. A pound mass attracts the earth with a force of one pound. The earth and the moon, by their mutual attraction, produce equal changes of momentum in each other; but the mass of the earth is 80 times that of the moon, and its acceleration is consequently sV as great. The earth and the moon, in fact, revolve in the same direction round their common center of gravity, which, as it divides the distance between the centers of the two bodies inversely as their masses, lies within the mass of the earth about noo mi. below the surface. (This motion of the earth has nothing whatever to do with its rota- tion on its axis.) The sun's attraction deflects the earth from a straight course by about one ninth of an inch in a second, while the earth is traveling nearly 19 mi. The mass of the earth is so great that the force required to produce even so slight a change of direction is incon- ceivable, being no less than 3,600,000 millions of millions of tons (36 with seventeen ciphers). The equal pull of the earth upon the sun moves it less than an inch in a month, the mass of the sun being 332,000 times that of the earth. If the invisible and unknown mech- anism of gravitation between earth and sun were replaced by a cable of the strongest steel, such as is used in suspension bridges, that cable would have to be 3000 miles in diameter, and even then would be strained to the breaking point. Rotation of the Earth. A spinning top is brought to rest by the resistance of the air and the friction upon the peg. The earth rotates on its axis without friction; hence its rate of rotation remains con- stant without the action of any force to maintain it. If the earth were fluid and were not rotating, the gravitation of its particles would cause it to assume the form of a perfect sphere. UNIVERSAL GRAVITATION 145 The rotation of a fluid planet would cause it to bulge at the equator and flatten at the poles, as a result of the greater centrifugal tendency in equatorial regions, where the velocity of rotation is greatest. This is illustrated in Fig. 123, which represents a section of the earth taken through the axis of rotation MN. (The departure from the spherical shape is greatly exaggerated.) The earth assumed its present form, disregarding minor inequali- ties, 'while still fluid; and, as a result of its rotation, the polar radius is nearly 13.5 mi. less than the equatorial. If the earth were F IG . I23 . to stop rotating, the Mississippi River would flow toward the north, for its mouth is farther from the center of the earth than its source is. Variation of Weight. All bodies on the earth must be acted upon by a centripetal force to carry them round with the earth in its rotation. A certain portion of gravity is thus employed, and only the remainder of it is sensible as weight. The necessary centripetal force increases from zero at the poles to ^-5 of gravity at the equator. Since 289 is the square of 17, and centripetal force varies as the square of the velocity, it follows that, if the earth rotated 17 times as fast as it does, bodies at the equator would be on the point of flying off at a tangent, and would weigh nothing. There is a further cause for decrease of weight as a body is taken toward the equator, namely, the increasing distance from the earth's center. For this reason alone, a given mass weighs ^-5 less at the equator than at either pole; and, in consequence of rotation and increase of distance together, it weighs about jiv less. PROBLEMS 1. (a) Would the variation of weight at different latitudes be. indicated by any form of balance by which the object weighed is balanced by "weights"? (6) Would it be indicated by an accurate spring balance? Give reasons for each answer. 2. What fraction of its weight would an object lose when taken from sea-level to a height of 4 mi.? 3. How does it follow from the law of universal gravitation that mass and weight, at any one place on the earth, are proportional? 146 DYNAMICS 4. The distance of the moon from the earth is 240,000 mi. (a) How does the force of gravity at this distance compare with its value at the surface of the earth? (6) What is the moon's acceleration toward the earth? (c) How far is the moon deflected from a straight course in one second by the earth's attraction? 6. Why does the atmosphere not offer resistance to the rotation or to the revolution of the earth? 6. Is the acceleration of a falling body due to the whole of the earth's attraction or to the part that we call weight? 7. What would be the subsequent motion of the moon and the planets if gravitation should suddenly cease to act upon them? 8. The average specific gravity of the whole earth is about 5.53. (a) How would gravity compare with its present value if the average density of the earth were equal to the density of water? (b) What would be the acceleration of a falling body in that case? 9. The diameter of Mars is 4230 mi. and its mass is approximately one- ninth of the earth's mass. How does gravity upon its surface compare with gravity upon the earth? 127. The Pendulum. Any suspended body that is free to swing to and fro, or vibrate, is called a pendulum. A pendulum consisting of a small sphere of some dense material, suspended from a fixed support by a slender thread or wire, is approximately a simple pendulum. Any pendulum having an appreciable portion of its mass else- where than in a compact mass or bob at the end is a com- pound pendulum. Pendulums for other than experimental purposes are always compound. A complete swing of a pendulum in one direction is called a vibration. The period of a pendulum is the time required for one vibra- tion, and is measured in seconds. The amplitude of a pendulum vibration is half the angle or half the arc through which it swings. The length of a simple pendulum is (very approximately) the distance from the point of suspension to the center of the bob. The length of a compound pen- dulum is denned as the length of a simple pendulum having THE PENDULUM 147 the same period. (It will be found by trial that this is greater than the' distance from the point of suspension to the center of gravity of the o pendulum and is less than its entire length.) The Motion of a Pendulum. After a pendulum has been drawn aside and released, the bob is under the action of its weight and the tension of the thread (friction being disregarded). As the pull of the thread acts always at right angles to the path of the bob, its only effect is a continuous change of direction. The weight of the bob may be resolved into two components, p and / (Fig. 124), at any point of the path. The component p causes a part of the tension on the thread, but does not affect the motion of the pendulum; the tangential component / accelerates the speed as the bob descends and retards it as the bob rises. It is evident that the tangential force decreases toward the lowest point of the path, where it is zero, and that it has equal values at equal distances on the two sides of this point. Hence, if this were the only force affecting the speed of the bob, it would rise exactly as far as it descends, and its vibration would continue indefinitely. It is brought to rest by the friction of the air and the friction at the point of support. 128. Laws of the Pendulum. The laws of the pendu- lum, as determined either by experiment or by mathemat- ical analysis, based on the second law of motion, are as follows : i. The period of a pendulum is the same (to an exceed- ingly close approximation) for all amplitudes less than 4;* for larger amplitudes, the period increases very slightly with increase of amplitude. 148 DYNAMICS 2. The period of a pendulum is not affected by its mass or the kind of material of which it is made. 3. The period of a pendulum is proportional to the square root of its length; or, the square of the period is proportional to the length. Let h and k denote the lengths of any two pendulums and t\ and k their periods; then ti'. h:: \/ li m . \/fe, or /i 2 : t?\\ l\\ lz. (17) 4. The period is inversely proportional to the square root of the acceleration of a falling body; or til k:: V gz'.-V gi- (18) Mathematical analysis shows that, for small amplitudes of vibration, the period of a pendulum is given by the formula t = ' in which IT denotes the ratio of the circumference of a circle to its diameter (= 3.1416). (Let the pupil derive the pro- portions 17 and 18 from this formula.) The effect of a change in the force of gravity was first recognized when, in 1671, a clock was taken from Paris to French Guiana, on the northern coast of South America, for use in astronomical observations. The clock lost two and a half minutes daily in its new location. The pendu- lum was shortened to correct its rate; but it had to be lengthened again when the clock was taken back to Paris. The effect of an increase in the force of gravity can be illus- trated experimentally with a pendulum having an iron bob. By holding an end of a strong bar magnet under and near the bob, its motion will be controlled by its weight and the attraction of the magnet, acting together, the lat- ter force being equivalent to an increase of gravity. With THE PENDULUM a small amplitude of vibration, the bob does not swing beyond the strong attraction of the magnet, and the period is consider- ably shortened. 129. Uses of the Pendulum. The principal use of the pendulum is to regulate the motion of clocks. The wheels of a clock are driven by a weight or a spring. The last or end wheel of the train is trie escapement wheel D (Fig. 125), the teeth of which come in contact with the projecting ends of a curved piece, called the escapement. As the pendulum swings, it rocks the escapement to and fro, permitting only one tooth of the wheel to pass at a time. Each tooth, as it passes, exerts a slight impulse on the escapement; and this im- pulse, transmitted to the pendulum, maintains its motion. A clock is regulated by means of a thread and nut at the lower end of the pendulum. The bob is raised or lowered by turning this nut. A compound pendulum of special construction is used in determining the acceleration due to gravity at different places. The length and the period of the pendulum are determined very accu- rately, and their values substituted in the pendulum formula, which the value of g is then computed. PROBLEMS 1. How would the expansion of the rod of a pendulum in summer and its contraction in winter affect the rate of a clock if the height of the bob were not adjusted to compensate the expansion and contraction? 2. What is the usual shape of the bob of a clock pendulum? What is the advantage of this shape? 3. What is the length of a pendulum that beats seconds (/ = i) at a place where the value of g is 980 cm. per second per second ? Suggestion. Substitute the values of / and g in the pendulum formula, and solve for /. 4. Account for the fact that the period of a pendulum is independent of its mass. FIG. 125. from DYNAMICS 6. Referring to Fig. 126, show why the period of a pendulum is in- creased by increasing its length. 6. Why does a pendulum of given length vibrate more rapidly at a place where the force of gravity is greater? 7. Why did the clock which was taken from Paris to Cayenne (Art. 128) lose time? 8. Find the lengths of the pendulums whose periods are .7 sec. and 1.5 sec. respectively. 9. Find the periods of pendulums whose lengths are 20 cm. and 250 cm. re- spectively. 10. If you have the opportunity to experiment with an old clock, study its mechanism and observe its behavior when the pendulum bob is removed and also when both the pendulum and the escapement are removed. FIG. 126. IV. WORK AND KINETIC ENERGY 130. Mechanical Work and its Effects. When a man carries a hod of bricks up a ladder, he does a certain amount of work. This amount is doubled if the size of the load is doubled, or if it is carried to twice the height. If both the load and the height to which it is carried are doubled, the amount of work done is increased fourfold. Work, in its common meaning as applied to physical labor and to the work done by machines, depends upon two factors, namely, the amount of force exerted and the distance through which it is exerted. It is therefore measured by the product of these factors. If the body acted upon does not move, no work is done upon it. A hod-carrier is not working, however long he may stand with a load of bricks on his shoulder. The product of force and the time dur- ing which it acts can not be taken as the measure of work. In carrying a load up a ladder the force exerted is the same WORK AND KINETIC ENERGY 151 whether the time occupied is 20 seconds or a minute; hence the product of the force and the distance measures the work done, regardless of the time. Daily life affords innumerable examples of mechanical work. In most cases this work consists in maintaining the motion of bodies against friction or against gravity, which tends to stop them. A horse in hauling a load does work against friction on a level road, and against both friction and gravity in going up grade. In going down grade, gravity does work against friction, relieving the horse of part or all of the task. It is a less evident fact that work must also be done in imparting and in destroying motion; for this work is com- monly insignificant in amount compared with the work done in maintaining motion over long distances, as in haul- ing loads. In some cases, however, the amount of work done in starting and in stopping a body is many times as great as the work done in the interval between, e.g. in throwing and catching a swift ball. The work done upon the ball during its flight, by its weight and the resistance of the air, is practically negligible. 131. Positive and Negative Work. Kinetic Energy. A force acting on a body in the direction of its motion is said to do positive work upon the bpdy; if acting in the opposite direction, it does negative work. Positive work, then, consists in maintaining motion against friction or other opposing force, or in imparting motion, or in doing both at the same time. Negative work consists in opposing and in reducing the motion of bodies. Positive work in excess of negative work upon a body increases its speed; negative work in excess of positive work decreases its speed; equal positive and negative work leaves its speed 152 DYNAMICS unchanged. Excess of positive work over negative work is stored up in the moving body, to be paid out again in the act of stopping. This is shown by the fact that the faster a body is moving the farther it will go before it is brought to rest by friction or other opposing force. The work stored in a moving body is called kinetic energy. Kinetic means pertaining to motion; and the kinetic energy of a body is the energy it possesses by vir- tue of the fact that it is a moving mass. Energy exists in many forms; and it is only by virtue of the energy it pos- sesses in one form or another that any body can do work. The energy of a bent bow is shown by its ability to project an arrow, and the energy of a coiled spring by its ability to run a clock. The energy of the wind enables it to turn windmills, propel ships, uproot trees, etc. The energy of coal, wood, and oilis utilized by the steam-engine in running mills, drawing trains, and propelling steamships. The kinetic energy of a body is equal to the excess of posi- tive work that has been done upon it from the instant of starting, and is also equal to the work that the body is capable of doing and will do in coming to rest. In considering the effect of work upon one body only, it is con- venient and customary to say that the work is done by a force; but we know that a second body is always necessary to exert the force, and, strictly speaking, it is this body that does the work. "Doing work" is, in /fact, a mutual transaction, like buying and selling. When a marble strikes another, the first loses and the second receives a certain store of kinetic energy; i.e. the positive work of the first marble on the second and the negative work of the second on the first consists in the transference of kinetic energy from the one to the other. 132. A Force Perpendicular to the Line of Motion Does No Work. Thus far we have considered only the work done by forces acting in the line of motion, i.e. in the direc- tion of motion or in the opposite direction. A force acting WORK AND KINETIC ENERGY 153 at right angles to the line of motion does no work; for mechanical work consists in increasing, decreasing, main- taining, or opposing the motion of bodies, and such a force produces none of these results. It is true that weight (a vertical force) indirectly opposes horizontal motion that the heavier a body is the greater is the force necessary to draw it. But the" opposing force is friction, a horizontal force, which happens to be propor- tional to weight. That friction is no part of weight is further evident from the fact that it can be almost indefi- nitely reduced by means of ball bearings, an even track to run on, etc. A centripetal force in curvilinear motion does no work, since a change in the direction of motion does not involve a transfer of energy. When a ball is revolved on a table at the end of a string, the inward pull can be supplied at the center of the circle without moving the hand; and the hand at rest does no work. 133. Oblique Forces. General Rule for the Measure of Work. To find the work done by a force whose direction is oblique to the line of motion, as the weight of a sphere rolling down an in- clined plane (Fig. 127), the force may be resolved into two components, one in the line of motion, /, and the other perpen- dicular to it, p. Only the first of these components does work. Thus if d denotes the length of the %J plane AB, the work done upon the ball by its weight, w, while it is rolling down the length of the plane is fd. If h denotes the height of the plane J5C, it follows from similar triangles that/ :w :: 154 DYNAMICS h : d, or fd = wh. Hence the work done is also measured by wh, i.e. by the product of the whole force and the dis- tance passed over in its own direction. The above relations are general, hence : The work done by an oblique force is measured either (i) by the product of the component force in the line of motion and the whole distance passed over while the force is applied, or (2) by the product of the whole force and the distance through which it is applied in its own direction. The choice between these two rules is a matter of convenience, to be determined by the nature of the problem. When the force and the actual displace- ment of the body are in the same or in opposite directions, the work done is simply the product of the two. If the force is variable, the average force exerted through the dis- tance considered must be taken. In general, the work done by a force /, applied through a distance d in its own line of direction, is given by the formula, Work = fd. (20) 134. Units of Work and Energy. The unit of work is the work done by unit force while its point of application moves unit distance in the line of the force. If the unit force is the pound and the unit distance the foot, the unit of work is called the foot-pound (ft.-lb.). Similarly we have the gram-centimeter (g.-cm.) and the kilogram-meter (kg.-m.) as the gravitational units of work in the C. G. S. system. The foot-pound is the unit of work employed by American engineers. The absolute units of work are the foot-poundal and the dyne-centimeter. (Define each of these units.) The dyne-centimeter is called an erg (from the Greek ergon, work). It is extremely small, and in actual practice a larger unit, equal to 10,000,000 ergs, is WORK AND KINETIC ENERGY 155 used. This larger unit is called a joule, in honor of the English physicist, James Prescott Joule. The units of work are also the units of kinetic energy. EXAMPLES. i. When a book weighing i Ib. is lifted 4 ft., the work done against gravity is 4 ft.-lb. This is equal to 128.64 foot- poundals, -5533 kg.-m., 55,303 g.-cm., 54,197,000 ergs, or 5.419? joules. 2. If the hammer of a pile-driver weighing 1000 Ib. falls 25 ft. and strikes a pile, its weight does 25,000 ft.-lb. of work upon it during its fall. Since the resistance of the air is negligible, this work is all stored in the hammer as kinetic energy, and the hammer does 25,000 ft.-lb. of work upon the pile. 3. A ball weighing 4 oz. is thrown vertically upward by a force of 10 Ib. acting through a distance of 3 ft. How high does it rise? The projecting force does 10 X 3 or 30 ft.-lb. of positive work upon the ball. Hence it will continue to rise until the negative work of its weight amounts to 30 ft.-lb.; and, since its weight is .25 Ib., the height is 120 ft. (= 30 -i- .25) from the point of starting, or 117 ft. from the point at which the ball is released. The unbalanced upward force exerted in throwing the ball is 9.75 Ib., and this imparts 9.75 X 3 or 29.25 ft.-lb. of kinetic energy. This kinetic energy is lost at the rate of .25 ft.-lb. for every foot of ascent, and enables the ball to rise 117 ft. above the point of release before coming to rest. 4. A horse pulls with a force of 190 Ib. through a distance of 10 ft. in starting a load, and the resistance of friction is 130 Ib. (a) How much work is done by the horse? (b) What does this work accom- plish? (c) In what distance will friction alone stop the cart, if the horse ceases to pull at the end of the ten feet? - (a) Work done by the horse = 190 X 10 = 1900 ft.-lb. (b) Work done against friction = 130 X 10 = 1300 ft.-lb. Work done by the unbalanced pull = kinetic energy imparted = 60 X 10 = 600 ft.-lb. (c) Since friction in stopping the cart must do 600 ft.-lb. of negative work, the distance = 600 + 130 = 4.61 ft. 135. Measure of Kinetic Energy. The kinetic energy of a body is equal to the positive work that has been done 156 DYNAMICS upon it, in excess of the negative work, from the instant of starting, and is also equal to the work that the body is capable of doing in coming to rest. Hence if we know either of these quantities of work, we know the kinetic energy of the body; but, since kinetic energy consists in the motion of mass, it can always be expressed in terms of mass and velocity, and is regularly thus expressed. The formula for kinetic energy (K. E.) can be derived by con- sidering either of the above-named quantities of work to which it is equal. Let / denote the unbalanced or accelerating force in the direction of motion, acting upon a mass m, and a the accel- eration which it produces. The work done by such a force consists in imparting K. E.; hence the K. E. imparted is measured byfd. But / = ma, (Equation 13). i) 2 and d = ; (Equation 6) from which fd = ma X = \ mv 2 . Hence K. E. = \ mv 2 (ergs or foot-poundals) (21) Since the formula / = ma holds only for absolute units of force (the dyne or the poundal), the above formula gives K. E. in ergs or foot-poundals, according as mass and velocity are expressed in English or metric units. With ma any gravitational unit of force, / = ; hence I K. E. = - - (g.-cm., kg.-m., or ft.-lb.) (22) In deriving the formula for K. E. by considering the amount of work that a moving body will do in coming to rest, the procedure is WORK AND KINETIC ENERGY 157 the same as above ; but in this case / is the force exerted by the body through the distance d in coming to rest, and the acceleration a is negative. 136. Power. The rate at which an engine or other source of energy is capable of doing work is called its power. Power can be measured in foot-pounds per sec- ond, foot-pounds per minute, kilogram-meters per second, etc. The customary unit of power is the horse-power (H. P.), which is equal to 550 ft.-lb. of work per second, or 33,000 ft.-lb. per minute, or 76 kg.-m. per second. A twelve-horse-power engine, working at three fourths of its full capacity, does work at the rate of 9 H. P., or 4950 ft.-lb. per second. The horse-power was denned by James Watt, the in- ventor of the steam-engine, and was his estimate of the rate at which a draft-horse is capable of working. The power of an average horse, for steady work, is about .8 H. P. PROBLEMS 1. What is the relation between the K. E. of a body and its mass? Between its K. E. and its velocity? 2. Two bodies have equal kinetic energy, but the velocity of the second is three times that of the first. How do their masses compare? 3. A body is thrown vertically upward, (a) What fraction of its initial kinetic energy remains after it has risen to one half the height to which it will ascend? (b) What fraction of its initial velocity remains? (c) What fraction of the initial kinetic energy and of the initial velocity remain after the body has risen to three fourths of the total height? 4. (a) A boy starting at rest coasts on a bicycle down a hill and up another. If there were no friction, how far would he ascend the second hill without pedaling? (6) How would the result be affected if either hill were steeper than the other? 6. A sphere weighing 20 Ib. is set rolling on a level surface with a veloc- ity of 24 ft. per second. The resistance to motion (friction) is i Ib. Find (a) the K. E. of the sphere at the start; (b) the loss of K. E. in rolling 40 ft.; (c) the distance the ball will roll before stopping. 158 DYNAMICS 6. A projectile weighing 500 Ib. is fired from a cannon with a velocity of 3000 ft. per second, (a) What is its kinetic energy? (b) What was the average force acting upon the projectile in firing it, if it moved a distance of 18 ft. before reaching the mouth of the cannon? 7. In whirling a body round the hand at the end of a string, the hand is moved in a smaller circle in advance of the body whirled (Fig. 128). Show how this imparts kinetic energy to the body. FlG I2g 8. A body weighing 100 Ib. slides 30 ft. down a plane inclined at an angle of 45. Friction is 30 Ib. Find (a) the accelerating force; (b) the K. E. at the end of the thirty- foot slide; (c) the work done against friction in that distance. 9. (a) Is a person doing work against gravity in carrying a load over level ground? (b) What is the measure of the work done when the load is carried uphill? 10. A stone weighing 1.5 Ib. is thrown vertically upward by means of a force of 20 Ib. acting through a distance of 3 ft. How high will it rise above the starting-point? 11. A body weighing 15 kg. falls vertically a distance of 20 m. What is its kinetic energy? 12. A bullet weighing 5 g. and having a velocity of 300 m. per sec- ond strikes a log and penetrates it a distance of 10 cm. What average resistance did the bullet encounter in penetrating the log? 13. (a) How much work is done in filling a reservoir that has a capacity of looo cu. m. if the water must be raised 12 m. to discharge it into the reservoir? (b) How long would it take a twelve-horse-power engine to fill the reservoir? 14. What is the power of an engine that is capable of drawing a train at the rate of 30 mi. per hour against a resistance of 6250 Ib.? 15. If a man pumps 250 Ib. of water per minute to a height of 15 ft., (a) how many foot-pounds of work does he do in an hour ? (b) At what rate in horse-power is he working? 16. Distinguish between momentum and K. E. Is the force that is required to stop a moving body proportional to its momentum or to its K. E., if the body is brought to rest in a given time? If it is brought to rest within a given distance? 17. The standard i2-in. gun of the United States Navy fires a projectile of icoo Ib. weight with a velocity of 2550 ft. per second. The new i4-in. gun MACHINES fires a i6oo-lb. projectile with a muzzle velocity of 2150 ft. per second. Find the muzzle energy of each projectile. 18. At what rate in ft.-lb. per second is a draft-horse working when exerting a pull of 175 Ib. in hauling a load at the rate of 3 mi. per hour? 19. Find by trial the rate in H. P. at which you can work against your own weight in going upstairs (a) as you usually do, (b) when running up as fast as you can. 20. Find the K. E. of the ball and the K. E. of the rifle of problem 27, p. 134. Account for the fact that the K.E. of the rifle is much less than that of the ball, while their momenta are equal. V. MACHINES 137. General Uses of Machines. Any contrivance used in doing work, from the simplest hand tool to the printing- press or the locomotive engine, is a ma- chine. In general terms, the purpose of a machine is either to secure some ad- vantage by its use when energy is to be transferred from one body to another, or to transform energy from one kind into another. For example, a horse can raise a load vertically, with the aid of a rope and two pul- leys to transmit the pull and change its direction (Fig. 129); and a crowbar en- ables a man to move a heavy object with comparatively little effort, the gain of force being attended by a corresponding loss of speed and distance (Fig. 67). A dynamo receives mechanical energy from an engine or a water-wheel and FIG. 129. Use of Fixed Pulleys. 160 DYNAMICS transforms it into electrical energy, for convenient trans- mission over a wire to some distant point, where it may again be transformed into mechanical energy by means of a motor, and utilized for running a street car or for other work. The principal advantages to be derived from the use of machines are the following: 1. They permit a gain of force at the expense of speed and distance. All machines for exerting great forces are constructed on this principle. Examples: The crowbar (Fig. 67), rope and pulleys (Fig. 142), the wedge (Fig. 148), the lifting jack (Fig. 150), the hydraulic press (Fig. 17), cranes, derricks (Fig. 141), etc. 2. They permit a gain of speed and distance, with a corresponding loss of force. Examples: The bicycle, the sewing-machine, lathes, saws, centrifugal machines, etc. 3. They permit any desired change in the direction of the applied force. Examples: Rope and pulley (Fig. 129), beveled gear-wheels (Fig. 159). 4. They enable man to utilize the various natural sources of energy. For example, the energy of winds by windmills, the energy of water by water-wheels (Figs. 156-160), the energy of fuel by means of steam, gas, and other engines (Figs. 202-211). 138. Useful and Wasted Work of Machines. A machine can transfer or transform energy only as it receives energy from some other body; it is never itself the original source of work or energy. An engine can run the machinery of a factory; but steam is necessary to run the engine, and coal or other fuel is necessary to generate the steam. An ideally perfect machine would do work without loss or waste within MACHINES 161 itself; and, as will be shown later, the work done by it would be the full equivalent of the work done upon it neither more nor less. This is a general principle of the greatest scientific and practical importance. No machine is perfect, in the above sense. The motion of the parts of a machine always develops friction; and friction results in loss of energy, or wasted work, within the machine itself. Further losses result from the bend- ing and vibration of the parts of a machine, and from the stiffness of ropes and belts. The useful work of a machine is thus the difference between the work done upon it and the work lost or wasted in it. 139. Mechanical Advantage and Efficiency of Machines. -The source from which a machine receives energy is called the agent. In hand labor the person who operates the machine is the agent. Steam is the agent that supplies energy to the steam-engine. .Strictly speaking, it is the agent, not the machine, that does the work. The body moved by a machine is called the load. The opposing force exerted by the load while it is being moved is called the resistance. If the work consists in lifting the load, the resistance is equal to its weight. The force exerted by the agent upon the machine, in moving the load with- out acceleration, is called the working effort. The force that would be necessary if the machine were perfect is called the static effort, for it is just the force that would be necessary to maintain equilibrium against the resist- ance. With a perfect machine, the static effort and the working effort would-be the same; with actual machines the working effort is necessarily the greater. The ratio of the resistance to the effort is called the mechanical advantage of a machine. If a crowbar is so 162 DYNAMICS adjusted that it overcomes a resistance of 450 Ib. when the effort applied is 75 Ib., its mechanical advantage for that adjustment is 6. In studying mechanical principles no account is taken of the imperfections of actual machines, and the mechanical advantage is consequently taken as the ratio of the resistance to the static effort. In actual practice it is the ratio of the resistance to the working effort. The mechanical advantage of a machine is de- termined by the relative size or number of certain parts of it, as we shall see. The ratio of the useful work done by a machine to the work done on it is called the efficiency of the machine. It is commonly expressed as a percentage. The efficiency of a perfect machine would be 100%. The efficiency of actual machines lies in most cases between 30% and 95%. Its determination in any given case is an experi- mental problem. Let E s denote the static effort and E w the working effort when the resistance offered by the load in uniform motion is R, and D e the distance through which the effort acts (the effort distance) in moving the load through the dis- tance D r (the resistance distance). Then the work done by the agent upon the machine is E w D e (the product of the working effort and the effort distance); and the work done by the machine upon the load is RD r (the product of the resistance and the resistance distance). Then, also, r> Mechanical advantage = -IT. (23) **j RD r Efficiency = - (24) E w D e = RD, + wasted work. (25) MACHINES 163 140. The Simple Machines. All machines, however complicated, are but modifications and combinations of one or more of the six simple machines. These are the lever, the wheel and axle, the pulley, the inclined plane, the wedge, and the screw. An understanding of the simple machines is, of necessity, the first step in mechanical knowledge. Each of the simple machines presents the following problems: 1. To find its mechanical advantage in terms of cer- tain dimensions of it. 2. To show that, as an ideal machine, it transmits energy without gain or loss, i.e. to show that E s D e = RD r . This is the general law of machines. 3. To become acquainted with some of its common forms and their uses. 4. To determine the efficiency of the machine provided for laboratory study. 141. The Lever. A bar or rod which turns about a fixed support or axis, in transmitting motion from one body to another, is called a lever, and the support is called the fulcrum. For convenience in discussing their use levers are grouped into three classes. In levers of the first class (Fig. 131) the fulcrum F is between the points of application of the effort E and the resistance R; in levers of the second class (Fig. 132) the resistance is applied between the fulcrum and the effort; in levers of the third class (Fig. 133) the effort is applied between the fulcrum and the resistance. Taking the classes in order, the fulcrum, the resistance, and the effort are respectively between the other two, a fact easily remembered from the initial letters FRE (as in the word free) . 1 64 DYNAMICS The mechanical principle of the lever, of whatever form, is that of moments of force in equilibrium (Art. 78). For the effort and the resistance always act in opposite direc- tions round the fulcrum; and, disregarding friction, their moments are equal, whether the lever is at rest or in uniform motion; that is, EA e = RA r , or R: E:: A e : A r , (26) A e being the arm of the effort and A r the arm of the resist- ance. Hence the mechanical advantage of the lever is deter- mined and measured by the ratio of the arm of the effort to the arm of the resistance. This ratio is commonly known as the leverage. The leverage can evidently be made as great or as small as is desired, by simply changing the position of the fulcrum or the points of application of the effort and the resistance. Applications of the Lever. The lever in different forms is adapted to various special uses. In most cases it enables a given force to overcome a resistance several times as great as itself. This advantage, which is known as a gain of A*- *-** *-- -A.- -4 _^_ __^ R FIG. 131. Levers of the First Class. FIG. 132. Levers of the Second Class. force j is afforded by forceps, pincers, wire cutters (Fig. 1316) and nutcrackers (Fig. 1326); all of which are double levers having the arm of the effort longer than the arm of the resistance. A lever may be curved or angular, as a claw- MACHINES 165 hammer when used in drawing a nail (Fig. 68). Many forms of the lever are designed with reference solely to *" - A : * FIG 133. Levers of the Third Class. convenience or adaptability to the work to be done, the relative value of the effort and resistance being unimpor- tant. Tweezers, coal tongs, sugar tongs, and scissors are examples. In certain applications of the lever the advantage secured is a gain of speed and distance, i.e. the point of application of the resistance moves faster and farther than the point of application of the effort. This is well exempli- fied by the movements of our bodies and the bodies of animals in general. The movable parts of the skeleton are levers; the joints are the fulcrums. The muscles are attached to the bones near the joints by means of tendons. A muscle acts by contracting or shortening. This causes the bone to which it is attached to move, and the farther extremity of the bone moves much faster and farther than the point to which the muscle is attached (Fig. 1336). Levers of this type are used in such instruments as the aneroid barometer to magnify small motions. 142. The Law of Work for an Ideal Lever. Suppose a load R (Fig. 134) to be raised through a vertical distance D r , by means of a perfect lever, while the effort E acts i66 DYNAMICS FIG. 134. The Law of Work. vertically through a distance D e . (The arms of the resistance and effort constantly change during the motion; but their ratio, A e : A r) remains constant. Why?) Since the lever is assumed to be perfect, the working effort is equal to the static effort and R:E::A e :A r , while the work is in progress. By geometry, D e : D r :: A e : A,. (Why?) From the two proportions, R: E:: D e : D r or ED e = RD r . But ED e is the work done by the effort and RD r is the work done upon the load. Hence we have proved that, in doing this piece of work, a perfect lever would transmit energy without gain or loss. It can be shown that this is always true of a perfect lever, whatever the character of the work. It follows that, under the most favorable con- ditions, whenever force is gained by means of a lever there is a proportionate loss of speed and distance, and whenever speed and distance are gained there is a proportionate loss of force. (Show this.) Evidently a lever, at least, can render no assistance as a part of a " perpetual-motion machine." PROBLEMS 1. In using scissors is greater force required when the cutting is done near the tips of the blades or near the handles? Why? 2. Classify the following levers, and state in each case whether the effort MACHINES 167 is greater or less than the resistance: the wheelbarrow, oar, fishing-rod, equal-arm balance, steelyard, nutcracker. 3. Use a pencil as a lever of the first class to move a book; also as a lever of the second class. 4. In which class or classes of levers is the effort necessarily less than the resistance? In which may it be either greater or less? FIG. 135. The Foot as a Lever. FIG. 136. 5. (a) If a stone offers a resistance of 850 lb., what leverage will be required to move it by means of a force of 125 lb.? (b) If the stone is moved by a crowbar 5 ft. long used as a lever of the first class, the effort and the resistance being applied at the ends, where is the fulcrum? 6. A person rises on his toes by the action of the calf muscles, which pull on a tendon attached to the heel bone (Fig. 135). To what class of levers does the foot belong when thus used? What relations hold between the three forces and the two distances? Which of the forces is equal to the person's weight? It may be of assistance to note that the three forces are parallel and in equilibrium. This action of the foot can be studied experi- mentally by means of two short boards, AC (Fig. 136), and a stout cord. The force exerted in thus lifting one's self can be measured with a balance, as shown in the figure. 143. The Wheel and Axle. The wheel and axle (Fig. 137) consists of a wheel and an axle or cylinder, turning as one body on the same axis. In the modified form known as a windlass (Fig. 138) the wheel is replaced by a crank, which serves the same purpose. The figures clearly indi- cate the use of this machine as an aid in manual labor. i68 DYNAMICS The effort is applied at any convenient point on the cir- cumference of the wheel or at the crank handle, and the resistance acts at the circumference of the axle. FIG. 137. Wheel and Axle. FIG. 138. Windlass. Mechanical Advantage. The wheel and axle may be regarded as a continuously acting lever (Fig. 139), the radius of the wheel being the arm of the effort, and the radius of the axle the arm of the resistance. If the machine were perfect, the moments of the effort and the resistance would be equal, both for equilibrium and for uniform motion; i.e. we should have E X radius of wheel = R X radius of axle, or R: E:: radius of wheel : radius of axle. (27) The mechanical advantage of the wheel and axle is, therefore, the ratio of the radius of the wheel to the radius of the axle. The Law of Work. During one complete revolution of the wheel the effort acts through a distance equal to its circumference, and the load is moved a distance equal to FIG. 140. The Capstan. MACHINES 169 the circumference of the axle. The ratio of the effort distance, D e , to the resistance distance, D r , is evidently the same for any number of revolution, and is given either by the ratio of the circumferences of the wheel and the axle or by the ratio of their radii; that is, D e : D r :: radius of wheel: radius of axle. From this proportion and the one above, we have for a perfect wheel and axle R: E:: D e : D r or ED e = RD r . That is, the wheel and axle, like the lever, transmits ener- gy without loss or gain, except in so far as there is loss due to friction. The loss usually amounts to from 10% to 20%, depending principally upon the condition of the bearings. Applications. The applications of the wheel and axle, in various modified forms, are numerous and important. The capstan (Fig. 140), used on small vessels for raising and lowering the anchor, has a vertical axle or drum, and the effort is applied at the end of hand spikes, inserted in holes at the top. Ratchets at the bottom prevent backward motion during any interruption of the work. The capstan or windlass is used^in connection with pulleys for moving houses, and, as part of the hoisting tackle of derricks, for lifting or lowering heavy weights (Fig. 141). Wheels of unequal size, interacting by means of cogs (Fig. 159) or connected with a belt or a chain, as in the sewing-machine, lathe, bicycle, clock, etc., are further applications of the same principle, the change of speed and of force being always in inverse proportion, barring losses due to friction. 170 DYNAMICS PROBLEMS 1. (a) The radius of a wheel is 40 cm. and the radius of the axle 12 cm. Neglecting friction, what effort is required to raise a load of 150 kg.? (b) Through what distance does the effort act in raising the load 35 m.? (c) How FIG. 141. Compound Windlass on a Derrick. FIG. 142. Derrick, with Hoisting Tackle of Rope and Pulleys. much work is done by the effort? (d) How much is done against the weight of the load? 2. What effort will be required to raise a weight of 200 kg. with a wheel and axle the efficiency of which is 90%, the radius of the wheel being 42 cm. and the radius of the axle 14 cm.? 3. The weight in the preceding problem is raised 25 m. Find (a) the work done upon the machine; (b) the work done against gravity; (c) the energy wasted. 4. Find the mechanical advantage of the windlass with gear-wheels shown in Fig. 144, if the length of the crank arm is 16 in., the radius of the axle or "barrel" round which the rope is wound 5 in., the number of cogs on the small wheel 15 and on the large wheel 75. If the efficiency of the machine is 75%, what effort is required to raise 1000 lb.? 144. The Pulley. A pulley-block or tackle-block holds from one to six pulleys, which are capable of turning indi- vidually* at unequal rates. A common form of hoisting tackle (Fig. 142) consists of a rope and two pulley-blocks, MACHINES 171 FIG. 143. The Fixed Pulley. FIG. 144. The Mov- able Pulley. one being attached to a fixed support and the other to the moving object. The pulleys of a fixed block are called fixed pulleys, and those of a movable block ; movable pulleys. Mechanical Advan- tage. A fixed pulley (Fig. 143) may be re- garded as a continu- ously acting lever of the first class, whose arms are radii of the pulley. The arms being equal, the static effort and the resistance are also equal. This is further evident from the fact that, except for a slight possible difference due to friction, the parts of the rope on the two sides of the pulley must be under the same tension. The only advantage gained by the use of one or more fixed pul- leys is a change in the direction of the effort (Fig. 129). The movable pulley (Fig. 144) is, in prin- ciple, a continuously acting lever of the second class, the arm of the effort being the diameter of the pulley, and the arm of the resistance its radius. Hence the mechan- ical advantage of a single movable pulley is two (R: E = 2). This follows also from the fact that the two equal upward pulls of the rope on the two sides of the pulley to- gether suppoj-t the load. FlG . I46 . FIG. 145. 172 DYNAMICS When pulleys are used in combination by passing a single rope alternately round the pulleys of a fixed and a movable block (Figs. 142, 145, and 146), the mechanical advantage is determined by assum- ing the tension to be the same in all parts of the rope. The load is thus sustained by as many equal parallel forces as there are parts of the rope running to and from the movable block; and each of these forces is equal to the effort. That is, if n denotes the number of times the rope passes to and from the movable block, then nE = R or R:E = n, and the mechanical advantage is n. The Law of Work. When a load is moved a certain distance D r with a rope and pulleys, the distance between the fixed and the movable pulley-blocks decreases by an equal amount, and hence each of the n parts of the rope extending between the blocks shortens by just that much. Since the whole rope is of fixed length, the free end of it, where the effort is applied, moves through a distance D e equal to n times the resistance distance, or nD r . Stated the other way about, the load moves through a distance only one n /A as great as that through which the agent acts. Thus, with perfect efficiency, gain of force by means of a set of pulleys is secured at the expense of a proportionate loss of distance. Expressed mathematically, R: E = n and D e : D r = n; hence R: E:: D e : D r , or ED e = RD r . Efficiency. When a load is raised by means of a rope thrown over a beam, the effort greatly exceeds the resist- ance of the load, on account of the sliding friction between the rope and the beam; and the two parts of the rope are thus under unequal tension. When the rope is passed over a pulley, friction is greatly reduced but not wholly elim- inated; hence there is a slight difference in the tension of the ropes on the two sides of the pulley when work is in progress. With a set of pulleys, there is loss of tension MACHINES 173 from the free end of the rope to the fixed end, at every point where it passes over a pulley; hence the working effort must be greater than the average tension in the rope and the work on the load is necessarily less than the work of the agent. The usual effi- ciency of a hoisting tackle of four to six pulleys is from 60% to 75%. PROBLEMS 1. Draw a diagram of a system of pulleys such that the mechanical advantage is seven; such that it is FIG. 147. A Steering-wheel and Gear. u t W, the wheel; B, the barrel; C, chains; S, S standards; R, the 2. What effort is required to raise Rudder-head; T, the tiller, a load of 500 Ib. with a set of pulleys arranged as in Fig. 146, the efficiency being 70 %? 3. The steering wheel and gear of a vessel shown in Fig. 147 is a com- bination of three machines, of which the last, T, called the tiller, is rigidly attached to the rudder-head, R. Describe the action of this compound machine, and express its mechanical advantage in terms of the appropriate dimensions of its parts. 4. Let the boys of the class investigate and report on the mechanism of a bicycle, with special reference to the ratio of gain of speed; and let the girls prepare a like report on the mechanism of a sewing-machine. Express in terms of the determining factors the distance that a bicycle goes with each revolution of the pedals, and the number of stitches that a sewing-machine takes for each up and down motion of the pedal. 145. The Inclined Plane. The principal use of the inclined plane is to raise heavy bodies that can be rolled. When a barrel of flour is placed in a wagon by rolling it up a heavy plank, the plank is used as an inclined plane. The laws of the inclined plane have already been fully considered (Arts. 74 and 133). What is the measure of its mechanical advantage? How has it been shown that, 174 DYNAMICS FIG. 148. disregarding friction, the work of the agent is equal to the work (against gravity) on the load? 146. The Wedge. The wedge is, in principle, a movable inclined plane. It is used in separating surfaces against great resistance, as in split- ting logs and timbers (Fig. 148). The motion of a wedge is opposed by great sliding friction against its faces, and its efficiency is consequently low. This friction is useful, however, as it keeps the wedge from slipping out of place during the intervals between the blows that drive it farther in. The wedge in actual use departs to such an extent from the conditions of a perfect machine that a study of the ideal relations is scarcely profitable. It is obvious, however, that the thinner the wedge the greater is its mechanical advantage. The ax, the knife, and the chisel are forms of the wedge adapted to special uses. 147. The screw is a cylinder about which ex- tends a spiral projection or thread. That the thread is a modified form of the inclined plane can be shown by winding a paper triangle about a pencil (Fig. 149). The distance between adjacent turns of the thread, measured parallel to the axis, is called the pitch of the screw. The nut in which the screw turns is provided with a spiral groove to re- ceive the thread. FIG. 149. The lifting-jack (Fig. 150) is a screw operated by means of a lever. It is used for lifting heavy bodies, as in replacing a derailed locomotive or in raising a house. During one complete turn of the lever, the load is raised a distance MACHINES 175 equal to the pitch of the screw, and the effort acts through the distance 2irA, A being the arm of the lever. Hence the work done by the agent during one turn is X 2irA, and the work done on the load is 'Rp. Assuming an efficiency of 100%, we should have Q.TT A E X 2-irA = Rp, or R :E = - (28) Thus with a lever arm 2 ft. long and a pitch of \ in., the mechanical advantage would be 302. But the friction is always great, in fact more than sufficient to hold the screw in place against any resistance. If we take the efficiency as i, the actual mechanical advantage with the above dimensions would be 100; under which conditions an effort of 100 Ib. would still be sufficient to raise a load of 5 tons. FIG. 1 50. Jack-screw. FIG. 151. Copying Press. The screw propeller (Fig. 162) is an efficient machine for doing work on a large scale. In many applications of the screw, such as the copying press (Fig. 151) and the vise, its purpose is to exert great static pressures; in others, as wood-screws, machine-screws, and bolts, its use is to hold the parts of bodies together. PROBLEMS 1. A block at rest upon a board 100 cm. long begins to slide when the board is inclined so that its higher end is 40 cm. above the other. Friction is what percentage of the pressure of the block on the plane? (The ratio of the sliding friction to the pressure between two surfaces is called the coefficient of friction for those surfaces.) i 7 6 DYNAMICS 2. Assuming an efficiency of 80%, what force is required to haul a load of 2 tons (including the weight of the wagon) up a grade such that the ascent is i ft. in a distance of 10 ft.? 3. A lifting-jack, the screw of which has a pitch of i in., is used to raise a load of 5 tons. The effort is applied at the end of a lever 3 ft. long. The efficiency is 25%. Find the effort. 4. The worm-wheel (Fig. 152) consists of an endless screw or worm and a gear-wheel. The worm drives the wheel, and is itself driven by a crank or a wheel. Express the mechan- ical advantage of this compound machine in terms of its dimensions. FIG. 152. Worm-wheel. 5. Show that the hydraulic press, regarded as a perfect machine, con- forms to the law that the work of the agent is equal to the work on the load. VI. ENERGY 148. Energy. The kinetic energy of a body may be denned as the measure of its capacity for doing work, by virtue of its mass and its motion. A body at rest may or may not be capable of doing work. A charge of powder behind a rifle ball has a definite capacity for doing work upon the ball which a mass of sand in the same situation would utterly lack. A bent bow can do work upon an arrow, but an unbent bow can not, although both are at rest. A locomotive with steam up has a certain capacity for doing work, which we know resides in the supply of steam in the boiler. Examples might be multiplied indefi- nitely of bodies capable of doing work, but not in motion. In all such cases the body is said to possess energy. En- ergy, then, depends upon various conditions and properties of bodies, and consequently exists in various forms. To discuss the different forms of energy and their rela- tion to one another would be nothing less than to summarize the whole of physics. The fuller knowledge of such mat- ters must therefore come in the regular progress of the sub- ENERGY 177 ject. Forms of energy other than mechanical are referred to in the present chapter only because they obtrude them- selves upon our attention as soon as we begin to consider energy at all. 149. Mechanical Energy. In bending a bow work is done against its elastic resistance, and the result is a state of strain (Art. 87) or a distortion of the elastic body. In recovering from distortion, the bow can do work equal in amount to the work done in bending it; and this amount of work is the measure of its energy of strain. All elastic bodies, solid or fluid, possess energy of this kind when in a state of strain. The coiled spring of a watch or a clock, the compressed air in an air rifle, and the steam in an engine boiler are familiar examples. A body so situated that it is capable of descending to a lower level possesses a definite store of energy by virtue of its mass, its elevation, and the earth's attraction. This is commonly known as energy of position. It is measured by the product of the weight of the body and its elevation. For example, if the hammer of a pile driver weighs 1000 Ib. and is raised to a height of 25 ft., its energy of position is 25,000 ft.-lb.; for, in falling from that height, its weight does 25,000 ft.-lb. of work upon it, imparting 25,000 ft.-lb. of kinetic energy to it, and enabling it to do 25,000 ft.-lb. of work upon a pile at the end of its fall. Water power is energy of this sort, and is utilized by means of water-wheels (Art. 157). Energy of strain and energy of position are two vari- eties of potential energy. In more definite terms, they are the two mechanical varieties of potential energy, for poten- tial energy, in the broadest sense, means static energy, as distinguished from kinetic, and includes such forms as the energy of coal, which is chemical energy, not mechanical. 178 DYNAMICS The essential conditions for potential energy of a mechan- ical nature are the existence of force tending to move the body or to cause relative motion of its parts, and room for such motion to take place. Energy of strain, energy of position, and the kinetic energy of moving masses are the different forms of mechan- ical energy. They are the result of mechanical work (the only kind of work yet con- sidered), and they are avail- able for doing mechanical work. 150. Energy of Rotation. Fly- wheels. The kinetic energy of a rotating body is frequently a matter of importance, particu- larly in the operation of machin- ery. The flywheel of an engine (Fig. 153) is an instructive ex- ample. It is always massive and is frequently of great size; and, when in rapid motion, it possesses a large store of kinetic energy. This, of itself, however, is of no importance, since the wheel can pay out no more energy than it receives from the engine. But the large energy capacity of the wheel enables it to pay out this energy at a practically con- stant rate throughout each revolution, while receiving it intermit- tently from the engine, with each stroke of the piston. This action changes what would otherwise be an unsteady, jerky mction of the engine and machinery into a steady motion; just as a water tank, while receiving its supply in spurts from an ordinary pump, main- tains a steady flow through an outlet pipe. The greater the kinetic energy of a flywheel the more effective will it be as an equalizer of motion. Now the energy of the wheel as a whole is merely the summed up energy of its parts; and the energy of any part, of mass mi, distributed in the form of a ring of radius r\ (Fig. 153), is \ miVi z (Formula 21). For a given number of revolu- FIG. 153. Flywheel. ENERGY 179 tions per second, Vi is proportional to the radius r\. (Why?) Hence the energy of a given portion of the mass, as mi, is proportional to the square of its distance from the axis. It follows that the effective- ness of a flywheel is increased by having as much of its mass as is possible at the greatest distance from the axis, i.e. in the rim; and the larger the diameter of the wheel the greater will be the advantage thus gained. Since kinetic energy is proportional to the square of the velocity, the energy of a flywheel increases as the square 'of the number of revolutions per second. Hence a small flywheel is effective if it is run at high speed. 151. Heat and its Relation to Mechanical Work. Fric- tion generates heat. The hands are warmed by rubbing them briskly together. A match is ignited by drawing it rapidly over a rough surface. The coaster brake of a bicycle, when applied on a steep grade, quickly becomes hot enough to burn the fingers. When the axle of a car wheel is not properly lubricated to diminish friction, a "hot box" results, and the temperature may even rise sufficiently to set the car on fire. When a cord or rope is grasped tightly and drawn rapidly through the hand, the heat generated quickly causes a burn. When a tool is ground on a dry grindstone, it becomes too hot to touch. In most cases of friction, it is true, there is no noticeable rise of temperature; but this is due to the fact that the heat is not generated rapidly and escapes readily to surround- ing bodies. The energy expended in overcoming friction always produces heat. In fact, the energy thus expended becomes heat; for, as will be more fully explained later, heat is a form of energy, and the amount generated by fric- tion is always equivalent to the amount of mechanical work done against the friction. When a moving body is brought to rest in the act of imparting motion to another body, its kinetic energy is i8o DYNAMICS transferred to the other body; when it is brought to rest by friction, its kinetic energy is transformed into heat, partly within itself and partly within the body or bodies that stop it. The same transformation of energy occurs when a body is suddenly stopped by impact against a body which it does not move. Thus a piece of lead can be noticeably heated by rapidly hammering it on an anvil, and bullets are often partly melted by the heat generated when they strike a steel target or a stone. The change of heat into mechanical energy which is accomplished by means of steam and other heat engines is further evidence that heat is itself a form of energy. Whether this energy is kinetic or potential and in what it consists are questions to be considered in a later chapter. 152. Other Forms of Energy. Heat is generated dur- ing many chemical changes, as in the burning of any fuel, the decay of vegetable matter, the slaking of lime, etc. Burning produces light, also a form of energy, as well as heat. The chemical changes in an electric battery produce electrical energy. Substances capable of generating heat, light, or other forms of energy by chemical change are said to possess chemical energy. This is a form of potential energy. All the movements of an animal involve an expenditure of muscular energy; for the movements are due to muscu- lar action, and in this action the muscles do work. The amount of potential energy stored in the muscles is great; a horse, for example, can do nearly two million foot-pounds of work per hour for several hours. It is evident, however, that the amount is not unlimited, for any animal becomes exhausted after prolonged exertion. The renewed supply of energy comes from the food eaten. Muscular energy is ENERGY 181 available for doing work only through chemical changes by which the muscles are in part consumed, much as fuel is consumed in a fire. Similar changes take place in all the organs of the body while they -are performing their special functions, and some of the energy is always liber- ated as heat. It is this heat that maintains the temper- ature of the body. The principal forms of energy with which physics deals are the forms of mechanical energy already considered, and the energy of heat, sound, light, and electricity. 153. The Conservation of Energy. Energy, as we have learned, is capable of transference from one body to an- other and of transformation from one form into another. Either of these changes may take place alone, or they may occur simultaneously. Is energy capable of other changes than these? Can energy cease to exist as energy, either by becoming something else or simply by ceasing to be? Is it possible to make or create energy, or by any device to increase energy as we do force? After long-continued observation and experiment, these questions were all answered in the negative about the middle of the last century, and later advance in science and invention has only confirmed the answer. All experience teaches that some portion of matter has lost whatever energy another portion of matter gains, and that energy in any form dis- appears only by transformation into an equivalent amount of energy in some other form or forms. This is known as the law or principle of the conservation of energy. Briefly, it asserts that energy is not created or destroyed in any phenomenon or process known to man. Energy is in this respect like matter; the total quantity of either in the uni- verse, so far as we know, remains constant. 182 DYNAMICS Assuming the principle of the conservation of energy, we can give a more complete account of phenomena. The following cases will serve to illustrate. When one highly elastic body strikes another, the greater part of the energy continues in the kinetic form in one or the other or in both of the bodies, depending upon conditions; the remainder is transformed into heat and sound. For example, when one ivory ball strikes another of the same size squarely (Fig. 3), there is a very nearly complete transfer of kinetic energy from one to the other. When a steel or an ivory ball is dropped upon the smooth, flat surface of a heavy block of steel or stone, the ball is slightly flattened and the block dented; and the kinetic energy of the ball becomes for the instant potential energy of strain in the two distorted bodies. All of the energy has been transformed by the impact, and some of it has been transferred to the block. In the immediate recovery of the bodies from distortion, the energy is again changed to the kinetic form, and the block transfers its energy to the ball. But the restoration of kinetic energy is not complete; for in both the impact and the rebound there has been a partial transformation into heat, the only effect of which is an imperceptible rise of temperature in both bodies. When either or both of two bodies is inelastic and one strikes the other, the kinetic energy is largely or wholly converted into heat, as when a stone falls to the ground or a lead bullet strikes a stone. The melting of the bullet in the latter case, as sometimes happens, is due to the fact that the amount of energy transformed is great in comparison with the mass. If the earth in its motion round the sun should collide "head on" with another body like itself and moving with an equal velocity in the opposite direction, the heat generated would convert the entire mass of both bodies into a white-hot vapor. 154. Availability of Energy. While energy is conserved in all its changes, it tends constantly to the condition of uniformly diffused heat, in which form it is no longer available to man for doing useful work. The wasted work of machines is mechanical energy lost as heat in the pro- cess of transmission. The work done in moving wagons, street cars, and trains is done against friction; and the DYNAMICS OF FLUIDS 183 energy thus expended is dissipated as heat. Coal, oil, gas, and electricity are valuable only by reason of their available energy; but, once used, this energy passes to the unavailable form. Economy of energy thus becomes quite as important a matter in the affairs of daily life as economy of materials. PROBLEMS 1. Show that, if the friction of the air is negligible, the sum of the kinetic and potential energies of a body thrown vertically upward remains constant during its flight. 2. Why is a brake not heated if it is applied with such force that the wheel slides along the ground or the track? Where will the heat then be generated? 3. What are the principal sources of energy utilized by man, and how are they made available? 4. Discuss the energy of a swinging pendulum. VII. DYNAMICS or FLUIDS 155. Pressure of Moving Fluids. The pressure exerted by a moving fluid in consequence of its mass and its motion must be distinguished from the pressure due to its weight. The gravity pressure of the air is between 14 and 15 Ib. per square inch, or over 2000 pounds per square foot, whether the air is at rest or in motion. This is a balanced pressure. It does not bend a twig or disturb the most delicate flower. Air in motion exerts a one-sided pressure, which, although small compared with its gravity pressure, is nevertheless responsible for all the familiar effects of winds. Experiments have shown that the pressure of a moving fluid (water or air) against a fixed surface varies approxi- mately as the square of its velocity. The pressure of the wind against a fixed surface which it strikes perpendicu- larly is approximately .004 V 2 Ib. per square foot, V being 184 DYNAMICS the velocity of the wind in miles per hour. In computing the necessary strength of tall buildings, architects allow for a maximum wind pressure of 30 Ib. per square foot, which, by the above formula, corresponds to a velocity of nearly 90 mi. per hour. Such a velocity is attained only in violent and destructive storms. 156. Action of Wind on Sails. When the wind strikes a surface at any oblique angle, the force that it exerts is perpendicu- lar to the surface, except for the tangential force of friction, due to the sliding of the air along the surface. This tangential force is relatively small, and may be disregarded. Let WO (Fig. 155) represent the direction of the wind against the sail S of a boat, and OP the total force exerted. OP is perpen- dicular to the sail, and is equivalent to the two forces OL and OF. The sideward component OL is opposed by the resistance of the water against the broadside of the boat, and produces little effect. The forward component OF is effective in propelling the boat. It is thus possible for a vessel to sail obliquely against the wind, as is clearly shown in the diagram, and, by tacking (sailing first to one side of the wind, then the other), to steer a general course exactly opposite to the direction of the wind. In sailing directly with the wind the pressure against the sail decreases as the speed of the vessel increases; and, if the speed should become equal to that of the wind, the pressure would vanish. But some pressure is always necessary to maintain the motion of the vessel; hence so great a speed is impossible. On the other hand, in sailing obliquely toward the wind, as shown in the figure, the wind would overtake the sail and exert pressure upon it even if the vessel was traveling much faster than the wind. This curious result is actually attained with ice-boats, which require little force to keep them going, the friction between their steel runners and the ice being slight. A record of 90 mi. an hour has been made in this manner. The action of the wind on the vanes of a windmill is the same as upon a sail. The vanes are oblique to the wind, and their motion is DYNAMICS OF FLUIDS 185 FIG. 156. Overshot Wheel. at right angles to it. (Draw a diagram similar to Fig. 155, show- ing the action of the wind on a vane, and explain in detail.) 157. Water-power. The energy of running water is utilized by means of water- wheels of various forms. The overshot wheel (Fig. 156) and the undershot wheel (Fig. 157) are the earlier and simpler types. They are still used to some extent where only a moderate amount of power is required and the supply of water is abundant for the purpose. The overshot wheel requires a considerable fall; and in order to utilize all the available power, its diameter must be equal to the height of the fall. The buckets on the circumference of the wheel fill with water at the top and empty at the bot- tom. The weight of this water turns the wheel. The under- shot wheel is used where there is little fall. The boards or buckets which project from its circumfer- ence at regular intervals dip into the running water, and are driven forward by the current. The available power of a stream, expressed in foot-pounds FIG. 157. Undershot Wheel. i86 DYNAMICS per second, is the product of the weight of water supplied in a second and the fall or "head" which can be utilized. (The horse-power is equal to this product divided by 550.) For a given power, the undershot wheel takes a relatively large flow of water, as its efficiency is only about 25%. Under favor- able conditions the efficiency of the overshot wheel is about 75%, but the height of fall which it can utilize is limited to the pos- sible diameter of the wheel. FIG. 158. Section of Turbine Wheel. G, guides; B, blades. 158. reached Turbine and Pelton Wheels. Water-wheels have a high degree of perfection in the turbine and the Pelton types. In the common form of tur- bine wheel (Figs. 158 and 159) the water is directed inward by fixed guides, G, so as to strike at the most efficient an- gle against the blades, B, of the wheel. The wheel is inclosed in an outer case, C (Fig. 159), to which the water is con- veyed from a higher level, through a supply pipe not shown in the fig- ure. This pipe is joined FIG. 159. Turbine Water-wheel. to the Case On the front DYNAMICS OF FLUIDS side, covering the large, circular opening. The water fills the supply pipe to the top; hence the height of the pipe determines the pressure at the wheel. L From the outer case the water forces its way between the guides, G, strikes against the blades of the wheel, and drops to the tail-race, having expended 80% or more ... . . FIG. 1 60. Pel ton Wheel. of its energy in turning the wheel. The shaft of the turbine is sometimes vertical, as in the figure, and sometimes horizontal. The turbine is placed above ground or at the bottom of a wheel pit, depending upon the location of the power house. In the latter case the waste water finds an outlet through a tun- nel, leading from the bottom of the pit. Turbine wheels utilize the available power to the full, whether the head is low or high, for the greater the head the greater will be the pressure at the wheel. The turbines of the Niagara Falls Power Company are located at the bottom of a wheel pit 136 ft. below the level of the supply. The shaft of each turbine extends to the power house above, and its upper end carries the rotating part of a 5000 horse-power dynamo. The turbines of the Great Western Power Company at Big Bend on the Feather River, California, work under a head of 525 ft., and each develops 18,000 horse-power. This is the greatest head which has been utilized by means of the turbine wheel, up to the present time (1911). The Pelton wheel (Fig. 160) is especially adapted for utilizing the power of small mountain streams, as it takes but little water and operates under any head, from 25 ft. up to the highest that nature provides. The water is 1 88 DYNAMICS conveyed to the wheel through steel pipes capable of with- standing great pressure, and issues from a nozzle in a small but powerful stream, which is directed against the lower buckets of the wheel. The buckets have a central parti- tion, which splits the stream, deflecting part toward each side. The water is thus caught and held by the buckets until nearly all its energy is imparted to the wheel. Standard Pel ton wheels range from 3 to 6 ft. in diameter; but it is the head under which they work, rather than their size, that determines their power. The power plant of the Pike's Peak Hydro- electric Company of Colorado Springs utilizes a head of 2150 ft., which is equivalent to a pressure of 935 Ib. per square inch. The water issues from the nozzles with a velocity of 22,300 ft. per min- ute, or 250 mi. per hour. The velocity of the wheel buckets is nearly half as great, or twice the velocity of the fastest express train. Each wheel develops 1500 horse-power. The development of electrical science within the past thirty years has enormously enhanced the industrial importance of water-power. Formerly water-power could be utilized only at its source, and all but an insignificant fraction of it ran to waste the world over. It has now become one of the most valuable natural resources. In mountain regions where high heads of water are available, electric power stations are being established in rapidly increasing numbers. Here torrents of water unceasingly deliver their store of energy to water-wheels, and these to dynamos generating electric currents, by which the energy is transmitted to distances of 100 to 200 mi. or more, and at small cost in most cases in comparison with steam. " The immeasurably vast resources of power available by this means open up in all directions new fields for enterprise, offering profitable employment for both labor and capital." 159. Resistance of the Air and Water. The resistance of still air to the motion of- a body through it varies as the square of the velocity of the body, just as wind pressure against a body at rest varies as the square of the velocity of the wind (Art. 155). The force, whether we call it DYNAMICS OF FLUIDS 189 a resistance or a pressure, is determined by the relative motion of air and body. The same is true of the motion of bodies through the water. When the speed of a train is not over 5 mi. per hour, the principal resistance to its motion on a level track is friction at the wheels; when its speed is 50 mi. per hour, the principal resistance is that of the air, which is 100 times as great as with a speed of 5 mi. per hour, while the friction at the wheel is practically unchanged. The high speed of modern express trains empha- sizes the importance of diminishing the air re- sistance as much as possible. The gasoline motor car shown in Fig. F '' rtl - Car bui " for 161 represents one solu- tion of the problem. It carries its own power, and its wedge-shaped end divides the air as the prow of a vessel divides the water. The resistance of the air is turned to account by an aeronaut when he drops from his balloon to the ground with the aid of a parachute (Fig. 112). Since the resistance of water is about 800 times as great as that of air (owing to its greater density), the proper design of vessels is a matter of the first importance, and has long been the subject of careful investigation. But while a good design diminishes the resistance at a given speed, it does not alter the fact that the resistance increases as the square of the speed. 160. Relation of Power to Speed. The work done in propelling a ship is the product of the resistance to the ship's motion and the distance that the ship travels. The work done in a second, or the i go DYNAMICS power, is the product of the resistance and the distance traveled in one second. But the resistance varies as the square of the speed, and the distance traveled in a second varies as the speed; hence the power necessary to propel a vessel varies as the cube of the speed. To double the speed of a vessel the power must be increased eight fold; to increase the speed from 20 knots to 25 knots the power must be nearly doubled (25* -5- 2o 3 = 1.953). 161. The Screw Propeller. The power of engines is applied in the propulsion of ships, tugs, gasoline launches, etc., by means of the screw propeller (Fig. 162). The mechanical action of the propel- ler is exactly opposite to that of the windmill. This is easily seen in the case of the electric fan, which is a screw propeller designed to FIG. 162. Twin Screw Propellers FIG. 163. Motor Ice-boat propelled of a Motor Boat. by an Aerial Screw. create a current of air. A windmill would accomplish the same result if it were run by an engine. But while the fan drives a current of air in one direction, the reaction of the air on the fan tends to drive it in the opposite direction. This reaction on the fan is transmitted to the bearings of the axle, and thence to the motor. If a motor and fan are mounted on a light carriage, the whole becomes a self-pro- pelling machine, running in one direction while setting up a current of air in the opposite direction. Similarly, the propellers of a steam- ship drive the water backward; and the reaction, transmitted by the shaft and its bearings to the vessel, drives the vessel forward. A propeller designed to work in water usually has three rounded blades, as shown in the illustration. An aerial screw, such as is used in propelling ice-boats (Fig. 163), dirigible balloons (Fig. 164), and aeroplanes (Figs. 165 and 166), has been found to be more efficient when constructed with only two blades or arms. The efficiency of DYNAMICS OF FLUIDS igi a ship's propeller is generally not above 50%, and reaches about 65% under the most favorable conditions. Much energy is necessarily lost in imparting motion to the water. 162. The Navigation of the Air. The well known spherical balloon, invented in France in 1773, was the first and, until recently, FIG. 164. The Baldwin Dirigible Balloon. the only successful device for navigating the air. Balloons of this type can only drift with the wind. The utmost that the aeronaut can do in determining his course is to choose the current of air in which his balloon shall float. This he does by throwing out ballast if he wishes to ascend, or by opening a valve to let some of the gas escape, if he wishes to reach a lower level. UPPER SUPPORTING PLANE SPROCKET WHEELS AND CHA.NS ELEVATING P SUPPORTING PLANE EXIBLE'END FIG. 165. The Original Wright Biplane. (The first successful flying machine.) The dirigible (steerable) balloon or airship is an invention of recent years, and is still being improved. Buoyancy is secured by means of a gas bag, as in the older type of balloon; but the bag is long and pointed, to diminish air resistance (Fig. 164). Motive power is provided by a gasoline engine, operating one or more screw propellers. The steering gear includes a horizontal rudder, placed 1 92 DYNAMICS in front, for steering up or down, and a vertical rudder at the rear, for steering to the right or left. A car or a long, rigid framework attached to the under side of the gas bag, carries the engine and propellers, together with tne aeronaut, passengers, and other load. The rigid framework of the Baldwin dirigi- FIG. i66.-The Bleriot Monoplane. (The aeroplane ble shown in the in which M. Louis Bleriot crossed the English Chan- figure also carries nel from Calais to Dover, July 25, 1909.) t h e rudders. This military dirigible was built by Capt. Thomas A. Baldwin for the United States Government in 1908. It is 94 ft. long and 20 ft. in diameter, is inflated with hydrogen, and carries two men. Several dirigible balloons about 450 ft. in length and 45 ft. in diam- eter, and having a carrying capacity of 12 to 20 persons, have been built and successfully operated by Count Zeppelin, of Germany. These huge airships have a car at the front and another at the rear, with an engine in each, and each engine drives two pro- pellers. " Zeppelin IV," which was destroyed by accident during its first long-distance flight, maintained an average speed of 34 mi. per hour. The latest form of aerial craft is the heavier-than-air flying ma- chine. Having no gas bag, these machines are supported in their flight only by the reaction of the air, like a bird or a kite. The only type of flying machine which has proved successful up to the present time (1911) is the aeroplane. Aeroplanes are classed as biplanes and monoplanes, the former having two principal planes or supporting surfaces (Fig. 165), the latter only one (Fig. 166). The biplane, as originally designed by the brothers Wilbur and Orville Wright, has three pairs of planes, constructed of a light wooden framework covered with muslin. The large central pair are the principal supporting planes. They are slightly convex, viewed from above, and slope downward toward the rear. As the machine is driven forward in its flight, the air, striking the under side of the planes, exerts an upward pressure upon them. This pressure does not exceed two or three pounds per square foot of surface when the machine is carrying two men. The horizontal planes at the forward DYNAMICS OF FLUIDS 193 end serve as a rudder to direct the machine upward or downward in its flight. (The forward planes were discarded in 1910, in favor of a single horizontal steering plane at the rear.) The pair of vertical planes at the rear serve as a rudder for horizontal steering, like the rudder of a ship. The machine is driven by a light but powerful gasoline engine, working two aerial screws. The success already achieved has led to the most extravagant expectations concerning the future of aerial navigation. Dirigible balloons and aeroplanes are now regarded as a necessary part of the military equipment of the great nations of the world, for use in scouting and carrying despatches. It is highly probable that they may render important service in other limited and special fields, as in the exploration of regions where travel on land is beset with great difficulties or dangers. But no form of aerial craft can ever serve as a practical means of transportation and travel in the ordinary circumstances of life. The Zeppelin airship is nearly as large as an ocean liner, yet its carrying capacity is no more than half that of an ordinary street car. It is, moreover, only a fair-weather vehicle, so far as safety in launching and landing is concerned. When at anchor in even a moderately strong wind it is helplessly buffeted about, owing to the enormous area subjected to wind pressure. The ordinary carrying capacity of the largest heavier- than-air flying machines is at present three men; and this is doubtless very near the possible limit. (Twelve have been carried for a short distance.) But the one decisive factor in limiting their field of usefulness is that they are and must always remain peculiarly hazardous. PROBLEMS 1. Assuming that the power necessary to propel a steamship varies as the cube of the speed, what is the relation between the speed and the energy expended in a given time? between the speed and the energy expended in a given distance? 2. A boy rides a wheel at the rate of 1 2 mi. per hour in a wind of 8 mi. per hour. What is the ratio of the air resistance against him when he is riding against the wind to the resistance when he is riding with the wind? 3. Discuss the action of a ship's rudder. Discuss the action of the for- ward horizontal rudder of a dirigible balloon or an aeroplane. CHAPTER VII THE MOLECULAR THEORY OF MATTER 163. Introduction. In the study of mechanics we have become acquainted with some of the most important phys- ical properties of matter and with the laws resulting from them. We have learned that all matter occupies space to the exclusion of other matter, that it possesses inertia or mass, is capable of storing energy in various forms, exerts an attractive force called gravitation, and possesses elasticity of volume. These general properties of matter are the material basis of the general laws and principles of mechanics, such as Newton's laws of motion and the law of gravitation. We have seen that the special or charac- teristic properties that distinguish the three states of matter from one another give rise to other less general laws and principles, as presented in the mechanics of solids, the mechanics of liquids, and the mechanics of gases. In all this work we have taken the properties of matter for granted, as facts of observation and experi- ment; we have not attempted to account for these properties. But scientific inquiry does not end with the attainment of such results as these, important as they are. Laws and principles relate, as it were, only to the surface of things. The question still remains: Why are the facts thus and so? Gases exert pressure and tend to expand indefinitely; but why do they? What is the minute invisible structure or the internal condition of a gas which causes this behavior? 194 THE MOLECULAR THEORY OF MATTER 195 In short, what is a gas? Liquids are slightly compress- ible, despite the fact that, even under the microscope, they seem to be absolutely continuous bodies, completely filling the space they occupy. We can easily see how a mass of loose earth or a piece of bread can be pressed into smaller compass; but how does it happen that liquids are compressible at all? Why does a solid, a liquid, or a gas expand when heated and contract again when cooled? How does heat convert ice into water and water into invis- ible vapor? To answer such questions as these about the effects of heat, we must not only know more about matter, we must know what heat itself is. Heat is a form of energy, we have learned, but what form? In what does it con- sist? These are only a few of the questions that arise con- cerning the physical properties of matter and physical phenomena; and the science of chemistry presents an equally formidable list, relating to chemical properties and chemical phenomena. To answer such questions we must know what matter is in its minutest structure ; and the eye is hopelessly incapable of giving us this information, even with the aid of the most powerful microscope. Similar difficulties confront the investigator in all departments of science, and they are always met in the same way. Where direct and final information fails, the investigator tries to imagine a cause that would give rise to the known results. The outcome of such a procedure is a theory or possible explanation of the facts under consideration. In trying to account for the facts presented in this chapter, physi- cists and chemists have formulated the molecular theory of matter and the kinetic theory of heat. That part of the the- ory of matter which relates to gases is known as the kinetic theory of gas. 196 THE MOLECULAR THEORY OF MATTER A theory is not necessarily true merely because it affords a satis- factory explanation of all the known facts to which it relates; for it is conceivable that the true cause may be very different from the one suggested. Indeed, it has happened repeatedly in the history of science that rival theories have been ably defended at the same time by different scientists of recognized authority. If, at any time, a new fact is discovered which is inconsistent with a theory, the theory must be modified to bring it into agreement with the fact, or, if this is impossible, it must be abandoned. Newly discovered facts have often served to distinguish between a true theory and a false one. The ancients believed that the earth was fixed in space, and that the apparent motions of the heavenly bodies were real. Their theory of the universe was in satisfactory agreement with the facts of astronomy then known; but in the light of present knowledge such a theory would be absurd. Only a few of the scientists of a hundred years ago dissented from the opinion then generally held that heat is a form of matter without weight. This opinion was reasonable then; it could not now be entertained for a moment by any intelligent person acquainted with the facts that have since been established. All the facts that a theory serves to explain, taken together, make up the evidence in favor of its truth. If this evidence finally becomes conclusive, as knowledge of the subject increases, the theory becomes an established fact. The Copernican theory of the solar system was true in the main; but the observations of Tycho Brahe and the mathematical researches of Kepler and Newton were necessary to correct and complete it. It then became an accepted body of scientific knowledge. I. THE STRUCTURE OF MATTER 164. The Physical Unit of Matter. Any substance can be cut, broken, or otherwise separated into parts, each of which can be separated into smaller parts, and so on. Can the subdivision of any kind of matter be continued indefinitely, or does it finally come to a definite end? A brittle solid can be crushed or ground to powder of such fineness that the individual particles are barely visible THE STRUCTURE OF MATTER 197 under the most powerful microscope; and such particles are less than one millionth as large as the smallest that can be seen with the unaided eye. But subdivision into even smaller particles than these is of common occurrence. A coin is visibly worn away after years of service; a knife loses its edge from continued use ; and a razor is noticeably dulled in shaving once with it. No microscope is capable of revealing the particles lost at any time in such cases. Still we have not reached a limit of divisibility. Water, standing in an open vessel, slowly disappears by evapora- tion ; but it continues to exist as water in the form of vapor, widely diffused in the air. A minute fragment of musk will continue for years to fill a room with its odoriferous particles, and at the end of that time will scarcely be diminished in weight. It would be impossible to determine from such facts as these whether there is a limit to the divisibility of a substance; but the chemist informs us that there is such a limit that there is, in fact, such a thing as the smallest possible particle of any substance. The facts that lend the strongest support to this conclusion belong to the sci- ence of chemistry, and can not be considered here. The smallest possible particle of a substance (other than an element) is called a molecule. All molecules of the same substance are exactly equal in size and weight, and are alike in every respect; but the molecules cf different sub- stances are unlike. The molecule is the physical unit of matter; it is the limit of physical divisibility. In all purely physical changes the molecule preserves its identity. When a grain of sugar dissolves in water, the molecules of which it is composed are separated from one another, but each of them exists as a molecule of sugar in the solu- tion. A molecule of ice exists as the same identical mole- 198 THE MOLECULAR THEORY OF MATTER cule after the ice is melted and the water evaporated. The size of molecules can only be roughly estimated; but, judging from all the evidence, there must be millions of them in the smallest microscopic particle. 165. The Chemical Unit of Matter. Since the molecules of one substance differ from those of another, the molecules themselves must change whenever a change of substance takes place, as is the case in every chemical process. For example, when alcohol burns it unites with oxygen from the air, forming carbon dioxide (an invisible gas) and water vapor. (The water condenses as visible moisture on the inner surface of a glass jar, held inverted over the flame.) In this process two substances unite chemically to produce two other substances. The molecules of alcohol and oxygen are broken up, and their constituents unite to form molecules of carbon dioxide and water. Chemical change thus involves particles of matter which are smaller than molecules, namely, the constituent parts of the molecules them- selves; and the facts of chemistry are explainable only upon the sup- position that these smaller particles or atoms, as they are called, continue in existence unchanged in all chemical as well as in all physical processes. The atom is thus the chemical unit of matter. The molecule of common (ethyl) alcohol consists of two atoms of carbon, six atoms of hydrogen, and one atom of oxygen, and its chemical formula is CzH b OH. In such formulas the initial letter indicates the kind of atom, and the figure placed after and a little below it the number of such atoms present in a molecule. A letter without a figure represents one atom. The oxygen molecule con- sists of two atoms of oxygen (Oz)', the carbon-dioxide molecule, of one atom of carbon and two atoms of oxygen (COz) ; and the water molecule, of two atoms of hydrogen and one atom of oxygen (HzO). In burning, one molecule of alcohol and three molecules of oxygen unite to form two molecules of carbon dioxide and three molecules of water. This is expressed by the chemical equation CzH.OH + zOz = 2C0 2 + 3 H 2 0, in which the numerical coefficients denote the number of molecules. The two sides of a chemical equation must show an equal number of THE STRUCTURE OF MATTER 199 each kind of atom, since no atoms are created, destroyed, or changed into anything else. 166. Elements and Compounds. Any substance whose mole- cules contain unlike atoms is called a chemical compound. Alcohol, carbon dioxide, and water are examples. Compounds are numbered by the thousands. In fact, nearly all substances belong to this class. By various methods all chemical compounds can be separated into their constituents. Substances are thus obtained which resist all attempts to decompose them further, and these are called elements. The presumption is that the molecules of an element really consist of but one kind of atom. Practically the only doubtful cases are some of the newer and rarer elements. The elements found in greatest abundance in rocks and soils are oxygen, silicon, aluminum, iron, calcium, magnesium, sodium, and potassium. The principal elements composing plant and animal tissues are carbon, hydrogen, oxygen, and nitrogen, together with small quantities of sulphur, phosphorus, potassium, sodium, and magnesium. Air is a mixture (not a compound) of oxygen, nitrogen, a little argon, still less of carbon dioxide, and a varying proportion of water vapor. Water is composed of hydrogen and oxygen, and table salt of sodium and chlorine. The metals iron, aluminum, copper, zinc, tin, lead, mercury, silver, gold, and platinum are ele- ments; brass, bronze, and German silver are alloys or mixtures of various metallic elements. In all about eighty elements are known. " It is impossible to state the number precisely, because, owing to the great rarity of some of them and the imperfections of our methods, there are always some whose elementary character is in doubt." Eighteen elements alone make up about 99% of the earth's crust, the ocean, and the atmosphere, taken together. 167. Molecular Attraction. The strength of a solid is due to the attraction of its molecules for one another. This attraction is called cohesion. The pieces of a brittle solid, as glass or china, do not unite again when they are fitted accurately to each other and pressed firmly together. This shows that cohesion acts .only at exceedingly minute distances. In fact, its greatest range is estimated at .00005 mm. (two million ths of an inch). 200 THE MOLECULAR THEORY OF MATTER t That cohesion only requires close contact is readily shown by means of two pieces of clean plate glass. .When pressed firmly together, they cohere with sufficient force to sustain the weight of one of them. Cohesion plates of metal, having surfaces accurately planed and polished, give the same results. Fragments of a plastic solid, as soft clay or putty, unite perfectly, when pressed together, because their surfaces are brought into intimate contact. In the process of welding, the two pieces of metal to be united are rendered plastic by heating. Their surfaces are then brought within the range of cohesion by hammering. Cohesion in liquids is shown by the fact that their par- ticles cling together in drops and in the form of thin films, as in soap bubbles. But the ease with which a liquid is broken up into drops is determined by the freedom of movement of the molecules (mobility) rather than by the strength or weakness of cohesion. Experiments have shown that water, from which the dissolved air has been driven by boiling, is capable of sustaining a tensile stress of 70 Ib. per square inch, while clinging to the walls of a clean glass tube. In general, however, cohesion in liquids is much weaker than in solids. The tendency of gases to expand was formerly regarded as evidence that the molecules of a gas repel one another. Omitting reasons for the present, we may say that there is no molecular repulsion, neither is there cohesion in a gas until it closely approaches condensation into a liquid. 168. Adhesion. There is no evidence of any difference in the nature of the attraction between molecules of the same kind and the attraction between molecules of differ- ent kinds; but the former is generally called cohesion and THE STRUCTURE OF MATTER 201 the latter adhesion. The distinction is convenient rather than important. Adhesion between solids is very common. The adhesive power of mud is sufficiently familiar. Butter adheres to a knife and to the bread upon which it is spread. The adhesion of metals is utilized in gold and silver plating. Ordinarily there is no adhesion between solids when brought in contact; but this is only because their surfaces are not sufficiently close together. Adhesion between solids and liquids is also common. In most cases, when a liquid and a solid are brought in contact, the liquid clings to the solid and wets it. This is because adhesion between the two is greater than cohe- sion within the liquid. The surface of the solid, by its superior attraction, tears a thin layer of the liquid from the remainder. But water does not wet a surface covered with grease or wax, and mercury wets but few substances. In such cases the liquid holds to- gether in somewhat flattened drops upon the surface of the solid (Fig. r FIG. 167. 167). This behavior does not prove that adhesion is wanting, but that cohesion within the liquid is the greater of the two attractions. Gases also adhere to solids, penetrating their pores and forming a very thin layer upon their surface. In setting up a barometer air adheres to the inner wall of the tube, and is driven off only by heating the mercury till it boils. 169. Cohesion and Gravitation Compared. We know the law of gravitation, but not its cause. We know neither the law nor the cause of cohesion; but it is evident that the law is very different from that of gravitation, for cohesion acts only at insensible distances, and within such distances it is enormously stronger than gravitation between the same masses. Hence the strength of bodies in general depends practically entirely on cohesion; and gravitation becomes 2O2 THE MOLECULAR THEORY OF MATTER appreciable only in bodies of very great size. The strength of the earth as a globe depends almost wholly on gravitation. If we im- agine the earth to be divided into hemispheres by any plane through its center, the gravitational attraction by which the hemispheres are held together is one hundred times as great as cohesion would be if the earth were made of solid steel. If there were a planet fifty miles in diameter having the same density as the earth and the cohesive strength of sandstone, gravitation and cohesion would be equally effective in keeping it together. II. MOLECULAR PROPERTIES OF GASES 170. Diffusion of Gases. If two bottles containing different gases are placed one over the other and mouth to mouth (Fig. 168), with the denser gas at the bottom, and are left standing thus for half an hour, they will be found to contain an equal mixture of the two gases. This can be readily shown if the lower bottle is filled with oxygen and the upper with coal gas; for a mixture of these gases is explosive, and an explosion occurs when a lighted match is held at the mouth of either bottle. The mixing would take place much more quickly if the denser gas were placed a ^ ^ ne ^P> being assisted by gravity; but the significant fact is that, with the denser gas below, mixing takes place although opposed by gravity. A like mixing of gases with the air is familiar in cases where the presence of the gas can be detected by its odor, as when illuminating gas is permitted to escape from a burner or a little ammonia is poured on the floor. The essential facts in such phenomena are these: (i) The gases mix without outside aid (such as gravity, stir- ring, or currents of air). The mixing is spontaneous; i.e. it takes place from internal causes causes depending FIG 1 68 MOLECULAR PROPERTIES OF GASES 203 on the nature of gases. (2) The final result, if sufficient time is allowed, is always a uniform mixture of the gases, regardless of their relative densities. (3) The less the den- sity of a gas the more rapidly does it mix with other gases. (4) Mixing is more rapid at higher temperatures. (5) All gases and vapors behave in this way. The spontaneous mixing of gases is called diffusion. Diffusion is explainable only upon the supposition that the molecules of a gas are in constant motion as individuals, darting hither and thither in all direc- tions at random, like bees or gnats in a swarm. Each molecule, according to this view, flies in a straight line till it hits another molecule or the wall of the con- taining vessel, when it rebounds like a perfectly elastic body. The following experiment lends emphasis to these conclusions. FIG. 169. Diffusion of Gases. A porous cup of unglazed earthen- ware is fitted with a stopper and a glass tube, and supported with the lower end of the tube in a glass of water (Fig. 169). A jar or a large beaker is held over the cup and filled with coal gas (or, better, hydrogen) through a rubber tube. The experiment has three stages, (i) Air is forced out through the lower end of the tube, showing an increase of pressure inside the porous cup. (2) On removing the jar, water rises rapidly in the tube, indicating a rapid decrease of pressure within the cup. (3) The water in the tube soon begins to fall, but more slowly than it rose, and finally reaches the same level as in the glass, showing that at- mospheric pressure has been restored. These phenomena are evi- dently due to the rapid diffusion of the coal gas and the less rapid diffusion of air through the microscopic pores of the cup. In the first 204 THE MOLECULAR THEORY OF MATTER stage of the experiment the coal gas diffuses inward more rapidly than the air inside escapes; hence the increase of pressure. Jn the second stage the coal gas within the cup escapes more rapidly than air enters, causing a decrease of pressure. Finally air diffuses inward until equilibrium of pressures is restored. There is then equally rapid diffusion of air inward and outward. Diffusion is a very different process from the flow of gases in currents. In diffusion the molecules move singly and at random; in currents they move collectively as one body. Diffusion supplements the action of winds in keep- ing the constituents of the air uniformly mixed. 171. Other Phenomena Explained by Molecular Motion. - The pressure of a gas is a necessary consequence of the motion of its molecules. The walls of a vessel containing a gas are subjected to an inconceivably rapid shower of blows from the flying molecules. The net result is a con- tinuous and constant pressure, like the pressure of a stream of water from a hose when directed against the side of a building. The pressure exists throughout the mass of gas, for the molecules in the interior collide with one an- other. Each molecule, knocking vigorously about among its fellows, tends to drive them farther away; hence the entire mass tends to expand indefinitely. If a given mass of gas is compressed to half its former volume, its density is doubled, and twice as many molecules strike each square centimeter of the wall of the vessel every second; hence the pressure is doubled. Thus molecular motion accounts for Boyle's law. From the known density and pressure of a gas, it is pos- sible to compute the average velocity of its molecules. For air this velocity turns out to be about eighteen miles per minute, or half the velocity of the swiftest cannon ball. If the molecules of what we call still air were moving with MOLECULAR PROPERTIES OF GASES 205 their actual velocity, but all in the same direction, the re- sult would be a wind blowing ten times as fast as the most violent hurricane. The average velocity of hydrogen molecules is about sixty-eight miles per minute, a veloc- ity sufficient to encircle the earth in six hours. To the thoughtful pupil such molecular velocities may well seem incredible; but any difficulty that may be experienced in picturing them to the mind is not to be regarded as evi- dence that they do not exist. It is demonstrated from astronomical observations and by actual experiment that the velocity of light is 186,000 miles per second, and it would be unreasonable to doubt the fact merely because we are unable to think it. It is evident from the very great compressibility of gases that, under ordinary pressures, their molecules occupy only an exceedingly small portion of the space allotted to them. The volume of a gram of steam under a pressure of one atmosphere is 1661 ccm., which is 1661 times as great as its volume as a liquid. A cubic inch of water makes nearly a cubic foot of steam, and that without any increase in the number or (so far as we know) in the size of the molecules. Even if the molecules were packed without space between them in the liquid (which is not true), the average distance between the molecules of steam, under a pressure of one atmosphere, would be nearly twelve times the diameter of a molecule, and there are good reasons for believing that it is much more than this. Oxygen has been subjected to a pressure of 3000 atmospheres, in which condition its density is greater than that of water, although it still remains in the gaseous state. The relatively wide separation of the molecules of a gas at ordinary tem- peratures and pressures is, as we have seen, the result of molecular motion. Thus molecular motion fully accounts 206 THE MOLECULAR THEORY OF MATTER for the great compressibility of gases, as well as for their great resistance to compression. Considering the great velocity of gas molecules, the comparatively slow rate at which two gases mix by diffusion requires a word of explanation. Diffusion progresses by the forward motion of the molecules of each gas between those of the other. But molecules are rather numerous, although relatively far apart; and it is esti- mated that, at ordinary temperatures and pressures, each collides with its neighbors some five thousand million times per second. Hence progression of the molecules in any given direction is frequently interrupted and is relatively slow. When a gas is admitted into a vacuum, there is no opposition to progressive motion, and expan- sion is practically instantaneous. 172. Heat and Molecular Motion. When a gas is heated it expands or, if expansion is prevented, its pres- sure increases. Some idea of the rate of expan- sion can be gained by means of a small flask containing air, and having a glass tube attached (Fig. 170). If the flask is inclosed in the hands while the end of the tube is held under water, the expansion will be shown by the escape of bubbles from the tube. If the flask is then allowed to cool while the tube remains under water, the water will rise in it, showing that the volume and the pressure of the air have both decreased. These effects can be due only to an increase of molecular velocity '' I7 ' with a rise of temperature and a decrease of molecular velocity with a fall of temperature. Thus, in general, when a gas receives heat its molecules move faster; when it loses heat they move more slowly. Since we know that heat is a form of energy, we might suspect, without further evidence, that the heat of a gas is the kine tic- energy of its molecules; and such is, in fact, the case. MOLECULAR PROPERTIES OF LIQUIDS 207 173. The Kinetic Theory of Gases. The theory upon which we have relied in the preceding pages to account for the physical properties of gases is known as the kinetic theory of gases. In its complete form, as presented in advanced texts, the theory applies the laws of dynamics to the individual molecules, and accounts definitely for all the laws of gases. There are many points of minor importance on which the theory is either silent or gives only a provisional answer; but in all its essential features (and we have considered only these) it is regarded by sci- entists as established fact. III. MOLECULAR PROPERTIES OF LIQUIDS 174. Diffusion of Liquids. If any two liquids that can be mixed with each other are placed in the same vessel, the denser at the bottom, and left undisturbed, they will mix by diffusion, the process being sim- ilar to the diffusion of gases. The progress of diffusion in liquids is visible in cases where it is accompanied by a change of color, as in the following experiments. A tall vessel (Fig. 171) is nearly filled with water colored with blue litmus. A little strong sulphuric acid is then admitted at the bottom through a thistle-tube. The acid is considerably denser than the water and supports it,, the surface separating the two being dis- tinctly visible. Since acid turns blue litmus red, the progressive change of color from blue to red, which FIG. 171. slowly takes place up the tube, indicates the height to Diffusion which the acid has risen by diffusion. A jar is partly filled with water, and a strong solution of copper sulphate is admitted at the bottom through a thistle-tube. The progress of diffusion is indicated by the very slow rise of the blue color of the solution. The process requires months for its completion. 208 THE MOLECULAR THEORY OF MATTER We see from these experiments that diffusion takes place in liquids, as in gases, without the aid of currents and in opposition to gravity. The explanation is therefore the same: the molecules of a liquid are in individual motion. The exceedingly slow rate of diffusion in liquids is due to the frequent collisions of the molecules, and is not to be taken as evidence that molecular motion is sluggish. The molecules of a liquid are always moving about among one another whether a second liquid is present or not; but in the latter case there is no direct evidence of the motion. 175. Evidence of Intermolecular Spaces in Liquids. - Since the molecules of a liquid are in motion, the question naturally arises whether they are moving fast enough to drive one another apart, leaving intermolecular spaces, or whether their mutual attraction (cohesion) is sufficient to hold them actually in contact, while permitting them to slip about among one another. Certain properties of liquids supply the answer. Compressibility. A pressure of 3000 atmospheres diminishes the volume of water by one part in ten and the volume of sulphuric ether by approximately one part in six. All liquids tested under great pressure have been found to be compressible in a greater or less degree. The reasonable inference is that the molecules of a liquid are separated by void spaces, and that in compression they are simply crowded more closely together. The only alterna- tive is that the molecules themselves are diminished in size by compression; but this view is discredited by other facts. Effect of Heat. The change of volume of a liquid when heated or cooled can be shown with a flask and tube (Fig. 170). When the flask is completely filled with cold water MOLECULAR PROPERTIES OF LIQUIDS 209 or other liquid and plunged into a vessel of hot water, the liquid rises rapidly in the tube. If the flask is now plunged into cold water, the liquid in the tube descends. When water is heated from the freezing to the boiling point, it increases in volume by four per cent. The expansion of liquids with heat, although much less than that of gases, is plainly due to the same cause, heat increases molec- ular motion, and the molecules drive one another farther apart. Loss of Volume in Mixing Liquids. When equal volumes of water and strong alcohol are mixed, it is found that the volume of the mixture is two per cent less than the sum of the original volumes. This is readily shown by filling a long test tube half full of water and adding alcohol care- fully, to avoid mixing, till the tube is nearly full. A rub- ber band round the tube conveniently marks the exact height. The tube is then closed with the finger, and the liquids are mixed by shaking. A similar shrinkage occurs in mixing strong sulphuric acid and water, and also when certain solids, as sugar or salt, are dissolved in water. There is, of course, no loss of matter in such phenomena; weight and mass are unchanged. Evidently the mole- cules are packed together more closely in the mixture than in the separate substances; and hence there must have been unoccupied spaces between the molecules before the mixing. Such phenomena as these lead with practical certainty to the conclusion that there are intermolecular spaces or pores in all liquids, as in gases, and that these spaces are maintained by decidedly vigorous molecular motion. This motion is principally an irregular oscillation; for the molecules are so close together that nearly all their time is spent in collisions with one another. They do, however, 210 THE MOLECULAR THEORY OF MATTER wander uncertainly about, as is proved by diffusion. Their mutual attraction (cohesion) is sufficiently strong to set a limit to expansion, without the aid of external pressure. 176. Surface Tension of Liquids. A pin or a needle that is slightly oily from contact with the fingers can be made to float on water, if carefully laid upon the surface. The floating pin lies in a depression of the surface which is considerably larger than itself (represented in cross-sec- tion in Fig. 172). This behavior can not be referred to the principle of buoyancy, since the pin is much if denser than water. If the pin is pushed O^jfMMJGf beneath the surface it immediately sinks. FIG. 172. The surface of the water behaves as if it had a certain degree of toughness and resisted tearing. Soap films, whether flat or in the form of bubbles, exhibit this property in unmistakable fashion. A bubble left upon a pipe slowly contracts, driving the air out through the stem. It behaves just as an inflated rubber balloon does when its tube is opened. The film of the bubble and the rubber of the inflated balloon are under tension and tend to contract. Since a spherical surface is smaller than any other that incloses an equal volume, both the bubble and the balloon assume this shape in shrinking as much as the pressure of the inclosed air or other gas will permit. A drop of any liquid when freed from the distorting effect of its weight, as in falling, is spherical. A drop of oil suspended in a solution of alcohol and water of its own density is an excellent illustration. The spherical form of a drop is due neither to the mutual gravitation of its par- ticles nor to cohesion acting throughout the mass, but to the tension of its surface. It is as if the drop were inclosed in a little rubber bag, tightly stretched. MOLECULAR PROPERTIES OF LIQUIDS 21 1 Every liquid may be regarded as bounded by a skin or film, which behaves like a stretched membrane. The ten- sion of this superficial film is called the surface tension of the liquid. The surface tension of different liquids has been deter- mined by experiment, and has been found to be greater for water than for any other liquid except mercury; hence the surface tension of water is diminished by mixing any other substance with it. This is readily shown by placing a drop of alcohol, ether, oil, or soap solution on the surface of a glass of water beside a floating sliver of wood, as a toothpick. The toothpick is quickly jerked away from the drop by the greater tension of the pure water on the other side. 177. The Cause of Surface Tension. Imagine a spheri- cal surface, whose radius is equal to the range of cohesion, to be described about any molecule A (Fig. 173) within a liquid. (This sphere is microscopically small; the figure repre- sents it enormously magnified.) Since all the molecules that are near enough to attract A are uniformly distributed about it within this sphere, it is equally attracted in all directions, and the resultant of these attractions is zero. But any molecule, B or C, whose distance from the surface is less than the range of cohesion, is more strongly attracted inward than outward, since most of the mole- cules within its sphere of attraction lie on the inside. Upon such molecules there is a resultant force of cohesion, acting inward; and this gives rise to surface tension. 178. Capillary Action. The free surface of a liquid is generally sharply curved where it comes in contact with a 212 THE MOLECULAR THEORY OF MATTER solid. The direction of the curvature depends upon the relative strength of cohesion in the liquid and adhesion between the liquid and the solid. If adhesion is the greater, the edge of the liquid is drawn up- ward against the surface of the solid. Water in contact with glass is a familiar example (Fig. 174). If cohesion in the liquid is the greater, the edge FIG. 1 74- of the liquid is drawn away from the solid, and Contact of * Water and the curvature is downward. This is true of 1SS ' mercury in contact with glass (Fig. 175). Surface tension plays an important part in all such phenomena. When a liquid creeps up over the surface of a solid, as from b to c (Fig. 176), its own surface is increased; but instead of turning at an angle, abc, it contracts into the curve ab'c. This curve has a definite shape, in which the attraction of the solid, the tension of the liquid surface, and the weight of the raised liquid are J IG ' I75- ~T u \ ' Contact of in equilibrium. If the liquid does not wet the Mercury solid, its surface is under tension even where a it comes in contact with the solid; and, in contracting as much as possible, it rounds off the edge. These and other effects of molecular forces, where a liquid comes in contact with a solid, are known as a capillary phenomena, because they are most con- spicuous in tubes of small or hair-like bore (Latin, capillus, a hair). Capillary action and capillarity are general terms for such phenomena FlG - I76 ' or for their cause. 179. Phenomena in Capillary Tubes. When a glass tube of small bore is placed with its lower end in water, the water rises in the tube above its level in the vessel, FIG. 177. Capillary Elevation. MOLECULAR PROPERTIES OF LIQUIDS 213 and comes to rest with its surface concave, viewed from above. With tubes of unequal bore, the water stands higher in the smaller and its surface is more sharply curved. The ele- vation of the water in the tube is merely an exaggerated instance of capillary action, as explained above. The water is drawn up at its edge by adhesion to the inner wall of the tube. The surface thus becomes curved, but surface tension tends to keep it flat by contraction; hence the entire surface rises. At a certain height the weight of the water col- umn lifted up balances the upward force due to adhesion and surface tension. As the size of the bore is reduced, the lifting force decreases, but the weight of water per unit length of the column decreases still more rapidly; hence a higher column is necessary to estab- lish equilibrium (Fig. 177). Mercury stands in small glass tubes below the level of its surface in the vessel, and the smaller the bore the greater the depression (Fig. 178). The surface is convex, for rea- sons stated above, and consequently exerts a downward pressure. The following laws have been established by experiment: i. If a liquid wets a capillary tube, its surface is concave and it is drawn up; if it does not wet the tube, Us surface is convex and it is depressed. FIG. 178. Capillary Depression. 214 THE MOLECULAR THEORY OF MATTER 2. The elevation or the depression in a capillary tube is inversely proportional to the diameter of the tube. Porous solids absorb liquids by capillary action in their minute openings. The absorption of water by a sponge or a towel, of ink by blotting paper, and of coffee by a lump of sugar are familiar examples. The flame of a lamp is fed by oil that is drawn up through the wick by capillary action. In dry weather moisture is drawn up from a depth of many feet through the pores of the soil, and evaporates at the surface. Cultivation of the soil increases the size of the pores, and consequently checks the rise of water through the cultivated layer, thus diminishing the loss by evaporation at the surface. IV. MOLECULAR PROPERTIES OF SOLIDS 180. Intermolecular Spaces in Solids. Many solids are visibly porous, either to the unaided eye or with the aid of a microscope. Paper, wood, leather, brick, and sand- stone are familiar examples. The absorption of water and other liquids by solids is due to capillary action in such pores as these. But the metals and many other solids do not absorb liquids, and appear to have a perfectly con- tinuous structure even under the microscope; nevertheless, there is direct experimental evidence that such bodies have invisible openings in and through them. Early in the seventeenth century, Francis Bacon, experimenting on the compressibility of water, hammered a shell of lead rilled with water. " The water exuded like perspiration through the pores of the lead. The Florentine Academi- cians tried the same experiment with a silver shell, but obtained the same result. They then tried to prevent the escape of the water by thickly gilding the shell, but again in vain." Under a pressure of 4000 atmospheres, MOLECULAR PROPERTIES OF SOLIDS 215 mercury has been forced through three inches of solid steel. In such cases as these it is highly probable that the pores are nothing more than spaces between the individual molecules of the solid. Further evidence of intermolecular spaces in solids is afforded by their change of volume with dhange of pres- sure or of temperature. All solids are compressible, al- though many are less so than liquids. The rolling and stamping to which silver is subjected in the process of coining causes a decrease of volume amounting to about four per cent. Glass is about one- twentieth as compress- ible as water. Nearly all solids expand with heat. The unequal expansion of a glass dish, when hot water is poured into it, strains it to the breaking point. The rails of a track are laid with a space between their ends to provide room for expansion in hot weather. Expansion is readily demonstrated with a small brass ring and u brass ball that will just pass through it when both are cold or both hot. If only the ball is heated it will not go through the ring. 181. Molecular Motion in Solids. Intermolecular spaces in solids can only be due to molecular motion, as in liquids and gases. Obviously the molecules of a solid are not free to wander about among one another; but we can imagine an irregular oscillation without change of position. This motion must be very energetic indeed to enable a solid to resist compression as it does, or to cause the observed expansion of a solid when heated. Under ordinary conditions the tendency of solids to expand with a rise of temperature is practically irresistible. To stretch a steel rod having a cross-section of one square inch as much as it expands of itself when it is warmed from the tempera- ture of melting ice to the ordinary temperature of a room (20 Centigrade) would require a tension of 8000 lb.; and 2i6 THE MOLECULAR THEORY OF MATTER it would take an equal pressure to prevent the expansion of the rod with this rise of temperature. The universally accepted view that heat is the energy of molecular motion necessarily implies that the molecules of all bodies solid, liquid, and gaseous are in motion; for we know that there is still heat in bodies at the lowest temperatures yet attained. But the consideration of such matters must be left to the next chapter. 182. Special Properties of Solids. Solids differ widely among themselves in many of their physical properties; indeed, it is by means of such differences that we distinguish substances from one another. Thus one substance is hard, another soft; one is brittle, another tough, etc. Special properties may be classed as mechanical, optical, magnetic, electrical, etc. It is only with the more important mechanical properties that we are now concerned. An elastic solid offers permanent resistance to change of shape; an inelastic or plastic solid does not. Elasticity of form has been considered at length in an earlier chapter. In the customary and narrow sense, a plastic substance is one that can be molded by a moderate pressure into any desired shape. Putty and wet clay are typical examples. In a broader sense, all solids that are not brittle are plastic beyond their elastic limit. Gold, silver, and copper are plastic at ordinary temperatures, when subjected to great pressures as in stamping coins. A force of 160 tons is applied in stamping a silver dollar; and under this force the cold metal behaves like butter in a butter mold. In some cases even brittle substances are plastic under the continued action of a moderate force. A stick of sealing wax is very brittle; but when supported at one end in a horizontal position, it slowly yields under the action of its own weight, and in the course of weeks becomes quite bent. The dividing line between plastic solids and viscous liquids is uncertain. Various substances, e.g. molasses candy, pitch, and shoemakers' wax, change imperceptibly to the liquid state by gradual softening, when heated. A substance is said to be malleable if it can be hammered or rolled into sheets, ductile if it can be drawn out into a wire. Ordinary temperatures are understood unless otherwise stated. Thus we say that glass is brittle, although it is very ductile when heated to redness. MOLECULAR PROPERTIES OF SOLIDS 217 / Gold, silver, copper, and platinum are the most ductile metals, gold the most malleable. Gold can be reduced to sheets having a thickness of rauforo of an inch. Our idea of what constitutes a hard body varies with the substance. It is easier to cut hard butter than soft wood; and we speak of soft iron and a hard pillow, although we should doubtless prefer the latter to sleep on. The meaning of these terms is variable in another respect. They are applied to quite different physical properties. Thus we say that soapstone or talc is soft, meaning that it is easily scratched; and also that rubber is soft, meaning that it offers little resistance to change of shape. Hardness may mean the opposite of softness in either of these senses. Where some degree of exactness is necessary, hardness is expressed in terms of certain standards. Thus the mineralogist's scale of hardness consists of a series of ten minerals, beginning with talc, which can be scratched with the finger nail, and ending with the diamond, the hardest of all substances. Hardness, referred to this scale, means resistance to scratching. A brittle or friable substance is one that is easily broken into frag- ments by a blow, e.g. glass and coal. Brittleness and toughness are opposite properties. The tenacity or tensile strength of a substance depends upon the cohesion of its molecules. It is one of the most important properties of building materials, and its measure is the force per unit area neces- sary to pull the body apart. Steel is the most tenacious of all sub- stances, a tension of 180 Ib. being required to break a steel wire hav- ing a cross-section of i sq. mm. The breaking strength of a copper wire of the same size is about 70 Ib., of lead wire 5 Ib., of glass 14 Ib., and of oak (in the direction of its fibers) 15 Ib. PROBLEMS 1. According to the molecular theory, how does heat convert a solid into a liquid? a liquid into a gas? 2. Why is ground damp under a board or a stone when it is dry all around? 3. The bristles of a paint brush or the hairs of one's head cling together when wet in air; but under water they spread apart as they do when dry. Explain. 4. A tray made of wire gauze will hold water to a slight depth, if it is poured in very gently. Explain. CHAPTER VIII HEAT I. NATURE OF HEAT 183. The Caloric Theory. From the days of the early Greek philosophers until about the middle of the nineteenth century, heat was very generally believed to be a substance. According to the accepted view of the eighteenth century, this supposed substance, then known as caloric, was an invisible, elastic fluid, without weight, whose particles repelled one another but were attracted more or less by the ordinary kinds of matter. With the overthrow of this theory, the word caloric has become obsolete; but several of its derivatives continue in use, e.g. the calorie is a unit quantity of heat, and the vessel in which substances are placed in measuring their gain or loss of heat is called a calorimeter. 184. Historical Experiments on the Nature of Heat. As early as the seventeenth century some of the ablest scientists, including such men as Boyle, Francis Bacon, Hooke, and Newton, believed heat to be molecular motion; but nearly two hundred years elapsed before this view again came into prominence, through the experiments of Benjamin Thompson (1753-1814). Although an Amer- ican, born in Massachusetts, Thompson is better known as Count Rumford, having received that title from the Elector of Bavaria, whose service he entered while still a 218 NATURE OF HEAT 219 young man. While engaged in boring cannon for the Bavarian government, he was surprised at the heat gen- erated, and, seeking further light on the phenomenon, he began to experiment. "He arranged apparatus so that the heat generated by the friction of a blunt steel borer raised the temperature of a quantity of water. In his third experiment, water rose in one hour to 107 Fahrenheit; in one hour and a half to 142; at the end of two hours and a half the water actually boiled. ' It is difficult to describe the surprise and astonishment,' says Rumford, ' expressed in the countenances of the bystanders, on seeing so large a quantity of cold water (18.75 lb.) heated, and actually made to boil without any fire.' The source of heat gener- ated by friction ' appeared evidently to be inexhaustible.' The reasoning by which he concluded that heat is not mat- ter, but is due to motion, we can give only in part. He says, ' It is hardly necessary to add that anything which any insulated body, or system of bodies, can continue to furnish without limitation cannot possibly be a material substance; and it appears to me extremely difficult, if not quite impossible, to form any distinct idea of anything capable of being excited and communicated in the manner in which heat was 'excited and communicated in these experiments, except it be motion.' "Rumford's conclusion regarding the nature of heat was vigorously attacked by the calorists, but it was confirmed in 1799 by Sir Humphry Davy. By means of clockwork he rubbed two pieces of ice against one another in the vac- uum of an air pump. Part of the ice was melted, although the temperature of the receiver was kept below the freez- ing point. From this he concluded that friction causes vibration of the corpuscles of bodies, and this vibration is heat." (Cajori's History of Physics.) 220 HEAT The views of Rumford and Davy were accepted by scientists here and there; but the caloric theory was too firmly established to be easily supplanted by its rival. The prevailing opinion continued in its favor until the middle of the nineteenth century; at which time the equivalence of the different forms of energy, including heat, and the principle of the conservation of energy were established by the experiments of J. P. Joule in England (Art. 242) and the discussions of Robert Mayer and von Helmholtz in Germany. 185. The Kinetic Theory of Heat. The molecular prop- erties of matter (Chap, vii) afford conclusive evidence that the molecules of all bodies solid, liquid, and gaseous are in motion, and that this motion is increased by heat. The production of heat from mechanical energy proves that heat is itself a form of energy, and that this energy is associated with the molecules of bodies. Since a moving molecule must possess kinetic energy by virtue of its mass and its velocity, as in the case of large masses, it follows that this kinetic energy of molecular motion must be heat. " Heat is not motion, for it is neither change of position, nor yet momentum; it is the energy of motion. Double the quantity of molecular motion, and you quadruple the molecular kinetic energy, that is, the heat." (Daniell.) "Cold," in the sense of something whose effects are opposite to those of heat, does not exist. Ice does not "give out cold"; it has none to give out. It cools sur- rounding objects by receiving heat from them ; and because it receives heat it melts. Some heat remains in all bodies at the lowest temperatures yet produced. II. TEMPERATURE * 186. Temperature. Our first ideas of temperature are derived from our bodily sensations of heat and cold; and these ideas are expressed in such terms as hot, warm, TEMPERATURE 221 tepid, cool, cold, etc. The warmer of two bodies is said to be at the higher temperature. The terms higher and lower suggest the possibility of exact measurement; and the use of the thermometer for this purpose is familiar to every one. When any two bodies whose temperatures differ are placed in contact, heat passes of itself from the hotter to the colder, until they reach the same temperature. The behavior of heat in passing from a body at higher to one at lower temperature is similar to the behavior of water in flowing from higher to lower level. Temperature may therefore be defined as that condition of a body on which its ability to impart heat to other bodies, or to receive heat from them, depends. It should be noted that two bodies at equal temperatures may or may not contain equal quan- tities of heat; just as two communicating vessels in which water stands at the same level may or may not contain equal quantities of water. The quantity of heat in a body at a given temperature varies as its mass, and depends also, as we shall see later, upon the material of the body. Our temperature sensations are often very misleading. In the first place, the sensation varies with the condition of our own bodies. A room may seem agreeably warm to a person entering it after a brisk walk on a cold day, while to another, who has not been exercising, it seems chilly. A cold room seems warm to one who is ill with a fever, and a warm room cold to one who has the "grippe." In the second place, the temperature sensation experienced on touching a body is largely determined by certain physical properties of the body itself. If there has been no fire or other source of new heat in a room for an hour or more, a thermometer would show the various objects in it to be at the same temperature; but to the touch they do not seem to be so. On a cold morning we find a rug or a carpet fairly comfortable to the bare feet, while the floor feels decidedly cold. A wash-bowl feels much colder to the hand than the air does, and the water in it still colder. 222 HEAT 187. Measurement of Temperature. Any instrument for measuring temperature is called a thermometer. Ther- mometers are of many kinds, and make use of various effects of heat, such as the expansion of a solid, a liquid, or a gas, the change of pressure of a gas, and the change of electrical resistance of a wire. A thermometer in which the expan- sion of a liquid is utilized is most convenient for general use. It is important that the liquid chosen should have a uniform expansion; i.e. equal quantities of heat should cause equal increases of volume at all temperatures. Mercury fulfils this condition better than any other liquid, and has the further advantage of remaining a liquid through a very wide range of temperature. The mercury- in-glass thermometer has therefore been adopted as the standard for all ordinary purposes. For temperatures below the freezing point of mercury, the alcohol thermom- eter is generally used, the freezing point of alcohol being 130 C. The hydrogen thermometer is the standard for accurate scientific work. In it a mass of hydrogen is kept at constant volume, and the pressure of the hydrogen measures the temperature. 188. The Mercury Thermometer has a capillary glass tube, called the stem, terminating in a bulb (Fig. 180). The mercury fills the bulb and more or less of the stem, according to the temperature; and its expansion or contrac- tion is measured by a scale engraved on the stem or at- tached to it. In making a thermometer the mercury is heated, to drive out all the air, before the stem is sealed at the top; hence the space in the tube above the mercury is a vacuum. Determination of the Fixed Points. Probably no two thermometers have bulbs of exactly the same capacity TEMPERATURE 223 and tubes of exactly the same bore; hence the readings of dif- ferent thermometers would be entirely inconsistent with each other if they were provided with scales of the same length. The correct position and dimensions of the scale must there- fore be determined separately for every thermometer. The first step in this process is to determine the two fixed points, called the freezing point and the boiling point. The freezing point is the temperature at which pure water freezes; but since this is exactly the same as the temperature at which ice melts, whatever the surrounding temperature may be, it is most conveniently found by inserting the bulb of the thermometer in a dish of melting snow or ice. The snow or crushed ice is packed about the bulb and stem, leaving the mercury just visible above it; and a mark is made on the stem at the top of the mercury column, after it comes to rest. The boiling point is the temperature of steam as it rises from water boiling under a pressure of one atmosphere. The temperature of boiling water is subject to slight vari- ations from different causes; but the temperature of the steam varies only with the pressure. The thermometer is therefore adjusted so as to be surrounded by the steam, as nearly as possible to the top of the mercury in the stem, and is not permitted to touch the water (Fig. 179). The height at which the mer- FIG. 179. cury stands, under these conditions, is marked on the stem as the boiling point. A correction must be applied to the 224 HEAT observed height of the mercury in the stem if the pressure of the steam is not 76 cm. (Art. 232). Centigrade and Fahrenheit Scales. The distance between the fixed points is divided into equal parts, called degrees. In the Centigrade scale the number of these divisions is 100, the freezing point being marked 212 . o and the boiling point 100. The Centigrade thermometer is used almost exclusively in scien- tific work. All temperatures referred to in this book are expressed in the Centigrade scale, unless otherwise indicated. In the Fahrenheit scale the freezing point is marked 32 and the boiling point 212, the interval between them being 1 80. The Fahrenheit thermometer is the one 17.8 - o i n general use in English-speaking countries. The scale of a thermometer may be extended to any desired distance beyond the fixed points. Temperatures below zero, on either scale, are FIG. 180. . j. 11,1 , indicated by the negative sign. Since the interval between the freezing and the boiling points is 100 Centigrade degrees or 180 Fahrenheit degrees, it follows that - i Centigrade degree = f Fahrenheit degree, and i Fahrenheit degree = f Centigrade degree. In changing a reading on either scale to the equivalent reading on the other, allowance must be made for the dif- ference in the zero points. Example: 50 C. means 50 Centigrade degrees above the freezing point. This is equal to 50 X I or 90 Fahrenheit degrees above the freezing point, or to 122 Fahrenheit. PROBLEMS 1. (a) According to the theory of heat, what would be the molecular condition of a body without heat? (6) Why could not such a body be a gas? CONDUCTION AND CONVECTION 225 2. Mention any familiar instances in which equal temperatures do not cause like temperature sensations. 3. (a) The reading of a thermometer gives the temperature of the thermometer. On what grounds do we assume that the reading of a ther- mometer in a liquid gives the temperature of the liquid? (b) Why do we not take the reading immediately on inserting a thermometer in a liquid to determine its temperature? 4. A living room is comfortable at a temperature of 67 F. What is this temperature on the Centigrade scale? 5. What would the Centigrade thermometer register in "zero weather "? 6. What is ''98 in the shade" according to the Centigrade thermom- eter? III. CONDUCTION AND CONVECTION 189. Conduction is the transmission of heat from hotter to colder parts of a body, or from a hotter to a colder body in contact with it, without change in the relative positions of the parts of the body. It is the only process by which heat travels in solids. The heating of the farther end of a poker, when one end is placed in a fire, and the heating of the handle of a spoon placed in a cup of hot tea are familiar examples. The kinetic theory suggests a mental picture of the process of heat conduction. When any part of a body is heated, its molecules are set in more rapid vibration. These molecules jostle their neighbors more violently, increasing the energy of their vibration. The disturbance thus spreads throughout the body without change in the rela- tive positions of the molecules themselves. In conduction, therefore, molecular energy is transmitted without the transmission of matter. The power of a substance to transmit heat by conduction is called its conductivity. This property varies greatly with different substances. Thus a burning match can be 226 HEAT held until the flame reaches the fingers, for wood is a poor conductor; but if one end of a pin is held in the flame of the match, the other end quickly becomes too hot to hold. The metals are the best conductors, although differing greatly among themselves; and other solids, with few excep- tions, are better conductors than liquids. Liquids, with the exception of mercury and molten metals, are poor conductors. Water can be boiled at the top of a test tube for several minutes, while at the bottom it remains cold (Fig. 181). But a number of solids, e.g. wood, paper, and wool, are poorer conductors than water (see table below). Gases are practically non-con- ductors. In testing the con- ductivity of liquids and gases, they must be heated at the top to prevent convection currents (Art. 191). The following table gives the conductivities of various substances referred to silver as the standard. TABLE OF CONDUCTIVITIES FOR HEAT (APPROXIMATIONS) FIG. 181. Silver 100. Copper 80. Brass 27. Iron 15. Ice 5 Marble 4 Glass 15 Water 14 Mercury 1.6 Wood 04 Writing paper ... . .012 Fresh snow 01 Felt 009 Air 005 Flannel 004 190. Temperature Sensations and Conductivity. Other Applications. We can now understand in part why different bodies, e.g. iron and wood, feel unequally hot, when actually at the same temperature (Art. 186). The difference in the sensations is largely due to the unequal conductivities of the substances; for the sensation is CONDUCTION AND CONVECTION 227 determined by the rate at which the hand receives or loses heat. Equally hot bodies, differing in conductivity, im- part heat to the hand at unequal rates, and thus make the hand unequally hot; equally cold bodies take heat from the hand at unequal rates, making the hand unequally cold. The unequal temperatures of the hand are commonly but mistakenly attributed to the bodies themselves. Temperature sensations are also determined in part by another property of bodies, called specific heat, which will be studied later. Materials of low conducting power are widely used both to keep heat from cold bodies and to prevent the loss of heat from hot bodies. The double walls of refrigerators and ice-houses are rilled in between with charcoal, sawdust, straw, or other loose, badly conducting mate- rial, to keep the heat out. A cooking box (Fig. 182) is packed with felt to keep the heat in. Food placed in such a box boiling hot will continue to cook for hours, with only a slight fall of temperature and without a further appli- cation of heat. The loss of heat from steam and hot-air pipes is greatly reduced by a FlG . l82 ._ Co okingBox. wrapping of asbestos, paper, or felt. The low conductivity of sawdust, straw, felt, fur, feathers, hair, and other poor conductors is largely due to the air spaces within them. Air is one of the poorest conductors; but to be effective it must be trapped in minute cells, which prevent its circulation. When not thus confined, it transfers heat by convection (Art. 191). 191. Convection. We have seen that water can be heated at the top in a test tube and boiled while it remains cold at the bottom. If the flame is applied at the bottom of the tube, the water quickly becomes heated throughout. This is evidently not the result of conduction; for conduc- tion is not affected by gravity, and takes place upward and downward with equal rapidity. When a portion of a 228 HEAT liquid is heated, it expands and becomes less dense than the remainder. If this heated portion is at the bottom of the vessel, it will be driven upward by the cooler and denser portions surrounding it, in agreement with the principle of buoyancy. As the colder liquid from above reaches the place where the heat is applied, it in turn becomes heated and is displaced by other portions. The process is called convection, and the streams of warmer and cooler liquid are called convection currents. When the heat is applied at the top, the expanded liquid is already in the position of equilibrium and it remains there. Convection takes place in air and other gases under the same conditions as in liquids. The strong ascending current above a bon- fire is indicated by the leaping of the flames and the rapid rise of sparks and smoke. The fire is fed by inward-flowing currents near the ground. They occupy much more space than the ascending current, and hence move more slowly and are less noticeable. Convection currents can be made visible in a beaker of water by means of sawdust or a little coloring matter (crystals of potassium permanganate) placed in the bottom of the beaker before the heat is applied. Convection currents in air are made visible by smoke, produced by burning filter paper which has been soaked in a solution of saltpeter and dried. 192. Applications of Convection. The draft of a chimney is a convection current. It is maintained by an upward force equal to the difference between the weight of the heated air in the chimney and the weight of a column of outside air of the same dimensions. (Why?) Hence a tall chimney has a stronger draft than a low one. A lamp chimney maintains a steady draft of air, which, entering be- low, supplies the flame with the oxygen necessary for combustion. If the chimney is closed at top or bottom, the flame at once begins to smoke from imperfect combustion, and is quickly extinguished. Convection is an essential process in the heating and ventilation of houses. A hot stove or radiator keeps the air of a room in constant circulation; for the pressure of the warm air about the stove is always less than that of the colder and denser air at the same level in other CONDUCTION AND CONVECTION 229 parts of the room. The cold air near the floor moves slowly toward this region of lower pressure, driving the warm air upward. As the current of warm air reaches the ceiling, it spreads out and crowds the colder air downward. The descending currents are strongest near the walls and windows, especially the latter, where the loss of heat is most rapid. Cold air, entering through cracks in windows and doorways, provides considerable ventilation, but generally less than there should be. A fireplace secures excellent ventilation by maintaining an outward flow of air through the chimney. Currents of air and water in the heating and ventilation of buildings are further considered in Arts. 238-241. 193. Winds are convection currents, due to the unequal heating and cooling of the atmosphere over different parts of the earth's surface. Unequal temperatures at different places cause unequal barometric pressures, and the unequal pressures cause atmospheric currents, or winds. The origin of winds is well illustrated by the sea and land breezes, which are of almost daily occurrence along ocean shores in temperate and tropical regions. "As the land heats and cools more quickly than the sea, it often becomes warmer than the adjacent water during the day and cooler at night, and it communi- cates its temperature to the lower part of the air. So by day the air above the sea is the denser and flows toward the land, and at night the cool air above the land flows toward the sea." Winds, in general, are very complex phenomena, and are influenced by other agencies besides temperature, e.g. the percentage of water vapor in the air (which affects its density), the direction and eleva- tion of coast lines and mountain ranges, and the rotation of the earth on its axis. As a rule, the direction and velocity of the wind at any particular time and place are determined by atmospheric conditions over an area many hundred miles in extent; and the general wind systems, due to the unequal heating power of the sun at different latitudes, cover the entire earth. A full account of such matters may be found in any physical geography. PROBLEMS 1. Is clothing a source of heat? What is "warm clothing"? 2. Tin teakettles, pots, and boilers are often made with bottoms of copper. What is the advantage of this? 230 HEAT 3. The conductivity of fresh snow is many times less than that of either ice or water (see table). What is the reason for this? 4. Water can be boiled in a tray made of writing paper, with a Bunsen flame playing directly against the bottom of it.- (Try it.) Why does the paper not burn? 5. Would convection currents be caused by cooling a liquid at the top? by cooling it at the bottom? Would a. piece of ice cool a pitcher of warm water more or less quickly, if kept at the bottom, than it does when floating? 6. What convection currents are set up when a door is left open between a warm and a cold room? 7. Does an open fireplace provide equally good ventilation whether there is a fire in it or not? 8. How does convection differ from diffusion? Define convection in your own language. 9. On some days smoke rises rapidly from chimneys, on others slowly. Account for the difference. 10. Criticize the statement "The air over a heated area expands and rises, while the air from the cooler surrounding regions rushes in to take its place." 11. Inspect at home the system of pipes connected with the hot- water tank. Note where the cold water is admitted to the tank, where the hot water is drawn off, and where the connections are made with the pipes lead- ing to and from the heating coil in the kitchen range. Why should the pipes be placed as you find them? IV. RADIATION 194. Radiant Energy. Near a large fire in an open grate the face becomes painfully hot, even in a cold room. If the hand or a sheet of paper is held before the face, the sensation of intense heat instantly ceases, and in a few seconds the face becomes cool. Evidently the heat is not received through the agency of the air, either by con- duction or convection, for the face becomes much warmer than the air in contact with it. Moreover, air is practically a non-conductor, and the only convection current of heated air from the fire passes up the chimney. Heating under such conditions is the result of processes wholly different from either conduction or convection. RADIATION 231 These processes take place on an enormous scale in the heating of the earth by the sun. Let us recall the condi- tions under which this heat is received. The average distance of the sun from the earth is 93,000,000 miles. Through all this distance, to the earth's atmosphere, there is neither solid, liquid, nor gas. The intervening space is a perfect vacuum, and where there are no mole- cules there can be no molecular energy, or heat. Clearly the sun's heat does not make the long journey to the earth as heat. It must, however, be transmitted as some form of energy; for energy in any form can become nothing else than energy in some other form. This solar energy passes through the greater part of the earth's atmosphere without warming it even to arctic temperatures, but, on reaching the earth, it is transformed into heat in enormous quantities. Similar conditions, on a miniature scale, are reproduced in the incandescent electric lamp. When lighted, the fila- ment is white hot. The bulb becomes hot from the hot filament; but there is nothing to convey heat from one to the other, for the bulb is exhausted to a nearly perfect vacuum. There is, then, a form of energy which heat becomes and which is again transformed into heat under certain conditions, and which can be transmitted through a vac- uum. This is called radiant energy or radiation. The process by which radiant energy is transmitted is also called radiation. Thus we say that "radiation takes place in a vacuum," that "a body loses heat by radiation," etc. Radiant energy is said to be emitted when given out by bodies, and absorbed when received by them and trans- formed into heat. Other forms of expression are com- mon, but they are not scientific (Art. 201). 2.3-2 HEAT 195. Light is Radiant Energy. At very high tempera- tures all bodies give out light, in addition to the radiant energy which they emit at lower temperatures. Light also travels through a vacuum, since it reaches the earth from the sun and the still more distant stars. During an eclipse of the sun its heating power and its light decrease together and are lost at the same moment ; as it reappears, its heat- ing power is restored. Accurate observation, with delicate instruments, shows that the radiant energy and the light of the sun travel through space with the same velocity (186,000 miles per second). The reflection of a beam of light by a plane mirror is familiar. Experiments prove that radiant energy is reflected in the same manner. Light and radiant energy both travel in straight lines through air and other transparent substances; both are brought to a focus by a concave mirror or a lens. At the spot where sunlight is brought to a focus by a large mirror or a lens, a hole is soon burned through a piece of paper or a thin board, and a match is quickly ignited. These are only a few of the many points of similarity between light and radiant energy. In short, they are alike in all their physical properties; they are one and the same form of energy. A small part of the radiation from very hot bodies affects the nerve of the eye, and produces the sensation of sight. This we call light or visible radiation. The radiation to which the eye is not sensitive is called invisible or dark radiation. Light is "visible" radiation only in the sense that it renders bodies visible. We do not see light; we see only luminous and illu- minated bodies. A sunbeam marks its course across a darkened room by illuminating a cloud of dust particles in its track; it is itself invisible, and in dust-free air there is no indication whatever of its presence. RADIATION 233 196. Nature of Radiation. The Ether. The forms of energy previously considered do not exist apart from matter. There can be no energy of motion where there is nothing to move, or energy of strain where there is noth- ing to be compressed or distorted, or heat where there are no molecules. It is impossible to conceive of radiant energy as an exception to this rule. Its existence depends upon the existence of something which can possess it. This something is called the luminiferous ether, or simply the ether. The ether is the only medium or " vehicle" by which radiant energy is transmitted ; hence it must be pres- ent wherever light travels. It fills all " empty" space, from the intermolecular spaces in ordinary matter to the "boundless depths of space" through which we receive light from the distant stars. A perfect vacuum is perfectly full of ether, and absolutely empty space is not found any- where. Since the ether is not perceived by any of the senses, its. properties can only be inferred from the phenomena to which it gives rise. It is thus found to have inertia or mass; and hence is properly called matter. But it has not the molecular structure of " ordinary" matter, and is not to be regarded as a highly rarefied gas. The change of matter from one state to another (from the solid to the liquid state, from the liquid to the gaseous, etc.) is very common; but, so far as we know, ordinary or molecular matter never becomes ether, nor does ether ever become ordinary matter. The ether can neither be hot nor cold; it has no temperature; it can not possess heat or transmit it. Our first ideas concerning the nature of radiation are derived from other phenomena which are in some respects similar to it, and about which we have more direct knowl- edge. A stone, dropped into a pond, starts a series of 234 HEAT waves, which travel out in circles from the center of dis- turbance. The energy of the stone is imparted to the water, and is transmitted by the waves to distant parts of the pond, perhaps to the margin, where it is expended in moving grains of sand, bending blades of grass, etc. In such phenomena the water serves as a 'medium through which energy is transmitted from one body to another by means of wave motion. A sounding body, e.g. a bell or a violin, is in rapid vibration, and sets up a disturbance in the surrounding air. This disturbance travels outward, or radiates, in all directions as a series of sound waves, which, falling upon the ear, produce the sensation of sound. Thus air serves as a medium for the transmission of energy from one body to another (from the vibrating body to the ear), in the form of sound waves. Sound also travels through many other substances, including solids and liquids; but it can not travel through a vacuum. The ether is not a sound medium. Similarly, it is believed that the vibrating molecules of ordinary matter set up a disturbance in the luminiferous ether, a disturbance which travels as a wave motion with inconceivable velocity in all directions. Light consists of ether waves which are of the right size or length to cause the sensation of sight, when they enter the eye and fall upon the retina. If a body is not hot enough to be lumi- nous, the ether waves radiated from it are too large or long to affect the retina, and we call them " in visible." Radiant energy, then, is energy transmitted through the ether -as a wave motion. These waves travel through the air and other transparent substances, but not by means of them. They are always waves in the ether, which fills the inter- molecular spaces in ordinary matter. By the velocity of light we mean the speed with which ether waves travel RADIATION 235 through space. They reach the earth from the sun in 8 m. and 20 sec., and from the moon, which is at a dis- tance of 240,000 mi., in 1.3 sec. 197. The Radiometer. Instruments of different kinds have been invented for detecting and measuring radiant energy by the heat effects that it produces. A mercury thermometer with a coating of lampblack over the bulb is sometimes used, but it is not very sensi- tive. The lampblack absorbs all radiation falling upon it; and the mercury in the bulb is heated above the temperature of the surround- ing air in proportion to the intensity of the radiation. The radiometer (Fig. 183) is a more sensitive instrument. It consists of four light vanes of mica or aluminum attached to a vertical axis, and inclosed in a glass bulb contain- ing air under very low pressure. One side of each vane is bright, the other is coated with lampblack. When the instrument is placed in the sunshine or in the path of other radiation, the vanes rotate with their bright side in advance, the rate varying with the intensity of the radiation. The rotation is explained as follows: The black surfaces absorb more radiation than the bright, and hence are warmer. The molecules of air that strike the black surfaces are heated, and, rebounding with an increased velocity, exert a greater pressure than the molecules that strike the bright sides. The black sides are consequently driven backward. If the air in the bulb were not highly rarefied, the collisions among the molecules would be so fre- quent as to equalize the pressures throughout the bulb, and the vanes would remain at rest. 198. Emission, Absorption, and Reflection of Radiant Energy. Bodies emit and absorb radiation at all tem- peratures. If -a body is warmer than its surroundings, it emits more radiation than it absorbs; if colder, it absorbs more than it emits; if at the same temperature, it emits FIG. 183. Radiometer. 236 HEAT and absorbs radiation in equal amounts. The hand, held near a fire, becomes hot because it receives and absorbs more radiation than it emits; if held near a large piece of ice, it becomes cold because it receives less radiation from the ice than the ice does from it. The rate at which a body cools by radiation varies approximately as the difference between its own temperature and that of its surroundings. This is Newton's law of cooling. For ex- ample, a cup of coffee cools five times as rapidly at 70 as it does at 30 in a room the temperature of which is 20. The rate of cooling of a body by radiation depends also upon the nature of its surface. Rough, blackened sur- faces are good radiators; bright and polished surfaces poor radiators. This can be shown by means of two vessels of the same size and shape, but having unlike surfaces, e.g. one nickel-plated and the other coated with lampblack. When filled with equal quantities of hot water at the same temperature and allowed to stand for some minutes, their unequal radiating power will be shown by the unequal cooling of the water in them. When radiation falls upon a body, part of it is absorbed, part is reflected, and, in many cases, part is transmitted through the body. Good radiators are good absorbers of radiation; and poor radiators, poor absorbers. Lampblack is the best radiator and the best absorber known. It absorbs practically all radiation, both visible and invisible, that falls upon it. Any polished metal reflects much the greater part of all radiation, and absorbs the remainder. The absorption of invisible radiation is shown by the heat- ing effects produced; the absorption of light by the color effects. A white surface reflects nearly all light that falls upon it; a black surface reflects almost none; a colored sur- face reflects part and absorbs part. Light becomes heat RADIATION 237 when absorbed; but its energy is, in general, very small compared with that of the longer waves of dark radiation. Since only the unabsorbed radiation can be reflected, it is obvious that good absorbers are poor reflectors, and vice versa. Reflected radiation does not affect the tem- perature of the body upon which it falls. 199. Selective Transmission and Absorption. Sub- stances that transmit light are called transparent or translucent; those that do not are called opaque. Some transparent substances also transmit the longer waves of dark radiation; others do not. Clear glass transmits all light waves and a considerable part of the radiation from bodies nearly red-hot, but almost completely absorbs the longer waves which radiate from bodies at ordinary temperatures. A large part of the sun's radiation passes through window- glass, and, in a sunny room, is absorbed by the objects upon which it falls. But, as the radiation from these bodies can not penetrate glass, the energy is trapped in the room, which may thus become much warmer than the air outside. Water is another highly transparent substance which transmits very little dark radiation. On the other hand, substances which transmit dark radiation are, in some cases, perfectly opaque to light. A solution of iodine in carbon disulphide is an example (Laboratory Exercise 26). Rock salt and pure, dry air are good transmitters of both light and dark radiation. Substances that transmit dark radiation are sometimes called diathermanous and those that do not, athermanous. The unequal absorption of ether waves of different lengths by the same substance is called selective absorption. 200. The Heating of the Atmosphere. The pure, dry air at high altitudes transmits solar radiation in enormous quantities and absorbs very little. Aeronauts find the atmosphere intensely cold 238 HEAT above a height of three or four miles. As the radiation approaches the earth, the rate of absorption rapidly increases, principally on account of the greater amount of water vapor; for experiments have shown that the absorbing power of air containing the average amount of water vapor is seventy- two times as great as that of perfectly dry air of the same density. The absorption at lower levels is further increased by the dust particles in the air. Elaborate investigations conducted at the Smithsonian Institu- tion, Washington, D.C., and at the solar observatory on Mt. Wilson, in southern California, have shown that, on the average, the atmos- phere transmits about two thirds of the solar energy. Much of this is absorbed by the surface of the land and the ocean; the remainder is reflected. The reflected radiation is partly absorbed on its way out through the atmosphere. The absorbed radiation warms the surface of the land, and this in turn warms the air in contact with it. This is especially noticeable on a hot day in summer, when, if there is no wind, the air close to the ground is many degrees warmer than at a height of a few feet. The heating of the air at the bottom causes convection currents (winds), by which the heat is carried to consider- able altitudes; but the temperature is necessarily the highest at the source of the heat, i.e. at the earth's surface. The earth is cooled at night principally by radiation. The loss of heat is rapid on clear nights, especially when the atmosphere is very dry. A moist or cloudy atmosphere absorbs the radiation, thus serving as a blanket to the earth. Hence clear nights are, as a rule, the coldest. At high altitudes, where there is but little hin- drance to radiation either by day or night, sheltered valleys are quickly warmed in summer by the early morning sunshine; and a sudden chill follows the disappearance of the sun in the evening, the nights being often cold enough for frost. Thus we see that the atmosphere, or rather the moisture in it, renders a very important service in moderating the intensity of solar radiation by day and in retaining the earth's heat by night. 201. Summary on the Transmission of Heat. Heat is transmitted from one portion of matter to another by con- duction only; it is transmitted from one place to another through matter by conduction, and with matter by convec- RADIATION 239 tion. A body loses heat by conduction and by the emis- sion of radiant energy; it gains heat by conduction and by the absorption of radiant energy. Before the true nature of radiant energy was known, dark radiation was supposed to be a form of heat, and was called " radiant heat." According to this older view, bodies " radiate heat" and " absorb heat" and "heat is transmitted by radiation." This is still the popular lan- guage of the subject, but it is disappearing from scientific literature. PROBLEMS 1. Why does snow melt more quickly when covered with a thin layer of earth? 2. Why is light-colored clothing more comfortable in summer than black? 3. Why is the difference between the temperature in the sunlight and in the shade greater upon the top of a mountain than at a low elevation? 4. Why must those who climb snow-covered mountains take special care to protect their faces? 5. Why is the heating power of the sun less at morning and evening than it is during the middle of the day? Why less in winter than in summer? 6. What is the "solar water-heater"? Explain its action. 7. The moon has no atmosphere and its days and nights are each two weeks long. What effect do you think these conditions must have on the temperature of lunar days and nights? 8. A Dewar flask (Fig. 184) for containing liquid air consists of "a double- walled glass vessel, the space be- tween the walls being exhausted as completely as possible. Traces of mercury vapor are left in this space; and at a low temperature this freezes, forming a metallic surface over the glass walls." The access of heat to liquid air in such a flask is almost completely 3 Dewar Flask, prevented. Explain. NOTE. The commercial form of the Dewar flask is known as the "thermos bottle" (Fig. i84a). The double-walled glass bottle is inclosed in a nickel-plated metal case. What useful FIG Ts _P ur P se is served by the nickel plating? It is claimed that the Thermos thermos bottle keeps liquids hot 24 hours in the coldest weather, Bottle, and ice-cold liquids ice cold 72 hours in the hottest weather. 240 HEAT V. CHANGES IN VOLUME AND PRESSURE 202. Linear Expansion of Solids. With few excep- tions, none of which are important, solids expand when heated and contract when cooled. The rate of expansion is different for bodies of different material; but it is in all cases so small that some special device must be employed in order to measure with any degree of accuracy the in- crease in the length of even a long rod when its tempera- ture is raised many degrees. The methods by which this is done are best studied in the laboratory. (See Labora- tory Exercise 27.) A solid expands in each of its three dimensions, and increase in any one of them is called linear expansion. In most cases it is only change of length that is important. The increase in length per unit length, for a rise of tempera- ture of one degree, is called the coefficient of linear expan- sion of the substance. This coefficient may equally well be regarded as the ratio of the whole increase in length to the whole length of the body, for a rise of temperature of one degree. For example, the coefficient of linear expansion of steel is .000012. This means that each centimeter of length of a steel bar or rod in- creases to i. 000012 cm. or each foot of its length to 1.000012 ft., with a rise of temperature of one degree Centigrade. The coefficient for aluminum is .000023, or approximately twice that of steel. The expansion of a solid with any rise of temperature can readily be computed when its length and its coefficient of expansion are known. This is a matter of great impor- tance to engineers and builders. EXAMPLE. What is the expansion of a thirty-foot steel rail between a minimum winter temperature of 30 C. and a maximum summer temperature of 40? The expansion per foot per degree is .000012 ft. Hence the total expansion is 30 X 70 X .000012 = .025 ft. or,3 in. CHANGES IN VOLUME AND PRESSURE 241 COEFFICIENTS OF LINEAR EXPANSION Hard rubber 000084 Platinum 0000088 Zinc 0000294 Glass 0000086 Lead 0000286 Oak parallel to grain 000005 Aluminum 000023 Steel alloyed with Brass QOQOiSS 36% nickel 0000009 Copper 0000172 Porcelain (Berlin) 0000027 Iron and steel 000012 Quartz, fused 0000004 203. Effects and Applications of Expansion. The expansion and contraction of solids with a change of tem- perature enters into the .affairs of daily life in many ways. Frequently it presents itself merely as a troublesome fact which must be taken into account. Thus it is often neces- sary to make special provision for expansion in designing metal structures and machinery. A steel truss bridge would wreck its foundations with its change of length be- tween winter and summer, if it were rigidly anchored to them ; hence one end is supported upon steel rollers. Bridge engineers allow for an expansion of one inch in each 80 ft. of length, in localities where the changes of temperature are extreme. The rails of tracks are laid with a small space between their ends, which provides room for expan- sion. The change of length of steam and hot- water pipes is provided for by inserting expansion joints, at which the end of one pipe is free to slip back and forth within the end of a larger one. Bends in the pipes can often be made to serve for the same purpose; for a bend will yield more or less as the length of the pipe changes. Referring to the table of expansion coefficients, it will be seen that the expansion of steel containing 36% of nickel is much the smallest in the list, being less than one fifth that of oak and about one tenth that of glass. This is a very valuable property. " Invar steel is a nickel steel in which by mechanical treatment the coefficient has been still 242 HEAT further reduced. It is used extensively for the construc- tion of standards of length, steel tapes, pendulums, etc." Expansion is usefully applied in various ways. The wooden wheels of vehicles have iron tires, which are made just large enough to slip on when heated; and, in cooling, they contract so as to fit tightly. Large cannon are some- times reenforced with an outer casing or jacket of steel, which is shrunk on like the tire of a wheel. Red-hot rivets are used in joining the steel plates of tanks and boilers; and, by their contraction in cooling, they draw the plates together with great force. Various mechanical devices have been invented, whose action depends upon the unequal expansion of different metals. The princi- ple involved can be illustrated with a compound bar consisting of a strip of brass and one of iron, riveted together in several places (Fig. 185). The bar is straight or nearly so at ordi- nary temperatures; but, when heated, it takes the shape of a circular arc, with the brass strip on the outside of the curve. The bending is due to the greater expansion of the brass (see table of coefficients). The bar becomes straight again on cooling, for the brass then contracts more than the iron. Metallic thermometers are constructed on this principle. Sometimes they are self-recording, as shown in Fig. 186. A compound strip of brass and steel is formed into a spiral, with the brass on the outside. When the temperature rises, the spiral becomes more curved and its outer end moves upward, raising the end C of a lever; when the temperature falls, the lever is depressed. A pen at FIG. 185. Compound Bar Showing Unequal Expansion. FIG. 186. Metallic Thermometer, Recording. the end of the lever records its movements on a paper wrapped round the drum D, which is moved by clockwork. CHANGES IN VOLUME AND PRESSURE 243 204. The Expansion of Liquids. In the expansion of liquids and gases it is increase of volume, or cubical expan- sion, with which we are concerned. The coefficient of cubical expansion of a liquid (or a solid) is the increase in its volume per unit volume, for a rise of temperature of one degree. As usually contrived (Fig. 170), experiments on the expansion of liquids give their apparent expansion, i.e. the difference between their true expansion and the expansion of the containing vessel. The true expansion of a liquid is the sum of its apparent expansion and the cubical expan- sion of the material of the containing vessel. Liquids differ from one another in their rates of expansion; and, in general, they expand more rapidly at higher temperatures. Mercury is exceptional in that its expansion is practically uniform between o and 100, hence the degree intervals on a mercury thermometer are equally spaced. The coefficient of cubical expansion of a solid is (very approximately) three times its coefficient of linear expan- sion, since the rate of expansion is the same in the three dimensions. Multiplying the linear coefficients in the table for solids by three, a comparison can be made with the coefficients for liquids given in the table below. It will be found that the expansion of liquids is, in general, much the greater. The total expansion of water between 4 and 100 is a little over 4%. COEFFICIENTS OF CUBICAL EXPANSION Ether 0.0018 Turpentine 0.0007 Alcohol (5 to 6) 0.00105 Glycerin 0.0005 Alcohol (49 to 50) 0.00122 Water (5 to 6) 0.000022 Acetic acid 0.00105 Water (49 to 50) 0.00046 Petroleum 0.0009 Water (99 to 100) 0.00076 Olive oil 0.0008 Mercury 0.00018 244 HEAT 205. Expansion of Water. The expansion of water is curiously irregular. It contracts as its temperature rises from o to 4. When heated beyond this point it begins to expand, at first very slowly, then more and more rapidly (see table). Hence the density of water is greatest at 4 C. (about 39 R). This behavior of water is of great importance in the economy of nature. In winter the water of a lake loses heat at the surface by contact with the cold air and by radiation. As the water at the surface cools, it becomes denser and sinks, displacing the water at the bottom. This continues until the water is cooled throughout to a temperature of 4. With further cooling of the surface layer, it expands and remains at the top. Hence freezing takes place at the surface, while the water at only a slight depth is at 4. Further loss of heat takes place only by conduction, which is slow in both water and ice; hence lakes and streams freeze only to a depth of a few feet, even in a long, cold winter, and the fish and other inhabit- ants of the waters are not destroyed. 206. Expansion of Gases. Laws of Gay-Lussac and Charles. We have seen that different solids and differ- ent liquids expand at very unequal rates when heated. The rate of expansion of gases, on the contrary, is found to be the same for all, to a very close approximation, at all temperatures and at any constant pressure, provided only that the temperature is considerably above that at which the gas liquefies under the given pressure. The volume of a given mass of any gas increases, under constant pressure, by ^js or ^ s volume at o C. for each degree of rise in temperature. This is known as the law of Gay-Lussac or the law of Charles, after two French physicists who shared in its discovery. The fraction ^js or -003665 is, according to the law, the coefficient of cubical expansion of all gases. This coefficient is much larger than that of liquids and solids in general. (See table.) When a gas is heated without being permitted to expand, its pressure increases by ^73- of the pressure at o C. for each degree of rise in temperature. This is more properly CHANGES IN VOLUME AND PRESSURE 245 called the law of Charles. It is a necessary consequence of the law of Gay-Lussac and the law of Boyle (Art. 54), taken together. For, if a gas is heated at constant pres- sure, it expands according to the law of Gay-Lussac. If it is then compressed to its original volume without change of temperature, the pressure increases according to Boyle's law. It is then in the condition in which it would be if it had been heated under constant volume; i.e. the net result is an increase of pressure, in agreement with the law of Charles. The following are numerical examples illustrating the laws: If a mass of gas is heated under constant pressure from o to 10, its increase of volume is ^W of its original volume, and its volume is then 1M of its volume at o. If it is cooled from o to 50, its loss of volume is 3% of its volume at o, and its volume is then ff | of its volume at o. If it is heated at constant volume from o to 100, its pres- sure becomes f y as great as at o. 207. Absolute Temperature and Absolute Zero. Let VQ denote the volume of a body of gas at o C., and vi its volume at any other temperature t\ under the same pressure. The increase in its volume is +5 then v\ VQ, and this increase is -- of the volume 273 at o; that is, 125 - 400 100 .- 375 373 75 350 50 325 25 .- 300 .- 275 273 3-25 250 ^ w-50 225 | o w |! -75 200 ^ 3-100 -125 _ 175 | 150 -150 126 -175 .- 100 -200 . - 75 -225 .- 50 -250 . - 25 -273 FIG . i 57- ) From which A 273 (i) Similarly, if vi 273 + h which reduces to - = ^ + fa - (3) The relation expressed by equation (3) has led to the adoption of a temperature scale whose degrees are the same as those of the Centi- grade scale but whose zero is at 273 C. This scale of temperature is called the absolute scale, and its zero the absolute zero. The freezing point is 273 Abs. and the boiling point 373 Abs. Any temperature on the Centigrade scale is changed to the absolute scale by adding 273. If we let T denote temperatures on the absolute scale, then TI = 273 + /i, and T% = 273 + / 2 , and equation (3) becomes That is: Under constant pressure the volume of any body of gas is pro- portional to its absolute temperature. This is the law of Gay-Lussac, stated in its simplest form. If the law held for all temperatures, it is evident that at absolute zero the volume of any mass of gas would be zero; but no substance exists as a gas at absolute zero. All known gases have been liquefied and all except helium reduced to the solid state at temperatures above absolute zero; and, as stated before, the law does not express the behavior of gases when near the point of condensation. If in the above equations we substitute pressures for volumes, we have the relations that hold between the pressure and the tem- perature of a gas, at constant volume; and we arrive at the conclusion that the pressure of a gas is proportional to its absolute temperature, the volume remaining constant. This law holds for any gas as it is cooled, until, at a certain low temperature, the molecules begin to cohere, when the loss of pressure becomes more rapid than the law indicates. If it were not for cohesion, we should expect the law to CHANGES IN VOLUME AND PRESSURE 247 hold at all temperatures, in which case the pressure would vanish only at absolute zero. Since pressure is due to molecular motion, the molecules would then be at rest, and the gas would have lost all its heat energy. By other lines of reasoning which belong to more advanced physics, it is proved that the absolute zero is indeed what its name indicates, namely, the temperature at which a body would possess no molecular kinetic energy, or no heat. No substance has yet been cooled to absolute zero; but this temperature has been more and more closely approached in recent years. By rapid evaporation in a vacuum, solid hydrogen has been cooled to -260 C. or 13 Abs., and in 1908 helium was liquefied at a temperature estimated at 5 Abs. PROBLEMS 1. The thinner a glass tumbler is, the less likely it is to break when hot water is poured into it. Why? 2. Why can not an air thermometer be used for measuring the lowest attainable temperatures? 3. In all accurate work the reading of a barometer must be "corrected for temperature"; i.e. its true height is taken as the height at which it would stand if the temperature of the mercury were o C. A barometer reading is 75.6 cm. at a temperature of 22; find its true or corrected height. 4. The steel cables of the Manhattan suspension bridge in New York City are about 1475 ft- long between the towers. How much does their length change in this span between winter and summer, allowing a minimum winter temperature of -25 C. and a maximum summer temperature of 35? 6. To what temperature must a gas be heated, under constant pressure, in order to double its volume, the temperature at the start being 30 C.? 6. A body of gas at 10 and a pressure of one atmosphere is inclosed in a vessel and heated to 300, none of the gas being allowed to escape. What is the pressure at that temperature? 7. If a balloon at the beginning of its ascent is fully inflated with gas at 20, what fractional part of the gas must be allowed to escape in rising to a height where the pressure is reduced one half and the temperature is -io? Suggestion. The pressure inside the balloon is practically the same as that of the surrounding air. Pressure and temperature really change together; but the final result is the same as if they took place separately. Hence compute first the effect of change of temperature, assuming constant pressure, then the effect of change of pressure. 248 HEAT VI. MEASUREMENT OF HEAT. SPECIFIC HEAT 208. The Heat Unit. Heat being a form of energy, it can be measured in terms of any of the units by which mechanical energy is measured (foot-pound, etc.) ; they are not used, however, as there are more convenient units for the purpose. Two heat units are in common use: one, the calorie, is the amount of heat required to raise the tem- perature of one gram of water one degree Centigrade ; the other is the amount of heat required to raise the tempera- ture of one pound of water one degree Fahrenheit. The calorie is almost exclusively used in scientific work, and is the only heat unit used in this book. The heat received or given out by any mass of water, when it is warmed or cooled through any range of tempera- ture, is measured by the product of its mass and its change of temperature. For example, to warm 10 g. of water one degree requires 10 calories; to warm 10 g. from 8 to 63 requires 10 X (63 8) = 550 calories. In cooling from 63 to 8, 10 g. of water would give out 550 calories. The amount of heat required to raise the temperature of one gram of water one degree is not exactly the same at all temperatures, but the difference is too small to be of importance except in the most accurate work. The numerical relation between the calorie and the units of mechanical energy is considered in Art. 242. 209. Specific Heat. When equal masses of hot and cold water are mixed, the resulting temperature is midway between the original (initial) temperatures of the separate masses; e.g. if the temperature of the cold water is 20 and that of the hot water 100, the temperature of the mix- ture will be 60. In cooling one degree, the hot water gives out enough heat to warm the equal mass of cold water one degree. MEASUREMENT OF HEAT. SPECIFIC HEAT 249 If a piece of hot metal is dropped into an equal mass of cold water, the resulting temperature is far below the average of the initial temperatures. If the metal is iron, for example, it will be found that the temperature of the iron falls about 9 degrees for each degree of rise in the tem- perature of the water. Plainly, therefore, iron gives out only ^ as much heat as an equal mass of water does during an equal fall of temperature. It follows further that only ^ as much heat is required to raise the temperature of a mass of iron a given number of degrees as is required to raise the temperature of an equal mass of water the same num- ber of degrees. When copper or zinc is used in the experi- ment, the ratio is found to be approximately ^T; with lead it is about 3*0- The ratio of the quantity of heat required to warm any mass of a substance one degree to the quantity required to warm an equal mass of water one degree is called the specific heat of the substance. (Compare with the defini- tion of specific gravity.) The specific heat of a substance is numerically equal to the number of calories required to raise the temperature of one gram of the substance one degree Centigrade. (Why?) The number of calories required to warm any mass of a substance through any number of degrees is measured by the product of the mass of the body, its rise of tempera- ture, and its specific heat. (Why?) The specific heat of water is unity, by definition; it is very large compared with that of most other substances, especially the metals, and is exceeded only by hydrogen. In the following table the substances are named in the order of their specific heats. Note that the specific heat of water changes with a change of state, its value for ice being .504 and for steam .48. 250 HEAT TABLE OF SPECIFIC HEATS Hydrogen (at constant pressure) 3.409 Aluminum 0.218 Water i.ooo Glass 0.198 Alcohol (o to 40) 0.597 Iron 0.113 Ice 0.504 Brass and copper 0.094 Steam 0.480 Zinc 0.094 Air (at constant pressure) 0.237 Mercury 0.033 Marble 0.216 Lead 0.031 210. Measurement of Specific Heat. The method generally employed for determining the specific heat of a substance is known as the "method of mixtures." It is illustrated by the following example: A brass calorimeter weighing 100 g. contains 400 g. of water at 18. Into this is put a roll of sheet iron, weighing 190 g. and heated to 1 00. After stirring, the temperature of the water is 22, and this is assumed to be the temperature of the roll of iron and the calorimeter. The specific heat of the calorimeter is given as .094. The specific heat of iron is to be found from the experimental data, and is denoted by s. The computation is as follows: Rise of temp, of calorimeter and water = 22 18 = 4 Heat received by the calorimeter = 100 X 4 X .094 = 37.6 cal. Heat received by the water = 400 X 4 = 1600 cal. Fall of temperature of the iron = 100 22 = 78 Heat given out by the iron = 190 X 78 X s =. 14,820 s cal. Assuming that the transfers of heat take place only among the calorimeter and its contents, it follows that the heat given out by the iron in cooling to the tempera- ture of the " mixture" is equal to the heat gained by the calorimeter and water in coming to the same temperature; that is, 14,8205 = 37.6 + 1600; from which s = 1637.6 -f- 14,820 = .no. MEASUREMENT OF HEAT. SPECIFIC HEAT 251 211. The Heat Equation. The above example illus- trates the method of treating the experimental data in all experiments in calorimetry (the measurement of heat) and in the solution of problems. The following summary of the method will therefore be of service now and later. 1. Find numerical or algebraic expressions for the sep- arate quantities of heat received or given out by the differ- ent bodies (including the vessel) during the equalization of temperature. 2. Write the sum of the quantities of heat given out equal to the sum of the quantities of heat received. This is the heat equation. 3. The heat equation contains as an unknown quantity the quantity sought (specific heat, heat of fusion, or heat of vaporization). To find this quantity, solve the equa- tion by the usual algebraic processes. 212. The Control of Heat in Calorimetric Experiments. Any transfer of heat between the contents of the calorimeter and the sur- rounding air or other bodies during an experiment is a source of error, and is to be avoided as far as possible. The calorimeter is sually nickel-plated and brightly polished to diminish radiation when it is warmer than the surrounding air, and to diminish absorption when it is cooler. It should stand on a poor conductor (wood) and should be touched with the hands as little as possible, to avoid conduction to or from the hand. A calorimeter is like a leaky vessel. By such precautions as these we endeavor to stop up the leaks. At the beginning of an experiment the water should be taken at such a temperature that it (and the calorimeter) will be colder than the air during a part of the time and warmer during a part, in order that the gain of heat by conduction and absorption at the lower temperature may be as nearly as possible equal to the loss by conduc- tion and radiation at the higher temperature. For accurate work other and much more elaborate precautions than these are necessary. 252 HEAT PROBLEMS 1. The specific heat of water is much greater than that of rocks and soils. How does this in part account for the fact that the change of temperature of the land between day and night and between winter and summer is much greater than that of the ocean? 2. Are equal quantities of heat required to raise equal volumes of different substances through equal changes of temperature? (Consult table of densi- ties and table of specific heats.) 3. What effect has the large specific heat of water on the sensation caused by putting the hand in hot or cold water? In general, how does the specific heat of a substance affect the sensation of heat or cold caused by it when touched (see Art. 190)? 4. Dry air at the temperature of boiling water does not cause a burn, and is not even painfully hot. Why not? 6. How many cubic feet of air can be warmed one degree by the heat given out by a cubic foot of water in cooling one degree? 6. Of what advantage is the high specific heat of water in the hot- water system of heating buildings? 7. A roll of lead weighing 800 g. is heated to 100 and placed in a brass calorimeter weighing 90 g. and containing 406.3 g. of water at 16.2. The final temperature is 21. Find the specific heat of lead. 8. A kilogram of mercury at 200 and a kilogram of water at o are mixed. Find the resulting temperature, no allowance being made for the vessel. 9. A piece of aluminum weighing 60 g. is heated to 63, and placed in a copper calorimeter weighing 50 g. and containing 103 g. of alcohol at 8. The temperature of the alcohol rises to 17. Find its specific heat, taking the specific heat of copper and aluminum from the table. VII. FUSION AND SOLIDIFICATION 213. Change of State. Among the various effects of heat none are more familiar than change of state. Solids become liquids and liquids gases, when heat is received in sufficient quantity; and the opposite changes of state occur when heat is given out. Water is the only sub- FUSION AND SOLIDIFICATION 253 stance which comes under ordinary observation in all three states. Various substances are familiar both in the solid and the liquid states, e.g. glue, wax, jelly, and butter; others in the liquid and the gaseous states (the volatile liquids and their vapors), e.g. ether, alcohol, and gasoline; and still others only as solids or only as gases. But most substances are capable of existing in all three states, at temperatures ranging very high in some cases and very low in others. The metals melt more or less readily in the heat of a furnace; and several of them, including iron, are known to exist as vapors in the atmosphere of the sun. The air and other gases, as already noted, liquefy and even solidify at temperatures approaching absolute zero. The tissues of plants and animals do not melt, but undergo chemical change at high temperatures, by which 'they are broken up into simpler substances. In studying the laws and principles relating to change of state, water is taken as the typical example, not only on account of its convenience and familiarity, but principally because the melting, freezing, evaporation, and condensa- tion of water are phenomena of the greatest importance in nature. 214. Melting of Ice and Freezing of Water. The term "ice cold" seems to imply that the temperature of ice is always the same. The fact is that ice loses heat and cools to the temperature of surrounding bodies, when their temperature is below zero; and, in receiving heat at all temperatures below zero, it becomes warmer, just as other substances do. The specific heat of ice is .504, or approxi- mately half that of water in the liquid state ; i.e. a gram of ice in losing .504 calories falls one degree in temperature, and in receiving .504 calories, at all temperatures below 254 HEAT zero, it becomes one degree wanner. But ice can not be heated above o C. When ice at this temperature is sur- rounded by warmer bodies, even when thrown into boiling water or placed on a hot stove, the heat received does not penetrate the ice, and only causes melting at its surface. The comparatively slow rate at which ice melts, even on a hot summer day, indicates that much heat is required to accomplish the change of state. Water cools as it loses heat until its temperature falls to o. With further loss of heat, it begins to freeze; but its temperature remains at o until it is all frozen. Ice melts and water freezes at exactly the same temperature, but under opposite conditions with respect to the transfer of heat. Ice melts only in proportion to the heat received at o, and water freezes only in proportion to the heat lost at o. With neither a gain nor a loss of heat, neither melting nor freezing can take place. 215. Melting Points. Every solid that can be melted has a constant melting point, which is also the temperature at which it freezes or solidifies. Among fusible solids, some, like ice, change abruptly from the solid to the liquid state. In such cases the melting point can be very accu- rately determined. Other solids, e.g. sealing wax, glue, pitch, and glass, gradually soften and pass by continuous change into the liquid state. In such cases the melting point is indefinite just to the extent that the distinction between the solid and the liquid state is indefinite. TABLE OF MELTING POINTS Alcohol -130 C. Paraffin 54 C. Mercury -39 Beeswax 62 Ice o Rose's metal (alloy of tin, Butter 33 lead, and bismuth) 96 FUSION AND SOLIDIFICATION 255 Sulphur -...'.... 115 C. Aluminum 657 C. Cane sugar 170 Copper noo Solder, soft 225 Glass 1000 to 1400 Lead 327 Iron 1200 to 1600 Zinc 420 Platinum 1775 216. Change of Volume during Fusion and Solidifica- tion. Most substances expand in melting and contract in solidifying, the change of volume in some cases being considerable. The contraction of beeswax or paraffin in solidifying leaves a considerable depression at the center of the top surface. Metals, with few exceptions, also contract in solidifying. Those that do are unsuitable for casting, as they take only an imperfect impression of the mold. Cast iron, bismuth, and type metal (an alloy of lead, tin, and antimony) are among the exceptions. Water expands in solidifying, the increase in volume amounting to about one eleventh. In consequence of this expansion, ice floats a fact of great importance in nature. If water contracted in freezing, ice forming at the surface of lakes and rivers would sink. Freezing would therefore continue rapidly throughout winter, or until the lakes and rivers were frozen solid ; and all animal life inhab- iting them would be destroyed. The expansion of water in freez- ing is responsible for the bursting of water-pipes in winter. That the force of expansion is practically irresistible was strikingly shown by the experi- ments of Major Williams, in Canada. " Having quite filled a thirteen-inch bomb-shell with water, he firmly closed the touch-hole with an iron plug weighing three pounds, and ex- posed it in this state to the frost. After some time the iron plug was forced out with a loud explosion, and thrown to a distance of 256 HEAT 415 ft., and a cylinder of ice 8 in. long issued from the opening. In another case the shell burst before the plug was driven out, and in this case a sheet of ice spread out all round the crack." (Ganot.) " Much of the destruction of rocks which is taking place on the earth's surface is due to the same quiet but intensely powerful action of freezing water. Rain sinks into the cracks and pores which all rocks are liable to contain, and when it freezes there, the crack is inevitably widened or the structure of the rock loosened. Thus room is made for more water, which acts in the same way when it freezes; and so by degrees immense masses of rock and earth are loosened from the mountainside, nor does the action end until the material is reduced to the finest soil." (Madan.) Substances that expand in solidifying have a crystalline structure in the solid state. The crystalline structure is plainly seen in the ice that first forms when water begins to freeze, in the frost that gathers on window panes, and in snow. In crystalline solids the molecules are arranged in clusters of definite shape, and may therefore occupy a greater space than they do when lying loosely side by side in the liquid state, just as a number of bricks occupy more space when arranged in patterns than they do when packed in layers. 217. Change of Melting Point due to Pressure. Experiments have shown that when water is subjected to great pressure its freezing point is lowered. It does not freeze at o because the expansion which normally accompanies freezing is retarded. In agreement with this it is also found that ice melts below o under great pressure; for the pressure tends to bring about the decrease of volume which accompanies melting. The melting and freezing points are equally lowered by a given pressure, as we should expect. Ice has been melted at 18 under a pressure estimated at several thousand atmospheres. The change in the melting point due to a pressure of one atmosphere is only .0072, and would escape detection by means of the thermometers used in elementary physics; yet the effects produced under certain conditions by moderate changes of pressure are very striking. Thus a loop of fine wire to which weights are attached slowly descends through a block of ice round which it has been passed (Fig. 189) ; yet, after it has passed completely through, the ice is one solid piece as at the beginning. The pressure of the wire very slightly lowers the melting point of the ice immediately beneath FUSION AND SOLIDIFICATION 257 FIG. 189. Wire Passing through Ice. it; and the ice melts, receiving the necessary heat from the water just above the wire. This water freezes in losing heat, since it is relieved from the pressure. The process is continuous; for the water from the melt- ing ice below the wire passes round and freezes above it. The three stages of the pro- cess are (i) melting under pressure, (2) change of posi- tion of the water, (3) regelation (refreezing) under diminished pressure. Snowballs are formed by partial melting and regela- tion of the snow under the pressure of the hands. The slow change of snow into the clear ice of glaciers is due to the same action under gravity pressure. This action continues even in the solid ice of the glacier, which, in con- sequence, slowly flows down the mountain valleys at a rate varying from a few inches to one or two feet per day. 218. Heat of Fusion. Since the melting of ice is slow, even in warm weather, it is reasonable to conclude that much heat is required to bring about the change of state. A more definite idea of the amount of heat required to melt a given mass of ice may be gained by applying a Bunsen flame to a beaker containing a mass of broken ice, and, at the same time, a similar flame to a second beaker contain- ing an equal mass of water, taken at the temperature of the ice. If the contents of the first beaker are constantly stirred until the ice is all melted, the temperature of the water will be but little above o, while the water in the other beaker will be found nearly boiling hot. The main point to be noted is that the change of state in the one case and the heating in the other are accomplished by approxi- mately equal quantities of heat. 258 HEAT By methods adapted to accurate measurement it has been found that 80 calories are required to convert a gram of ice at o into water at the same temperature. When the opposite change of state occurs, an equal quantity* of heat is given out, i.e. a gram of water at o freezes only on losing 80 calories. This quantity is called the heat of fusion of ice. The heat of fusion of any substance is the number of calories required to melt one grym of ity-a.ft.er it has reached its melting point, or the number of calories given out by one gram of the substance in solidifying, with- out a change of temperature in either case. The heat of fusion of water is much larger than that of most substances. TABLE or HEATS OF FUSION Calories Calories Ice 80.0 Tin 14.25 Paraffin 35.1 Sulphur 9.37 Zinc 28.1 Lead 5.86 Iron 23 to 33 Mercury 2.83 219. Transformations of Energy during Fusion and Solidification. Since the heat received by a solid while it is melting does not affect its temperature, we conclude that this energy has ceased to exist as heat in producing the change of state. Melting may be said to consist in overcoming the cohesion which binds the molecules of a solid together. In doing this internal work, heat becomes molecular potential energy the energy of an altered molecular condition. This energy is recovered as heat during the opposite change of state, as the principle of the conservation of energy would lead us to expect. These transformations of molecular energy may be illustrated by means of two balls connected by a rubber band, the balls representing molecules and the rubber band, cohesion. In pulling the balls apart work is done against the force FUSION AND SOLIDIFICATION 259 which tends to hold them together. This work is stored as potential energy, and is recovered when the balls are permitted to come together again. According to the caloric theory, heat always remains heat, being (as was supposed) a form of matter. In the language of this theory, the heat that disappears during fusion and vaporization was called "latent," i.e. inactive or hidden; and heat, properly so called, was known as "sensible" heat. The trio of misnomers "sensible heat," "latent heat," and "radiant heat" are only now falling into dis- repute, half a century and more after the overthrow of the theory that gave rise to them. 220. Heat of Solution. Freezing Mixtures. Work is done in overcoming cohesion in a solid when it is dissolved as well as when it is melted ; and in many instances there is direct experimental evidence that heat disappears in the process, proving that this work is accomplished by heat.* Thus when ammonium chloride or ammonium nitrate is dissolved in water, there is a fall of temperature of several degrees ; for the heat required to dissolve the solid is taken from the nearest available source, i.e. the water. Solution differs from fusion in that it can take place within a wide range of temperatures; hence the temperature continues to fall (unless heat is received from outside sources) until all the solid is dissolved or until the solution is saturated. A mixture of one or more solids and a liquid, or of two solids, is called a freezing mixture if the solution or the liquefaction of the solids causes a fall of temperature below zero. The following are examples of freezing mixtures: i. One part by weight of ammonium chloride and one of potassium nitrate or ammonium nitrate, powdered together and dissolved in two parts of water. Fall of temperature about 20. * When chemical action accompanies solution, it may result in a rise of temperature, the heat generated by the chemical action being greater, in such cases, than the heat lost in solution. 260 HEAT 2. One part of table salt and two parts of snow or crushed ice. Fall of temperature to about 18. The strong attraction of salt for water causes the ice to melt rapidly, and at a temperature below its normal melting point. The heat required to melt the ice and to dissolve the salt is taken first from the ice and salt, then, by conduc- tion, from surrounding bodies. This freezing mixture is well known from its use in making ice cream. 3. One part each of crystallized calcium chloride and snow or crushed ice. Fall of temperature to about 40. PROBLEMS 1. What determines whether, in a mixture of ice and water, both at o, the ice will melt or the water freeze? 2. The heat of fusion of iron being much less than that of ice, how does it happen that iron does not readily melt and ice does? 3. What purpose is served by vessels of water placed in a cellar where vegetables are stored or in a greenhouse on a frosty night? 4. (a) Do freezing and thawing take place more or less rapidly than they would if the heat of fusion of ice were less? (&) Of what importance is this in the economy of nature? 5. How much heat is required to convert 750 g. of ice, taken at -20, into water at 50? 6. How many grams of ice at o can be melted by 500 g. of water at 6o? 7. A piece of aluminum weighing 250 g. and heated to 100 is placed in a dry cavity in a block of ice, and melts 68,8 g. of the ice. Find the specific heat of the aluminum, taking the heat of fusion of ice as 80 calories. 8. The quantity of heat that melts one gram of ice, taken at o, would melt how many grams of lead, also taken at o? 9. Does ice mixed with salt melt more or less rapidly than ice alone? (Try the experiment.) Account for the result. If the temperature of the freezing mixture is -18, do we infer that the ice is melting at this tempera- ture? 10. Does ice in a refrigerator serve its purpose by its mere presence or by melting? Explain. VIII. VAPORIZATION AND CONDENSATION 221. Vaporization. The gaseous form of a substance that exists as a liquid or a solid at ordinary temperatures VAPORIZATION AND CONDENSATION 261 is called a vapor, and the change of a liquid or a solid to the gaseous state is called vaporization. Vaporization may take place at the free surface of a liquid or within its mass. In the first case it is generally called evaporation; in the second case, boiling. A volatile liquid is one that evapo- rates readily, e.g. gasoline, alcohol, and ether. Liquids boil at definite temperatures, for reasons which are considered later; they evaporate at all temperatures, but more rapidly as the temperature rises. A damp cloth dries slowly in a cold room, more quickly in the warm sun- shine, and very quickly before a hot fire. Evaporation is due to molecular motion. Some of the molecules of a liquid, in their irregular and unequal motion, reach the surface with a sufficient upward velocity to carry them into the space above, out of the range of cohesion, where they exist as a gas or vapor. With a rise of tempera- ture the velocity of the molecules is increased, and more of them are able to escape from the liquid in a given time. Evaporation takes place on the largest scale from the surface of the oceans, lakes, ponds, and streams, and from damp soil ; in consequence of which the air always contains a greater or less amount of water vapor. 222. Saturated Vapor. A liquid kept in an open vessel continues to evaporate until, in the course of time, it en- tirely disappears. In a closed vessel this does not happen. A small quantity of such a volatile liquid as ether can be kept indefinitely in a large bottle, if only it is tightly stop- pered. Evaporation takes place for a time into the closed space above the liquid, and then apparently ceases. The vapor is then as dense as it will become at the existing temperature, however much or little of the liquid may still remain, and however long it may stand. Any vapor in 262 HEAT this condition is called a saturated vapor. When the con- dition of a vapor is such that further evaporation of the liquid into the same space is possible, it is said to be unsat- urated or superheated. Saturation is explained by the kinetic theory of matter as follows : Whenever a liquid and its vapor are in contact, there is a constant exchange of molecules between them. Molecules of the liquid, breaking away from the surface, become a part of the vapor; and molecules of the vapor, striking the surface of the liquid, are captured and become a part of the liquid again. As evaporation into a closed space continues, the density of the vapor increases, and an increasing number of its molecules return to the liquid in a given time. Finally, the condition is reached in which there is an equal exchange of molecules between the liquid and its vapor. The two are then in equilibrium with each other, and the vapor is saturated. 223. Vapor Pressure. A vapor, like any gas, exerts a certain pressure which, at a constant temperature, is proportional to its density; but the behavior of vapors differs from that of other gases in important respects, as shown by the following experiment. A simple barometer is set up, and a drop or two of ether introduced into the bottom of it, by means of a curved dropping tube, care being taken to let no air enter (Fig. 190). The ether evaporates as it rises through the column of mercury, and the pressure that it exerts as a vapor causes a depression of the mercury column. When more ether is introduced, it rises without evaporating and remains as a liquid above the mercury. The space above the liquid is now filled with its saturated vapor. The pres- sure exerted by the vapor, expressed in centimeters of VAPORIZATION AND CONDENSATION 263 mercury, is measured by the difference between the present height of the mercury column and its height before the ether was introduced. When the tube is inclined, the space occupied by the vapor becomes smaller; but the vertical height of the mercury column remains the same as before, showing that the vapor pressure is unchanged. This holds true as the tube is further inclined, until the space above the liquid entirely disappears (provided all air has been excluded from the tube). In inclining the tube the vapor is evidently not compressed and made denser; for in that case it would exert an increased pressure. The fact is that the vapor condenses as rapidly as its volume is diminished, and the density of the remainder is unchanged. This behavior is char- acteristic of all saturated vapors. The density and pressure of a saturated vapor are both at a maximum for the existing temperature. Any at- tempt at compression only results in condensa- tion, which takes place in exact proportion to the decrease of volume. (Contrast this behavior with that of "perfect" gases, as expressed in ^ , , FIG. 190. Boyle s law.) When the tube in the above experiment is returned to the vertical position and the ether warmed by clasping the tube in the hands, the mercury descends further, show- ing an increase of vapor pressure with a rise of temperature. This is partly due to the heating of the vapor already in the tube, but chiefly to the evaporation of more ether. The density and pressure of a saturated vapor increase with the temperature. If there were no more ether in the tube to evaporate, heat would cause expansion of the existing vapor, and it would become less dense, and unsaturated. 264 HEAT Similar results are obtained throughout when alcohol is substituted for ether in the experiment; but they are all on a greatly reduced scale, for the maximum pressure of alcohol vapor is much less than that of ether at the same temperature. With water the results are very slight. At 20 the maximum vapor pressure of ether is 43.28 cm. (of mercury), that of alcohol is 4.45 cm., and that of water 1.74 cm. At the same temperature the maximum pressures of different vapors are unequal. The behavior of unsaturated or superheated vapors is approximately like that of perfect gases, as expressed in the laws of Boyle, Gay-Lussac, and Charles. 224. Mixtures of Gases and Vapors. In a vacuum evaporation is very rapid, and the space is almost immedi- ately filled with the saturated vapor of the liquid. In the presence of air or any other gas, evaporation takes place much more slowly; but it does not cease until any inclosed space above the liquid contains as much of the vapor as it would if the other gas were not present, (i) The quantity of vapor which saturates a given space is the same, at the same temperature, whether this space contains a gas or is a vacuum. (2) The pressure exerted by a mixture of one or more gases and vapors is equal to the sum of the pressures which each would exert if it occupied the same space alone. The most familiar example of such a mixture is the atmos- phere. The statements in italics are known as Dalton's laws. 225. Loss of Heat in Evaporation. Common observa- tion teaches that evaporation is a cooling process. The skin is cooled by the evaporation of water or perspiration from it. This is especially noticeable in a draft, which causes more rapid evaporation by carrying the vapor VAPORIZATION AND CONDENSATION 265 away as fast as it is formed. Damp earth is considerably cooler than dry earth on a dry, hot day, when evaporation is rapid. The very rapid evaporation of ether and other highly volatile liquids causes much greater cooling. The heat lost during evaporation is required to produce the change of state, just as heat is required to melt or dissolve a solid. In evaporation, as in solution, this heat is taken from the nearest available source first the liquid itself, then adjacent bodies. The nature of the work done by heat during vaporization is discussed in Art. 234. 226. Conditions Affecting the Rate of Evaporation. The rate of evaporation of a liquid is affected by various conditions, as follows: Temperature. The rate of evaporation increases with a rise of temperature (Art. 221). Density of the Vapor. The evaporation of a liquid decreases as the space about it approaches saturation by its own vapor, and ceases when that space is saturated (Art. 222). Presence of Air or other Gas. The rate of evaporation increases as the density of the air or other gas surrounding the liquid is diminished. It takes place most rapidly in a vacuum (Art. 224). This may be readily shown as follows: A small beaker is partly filled with ether and left exposed to the air of the room; and a second beaker con- taining ether is placed under the receiver of an air pump. The ether in the exhausted receiver evaporates so rapidly that its temperature falls several degrees below zero in a few minutes, while the ether in the other beaker is cooled comparatively little by the much slower evaporation into the air. The beaker under the receiver should stand on wood or cork to prevent conduction from the metal plate 266 HEAT of the pump. If the support of the beaker is wet with a few drops of water, the beaker will be frozen to it. Changes of barometric pressure are not sufficiently great to affect the rate of evaporation in the open air to any appreciable extent. Air Currents. The rate of evaporation in air in- creases with a more rapid change of air about the liquid. Currents of air carry the vapor away from the space about the liquid; and the stronger the currents are the farther will this space be from saturation. It is a familiar fact that moisture quickly disappears in a dry, hot wind. Area of Free Surface. Evaporation increases with an increase of the free surface of the liquid. Water evapo- rates slowly from a cup, more rapidly from a broad and shallow dish, and still more rapidly from wet clothes hung on a line. A small quantity of ether in a beaker is quickly cooled below zero by a current of air forced through it from a small bellows (Fig. 191). The ether evaporates into the bubbles of air as they rise through the liquid. The area of the evaporating surface is thus greatly increased; and there is a constant renewal of unsaturated space, into which evaporation can take place. Nature of the Liquid. Under the same conditions the rate of evaporation differs with different liquids. 227. Water Vapor in the Atmosphere. The atmos- phere is a mixture of several gases, principally nitrogen and oxygen. The only other constituents of importance are carbon dioxide and water vapor. All of the constitu- ents of the atmosphere except water vapor are practically constant in amount; the water vapor varies from an inap- VAPORIZATION AND CONDENSATION 267 preciable fraction to about 2 % of the whole, the average amount being not far from i %. The condition of the water vapor in the air with respect to saturation is not in the least affected by the presence of the other gases (Dal ton's first law), and depends only upon its own density and temperature (which, of course, is the temperature of the air) ; yet common forms of expres- sion seem to imply that the presence and condition of the vapor are due to some action of the air. Thus when the water vapor in the air is saturated, we say that the air is saturated or that the air has all the moisture it can hold; although, strictly speaking, it is the space that has all the water vapor it can hold at the existing temperature. There is perhaps no objection to the use of such expressions when their true meaning is understood. The air is generally not saturated; it is evidently not saturated whenever further evaporation can take place. Unsaturated air may become saturated (i) by further evaporation, (2) by a fall of temperature, or (3) by the two processes combined. Saturation results from a sufficient fall of temperature because the density of a saturated vapor is less at lower temperatures (Art. 223, third paragraph). Consequently when the quantity of water vapor in the air is less than that required for saturation at the existing temperature, it is sufficient to cause saturation at a definite lower temperature, called the dew-point. 228. The Dew-point. The temperature at which the water vapor present in the air at any time would be satu- ratecl is called the dew-pnint of fbg air a,t that H** When any body of air is cooled to its dew-point, condensation of water vapor begins, and it continues as long as the tempera- ture continues to fall. The moisture that gathers on the 268 HEAT outside of a pitcher of ice- water is a familiar example. This moisture comes from the surrounding air, which is cooled by contact with the pitcher. The temperature usually falls several degrees below the dew-point, causing a considerable deposit which runs down the sides. (What error is implied in calling this phenomenon "sweating"?) The dew-point can be determined experimentally by slowly cooling the contents of a vessel till the first trace of moisture appears on its surface. The temperature of the vessel and contents when this occurs is the dew-point of the surrounding air. The vessel should be one upon which a thin film of moisture can easily be seen, such as a nickel-plated calorimeter. It can be cooled with water to which ice or ammonium chloride is added, or ice and salt if the dew-point is below zero, or with ether, cooled by evaporation. .The dew-point varies between wide limits. In winter it is often many degrees below zero. It is always consider- ably below the temperature of the air when the air is not noticeably damp, and is as high as the temperature of the air only when the air is saturated. 229. Humidity. The ratio of the amount of water vapor present in the air at any time to the whole amount that it would contain if saturated at the existing tempera- tur*e is called the relative humidity or, simply, the humidity of the air at the time. This ratio is usually expressed as a percentage. Thus the humidity is 75 % when the air contains three fourths as much water vapor as it would if it were saturated at the same temperature. Humidity is simply the measure of the dampness of the air. It is high when the air is damp and low when it is dry. The humidity of the air varies not only with the amount VAPORIZATION AND CONDENSATION 269 of water vapor in it, but also with its temperature. As air cools the humidity rises, until at the dew-point it is 100 %. Conversely a rise of temperature lowers the humidity; for the quantity of vapor actually present is not changed, while the capacity of the air for vapor is increased. Thus damp air in a cold room is dried by heating it, although there is no less water vapor in the room after the heating than there was before. The temperature, pressure, and humidity of the air and the direction and velocity of the wind are the principal atmospheric conditions which determine the weather. These conditions are regularly measured and recorded at all stations of the Weather Bureau, and the weather fore- cast is based upon them. The humidity can be determined in various ways, by means of instruments called hygrom- eters (Greek hygros, moist, and metron, measure). A dew-point hygrometer is merely an instrument for the con- venient determination of the dew-point. Knowing the temperature of the air and the dew-point, the humidity is computed with the aid of a table of densities of satu- rated water vapor. Suppose, for example, that the tem- perature is 30 and the dew-point 20. There is then enough moisture in the air to saturate it at 20; and it is found from the table that a cubic meter of saturated air at this temperature contains 14.3 g. of water vapor, while, if saturated at 30, it would contain 26.2 g. The humidity is therefore 14.3 -f- 26.2 or 55% nearly. i The wet-and-dry-bulb hygrometer -(Fig. 192) is more convenient, and hence is more generally used. It consists of two thermometers, the bulb of one of which is kept moist by means of a cotton wick surrounding it and dipping into a vessel of water. The constant evaporation about the bulb of this thermometer lowers its tempera- ture more or less, according to the rate of evaporation; and this 270 HEAT depends upon the humidity and temperature of the air. In saturated air the readings of the thermometers are equal. (Why?) Before reading the wet-bulb thermometer, it is whirled rapidly through the air, or a current of air is driven over the bulb. (Why?) The hu- midity corresponding to the observed temperatures of the wet and the dry bulbs is found directly from a table. The greater or less humidity of the air affects our bodily comfort through its influence on the evaporation of mois- ture from the skin and clothing. Generally speaking, a medium humidity is most agreeable, for the skin is then neither too moist nor too dry. A very damp atmosphere adds greatly to the discomfort of cold weather, and in hot weather it endangers health and even life itself. The heat of the body is generated by chemical action, which is constantly going on within it. In winter this heat is retained by heavy clothing of poorly conducting materials; in summer we wear light garments of cotton and linen to permit its ready escape. But on the hottest days this alone is insufficient. In- deed, when the temperature of the air is equal to or higher than that of the body (98 F.), loss of heat by conduction ceases, FIG. 1 92. an d nature provides a substitute in abundant Hygrometer. . . , f perspiration, especially during active exercise, when the generation of heat in the body is most rapid. But perspiration is of no avail unless it evaporates, for the cooling effect is due to the heat taken from the body in evaporation. Hence hot weather is especially oppressive and dangerous when evaporation is retarded by excessive humidity. In the very dry atmosphere of deserts there is comparatively little danger of sunstroke even at temperatures above 100 F. VAPORIZATION AND CONDENSATION 271 230. Condensation of Water Vapor in the Atmosphere. - Water vapor is always invisible. The visible forms of moisture in the atmosphere, such as fog, clouds, and the so-called " steam" near the spout of a kettle in which water is boiling, consist of minute particles of liquid water, due to the condensation that accompanies a fall of tempera- ture after the dew-point is reached. Dew, frost, rain, sleet, hail, and snow are the various forms in which the water vapor in the air is condensed and precipitated. Dew is formed by condensation of vapor from the air immediately surrounding the bodies on which it appears. This occurs when air that is saturated, or nearly so, is cooled below the dew-point by con- tact with colder objects. Dew forms most abundantly on the coldest objects, which are in general the best radiators and the poorest con- ductors; for such objects lose heat by radiation more rapidly than they receive it from the earth by conduction, until they become several degrees colder than the air. Grass, leaves, and boards are good ex- amples. Dew forms only at night, and most abundantly toward morning, when, by cooling, the air has become nearly saturated. It forms only on calm, clear nights; for on clear nights cooling is most rapid (Art. 200), and it is only on calm nights that any portion of the air remains long enough in contact with cold surfaces to be cooled to the dew-point. Frost. When the dew-point is below zero, condensation takes place in the form of frost, under conditions otherwise the same as are necessary for the formation of dew. The water vapor then crys- tallizes in the solid state as it condenses, without passing through the intermediate state of a liquid. Clouds and Fog. The condensation of vapor in the air near the earth produces a fog; at higher altitudes it forms a cloud. A fog consists of minute globules of liquid water. A cloud is composed of liquid particles, like a fog, or of ice particles, depending upon the temperature. The cooling of the air by which clouds are formed may be brought about by radiation, by the contact and partial mixing of a vapor-laden current with a current of colder air, or by the expan- sion of ascending currents. The last-named process requires a few words of explanation. Gases are always cooled by expansion (Art. 272 HEAT 235); and ascending currents of air expand, in consequence of the diminished pressure at higher altitudes. When moist air is thus cooled below the dew-point, towering masses of cloud, known as " thunder-heads," are formed. The cooling of air by expansion, and the resulting condensation of vapor, are beautifully shown in the following experiment. A large flask containing a little warm water is tightly closed with a rubber stopper, and vigorously shaken to saturate the inclosed air. It is then placed under the receiver of an air pump. On exhausting the receiver, the stopper is driven out by the pressure within the flask; and the flask is instantly filled with a dense fog, formed by the sudden expansion and cooling of the saturated air. Rain, Sleet, Snow, and Hail. As the particles of a cloud grow by further condensation or by uniting with one another, they may become too large to be sustained in the air, and they then fall as rain. When rain freezes in falling through a layer of colder air, it is called sleet. Snow is formed by the condensation of vapor in the atmosphere at temperatures below zero. The vapor passes directly into the solid state, as in the formation of frost. Hailstones are masses of ice, or of ice and snow, which are sometimes an inch or two in diameter. They are often made up of several layers or shells of ice and snow, showing that they have passed through a variety of atmospheric conditions. Hailstones of large size are formed only in violent storms; but the exact manner of their formation is not known. PROBLEMS 1. Give two reasons why a liquid evaporates more rapidly in a wide and shallow vessel than it does in an unstoppered bottle. 2. (a) Why does the breath often form a visible cloud on a cold day? (b) Is it more likely to do so when the humidity of the air is high or low? 3. The moisture from the spout of a kettle of boiling water is invisible for a few inches beyond the spout then for some distance farther it forms a cloud; still farther it is all invisible again. Account for these facts. 4. Why does frost form on the inside of a window-pane but not on the outside? 5. Is frost frozen dew? 6. Why does frost form on board walks when it does not on cement walks? VAPORIZATION AND CONDENSATION 273 231. Boiling. When fresh water is heated, dissolved air is given off in the form of minute bubbles, which begin to form on the sides and bottom of the vessel as soon as the water has become slightly warm. These bubbles often rise to the surface in large numbers, where the air that they contain escapes. After the water has become hot, much larger bubbles begin to form at the bottom where the heat is applied. These are bubbles of steam, or water vapor. They rise rapidly, but disappear before reaching the surface, being condensed by the cooler water near the top. It is the collapse of these first bubbles of steam that causes the singing of a kettle of water shortly before it begins to boil. As the temperature of the water approaches 1 00 C., the bubbles rise higher, until finally they burst at the surface, throwing the water violently about. The water is then boiling. In general, any liquid is said to boil when bubbles of its vapor form within it, rise to the surface, and break. The temperature of a liquid, when boiling in an open vessel, varies only slightly, if at all, however slow or rapid the boiling; and the temperature of the escaping vapor is constant. Heat energy is required to produce the change of state, and a more abundant supply of heat merely causes the change to take place more rapidly. Boiling is in this respect like melting. A bubble of vapor within a liquid sustains a pressure which is made up of the gravity pressure of the liquid and the transmitted pressure of the air, or other gas, upon its free surface. If the pressure of the saturated vapor of the liquid is less than this at the existing temperature, internal vaporization can not take place, and if bubbles of the vapor are present they can not withstand the pres- sure, and are immediately condensed. This is what hap- 274 HEAT pens, as we have seen, when the first bubbles of steam rise into cooler water at the top, shortly before water begins to boil. At small depths in a liquid the gravity pressure is small, compared with the pressure of the air, and may be disregarded. The boiling point of a liquid is therefore defined as the temperature at which the pressure of its satu- rated vapor is equal to the pressure upon the free surface of the liquid, a pressure of one atmosphere being understood unless otherwise stated. BOILING POINTS UNDER A PRESSURE OF ONE ATMOSPHERE Ether 34.6 Turpentine 160 Chloroform 61.2 Glycerin 290 Alcohol . . 78.4 Mercury 357 Water . . 100. Sulphur 445 232. Effect of Pressure upon the Boiling Point. An increase of pressure upon the surface of a liquid raises its boiling point, for the vapor bubbles within the liquid must exert the increased pressure, and this is possible only at a higher temperature. Thus in engine boilers the tempera- ture of the boiling water and the steam steadily rises during the process of " getting up steam." When the steam gage registers a pressure of 150 Ib. (per square inch) the tempera- ture is 185 C. (See table FIG. 193 Franklin's Experiment. ^^ Conversely a dec rease of pressure lowers the boiling point. This is readily shown by either of the following experiments, (i) An VAPORIZATION AND CONDENSATION 275 open flask containing water at 50 to 60 is placed under the receiver of an air pump and the air exhausted. When the pressure is sufficiently reduced, the water boils rapidly. (2) Water is boiled in a round-bottomed flask until the air is expelled by the steam. The flask is then quickly closed with a rubber stopper and inverted (Fig. 193). When cold water is poured over the flask, the water within it boils violently. This may be repeated till the water in the flask is barely warm. The cold water condenses some of the vapor, thus decreasing its pressure upon the liquid. Owing to the diminished pressure of the atmosphere at high altitudes, the boiling point of a liquid is consider- ably lower upon a mountain than it is near sea-level. On the summit of Mont Blanc water boils at 84. The following table gives the pressure of saturated water vapor, and hence also the pressure under which water boils, at various temperatures. TEMPERATURE PRESSURE IN CM. OF MERCURY TEMPERATURE PRESSURE IN ATMOSPHERES .46 100 1. 00 20 1-74 1 20 1.96 40 5-49 140 3-58 60 14.9 1 60 6.12 80 35-5 1 80 9.92 100 76.0 200 15-35 233. Distillation. A liquid can be separated from non- volatile impurities by boiling it in a closed vessel, and condensing the vapor as it passes off through a tube con- nected with the vessel. The process is called distillation, and the apparatus a still. The vapor is condensed by in- 276 HEAT closing a portion of the tube through which it passes within a larger tube or a vessel, in which it is surrounded by a continuous supply of cold water (Fig. 194). The process may be illustrated by distilling pure water from a solu- tion of some substance whose presence is shown by its color, e.g. copper sulphate or potassium permanganate. FIG. 194. Distillation. Two or more liquids whose boiling points differ by several degrees can be separated from one another by distillation. When such a mixture is slowly boiled, the vapor that passes off contains a much higher percentage of the more volatile constituent than the original mixture does. Some of the less volatile liquid also passes off, and complete separation can be effected only by repeated distillation. This process is known as fractional distillation. It is employed on a large scale in separating the constituents of crude petroleum, such as gasoline, naphtha, benzin, kerosene, lubricating oils, paraffin, etc. 234. Heat of Vaporization. We have seen that heat energy disappears during vaporization, whether by evap- oration at the surface (Art. 225) or within the liquid (Art. 231). This energy is recovered as heat when a vapor condenses, just as the heat of fusion is recovered when a VAPORIZATION AND CONDENSATION 277 liquid freezes. In the steam-heating system, buildings are warmed by the condensation of steam in the radia- tors. Each gram of steam, in condensing to water at 100, gives out more than five times as much heat as a gram of water does in cooling from the boiling to the freezing point. The heat generated by the condensation of water vapor in the atmosphere is the principal cause of the milder temperatures which herald the approach of rain or snow in winter. What becomes of the heat energy required to vaporize a liquid? In the first place, work must be done against atmospheric pressure in providing the additional space which the substance occupies as a vapor. Imagine a gram of water to be placed in a long tube", closed at one endjind having a cross-section of i sq. cm. .Suppose- further that the tube is fitted with an air-tight piston, which moves without fric- tion (Fig. 195). If the water is heated, it will begin to .vaporize at 100, and in doing so will push, the piston up. When vaporization is complete the piston will have been moved upwarcl^a distance of 1660 cm. against the pressure of the atmosphere. At normal pressure this amounts to 1033.3 g.; and the steam must evidently exert an equal force against the FlG - piston. Hence in making room for itself the gram of steam does 1033.3 X 1660 = 1,715,278 g.-cm. of work. The energy thus expejided is a part of the heat energy reqinmLjo vaporize the water; and, as we shall see later, it is the equivalent of 41 calories. The same ^amount of work must be done against atmospheric pressure when a gram of water is boiled away in the open air. The sup- posed tube and piston are merely an aid in explaining the process. 2 7 8 HEAT Experiment shows, however, that 537 calories are actu- ally required to vaporize a gram of water at its boiling point. The additional 496 calories are required to do the work of separating the molecules against their mutual attractions, i.e. in overcoming cohesion^ This internal work^is stored in Jtiie vapor as molecular potential energy. The work done against atmospheric pressure is called external work. Both are fully recovered as heat when the vapor condenses. of any liquid at its boiling point is called the heat ofjvapor- The heat of vaporization of water' is 537 calories, as stated above, and is greater than that of any other liquid. For ammonia it is 295 calories; for alcohol, 209 calories; for ether, 90 calories; and for mer- cury, 62 calories. 235. Heat and Work in the Compression and Expan- sion of Gases. A gas does external work in expanding against pressure, just as a vapor does in mak- ing room for itself. In doing this work, the gas loses an equivalent amount of heat energy, and is cooled, unless it receives an equal supply of heat from without. This cooling effect is shown by the condensation of moisture when saturated air expands (see Art. 230, under Clouds and Fog). It can also be shown by inserting the bulb of a ther- mometer into a short rubber tube through FIG. 196. Fire which a jet of air is escaping under consider- able pressure from a tank. Conversely, a gas is heated by compression, the mechan- ical energy expended upon the gas being transformed into heat within it. This is shown in the heating of a bicycle VAPORIZATION AND CONDENSATION 279 pump when in vigorous use, for the very considerable rise of temperature is due to the heat received from the compressed air. When air is suddenly compressed as much as possible in a fire syringe (Fig. 196), it becomes hot enough to ignite a small piece of tinder, at the bottom of the cylin- der, or vapor of carbon disulphide, mixed with the air. The burning of the vapor is shown by a flash of light. 236. The Liquefaction of Gases. Critical Temperature. - Within recent years all gases have been liquefied, and all but helium reduced to the solid state. Gases become vapors before liquefaction takes place, and they then be- have like other vapors. A gas can therefore be liquefied (i) by cooling, at atmospheric pressure, to a definite tem- perature, which varies with different gases, (2) in some cases, by compression at ordinary temperatures, and (3) by cooling and compression together. Sulphur dioxide (the gas formed by burning sulphur) is liquefied under atmospheric pressure by a freezing mixture of ice and salt, its boiling point being -10.5. Under a pressure of 3 atmospheres, it liquefies at 15. Carbon dioxide liquefies 'at -80 under a pressure of i atmosphere, and at 15 under a pressure of 52 atmospheres. For every gas there is a certain temperature, called the critical temperature, above which it cannot be liquefied , however great the pressure (see table below). The further a gas is cooled below its critical temperature the less is the pressure required for its liquefaction. In the manufacture of liquid air, the air is compressed to about 200 atmospheres, usually in two or more stages to avoid excessive heating. After each compression the air is cooled by passing it through coils surrounded by water. It is then led to the liquefier, in which it flows downward through the inner tube of a double coil. At the bottom of this coil the air escapes through a small opening 280 HEAT into a closed space, and in expanding becomes very cold. It then passes upward through the outer tube of the coil, in which it surrounds and cools the air in the high-pressure tube. After a time the tempera- ture falls so low that some of the compressed air liquefies as it issues from the opening. In liquefying hydrogen the gas is cooled with liquid air, while it is under great pressure. The further cooling due to sudden expansion causes it to liquefy. In 1908, Professor Onnes, of the University of Leyden, succeeded in liquefying 60 ccm. of helium at 5 absolute, 70 liters or more of liquid air and 20 liters of liquid hydrogen being consumed in reducing the helium to this temperature. SUBSTANCE CRITICAL TEMPERATURE BOILING POINT FREEZING POINT CENT. ABS. Helium -2 4 2C. -146 140 -119 + 31 130 156 365 -268 -252 -195 -191 -184 - 79 -33-7 -10.5 100 5 21 78 82 8 9 194 239-3 262.5 373 -2 5 8C. 210 -227 - 6 5 - 77 - 76 o Hydrogen Nitrogen Air Oxygen Carbon dioxide Ammonia Sulphur dioxide Water 237. Ice Manufacturing and Cold Storage. The rapid evapora- tion of a highly volatile liquid cools not only the liquid, but surround- ing bodies as well (Arts. 225 and 226). A little water in a test tube, placed in a beaker of ether, is easily frozen by evaporating the ether as shown in Fig. 191. This principle is utilized in manufacturing ice on a commercial scale. For practical use the freezing agent must be a liquid whose boiling point is below zero, but which does not require an excessive pressure to liquefy it at ordinary temperatures. Carbon dioxide, ammonia gas, and sulphur dioxide answer these requirements in different degree. (See table, Art. 236.) Sulphur dioxide has been tried, but is inferior to either of the others. Carbon dioxide is sometimes used-, although a pressure of 800 to 900 pounds per square inch is required to liquefy it at ordinary temperatures. Ammonia gas is generally preferred. VAPORIZATION AND CONDENSATION 281 The ammonia sold at drug stores is ammonia water, i.e. water in which ammonia gas is dissolved. When the water is heated, the gas (NHs) is driven off in large volumes. The pure ammonia exists as a liquid under atmospheric pressure only at or below ~33-7 C. It liquefies at 20 C. when subjected to a pressure of 124 Ib. per square inch, or about 8.5 atmospheres. Ice-making plants differ from one another in more or less im- portant details. A simplified diagram of one system is shown in Fig. 197. The ammonia circulates through a system of pipes which begins and ends at a storage tank (not shown in the figure) contain- ing a supply of liquefied ammonia at the temperature of the room and a pressure of about 10 atmospheres. From this tank the ammonia Gold Water trickling over the ammonia pipes to condense the compressed gas Expansion valve ae pump FIG. 197. Artificial Ice and Cold Storage Plant. flows to the expansion or refrigerating coils, immersed in a tank of strong brine. A regulating valve in the pipe between the tank and the coils permits only a limited flow; and the ammonia, on entering the coils, rapidly vaporizes under the low pressure which is always maintained in them. As the evaporation takes place many degrees below zero and requires a continuous supply of heat, the surrounding brine is also cooled below zero, and large cans of fresh water immersed in it are frozen in from 24 to 36 hours. The ammonia vapor is pumped from the refrigerating coils as fast as it forms, and is driven under a pressure of about 10 atmospheres through a set of condensing coils. The compression of the ammonia raises it to a high temperature, and it must be cooled before it will liquefy. This is accomplished by allowing water to trickle over the condensing coils. From the condenser the liquid ammonia is carried by gravity to the storage tank, to be used again. 282 HEAT Briefly stated, the purpose of the ammonia is to absorb heat at a low temperature from the brine and to give it out at a high tempera- ture in the condenser. This requires an expenditure of energy, derived from the engine which runs the pump. The brine used is sometimes a strong solution of table salt, but as this is liable to freeze from overcooling while a brine of calcium chloride is not, the latter is preferable. Artificial cooling or refrigeration is employed on a large scale for the purpose of maintaining low temperatures in cold-storage ware- houses, in which are kept dairy products, eggs, meat, fish, fruits, etc. Ammonia is used as the refrigerating agent in either of two ways. In the brine system the ammonia cools a tank of brine, as in ice-making, and the brine is pumped through coils of pipe, placed on the sides or ceiling of the room to be cooled. The circulating brine continually absorbs heat from the cold room and gives it to the ammonia, and the ammonia carries it to the condenser. "In the direct-expansion system the ammonia is admitted directly into the coils in the rooms to be refrigerated. The heat of the cold room is taken up by the ammonia directly, and the intermediate agent, brine, is not employed." PROBLEMS 1. Is rain water distilled water? Is it perfectly pure? 2. Water kept in porous earthenware jars in warm, dry weather remains several degrees below the temperature of the air. Explain. 3. At what depth in water, exposed to atmospheric pressure, would the boiling point be 120? 4. It takes more heat to raise the temperature of a mass of gas a given number of degrees, under constant pressure, than it does at constant volume. Why? 5. What quantity of heat is required to convert 850 g. of ice at 20 into steam at ioo? 6. How much heat is given out by 500 g. of steam at 100 in condensing and cooling to water at 30? 7. A room 4 m. by 5 m. and 3 m. high is warmed by a steam heater. Assuming no loss, what weight of steam must be condensed in the heater to warm the room from 10 C. to 18 C.? (Density of the air = 1.25 g. per cu. dm.; specific heat of air = .237.) HEATING AND VENTILATION OF BUILDINGS 283 8. The nitrogen and oxygen of liquid air evaporate at unequal rates. Which evaporates more rapidly, and why? 9. The temperature of liquid air does not rise above 184 in an open vessel. Why not? How can it be made warmer? What is the highest temperature to which it can possibly be raised as a liquid? 10. Why are icebergs often enveloped in fog? IX. HEATING AND VENTILATION OF BUILDINGS 238. Temperature and Ventilating Requirements in the Home. It is very generally agreed that an indoor temperature of 66 F. during the winter season is best for persons in health. Some prefer a temperature as low as 64 and others as high as 70. Generally speaking, these may be regarded as reasonable limits. No system of ventilation will keep the air in an occupied room as pure as outdoor air; but it is greatly in the interests of health and comfort to keep it as pure as circumstances will permit. The importance of this is more generally understood now than ever before. "We are at last coming to the conclusion that we might better pay the coal dealer for the energy to produce heat, ventilation, and comfort than to pay our physician for doctoring the ills resulting from our carelessness." Competent authorities are agreed that the fresh-air supply of rooms should not be less than 1800 cu. ft. per hour for each occupant. For a living room of moderate size, say 12 by 16 ft., this would require a complete change of air every hour, with only one person in the room, or as many complete changes per hour as the number of occu- pants. In rooms having two outside walls, the leakage through the cracks of doors and windows would ordina- rily amount to one or two changes of air per hour, varying with the number of windows, the direction and velocity of the wind, the difference between inside and outside tern- 284 HEAT peratures, etc. Clearly when a room of this size is occu- pied by more than two persons, the leakage should not be depended upon as the sole means of ventilation. The ventilation of bedrooms is satisfactorily provided for by means of open windows. They should be wide open except in freezing weather; for the purer the air the better, and a low temperature at night is in no wise harmful. 239. Fireplaces and Stoves. An open fire in a fireplace is a very effective and agreeable means of ventilating a living room. The fire maintains a steady outflow of air by way of the chimney, and this quickens the inflow through the cracks of doors and windows. As a means of heating a room in really cold weather, a fireplace is very inefficient and wasteful. From 80 to 90% of the heat passes up the chimney; and the small remainder is of little service except directly in front of the fire, where the radiation is most intense. More- over, the result is at best a compromise between two uncomfortable extremes, a torrid heat on one side and an arctic chill on the other; for heating by radiation is distinctly a one-sided affair. Owing to these defects of the open fire and the increasing cost of fuel, the fire- place has fallen into disfavor as the principal means of heating, even in regions where the cold of winter is not severe; but it justifies itself as an auxiliary source of heat and a means of ventilation, aside from its attractiveness. The coal or wood stove is an efficient and sanitary means of heat- ing single rooms, and offers the only practical solution of the heating problem for small country and town houses. It diffuses heat both by radiation and convection, and provides a fair amount of ventilation by the action of the draft, which withdraws air from the room and carries it off with the products of combustion. Gas and oil stoves, from which the products of combustion pass off into the room, are highly objectionable, even for occasional use. They are not an aid to ventilation, and consume as much oxygen as a dozen or more persons. Where necessity compels their use, a window should be left partly open for ventilation, and the air should be com- pletely renewed at frequent intervals by throwing doors and windows wide open. HEATING AND VENTILATION OF BUILDINGS 285 240. Furnace Heating. In the hot-air system of heating, the heater or furnace is located in the basement, and is inclosed in a small brick chamber or a casing of sheet iron. A constant supply of out- door air is brought to the lower part of this chamber through a large pipe or duct, and, in passing over the hot surfaces of the furnace, is heated. It then rises through the warm-air pipes at the top of the chamber, and is discharged through the registers into the rooms above. In dwellings and other buildings of moderate size, the warm-air registers are commonly placed in the floor, or in the walls near the floor, and cracks in doors and windows are depended upon to provide the necessary outlet (Fig. 198). The circulation is maintained as a natural draft or convection cur- rent, due to the unequal densities of hot and cold air. This meets all re- quirements except in windy weather, when there is very likely to be trouble in heating rooms on the windward side of the house; for the wind blows in through the FIG. 198. Hot-Air Heating System. window crevices, and retards if it does not entirely prevent the inflow of warm air from the registers. In schoolhouses and other large buildings the circulation is main- tained as a forced draft by means of a centrifugal fan or blower, placed in the cold-air duct. A ventilating flue leading from each room provides an outlet which is not affected by the wind. Within the room the air circulates by convection, the direction and extent of the currents being determined by the position of the inlet and the outlet. The best results are obtained with the inlet near the top and the outlet at the bottom on the same side of the room. The hot-air system of heating is a great improvement over the use of stoves. It is comparatively inexpensive, and, on the whole, is satisfactory in mild climates. One of its great advantages is that it insures a constant supply of pure air. 286 HEAT 241. Hot-water and Steam Heating. Hot-water heating is, on the whole, most satisfactory for residences and other buildings of moderate size, and has been developed to a high degree of perfection. The more important parts of the apparatus are the heater, located in the basement, the radiators in the various rooms to be warmed, and a system of iron pipes, form- ing a continuous circuit from the heater to the radiators and back (Fig. 199). An expansion tank, placed above the highest radiators, provides for the expansion and contraction of the water. The pipe lead- ing from the heater connects with the top of it, and the return pipe with the bottom, as shown in the figure. By this arrange- ment a gravity circulation (convection cur- rent) is constantly maintained. Owing to the great specific heat of water, it is admirably adapted to serve as a medium for conveying heat from one place to another. A pound of water in cooling through 20 F. (which is about the usual fall of temperature in the radiator) gives out enough heat to raise the tem- perature of 1 1 oo cu. ft. of air one degree. Water is also well adapted to meet the varying demands of warmer and colder weather, as it can be heated to any desired temperature up to the boiling point. The hot-water system shown in the figure makes no provision for ventilation. One way of meeting this requirement is shown in Fig. 200. Outdoor air is admit- ted through a duct leading to the base of the radiator, whence it rises by convec- tion between the radiator sections, becoming warm before it escapes into the room. In steam heating the water compartment of the heater is only Fig. 199. Hot- Water Heating System. HEAT AND OTHER FORMS OF ENERGY 287 FIG. 200. Heating with Ventilation. partly filled, and the water is boiled. The steam forces its way through the pipes to the radiators, where it gives out heat and con- denses, returning as hot water to the boiler. A safety valve at the boiler guards against excessive pressure, taking the place of the expansion tank in hot-water heating. The same pipe can be made to serve both for the flow of the steam and the return of the hot water, the steam taking the upper side of the pipe and the water the lower. The weight of steam required to deliver a given amount of heat to the radiator is only about one fiftieth as great as the weight of water re- quired to accomplish the same result by the hot-water system; for each gram of steam in condensing gives out 537 calories, while a gram of hot water in cooling through 20 F. gives out only n calories. Steam radia- tors are about two thirds as large as hot- water radiators of the same heating capacity; for steam keeps the radiating surfaces at 212 F., while with water the average is about 170 F. The relative merits of steam and hot-water heating depend largely upon the requirements to be met in any given case. In general, hot- water heating is preferable for residences and other buildings of moderate size, and steam heating for large apartment houses, busi- ness buildings, churches, public halls, etc. X. HEAT AND OTHER FORMS OF ENERGY 242. The Mechanical Equivalent of Heat. If energy is never created or destroyed (Art. 153), a definite numer- ical relation must exist between any given amount of energy of one kind and the equivalent amount of energy of an- other kind. Thus a certain number of gram-centimeters of mechanical energy must always be required to generate one calorie of heat, in whatever way the transformation may be brought about. This number is called the mechan- ical equivalent of heat. It is of very great scientific and 288 HEAT practical importance, and has been carefully determined in various ways by different physicists. James Prescott Joule, of Manchester, England, was the first to establish the fact that such a relation exists and to determine its value. His numerous experiments extended over several years (1843 to ^So), and embraced several different methods. In each case a measured quantity of mechanical or electrical energy was converted into a measured quan- tity of heat. These notable experiments settled the long dispute concerning the nature of heat (Art. 184), and af- forded a sure foundation for the doctrine of the' conserva- tion of energy. The main features of the method which Joule preferred are as follows: By an arrangement of wheels and axles shown in Fig. 201, two heavy weights, e and e, turn a set of paddles in a calorimeter filled with water. Stationary projections, extending inward between the paddles from the sides of the calorimeter, prevent the water from revolving bodily with the paddles and keep it in violent agitation. The work done by the weights in falling is converted into heat by the internal friction of the water. This work is measured in gram-centimeters by the product of the force of gravity upon the weights and the distance through which they descend. The number of calories generated is computed from the weight of the water and the calorimeter, the specific heat of the calorimeter, and the rise of temperature. Hence, after making necessary allowances for conduction, radiation, etc., the equivalent of a certain number of calories is obtained in gram-centimeters of mechanical energy; from which the equivalent of one calorie is computed. The value now accepted for this equivalent, after re- peated determinations by different physicists, is 42,700 FIG. 201. Joule's Apparatus. HEAT AND OTHER FORMS OF ENERGY 289 g.-cm.; that is, 42,700 g.-cm of mechanical energy will generate one calorie when wholly transformed into heat, and vice versa. 243. Sources of Heat. All other forms of energy can become heat, directly or indirectly, by various transforma- tions, many of which have already received attention (Arts. 151, 152, 184). There is, as we have seen (Art. 154), a natural tendency for such transformations to take place ; but in most cases the heat generated is of no impor- tance as heat and is not available for doing useful work. The only useful sources of heat, on a large scale, are (i) solar radiation, (2) the chemical energy of fuels and foods, and (3) electrical energy. Electrical energy as a source of heat will be considered under the general subject of electricity. The heat ob- tained from fuels is generated during chemical change, in which the molecules of the burning substance break up, and their constituent parts (atoms) unite with oxygen from the air. New substances are thus formed, principally gases, whose molecules are generally less complex than those of the fuel and always possess less energy. 244. Solar Energy and its Transformations. The sun is the original source of practically all available energy, and this energy is directly or indirectly the cause of nearly all terrestrial phenomena. The only exceptions of any magnitude are the tides and phenom- ena due to the slow shrinking of the earth and to the heat of its interior, such as volcanic action, earthquakes, the formation of mountains, etc. Although the interior of the earth is intensely hot, the heat is conducted to the surface so slowly that its effect upon the temperature of the atmosphere is inappreciable. The energy of winds is directly traceable to the sun, for the winds are due to the unequal heating of different portions of the earth's surface. Water-power has the same ultimate source; for it is by means of solar radiation that water is evaporated from the oceans and carried, through the agency of winds, to the highest mountains. 2 go HEAT Plants take the materials necessary for their growth from the earth and air; but their energy is received directly from the sun. The leaves absorb carbon dioxide from the air, and, under the action of sunlight, separate it into its constituents (carbon and oxygen). The carbon unites with water sent up from the roots, forming starch; the oxygen is given off to the air again. From the starch and various earthy materials absorbed with water from the soil, the more complex substances are formed which are needed for the growth of the plant. These substances possess energy which comes from sunlight absorbed by the green coloring matter in the leaves (the chlorophyll). Radi- ant energy is thus transformed and stored as chemical potential energy in the substance of the plant itself. The energy of coal is also stored solar energy; for the coal-beds are the remains of immense forests that grew long before man appeared upon the earth. Animals, as already noted (Art. 152), derive the energy for all their bodily activities from their food, and hence originally from the \ sun, whether the food be of vegetable or of animal origin. 245. Amount of Solar Radiation. From a law of radiation it is known that the amount of energy radiated from any portion of the sun's surface is 46,000 times as great as that received by an equal area of the earth's surface in the same time. The energy received has been approximately measured; hence the rate at which the sun is giving out energy is known to the same approximation. This amounts to nearly 100,000 horse-power per square meter of the sun's surface, acting continuously. To maintain this rate of radiation by combus- tion " would require the hourly burning of a layer of the best anthra- cite coal from sixteen to twenty feet thick over the sun's entire surface, a ton for every square foot of surface, at least nine times as much as the most powerful blast furnace in existence. At that rate the sun, if made of solid coal, would not last 6000 years." * Of this enormous output of energy the earth receives only one part in twenty-two hundred million. 246. Source of the Sun's Energy. The source of the sun's energy is a question of the greatest scientific interest. A direct answer not being obtainable, various theories have been suggested, all of which recognize the principle of the conservation of energy. * Young's General Astronomy. HEAT AND OTHER FORMS OF ENERGY 291 The sun's heat can not be maintained by combustion, for in that case it would have been burned out long ago. " Nor can it be simply a heated body cooling down. Huge as it is, an easy calculation shows that its temperature must have fallen greatly within the last 2000 years by such a loss of heat, even if it had a specific heat higher than that of any known substance." * The theory generally accepted is known as Helmholtz's theory of solar contraction. This is that " the heat necessary to maintain the sun's radiation is principally supplied by the slow contraction of its bulk, aided, however, by the accompanying liquefaction and solidification of portions of its gaseous mass. When a body falls through a certain distance gradually, against resistance, and then comes to rest, the same total amount of heat is produced as if it had fallen freely, and been stopped instantly. If, then, the sun does contract, heat is necessarily produced by the process, and that in enormous quantity, since the attracting force at the solar surface is more than twenty-seven times as great as terrestrial gravity, and the contracting mass is immense. Helmholtz has shown that, under the most unfavorable conditions, a contraction in the sun's diameter of about two hundred and fifty feet a year would account for the whole annual output of heat." * At this rate, a period of 9000 years would be required for a total contraction sufficiently great to be detected by measurement with the best astronomical instruments. PROBLEMS 1. Compute the rise of temperature that would be obtained with Joule's apparatus under the following conditions: Mass of the weights used = 30 kg. Distance through which the weights descend = 25 m. Weight of water in the calorimeter = 2 kg. Weight of the copper calorimeter = i kg. 2. (a) A mass of iron weighing i kg. falls upon a stone from a height of 100 m. How much heat is generated, assuming that the energy is all transformed into heat? (b) If half of the heat is generated in the mass of iron, what is its rise of temperature? 3. From what height must a mass of iron fall in order that the heat gen- erated when it strikes shall be sufficient to raise its temperature one degree, assuming that the energy is all transformed into heat in the iron itself? 4. A lead bullet strikes a target with a velocity of 300 m. per second. Assuming that 90% of its energy is transformed into heat in itself, what is its rise of temperature? * Young's General Astronomy. 292 HEAT XI. HEAT ENGINES 247. Fundamental Principle. The transformation of heat into mechanical energy takes place on a large scale only in the vaporization of liquids (Art. 234) and in the expansion of gases and vapors (Art. 235). Any machine or device by means of which this transformation is con- trolled and made to do useful work is called a heat engine. In this general sense a cannon is a heat engine. When a cannon is fired, the chemical energy of the powder becomes heat energy in the gases generated by the explosion. At the instant of the explosion the gases are intensely hot; but, in expanding under great pressure, their heat is largely transformed into the kinetic energy of the projectile, and they issue from the mouth of the cannon reduced in temperature. Mechanical energy can be wholly converted into heat; but the reverse transformation is always partial. The best that any heat engine can do is to utilize the heat given out by steam or gases in expanding and cooling through a greater or less range of temperature. 248. The Steam Engine converts heat into mechanical energy through the expansive power of steam generated under pressure. A simplified diagram of its working parts is shown in Fig. 202. While the engine is running, the steam chest receives a constant supply of steam through a pipe leading from a boiler. This steam is admitted into the cylinder through narrow passages, called ports, which are alternately opened and closed by a slide-valve. In the position shown in the figure, the steam enters at the upper end of the cylinder, and pushes the piston and the piston- rod down. Meanwhile the slide-valve rises, shutting off the steam from the upper end of the cylinder, while the HEAT ENGINES 2 93 piston is still moving downward, and admitting it into the lower end just as the stroke is completed. The piston is now driven upward, while the slide-valve descends. At the end of this stroke the valve and the piston are again in the positions shown in the figure. As the valve moves to admit steam at either end of the cylinder, it connects the port at the other end with an exhaust pipe, through which the spent steam on that side of the piston escapes, under reduced pressure, into the open air or into a condens- ing chamber (Art. 250). The to-and-fro motion of the piston rod causes the connecting rod to turn a shaft. The shaft carries a flywheel to maintain uniform rotation (Art. 150), and is connected, generally by means of a belt, with the machinery to be operated by the engine. The shaft also carries an eccentric, which imparts the necessary to-and-fro mo- tion to the valve-rod. FIG. 202. Section of Steam Engine. 249. Efficiency Gained by an Early Cut-off. If the steam were admitted to the cylinder of an engine during the entire stroke of the piston, only a fraction of its avail- able energy would be used; for it would still be capable of expanding and doing work. The greatest efficiency under ordinary conditions is secured by adjusting the slide- valve so as to cut off the supply of steam at or near the first quarter of the stroke. The steam then in the cylinder works expansively to the end of the stroke; and, in expand- ing, its temperature and pressure fall rapidly. For example, 294 HEAT if the steam is admitted under a pressure of 6 atmospheres and discharged into the air, its temperature falls from about 160 C. to 100 (see Table, Art. 232). The heat lost by the steam in expanding is converted into mechanical en- ergy in driving the piston. (There is, of course, some loss of heat to the walls of the cylinder.) 250. Condensing Engines. The effective or resultant pressure against the piston of an engine is the difference between the pressure of the new steam on the one side and the back pressure of the exhaust steam on the other. When the steam is exhausted into the air, this back pressure is necessarily somewhat greater than that of the air, i.e. 15 Ib. or more per square inch. Thus if the average pres- sure of the working steam during a complete stroke is 60 Ib. (absolute) per square inch, the average effective pressure against the piston is less than 45 Ib. per square inch, and one fourth of the energy that would be available if the steam were exhausted into a vacuum is lost in working against atmospheric pressure. To avoid this waste, the exhaust pipe of stationary and marine engines usually leads to a condensing chamber (Fig. 203), in which a partial vacuum is maintained by means of a pump. As the exhaust steam enters this chamber it is quickly condensed by a spray of cold water, as shown in the figure, or by water surrounding the chamber, ing Chamber. The vacuum mamta ined in the con- denser reduces the back pressure of the exhaust steam by approximately one atmosphere, and hence increases the effective pressure against the piston by that amount. FIG. 203. Condens- HEAT ENGINES 295 251. Compound Engines. The available energy of steam in- creases with its pressure. At a gage pressure of 10 Ib. (i.e. 10 Ib. per square inch above atmospheric pressure), a perfect non-condensing engine uses 69 Ib. of steam per horse-power per hour. It does an equal amount of work with 26 Ib. of steam at 40 Ib. pressure, with 18 Ib. at 80 Ib. pressure, with 16.3 Ib. at 100 Ib. pressure, or with 13.6 Ib. at 150 Ib. pressure. The additional amount of heat required to generate steam at high pressures is relatively small, compared with the gain of available energy. Hence there is economy of fuel as well as of steam in work- ing the steam at high pressure. But the highest efficiency of steam at any pressure is secured only when the full expansive power of the steam is utilized; and the greater the initial pressure the greater the possible expansion. When the pressure is above 100 Ib., the greatest efficiency is obtained by using the steam successively in two or more cylinders (Fig. 204), with not more than a threefold or fourfold FIG. 204. Section of a Duplex Compound Engine. expansion in each. The exhaust from the first or high-pressure cylinder passes into the larger low-pressure cylinder, in which it drives a second piston. Such engines are called compound. Large marine and stationary engines, working under the highest pressures (200 to 225 Ib.), use the steam successively in three or four cylinders. These are called triple-expansion and quadruple-expansion engines respectively. The cylinders of a compound engine are sometimes placed side by side, as in the accompanying illustration, sometimes end to end. 296 HEAT The latter arrangement is common with two-cylinder engines, and is known as the tandem compound. In such. engines the two piston heads are attached to the same rod. A twin-tandem compound consists of two tandem compounds, built as one unit for driving the same shaft. Compound engines are usually condensing engines. The various types of compound condensing engines represent the highest development of the modern steam-engine. They furnish the power for propelling steamships and for the heaviest duty in manufacturing establishments. Engines ranging in power from 1000 to 5000 horse-power are common. The largest steam-engine in the world is installed in a rolling-mill at Sharon, Pennsylvania. It is a twin-tandem engine, having high-pressure cylinders 42 in. in diameter and low-pressure cylinders 70 in. in diameter. It weighs 550 tons without foundation plates or flywheel, and is rated at 25,000 horse-power. 252. The Locomotive. The main body of a locomotive consists of the long, horizontal boiler, the cab back of it, the fire box under its rear end, and the smoke box in front of it. The boiler contains from 300 to 400 tubes, extending from the fire box to its front end (Fig. 205). These convey the hot gases from the fire to FIG. 205. Tubular Boiler. , the smoke box, and pro- vide a large heating surface, being entirely surrounded by the water in the boiler. A locomotive has at least two cylinders, one on each side, below the level of the boiler at the forward end. The piston working in each cylinder acts on the driving wheels on its own side. The cylinders are supplied with steam through a pipe which, starting within the steam dome, runs forward through the boiler and down through the smoke box. The throttle valve at the entrance to this pipe serves to regulate the pressure of the steam admitted to the cylinders. The ports of each cylinder are opened and closed by means of a slide valve, as in stationary engines. The adjustment of this valve is under the immediate control of the engineer, through a system of eccentrics and levers, ending with the reversing lever in HEAT ENGINES 297 the cab. When this lever is in a forward position the engine runs forward; when it is inclined to the rear the engine runs backward; when it is in a vertical position the valve remains at rest with both ports closed, and the engine coasts, without working. The point of cut-off of the steam is determined by the greater or less inclination of the lever. For most economical working the cut-off occurs at quarter stroke; for greater power it occurs at half stroke. Locomotives are non-condensing engines. The exhaust takes place into the smoke box, just below the smokestack. This maintains a powerful draft for the fire, and is the cause of the familiar puffing sound. Express locomotives, being built for speed, have large driving wheels from 6 to 7 ft. in diameter; freight locomotives are designed for great tractive power, their drivers having a diameter of 5 or 6 ft. The most powerful locomotives are compound and have from FIG. 206. Diagram of a Locomotive Engine. 8 to 1 6 driving wheels. The largest are capable of exerting a tractive effort or pull at the draw-bar, of 80,000 to 100,000 Ib. The trac- tive effort of the average express locomotive is from 16,000 to 25,000 Ib.; of the average freight locomotive, from 30,000 to 45,000 Ib. The tractive effort averages about one fifth of the weight carried by the drivers. If it is greater than one fourth, the wheels are liable to slip. The usual boiler pressure of locomotives is about 200 Ib., the maximum 225 Ib. . 253. Efficiency of the Steam-Engine. The various modern types of the steam-engine are highly perfected machines; and yet 298 HEAT the entire process of converting the energy of fuel into useful work through the agency of steam is a very wasteful one, only 1 2% of the energy being thus converted under the most favorable conditions. In the first place, " from one fourth to one fifth of the heat produced by combustion in the furnace is lost or carried up the chimney in the gases at the high temperature, besides the waste due to radiation, and smoke imperfectly burned." But this is not the greatest waste. At least three fourths of the energy of high-pressure steam is carried away in the exhaust, when this loss is reduced as much as possible by means of a condenser. This is the heat of vaporization, and it can not be utilized in any engine. The additional energy put into steam by generating it under great pressure and by superheating it is all available for doing work, and adds greatly to the efficiency of an engine. Steam is superheated, after leaving the boiler, by passing it through pipes which are sur- rounded by the hot gases from the furnace. . Ordinary or saturated steam begins to condense as soon as it begins to expand in the cylinder of an engine; superheated or unsaturated steam does not (Art. 222). The thermal efficiency of an engine is the ratio of the work done upon the piston to the energy supplied in the steam. The best compound condensing engines, when using highly superheated steam, have a maximum thermal efficiency of about 17%. Lastly, from one tenth to one fifth of the work done on the piston is lost or absorbed by friction of the engine mechanism, and only 80 to 90% of it is transmitted to the driving shaft. The percentage of work transmitted is called the mechanical efficiency of the engine. The highest economic efficiency of an entire steam plant, including furnace, boiler, and engine, is therefore about .8 X .17 X .9 = 12%, and 6% is a fair average value. The efficiency of locomotives is considerably less, rarely if ever exceeding 6%. 254. Historical Notes on the Steam-Engine. The earliest steam-engine having a cylinder and piston was invented in 1705 by Thomas Newcomen, an English blacksmith. The cylinder was vertical, and its upper end open. The piston was driven up, against atmospheric pressure, by steam admitted into the lower end of the cylinder. The steam in the cylinder was then condensed by a jet of water. A partial vacuum was thus formed under the piston, which was then driven down by atmospheric pressure. The supply HEAT ENGINES 299 of steam and water was controlled by opening and closing stop cocks, which were at first operated by hand. Newcomen engines were successfully used for pumping water from mines. Such was the steam-engine when James Watt, instrument maker to the University of Glasgow, began its improvement in 1768. He introduced a separate condenser, admitted steam to both ends of the cylinder, and, during a period of thirty years or more, made many other improvements which greatly increased its efficiency. Many inventors have taken part in the development of the steam-engine since the time of James Watt; but to him, more than to any other man, is due the honor of having made it one of the great factors in the industrial progress of the world. The first self-moving steam-engine was built in France in 1769; the first in America was built in 1790. Both were designed to run on common roads. Railroad locomotives were first built and successfully operated in England. The first " prac- tical " American locomotive was the "Tom Thumb," built by Peter Cooper, of New York, in 1831. "That engine developed about one and one half horse-power, and the chief objection raised * ^^5^ FIG. 207. Locomotive of 1839. was that it was not powerful enough." That has been a common objection to the most powerful engines ever since. Navigation by steam power began to be a success in 1807, when the little steamboat Clermont, constructed under the direction of Robert Fulton, made its first trip from New York to Albany. 255. The Internal Combustion Engine. The problem of devising an engine that could be made to do useful work by the combustion of fuel within its cylinder was attempted even before the invention of the steam-engine. It was first proposed to drive the piston by the explosion 300 HEAT of gunpowder; but the first practical success was achieved in 1860, with coal gas as fuel. Since then improvements have followed in rapid succession, the most important being the adoption of the four-stroke cycle by the German inventor, Dr. Otto, in 1876. Modern gas and gasoline engines are principally of the Otto type, and are known as four-cycle, or, more correctly, four-stroke-cycle engines (Fig. 208). The Otto cycle consists of a series of operations which take place during four successive strokes of the piston, or two complete revolutions of the flywheel. These opera- tions are as follows: First or suction stroke; charging. The piston moves forward or out- ward (downward in vertical engines), drawing in a charge of air and gas through the inlet valve /. The exhaust valve remains closed. Second stroke; compression. During the return or inward stroke, both valves are closed, and c, Working stroke; D, Exhaust; the charge admitted during /, Inlet Valve; E, Exhaust Valve. -,...- , . the first stroke is compressed. Third or working stroke; explosion and expansion. The compressed mixture is ignited by an electric spark or other device. It explodes, forming other gases at a very high temperature and a correspondingly high pressure. These gases expand, driving the piston outward, and the work done on the piston is transmitted through the con- necting rod to the crank-shaft. This is the only part of the cycle in which heat is converted into work; hence it 1 HEAT ENGINES 301 is called the working or power stroke. Fourth stroke; ex- haust. During the second inward stroke the piston drives out the products of combustion through the exhaust valve E. This completes the cycle. The piston of an internal combustion engine is generally driven from one side only, the forward end of the cylinder being open to the air. Since there is only one working stroke in four, the flywheel must be very massive in order to maintain steady motion. There are usually two flywheels, one on each side. Small engines are started by hand, large ones usually by compressed air. The valves are circular and fit circular openings, against which they are tightly held by spiral springs. They open with an inward motion, and are operated from the crank-shaft by a rather complicated mechanism of gear-wheels, rods, cams, and levers. Since the cylinder is directly exposed to the intense heat of the burning gases, some effective means must be provided for cooling it. The cylinders of very small engines are cast with numerous projecting flanges or ribs, which present a large cooling surface to the air. In the larger sizes the cylinder is surrounded by a jacket, through which water is kept in constant circulation. 256. Types of Internal Combustion Engines. There are many forms of the internal combustion engine, varying with the kind of fuel used, the power of the engine, and the purpose for which it is intended. Stationary Gas Engines for general power purposes are both ver- tical and horizontal. Those of moderate power have one cylinder; the largest are tandem or twin-tandem engines, having two and four cylinders respectively, and ranging from 500 to 5000 horse-power. The smaller engines are run with illuminating gas, the larger with a cheaper gas, generated in " gas producers " from crude oil, inferior grades of coal, or other inexpensive fuels. The gas producer is to the gas engine what the boiler is to the steam-engine. A power plant, consisting of a modern gas engine and gas producer, converts from 20 to 25% of the heat energy of the fuel into useful work, the average efficiency being two or three times as great as that of a steam-engine plant of the same capacity. The Marine Gas Engine. The use of the gas engine for propelling 302 HEAT ships is still in the experimental stage, but success in the near future is very probable. The principal advantages offered by the marine gas engine are a higher efficiency, a considerable saving of space and weight, and the fact that smokestacks would be done away with. The Gasoline Engine (Fig. 209) is driven by an explosive mixture of air and gasoline vapor. In con- struction and action it is es- sentially the same as the gas engine. The liquid fuel is vaporized in a device called a carbureter, FlG - 209 '~ A Gasoline placed near the inlet valve of the cylinder. The carbureter thus corresponds to the producer of the producer-gas engine. The special advantages of the gasoline engine are that it is a com- plete power plant in itself, is small and light, and is simple and con- venient to operate. It supplies the power for motor cycles, motor boats, gasoline launches, automobiles, and flying machines, and is extensively used in the form of stationary and portable engines for running light machinery. The automobile engine usually has four cylinders, sometimes six, operating the same shaft. The power strokes of the different cylinders occur one at a time; hence with the four-cylinder engine there is one power stroke to each half turn of the shaft, as in the case of the single-cylinder steam-engine. With a six-cylinder engine there are three power strokes to each revolution of the shaft, and the motion is very steady. 257. The Steam Turbine. All engines having a cylin- der and piston are classed as reciprocating engines, from the to-and-fro or reciprocating motion of the piston and the parts intervening between it and the crank-shaft. The sudden starting and stopping of these parts with every stroke causes vibration and loss of power; and this type of motion is objectionable in other respects. HEAT ENGINES 303 These difficulties are wholly avoided in the steam tur- bine, in which the only moving parts are a revolving wheel and shaft. The wheel of a simple steam turbine the De Laval is shown in Fig. 210. It carries a single set of concave blades upon its circumference, and is driven by jets of steam, issuing from nozzles and directed at the proper angle against the blades. In the illus- tration one of the nozzles is represented as transparent, to show the diverging outlet. On account of this divergence the steam expands in passing through the nozzle, and in FIG. 210. De Laval Steam Turbine. FIG. 211. De Laval Steam Turbine and Dynamo. expanding it acquires a high velocity, its potential energy thus becoming kinetic. The velocity of the steam as it strikes the blades is in some cases as high as 4000 ft. per second or 45 mi. per minute. The wheel rotates at speeds ranging from 10,000 to 30,000 revolutions per minute. 304 HEAT This speed is too great for direct utilization, and is reduced by gear-wheels in the ratio of 10 to i. The De Laval steam turbine is used for operating dyna- mos, centrifugal pumps, blowers, etc. A 3o-horse-power turbine-dynamo unit is shown in Fig. 211. The turbine wheel is inclosed in the short cylindrical case at the right. The large cylindrical case at the center incloses the gear-wheels, by which the speed is reduced in transmitting the motion to the armature shaft. By means of a multiple turbine wheel greater power can be developed and at slower speeds. The Parsons and the Curtis turbines are of this type. The wheel or rotor is a long cyl- FIG. 212. Diagram of inder, around which there are many in,, Tu i b - ne rows of blades, with spaces between Wheel. M, Revolving Blades; s, stationary the rows. These spaces are occupied by rows of fixed blades, which project inward from the cylindrical case. The arrangement is shown in Fig. 212. The fixed blades serve to direct the steam against the successive sets of moving blades. The steam turbine is the latest type of heat engine to achieve success. Its earliest use dates from 1883. Since then it has been greatly improved and adapted to many different kinds of work. The efficiency of the earlier turbines was very low. The modern turbine is superior to the best reciprocating steam-engines in this respect. It gives the best results when operating as a condensing engine, with superheated steam at high pressure. The steam turbine has the further advantage of being much smaller than a reciprocating engine of the same power. Many steamships, cruisers, and other vessels built in recent years have been equipped with turbine engines of the Parsons and the Curtis types; and electricity for lighting and power purposes is now generated in large centers of population by turbine- dynamo units of 5000 to 18,000 horse-power. CHAPTER IX SOUND 258. Introduction. The five senses sight, hearing, smell, taste, and touch are the channels through which the mind receives impressions from the outer world. These impressions furnish the raw materials out of which the mind, by processes of reasoning, constructs scientific knowledge. Through the sense of touch we receive certain impressions by which we know whether a, body is hot or cold; but the nature of heat was discovered only by reasoning, based upon experiment and a general knowledge of physical prin- ciples. Through the sense of hearing we receive a class of impressions called sensations of sound. Experience teaches that these sensations are due to vibrating bodies, which may be near or distant. Obviously some invisible action takes place across the intervening space between the vibrating body and the ear. This action we call sound. It is the physical cause of the sensation, but its nature is not revealed by ordinary experience. We might hear sounds all our lives without learning what sound is. We describe different sounds in terms of the sensations which they produce, calling them musical, unmusical, loud, faint, shrill, high, low, deep, sweet, melodious, hol- low, harsh, discordant, etc. These wonderfully varied impressions must be due to certain physical differences between one sound and another; but the nature of these differences and the nature of sound itself can be deter- mined only by reasoning and experiment. 305 3o6 SOUND In the study of sound as a branch of physics, we are principally concerned with such questions as these, which relate to physical processes rather than to the sensations produced by them. It should be noted at the outset that these processes are mechanical, and are in perfect agree- ment with mechanical laws and principles. I. ORIGIN AND TRANSMISSION OF SOUND 259. Sounding Bodies. It can be shown in various ways that a sounding body is in a state of vibration. In many cases this is evident from the appearance of the body, although the motion is always too rapid to follow with the eye. A string of a musical instrument has the appearance of a gauzy spindle when sounding; and the prongs of a tuning fork become indistinct and appear to widen out at the free ends, where the motion is greatest. The vibrations can generally be felt, and, unless very weak, can be shown 1.3.- Vibrating spring. by ^ mechankal effectg For example, a shower of spray is thrown up when the prongs of a vibrating fork are dipped into water; and a pith ball or a bit of cork, tied to the end of a thread, is driven away when suspended so as to touch the edge of a sounding bell. Sound may also be produced by a vibrating body of liquid or gas. The sound of running water and the notes of wind instruments are familiar examples. Sounding bodies differ from one another in their modes of vibration, and the same body may vibrate differently under different conditions. One of the simplest cases is that of a long, straight, steel spring, rigidly fastened at a ORIGIN AND TRANSMISSION OF SOUND 307 greater or less distance from the vibrating end (Fig. 213). As this end is gradually shortened the vibrations become more and more rapid, until, at a certain rate, a low, musical note is heard. Incidentally we may observe that the spring sounds a higher note as its length is further decreased, illustrating the fact that the pitch of a sound is determined by the rate of vibration of the sounding body. This im- portant relation will be studied later. Our present purpose is to observe the manner in which the spring vibrates. The character of the vibration remains the same whether it is slow or rapid; it can therefore be studied by adjusting the spring to one or two swings per second. It will then be seen that the motion is similar to that of a pendulum. During the first half of each swing the motion is acceler- ated, during the latter half it is retarded; it is most rapid at the mid-position D. The vibratory motion is main- tained by the elastic force of the spring, which plays the same part as the force of gravity upon a pendulum bob. In the study of sound, one vibration includes a swing both ways (from D' to D" and back to D f ). The amplitude of vibration is the extent of motion on either side of the posi- tion of rest. The rate of vibration of sounding bodies is independent of the amplitude; the vibration is therefore regular or periodic. This can be shown in the case of the steel spring by counting the number of vibrations in a given time, with different amplitudes; but it is proved for all sounding bodies by the fact that a sound does not change in pitch as it becomes fainter and gradually dies away. The rate of vibration is measured by the number of vibra- tions per second. Tuning forks are so frequently used in experiments in sound that it is important to know how a fork vibrates. A quick blow upon one prong, in the direction of the other, sets both prongs in vibration. 308 SOUND Their motion is always toward each other and from each other in succession (Fig. 214). The transverse vibration of the prongs is accompanied by a vibration of the stem in the direction of its length (longitudinal vibration) , which can be distinctly felt by placing the stem of a sounding fork against the teeth. A sounding fork is always held or supported by the stem; touching a prong quickly stops it. 260. Sound Media. In ordinary circumstances FlG 2I4 sound is conveyed to the ear through the air. Does the air play an essential part in the process? We know that radiant energy is transmitted through the air, but not by means of it. A vacuum serves the purpose quite as well or even better (Art. 194). That the air does take an active and necessary part in the transmission of sound is read- ily shown with an electric bell, stand- ing on a soft cushion or suspended by fine wires within the receiver of an air pump (Fig. 215). Before the air is exhausted the bell can be heard distinctly; but the sound grows con- tinually fainter as the exhaustion proceeds, and ceases when a good vacuum is secured. It is restored when air or any other gas is admitted into the receiver. Sound is also transmitted by liquids and solids. Its transmission through the walls and floors of buildings is familiar, and the faintest tapping or scratching at one end of a long board or table can be heard at the farther end wnen the ear is pressed against it. A swimmer, with his head under water, hears a loud sound when two stones are struck together, also under water, at a long distance from him. FIG. 215. ORIGIN AND TRANSMISSION OF SOUND 309 Whether all solids transmit sound equally well can be readily determined by experiment. When the stem of a sounding fork is pressed against the top of a table, the sound becomes much louder. The stem in vibrating strikes the table with a rapid succession of blows, and the impulses thus imparted cause the table to vibrate in unison with the fork. The table thus becomes a sounding body. The sound from the table is almost if not quite as loud when con- nection between it and the sounding fork is made through a meter stick, or other long rod of wood or metal; but little or no sound is heard when a rubber stopper or a roll of cot- ton-wool is placed between the fork and the table. All highly elastic (rigid) bodies transmit sound well; soft and yielding solids transmit it poorly. The latter are said to "deaden" the sound. From such experiments as the above we learn that sound can exist only in ordinary matter, and that its transmis- sion depends upon the elasticity of the substance through which it is passing. Any substance that transmits sound is called a sound medium (plural, media). 261. Wave Motion. Taking together the facts that sound originates at vibrating bodies and that it is trans- mitted by the surrounding air and other elastic media, the inference is plain that it must be some kind of dis- turbance produced in the medium itself. Since this disturb- ance is invisible, we can better understand what it is like, how it is produced, and how it is propagated after a pre- liminary study of a few types of visible motion which are in some respects like it. When a stretched rubber tube or a spiral spring, three or four meters long, is struck a sharp blow near one end, a distortion is produced which travels rapidly as a wave to 310 SOUND the other end. By tying strips of cloth to the tube at different points, it can be seen that, as the wave passes any point, that point moves quickly out in a direction at right angles FIG. 216. Longitudinal Vibration in a to the length of the Spiral Spring. ,11 r tube and returns. The curved form that we call the wave is, in fact, passed from point to point along the tube by the transverse vibration of successive portions of the tube. A distortion of a differ- ent character is started by stretching a portion of the tube near one end either considerably more or less than the remainder and suddenly releasing that portion (Fig. 216). The strips of cloth now indicate a to-and-fro or longitu- dinal vibration as the disturbance passes. The waves that travel over a field of grain when dis- turbed by the wind are formed of the swaying stalks, as they bend forward and spring back. The wave form pro- duced by this motion of the stalks sweeps over the field in the direction of the wind. A water wave is transmitted by a vibratory motion of the water particles. This is mainly an upward and down- ward or transverse vibration, as is shown by the rise and fall of any floating object as the waves pass under it. A wave is not formed of the same water as it travels ; it is the disturbance that travels, not the water. A pebble dropped into a pool produces a train of circular waves, which travel outward from the point where the pebble strikes. The waves are circular because the disturbance is trans- mitted with equal velocity in all directions over the sur- face; they are concentric because they all start at the same point. Each wave consists of a crest and a trough. The length of a wave is the distance from crest to adja- ORIGIN AND TRANSMISSION OF SOUND 311 cent crest or from trough to adjacent trough, measured at right angles to the line of the crest. The waves rapidly decrease in height as they travel because their energy is handed on to an ever-increasing body of water. A wave of any sort is made up of moving and distorted parts of the medium in which it is traveling, and hence possesses energy. This energy goes with the wave, the net result of the process being a transmission of energy, but not of matter. 262. Sound Waves. If the action upon the air were the same on all sides of a sounding body at the same in- stant, a sound wave would have the form of a hollow sphere, with the sounding body at its center. Generally the waves are only greater or less fractions of such spherical shells, depending upon 'the particular manner in which the sound- ing body vibrates; but the simplest case is that of a com- plete spherical wave, such as is produced by the explosion of a firecracker in the air. Let us try to form a mental picture of such a wave. The gas generated by the explo- sion of the powder in the firecracker instantly expands, forcing the air violently away on all sides. This shell of outward moving air, on account of its momentum, goes farther than is necessary to restore equilibrium, leav- ing a partial vacuum at the center of disturbance. There is now an unbalanced pressure toward the center, which drives the air back again. While the air particles are moving outward they form a spherical shell of increased density, represented by CiC 2 C 3 C 4 (Fig. 217). During their inward motion they form a shell of diminished density, represented by RiR 2 R 3 R t . The former is called a compres- sion, the latter a rarefaction. The two together consti- tute a sound wave. The air particles in the compression 312 SOUND transmit their forward motion to the particles just ahead, and themselves come to rest. The compression thus travels outward through the air. At the rear (inside) of the com- pression the forward motion of the particles has just ceased, and they are on the point of moving backward into the FIG. 217. Section of a Sound Wave in Air. rarefaction. This backward motion of the air particles transmits the rarefaction forward, following the compres- sion. The forward and backward motion (longitudinal vibration) of the particles of the medium should be care- fully distinguished from the steady onward motion of the wave through the medium. The mechanical action of the air particles in transmitting a sound wave is similar to that of a row of balls in transmitting an impulse from one to another along the row. This action can be shown with .a number of balls, placed close together in a groove (Fig. 218). When a ball is rolled against one end of the row, each one in the row collides with the next in rapid succession, and the last one is driven away. The motion of the others is scarcely perceptible, although each in ORIGIN AND TRANSMISSION OF SOUND 313 turn receives and transmits the energy that is imparted finally to the one at the end. It should be noted that the last ball flies away seemingly at the very instant when the first one is struck. The speed of the disturbance is many times greater than that of any of the balls. FIG. 218. 263. A Musical Sound consists of a continuous series or train of waves of the same character. The sound of a tuning fork is a good example. As either prong of a fork advances, it drives the air rapidly before it, producing a compression (Fig. 219, I and III); as it retreats, the air follows it up, producing a rarefaction (II and IV in the figure). A wave is thus produced by each complete vibra- tion. (The waves spread out as they travel. The sec- tions represented in the figure are only partial.) FIG. 219. Propagation of Sound Waves. The air particles at the fork vibrate with the amplitude and velo- city of the prong. This amplitude is rarely more than i mm., and it grows less at increasing distances from the fork, owing to the increase in the size of the waves. At the fork the air particles move to and fro with the prong; but as the distance from the fork increases the vi- brations of the particles lag more and more behind, owing to the fact 314 ' SOUND that it takes time for the disturbance to travel. This can be shown with the aid of the figure. While the prong is making its first out- ward swing the disturbance advances to a\ in I, a point at a distance of about two feet if the pitch ot the fork is "middle C." The prong and the adjacent air particles at c\ complete their forward motion at the same instant. The particles at b\ are then at the mid-point of their forward swing, and those at a\ are just starting. During the backward swing of the prong, the front of the compression advances to az in II, and its rear to cz. Meanwhile the retreat of the prong has produced a rarefaction, which extends from ez to a. At ez the backward swing of the air particles has just ceased, with that of the fork; at dz it is half completed; at cz it is just beginning. Parts III and IV of the figure represent the state of things at the end of the second forward and the second backward swing of the prong respectively. A sound wave consists of a compression and the follow- ing rarefaction A wave front is the surface bounding the front of a compression. The length of a sound wave is the distance from wave front to wave front. A wave travels its own length while the sounding body is making one complete vibration; and its length remains the same however far it may travel. 264. Intensity and Loudness of Sounds. The inten- sity of a sound refers to the mechanical energy of the waves. It is a physical quantity. The loudness of a sound refers to the sensation produced when the sound is heard. It depends in part upon the condition of the ear and in part upon the pitch of the sound, a shrill sound being more dis- tinctly heard than one of equal intensity but of low pitch. In general, however, the loudness of a sound increases with its intensity. In physics we are concerned with the conditions which determine the intensity of a sound at its source and its loss of intensity in transmission. ORIGIN AND TRANSMISSION OF SOUND 315 265. The Intensity at the Source is proportional to the rate at which the air or other medium receives energy from the sounding body; and this depends upon the ampli- tude of vibration of the body, the area of the vibrating surface, and the density and elasticity of the medium. Effect of Amplitude. The gradual dying away of the sound of a bell, a piano wire, a tuning fork, etc., is due to the diminishing amplitude of vibration as the body ap- proaches a state of rest. With a decrease of amplitude the blows of the vibrating surface against the surrounding air grow less vigorous, and the sound waves are correspondingly weaker. Effect of the Area of the Vibrating Surface. A narrow vibrating surface cuts through the air, producing little effect; the air slips round it. A broad surface catches the air and carries it bodily forward. This is the reason why the sound of a tuning fork is very faint when it is held in the hand and loud when it is touched to a table. In the latter case the vibrations are transmitted to the air almost entirely by the vibrating table. The music of a violin or guitar comes practically entirely from the body of the instrument, and the music of a piano from the sound- ing board on which the wires are strung. (The bodies of stringed instruments are capable of taking up the vibra- tions of any number of the strings at the same time.) Effect of the Density and Elasticity of the Medium. In the experiment with the sounding body under the receiver of an air pump, it was observed that the sound grows fainter as the exhaustion continues, i.e. as the density of the remain- ing air is diminished. The reason is obvious: there is less matter in motion in the wave of rarefied air, hence there is less energy kinetic energy being proportional to the SOUND mass of the moving body. "On high mountains, where the air is much rarefied, it is necessary to speak with some effort in order to be heard, and the discharge of a gun produces only a feeble sound. " It has already been shown in a number of experiments that sound is louder when transmitted through elastic solids than it is when trans- mitted through the air. We are not to infer that solids are better transmitters of sound than air is, but rather that the intensity of the sound is greater at the source when the medium is a solid. For example, a vibrating fork produces waves of greater intensity in the top of a table than it does in the air directly; for the rigid wood offers much greater resistance to the blows of the fork than the air does, and consequently receives a greater amount of energy with each vibration. 266. Loss of Intensity in Transmission. It is well known that a sound grows fainter with increasing distance from its source. The definite relation be- tween intensity and dis- tance follows from the principle of the conser- vation of energy, and holds for light as well as for sound. This im- portant law is derived as follows: As a sound wave travels in the open air, its distance from the source is the radius of its surface (the spherical wave front). Now it is proved in geometry that the sur- faces of two spheres (or equal fractions of their respective surfaces) are proportional to the squares of their radii. Hence, since the length of a wave remains constant, its volume and the mass of air composing it are proportional to the square of the distance it has traveled. FIG. 220. Relation of Volume to Distance. ORIGIN AND TRANSMISSION OF SOUND 317 This is shown in Fig. 220 which represents a section of a spherical wave at a distance d\ from the source and again at twice that dis- tance, or fa. If vi denotes the volume of the section at the first distance, and % its volume at the second distance, then v z : vi :: 4 : i, or, in general, vz :vi :: ck 2 : d?. Assuming that the total energy of a sound wave remains unchanged as the wave travels, the intensity of the sound (or the amount of energy per unit volume of the medium) varies inversely as the volume of the wave, and hence, in the open air, it varies inversely as the square of the distance from the source. This relation is not strictly true, for the energy of sound is more or less slowly transformed into heat by internal friction in any medium. In the end it is all dissipated in this manner. Hence the intensity decreases somewhat more rapidly than the law indicates. 267. Confined Sound Waves. Sound travels long distances in elastic media with very little loss of intensity when the waves are prevented from increasing in size. This is the principle of the speak- ing tube, which is a long metal tube of small diameter, used for Communication between different rooms of a building or between some room and the street door. The tube does not readily take up the vibrations of the air, hence the sound is almost completely con- fined within it. Practically the only loss is that due to friction. Solids, such as rods, stretched wires and strings, the rails of a track, etc., transmit sound with comparatively little loss of intensity for the same reason, the vibrations being largely confined in the solid medium. This fact is utilized in the acoustic or string telephone, which is capable of rendering practical service over short distances. 268. The Velocity of Sound in Air. It is a matter of common observation that an event which takes place at a distance is seen before the sound produced by it is heard. At a distance of a few hundred feet the blow of a hammer is heard after the hammer is raised for the next stroke; 3i8 SOUND the cloud of "steam" issuing from the whistle of a dis- tant locomotive is seen and may even disappear before the whistle is heard; the thunder that accompanies a flash of lightning is often delayed many seconds in reaching the ear. Now the time required for light to travel terres- trial distances is inappreciable (the velocity of light being 186,000 mi. per second); hence the interval between the sensations of sight and hearing in such phenomena as these is the time occupied by the sound in traveling from the sounding body to the observer, and if the time and the distance are known the velocity of sound can be computed. Observations have repeatedly been taken for this purpose by firing a cannon at each of two stations several miles apart, and noting the time between the flash and the re- port as observed at the other station. By taking observa- tions at each of the stations alternately, a correction can be made for the effect of the wind. The average of the best determinations is 332 m. or 1090 ft. per second at o C. At 20 the velocity is 344 m. or 1129 ft. per second. That the velocity of sound is independent of its pitch and intensity is proved by the fact that all notes sounded together by an orchestra are heard together at all distances. 269. Causes which Determine the Velocity of Sound. - The transmission of waves of any kind in ordinary matter is a mechanical process, and is in agreement with the general laws of dynamics. The mathematical physicist, starting with these general laws and the physical proper- ties of gases, can compute the velocity of sound in air with- out resorting to experiment ; but the factors upon which the velocity of a wave depends can very well be shown in a general way by means of two long rubber tubes of the same size, one of them filled with shot or sand. With the empty ORIGIN AND TRANSMISSION OF SOUND 319 tube, a wave started by striking it sharply near one end travels rapidly back and forth from end to end, the speed increasing with the tension of the tube. In transmitting the wave, successive portions of the tube swing out to one side and are jerked back again by the elastic resistance developed in the stretched rubber. This transverse vibra- tion is quickened by increasing the tension, in accordance with the law that the acceleration of a given mass is pro- portional to the force. When the weighted tube is stretched equally with the other, the waves travel much more slowly in it, according to the law that, with a given force, the acceleration varies inversely with the mass. Thus in general the two factors which control the trans- mission of a disturbance in any medium are force and mass. In the case of sound waves the force is due to the elasticity of volume of the medium (expansive force), and the other factor is the mass per unit volume, or the den- sity. The exact relation is that the velocity varies directly as the square root of the elasticity and inversely as the square root of the density. The velocity of sound in water has been found by experiment to be 1435 m. per second at 8, a velocity more than four times as great as in air. Thus, in comparison with air, the retarding effect of the greater density of water is more than offset by the acceler- ating effect of its still greater relative elasticity (rigidity) . This is true in even greater degree of solids, the velocity in glass and steel being about fifteen times as great as in air. The velocity of sound in air increases with the temperature, be- cause air expands with a rise of temperature, and its density diminishes while its elasticity remains unchanged. The gain of velocity is i ft. per second per degree Fahrenheit, or .6 m. per second per degree Centigrade. An increase of pressure increases the elasticity and the density in the same ratio; hence a change of barometric pressure does not affect the velocity. 320 SOUND 270. Reflection of Sound. Echoes. When sound waves strike a large surface, as a cliff or the side of a building, they are reflected. The re- flected sound is called an echo when it reaches the ear long enough after the original sound to be distin- S uisned from it: This re- quires about one fifth of a second at least, during which time sound travels 68 m. Hence a distinct echo FIG. 221. Reflection of Sound Waves will not be heard unless the from a Plane Surface. ~ ... reflecting surface is at a distance of 34 m. or more from the source of the sound. At less distances the direct and the reflected sounds blend more or less perfectly together. When the reflect- ing surface is within a few meters of the source, the two sounds are heard as one, and the only effect of the reflection is a greater intensity. A good example is the reflecting surface at the rear of a band stand. When the distance is nearly great enough for an echo, the direct and the reflected sound are mixed confusedly, causing indistinctness. This is often noticeable in large halls. When sound waves are reflected from a vertical plane surface, their curvature is reversed as shown in Fig. 221, in which is the source, AB the reflecting surface, and 0' the center of the reflected waves. The reflected waves behave in all respects as if 0' were their real source. They increase in size and decrease in intensity in pro- portion to the square of their distance from 0' '. The energy trans- mitted along any radius from O is transmitted after reflection along the corresponding radius from O', as OA and AD, OB and BE, etc. After reflection from a large concave surface, sound waves increase less rapidly in size and decrease less rapidly in intensity than they ORIGIN AND TRANSMISSION OF SOUND 321 do from a plane surface. The behavior of the reflected sound in such cases is similar to that of light when reflected in a strong beam from the concave mirror of the headlight of a locomotive or a street car. To secure' this effect the reflecting walls at the rear of band FIG. 222. Sound Wave Reflected from a Concave Surface, MN . source of the sound; B, the sound focus. A, the stands are made concave. When a sounding body is beyond a certain distance from a concave surface, the reflected waves are concave (Fig. 222). Such waves decrease in size and increase in intensity as they travel toward a point. This is the principle of "whispering galleries," which owe their remarkable effects to curved walls or ceilings. " Sails of ships are sometimes inflated by the wind so that they act as concentrating reflectors of sound. Arnott says that in coasting off Brazil he heard the bells of San Salvador from a distance of no mi., by standing before the mainsail, which happened at the time to assume the form of a concave reflector, focusing at his ear." PROBLEMS 1. By what force are waves transmitted along a stretched rubber tube? By what force are water waves transmitted? 2. Sound waves consist of compressions and rarefactions in all media, solid, liquid, and gaseous. Is their transmission due to the elasticity of form or volume of the medium? 3. Would a sounding body continue to vibrate longer in water or in air? in the air or in a vacuum? Why? Find by trial whether a tuning fork vibrates longer when held in the hand or when its stem is in contact with a table. Explain. 322 SOUND 4. How does the intensity o f sound at a distance of 5 m. from the source compare with its intensity at 10 m.? at 15 m.? at 20 m.? 5. At what distance is the intensity of sound one fourth as great as at 100 m.? one half as great? 6. How would music be affected if sounds of different pitch or different intensity traveled with different velocities? 7. If a cannon ball goes 6 mi. at an average speed of 2500 ft. per second, how does its time of flight compare with the time required for the sound of the firing to travel the same distance in air at 20? 8. Referring to Fig. 221, account for the reversal of curvature of sound waves when reflected from a plane surface. Where is the center of the reflected waves with respect to the reflecting surface and the position of the source? Account for this. 9. A rifle is fired on one side of a canon and 3.2 sec. later the echo is heard from the opposite side. The temperature is 20. What is the width of the canon? 10. A flash of lightning is seen 12.5 sec. before the thunder is heard. At what distance did the lightning occur, the temperature being 20? 11. The mean distance of the sun from the earth is 93,0x30,000 mi. How long after an explosion occurs upon the sun would we hear it if air at o were provided as a medium for the transmission? (Light reaches us from the sun in 499 sec.) II. PROPERTIES OF MUSICAL SOUNDS 271. Musical Sounds and Noises. All sounds may be classed as musical sounds and noises, although the divid- ing line between the two classes is rather indefinite. We distinguish a great variety of musical sounds and a still greater variety of noises, and employ a considerable vocab- ulary of adjectives in attempting to describe them. But with all their variety these descriptive terms, at least so far as they relate to musical sounds, may be grouped under three heads, namely: 1. Loudness, as loud, soft, faint, powerful, weak; 2. Pitch, as high, shrill, piercing, low, deep, grave; 3. Quality, or character (in the narrower sense), as melo- dious, harsh, nasal, hollow, rich, full, sweet, mellow, dis- PROPERTIES OF MUSICAL SOUNDS 323 , cordant. Language fails to express the subtle variations of quality which the ear easily recognizes. It is by differ- ence in quality that we distinguish the notes of one musical instrument from those of another, or the sound of a familiar voice from a thousand others. Such terms as these evidently refer to the impressions which different sounds produce upon the hearer; they describe sensations, not their physical cause. It is clear, however, that the different sen- sations must be due to differences of some sort between the sound waves themselves; and it is with these char- acteristics of the waves and the man- ner in which they are produced by the sound- ing body that we are directly concerned. The first point to be determined is the difference between a musical sound and a noise. This difference is plainly shown in the following experiment. A disk of wood or metal is provided with two circular rows of pegs, equally spaced in one row at intervals of about i cm., and an equal number unequally spaced, at various irregular intervals, in the other (Fig. 223). The disk is mounted on a rotator and rapidly whirled, while the edge of a small card is held lightly against the regular row of pegs, then against the other row. In contact with the regular row the card is forced to execute regular periodic vibrations; and it then gives out a musical sound of definite pitch, which varies with the speed of the disk. This musical note is readily distinguished from the rattling noise which accompanies it. In contact with the irregular FIG. 223. 324 SOUND row the vibration of the card is also irregular, and only a noise is produced. (A disk provided with two rows of holes instead of pegs serves equally well for the experiment, the sound being produced by directing a jet of air against the holes.) A musical sound is one that continues of uniform loud- ness, pitch, and quality for an appreciable length of time, and does not change irregularly. Uniform loudness or intensity is due to a constant amplitude of vibration (Art. 265); uniform pitch, to a constant rate of vibration; and uniform quality, to a constant mode of vibration, i.e. to vibrations which are all simple or all of the same complex character. (We shall presently learn that the vibrations of a sounding body rarely consist of a simple to-and-fro motion, like that of a pendulum bob. They are generally very complex indeed, and this complexity is reproduced in the vibrations of the air particles in the sound wave.) A musical sound is often called a tone or note. A noise consists of a confused mixture of sounds, due to an extremely complex and rapidly changing vibration of the sounding body. Owing to the unsteady and non- periodic character of the vibrations, a noise has no definite pitch or wave length. 272. Relative and Absolute Pitch. Musical Intervals. The pitch of a note may be expressed either relatively or absolutely. It is expressed relatively by stating its rela- tion to some other note, generally the keynote of the mu- sical composition in which it occurs. The correspondence between relative pitch and relative rates of vibration is easily shown by means of the disk siren. This is a disk pierced with two or more circular rows of holes, equally spaced in each row. It is customary to provide four rows, having respectively 24, 30, 36, and 48 holes. While the disk is revolved at constant speed, a jet of air is directed against the different PROPERTIES OF MUSICAL SOUNDS 325 rows of holes in succession (Fig. 224). A note is produced by the regularly recurring puffs of air which escape through the holes, each puff producing a sound wave. The notes from the four rows bear to one another the pitch relation expressed by the syllables do, mi, sol, do', the interval between the first and the last being an octave. As the speed of the disk is increased the pitch of all the notes rises, but their relation to one another remains unaltered. Thus we learn that, if the interval between two notes is an octave, the vibration rate of the higher is always twice that of the lower, and, further, that the notes do, mi> sol, do' are produced by vibrations whose ratios to one another are expressed by the numbers 24, 30, 36, and 48, or by the smaller numbers 4, 5, 6, 8. The absolute pitch of a note is measured by the number of vibra- tions of the sounding body per sec- ond. This is called the vibration number or frequency of the note. To find the frequency of any note of the siren we have only to multiply the number of holes in the row by the number of revolutions of the disk per second. The vibration number of a fork can be determined by causing the fork to make a perma- nent record of its vibrations, as in the accompanying labo- ratory exercise. The frequency of the C fork corresponding to middle C of the piano or organ is 256, i.e. the prongs of this fork swing outward and inward 256 times in a second. (The middle C of musical instruments is slightly higher than this.) The experiment with the siren teaches that the relative pitch of any two notes is determined by the ratio of their frequencies. The ratio of the greater frequency to the less FIG. 224. Disk Siren. 326 SOUND is called the interval between the notes. For example, the octave interval is always 2 and the do-sol interval f or f . A person with a musically trained ear recognizes relative pitch with great accuracy, not as a numerical ratio, but as a direct mental impression. 273. The Major Diatonic Scale. When two or more notes are sounded together the effect upon the ear may be either pleasant or disagreeable. If the keys of a piano are struck at random, the effect will certainly be disagree- able. This shows that the pleasing or harmonious combina- tions of sounds are comparatively few in number, while the discordant combinations are limitless. The harmonious musical intervals can all be expressed as the ratio of small whole numbers, such as f , f , J, etc. These are called simple ratios, as distinguished from the ratio of larger numbers, e.g. -j-f . The most perfect harmony is that of a note and its octave, and the interval in this case is the simplest possible. Any three notes whose frequencies are in the ratio 4, 5, 6 form a major triad. This combination is especially pleasing, and is the basis of the major diatonic scale, which is made up of three such triads, as follows: do mi sol fa la do' sol si re' If we reduce the last note, re' , one octave, to bring it within the same octave as the others, we have the complete scale. The following table gives first the relative fre- quencies of the notes in terms of the smallest whole num- bers in which they can all be expressed, second the interval between each note of the scale and the first or keynote, third the interval between each note and the preceding PROPERTIES OF MUSICAL SOUNDS 327 one. This table should be memorized, as a knowledge of it is presupposed in the discussion of later topics. do re mi fa rf la si do' 1. 24 27 30 32 36 40 45 48 2. i 9 t f f 5 3 2 3. i t. H t f if It will be observed that the intervals between successive notes are of three different magnitudes, namely f , -9-, and |f. The first two are only slightly unequal and are called whole tones; the third is considerably smaller and is called a semitone. The eighth note completes the octave, and is at the same time the first of another series of eight notes, each of which is an octave above the note of the same name in the preceding series. The scale may thus be repeated both upward and downward, through as many octaves as is desired. The intervals remain the same whatever the absolute pitch of the keynote may be. When middle C (as sounded by a tuning fork) is taken as the keynote, the letter names and the vibration numbers of the notes of the scale are as given below. Position on the staff, Letter names, c' d' e' f g' a! V c' Syllable names, do re mi fa sol la si do' Vibration numbers 256 288 320 341^ 384 426! 480 512 Vibration ratios, i f 4 I I ; t 2 Intervals between successive notes, I If t I If 274. The Equally Tempered Scale. The white keys of a piano or an organ have the semitone intervals in the right place for musical compositions written in the "key of C." For compositions in any other key one or both of these intervals would be out of place, and a whole tone interval would come where a semitone belongs. It is 328 SOUND therefore necessary to introduce other notes within the octave. These notes are called sharps or flats, and are played by means of the black keys. Five notes are thus added in the octave, forming the chromatic scale CC#DD#EFF8GG#AA#B C It is further necessary to change the intervals slightly, so as to make them all exactly equal; otherwise the interval V would sometimes occur where the interval should be, and vice versa. The scale thus modified is called the equally tempered scale, to dis- tinguish it from the natural scale first described. The tempered scale is a compromise, necessitated by the practical requirements of musical instruments. In the physical study of sound, pitch is always referred to the natural scale. 275. Limits of Audibility. Range of Pitch Used in Music. The human ear is not sensitive to all rates of vibration. If the rate is below a certain minimum, either no sound is heard or only separate pulsations. This minimum varies with different persons, but generally lies be- tween 1 6 and 30 vibrations per second. ' IG ' 2 WhiTtk alt n ' S lt is an inter esting fact that the muscles vibrate when in action, sounding a note near the lower limit of audibility. This note can be distinctly heard as a rapid pulsation by pressing the palms of the hands firmly over the ears. The sound comes from the muscles of the arms, which are then contracted. The lowest note of a piano (A of the fourth octave below middle C) is near the limit of audibility, its frequency being 26.6. There is also a higher limit of frequency above which the ear is not affected and no sound is heard. This upper limit varies much more widely with different persons than the lower, ranging, as experi- ment shows, from 12,000 to 30,000 vibrations per second. The shrill cry of a bat and the high-pitched noises of many insects are inaudible to some whose ears are normal for sounds of ordinary pitch. The upper limit of audibility is determined by gradually raising the pitch of a Gal ton whistle (Fig. 225) until the note is no longer heard. When the experiment is tried before a class it will always be found that the note is clearly heard by some after it has become inaudible to others. The highest note of a piano is the fourth C above middle PROPERTIES OF MUSICAL SOUNDS 329 C, and its vibration number is 256 X 2 4 = 4096. Pitch is not easily or definitely appreciated beyond this limit. A sound of any pitch will be inaudible if the amplitude of vibra- tion is too small. Lord Rayleigh found that the faintest audible sound has an amplitude of vibration of the air particles of about one millionth of a millimeter. "In an extremely loud sound, such as that of a steam whistle heard close at hand, the amplitude of vibration is probably less than i mm." H attack. 276. The Relation between Pitch, Wave Length, and Velocity. Consider what happens while a sounding body vibrates for one second. If n denotes the frequency of the body, a train of n waves will be sent out during the second. At the end of the second the last of these waves is on the point of leaving the body, and the first, having traveled for one second, has gone a distance equal to the velocity of sound in the medium. Let v denote this veloc- ity and / the length of the wave; then, since the n waves extend over the distance v, the length of each wave is one ii h of v; i.e. I = ~, or v = In. It follows that the wave length in a given medium is inversely proportional to the frequency, e.g. raising the pitch of a sound one octave reduces its wave length one half. It follows also that the wave length of a sound of given pitch is proportional to the velocity of sound in the medium. Thus the wave length of middle C in air at o is 1090 -f- 256 = 4.26 ft. ; in water it is about 4.3 times as great, and in steel 15 times as great, or 64 ft. 277. Interference of Sound. Under ordinary circum- stances, when many bodies are vibrating at the same time, the sound from each is heard just as if the others were silent. The leader of an orchestra recognizes the notes from each instrument, though all are sounding together. 330 SOUND It is evident, therefore, that the same body of air may, at a given instant, be taking part in the transmission of any number of sounds in any and all directions, regardless of the relative lengths and intensities of the different waves. The actual motion of each air particle is, of course, the resultant of all the component motions that it would have at the instant in transmitting the waves individually. The union of two or more sets of sound waves often pro- duces certain special effects, which differ under different conditions. These effects include harmony, discord, and quality, the causes of which remain to be considered, and the phenomena of interference, beats, and resonance. The simplest phenomenon of this class is the mutual destruction of two trains of sound waves in cer- tain regions, producing silence in those regions. This is called de- structive interference. Any tuning fork affords an excellent example. When a vibrating fork is held close before the ear and slowly ro- tated about the stem as an axis, the sound swells to a maximum and dies away to silence four times during one complete rotation. With the fork held in a position of silence, the sound is restored by covering either prong with a small paper cylinder, care being taken not to touch the prongs, as this would stop the vibration. The phenomenon can be shown before a class with the aid of a resonance jar, tuned in unison with the fork (Fig. 226). These curious effects are explained with the aid of Figs. 227 and 228, which represent the waves about a sounding fork, the prongs pointing toward the observer. As the prongs move apart a compression is set up on the outside of each, and a rarefaction between them; as they move toward each other the effects are inter- changed. The fork thus sends out four trains of waves of equal wave FIG. 226. Interference. PROPERTIES OF MUSICAL SOUNDS 331 length and intensity, but with the waves of adjacent trains in opposite phase, i.e. the waves of one train are half a wave length in advance of FIG. 227. Component Waves FIG. 228. Resultant Waves about a Tuning Fork. about a Tuning Fork. the waves of the adjacent train. If the adjacent trains had no effect upon each other, the compressions of one would travel outward side by side with the rarefactions of the other and vice versa, as repre- sented in Fig. 227. But in the region where they meet, their opposing tendencies constantly neutralize each other; for the compression would be transmitted by a forward motion of the air particles and the rarefaction by an equal backward motion at the same time. Hence the air in this region remains at rest and there is neither compression nor rarefaction. Silence is thus the result of the interference of the waves with each other. 278. Periodic Interference or Beats. A more impor- tant case of interference is that of two trains of waves of very slightly unequal wave length. Two forks of the same pitch will serve to illustrate. If they are of exactly the same pitch, they sound together as one; but if there is a very slight difference (as when the prongs of one are loaded with a bit of soft wax), their united sound periodically swells and dies away in strongly marked pulsations. This is explained as follows: We will suppose that two middle C forks are used and that, by loading with wax, the fre- quency of one is reduced from 256 to 255. At intervals 332 SOUND of one second, the forks "keep step" in their vibrations; and the waves that they then set up approximately coin- cide, compressions with compressions and rarefactions with rarefactions, as at A and C (Fig. 229), in which the compressions are represented as crests and the rarefac- tions as troughs. These waves unite in resultant waves of increased intensity, as represented at X and Z. Half a second after each coincidence the forks vibrate oppositely, FIG 229. Interference of Two Trains of Sound Waves. the compressions produced by each approximately coin- ciding with the rarefactions produced by the other, as at B; and the resultant waves are of diminished intensity. A complete set of intensified and weakened resultant waves is thus sent out from the forks during each second. This constitutes one beat. Beats may therefore be defined as regularly recurring pulsations of sound caused by the successive reinforcement and interference of two trains of sound waves differing slightly in wave length or pitch. The number of beats per second is equal to the difference between the frequen- cies of the two sounds. Beats are produced when two wires of a sonometer, tuned nearly to. unison, are sounded together. They become less frequent and finally cease as the unison is made more nearly perfect. 279. Beats the Cause of Discord. When beats become too frequent to be separately distinguished by the ear, the PROPERTIES OF MUSICAL SOUNDS 333 constituent notes blend into a rough, unpleasant sound, or discord. This can be shown by means 'of a sonometer (Fig. 230) . Its two wires are first tuned to unison, then the pitch of one is gradually raised, either by shortening the vibrating portion with a movable bridge or by increas- ing the tension. The beats grow more rapid as the inter- val between the notes increases, and presently merge into a discord. As the interval is further increased, the sound becomes less discordant, then harmonious, then again dis- cordant, etc. The first harmonious interval is f (mi-sol), the second f (do-mi), the third f (do-fa), etc., as we have already learned in studying the musical scale. That discord is always due to beats was shown by the investigations of the noted German physicist, von Helm- holtz. A rapidly pulsating sound is disagreeable to the ear, just as a flickering light is to the eye. But if beats cease and the sound becomes steady when the interval between two notes is increased to f , how is the recurrence of discord at greater intervals to be accounted for? The answer is that the notes of all musical instruments are complex. The principal constituent of a note is always accompanied by one or more generally several higher constituents, and beats are possible between these higher constituents of different notes. We shall have occasion to refer to this again when we take up the study of the com- plex character of sounds. 334 SOUND 280. The Frequency of Vibrating Strings. The notes of nearly all musical instruments are produced by the vibra- tion of strings (including wires) or of columns of air. The laws of vibration of strings are therefore of special impor- tance in the study of musical sounds, and now demand our attention. Although the sound of a stringed instrument comes almost wholly from the body of the instrument, or from some part of the body, the pitch of the notes is determined by the rate of vibration of the strings. It is well known, in a general way, that the pitch of a string is raised by shorten- ing the part that vibrates or by increasing its tension, and that a light string sounds a higher note than a heavier string of the same length and under the same tension. The effect of length is illustrated by the different notes obtained from the same string of a violin, mandolin, or guitar, by vary- ing the length of the vibrating portion with the finger; the effect of tension, by the use of the tightening pegs in tuning the strings ; and the effect of mass, by the different strings of the instrument, the heaviest giving the lowest note. Definite information concerning these effects is readily obtained with the aid of a sonometer (Fig. 230). The effect of length is particularly important for our pur- pose and must be carefully noted. Effect of Length. If the two wires of a sonometer are tuned to unison and the bridge is then placed under one of them at its mid-point, it will be found that the note of this wire has been raised exactly an octave. (In this and similar experiments the interval may be judged by the ear, the two notes being sounded together or in quick suc- cession, or the pitch of each note can be determined by com- parison with a set of forks.) If the note sounded by the whole length of the wire is taken as the first note of the scale, IjROPERTIES OF MUSICAL SOUNDS 335 or do, re is sounded by | of the length, mi by -f , fa by f , sol by |, la by f , and si by -f%. These length ratios are the reciprocals of the intervals between the corresponding notes (Art. 273). The relation is general: The tension re- maining the same, the frequency of a string or wire varies inversely as its length. Effect of Tension. If the sonometer is provided with some means of measuring tension, it will be found that the pitch of a string is raised one octave by increasing its tension fourfold. The general law is that, other conditions remaining constant, the frequency of a string varies directly as the square root of Us tension. Effect of Mass. If a sonometer is strung with piano wires whose diameters are in the ratio f, and the lengths and tensions of the wires are the same, the note of the smaller will be an octave above that of the larger. Since the cross-sections of the wires are as the squares of their diameters, their masses are also as the squares of their diameters. Hence the masses of the wires are in the ratio f ; and since their frequencies are in the ratio i, it follows that their frequencies are inversely proportional to the square roots of their masses. In general: The lengths and tensions of two strings being the same, their frequencies are inversely proportional to the square roots of their masses. With strings of the same material, their frequencies are inversely proportional to their diameters. Since the laws of strings are only special cases of accelerated motion, it is clear that they must be in accord with the general laws of dynam- ics. The proof of this involves mathematical work beyond the scope of elementary physics. It will be an instructive exercise, however, to account in a general way for the increase of frequency with decrease of length, increase of tension, and decrease of mass per unit length. (This is left as an exercise for the pupil.) PROBLEMS 1. Why is the sound of a fork restored in the position of silence when one prong is covered? Would you expect to find new positions of silence about the single prong? Text your conclusion. 2. Why is the pitch of a fork lowered by loading its prongs with a piece of wax? 336 SOUND 3. Compute the wave lengths of the following notes in air at 20: Ci ( = 32), C ( = 64), c v (= 4096), and the highest audible sound, assuming it to be 30,000. 4. A string i m. long sounds C (= 64) under a certain tension. Com- pute the frequency of the notes sounded by f , f , i, |, |, fc and | of its length, the tension remaining the same. All these notes but one are notes of the diatonic scale in the first three octaves above the note of the whole wire. Calling the lowest note do, what are the syllable names of the others? 5. The pitch of the whistle or bell of a passing locomotive or gong of a trolley car drops as the source of sound passes the observer and changes from approaching to receding. This phenomenon is called, after its dis- coverer, the " Doppler effect." Explain it. 6. When sound travels from colder to warmer air, does its wave length change? Does its frequency change? Explain. 7. Examine the wires of a piano and find in what respects they differ from one another. What means are employed to obtain the wide range of pitch from the lowest note to the highest? 281. Fundamental Tone and Overtones. When a string swings to and fro as a whole, it sounds its lowest note for the given length and tension. This is called its funda- mental tone. A string can also vibrate in two or more equal segments; and the notes thus produced are higher than the fundamental, and are called overtones or harmon- ics. This mode of vibration can be studied to advantage by means of a long rubber tube or a coil of wire. When such a tube is made fast at one end and stretched in either a vertical or a horizontal position, it can be thrown into vibration as a whole by impulses properly timed with the hand at the free end (Fig. 231). This is the motion of a string when sounding its fundamental tone. When the rate of the impulses is doubled, the tube vibrates in two equal segments (Fig. 232). This is the motion of a string when sounding its first overtone; and, by the law of lengths, the pitch of this tone is the octave above the fundamental. Similarly, under impulses which are three times as rapid PROPERTIES OF MUSICAL SOUNDS 337 as at first, the tube vibrates in thirds; under impulses four times as rapid it vibrates in fourths, etc. In all cases the FIG. 231. Vibration as a Whole. FIG. 232. Vibration in Two Segments. vibrating segments are separated by points, called nodes, which remain approximately at rest; and adjacent seg- ments are constantly in opposite phases of their motion, i.e. when any segment is moving down, the next one on either side is moving up, etc. The frequency varies inversely as the length of the segments, or directly as the number of segments. The wire of a sonometer can easily be made to vibrate as a whole or in any number of equal segments up to eight or ten. When it is bowed or plucked near one end and, at the same time, is lightly touched at its mid-point with a FIG. 233. String Sounding its Second Overtone. feather or the tip of the finger, it vibrates in halves, sound- ing the octave above its fundamental, even after the bow 338 SOUND and the finger are removed (the finger being removed an instant after the bow). When it is touched at one third its length from the end that is bowed, it vibrates in thirds (Fig. 233); when touched at one fourth its length, it vi- brates in fourths, etc. In every case the entire string vibrates. Its division into segments can be shown at a distance by placing upon it small folded pieces of paper. These " riders" are thrown off by the vibration except at or very near the nodes. The overtones are numbered in order, beginning with the lowest. If we call the fundamental tone do\, the series of possible overtones, up to and including the seventh, is as follows: OVERTONES FUNDA- MENTAL ISt and 3rd 4 th 5th 6th 7th No. of segments I 2 3 4 . 5 6 7 8 Frequency n 2n 3n 4n 5n 6n ?n 8n Relative pitch dot do 2 soh dos mia soh do* 282. Simple and Complex Sounds. Quality. The question remains whether a string can vibrate as a whole and in segments at the same time. This is answered by the following experiments. Let the wire of a sonometer be bowed or plucked at about one fourth its length from one end. It sounds its fundamental, and is therefore vibrating as a whole. If, while still vibrating, it is lightly touched at the center with the finger or a feather, the fun- damental ceases and the first overtone is heard; hence it must be vibrating in halves. The touch at the center destroys the vibration as a whole, but does not interfere with the vibration in halves, since this is the position of PROPERTIES OF MUSICAL SOUNDS 339 the node. The overtone is present with the fundamental; but, being relatively weak and blending perfectly with the lower note, it can be recognized only by fixing the atten- tion upon it, and then only by a trained ear. If the wire is very lightly touched at the center when plucked at the quarter, it is possible, by skilful manipulation, to sound the fundamental and the overtone with nearly equal inten- sity, and both can then be distinguished with little diffi- culty. The behavior of a string or wire when sounding FIG. 234. Complex Vibration of a String as a Whole and in Halves. its fundamental and first overtone is shown by the full lines of Fig. 234. The vibration as a whole is represented by the dotted lines. Upon this is superposed the vibra- tion in halves. When the wire is plucked at the center, then touched at that point, the octave is not heard. The reason is that vibration in halves requires a node at the center, and a node can not exist at the point where the string is drawn aside. When the wire is plucked at one sixth its length 340 SOUND from one end and touched at one third its length, the fundamental is quenched and the second overtone is heard. Higher overtones than the first or second are also present in every case. This can be shown as follows. Pluck the wire near one end, and immediately touch it at the center. The first overtone is heard. Again pluck it, exactly as before, and touch it at one third its length. The fundamental and the first overtone are quenched by the touch, and the second overtone is heard. Repeat, touching the wire at the quarter. If the third overtone is present it will now be heard, as all lower notes are quenched. Similarly the presence of any overtone can be determined by touching the vibrating wire at the point where a note would occur for that overtone. As a rule the first six or eight overtones can be detected in this manner; but their relative intensity varies greatly with the place where the wire is struck, plucked, or bowed, and also with the character of the impulse. When it is struck with a soft mallet or plucked with the finger near the center, the fundamental is loud and all the overtones weak; when it is struck or plucked near the end the higher overtones are much stronger, and especially when it is struck with a hard body or plucked with the finger-nail. In listening to the note of the wire in the above experi- ments, the pupil has doubtless observed that it sounds differently when produced in different ways. It may be equally loud in the different trials, and it does not vary in pitch, for the pitch is always that of the fundamental. The difference is a difference of quality (Art. 271). When the wire is struck with a rubber mallet or plucked near its center, the tone is soft and mellow; when struck or plucked near one end, the tone is described as full, rich, or bright; when struck with a hard body or plucked with the finger- nail close to one end, the tone loses its musical character and becomes a sharp, discordant jangle. These various qualities are plainly due to the overtones. The first five overtones are in harmony with one another, as well as with the fundamental (see table). When the wire is sounding PROPERTIES OF MUSICAL SOUNDS 341 only these, the effect is pleasing. The first discordant overtone is the sixth of the series, which forms a discord with both the fifth and the seventh. The seventh and ninth overtones are in harmony with each other and with all below the sixth; the eighth and the tenth are discordant. As we go higher in the series, the discordant overtones are found in increasing number. Sounds are in general complex, and their quality is deter- mined by the pitch and relative intensity of the waves of higher frequency which accompany the fundamental, A sound without overtones is called a simple or pure tone. The only perfect example is the note of a tuning fork. The overtones of a fork are very high. They sound as a clear, shrill note when a fork is struck; but they quickly die away, leaving only the fundamental. The tones of the diapason or stopped pipes of an organ are the nearest approach to simple tones used in music. Their overtones are few and weak and affect the quality but little. In general, musical sounds are rich in overtones, or harmonics, as they are usually called; but the particular combination of overtones and their relative intensities vary with differ- ent instruments. " In the sound of a violin the upper harmonics are loud and piercing; the nearer harmonics are feeble, and the fundamental tone stands apparently alone, but rendered penetrating in quality by the high mass of harmonics. In a piano string struck by an elastic soft ham- mer the harmonics up to the sixth are present; the seventh is obliter- ated, or nearly so, by the hammer being made to strike the string at a spot one seventh of its length from the end of the string; and the components beyond the seventh are feebly represented." Daniell. 283. Harmony Destroyed by Discordant Overtones. Two notes sounded together are discordant when beats are produced by the fundamentals of the notes, by either fundamental and an overtone of the other, or by an overtone of each (Art. 279). It can be shown 342 SOUND that these numerous possibilities of discord limit the harmonious combinations to notes whose relative frequencies are expressed by simple ratios (Art. 273). 284. Vibrations of Bells. A body sounds its fundamental tone when vibrating in the least number of segments possible. With strings this is one segment; but the number varies with different bodies. 1 With bells and bell- shaped bodies, such as glass tumblers and bowls, it is four. The motion of the rim of a bell is shown in Fig. 235, the point where the clapper strikes being the middle of a segment. The over- tones of bells are not, as a rule, multiples of the FIG. 235. Seg- fundamental, and their musical quality is conse- S^undin^ B it! \e precautions are taken to secure accuracy. Manufacturers issue catalogues giving the results of these experimental tests, so that buyers may be able to select the lighting equipment that best suits their needs. PROBLEMS 1. What does the act of aiming a gun assume concerning light? Explain. 37 2 LIGHT 2. The shadow of a flagpole is 80 ft. long upon the ground when the shadow of a vertical stick 4 ft. high is 2.5 ft. What is the height of the flagpole? 3. Why are shadows as we see them in nature not perfectly dark? 4. What is the apparent shape of the moon two or three days after new moon? at the first quarter? between the first quarter and full moon? Account for these different appearances. 6. Assuming the diameter of the sun to be 866,000 mi., the diameter of the earth 8000 mi., and the distance between the sun and the earth 93,000,000 mi., find the length of the umbra of the earth's shadow. 6. State and account for the change in the size, brilliance, and sharp- ness of outline of a pinhole image (a) when the screen upon which it is caught is moved farther from the opening; (b) when the size of the opening is increased. i 7. What determines the ratio of the length of a pinhole image to the length of the object? 8. In what ratio does the illumination upon the page of a book change as the book is moved from a distance of 2 m. to a distance of 60 cm. from a lamp? 9. An incandescent lamp at a distance of 150 cm. and a standard candle at a distance of 21 cm. equally illuminate the screen of a photometer. Find the candle power of the lamp. III. REFLECTION OF LIGHT 307. Regular and Irregular Reflection. The behavior of light after reflection from mirrors and other surfaces can FIG. 2630. Regular Reflection. FIG. 2636. Irregular Reflection. be studied to the best advantage by means of a beam of sunlight admitted into a darkened room. Chalk dust in the air makes the path of the light plainly visible. Under REFLECTION OF LIGHT 373 such conditions it will be observed that the reflection of a beam by a plane mirror merely changes its direction (Fig. 2630). It is still a beam of light. This is a case of regular reflection. When the beam falls upon a sheet of paper, the light is scattered or diffused in all directions by the minute irregularities of the surface (Fig. 2636). This is called irregular or diffuse reflection. It will be observed that the paper is brilliantly illuminated by the sunbeam and can be distinctly seen from all parts of the room; but the surface of a mirror is nearly invisible, even when the eye is in the path of the reflected beam. Light is regularly reflected only from surfaces (either plane or curved) which appear smooth even under the micro- scope, such as the surfaces of still water, glass, polished metals, and ordinary mirrors. An unpolished surface is made up of minute areas inclined at all angles to one an- other, and each one acts as a separate reflector. Any small spot or " point" of such a surface includes a sufficient number of these irregularities to reflect light in all direc- tions, and thus in effect becomes a new source of light waves. The light received from the sky during the day is sunlight that has been diffused by minute particles of dust in the air. This diffusion takes place principally in the lower regions of the atmopshere, where the dust particles are largest and most numerous; hence the sky is much darker upon high mountains than at low altitudes. Shadows cast by objects in the sunlight are only relatively dark, as they are generally quite strongly illuminated by diffused light from the sky and from surrounding objects. 308. Visibility of Objects. Objects which are not themselves luminous are seen only by the light which they diffuse. Regular reflection produces images. The light appears to come from the image of the source, not from the 374 LIGHT reflecting surface. A good mirror with a clean surface is nearly or quite invisible, a fact which is often turned to account in producing stage illusions. When an object is viewed directly (i.e. without the aid of mirrors of lenses), the light by which each point is seen comes straight from the point to the eye as a slender diverging cone, and the point is seen in its true position at the vertex of FIG. 264. The Eye Receives a Slender , v /-r^' n \ Cone of Light from each Visible Point. thlS COne V Fl S' 26 4)- 309. Laws of Reflection. When light falls upon any surface the phenomenon is called the incidence of light, and the light before it reaches the surface is called inci- dent light. In Fig. 265 MN represents a section through a mirror whose surface is perpendicular to the plane of the paper; CP' represents a slender beam (ray) , falling upon the mirror ** at P (called the point of inci- dence), and P'D the reflected beam; and PP' is the perpendi- FIG. *6 S . -Reflection, i, Angle cular or normal to the reflecting of incidence; r, Angle of .. Reflection. surface at the point of incidence. The angle i between the incident beam (or ray) and the perpendicular to the reflecting surface at the point of inci- dence is called the angle of incidence, and the angle r between the reflected beam and this perpendicular is the angle of reflection. Direct measurement establishes the following laws of reflection: (i) The angles of incidence and reflection are equal; and (2) they are in the same plane. Since this plane contains the normal, it is always perpendicular to the re- flecting surface. REFLECTION OF LIGHT 375 These laws are readily accounted for by considering what happens to light waves during reflection from a plane surface. Let 6* (Fig. 266) be a point source radiating light waves which fall upon the mirror MN. The shortest distance from ,5* to the mirror is along the FIG. 266. Position of a Point Image. perpendicular from 5 to the mirror, or SP', hence each wave strikes the mirror first at P, and its reflection begins at that point. While the part A of the wave APB is advancing to A' and the part B to B', the part at P would advance to P', if the mirror were not there, and the wave would occupy the position A' P' B' '; but with the mirror in place, the part at P is reflected and travels back an equal distance PP" in the same time, and the entire wave is reflected so as to occupy the position A'P"B f . Now from the geometry of the figure it is evident that the arcs A' P' B' and A' P"B' are equal arcs of equal circles; hence their radii are equal. But the radius of A 1 P'B' is P'S; hence, laying off an equal distance P"I along the perpendicular, we find 7, which is the center of the arc A'P"B'. I is therefore the center of the reflected waves. This means that the reflected light behaves in all respects as if it radiated from a source at 7. The direction of all reflected rays is 376 LIGHT from / as a center; e.g. light traveling along SB' is reflected along B'C. Further, it is easily proved from the geometry of the figure that the angle of incidence i is equal to the angle of reflection r, and these angles evidently lie in the same plane (the plane of the paper). (The proof that angle i = angle r is left to the pupil. Remember that, by construction, SI and QB' are each perpendicular to M N, that PP" = PP', and that SP' = IP".) 310. The Image of a Point in a Plane Mirror. It is found by experiment that the image of a point in a plane mirror (from whatever position it may be viewed) is on the perpendicular from the point to the mirror, and is as far behind the reflecting surface as the point is in front of it. Now in Fig. 266 SP' = IP" and PP' = PP"; hence SP' PP' = IP" PP", or SP = IP. That is, the cen- ter / of the reflected waves is on the perpendicular from the point source S to the mirror, and is as far behind the mirror as the source is in front of it. It follows that the point image is at the center of the reflected waves. The im- age is, in fact, nothing else than the apparent source of the reflected light. In looking at the image the eye receives a cone of light whose vertex is at the image; and the impression received is just the same as if this vertex were the real source. The eye takes no account of the original direction of the light. In all diagrams representing optical phenomena the actual path of light is represented by full lines and the apparent path by dotted lines, as in the above figure. 311. Image of a Body Object Formed by a Plane Mir- ror. In the discussion of images we shall call an object having visible form and dimensions a body object, to dis- tinguish it from a point object or point source of light. The image of a body object is made up of point images of the corresponding points of the body; i.e. the light radiated or REFLECTION OF LIGHT 377 diffused from each point of the body falls upon the mirror, is reflected, and after reflection travels from the direction of its point image, just as if no other M points and no other light were there. c , It follows, therefore, that the line join- ing any point of a body and the image of j ^ that point, formed by a plane mirror, is %-- N FIG. 267. perpendicular to the mirror and is bisected by it (Fig. 267). Images formed by plane mirrors are evidently of the same size as the objects. They are erect (unless the mirror is horizontal, as the surface of still water); but object and image differ as the right hand differs from the left. To locate such an image in a diagram perpendiculars are drawn from a sufficient number of points of the object to the mirror, and ex- tended equal distances behind it, as shown in the figure. The ex- tremities of these perpendiculars are the corresponding points of the image. To construct the path of light to the eye for any point of the image, as ZX, we draw a line from that point to the eye, and from the point of intersection of this line with the mirror we draw a line to the corresponding point of the object. This construction makes the angles of incidence and reflection equal without the trouble of meas- uring them. Such diagrams are com- monly simplified by drawing only a single line to represent the cone of light that enters the eye from any point. 312. Images by Multiple Reflec- tion. When two mirrors are placed facing each other, either parallel or at an angle, a part of the light from any source between them is reflected from one to the other, giving rise to a series of images. Thus with mirrors AB and CD (Fig. 268), placed as shown, three images of the point source O are formed as follows: A single reflection from FIG. 268. 378 AB forms the image LIGHT FIG. 269. Kaleidoscopic Pattern Formed by Three Mirrors at Angles of 60. Some of this reflected light falls upon the mirror CD, and is again reflected, forming the image /2. The position of this image is the same as it would be if /i were the actual source of the light.- (Why?) A part of this light again falls upon AB, forming the image 7 3 , just as if /2 were its source. With the given angle be- tween the mirrors, /a is the last image of the series; for /3, which is now the center of the reflected waves, lies behind the plane of the mirror CD, and no light from this point or from its direction falls upon CD. The figure shows the path of the light by which /2 is seen from the position E. There is also a second series of three images, formed by reflection first from CD, then from AB, then from CD again. This series is not shown in the figure. The smaller the angle between the mirrors the greater is the num- ber of images (Fig. 269). When the' mirrors are parallel the series is indefinite, being limited only by the gradually failing intensity of the light. PROBLEMS 1. What is the length of the shortest plane mirror in which a man 6 ft. tall can see his full-length image? 2. If a person stands erect before a plane mirror inclined forward at an angle of 30, at what angle to the vertical is his image? 3. (a) In what respects does a pinhole image differ from an image formed by a plane mirror? (b) What purpose is served by the screen upon which a pinhole image is caught? (c) Can an image formed by a plane mirror be caught upon a screen? 4. Why are stars invisible by day? 5. Copy Fig. 268 and locate the other series of images. Construct the path of light to the eye for the third image of either series. REFLECTION OF LIGHT 379 6. Draw a figure of mirrors at an angle of 90, locate the images of a point object, and construct the path of light to the eye for each image. 7. Make a similar diagram with the mirrors at an angle of 60, and the object at unequal distances from the mirrors. Prove that a point object and its images lie on the circumference of a circle. 8. If a kaleidoscope is provided, examine it and account for the geo- metrical patterns observed in it. 9. Show by means of a diagram that in the case of parallel mirrors a point and its images lie on a straight line. 313. Reflection from a Concave Spherical Mirror. The reflecting surface of a concave spherical mirror is a portion (usually a very small portion) of a spherical surface, the T K FIG. 270. Formation of a Real Image by a Concave Mirror. reflection from which takes place on the inner or concave side. Such a mirror is represented in a section diagram by an arc of a circle, as MA 7 " (Fig. 270). The center of curva- ture of the mirror, C, is the center of the sphere of which the mirror is a part. The radius of this sphere is called the radius of curvature of the mirror. The straight line AC, which passes through the center of curvature and the center of the reflecting surface, is called the principal axis of the mirror; any other line passing through the center of curvature to the mirror is a secondary axis. When a point source of light, S (Fig. 270), is beyond a certain distance from a concave mirror, the divergent cone of incident light, MSN, is reflected as a converging 380 LIGHT cone, MIN. The reflected waves are concave, with /, the vertex of the cone, as their center. All the reflected light from the point 5 comes to the point /, where it is said to be brought to a focus. Beyond / the waves are convex, with 7 as their center, just as if this point were their original source. The reflected rays continue in straight lines through 7, where they all meet and cross. When the reflected light falls upon a screen, it illuminates .a circular area, which is the cross-section of the cone at that place. As the screen is brought nearer to 7, the illuminated area becomes smaller and brighter; at 7 it is a bright point, and the light is then said to be focused upon the screen. This point of light is the real image of the point source S. The image is real in the sense that it is formed by light which actually travels to it and from it; and it exists at 7 whether caught upon a screen or not. It can be viewed without the aid of a screen from any position within the cone KIL, and appears to be out in space where it really is, when the eyes are directed toward it (not toward the more distant mirror). If the point source is at 7 its image will be at 5; for the path of any incident and reflected ray from S to 7 is also the path of a ray from 7 to S. Any two points which, like S and 7, are so situated with respect to a concave mirror that light radiating from either converges to the other are called conjugate foci. 314. Conjugate Foci on the Principal Axis. The Prin- cipal Focus. The relative positions of a point source and its image can best be accounted for and most easily remem- bered by considering the angles of incidence and reflec- tion of the rays. These angles are equal for any ray, as REFLECTION OF LIGHT 381 is always the case in regular reflection, whether from plane or curved surfaces. The perpendicular at any point of a concave mirror is always a radius of the spherical surface, and passes through the center of curvature, as M C and NC (Fig. 271). Hence the angle of incidence of the ray SM is SMC, and the angle of reflection is CMG. The cone of light MSN from a point source, 5, on the principal axis and beyond the center of curvature, is reflected FIG. 271. Conjugate Foci on the Principal Axis. as a converging cone MIN, whose vertex /, is also on the principal axis and on the opposite side of the center of curvature. / is the focus of the reflected light and is the image of S. Its exact position is determined in the figure by constructing any two incident and reflected rays, as SMG and SNE. Any two will serve, since all intersect at the same point. We might regard SA as one of the two, for it is incident along the radius CA and is reflected back along the same path, as is always the case with per- pendicular incidence. If the source S is moved away from the center of curvature along the principal axis, the angle of incidence i increases; and since the angle of reflection r is always equal to i, it is evident from the figure that the focus / moves away from the center of curvature toward the mirror. When 5 is moved to a relatively great distance (100 times the radius of curvature or more), the incident rays are all sensibly parallel to the principal axis, and their focus is called the principal focus of the mirror. 382 LIGHT This can be shown in a striking manner with a broad beam of sunlight, falling upon a concave mirror in a darkened room. The reflected light converges to a small spot of intense brightness, which lies to one side of the prin- cipal axis when the mirror is oblique to the incident beam (Fig. 272). As the mirror is turned round until its prin- cipal axis is parallel to the incident rays, the focus F' moves with it to the point F on the principal axis. This point is the principal focus. FIG. 272. Focus of* an Oblique Beam. o* ^i i i Strictly speaking, however, the sun is only approximately equivalent to a point source, since rays from opposite sides of it form an angle of half a degree at the distance of the earth. Hence the reflected light focuses as a small round spot, which is an image of the sun and is larger than a point. The principal focus lies on the principal axis midway be- tween the mirror and the center of curvature. Its distance from the mirror is called the principal focal distance or the focal length of the mirror, and is equal to half the radius of curvature. This is the least possible distance of a real image. The position of the principal focus can be determined by geometry as follows: Let BM (Fig. 273) be any incident ray parallel to the principal axis, F the point where the reflected ray cuts the axis, and C the center of curvature. Angles i and e are equal (alter- nate-interior angles of parallel lines), and angles i and r are ^FOCUS.^ equal (law of reflection). Hence angles r and e are equal, the triangle MFC is isosceles, and MF = FC. If MA is not more than a tenth of the radius "of curva- ture (which is necessarily the case if the mirror is to give distinct REFLECTION OF LIGHT 383 images), AF is very nearly equal to MF, and hence also to FC; i.e. AF = PC, approximately. For points of incidence at and immedi- ately about A, the equality is exact; and this determines the true position of the principal focus. Since a point source and its real image have interchange- able positions, the above discussion may be summarized and extended as follows: The image of a point source at any relatively great distance on the principal axis is at the principal focus. As the source moves up along the prin- cipal axis to the center of curvature, its image moves from the principal focus to the center of curvature, where source and image coincide. (Why?) As the source moves up from the center of curvature to the principal focus, its image moves away from the center of curvature to an indefinite distance, the reflected rays being then parallel to the prin- cipal axis. As the source moves up from the principal focus to the mirror, the reflected rays become more and more divergent, and the image is virtual, as in a plane mirror (Art. 316). 315. Conjugate Foci on Secondary Axes. Real Image of a Body Object. A point source A (Fig. 274 a, b, c, and d) and its image A' always lie on the same axis, i.e. the same straight line through the center of curvature of the mirror; and the source and its image have the same rela- tive positions on a secondary axis as on the principal axis. In locating the image of a point source in a diagram, it is sufficient to find the point of intersection of any two reflected rays from the given point. When the point is not on the principal axis, the measurement of angles can be avoided by choosing any two of the following rays: (i) The incident ray passing through (or from the direction of) the center of curvature, which is reflected back along 384 LIGHT the same path; (2) the incident ray parallel to the princi- pal axis, which is reflected through the principal focus; FIG. 274. Do, Distance of Object; Di, Distance of Image; /, Focal Length of Mirror. (3) the incident ray passing through the principal focus, which is reflected parallel to the principal axis. This method of construction serves for both real and virtual images, and is followed in Figs. 274, 275, and 277. In constructing the image of a body object (usually represented by an arrow), we locate the image of top and bottom by the above method. The line con- necting these points represents the size and position of the image in true proportion. In Fig. 274 a, b, and d, this construction is shown for the image of A only, to avoid confusion. REFLECTION OF LIGHT 385 Real images formed by concave mirrors are always, inverted, since each point of the object and the image of that point are on the same axis and on opposite sides of the center of curvature, and all axes cross at this center. From the similar triangles ACB and A'CB' of Fig. 275, it follows that the size (length) of the image is to the size (N/ FIG. 275. of the object as the distance of the image from the center of curvature is to the distance of the object from this cen- ter. It can also be shown that these distances are always in the same ratio as the distances of image and object from the mirror. Hence the image is always larger than the object when it is at the greater distance from the mirror, and smaller than the object when it is at the less distance. 316. Virtual Images by Concave Mirrors. When light from a point source at the principal focus is j^ected by a concave mirror, the reflected waves are planeMhe rays parallel. The light does not come to a focus^RT no di tinct image is formed. When the source is at less the focal distance, the reflected waves are convex and the rays divergent (Fig. 276), but less so than when the reflec- tion is from a plane surface. The curvature of the mirror has the effect of decreasing the curvature of the reflected waves. The center of the reflected waves is their virtual 3 86 LIGHT FIG. 276. Formation of a Virtual Image by a Concave Mirror. focus, and the virtual image, i, of the sources is at this point. A point source and its virtual image are on the same axis, as A and A' (Fig. 276). The image can be located in a dia- gram by producing backward any two lines representing rays of reflected light. Their point of intersection is the position of the image. By following the method of con- struction described for real images the necessity of measur- ing angles is avoided (see Fig. 277). As shown in the figures, the distance of a virtual -image is always greater than that of the object. (Why?) The virtual image of a body object is erect and magnified (Fig. 277). As the object approaches the mirror, the image also approaches it and grows smaller. 317. Spherical aberration. Parabolic Mirrors. As we have previously noted, the reflecting surface of a concave mirror is usually only a very small portion of a spherical surface. This must be so if the mirror is to form distinct images. As a rule, the angle MCN (Fig. 275) is less than 10. This angle at the center of curvature, formed by radii extending to opposite sides of the mirror, is called its angular aperture. When the aperture is large the images are blurred; for only a part of the reflected light from each point of the object is brought to the corresponding focus. The remainder is more or less widely scattered. This effect is known as spherical aberration (aberration = a wandering away). It is illustrated in Fig. 278 for a beam of light parallel to the principal axis. Only the REFLECTION OF LIGHT 387 M central portion of the beam is focused at F; and the marginal rays " wander " far from this point. There is no curved reflecting surface of any shape that will accurately focus all reflected light from a point source in any and all positions; but a parabolic mirror (Fig. 279) does accurately focus a beam of light from a distant point; and conversely, light radiating from its focus is reflected as a beam, however large the aperture of the mirror. Parabolic mirrors have two impor- tant uses: (i) in headlights, search- lights (Fig. 280), etc., to reflect a strong beam in a definite direction; (2) in reflecting telescopes, to gather and focus the light of heavenly bodies. The largest telescopes ever constructed have been reflectors, the greatest having a mirror 6 ft. in diameter, with a focal length of 60 ft. Such J S^ c /KK* 7/60\ / I \ ^/v \ , f\\ \ /XXj \ /CNo^ / / ' \ 22S zzz^ \ / ^ji~ S^*^^""""" \ ^ I^EfE ^TY^^ / ^"^ \ -H N ^ / 33 / < \A(i / W V A / V / \ \ / \\ a/ >X/60 ^^.^^^ M' FIG. 278. Spherical Aberra- tion. FIG. 279. Parabolic Mirror. FIG. 280. Search-Light of the United States Battleship "Connecticut." Its mirror is 5 ft. in diameter. mirrors have very little curvature, but they must be exceedingly accurate. 318. The Convex Spherical Mirror. By reflection from a convex 3 88 LIGHT mirror the plane waves of a beam become convex, and the beam is reflected as a diverging cone (Fig. 281). The center of the reflected waves is their virtual focus. The convex waves from a near point source are made still more convex by such reflection; hence the image is virtual and nearer the mirror than the source (Fig. 282). A convex mirror always produces or increases diver- gence of the light, and hence forms only virtual images. The images are and smaller than the object. The midway between the mirror and FIG. 281. Reflection of Plane Waves a Convex Mirror. from 282), s FIG. 282. Formation of an Image by a Convex Mirror. always erect (Fig. image of a distant object its center of curvature, at the principal focal distance, and it is very small. As the object approaches the mir- ror, its image also approaches the mirror and grows larger. The definite geometrical relations between the radius of a convex mirror, its prin- cipal focus and focal length, and the relative size and distance of object and image can all be readily established on the same principles and by the same methods as for the concave mirror. It should not be difficult for the pupil to solve any of these problems in which he may chance to be interested. PROBLEMS 1. (a) Account for the difference in the brilliance and distinctness of a pinhole image and a real image formed by a concave mirror. (6) A pinhole image is an imperfect real image. Why real? Why imperfect? 2. A pinhole image can be seen only when caught upon a screen. Why can it not be seen in the air like a real image formed by a concave mirror? 3. (a) Prove that the divergence of a cone of light is not changed by REFRACTION OF LIGHT 389 reflection from a plane mirror; (6) that it is always increased by reflection from a convex mirror. 4. Why do plane and convex mirrors form only virtual images? 5. What are the essential characteristics of a virtual image? of a real image? IV. REFRACTION OF LIGHT 319. Refraction. When light falls upon the surface of still water, part of it is regularly reflected, forming an image of the source as in a plane mirror. The remainder of the light passes into the water, and it is with this part that we are now concerned. The path of a beam of light in water is plainly visible in a darkened room, if the water is clouded with a small quantity of an alcoholic solution of mastic or a little milk. A rectangular vessel should be used, to afford a view of the water through a flat surface. It will then be observed that, when a beam passes obliquely into the water, its direction is changed at the surface, as shown in Fig. 283. The direction of the bending is toward the perpendicular to the surface at the point of incidence. The amount of bending is less when the in- FIG. 283. ^-Refraction. cident beam is more nearly perpendicular to the sur- face; and when the incidence is exactly perpendicular, the path is one straight line. A similar abrupt change of direction takes place when light passes obliquely from almost any transparent medium into another, and the phenomenon is called refraction (breaking). The direction of bending is stated with ref- erence to the perpendicular to the surface at the point of 390 LIGHT FIG. 284. incidence. On passing obliquely from air into water, light is refracted toward the perpendicular MN (Fig. 284). The angle i between this perpendicular and the incident ray is the angle of incidence; the angle r between the per- pendicular and the refracted ray is the angle of refraction. The change of direction of a ray, due to refraction, is called its deviation (angle FOE in the figure). The path of light is reversible in refraction as in reflec- tion. For example, if the ray EO (Fig. 284) is refracted in the direction OH on passing from air into water, then a ray passing from water into air and incident along HO will be refracted in the direction OE. Hence a ray passing obliquely from water into air is refracted away from the perpendicular. This can readily be shown by placing a mirror in the bottom of the water tank in the above experiment, to reflect the beam upward through the water to the surface again. 320. The Cause of Refraction. When we look into a vessel of water, the bottom of the vessel appears to be raised above its true position; and any object similarly viewed appears to be at a less depth than it really is, the ratio of apparent to real depth being approximately f when the line of sight is vertical or nearly so. It is evident, there- fore, that the waves of light from any point under water, as 5 (Fig. 285), must change in shape as they pass out into the air; for their center /, as they travel through the air to the eye, is the apparent position of the point, and their center while they are still in the water is its true position S. Further, since a point under water appears to be above its true position, it is evident, as shown in the figure, that REFRACTION OF LIGHT 391 the curvature of the waves must increase as they pass out into the air. This means that a wave must travel the dis- tance BB' in air while it travels the less distance A A' in water; hence the velocity of light in air must be greater than it is in water, the ratio of the velocities being BE' :AA f . It can be shown that this ratio is equal to the ratio of the real depth of the point to its apparent depth (SB: IB) when the point is viewed perpendic- ularly to the surface; and it is found by measurement that this ratio is approx- imately f. Hence the velocity of light in air is f as great as it is in water, or the velocity in water is f as great as in air. This agrees with the result obtained by direct measurement of the velocity in water, an experiment first per- formed by the French physicist, Foucault, in 1850. Foucault's experiment, it may be remarked, was the final test between the two theories of light; for the emission theory involved the assumption that light travels faster in water and other refractive media than it does in air. The figure explains the refraction of light from the per- pendicular in passing obliquely from water into air. It shows further that rays perpendicular to the surface, as SB, pass into the air without deviation. Hence an object under water is seen in its true direction (but not at its true distance) when the eyes are vertically above it. It is not seen in its true position from any point of view. What FIG. 285. Refraction Caused by a Change of Speed. 392 LIGHT we really see is not the object, but its virtual image formed by refraction. The refracted waves are not exactly spherical (Fig. 286). If they were, the apparent position of any object under water would not change when it is viewed more and more obliquely; but we find that it appears to rise toward the surface, as shown in the figure. From the position E it is seen at 7i, from F it appears to be at 7 2 , etc.; hence the curvature of such a wave must be slightly greater near its margin. One in- teresting consequence of this FIG. 286. Refraction at Different Angles. is that the bed of a lake which is really of uniform depth has the appearance of a deep basin surrounded by a shoal, to an observer looking down at it from a boat. Standing at the margin of a pool, the water may appear to grow shallower at some distance off shore, where it is really deeper, a deception which has doubtless led many a boy beyond his depth. 321. Refraction in Different Media. Light is always refracted on passing obliquely from one transparent me- dium into another, except in the rare case when its velocity is the same in the two media. The greater the ratio in which the velocities differ in the two media the greater will be the refraction or deviation of a ray, for a given angle of incidence. For example, the refraction is considerably greater with glass and air than it is with water and air, the velocity of light in glass being only about two thirds as great as it is in air, while in water it is three fourths as great. A printed page, viewed through a piece of thick glass lying upon it, appears to be raised one third the thick- REFRACTION OF LIGHT 393 M Water Fig. 287. Refraction from Air into Water. ness of the glass, owing to refraction at the upper surface. In general, the less the velocity of light in any medium the greater is its refractive power. When light passes from one medium into another in which its velocity is less, as from air into water or from water into glass, the curvature of the waves is diminished and the rays are bent toward the per- pendicular (Fig. 287). To an observer under water an object in the air appears to be at a greater distance than it really is. The velocity of light is less in ordinary matter of any kind than it is in the ether or empty space; hence the refraction is toward the per- pendicular when light passes into any substance from a vacuum, and from the perpendicular when it passes from any substance into a vacuum. Air is practically equiv- alent to a vacuum in all phenomena of refraction where the second medium is a liquid or a solid. 322. Laws of Refraction. The incident and refracted parts of a ray always lie on opposite sides of the perpendic- ular at the point of incidence; and the angles of incidence and refraction are in the same plane. The angles are un- equal, except in the special case where both are zero (the incidence being perpendicular), and in the very unusual case where .light travels with equal velocity in the two media. In either of these cases there is, properly speaking, no refraction at all. As the angle of incidence increases 394 LIGHT the angle of refraction also increases, for any two given media; but they do not increase in the same ratio. The exact relation between these angles is known as the law of refraction. It is a very simple relation, but troublesome to state, except in the language of trigonometry. Let AOB and A'OB' (Fig. 288) be the path of any two rays from one medium into another, e.g. Air <\r Glass from air into glass. M N is the surface at which the re- fraction takes place; KL the perpendicular at the point of incidence. Equal distances OC, OC, OE, and OE' are laid off along the incident and refracted rays. This is conveniently done by describing a circle about the point of incidence as a center. From the points C, C ', E, and E f perpendiculars are drawn to KL. It is found by experiment that the ratio of CD to EF is equal to the ratio of CD' to E'F; CD C'D' This ratio differs with different media, FIG. 288. The Law of Refraction. I.e. a constant. EF E'F' but is constant for all angles of incidence with the same two media. In trigonometry the ratio of a side of a right triangle to the hypothe- nuse is called the sine of the angle opposite to that side; i.e. the sine (~>T\ Tfjf of angle i (abbreviated to sine i ) is -7^ and sine r = Now CZ> = CD + CO _ sine I EF = CO " EO Hence the law of refraction: Whatever EF + EO siner the angle of incidence, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant, for the same two media. It can be shown that this ratio is equal to the ratio of the velocities of light in the two media. When light passes from a vacuum (or air) into any substance, the REFRACTION OF LIGHT 395 Air .M ratio of the sine of the angle of incidence to the sine of the angle of refraction is called the index of refraction of the substance. This ratio measures the refractive power of the substance. It is constant for all angles of incidence, in agreement with the law of refraction; but it differs very slightly with light of different colors (wave lengths), as we shall see later. The refractive power of air or other gas, although very slight, increases with its density; but the refractive powers of different media stand in no relation whatever to their relative densities (see table following). The index of refraction of a sub- G / Water FIG. 289.. Construction for the Refracted Ray. stance is equal to the ratio of the velocity of light in a vacuum (or air, approximately) to its velocity in the substance. In fact relative velocity is at the root of the whole matter of refraction. INDICES OF REFRACTION SUBSTANCE INDEX OF FRACTION DENSITY (G. PER CCM.) Diamond 2.47 to 2.75 3-5 Carbon bisulphide .63 1.29 Glass, flint . . . .^8 tO 1.7^ VOO tO V?2 Glass, crown .$2 tO I."j6 2.S Alcohol .36 0.8o Ether, sulphuric .36 0.72 Water 33 I. CO Air ... .00029 0.00129 Vacuum . .00 323. Construction, for the Refracted Ray. Knowing the index of refraction of a substance, we can determine the angle of refraction corresponding to any given angle of incidence by the following method. Let EO (Fig. 289) be a ray of light passing from air into water at 0. Taking f as the index of refraction of water, the construction for the refracted ray is as follows: Draw KL per- 396 LIGHT pendicular to the surface M N at the point of incidence; and from any convenient point, C, on the incident ray draw CD perpendicular to KL. Lay off on M N a distance OF equal to f of CD, and construct FB perpendicular to MN and parallel to KL. With O as a center and a radius equal to OC, describe an arc cutting FB at G. OG is the refracted ray. (Prove it.) To construct the refracted ray for light passing from air into any substance, take OF of such length that CD : OF is equal to the index of refraction of the substance. To construct the refracted ray for light passing from any substance into air, take OF of such length that OF: CD is equal to the index of refraction of the substance. PROBLEMS 1. Draw a figure showing and accounting for the appearance of a straight stick, partly immersed in water in an oblique position. 2. Stones and other objects lying in the bed of a brook or a pond appear to dance about with a jerky, irregular motion and constantly to change in shape, as small waves pass over the surface of the water. Explain. 3. Look through common window glass at a distant building. Why do the straight lines of the building not appear straight? Note the appear- ance of these lines as you move the head from side to side. Explain. 4. Upon what factors or conditions does the deviation of a ray in refrac- tion depend? NOTE. For the following constructions take f as the index of refraction of crown glass, f as the index for water, and f (which is f -f- f ) as the relative index of refraction 'from water into crown glass. 6. Construct the path of a ray of light from air into water for an angle of incidence (a) less than 30; (b) for an angle between 40 and 50; (c) for an angle between 80 and 90. 6. Construct the path of a ray (a) from crown glass into air; (6) from air into crown glass. 7. Construct the path of a ray (a) from water into crown glass; (6) from crown glass into water. V. REFRACTION IN SPECIAL CASES. TOTAL REFLECTION 324. Refraction through a Plate having Parallel Sur- faces. Certain cases of refraction are of special interest REFRACTION IN SPECIAL CASES 397 and importance, owing to their frequent occurrence or their useful applications. These cases are of three types, repre- sented by the plate, the prism, and the lens. In a plate the refracting substance is bounded by two parallel plane surfaces, in a prism by two plane surfaces inclined at an angle to each other, and in a lens by curved surfaces, which are usually spherical. When a ray of light EOO'F (Fig. 290) passes through a transparent body having parallel plane surfaces AB and CD, the angle of internal inci- dence at O f is equal to the angle of refraction at O; hence the angle of refraction into the air is equal to the first angle of incidence. The emer- gent ray O'F is therefore par- allel to the incident ray EO } but the two are not in the same straight line. The re- sultant effect of the two refractions is a displacement of the ray to one side. This lateral displacement in- creases with the thickness of the plate, with its index of refraction, and with the angle of incidence. It is zero when the angle of incidence is zero, and is small for all angles of incidence with thin plates, such as a window- pane. The displacement of the rays when an object is viewed obliquely through a glass plate causes an equal apparent displacement of the object. Thus, an eye at F would see the source of the ray at some point on the line FH. A beam of light is transmitted as a beam by a plate; for the incident rays are parallel, and having equal angles of incidence, their angles of refraction are also equal, and the Air FIG. 290. Refraction through a Parallel Plate. Refraction through a Prism. 398 LIGHT refracted rays are parallel. This means that plane waves are refracted at the plane surfaces of the medium without becoming convex or concave (see figure). 325. Refraction through a Prism. When light passes through a refracting medium in the form of a prism (Fig. 291), its direction is necessarily oblique to one of the sur- faces, and it gener- ally is to both. In the case shown in the figure, the path of the ray is EFGH. The first refraction is toward the perpendicular MN, the second from the perpendicular M 'N f ; but they are in the same direction in space, and the resultant deviation is their sum, or the angle KPH. The angle A between the surfaces through which the light passes is called the refracting angle of the prism. It should be remembered that the resultant deviation is always away from this angle. This is a necessary consequence of the retarding effect of the medium on the light waves (Fig. 292). That part of a wave which is farthest from the refracting angle travels the greatest distance in the prism, and consequently is most retarded (the velocity in the prism being less than in air). This is true for all angles of incidence, hence the waves must always swing round from the refracting angle. FIG. 292. Change of Speed the Cause of Refraction. REFRACTION IN SPECIAL CASES 399 The deviation increases with the refracting angle of the prism, and with its index of refraction; it also varies with the angle of inci- dence, being least when the angle of incidence is such that the angle of emergence is equal to it. The deviation varies slightly for light of different colors, producing effects which are considered later. The apparent source of the ray GH is some point on the line HL\ hence an object viewed through a prism is apparently displaced in the direction of the refracting angle of the prism. 326. Partial and Total Reflection. When light in any medium strikes the surface of another transparent medium which has either a greater or a less refractive power than the first, a part of the light is always reflected, and under cer- tain conditions all of it is. In the first case the reflection is partial; in the latter it is called total reflection. The partial reflection in air at the surface of water has already been mentioned; it forms the images we see in still water. Partial reflection in water, at its upper surface, can be shown in a darkened room with a beam of sunlight, reflected upward from a mirror at the bottom of a tank of water, as in a former experiment in refraction (Fig. 283). The greater part of the light is in the refracted beam EF; but a reflected beam, EG, is also visible. When a beam is directed obliquely up through the side of the jar, so that the angle of incidence exceeds 48.5, all the light is reflected (Fig. 293). This is a case of total reflection. That partial reflection takes place, both externally and internally, at the surface of glass is readily shown by reflect- ing a beam of sunlight to the ceiling from a piece of colored glass. Two spots of light will appear upon the ceiling, one white, the other having the color of the glass. The white FlG> 293 ' light is reflected at the front surface of the glass; the col- ored light shows by its color that it has traveled through 4oo LIGHT the glass, and must therefore have been internally reflected at the rear surface. Internal reflection, both partial and total, can be admi- rably shown by means of a glass prism, preferably one hav- ing an angle of 90 and two angles of 45. When the prism is held in the path of a sunbeam in a darkened room, in the position shown in Fig. 2940, the light falling upon the rear surface at B is partially reflected in the direction BC and partially refracted along BD. Both beams can be directed toward the ceiling, where their relative intensities are indi- cated by the relative brightness of the two spots of light. (The refracted beam pre- sents the colors of the rainbow, but this effect does not con- u- u FIG. 294. Partial and Total Reflection. cern us at present.) As the prism is turned into the second position shown in the figure, the inten- sity of the refracted beam decreases, as its angle of refraction approaches 90, while the intensity of the re- flected light increases. Finally, in the position shown, the refracted beam disappears, and all the light is reflected in the direction B'C f . It should be noted that total reflec- tion has been brought about by increasing the angle of internal incidence i' , which is now about 45, and that the refracted ray BD disappears when the prism is turned beyond the point where the angle of refraction r\ is 90. Total reflection can take place 'only when the incident light meets a less refractive medium than that in which it is traveling. It can take place in water at a surface bounded by air, but not at a surface bounded by glass. A REFRACTION IN SPECIAL CASES 401 FIG. 295. further necessary condition is that the angle of incidence must be greater than that for which the angle of refraction is 90. The angle of incidence in the more refractive medium for which the angle of refraction is 90 is called the critical angle. When the second medium is not mentioned, it is assumed to be air. The critical angle for water and air is 48.5, for crown glass and air it is about 41, for flint glass and air 38, for diamond and air 24. When a face of a prism is viewed internally at such an angle that the eye receives light from it by total reflec- tion, it has the brilliant, silvery appear- ance of a perfect mirror. In fact, the most perfect mirrors that can be made are total-reflecting prisms, and on this account they are much used in optical instruments. For example, a right angled prism at the eye end of an astronomical telescope changes the direction of the light by 90 (Fig. 295), and enables the observer to look obliquely downward in viewing the heavenly bodies, thus avoid- ing the tiresome position that must be assumed in looking upward in the direction of the object. Total reflection is usefully applied in the natural and artificial lighting of buildings. Where windows of a store or office face a narrow court, the amount of light admitted through ordinary window glass is often insufficient. In such cases the lighting is greatly im- proved by using luxfer prism glass, the outer side of which is formed into angular ridges or prisms, running horizontally. In a vertical FIG. 296. Luxfer Prism Glass. 402 LIGHT cross-section these ridges appear like saw-teeth (Fig. 296). The light coming from a nearly vertical direction strikes the upper surface of the prisms less obliquely than it would upon the vertical surface of plane glass. This diminishes the loss due to external reflection. The light that penetrates the glass is refracted and internally re- flected as shown in the figure, being thus directed toward the Extensive Reflector. walls and ceiling, while with ordi- Intensive Reflector. Focusing Reflector. FIG. 297. Types of Holophane Reflectors. nary glass it would fall directly on the floor and be almost wholly lost. Prismatic reflectors, shades, and globes of the so-called holophane type are very effective as a means of distributing and diffusing arti- ficial light. By refraction and internal reflection the light is directed downward, and more or less concentrated, according to the shape of the reflector. Three types of distribution are shown in Fig. 297. Similar results are obtained with holophane globes, the character of the distribution being determined by the shape of the prisms. 327. Atmospheric Refraction. Although the refractive power of the air is small, it is responsible for certain rather curious and interesting phenomena. Objects seen through currents of heated REFRACTION IN SPECIAL CASES 403 air rising from a bonfire or a hot stove, or from the ground on a hot summer day, seem to quiver and to shift about with a slight, unsteady motion. Tips anngaj^nceis dii_tQ the (jpnst.fl.ntly rhan- gn refra.rt.inn of Tjfre. ig^t as. it passes through the unequally for the density of the air varies with its temperature, and its refractive, .power varies with its densjty. The unsteady condition of the air itself can be seen when the .light isJavorabJe. The twinkling of the stars is an atmospheric phenomenon. The stars themselves are fixed and shine with a steady light. The twinkling is caused by the changing refraction of the light as currents of air of varying density cross the line of sight. As a beam of light passes through successive layers of air, the refraction at their irregular boundaries may cause either a slight convergence or divergence of the rays. Convergence increases the intensity of the beam, diver- gence diminishes it; and the twinkling is largely due to the rapid alternation of these effects. Stars near the horizon, the light from which traverses a greater stretch of atmosphere, twinkle more than those overhead. The twinkling also differs greatly on different nights, according to the steadiness of the air. The inconstant and irregular refraction to which the twinkling of the stars is due is small in comparison with the regular atmospheric refraction, due to the increasing density of the atmosphere from its upper limit to the earth's surface. Light traveling obliquely down- ward through the atmosphere is bent continuously toward the per- pendicular (Fig. 298). The total deviation varies from zero, for heavenly bodies directly overhead, to a little more than half a degree at the horizon. (It is greatly ex- aggerated in the figure.) Since the angular diameter of the sun at the earth is about half a degree, the sun is really just below the hori- FlG - 2 9 8 - Atmospheric Refraction. i i (Much exaggerated.) zon when it appears to be just above it. Thus, on account of atmospheric refraction, sunrise occurs from two to four minutes earlier than it otherwise would (varying with the angle that the sun's path makes with the horizon), and sunset is retarded by the same amount. The mirage is a most interesting optical illusion, due to atmos- 404 LIGHT pheric refraction. It is most frequently observed in hot, desert regions, where it presents the appearance .of a tranquil lake in the distance, in which the traveler sees the reflection of the sky and the scattered trees or other objects of the landscape. But no water is FIG. 299. The Mirage. there; the reality is th^ hot, o,f the The reflection of the light takes place in the lower layers of heated air near the ground^ The lowest layers are the hottest, and, having expanded the FIG. 300. Looming. e less_ dense and less refractive than the air abov.e. Hence a ray of light, ADE (Fig. 299), traveling obliquely downward is refracted from the perpendicular, as it passes through successive layers LENSES 405 of air near the ground; and if its course is nearly horizontal, it will fi- nally meet a layer.of less refractive air at an angle of inqjjence greater an fl1 p i and will then be totally reflected. The ray is refracted toward the perpendicular as it returns through the denser air above. By this refraction and total reflection images are formed like those seen in the surface of still water. The sky and other objects are also seen, at the same time, erect and in their true posi- tion^ hy light that rnmpfi straight to the eye; hence the illusion Js perfect. A similar phenomenon, called looming, is occasionally seen over the sea in still, hot weather. The image of a distant ship appears in the sky, sometimes inverted, sometimes upright. In such cases the total reflection takes place where the light meets an upper layer of warm, still air (Fig. 300). VI. LENSES 328. Forms of Lenses. A lens is a transparent body bounded by two curved surfaces, or by a plane and a curved surface. Lenses are usually made of glass, and their curved surfaces are usually spherical. There are six forms of spherical lenses, sections of which are shown in Fig. 301. a VI L\ FIG. 301. a, b, and c, are Converging, and d, e, and/, Diverging Lenses. The first three are of the type known as convex or conver- ging lenses; the last three are concave or diverging lenses. Convex lenses are all thickest at the middle, concave lenses thinnest at the middle. The different forms of lenses are adapted to different special uses in optical instruments; 406 LIGHT FIG. 3026. but the double convex lens, a, and the double concave lens, d, are typical and serve for experimental work. 329. Effects of a Convex Lens on Light. When light passes through a convex lens, the central part of each transmitted wave is most retarded, since it passes through the greatest thick- ness of glass. From the center out to the margin of the lens, on all sides, the retardation of the wave grows less as the length of its path through the glass decreases. This action of the lens produces the following results: (i) Plane waves are changed to concave waves, which converge to a real focus (Fig. 3020). (2) Convex waves, if their curvature is not too great, are also changed to concave waves, but of less curvature than in the first case, and the focus is at a greater distance (Fig. 3026). (3) Convex waves of a certain degree of curvature are refracted as plane waves. This is the first case with the direction in which the light is traveling reversed. (4) When the curvature of the incident waves is still greater, the refracted waves are convex, but less so than FIG. 3o2c. FIG. 302. Effects of a Converging Lens. LENSES 407 the incident waves (Fig. 302^). In this case the focus is virtual. These effects will be recognized as identical with those produced by a concave mirror. A convex lens forms a real or a virtual image of the source of light, depending upon the converging power of the lens and the distance of the source from it. The behavior of the light in forming these images is the same as with mirrors; but the action of the lens in causing this behavior is different from that of mirrors and presents a new prob- lem. The geometrical relations involved are more simply presented by considering the rays of light rather than the waves. 330. Conjugate Foci on the Principal Axis. The Prin- cipal Focus. The straight line, XY (Fig. 302), through the centers of curvature, C and C', of the spherical surfaces of a convex lens is called its principal axis. This line also passes through the center of the lens, O. A ray of light incident along the principal axis continues, as an emer- gent ray, along that axis; for it meets both surfaces of the lens perpendicularly and is not refracted. Hence when a point source S (Fig. 302 a, 6, and c) is on the principal axis, its image /, whether real or virtual, is also on that axis. In Fig. 3020 the point source is at a relatively great distance, and the incident rays are parallel to the principal axis. The position of the point image in this ease is called the principal focus, and its distance from the lens is called the principal focal distance or the focal length of the lens. This case is approximately shown for the lens, as it is for the concave mirror, when a sunbeam is inci- dent parallel to the principal axis; for the light converges to a small round spot (the image of the sun) at the princi- 408 LIGHT pal focus. (There is a principal focus at the same dis- tance on each side of a lens.) A point source and its real image formed by a lens are at conjugate foci : light radiating from either point converges to a focus at the other. When the point source is at a relatively great distance on the principal axis, its image is at the principal focus (Fig. 3020). As the source moves toward the lens, the image recedes from it. (Why?) When the source is at twice the focal length, the image is also at twice the focal length. As the source moves up to the principal focus, its image re- cedes to an indefinite distance. When the source is nearer than the principal focus, the refracted waves are convex and the image is virtual (Fig. 302^). If the distance of the source is only very slightly less than the focal length, the refracted waves are very nearly plane, and the virtual image is at a great distance. As the source moves up from the principal focus to the lens, its virtual image moves up from a very great dis- tance to the lens; but it is always at a greater distance than the source. (Why?) The real image is always, on the opposite side of the lens from the source, and the vir- tual image is on the same side. The focal length of a lens depends jointly upon the cur- vature of its surfaces and the index of refraction of the glass. Experiment shows (and it can be proved mathe- matically) that, when the index of refraction is 1.5 and the faces of the lens have equal curvature, the principal focus is at the center of curvature of either face. Since this is FIG. 303. Thick and Thin Lenses. LENSES 409 approximately the index of refraction of crown glass, it may be assumed in constructing diagrams that the prin- cipal focus is at the center of curvature of either surface. The converging power of a lens increases as its focal length decreases. In popular language lenses are "stronger" or "weaker" according to their greater or less converging power. We can roughly estimate the focal length of a lens of given diameter from its thickness (Fig. 303). 331. Conjugate Foci on Secondary Axes. Real and Virtual Images of a Body Object. Any straight line through the center of a lens, other than the principal axis, is called a secondary axis, as A A' (Fig. 304). A ray of light incident along a secondary axis strikes the surface of the lens obliquely and is re- fracted, but on emerging from the lens it is equally refracted in the Opposite direction, just F ^ 304--Ray through the Center of a as if it had passed through a "plate." The lateral displacement of the ray is slight, especially if the thickness of the lens is only a few millimeters, and in the elementary treatment of lenses it is disregarded. A point and its image, whether real or virtual, are therefore on the same axis (Figs. 305 and 306). Since all axes cross at the center of the lens, real images, being on the opposite side of the lens from the object, are inverted, and virtual images, being on the same side of the lens, are erect. A real image can be seen by focusing it on- a screen. It is also directly visible, in its true position in space, when LIGHT the eyes of the observer are within the path of the light diverging from it. The observer must look toward the lens, but at the image, which is nearer than the lens. A' FIG. 305. Formation of Real Image by a Convex Lens. A virtual image is seen by looking through the lens. In unscientific language it is termed the "magnified object." The image of a point source is located in a diagram by constructing the path of two refracted rays from the point. If the rays are convergent, the image is at their point of intersection; if they are divergent, it is at their apparent source. In the latter case the lines representing the refracted rays are produced backward to their point of inter- section. The necessity of measuring angles of incidence and refraction can be avoided by choosing any two of the follow- ing rays, (i) The ray along the axis which passes through the point source. This continues in the same straight line. (2) The ray paral- lel to the principal axis. FIG. 306. Formation of a Virtual Image by This paSSCS through the Convex Lens. . . . . . principal focus after re- fraction. (3) The incident ray passing through the principal focus on the same side as the object. This is refracted parallel to the principal axis. This method of construction is illustrated by Figs. 307, 308, and 309. LENSES 411 332. Relative Size and Distance of Object and Image. The relative size of image and object is a matter of the first importance in the use of lenses in optical instruments. The fol- lowing relations should therefore be carefully . . , , FIG. 307. Image of a Distant Object. noted and remembered. From the similar triangles AOB and A' OB' (Figs. 308 and 309), A'B' : AB : : D'O : DO; i.e. the size (length) of the image (real or virtual) is to the size of the object as the distance of the image from the lens is to the distance of the object from the lens. For a given object at a given distance, the size of the real image increases with the focal length of the lens; since, under these conditions, the greater the focal length the greater is the distance of the image from the lens. (Illustrate with two diagrams, constructed for lenses of unequal focal length.) An important special case, relating to the use of the telescope, is that of a distant object (Fig. 307). In this case the distance of the image is the focal length of the lens, and the size of the image is proportional to the focal length of the lens. (Draw figures to illustrate.) When a lens is used as a simple microscope or a "magni- fying glass" in A r * ^^-^^^ looking at small objects, the ob- ject and the lens are so adjusted that the distance of the magnified virtual image is about 12 or 14 in., or the distance at which a book is held for reading. Hence in studying the FIG. 308. Real Image. 412 LIGHT effect of greater or less focal length on the size of the virtual image, we are interested only in the case where the distance of the image is the same with the different lenses. With this adjustment, the shorter the focal length of the lens the larger is the virtual image. (Draw figures to illustrate.) An important general fact, then, is this: Other conditions remaining the same, larger real images are formed by convex lenses of greater focal length, larger virtual images by con- vex lenses of shorter focal length. 333. Relation between Conjugate Focal Distances and the Focal Length of a Convex Lens. From the similar triangles AOD and A'OD' (Fig. 308), AD-.A'D' = OD-.OD'. From the similar triangles EOF and A'D'F, EO'.A'D' = OF-.FD'. Since AD = EO, we have from these proportions OD:OD' = OF'.FD'. Let OD be denoted by D (object distance), OD' by D { (image distance), and OF by f (focal length); then the last proportion becomes From which DJ = D Q D { - DJ. Transposing, DJ + DJ = D D { . Dividing by D DJ, jr + = 7- (Formula for real images.) UQ L)\ J By means of this formula we can find any one of the LENSES 413 three quantities, D , D v and / when the other two are known. The formula for virtual images is derived as follows: From the similar triangles A'EA and A'FO (Fig. 309), A'A-.A'O = AE-.OF. From the similar triangles AOD and A'OD', A'A-.A'O = D'D:D f O. Hence D'D: D'O = AE : OF. Representing the distances of object and image and the focal length by the letters D , D { and / respectively, the last proportion becomes (A- A>):A = A>:/. From which DJ - DJ = D D { . Dividing by D D-J, j^ -- jr 7- (Formula for virtual images.) J 334. The Concave or Diverging Lens. When light passes through a concave lens, the central part of each transmitted wave is least retarded, since it passes through the least thickness of glass. From the cen- ter out to the margin the retardation of the B wave increases with the increasing thickness of the lens; hence, in passing through the lens, the marginal portion of a wave lags behind its center. Plane waves are thus changed to con- vex waves (Fig. 3100), and convex waves are made more convex (Fig. 3106). When the incident light is a beam FIG. 309. Virtual Image. i 414 LIGHT parallel to the principal axis, the center of the refracted waves is a point on the principal axis, and is called the principal focus of the lens. It is, of course, a virtual focus. Whatever the position or distance of a point source, its image formed by a concave lens is virtual. It is on the same \ side of the lens \ Ju and on the same axis as the source, and is at a less dis- tance from the lens. As the source moves up from a great distance to the lens, the image moves up from the principal focal distance to the lens. The image of a body object is erect and smaller than the object (Fig. 311), like the images formed by convex mirrors. Concave lenses are used in combination with convex lenses in optical instruments, and are worn as eye-glasses to correct short sight (Art. 340). 335. Spherical Aberration. No single lens, whatever its shape, brings all the light that passes through it from a point source exactly to the conjugate focus. There is always an imperfection of focusing, called chromatic (or color) aberration; and with spherical lenses there is an- other imperfection, known as spherical aberration. We are at present concerned only with the latter. FIG. 310. Effects of a Concave Lens. LENSES 415 FIG. 311. Image by a Concave Lens. The outer or marginal part of a spherical lens refracts the light to a nearer focus than the central part of the lens (Fig. 312)., This causes a blurring of the image A and loss of detail, which is especially marked with lenses of short focal length. The defect can be remedied in three ways, (i) By using an opaque diaphragm with a small circular opening, which admits light only to the central part of the lens. This device is often resorted to where only a small amount of light is needed, as in small and inexpensive cameras. (2) By decreasing the curvature of the surfaces of the lens near the edge. This is a method adopted for the large lenses of astronomical telescopes. The grinding and polishing of such surfaces must be done by hand. This requires exceptional skill, and is a very slow and costly process. (3) By using a set of two or FIG. 312. Spherical Aberration. more lenses to do the work of one. Such sets can be constructed so as to correct both spherical and chromatic aberration, and are regularly used in first-class cameras, microscopes, etc. PROBLEMS 1. State and account for the points of resemblance and of difference between the real image formed by a lens and a pinhole image. 416 LIGHT 2. Show from the lens formula that (i) when D is very great, D{ = /; (2) when DO = f, Di is very great; (3) when D = 2/, D{ = 2/5 (4) that DI increases as D decreases, and vice versa, for real images; and (5) that Di decreases as D decreases, for virtual images. 3. An object 2 cm. long is at a distance of 50 cm. from a lens whose focal length is 15 cm. Find the distance of the image and its length. 4. An object i cm. long is at a distance of 1.7 cm. from a lens whose focal length is 2 cm. Find the distance of the image and its length. VII. THE EYE 336. The Eye as an Optical Instrument. The fore- going optical principles are beautifully exemplified in the structure and action of the eye. The human eye (Fig. 313) is a nearly spherical ball some- what less than an inch in diameter. Its thick outer coat or wall is opaque and white, except the part in front, which is transparent. This part is called the cornea. Behind the cornea there is a thin mus- cular diaphragm, called the iris, which is visible in the living eye as its colored part. The iris is circular in form and has a circular opening, called the pupil, at its center. It regulates the amount of light that enters the eye by involuntary mus- cu j ar ac tion, which enlarges FIG. 313- -Horizontal Cross-section of the pu y when more light the Right Eye. . is needed and contracts it when the light is too strong. Just behind the iris is the crystalline lens, a double-convex, transparent solid, made up of concentric layers which increase in density and refractive power toward the center. The cavity between THE EYE 417 the cornea and the lens is filled with a watery liquid, called the aqueous humor. The large cavity back of the lens is filled with a transparent, jelly-like substance, called the vitreous humor. The rear half of the eyeball is lined with the retina, a semi-transparent membrane, which contains a network of nerve fibers branching from the optic nerve. A thin, black membrane, called the choroid coat, underlies the ret- ina and extends forward to the iris. This membrane ab- sorbs all light transmitted by the retina and all diffused light within the eye, making the eye a dark chamber. Light on entering the eye is refracted by the cornea and aqueous humor as by a convex lens. The crys- talline lens adds to this effect, since it is a more refractive medium than either the aqueOUS Or the FlG - 3i 4 - -The Rednaljmage is Real and vitreous humor. The re- sult is that the light is focused on the retina, forming real, inverted images of external objects (Fig. 314). These images in some way affect the retina, and the optic nerve carries the impression to the appropriate brain center, pro- ducing the sensation of sight. The purely physical part of the process is the formation of the image upon the retina. If the focusing is exact and the optic nerve is in normal con- dition, vision is perfect; if for any reason the image is more or less out of focus, vision is correspondingly imperfect. The question naturally arises how we see objects erect when the images in the eye are inverted. This is only a part of the larger question how an image within the eye produces the impression of an external object, whether erect or inverted, or of the still larger question how the image causes vision at all. The physiologist examines and 4i8 LIGHT describes tne minute structure of the retina and traces the course of the optic nerve to the brain; but the question remains unanswered. We can only say that experience teaches us to locate each point of an object on the axis of the cone of light which enters the eye from it (see figure). Seeing objects erect is thus a necessary consequence of the fact that they appear to be out in space and not inside the eye. 337. The Field of Distinct Vision. The entire region that is visible when the eyes are held in a fixed position is called the field of vision. This field extends at a very wide angle from the eye; but throughout nearly the whole of it objects are seen very indistinctly. This can be readily tested by looking steadily at one word of a printed page, while trying to read the words round about it. If the words are short, perhaps three or four can be made out with certainty, but not more. The field of distinct vision is surprisingly small. We are seldom conscious of the fact, however, for we are accustomed to fix the attention wholly on the spot at which we are directly looking. In looking attentively at a large object, a rapid shifting of the eyes brings successive portions of it into distinct view. When we look directly at a small object, its image falls upon a small central area of the retina, which is more sensitive to light than the rest of it. This part of the retina is known as the yellow spot, on account of its yellowish color. i 338. Adaptation of the Eye to Different Distances. We know that, as a distant object approaches a convex lens, its real image recedes from the lens, at first slowly, then more and more rapidly as the distance of the object becomes relatively small. The perfect eye, when at rest, forms distinct images of distant objects upon the retina. If the eye were not capable of some form of adjustment, the focus- ing would remain sensibly perfect for shorter distances down to 20 ft.; but for distances less than this the light would be focused behind the retina, and objects would appear less and less distinct when brought 'hearer the eye. THE EYE 419 Since we are able to see both near and distant objects distinctly, it is evident that the eye is capable of adjust- ment for distance. This adjustment is known as accom- modation. Observations upon the eye, such as oculists CILIARY MUSCLE FAR NEAR CILIARY PROCESS FIG. 315. Accommodation. are able to make, have shown that accommodation is ef- fected by the crystalline lens, the front surface of which moves forward and becomes more convex when near objects are viewed (Fig. 315). This diminishes the focal length of the lens, and increases the distance of its center from the retina, both of which changes assist in bringing the image forward to the retina. The adjustment of the lens for near vision is controlled by the involuntary action of the ciliary muscle, which surrounds the lens in the form of a ring (shown in cross-section in the figure). The power of accommodation is limited. When an ob- ject is at less than a certain distance, the effort to focus the eye upon it becomes tiresome; and at still shorter dis- tances focusing is impossible. These distances vary con- siderably for different eyes. For perfect eyes the least distance that is at all comfortable or suitable is about 25 cm. or 10 in. This is commonly taken as the dis- tance of most distinct vision in computing the magnifying power of microscopes. 339. Angular Size of an Object. As the distance of 420 LIGHT an object decreases, its image on the retina grows larger (Fig. 316), and smaller details of the object are reproduced A A , in it just as a large photo- graph shows more detail than a small one. It is _ E , owing to the greater size FIG. 316. -Angular Size of an Object. of the retinal ima ge, and not to more exact focus- ing, that we see an object more distinctly as its distance decreases. The size of the retinal image of an object is proportional to the angle within which the object is seen, as angle AOB or A' OB' (Fig. 316). This angle is called the angular size of the object. For small angles, the angular size of an ob- ject varies directly as its actual size and inversely as its distance from the observer. The angular size of the sun, as seen from the earth, is approximately half a degree. The angular size of a copper cent at a distance of 7 ft. is the same. Hence, at these respective distances, the ret- inal images of the sun and the cent are of equal size. When an object is brought within a few inches of the eye, the advantage of the larger retinal image is more than offset by the disadvantage of imperfect focusing. An object is seen most clearly when its retinal image is as large as it can be, while still in perfect focus. It is then at the least dis- tance of distinct vision, which we have assumed to be 25 cm. 340. Optical Defects of the Eye. In some eyes the image of distant objects is formed in front of the retina, the eyeball being too long or the curvature of the cornea' or the lens too great. Such eyes are said to be near-sighted, for the image is in focus upon the retina only when the object is very near. This defect is corrected by wear- ing concave glasses, which offset the excessive convergence within the eyes by increasing the divergence of the incident light. THE EYE 421 In some cases the eye is too short or the crystalline lens not suffi- ciently converging, and the focus, even for distant objects, would fall behind the retina if the power of accommodation were not exer- cised. Such eyes are far-sighted, and cannot be focused on near objects without fatiguing effort, if at all. Convex lenses correct the defect by supplementing the deficient convergence within the eyes. In old age the crystalline lens loses its elasticity and becomes in- capable of accommodation for near vision. Hence old people, whose eyes were perfect in earlier years, see distant objects distinctly, but require convex glasses for reading. An unequal curvature of the cornea or of the crystalline lens in different planes is called astig- matism. Owing to this defective curvature, the light from any point of an object does not con- verge to a point on the retina, but forms a line instead. An astigmatic eye can not be exactly focused for vertical and horizon- tal lines at the same time; hence Fig. 317 presents a simple test for this defect. The radiating lines are all alike; but to most persons they will appear unequally distinct, for there are very few eyes that are not astigmatic in some degree. If the difference in distinctness is very marked, the eyes are strongly astigmatic, and glasses should be worn, especially for reading and other close work. Astigmatism is corrected either by sphero-cylindrical or toric lenses. A sphero-cylindrical lens has a spherical curve on one side and a cylindrical curve on the other. The curvature of a cylindrical surface varies from zero in the direction of the axis of the cylinder to a maximum at right angles to the axis, and thus offsets the un- equal curvature of an astigmatic eye. One disadvantage of this lens is that it is practically flat, like c and e of Fig. 301, whereas a deeply curved or periscopic lens, like b and / of the figure, gives a much better field of view, owing to the fact that its entire surface FIG. 317. Test for Astigmatism. 422 LIGHT is nearly perpendicular to the line of sight. The toric lens is a periscopic lens, like b and/ of Fig. 301, ground to correct astigmatism. The curvature of one surface is spherical, while that of the other varies, the minimum and maximum curvatures being in planes at right angles to each other. The grinding of such surfaces requires special machines, which have only recently been perfected after many years of effort. 341. Care of the Eyes. Defective eyesight is very common, and yet to a large extent avoidable. When any defect is known to exist or is suspected, an oculist should be consulted. If it is found that glasses are needed, they should be worn; for an uncorrected defect tends to become aggravated, especially when the eyes are much used for near work, as in reading. Even with perfect eyes, it is necessary to exercise intelligent care in order to keep them so. In reading, the distance of the printed page should not be less than 14 in. If it must be held closer than this to be seen distinctly, the eyes are near-sighted, and they should be fitted with glasses for constant use. When the object viewed is held too near, the eyes are turned toward each other at an excessive angle of convergence, and the muscular tension necessary to hold them in this position gradually pulls the eyeballs out of shape, making the sight still more defective. To one who reads much, the proper illumination of the printed page is a very important matter, ignorance or neglect of which is often responsible for serious and permanent injury to the eyes. Direct sunlight is too intense for reading, and should be avoided. Artificial sources of light should be such as to give a constant and uniform illu- mination, neither too faint nor too bright. A flickering light is very fatiguing. The printed page is seen only by the light that it diffuses. The bright glare from smooth, glossy paper is due to regular reflection, and is a hindrance to clear seeing, as experience teaches. Glare is disastrous to the eyes, and should not be tolerated for a moment. . It can be avoided by holding the page at such an angle that the regularly reflected light is thrown off to one side. One should not sit facing the source of light, even if it is covered with a shade; and, if it is bare, it should by all means be out of the range of vision. It is difficult to get a good distribution of light, either for reading or writing, from a table lamp of any description. THE EYE 423 An electric lamp, pointed directly downward, or nearly so, from a low chandelier or a wall bracket answers all requirements. The lamp should be covered with a reflector or a shade, designed to con- centrate the light in a downward direction. Lastly, the eyes should be exposed as little as possible to the undiffused light from brilliant sources, such as incandescent lamps, arc lights, and Welsbach burners. Strong light should always be diffused and softened by the use of frosted bulbs and globes, reflec- tors, shades, etc. 342. Binocular Vision. The ordinary use of both eyes at the same time is called binocular vision. A person who has two eyes rarely uses one alone, except for some special purpose, as in aiming a gun; and he is therefore likely to be wholly unaware of the interest- ing and important differences between vision with one eye and with both. The principal differences are shown in the following simple experiments. Hold a pencil, point up, at a distance of 12 or 14 in. from the eye, and a second pencil, point down, at arm's length. With one eye closed, bring the points of the two pencils into line with the other eye. They now appear to touch each other, although they are nearly a foot apart. The single eye conveys no impression of the unequal distances. Look with both eyes, and you at once receive the true impression of distance. Holding the pencils as before, in line with one eye, close that eye and look with the other. The pencils no longer appear to be in line. With the two eyes we see the same object from two slightly different positions, and hence in slightly different directions. To study this further, hold up a finger before your face and look beyond it, with both eyes, at a wall. The finger appears double and transparent. (Explain.) Distances to right and left and distances up and down are per- ceived with one eye as well as with two; but the impression of distance along the line of sight is much more vivid and accurate when both eyes are used. This is mainly due to the fact that the eyes are turned toward each other more or less, according to the distance of the point at which they are directed (Fig. 318). By experience we learn unconsciously to base our estimate of distance on the greater or less convergence of the lines of sight, BA and CA, of the two eyes. 424 LIGHT Another reason why the impression of distance is more definite in binocular vision is that, with the two eyes, we have two slightly different views of an object at the same time. In looking at a small cube, for example, we can see the fr nt anc * tlie "S^t side with the right eye and, at the same time, the front and the left side with the left eye. It is to these dissimilar views that we owe the mental impression of solidity or of form in three dimensions. Let us try to analyze this impression. When we look at an object with both eyes, the point of it to which the eyes are at any instant directed appears single, while all the rest of it appears double. Ordinarily we are not conscious of this doubleness, for the attention is fixed on the point under direct observation; but if we direct the attention to the whole object while looking steadily at one point, the apparent doubling is very conspicu- ous. Thus when we look at a long pencil, held in the hand with the sharp end pointing toward the chin, it appears as shown in a, b, or c of Fig. 319, according as the eyes are directed toward the nearer end, the middle, or the farther end of it. The impression of distinct vision for the point under direct observation and the impression of indistinct and double vision for the remainder of the object together make up the impression of solidity or of form in the three dimensions of space. With a single eye there is only an imperfect suggestion of the third dimension, or distance from the observer, as in a photograph. 343. The Principle of the Stereoscope. If we present to each eye a picture of an object taken from its point of view, and direct the eyes so that these slightly dissimilar pictures seem to coincide, the appearance of solidity will be perfectly reproduced. This can be shown with Fig. 320. The two pictures of the tunnel represent it as it is seen by the two eyes separately. Hold the book so that one picture is immediately in front of each eye, with a card between them and perpendicular to the page, so that each eye can see only the picture on its own side. Now direct the eyes as if you were looking through the book at a point some distance behind it. When OPTICAL INSTRUMENTS 425 this is done, the pictures will seem to move together and unite into a single view, which has the appearance of real depth extending 2 ft. or more behind the book. This stereoscopic picture will FIG. 320. Stereoscopic Pictures. appear blurred for half a minute or more while the eyes are strug- gling to bring it into focus, but it finally becomes perfectly clear. With a little practice, the card between the pictures can be dispensed with; but two ad- ditional tunnels will then be indistinctly seen, one on either side, the one on the left being the left picture as seen by the right eye and the other the right picture as seen by the left eye. The stereoscope is an instrument designed to aid the eyes in uniting into one view two slightly dissimilar photographs of the same scene. These photographs are taken with a double camera, and represent the scene just as it would appear to the two eyes of an observer. They are mounted on the same card, AB (Fig. 321), and are viewed through the half lenses M and N. The magnified virtual images of the two pictures coincide at A' B' . The partition P prevents each eye from seeing the picture intended for the other. VII. OPTICAL INSTRUMENTS 344. Magnification. In the elementary study of opti- cal instruments, such as microscopes, telescopes, and opera -*' /> p i j> 1 | i \ i i \ \ j \ r \ i \ 'A ' R A\ /? - / \ \ ' / I / P i \ i l 1 1 i> ' t scope is approximately rj? when the unit of length is the v * centimeter, and -j when the unit is the inch. The usual range of magnifying power for single lenses is from 5 to 10 diameters. With higher powers the image is badly distorted and colored, owing to spherical and chromatic aberration. Doublet magnifiers give good results with powers as high as 24 diameters. These have two lenses placed a short distance apart. 346. The Compound Microscope. In viewing very minute objects a higher power is required than is possible with a simple magnifier. The instrument designed for this purpose is called a compound microscope. In its ele- mentary form it consists of two convex lenses of short focal length, called the objective, O, and the eye-lens, E (Fig. 323). The objective is placed at a distance only slightly greater than its focal length from the object, AB } in order 428 LIGHT to form a magnified real image, A'B', at a much greater distance on the other side. The eye-lens serves the pur- pose of a simple microscope, through which the observer sees a magnified virtual image, A"B" , of the real image. Let DI denote the distance of the real image from the objective. The distance of the object may be taken as FIG. 323. Diagram of the Compound Microscope. the focal length, / , of the objective. The magnification due to the objective is the ratio of the size of the real image to the size of the object, and this is equal to the ratio -- 1 . The magnifying power of the eye-lens is , / e being Jo . 7e ' its focal length. The magnifying power of the two lenses together is the product of their separate magnifying powers, or f f ', all distances being expressed in inches, yeyo The distance D\ is determined by the length of the microscope tube, at the ends of which the lenses are inserted (Fig. 324). In standard instruments it is approximately 6 in. As shown in the formula, the magnification varies inversely as the focal length of either the objective or the eye-lens. Hence the shorter the focal length of either, the greater is the magnification. In ordinary OPTICAL INSTRUMENTS 429 practice the objectives vary in focal length from f in. to i in.; and with these, both two-inch and one-inch eyepieces are used. With a objective and a one-inch eyepiece, the magnification is = 360 diameters. Higher powers, up to 2000 diameters, 1 X i are possible. The images formed by a compound microscope consisting of two single lenses are distorted and indistinct, and are colored by chromatic aberration. To avoid these defects the objective is built up of from four to ten lenses (Fig. 339), and the eyepiece of two. 347. The Astronomical Telescope. The telescope serves the same purpose in viewing a distant object that the microscope does in viewing a small one. Both instruments form images which have a larger angular size than the objects. There are various types of tele- scopes, including such extremes of size and use as the astro- nomical telescope and the opera glass. The astronomical telescope, in its simplest form, consists of two convex lenses, which serve as objective and eye-lens respect- ively, as in the compound micro- scope. Since the object viewed is always a distant one its real image, ab (Fig. 325), is at the FlG - principal focal distance of the objective. The eye-lens forms a magnified virtual image, a'b', of the real image. Both the real and the virtual images are inverted with respect to the object; but this is not a disadvantage in viewing the heavenly bodies. Mi ' 430 LIGHT The object is not shown in the figure, since, in its true relative position, its distance would be enormously greater than the focal length OC. The parallel lines JK and NO represent rays from a point at the lower side of the object. All such rays converge to the corresponding point b of the FIG. 325. The Astronomical Telescope. image. Similarly the parallel rays, such as MO and HI, from a point at the top of the object converge to a. The angular size of the object is the angle MON, or the equal angle aOb. The angular size of the image, as it appears in looking through the telescope, is the angle a'Eb f , or the equal angle aEb. Hence the magnification is, by angle aEb definition, - Since these angles are subtended angle aOb by the same line ab, they are (for small angles) inversely proportional to the distances of their vertices from this line; i.e. - = Now OC is the focal length, angle aOb CE / , of the objective, and CE is sensibly equal to the focal length, /j, of the eye-lens. Hence the magnifying power of the telescope is A or the ratio of the focal length of 7i the objective to the focal length of the eye-lens. It should be noted that this ratio is increased either by in- creasing / or by decreasing f { . The objectives of the most OPTICAL INSTRUMENTS 431 powerful telescopes have focal lengths ranging from 40 to 60 ft. or more. The eyepieces are the same as in com- pound microscopes, and serve exactly the same purpose. An objective of great focal length must also have a large diameter; for a highly magnified image will not be as bright as is necessary unless it is formed by a proportionately great amount of light, and the light-gathering power of the objective is proportional to its area. The great telescope of the Lick Observatory, at Mt. Hamilton, California, has a diameter of 36 in. and a focal length of 57 ft.; that of the Yerkes Observatory, at Williams Bay, Wisconsin, a diam- eter of 40 in. and a focal length of 62 ft. The largest telescope objective ever constructed is that of the Carnegie Solar Observatory, on the^ummit of Mt. Wilson, near Pasadena, California. This lens is 60 in. in diameter and is 8 in. thick at the center. Three years were spent in grinding and polishing its surfaces. The great telescope of the Paris Exposition, in 1900, was 180 ft. long, and its objective 47 in. in diameter. Owing to its enormous size, it was mounted in a fixed horizontal position, and the light from the heavenly bodies was reflected into it by a large mirror. With such telescopes as these the surface of the moon is shown in great detail, objects less than half a mile in diameter being visible, and the earth's nearer neighbors Mars, Jupiter, and Saturn appear as large and beautiful orbs. The fixed stars, however, are at such enormous distances that they are seen only as points of light; but the telescope increases their brightness and the apparent distance between them, and thus brings into view millions of stars which are never seen with the naked eye. Even an opera glass greatly increases the number of visible stars. 348. The Terrestrial Telescope has two additional lenses to rein- vert the image and make it erect. A single lens, L (Fig. 326), would accomplish this result, but the image would be imperfect. The objective (not shown in the figure) forms the inverted real image ab. The lens L, placed at twice its focal length from this image, forms a real and erect image, a'b', at an equal distance on its opposite side; and the eyepiece forms an erect virtual image of a' b'. A small terrestrial telescope is called a spy-glass. 432 LIGHT 349. Opera and Field Glasses. In the common opera glass or field glass the part for each eye is a complete tele- scope of the type shown in Fig. 327. The objective is a convex lens, having a focal length, OC, of about 4 in. in FIG. 326. The Terrestrial Telescope. opera glasses and 5 to 7 in. in field glasses. The eye-lens is concave and of very short focal length, EC. The dis- tance between the lenses is equal to or very slightly less than the differences between their focal lengths. The objective would form an inverted real image, ab, of a distant object if the light were not intercepted by the eye- FIG. 327. The Galilean Telescope. lens; but, with this lens in position, the converging cone of light from any point of the object is changed to a slightly diverging cone, and forms a virtual image of the point. Thus all rays converging toward a focus at a appear to OPTICAL INSTRUMENTS 433 come from a', and all rays converging toward b appear to come from b'. Hence the only image actually formed is the erect virtual image a'b r . The angular size of this image is the angle a'Eb' or its equal a6,.and the angular size of the object is MON or its equal aOb. The magnifying power of the instrument is therefore angle aEl ', which is equal to**?, or the ratio of angle aOb EC the focal length of the objective to the focal length of the eye-lens, as for the astronomical telescope. For equal power the opera glass is the shorter instrument by twice the focal length of the eye-lens. It has a still greater advantage in this respect over the terrestrial tele- scope, since the lenses in the latter for erecting the image increase the length without increasing the power. The earliest telescopes were constructed on the principle of the opera glass; but they were single-tube instruments, for one eye only. The first authentic record of such an instrument is of one made in Holland in 1608. Galileo, hearing of this invention, took up the prob- lem, and was soon making telescopes of considerable power. With their aid he made several astronomical discoveries which occasioned a great sensation at the time and brought him great renown. " He turned his telescope toward the moon and discovered mountains and craters; he turned it to Jupiter and saw its satellites; he pointed it at Saturn and saw the planet threefold (now known to have been due to an imperfect view of the ring) ; he examined the sun, saw its spots moving, and concluded that the sun rotates." The early history of the telescope is therefore closely associated with the name of Galileo; and the telescope with a concave eye-lens is commonly known as Galileo's telescope. 350. The Prism Binocular. There is one important matter concerning telescopes of all kinds which remains to be considered, namely, the size of the field of view. The field of an opera glass of the common, or Galilean, type is large enough to include only a small 434 LIGHT group of actors, and it is necessary to turn the glass about in order to see the different parts of the stage. With different telescopes of the same type the angular diameter of the field of view varies inversely as the magnifying power. In doubling the power the diameter of the field is reduced one half, and its area is reduced fourfold. In considering the relative merits of the different types of instruments, the magnify- ing power and the size of the field must both be taken into account. A terrestrial tele- scope has a much larger field than a Galilean telescope of the same power, but it is much less con- venient on account of its greater length and weight. A new type of instrument has recently come into use, which combines the advantages of a large field and a short, compact form. This is the prism bin- FIG. 328. The Stereo-Prism Binocular. FIG. 329. Magnifying Power. ocular (Fig. 328). The large field is secured by its converging eye- piece, which is the same as in astronomical and terrestrial telescopes. Its small size is due to the use of two total-reflecting prisms in each tube. A ray of light is reflected twice in the same plane at one prism, OPTICAL INSTRUMENTS 435 and twice in a plane at right angles to the first at the second prism. The angle of incidence at each reflection is 45; hence the image is turned through an angle of 90 at each reflection. These four reflec- tions, therefore, reinvert and reverse the image (interchanging top and bottom, and right and left sides), thus accomplishing the same result as the erecting lens, L (Fig. 326), of the terrestrial telescope. But while the erecting lens increases the length of the instrument, the prism shortens it nearly two thirds; for the tube, although comparatively long, is folded into three parts lying side by side. A prism binocular having a field of the same size as that of an old-style glass has three times the magnifying power (Fig. 329). In the stereo-prism binocular, shown in Fig. 328, the objectives are considerably farther apart than the eyes. This wider separation of the two points of view enhances the stereoscopic effect of binocular vision. 351. The Projection Lantern. With this instrument highly magnified images of transparent photographs and drawings are pro- jected on a white screen in a dark room. Its essential parts are a strong source of light, A (Fig. 330), a condensing lens or pair of lenses, FIG. 330. The Projection Lantern. L, for concentrating the light upon the picture or " slide " P, and an achromatic objective, L f , placed at a little more than its focal length from the slide. The image is formed at a relatively great distance, and is correspondingly magnified. In order that it may be erect, the slide is inverted. 352. The Biograph, or moving-picture machine, is a projection lantern with a mechanism by which a series of pictures, printed on a 436 LIGHT long film, can be thrown upon a screen in rapid succession. The pictures follow one another at the rate of about 15 or 20 per second; and, -as each comes into position in the instrument, a flash of light projects it upon the screen. During the intervals between, while the film is shifting from one picture to the next, the light is cut off by a diaphragm, and the screen is then dark. The picture appears to be continuously present on the screen, for the visual impression persists long enough to bridge over the interval of darkness. (It is owing to this " persistence of vision " that falling rain-drops are seen as long streaks, vibrating strings as gauzy spindles, etc.) Moving picures are taken with cameras provided with a mecha- nism which opens and closes the shutter at regular intervals, and which, while the shutter is closed, jerks the ribbon film into place for the next impression. As the pictures taken are of moving objects, they record successive positions of the objects at equal time inter- vals; and, when they are projected upon a screen, the objects appear to move from each position to the next, owing to the persist- ence of the visual impression. PROBLEMS 1. What is the magnifying power of a simple microscope whose focal length is (a) 2 in.? (b) .5 in.? 2. Show from the formula of Art. 333 that the magnifying power of a simple microscope is inversely proportional to its focal length. Draw figures to illustrate. 3. Assuming that the real image is formed 5 in. from the objective, find the' magnifying power of a compound microscope (a) with a -5-in. objective and a 2-in. eyepiece; (b) with a |-in. objective and a -in. eyepiece. 4. The great telescope of the Lick Observatory is 57 ft. long. What is its magnifying power (a) when fitted with a 2-in. eyepiece? (b) when fitted with a ^-in. eyepiece? 6. The objective of a field glass has a focal length of 7.5 in., and the eyepiece a focal length of 1.25 in. Find the length of the instrument and its magnifying power. 6. What are the essential parts of a photographic camera, and what purpose do they serve? What adjustment is necessary for the distance of the object and why? How is the size of the image of any object affected OPTICAL INSTRUMENTS 437 by the distance of the object from the camera? Why does a large camera take a larger picture of a distant object than a small camera does? How is the necessary time of exposure affected by "stopping down" the lens opening with the diaphragm? Suggestion. If you have a camera you will be interested in trying "pinhole photography." For this purpose remove the lens and cover the opening with tinfoil. In the center of the foil make a minute hole with a fine needle. The edge of the hole should be as smooth and thin as possible. How would you compute the proper time of exposure, knowing the time necessary with the lens and a given adjustment of the diaphragm? VIII. DISPERSION OF LIGHT. COLOR 353. The Composite Character of Sunlight. Disper- sion. When a beam of sunlight is admitted through a small opening into a dark room and allowed to fall on a white screen, at a distance of several meters from the open- ing, it forms on the screen a brilliant pinhole image of the sun. A prism placed in the path of the beam deflects it to one side. When this deflected beam falls on the screen, it appears as a band of light, VR (Fig. 331), rounded at the ends and brilliantly colored. Violet, blue, green, yellow, orange, and red are all present, in the order named. These beams of colored light can be brought together again on the screen by means of a lens, placed at conjugate focal distances from the prism and the screen (Fig. 332), or by means of a second prism, placed near the first but with its refracting angle in the opposite direction (Fig. 333) ; and the round spot on which the light falls is white. What -is the lesson conveyed by these strikingly beauti- ful experiments? Obviously a prism or a lens of colorless glass can not make or destroy colored lights. The plain inference is that these colored lights are present in the sun- beam and are separated by unequal refraction in passing 438 LIGHT through the first prism, and that the second prism or the lens brings them together again in the same condition as at first. The white light of the sun consists of these various colored lights, all traveling together. Their separation by FIG. 331. Dispersion by Prism; Impure Spectrum. the prism is called dispersion, and the prism is said to analyze the sunbeam into its constituent colors, forming the solar spectrum. Similarly, when the light from any source is separated into its constituents by a prism, or by FIG. 332. Colors of Spectrum Recombined by Lens. other means, the resulting colors, taken together, are called the spectrum of that light. The solar spectrum, when formed as described above, consists of an indefinitely great number of colored images of the sun, overlapping one another. If a piece of red (ruby) glass is placed in front of the opening, it trans- mits only the red light and absorbs the other colors. The DISPERSION OF LIGHT. COLOR 439 spectrum will then consist of a circular red spot, which is the red image of the sun. There are thousands of such images, of as many different gradations of color, in the complete spectrum, and there is consequently much over- lapping and mixing of the colors. A spectrum of this char- acter is said to be imperfect or impure. On looking through the prism, with the eye close to it and in such a position as to receive all the colors, the ob- server will see a virtual /\r^^ spectrum, V'R' (Fig. s - ^^^^=== ==== ) 331), which is nearly pure. This is quite ^ IG- 333> Colors of Spectrum Recombined by Second Prism. satisfactory for indi- vidual observation in the laboratory (see Lab. Ex. 56); but for experimental work in the class-room the spectrum must be real. It should also be large, brilliant, and ap- proximately pure. 354. Formation of a Pure Spectrum. Fraunhofer's Lines. To obtain a wide spectrum the sunlight is admitted through an opening 2 or 3 cm. long. To prevent overlap- ping of the colored images the opening must be narrow. A long, narrow slit fulfils both these requirements. It should be vertical and the sunbeam horizontal. To obtain a pure spectrum the illuminated slit must be treated as the source of light, not the sun; and the colored images of the slit must be exactly focused on the screen by means of a lens. To secure this adjustment the screen XY (Fig. 334), is placed at a distance of several meters from the slit, and the lens is adjusted so as to form a magnified im- age of the slit upon it, the prism being removed. The prism is next placed in line, near the lens (on either side of it), and the screen moved into the position X'Y f , in line with 440 LIGHT the deflected light and at the same distance from the lens as before. The spectrum thus formed consists of a series of narrow and only slightly overlapping images of the slit. A prism of flint glass produces a wider separation of the colors and a longer spectrum than one of crown glass, and a bottle prism containing carbon bisulphide is still better. With a suitable adjustment of the apparatus, the carbon bisulphide prism gives a spectrum a foot wide and two feet or more in length; and in a perfectly dark room the FIG. 334. Pure Spectrum by Means of Lens and Prism. spectrum is very brilliant. Under such conditions the experiment is the most beautiful in the whole range of elementary physics. With a bisulphide prism, a very narrow slit, and a lens exactly in focus, the spectrum is very pure and is crossed by many dark lines at right angles to its length (i.e. in a direction parallel to the slit) . These lines are called Fraun- hofer's lines, after the celebrated optician of Munich who first studied and gave a detailed description of them. They represent missing images of the slit, and indicate that the light which would occupy these positions in the com- plete spectrum is missing from sunlight. When the slit is a little wider or the lens slightly out of focus, the dark DISPERSION OF LIGHT. COLOR 441 lines are obliterated by the overlapping of the adjacent images. (A detailed account of the Fraunhofer lines is given in Arts. 492-495.) 355. The Nature of Color. When we call one part of the spectrum red, another part green, etc., we are only naming the color sensations which we experience when these different lights stimulate the optic nerve. The character of the sensation is determined by the wave length of the light. In fact, color, as a property of the light itself, is nothing else than its wave length or the frequency of the ether vibrations. The sensation of -color bears the same relation to the wave length of light that the sensation of pitch does to the length of sound waves. The measurement of the wave lengths of light is based on principles and methods which belong to advanced physics. By such measurements it is found that red light consists of the longest visible ether waves, and violet of the shortest. In the complete spectrum all possible wave lengths between these limits are present, decreasing in an unbroken series from the extreme red to the extreme vio- let. Corresponding to these innumerable wave lengths, there is a continuous gradation of color from one end of the spectrum to the other. The most dissimilar colors are violet, blue, green, yellow, and red. To the normal eye these bear no resemblance to one another. In addition to these we recognize the intermediate colors, violet-blue or indigo, between the violet and the blue; greenish blue and bluish green, between the blue and the green; yellowish green and greenish yellow, between the green and the yellow; and yellowish red, or orange, between the yellow and the red. As the temperature of a body rises, the vibrations of 442 LIGHT its molecules become more and more rapid, and shorter waves are set up in the ether. At about 525 C. some of the molecules vibrate with sufficient rapidity to give out red light. As the temperature continues to rise, additional colors are given out in order from red to violet, and the color of the body changes from red through orange and yel- low to white. Sunlight, and the light from white-hot bodies in general, is a mixture of all or practically all wave lengths within the range of visibility. Such light may be compared to an utterly discordant chaos of sound of every audible pitch. 356. The Cause of Dispersion. The dispersion of light is the result of unequal refraction of its constituent colors. As -shown in the figures, the violet or shortest waves are refracted most, and the red or longest waves least. Accord- ing to the wave theory, this can only be due to a greater retardation of the shorter waves in passing through a re- fractive medium. In a vacuum waves of all lengths travel with the same velocity. 367. Invisible Portions of the Spectrum. The wave lengths of light vary from .00077 m m- for the extreme red to .00039 mm. for the extreme violet. The interval between these extremes, expressed as in music, is slightly less than one octave. (Read again Art. 297.) The solar spectrum includes two octaves of ether waves beyond the violet end, and more than six octaves beyond the red end. The part beyond the violet is called the ultra-violet spectrum. Although invisible, it can be photographed. In fact, the photographic plate is more sensitive to ultra-violet waves than it is to the visible spectrum. The part of the spectrum below the red is called the infra-red spectrum. Only a small portion of it, at its upper end, has ever been photo- graphed. It has been studied mainly by its heating effect. For this purpose the late Professor Langley invented an instrument by DISPERSION OF LIGHT. COLOR 443 which he could detect a variation in temperature of one hundred- millionth of a Centigrade degree. The distribution of energy in the solar spectrum has been made the subject of careful investigation. It is determined by measuring the heating effect of the waves, when absorbed by a black surface. The region of maximum energy is found to be in the yellow and green; but the energy of all the infra-red waves, taken together, greatly exceeds that of the visible spectrum. The blue waves possess very little energy, the violet and the ultra-violet still less. In the spectra of even the most efficient artificial sources of light, such as the electric arc, the region of maximum energy lies far below the red; and the energy of all the luminous waves together is only a very small per- centage of the whole radiation. The great problem of artificial illu- FIG. 335. Chromatic Aberration. mination is to discover some means of producing a good white light which represents a reasonable fraction of the energy expended in producing it. Great advance has been made in this direction in recent years, especially in the newer forms of electric lamps. Much longer ether waves than those of the infra-red radiation from hot bodies can be produced by electrical means. These vary in length from a few millimeters to many meters. It is such waves that are made use of in wireless telegraphy. 358. Chromatic Aberration. Achromatic Lenses. In studying lenses and their use in optical instruments we had occasion to note a certain imperfection of focusing, called chromatic aberration (Arts. 335, 345, 346, FlG - 336. Achro- and 347). The nature of this imperfection is shown in Fig. 335. The constituent colors of the incident light are refracted unequally, violet most and 444 LIGHT red least, just as with prisms. Hence the violet waves of white light from a point source, S, are brought to a A focus at one point, v, and the red waves to a more distant focus, r, while the other colors are focused at intermediate points. When a screen is placed at r, we see a red image of the point surrounded by a circle of blue and violet light; when the screen is at v, we see a blue-violet image surrounded by orange and red. The dispersion of the colors in- P IG . 337. Action of the creases from the central portion of greater refraction and greater dispersion go together. Hence a diaphragm, with a small circular opening for the central rays, serves the double purpose of diminishing both spherical and chromatic aberration. But this is only a partial remedy at the best, and it will not serve at all where a large amount of light is re- quired. A century and a half elapsed after the telescope and the microscope were in- vented before it was discovered that chromatic aberration could be corrected by combining a convex lens of crown glass with a concave lens of flint glass (Fig. 336). The refractive power of flint glass is only slightly greater than that of crown, while its dispersive power is nearly twice as great. Hence if the focal length of the concave lens is about twice that of the convex lens, it will produce an equal and opposite dispersion, with only half the FIG. 338. High- grade Camera Lens. DISPERSION OF LIGHT. COLOR 445 deviation (Fig. 337, A and B), and the light after passing through both will still be convergent, but not dispersed (Fig. 337C). Double lenses constructed on this prin- ciple are called achromatic lenses. They are used as objectives in telescopes, opera glasses, and microscopes, also in projection lanterns, cameras, etc. Achromatic eyepieces are somewhat differently constructed, but accom- plish the same result. Chromatic aberration can not be wholly avoided with two lenses only. Some of the colored rays are brought exactly together again, but not all. The best objectives for lanterns cameras (Fig. 338), and microscopes (Fig. 339) have three or more lenses. Such com- binations are not only perfectly achromatic, but are free from other imperfections of focusing as well. 359. The Rainbow is a solar spectrum, formed by the dispersion of sunlight by drops of water in falling rain, and in the spray of fountains, waterfalls, etc. Sometimes one bow is seen, sometimes two. They are always arcs of circles; and, when two are formed, they are concentric (Fig. 340). The inner or lower one is called the primary bow, and the other the secondary bow. The primary bow is always much the brighter. In it the red is on the outside, the violet on the inside. In the secondary bow the order of the colors is reversed. Rainbows * , . . FIG. 339. Microscope are always seen in the direction op- objective, One-Sixth posite to the sun, with the sun, the Inch - observer, and the center of the circular arc in the same straight line, EO. This line is called the axis of the bow. The action of the individual drops in forming a rain- bow can be shown on a large scale, with the aid of 446 LIGHT a slender sunbeam in a darkened room and a globe filled with water. A round-bottomed flask will answer the purpose very well. When the globe is held in the path of the beam, the greater part of the light passes through it; the remainder undergoes one or more internal reflections before it is refracted out. The primary bow is formed by light that is reflected once (Fig. 341), when the angle of incidence at A is approximately FIG. 340. Primary and Sec- 59. With this adjustment of the globe and a white screen, XY, in ondary Rainbows. the path of the emergent light, a curved spectrum is seen, with the violet at its inner edge. Part of the dispersion is due to each refraction, as shown in the figure. When the beam is incident less obliquely, the different colors are spread out, without separation, as a broad band of faint white light, and no spectrum is formed. When the angle of incidence is slightly greater, as at A in Fig. 342, the light that is twice internally reflected forms the secondary spectrum, y This spectrum is faint, owing to the additional loss of light at the second reflection. With an incident beam large enough to cover the globe, the primary spectrum forms a complete spectrum on the screen. In the strong illumination the secondary spectrum is very faint, if not invisible. In looking at a rainbow (Fig. 340), the eye receives only FIG. 341. Dispersion in the Primary Rainbow. DISPERSION OF LIGHT. COLOR 447 a single color from any one drop; and this color is the same for all drops which are at the same angular distance from the axis of the bow. Hence the bow is circular. The inner edge of the primary bow is at an angle of 40 from 54 51' the axis and its outer edge at 42. The edges of the secondary bow are at angles of 51 and 54 respectively. At sunrise or sunset a rain- bow, if complete, appears as FlG ' a semicircle (Fig. 340), its axis being horizontal. Since the center of the bow is always at the same angle below the horizon that the sun is above it, the higher the sun is the shorter will be the arc of the bow. When the sun is more than 42 above the horizon, only the secondary bow can be seen. 360. Color of Bodies. What we commonly regard as the natural color of an object is really the color of the light that the object transmits or reflects when white light falls upon it. A body that is transparent to light of all wave lengths is colorless, e.g. window glass and water. A body that reflects light of all wave lengths in equal proportions is white if it has a high reflecting power, gray if its reflecting power is rather low, and black if it reflects almost no light. A body absorbs the incident light that it does not transmit or reflect. If it transmits none, it is opaque; if it reflects practically none, it is black. If a body transmits waves of different lengths in unequal proportions, or transmits some and wholly absorbs others, the transmitted light is colored, and its color is called the 448 LIGHT color of the body. Colored glass and colored liquids of various kinds are familiar examples. A colored opaque body owes its color to the fact that it reflects waves of dif- ferent lengths in unequal proportions, or else reflects some and wholly absorbs others. Whatever the color of a body, the analysis of the light transmitted or reflected by it shows that this light is com- posed of certain spectral colors (colors of the spectrum) in certain porportions. There are no simple, indivisible, or elementary colors other than those of the complete spectrum; and any composite color can be analyzed into its elementary or spectral constituents by means of a prism. This analy- sis gives a spectrum in which the constituent parts are arranged as in the solar spectrum, the only difference being that certain parts of the complete spectrum are missing and others, perhaps, relatively weak. In analyzing the light from any body we may make use of either the real or the virtual spectrum. The study of virtual spectra is adapted to individual laboratory work (see Lab. Ex. 56). For class observation the spectrum is projected on a screen in a darkened room, as in the following experi- ments. Let a pure solar spectrum be projected on a screen, as in Art 354, and one end of the slit covered with a piece of red (ruby) glass. The solar spectrum and the spectrum of the light transmitted by the glass will appear upon the screen, one above the other. The latter consists of red, with perhaps a little orange. The other constituents of white light are absorbed by the glass. Substituting blue (cobalt) glass, the spectrum of the transmitted light will be found to consist of violet, blue, green, and a little red. Yellow glass transmits red, orange, yellow, and green. Colored solutions of chemicals, such as copper sulphate DISPERSION OF LIGHT. COLOR 449 and potassium bichromate, can be tested in the same way. A flat bottle will serve for holding the liquid/but a narrow tank of plate glass is better. When two transparent bodies of different color are placed before the slit, one in front of the other, the first absorbs certain constituents of the incident light, the second absorbs certain other constituents; and the spectrum of the transmitted light consists only of the color or colors which are common to the light transmitted by the two separately. Thus green is the only spectral color trans- mitted by either blue or yellow glass that is also trans- mitted by the other ; hence the two together appear green. Similarly the combination of red and blue, red and green, or orange and blue glass is very nearly opaque, since no color that they separately transmit in considerable quantity is common to both. The light reflected by a colored opaque body can be analyzed in a very interesting way by noting the appear- ance of the body when it is held successively in the different colors of a large solar spectrum, projected on a screen. A white card held in the violet light appears to be violet; in the blue it appears blue; in the green, green, etc. Whatever the color of the light by which it is illuminated, it appears to be of that color. In white light it appears of its natural color white for it re- flects all the constituents of white light in equal pro- portions, and absorbs but little of any. A piece of green paper will appear black in the violet, indigo, orange, or red; in the blue it will probably appear to be a dark blue, and in the yellow a dirty yellow, due to the reflec- tion of a little of these colors; in the green it will appear at least very nearly of its natural color. Similarly we can determine the spectral colors that any colored body 450 LIGHT is capable of reflecting; and these will be the constitu- ents of the light that it reflects when it is illuminated by white light. Most artificial lights are deficient in violet and blue, and hence are more or less yellowish. In such a light, white has the appearance of pale yellow, and blue is often mis- taken for green. The greenish appearance of blue is due to the fact that blue pigments reflect violet and green as well as blue light, and green predominates in the light that they reflect when the incident light is weak in the violet and blue. The light transmitted or reflected by colored bodies in general is composite, consisting in many cases of fully half of the complete spectrum. The composite character of light can not be detected by the unaided eye; for the eye is absolutely wanting in the power of analysis. If the ear were similarly deficient, we could not distinguish the constituents of a complex sound. The notes sounded simul- taneously by an orchestra would produce the sensation of a single note of average pitch, and harmony and discord would alike be unknown. 361. Colors of the Sky. A gas or a liquid which, of itself, is colorless becomes colored when it contains finely divided matter in suspension. An excellent example is the sky-blue liquid obtained by adding to water a very small proportion of milk or an alcoholic solution of mastic, or by mixing a few drops of dilute nitrate of silver with a quantity of water in which a little table salt has been dis- solved. (In the last case chloride of silver is formed, which is insol- uble in water, but remains suspended in the form of extremely minute solid particles. The same is true of the mastic.) These liquids appear blue by reflected light; but are yellow or orange when viewed by transmitted light. This is due to the fact that the suspended particles reflect a considerable part of the violet and blue light, but reflect less and less of the other colors toward the red end of the spec- trum. Thus violet and blue predominate in the reflected light, and red, orange, and yellow in the transmitted light. The blue color of the sky is similarly explained, the air being ren- DISPERSION OF LIGHT. COLOR 451 dered visible against the dark background of black space by sunlight reflected from its fine suspended dust or water particles; while the light transmitted directly from the sun is always more or less yellowish, and, in the afternoon and evening, when sunlight comes to us through a greater thickness of the dusty layers, verges toward orange or even red. 362. Color Sensation. Complementary Colors. We have seen that the same color sensation may be produced by light of one wave length, selected from the spectrum, or by composite light of many different wave lengths. Thus the light transmitted by yellow glass may appear to the eye exactly like the yellow of the spectrum, although, when analyzed, it is found to consist of red, orange, yellow, and green waves. More curi- . . . Purple ously still, a given color sensa- tion may be due to composite light in which the wave length corresponding to that color is wholly wanting. For example, a mixture of spectral red and spectral yellow produces the sensation of orange, and a mix- ture of spectral violet and green produces the sensation of blue. FlG ' 343 ' " pposite Colors Com - plementary. In general the mixture of any two spectral colors appears to the eye to be identical with the color named midway between them in the accompanying chart (Fig. 343). The mixture of red and violet lights is purple, a color not found in the spectrum. The mixture of any pair of spectral colors named on opposite sides of the chart appears white, when the two lights are taken in the right proportion. With the un- 452 LIGHT aided eye these different white lights can not be dis- tinguished from one another or from ordinary white light, which is a mixture of all the spectral colors; but the prism would instantly reveal their differences. Any two colored lights which together produce the sensation of white are called complementary colors. All possible color sensations, including white, can be produced by combining spectral red, green, and violet lights in different proportions. Red, green, and violet are therefore called the primary color sensations. In studying mixtures of colored lights, the selected colors of the spectrum can be focused by a lens or reflected by mirrors to the same spot on a white screen. The same effect is more conveniently pro- duced by means of colored disks, each of which is slit along a radius, thus permitting any desired amount of overlapping when two or more of the disks are placed together on an axis through their common center (Fig. 344). When the disks are rapidly rotated about the axis, only one color is seen, and this covers the entire circular area. This result is due to the fact that the sensation of sight continues for a fraction of a second after the light ceases to enter the eye or ceases to fall upon the same part of the retina. The rapid rotation causes the different colors to come from all parts of the disk in such rapid succession that each color produces a con- tinuous impression for the entire circular area, just as if it were reflected by the en- tire area. The colors of the disk are com- posite; but their effect in a mixture is the same as if they were simple spectral colors. In most cases, however, the total amount of reflected light is so small that the resultant color is deficient in brightness. Com- plementary colors, for example, generally yield a dark gray instead of white. 363. The Young-Helmholtz Theory of Color Sensation. Color Blindness. The most probable theory of color sensation is that proposed by the English physicist, Dr. Thomas Young, and further DISPERSION OF LIGHT. COLOR 453 amplified by Helmholtz. .According to this theory the normal eye is provided with three sets of nerves, which are most sensitive to red, green, and blue-violet light respectively. A primary color sensa- tion is due mainly, if not wholly, to the stimulation of one set of nerves. All other color sensations are resultant effects, due to the stimula- tion of two or of all the sets in different degrees. For example, the sensation of yellow is produced when the "red" and the "green" nerves are equally stimulated, whether by a mixture of red and green lights or by spectral yellow. A person who does not see all the colors of the spectrum as they appear to the normal eye is said to be color blind. Usually it is the red that is seen abnormally; in rare instances it is the green or the violet. This defect of vision is supposed to be due to the absence or inactivity of the corresponding set of nerves. In red blindness red is perceived by a weak stimulation of the "green" nerves, and it is distinguished from green only as a darker shade of the same color. The absence of the red sensation modifies the other color sensations more or less, with the probable exception of blue and violet. Extensive tests have shown that three or four per cent, of all per- sons are color blind. The defect usually exists from birth, and doubtless in most cases is never discovered. For the person thus afflicted learns in childhood to call colors by their right names, with- out having any reason to suspect that his color sensations are differ- ent from those of his companions. He might sometimes wonder why other children could spy out ripe strawberries among the green vines more readily than himself; but he would most assuredly not hit upon the true reason. However, a simple test has been devised by which color blindness can be detected with certainty. The per- son undergoing the test is directed to assort a large number of vari- ously colored skeins of wool, placing together all that resemble each other. The colors are so chosen that one who is color blind will be sure to make mistakes. Mariners, soldiers, and railway employes are thus examined as a test of their ability to distinguish colored signals. 364. Colors of Mixed Pigments. A mixture of blue and yellow lights is white; but the light transmitted through blue and yellow glass in succession is green. The results 454 LIGHT are not inconsistent, for they are obtained by wholly dif- ferent processes. The first is a case of addition, the sec- ond a case of double subtraction, as already explained. The mixture of blue and yellow paints or powders is green. This is also a case of double subtraction. The blue paint absorbs the red, orange, and yellow of the incident light, and the yellow paint absorbs the violet and blue. Green is the only color not strongly absorbed by one or the other, and hence is the principal color in the light reflected by the mixture. In general, the light reflected by mixed pigments consists of the colors which are not absorbed by any of the constituents. If the light reflected by one pigment has no constituent in common with the light reflected by a second pigment, a mixture of the two is black. This is the case with vermilion (a bright red) and ultra- marine (a deep blue). The artist mixes his pigments before applying them to the canvas. In making colored prints the different pigments are laid on, one after the other, in separate impressions. The inks are differently distributed, so that the final color is in some places due to one only, in other places to two, and in still others to all three in varying proportions. Thus a surprising number of delicate tints and shades are produced. 365. Color by Interference. White light becomes colored whenever it loses one or more of its constituents, whether by selective absorption or any other process. The rainbow colors of soap bubbles and of thin films of oil floating on water are due to the loss of certain wave lengths by interference. The phenomenon is similar to the interference of sound waves (Art. 277), and takes place under similar conditions, as in the following experiment: Let two pieces of clean plate glass be pressed firmly together with a small clamp, and held so that a strong light falls upon them. From the illuminated side, brilliant colored bands will DISPERSION OF LIGHT. COLOR 455 be seen, surrounding the point of closest contact. When pres- sure is applied at other points with the fingers, the bands become wider and shift into new positions, showing that the color varies with the distance between the plates. The nature of these color effects can be under- stood, in a general way, with the aid of Fig. 345. MM and NN are sections of the glass plates, the distance between them being greatly exaggerated. Light incident along the path AB is partially reflected at C ^Jjj Jjp from the lower surface of the upper plate, FlG and also at E from the upper surface of the lower plate. Some of the light reflected at E is transmitted through the upper plate, parallel to and nearly coincident with the light reflected from C. But, in twice crossing the space between the plates, the waves reflected at E fall behind the corresponding waves reflected at C, and waves of a certain length in the two sets will meet in opposite phase and destroy each other. The re- flected light is complementary to the color lost by interference; and the latter differs at different places, depending upon the distance between the plates. Interference colors are produced whenever light is reflected from the two surfaces of a very thin film or plate of any transparent sub- stance. They are most common with liquid films, such as soap bubbles and films of oil on water. In the above experiment the film is the thin sheet of air between the plates. Solids are rarely thin enough to exhibit such colors, mica in very thin flakes being the only common example. Interference also occurs, with even more beautiful color effects, when light is reflected from surfaces covered with minute parallel grooves and ridges, called striations, as in mother-of-pearl and the plumage of many birds. As such a surface is turned about, so that the light falls upon it at changing angles, brilliant rainbow colors sweep over it in rapid succession. This beautiful " play of colors " is often seen on the breasts of humming birds, as they dart about in the sunshine. Bodies that exhibit interference colors are said to be iridescent. 456 LIGHT PROBLEMS 1. What is the function of the lens in producing a pure spectrum? 2. Why is it not possible to correct the chromatic aberration of a lens by any change in the form of its surfaces? 3. (a) State all the conditions necessary for a rainbow. (6) Do two observers see exactly the same rainbow? 4. Prove that the reflection within a rain-drop takes place at less than the critical angle, and is therefore not total. CHAPTER XI MAGNETISM I. PROPERTIES OF MAGNETS 366. Natural and Artificial Magnets. A magnet is distinguished from other bodies by its power of attracting pieces of iron. This power is a property of the material composing the magnet, and may be either temporary or permanent. Natural magnets were known to the ancients. They are black stones, consisting of a certain iron ore called magnetic oxide of iron or magnetite (FesO^. The word magnet is derived from Magnesia, the name of the city in Asia Minor near which these magnetic stones were first found. After the discovery, in the eleventh or twelfth century, that suspended magnets always point in a defi- nite direction, they came into use for determining direc- tions on land and sea, and were called lodestones (leading stones). When a piece of highly tempered steel is rubbed with a magnet or in any other way subjected to strong magnetic action, it acquires permanent 'magnetic proper- ties, and becomes a manufactured or artificial magnet. Magnets are made of various shapes, which are adapted to different uses (Figs. 346, 347, and 348). 367. The Poles of a Magnet The different H G rs 3 e 4 s 6 h ~ parts of a magnet have very unequal power of Magnetand attracting iron, as is shown by the distribution or Arma- of the mass of filings or small tacks which the ture> 457 458 MAGNETISM magnet can hold (Fig. 347). The quantity is greatest at and near the ends, and diminishes rapidly toward the middle portion, which is generally bare. The regions near the ends where the FIG. 347. magnetic action is strongest are called the poles of the magnet. A magnet has regularly two poles, one at each end, whatever its shape. By irregular magnetization additional poles can be developed; but such cases are unimportant and need not be considered. When a straight magnet is suspended or supported so that it is free to turn in a horizontal plane, and is subjected only to the magnetic action v , , , S ^ N Mn(mft ^ of the earth (as will be ex- plained later), it always comes to rest with the same end pointing in a northerly *^ * FIG. 348. Magnetic Needle. direction. This end of the magnet is called the north pole (i.e. the north-seeking pole), and the other end is called the south pole. A slender magnet, balanced on a pivot (Fig. 348) or suspended at its center by an untwisted fiber is called a magnetic needle. 368. Magnetic Attraction and Repulsion. When the north pole of a bar magnet is brought near the north pole of a magnetic needle, the latter is driven away, or repelled. Similarly the south pole of the needle is repelled by the south pole of the magnet. But the north pole of the needle is drawn toward the south pole of the magnet, and its south pole toward the north pole. If the bar magnet is suspended so that it also is free to move, or if the experi- ment is tried with two magnetic needles, it will be found PROPERTIES OF MAGNETS 459 that both are attracted or both repelled at the same time. A magnetic force is always a mutual action between two bodies, in accordance with Newton's third law of motion (Art. 118). Such experiments show that there is a real difference between north poles and south poles, and that like poles repel and unlike poles attract each other. 369. Force Exerted between Two Magnetic Poles. It is a well known fact that magnets differ in strength or attracting power. Greater strength in any given case may be due either to greater size, or to a greater degree of magnetization, or to both causes together. The force exerted between two poles varies as the prod- uct of the strengths of the poles, and also varies with the distance between them. If the poles are small compared with the distance, the attraction or repulsion varies inversely as the square of the distance. With ordinary magnets, how- ever, this relation does not hold very closely. The effect of distance is seen in the tendency of two magnets to move bodily toward each other when their nearer poles are un- like, and away from each other when these poles are alike. In the first case the attraction between the adjacent poles exceeds the repulsion of the more distant ones; in the second case the repulsion exceeds the attraction, for the same reason. The strengths of magnetic poles and the forces which they exert upon one another are measurable quantities; but their measurement is unnecessary in elementary physics. We are concerned, only in a general way, with their relative magnitudes. Greater or less mag- netic attraction is shown by the more or less rapid vibration of a mag- netic needle. Thus when a pole of a magnet is brought slowly toward a magnetic needle, the unlike pole of the needle swings round and vibrates before it, more and more rapidly as the distance decreases. 460 MAGNETISM 370. Magnetic and Non-magnetic Substances. A sub- stance which is attracted by a magnet and which can itself be magnetized is said to be magnetic. Substances which do not possess these properties in any appreciable degree are usually classed as non-magnetic. Iron in its different forms, such as cast iron, wrought iron, and steel, is the most strongly magnetic material known. The next in order are cobalt and nickel. These are also quite strongly magnetic, but much less so than iron. All other substances are practically non-magnetic. The magnetic properties of iron are of very great impor- tance, being usefully applied in the telegraph, the tele- phone, the dynamo, the motor, and many other electrical machines and instruments. 371. Magnetic Induction. The law of magnetic action between the poles of two magnets does not seem to hold for the attraction between a magnet and an unmagnetized piece of iron; for unmagnetized iron has no poles. Let us see what further may be learned concerning this apparent exception. In the first place we can determine whether a piece of iron is magnetized by trying to pick up iron filings with it. If none cling to it, it is not appreciably magnetized. A rod of soft iron, tested in this manner, will be found to be un- magnetized. Let the test be re- Is ~N] pea ted at one end of the rod, while a pole of a magnet is held against or very near the other end. It now gathers a considerable tuft of filings (Fig. 349), which drop off as soon as the magnet is removed. FIG. 349. Temporary Mag- . . , . . ir netic induction. Evidently the rod is itself a magnet PROPERTIES OF MAGNETS 461 while the permanent magnet is near it, and ceases to be one when the magnet is removed. This action by which iron or steel becomes magnetized, when subjected to magnetic forces, is called magnetic induction. In the case considered the induced magnetism is only temporary. The north and south poles induced in the iron rod can be determined by means of a magnetic needle. Thus when the north pole of the magnet is in contact with the rod at one end, the other -end repels the north pole of the needle, and hence must itself be a north pole. The end in contact with the magnet is then a south pole, as shown in the fig- ure. Hence we find that the attraction of the magnet for the supposed unmagnetized rod is really an attraction between the unlike poles of a permanent magnet and a temporary one. In all cases where unmagnetized iron or steel is brought near enough to a magnet to be attracted by it, the attrac- tion is the result of magnetic induction. 372. Temporary and Permanent Magnets. When soft iron, hard iron, and tempered steel are subjected to equal inductive action, e.g. by contact with the same magnet, a test with iron filings will show that the soft iron becomes most strongly magnetized and the tempered steel the least. But the soft iron loses its magnetism almost completely, as soon as it is removed from the influence of the magnet, while the tempered steel retains its magnetism indefinitely. Thus by magnetic induction a piece of soft iron becomes a temporary magnet, and tempered steel a permanent one. Hard iron and untempered steel retain a considerable part of their induced magnetism, and are called subpermanent. A piece of highly tempered steel can be made a permanent magnet by rubbing it repeatedly from end to end, with another magnet, or 462 MAGNETISM from the center toward both ends with unlike poles of two magnets. A more effective method, depending on the use of an electric cur- rent, will be described later. Permanent magnets can be demag- netized or magnetized with opposite polarity, by sufficiently strong inductive action in the opposite direction. Soft iron, as a magnetic material, has more numerous and more important uses than tempered steel. It plays a necessary part in the generation of electric currents by dynamos, and also in a great many applications of electricity. 373. Magnetic Action through Bodies. Permeability. Mag- netic action takes place through non-magnetic bodies without hin- drance or modification of any sort. For example, a magnet attracts or repels a magnetic needle through a board, a book, or a plate of glass, just as if nothing intervened. But when a sheet of iron is thrust between them, the needle is only slightly affected by the presence of the magnet, if at all. The sheet of iron, especially if large, acts as a screen to cut off magnetic action from the side oppo- site to the magnet. This effect is due to induction in the iron, by which it becomes magnetized; and the mange tic action is carried off to the edges of the sheet. This can be shown by bringing the needle up to the edge, where it will be attracted or repelled as by an ordi- nary magnet. A rod of soft iron, placed lengthwise between the mag- net and the needle, intens'fies the action of the needle, just as if the magnet had been brought up closer to it. The rod, by induction, serves as a carrier of the magnetic action. Only magnetic substances can thus deflect, extend, and intensify the action of a magnet. It is as if the magnetic forces found an easier path through the magnetic substance than that afforded by the air or other non-magnetic substance; and the material which affords the better path is said to have greater magnetic permeability. The permeability of air and other non-magnetic substances is taken as unity. Magnetic forces act with equal intensity through all of them. Nickel and cobalt are highly permeable, steel is much more so, and soft iron most of all. The greater the permeability of a substance the greater will be the magnetic induction in it, when subjected to a given magnetizing force. 374. Magnetism is a Molecular Property. The same magnet may be used to magnetize any number of pieces of PROPERTIES OF MAGNETS 463 steel, without itself becoming weaker. Evidently, there- fore, the induced magnetism is not something transmitted from the magnet to the body magnetized. On the con- trary, it is a molecular condition developed within the body itself. The probable nature of this condition is suggested by the following experiments. When a magnet is broken, unlike poles are produced at the broken ends, and each piece becomes a complete magnet (Fig. 350). (A magnetized sewing or knitting FIG. 350. Poles of a Broken Magnet. needle is convenient for the experiment.) A test with iron filings will show that the new poles are as strong as the original ones. The magnet may be broken into smaller and smaller pieces indefinitely, and each piece will still have a north and a south pole. Since the act of breaking is not a magnetizing process, it follows that a magnet is magnetized throughout its entire length. The absence of attracting power at any point, as at the center, may be regarded as due to the equal and opposite action of a north and a south pole at that place. When these poles are sep- arated by breaking, they no longer neutralize each other. A magnet may there- fore be regarded as composed of a multi- tude of little mag- & ~ sw_. s nets, with their like FlG - 35I- A Magnet is Virtually Composed of a Multitude of Smaller Magnets. poles pointing in, the same direction (Fig. 351). If the intensity of magnetiza- tion were the same at all points, the adjacent unlike poles SN' n s n s n s n s\n s n s n s n s n s n s n s n s\n s n s n s L ?l S n s n s n s n s\n s n s n s n s 464 MAGNETISM of these little magnets would all neutralize one another except at the very ends. In reality, however, the mag- netization grows weaker toward the ends, and consequently the poles extend some distance .back from them. Undoubtedly the smallest visible fragment of a magnet is itself a complete magnet, having a north and a south pole; and this is probably true of the individual molecules, for any action that is known to affect the molecules of a body also affects the magnetism of a magnet. Thus the strength of a magnet is diminished by heating it, until, at a bright red heat, it is completely demagnetized. A mag- net is also weakened by any mechanical disturbance of its molecular arrangement, as in striking, bending, or twist- ing it. The effect of bending and twisting is easily shown with a magnetized piece of iron wire. On the other hand, a piece of steel becomes more strongly magnetized if it is hammered, or heated and allowed to cool, while it is near a magnet. These facts and others of a similar character have led to the theory that each molecule of a magnetic substance is a permanent magnet. In an unmagnetized body these molecular magnets point indis- FIG. 352 . - Arrangement of Molecules in Criminately in all Unmagnetized Iron or Steel. tionS (Fig. 352), SO that they neutralize each other's external magnetic effects. In a magnetized body the greater number of the mole- cules lie with their like poles pointing in the same general direction (Fig. 353). In the act of magnetizing a body the molecules are turned around, more or less completely, into one particular direction. If all the molecules were turned in the same direction, the limit of possible magneti- PROPERTIES OF MAGNETS 465 zation would be reached. Soft iron is more readily mag- netized than steel because its molecules are more easily turned about, and it loses its magnetism more readily for the same reason. According to this theory, heat weakens a magnet because it increases molecular motion, and the mole- cules jostle one another out of position. Ham- mering, bending, and FIG. 353- - Arrangement of Molecules in twisting also disturb Magnetized Iron or Steel. the molecular arrangement. On the other hand, any dis- turbance of the molecules in the presence of a magnetizing force helps to turn them round into line with that force. These effects may be illustrated by an experiment with steel filings, in which each particle represents a molecule on a greatly magnified scale. A test tube loosely filled with the filings is held in a horizontal position, while the filings are .jarred toward one end by repeatedly tapping that end with a pole of a magnet. Testing with a magnetic needle will show that the mass of filings now has a pole at each end, like a bar magnet. This is due to the regular arrangement of the magnetized particles. Shaking the tube destroys this arrangement, and the mass as a whole " loses its magnetism," although each individual particle is still a magnet. PROBLEMS 1. (a) When a pole of a strong magnet is brought toward the like pole of a magnetic needle, repulsion may be followed by attraction as the mag- net is brought closer. Explain. (6) The same may happen when an end of a weakly magnetized piece of iron is brought toward a needle. Explain. 2. Why should decision as to the polarity of a magnetized body be based on repulsion of the magnetic needle rather than on attraction? 3. In what different ways may an unmagnetized magnetic substance be distinguished from a magnet? 4 66 MAGNETISM II. THE MAGNETIC FIELD 375. Magnetic Lines of Force. When a magnetic needle is near a magnet, as at O (Fig. 354), its north pole is attracted by the south pole of the magnet and re- pelled by the north pole. These forces are represented in magnitude and direction by OB and OA respectively, FIG. 354. Magnetic Line of Force. . . . . O being the position of the north pole of the needle. The attraction is the greater force, since it is due to the nearer pole. The resultant of these two forces is represented by OR (found by constructing the parallelogram of forces). Hence the magnetic needle behaves as if its north pole were acted upon by the single force OR. If the south pole of the needle is at 0, the result- ant force upon it is equal and opposite to OR. Obviously the two poles of the needle can not be at at the same time ; but if the needle is very short and its center is at 0, the forces acting on its poles are approximately as stated, and the needle will come to rest with its north pole pointing in the direction OR, this being the position of equilibrium. If the needle is moved constantly in the direction in which its north pole points, it will trace the curved path OCS. Starting from N, the entire curve NOCS can be traced in this manner. This curve is called a magnetic line of force. Going from N toward S, its direction at every point is the direction of the resultant magnetic force at that point upon the north pole of the needle. Going from S toward N, its direction at every point is the direc- tion of the resultant magnetic force at that point upon the south pole of the needle. THE MAGNETIC FIELD 467 Lines of force are of great importance, and we shall meet with them frequently in the study of electricity. To save words, it is always understood that the expression the direction of a line of force means the direction of the magnetic force upon the north pole of a magnetic needle at any point along the line. Thus, in the present in- stance, the direction of the line of force is from N toward S. Any number of lines of force can be traced about a magnet, in the manner above described. In general, the direction of the line of force passing through any point within the range of action of a magnet is the direction in which the north pole of a short magnetic needle points at that place. To explore the entire space about a magnet by this method is a long and tedious process; but the same information can be obtained in a very simple and striking manner by means of iron filings. The filings are sifted upon a sheet of cardboard, laid over the magnet. Each particle of iron becomes magnetized and tends to place itself lengthwise along a line of force. Tapping the cardboard with the finger assists the magnetic forces by overcoming friction. The fil- ings cling together in somewhat irregular, broken lines, which never- theless ind ; cate the lines of force very clearly (Fig. 355^). The lines of force are really smooth, unbroken curves, and are continuous with lines of magnetic induction within the magnet (Fig. 3556). Each line of force and the corresponding line of induction together form a closed curve. (Lines of force are often represented by dotted lines, as in some of the diagrams that follow.) 376. The Magnetic Field. Any space within which magnetic forces act, when magnetic material is present to be acted upon, is called a magnetic field. Every magnet is surrounded by a magnetic field, which extends indefi- nitely in all directions. Practically, it is regarded as extend- ing only as far as the magnet noticeably affects a magnetic needle. The region within which the field is strong enough to turn iron filings into line is smaller than this. 468 MAGNETISM There are other magnetic fields than those of magnets. We shall find later on that a current of electricity always produces a magnetic field about the wire or other conduc- FIG. 3550. Lines of Force in the Field of a Bar Magnet. tor in which the current is flowing. Everywhere upon the earth's surface a magnetic needle sets itself in a definite direction, when no magnet or electric current is near. This behavior shows that the earth is surrounded by a magnetic field, as if it were a huge magnet. This field is much too weak to direct iron filings, and so does not FIG. . Lines of Force and Lines of Magnetic Induction. interfere with their use in studying the fields of magnets. A magnetic field is to be regarded as having an actual physical existence. The portion of space that it occupies possesses properties which other space does not. These THE MAGNETIC FIELD 469 AA/fAAAAAAAAAA'AAA/f il' IAI I j A B FIG. 356. Magnetic Ac- tion in a Uniform Field. properties are often considered without reference to the ori- gin of the field. Thus we say that a magnetic needle, when placed in a magnetic field, tends to set itself parallel to the lines of force of the field. The properties of a magnetic field with which we are principally con- cerned are its intensity, or strength, and the direction of its lines of force. A diagram or map of a field indi- cates the relative intensities of its different parts by the relative distances between the lines of force. Where the lines run closer together the field is stronger; where they are more widely separated it is weaker (Fig. 3556). In a field of uniform intensity the lines are straight and parallel, and are equally spaced. Any limited portion of the earth's field is a good example (Fig. 356). In such a field the forces act- ing upon the poles of a magnetic needle are ex- actly equal and opposite. Together they form a couple, which causes rota- tion when the needle is at an angle with their lines of action; but they do not tend to move the needle as a whole in either direction. 377. Other Properties of Magnetic Fields. - When the north pole of one magnet is placed near the south pole of FIG. 357. Magnetic Attraction along Lines of Force. 470 MAGNETISM another, many of the lines of force extend across be- tween them (Fig. 357); when like poles are adjacent the lines in one field turn away from those in the other (Fig. 358). These are typical cases. In general, we find lines of force extending from the north pole of a magnet to the south pole of the same or to the south pole of an- other magnet; but there are no lines connecting like poles. Neither do lines of force ever cross each other; for at any point of inter- section the magnetic forces would have two resultants, which is not true of any set of forces. Any magnetic body placed in a magnetic field modifies the field and alters the dis- tribution of the lines of force. The effect of a soft iron bar in a uniform field is shown in Figure 359. The lines of force crowd together, entering the iron at one end and leaving it at the other. It is as if the iron afforded an easier path for the lines than air does; i.e. the iron has greater magnetic permeability than air (Art. 373). Another way of stating it is that the iron becomes mag- netized by induction (with the polarity shown in the figure), and adds its own field to the original one. 378. Theory of Magnetic Action. When an object is pushed with a stick or pulled with a rope, the mechanism of the action is clear. The stick or the rope serves to transmit the force from the m FIG. 358. Magnetic Repulsion across Lines of Force. THE MAGNETIC FIELD 471 hand to the body acted upon. The stick sustains a compressive stress, which tends to bend it; and the rope a tensile stress, which tends to pull it apart. Magnetic forces act between bodies at a dis- tance, without the visible aid of any intervening medium, and this action takes place in a vacuum as readily as in air. In these respects mag- netic action is like gravitation; and both are inversely proportional to the square of the distance. In other respects they are very dif- ferent. The force of gravitation is always an attraction, and it acts on all masses irrespective of their material. Magnetic action is lim- ited to magnetic materials, and the force may be either an attrac- tion or a repulsion. Magnetic action can be cut off by a magnetic screen. Gravitation is unaffected by any intervening medium. Gravitational attraction is excessively small between masses of ordi- nary size. Magnetic forces are relatively enormous. No satisfactory theory of gravitation has yet been proposed; but it is very probable that the attraction takes place through the medium FIG. 359. Effect of Soft Iron in a Magnetic Field. of the ether. That the ether is the medium through which magnetic forces act is hardly a matter of doubt. The action appears to be in the nature of a tension along .the lines of force and a pressure at right angles to them, as if the ether were an elastic solid (Art. 296) in a state of strain. The ether in this condition may be compared to a stretched piece of rubber, which tends to shorten and to become thicker. The tension along the lines of force draws unlike magnetic poles together (Fig. 357), and the pressure at right angles to the lines pushes like poles apart (Fig. 358). 472 MAGNETISM III. THE EARTH'S MAGNETIC FIELD 379. Magnetic Meridians and Declination. Every- where upon the earth's surface a magnetic needle, when removed from all magnetic bodies, comes to rest in a defi- nite direction, clearly indicating that it is controlled by a magnetic field. This is the magnetic field of the earth. Its cause is not very well understood; but it is, probably due to electric currents circulating round the earth. Large masses of iron ore produce local variations in the field, but they are evidently not its primary cause. FIG. 360. Magnetic Meridians. A line extending over the earth's surface and having at every point the direction of the magnetic needle is called a magnetic north-and-south line, or a magnetic meridian. Magnetic meridians are represented in Fig. 360 by the heav- ier lines. They are more or less irregular, and are nearly everywhere at a considerable angle with the geographical meridians, or the true north-and-south lines. This angle is called the magnetic decimation, or, simply, the declina- tion. In the eastern part of America the declination is toward the west; in the western part it is toward the east. THE EARTH'S MAGNETIC FIELD 473 A line connecting all points where the declination is the same is called an isogonic line (Greek isos, equal, and gonia, angle). Such lines are irregular curves (Fig. 361). Magnetic declination is subject to daily and annual variations, amounting, however, only to a small fraction of a degree. There is also a slow but continuous change in one direction from year to year. At London, England, the declination in 1580 was 11 east; in 1800 it was 24 west. Since the latter date the change has been in the opposite direction. 150 120 80 60 80 FIG. 361. Chart of Isogonic Lines. 380. Magnetic Inclination or Dip. Magnetic Poles of the Earth. The ordinary magnetic needle is free to turn only in a horizontal plane. Since its center of gravity is below the point of support, its weight opposes any downward tilting of either end. Hence we can not tell from the behavior of such a needle whether the lines of force of the earth's field are horizontal or inclined. For this purpose we require a dipping needle, which is a magnetic needle mounted on a horizontal axis through its center of gravity, and sometimes also 474 MAGNETISM suspended from an untwisted fiber, which serves as a vertical axis (Fig. 362). A needle thus mounted is free to assume any direction, and its direction is wholly unaffected by gravity. It will, therefore, come to rest parallel to the lines of force of the earth's field. The angle between the direction of the dipping needle and the hori- zontal is called the magnetic inclination or dip. The irregular lines extending across Fig. 363 are lines of equal dip. The line of no dip is called the magnetic equator. North of the magnetic equator the north pole of the needle is depressed, and south of it the south pole. Arctic explorers have found a place where the dip is 90. This is the north magnetic pole of the earth (so called from its geographical position; its polarity is like that of the south pole of a magnet). It is nearly 1400 mi. from the geographical north pole, and is shown in Fig. 360 as the point in the northern hemisphere to which the magnetic me- ridians converge. It is situated in latitude 70 5' N. and longitude 96 43' W. The south magnetic pole was discovered in 1908 by an expedition under the FlG 62 command of Lieutenant Shackleton of the British The Dip- navy. It lies in latitude 72 25' S. and longitude 154 ping Needle. E strictly speaking, the magnetic poles of the earth lie far below the surface. 381. Intensity of the Earth's Magnetic Field. Induct- ive Action. The magnetic field of the earth is relatively very weak, that of an ordinary magnet being thousands of times stronger; but its inductive action is sufficient to pro- duce considerable magnetization in iron and steel. This is readily shown with a long rod of soft iron (Norway iron). While the rod is held in a north-and-south line, or, better, at the angle of dip, it will be found to be magnetized, with a north pole at its north or lower end. On reversing the rod, its polarity is also instantly reversed, provided the iron is very soft; otherwise it may be necessary to strike the rod on the end while it is held in position. Any mass of iron or steel that remains in one position for a time becomes magnetized by the earth's inductive action, THE EARTH'S MAGNETIC FIELD 475 especially if it is subjected to jarring, as in railroad tracks and bridges. The magnetism of lodestones has doubtless been produced by the same cause. 382. Importance of the Earth's Magnetic Field. Mag- netic Surveys. The earth's magnetic field is of the great- est importance, since the use of the compass in determining directions on land and sea depends upon it. A compass is a magnetic needle suitably mounted within a box, to- gether with a compass card or dial. In the mariner's com- FIG. 363. Lines of Equal Dip. pass the card turns with the needle, so that at all times it correctly indicates the directions marked upon it (Fig. 364), allowance being made for the declination. In the surveyor's compass the dial is in the bottom of the box and the needle moves over it. The true north can be determined with a compass only when the declination at the place is known. This is given by a declination map or chart of the region, which should be as accurate as possible. Owing 476 MAGNETISM to the continuous change in the earth's magnetism, new magnetic charts must be constructed from time to time; and in order to obtain the necessary information for this purpose, the civilized nations of the world are constantly making magnetic surveys on land and sea. In the United States this work is done by the Division of Terrestrial Magnetism of the United States Coast and Geodetic Survey. In a report recently published the Survey gives maps and tables constructed from observations made at over 3300 stations over two thirds of which were occu- pied by the Survey from 1899 to 1906. The Carnegie Insti- tution at Washington, D. C., has undertaken a series of sur- veys to determine the magnetic conditions over all the oceans. For the greatest accuracy the vessel in which such work is carried on must be as nearly non-magnetic as possible; for all iron and steel parts of a ship become magnetized by the induction due to the earth's field, and this magnetism affects the compass needle more or less. A ship has been built especially for this ser- vice (1909). It is constructed entirely of non-magnetic materials, with the exception of certain parts of the engine. The fasten- ings consist of locust-wood nails, copper and bronze bolts, and composition spikes. All metal deck fittings and metal work on spars and rigging are of bronze, copper, or gun-metal. FIG. 364. The Compass Card. Recit- ing the names of the thirty-two points is called by sailors "Boxing the Compass." CHAPTER XII ELECTROSTATICS 383. Introduction. This is often called the electrical age, and with good reason; for electricity is now doing a large part of the work of the world, and that part is increas- ing rapidly from year to year. The use of electrical energy in transportation, in driving the machinery of shops and factories, in lighting buildings and city streets, in trans- mitting messages by telegraph and telephone, etc., is more or less familiar to every one. All this has been accomplished within the past century, through the discovery and appli- cation of the laws of electrical action, but without a knowledge of what electricity really is. The earlier ideas concerning the nature of this wonderful agent have been discarded. In recent years rapid progress has been made in the development of a new theory, which is supported by such an array of facts that it promises to be final. Of this we shall have something to say in the concluding chapter. Meanwhile we shall be mainly concerned with matters of fact with electrical phenomena, their laws, and their applications. The subject of electricity is divided into two parts, electrostatics dealing with electricity at rest, or in equi- librium, and electrodynamics, dealing with electricity in motion. We shall begin with electrostatics, which is the older branch of the science, the period of its greatest development being the eighteenth century. 477 478 ELECTROSTATICS 384. Electrification by Friction. When a vulcanite (hard rubber) rod is rubbed with fur or flannel, it acquires the power of attracting bits of paper or pith, and other light bodies. The same results are obtained in greater or less degree with many different substances, e.g. with sealing wax, resin, or sulphur when rubbed with fur or flannel, and with a glass rod when rubbed with silk. In all cases the bodies must be dry, and the drier the atmosphere the better. It was known to the ancient Greeks that amber possesses this power of attraction when rubbed; but Dr. Gilbert, an English physician and scientist of the sixteenth century, seems to have been the first to make a systematic study of the phenomenon. In his great work on magnetism, published in 1600, he called all substances which he had found to exhibit this property of amber "electrics," after elektron, the Greek name for amber. He described the condition of the rubbed body as a state of electrification, and called the force exerted by it electric attraction. The agent to which electrostatic phenomena are due became known as electricity. A body in a condition to exert electric at- traction is said to be electrified or to have an electric charge, or to be charged. 385. Positive and Negative Electrification. An elec- trified vulcanite rod suspended by a thread turns away when another elec- trified vulcanite rod is brought near it showing that it is repelled (Fig. 365). Two electrified glass rods also exhibit FIG. 365. Electrostatic" repulsion when tested in the same manner; but a vulcanite rod rubbed with fur and a glass rod rubbed with silk attract each other. ELECTROSTATICS 479 Any electrified body either attracts the electrified glass rod and repels the electrified vulcanite, or it repels the glass and attracts the vulcanite. From this we learn that there are two, and only two, states of electrification or two kinds of electric charges. The electrification of glass when rubbed with silk, is called positive, and the glass is said to have a positive charge, or to be positively eletri- fied. A negative charge is one like that of vulcanite when rubbed with fur or flannel. The law of electrostatic action, as shown by the experi- ments described above, is that like charges repel and unlike charges attract each other. The force, whether of attrac- tion or repulsion, becomes less as the distance between the charges is increased. 386. Conductors and Non-conductors or Insulators. Gilbert was unable to electrify the metals and some other substances, and he therefore called them "non-electrics." It was later discovered that such bodies are conductors of electricity, and permit the charge to escape through the body of the experimenter to the earth as fast as it is formed. Vulcanite and other substances which retain their charges are called non-conductors, or insulators. A metal rod or other conducting substance can be electrified if the precau- tion is taken to interpose some insulating body between it arid the hand. All substances may be roughly classified as conductors or non-conductors of electricity; but there is no sharp divid- ing line between them. The two classes merge impercep- tibly into each other when the substances are arranged in the order of their electrical conductivity, just as in the case of good and poor conductors of heat. At the one extreme we have the best conductors, among which are the metals and 480 ELECTROSTATICS solutions of salts and acids in water; and, at the other ex- treme, the best insulators, such as vulcanite, rubber, sul- phur, shellac, glass, paraffin, sealing wax, silk, and air. Wood, cotton, and various other substances occupy an intermediate position. If the charged bodies in electrostatic experiments are not themselves good insulators, they must have a non- conducting support, to prevent the escape of the charge to neighboring bodies. A body thus supported is said to be insulated. It should be noted that a given material may be a sufficiently good insulator under certain con- ditions but not under others. This is familiar in the vari- ous uses of the electric current. A cotton covering suffices for the wire used in the circuits of electric bells, telegraph instruments, and the like. Better insulation is afforded by a thick covering of cotton and rubber, as in electric- light circuits. In spark coils and similar appliances, where the conditions are very exacting, the wire is covered with silk. In controlling electric charges the best insu- lators are required, such materials as cotton and wood being wholly inadequate. 387. Charging and Discharging by Conduction. TJie bits of paper or other light bodies that cling to an electri- fied rod often dart away after brief contact. While the paper is in contact with the rod, it receives a portion of the charge by conduction, and is then driven off by the repulsion between its charge and the like charge of the rod. Since the rod is a non-conductor, it parts with its charge very slowly, even at points of actual contact. On this account the papers that chance to touch at only a few points are not repelled. This action is shown to better advantage with a pith ELECTROSTATICS 48 1 ball, suspended by a silk thread. The ball swings out toward an electrified rod, and rolls about over it, taking up the charge, until it is repelled (Fig. 366). The repul- pulsion continues, since the charge on the ball can not escape through the thread; but, if the ball is permitted to come in contact with the metal or wood sup- port from which it hangs, or is touched with the fin- FlG 366 _ pith Ball a Attracted ger, its charge is Conducted before Touching Electrified Rod; b, ... . Repelled after Touching. away and it is again at- tracted by the rod. If the ball is suspended by a cotton thread, it is not repelled at all, for the cotton conducts the charge away as fast as the ball receives it. An electrified non-conductor is discharged by bringing every part of its surface in contact with a conductor. Wiping the surface with the hand is sufficient. 388. Electroscopes. An instrument which shows whether a body has an electric charge, and, if so, whether the charge is positive or negative, is called an electroscope. The simplest form of electroscope is a pith ball suspended by a silk thread. The ball is first given a charge of known kind, either positive or negative, as may be desired, and the body to be tested is then brought near it. If the ball is repelled, the body is electrified, and its charge is like that of the ball. Attraction is not a reliable test; for an un- charged body will attract the ball, as may be shown by bringing a finger near it. The gold-leaf electroscope is a much more sensitive instrument (Fig. 367). Its conducting parts consist of a 482 ELECTROSTATICS metal rod, with a knob or a disk at the top and two leaves of gold foil at the lower end. The leaves are inclosed in a box or a flask, to protect them from currents of air and from mechanical injury. The stopper through which the rod passes is of some non-conducting mate- rial, and serves as an insulating support. When the leaves are charged they spread apart, owing to their mutual repulsion, for FIG. 367. -Electroscope. ^ charges ^ necessarily both positive or both negative. The use of this instru- ment depends upon electrostatic induction. 389. Electrostatic Induction. As a charged rod is brought toward the knob of an electroscope, the leaves diverge more and more; when the rod is removed, they drop together again. The presence of the charged rod (without contact) produces an unlike charge on the knob of the electroscope and an equal like charge on the leaves (Fig. 368). This action is called electrostatic induction, and the resulting charges are called induced charges. When the rod is removed these charges disappear, and the leaves no longer repel each other. Induction and electrostatic phe- nomena in general can be explained if we adopt the theory that an unelectrified body possesses equal quantities of positive and negative electricity, which, being equally distributed over the body, exactly neutralize each other. When a negatively charged rod is FIG. 368. Induction in Knob and Leaves. ELECTROSTATICS 483 brought near an electroscope, as in the above experiment, it attracts the positive electricity to the knob and repels the negative to the leaves. As soon as the rod is removed the induced charges are brought together again by their mutual attraction. When the inducing charge is positive the negative electricity is attracted to the knob and the positive repelled to the leaves. The attraction of unelectrified \ bodies, as pith balls, bits of \ paper, etc., is due to induction \ ,T^<^ (Fig. 369). Since the unlike in- (i** duced charge is the nearer, the attraction is greater than the re- pulsion, and the resultant force FIG. 369. inductive Action on ,., . Pith Ball. is an attraction. Equal and op- posite induced charges always appear upon an insulated conductor in the presence of a charge on a neighboring body. According to present theory, there are really two kinds of elec- tricity, but, in a solid conductor, only the negative electricity is free to move. If this view is correct, a body becomes positively charged by losing some of its negative electricity, and negatively charged by receiving an excess of negative electricity. It would seem more appropriate, in the light of present knowledge, to call negative elec- tricity positive and vice versa, since the negative electricity is appar- ently the freer and more active agent; but, as the terms are purely arbitrary any way, it does not matter. 390. Charging by Induction. In the above experiment with the electroscope the opposite charges induced on the knob and the leaves are temporary, in the sense that they disappear as soon as the inducing charge is removed. The electroscope, or in fact any insulated conductor, can be permanently charged by induction, the charge being per- manent in the sense that it continues after the inducing 484 ELECTROSTATICS charge is removed. To charge the electroscope by induc- tion we proceed as follows: An electrified rod, having, let us say, a negative charge, is brought near the knob. The induced charge on the knob is positive, that on the leaves negative; and the leaves diverge (Fig. 368). While the inducing charge is still present, the knob is touched with the finger. The leaves instantly fall together, show- ing that their negative charge is lost. It has, in fact, been conducted away through the finger and body of the experi- menter to the earth. Owing to the repulsion of the nega- tive inducing charge, the negative charge of the leaves has been driven away as far as possible. The positive charge of the knob can not escape, although a conductor is provided, for it is held or " bound" by the attraction gf the negative charge on the rod. The finger is now re- moved, and afterward the rod. As the rod is removed the leaves again diverge. They are now positively charged ; for the positive charge of the knob, when freed from the attraction of the inducing charge, is shared with the leaves. The electroscope now has a permanent positive charge. This fact can be tested by again bringing up the nega- tively charged rod; for, as the rod approaches, the leaves gradually drop, showing that their positive charge has again been attracted to the knob. On the other hand, if a positive charge is brought up, the divergence of the leaves increases. (Why?) 391. Use of the Electroscope in Testing Charges. This behavior of a charged electroscope serves as a ready means of determining the kind of charge on any body. For this purpose the charge on the electroscope must be of known kind, either positive or negative. If the charge to be tested causes greater divergence of the leaves, as it is brought ELECTROSTATICS . 485 near, it is of the same kind as the charge of the electro- scope; if it decreases the divergence of the leaves, it is of the opposite kind. (Why?) 392. Positive and Negative Electricities always Pro- duced in Equal Quantities. As already stated (Art. 389), the inductive action of a neighboring charge always pro- duces equal positive and negative charges on an insulated conductor. The equality of the charges is shown by the fact that they exactly neutralize each other when they reunite, after the inducing charge is removed. If the conductor is not insulated, the two electricities are still produced in equal quantities, but the repelled charge is conducted away and lost, e. g. when the electroscope was touched with the finger in the presence of the charged rod. Friction also produces both kinds of electricity and in equal quan- tities. In order to show this, precautions must be taken to prevent the escape of the charge from either of the two bodies which are rubbed together. When a glass rod is rubbed with silk and each is then tested by bringing it near a charged electroscope, it is found that the glass has a positive charge and the silk a negative one. The silk being a non-conductor, retains much if not all of its charge; but it is necessary to proceed somewhat differently if we wish to prove that the charges are equal. It is also better to use vulcanite and flannel, since these materials are very easily electrified. A small cap of flannel is made to fit over the end of the rod, and a silk thread is attached to serve as an insulating handle (Fig. 370). When the end of the rod is twisted about FIG. 370. in the cap, and is then brought near a charged electroscope with the cap still on it, it produces no effect; but when the cap is removed by means of the thread, a positive charge is found on it and a negative charge on the rod. Since the two charges exactly neutralize each other before they are separated, they must be equal as well as opposite. 486 % ELECTROSTATICS It is found by similar tests that any two insulated substances become oppositely electrified when rubbed together, and that the same substance receives a positive charge when rubbed with cer- tain substances and a negative charge when rubbed with certain others; e.g. glass is positive if rubbed with silk, but negative if rubbed with fur. 393. Electrical machines are devices for producing and collecting electric charges more conveniently and more rap- idly than is possible by the methods already described. They are of two types, one depending upon friction, the other upon induction. One form of friction machine is shown in Fig. 371. A positive charge is de- veloped on a large revolving glass disk, A, by the friction of leather pads, B. The charge is collected on each side by a num- ber of points which FIG. 371. -Friction Machine. from rod, Fj and nearly touch the disk. The rods carry the charge to an insulated brass cylinder, C, from which it can be drawn off as a spark discharge by bringing the finger or any other conductor near it. A spark a centi- meter or more in length can be obtained in this manner from a machine in good condition. Friction machines of various forms were invented during the eighteenth century; but they are greatly inferior to the more modern induc- tion machines, and are no longer used to any extent. The simplest and earliest form of induction machine is the electrophorus (Fig. 372). It consists of a disk of vul- canite or resinous material, and a metal disk or cover" of ELECTROSTATICS 487 slightly smaller diameter, provided with an insulating handle. The vulcanite is negatively electrified by strik- ing or rubbing it with cat's fur or flannel, and the cover is then placed upon it. Since the vulcanite is a non-con- ductor and is in actual con- tact with the metal at only a few points, its charge does not pass to the cover. But the inductive action of the charge produces an opposite or positive charge on the lower side of the cover, and a negative charge on the upper side (A, Fig. 373). The negative charge is repelled by the inducing charge, and is permitted to escape by touching the cover with the finger; while the positive FIG. 373. Action of the Electrophorus. charge is retained by the attraction (5, Fig. 373). When the cover is removed, the positive charge spreads over its entire surface, and can be drawn off at any point by bring- ing the finger or other conductor near it. Sparks a centi- meter long can be obtained in this manner. The cover can be repeatedly charged and discharged without again rub- bing the vulcanite. (Why?) Various forms of induction machines have been invented, which are continuous and automatic in their action and are much more powerful than the electrophorus. One of these, 488 ELECTROSTATICS called the Toepler-Holtz machine, is shown in Fig. 374. It has a revolving glass disk, D, and a stationary one, D f . To both are attached small disks and strips of tin-foil, F, which serve as conductors. While the machine is in action, fixed positive and negative charges accumulate on the metal-covered parts of the stationary disk; and these act FIG. 374. Toepler-Holtz Induction Machine. inductively on the metal parts of the revolving disk, producing a positive charge on one side of the axis and a negative charge on the other. These induced charges are collected by projecting metallic points, as in the friction machine, and accumulate on insulated conductors (rods, knobs, and Leyden jars), until finally a, spark discharge occurs across the air space between them. Sparks from 5 to 10 cm. in length can be obtained from machines of moderate size. Full descriptions of these machines are to be found in larger works on the subject. ELECTROSTATICS 489 394. Distribution of a Charge on a Conductor. Since the dif- ferent parts of an electrical charge are of like kind and repel each other, it is reasonable to suppose that, if the charged body is a conductor, this mutual repulsion of the parts will drive the entire charge to the outer surface, where it will be distributed in a definite manner depend- ing on the shape of the conductor. Experiment shows that this is actually the case. To illustrate, let a metal vessel, such as a tin can or a calorimeter, be placed on an insulating support and strongly charged from an induction machine. When a proof plane (consisting of a small metal disk with an insulating handle) is touched to any part of the outside of the vessel and is then presented to an electroscope, it will be found to be charged. A like test shows that the proof plane is not charged by contact with the inside of the vessel; hence the entire charge of the vessel is on its outer surface. On the surface of a spherical conductor a charge is distributed uni- formly; on a conductor of any other shape the distribution varies with the curvature of the different parts of the surface. This can be shown by testing, with an electroscope, the strength of the charge received by a proof plane, when touched to the conductor at different points. With a conductor shaped as in Fig. 375 it will be found that the charge is greatest at A, where the curvature is greatest, less at B, and least at C, where the curvature is least. The quantity of electricity per unit area of a charged body is FlG - 375 -Insulated n t , - , , T Conductor. called the surface density of the charge. In gen- eral, the surface density on a charged conductor increases as we go from places of less to places of greater curvature. At sharp projecting points the electric density is very great. 395. Discharge from Points. An insulated conductor in a dry atmosphere retains a charge for a long time, provided its surface is everywhere smooth and gradually curved; but at any sharp point there is a continuous and rapid loss of the charge to the surrounding air. The nature of the action in detail is a theoretical question which need not be considered here. The net result is that the charge passes off to the neighboring air particles, which are then repelled. These charged particles, streaming away from the point, form a cur- rent in the air, known as an "electrical wind." 490 ELECTROSTATICS This action can be shown by attaching a pointed wire to the knob of an electrical machine. The discharge at the point is accompanied by a hissing sound, and, in a dark room, a fine jet or brush of pale blue light is visible about the point. The wind can be felt by the hand, and the flame of a candle, held near the point, is blown aside (Fig. 376). A conductor may be charged as well as discharged FIG. 376. Electrical Wind Due to Discharge from a Point. by the action of points on its surface. This is the pur- pose of the rows of points which extend toward the revolving disk of an electrical machine. The charge developed on the disk passes across through the air to the points. 396. Energy of a Charge. Electrical Potential. There is a definite amount of energy associated with every elec- trical charge. This energy is manifested in various ways. When a spark discharge occurs, the energy of the charge is converted into heat, light, and sound. The heat of even a short spark is sufficient to light a gas jet; the light and sound are directly evident to the senses. The energy of a charge is further shown by mechanical effects. For example, a piece of cardboard, placed between the knobs of an electrical machine, is punctured by a heavy spark. The energy of a charge is a form of potential energy; and, like the potential energy of a mass raised above the earth, its value is determined by two factors. The energy of a tank of water, A (Fig. 377), standing on the ground, is measured by the product of the weight, w, of the water and its average height. If the height of the surface is denoted by h, the average height of the whole body of water is ^ h y ELECTROSTATICS 491 and its potential energy, with respect to the level of the ground, is \ wh foot-pounds. If the water is drawn off through a pipe at the bottom of the tank, its pressure will steadily decrease from a maximum at the start to zero, as the last water flows out. The average pressure of the water is half the pressure at the start, or the gravity pressure at half the original depth. We may just as well take this average pressure as one of the energy factors, instead of the average height. It is instructive to consider the reverse process. When water is pumped into the tank through a pipe at the bottom, it must be forced FIG. 377. Electrical Poten- tial Corresponds to Water Level. in against an opposing pressure which steadily increases as the level of the water rises; and the amount of work required to pump in each succeeding pound of water increases in the same ratio. The work done in filling the tank is pro- portional to the average value of this pressure. These energy relations are exactly paralleled by those involved in charging and discharging a conductor. Sup- pose, for example, that an insulated tin can, or other sim- ilar conductor, is charged by carrying unit quantities of electricity to it on charged pith balls, which are lowered within and touched to its inner surface. As each new por- tion of the charge is brought up from a distance, it is re- pelled by the charge already on the conductor (until it is 4Q2 ELECTROSTATICS lowered within), and work must be done in overcoming this repulsion. The repulsion increases in proportion to the charge on the conductor; and consequently the amount of work that must be done against the repulsion in bringing up a unit charge increases in the same ratio. The work done against the repulsion of the charge in bringing up a unit quantity of electricity to it is called the potential of the charge, or the electrical potential of the conductor. The potential increases in proportion to the charge, as stated in other terms just above. Since it is zero when the charging begins, its average value during the process is half its final or maximum value. The product of this average potential and the quantity of the charge measures the total work done against the repul- sion in producing the charge; and this is also the energy of the charge. Thus the two factors of electrostatic energy are quantity of the charge and potential of the charge; and these correspond respectively to weight and height in the case of the tank of water. Electrical potential may also be regarded as corresponding to fluid pressure, as shown below; indeed, it is often called electrical pressure. If the two tanks A and B (Fig. 377) are connected by a pipe at the bottom, water will flow from A to B until the surface stands at the same level in both. It is difference of level that determines the flow, and not the relative quan- tities of water in the two tanks. Similarly, when two posi- tively charged conductors are connected by a wire, the one at the higher potential loses a part of its charge to the other by conduction through the wire. This lowers the poten- tial of the first conductor and raises that of the second until they become equal. Just as water tends to flow from higher to lower levels, so positive electricity tends to flow from places at higher to places at lower electrical potential. ELECTROSTATICS 493 Elevation is measured from the level of the ground, or, on a large scale, from sea-level. Taking the level of the ground as the zero of elevation, the level of the water is positive in the tanks A and B (Fig. 377) and negative in the wells C and D. If the wells were connected, the water would flow from C to D, i.e. from the one in which the negative elevation is less to the one in which it is greater. Similar relations hold between positive and negative po- tentials. The electrical potential of the earth is taken as zero. The earth is so large that its potential is not mate- rially changed by any positive or negative charge that may be imparted to it; just as the level of the sea is not percep- tibly changed by the flow of rivers into it. Any conductor is at zero potential if, when electrically connected with the earth, there is no flow of electricity from either to the other. Its potential is positive if, when thus connected, positive electricity flows from it to the earth, and negative if positive electricity flows from the earth to it. A positively charged body has a positive potential, a negatively charged body a negative potential, and an uncharged body a zero potential, provided there is > no other charged body in its vicinity. The greater the positive charge on a body the higher is its potential; the greater the negative charge on a body the lower is its potential. As long as the level of the water in the tank A is higher than the level in B, the pressures at the two ends of a con- necting pipe will be unequal; and the difference between these pressures is the immediate cause of the flow. (When we say that the difference of level is the cause of the flow, we go one step farther back in the sequence of cause and effect; for the difference of pressure is due to the difference of level.) Similarly a flow of positive electricity from 494 ELECTROSTATICS higher to lower potential may be attributed to a difference of electrical pressure, acting in the direction of the flow. The meaning of electrical potential is presented only in part in the above discussion, but enough has been said to serve our purpose. We do not need to concern ourselves with definitions of unit charge, unit potential, and the various other electrostatic units, or to discuss the methods by which electrostatic quantities are measured. The study of electrostatics is principally valuable as an introduction to the study of electric currents; and the units employed in the latter branch of the subject are different from the electrostatic units. We may anticipate matters by saying that the practical unit of potential is called the volt and when potential is measured in terms of this unit, it is called voltage. As the volt is more or less familiar from the industrial uses of the electric current, we shall adopt it in advance. 397. Potential and Insulation. Sparks. As a gas is pumped into a closed vessel the pressure steadily rises until the vessel bursts or begins to leak. This maximum pres- sure depends on the strength of the vessel, and not on its size. Similarly, as the charging of a conductor progresses, the potential rises, until finally the resisting power of the air or other insulating medium is overtaxed, and the charge either leaks off gradually or escapes suddenly in a spark discharge. The potential at which either form of dis- charge occurs depends upon the strength of the insula- tion, and not upon the material of the conductor or to any very great extent upon its size. (Shape is more or less important. Recall the effect of sharp points.) It has already been mentioned that the value of differ- ent materials as insulators depends upon the particular use to which they are put (Art. 386). It will now be under- stood that the one determining factor is the potential of the electricity. A cotton covering is quite sufficient to prevent the loss of electricity from wires carrying low- ELECTROSTATICS 495 voltage currents, such as are used in ringing electric bells; but when a piece of this wire is connected with an induc- tion machine, sparks can be drawn off through the insula- tion as readily as from the knob of the machine. Even the heavy cloth and rubber covering of an electric light wire makes a very poor showing when put to the same test; for sparks of considerable intensity are obtained through it. Such tests as these call our attention again to the fact that only the strongest insulation is effective in preventing the escape of electric charges. (Note that the charged parts of an electrical machine are all insulated by several centimeters of vulcanite.) Evidently the ordinary poten- tials of electric charges are enormously high compared with the ordinary potentials of electric currents. The potential of a charged conductor can be roughly estimated at 25,000 volts for each centimeter of length of the spark that passes between it and a second conductor at zero potential. If the second conductor is also charged, the length of spark is determined by the difference of potential between the two. A potential difference of 100,000 to 200,000 volts between the knobs of an induction ma- chine is not uncommon. In fact, the potential to which it is possible to charge the machine or any other body depends only upon the strength of the insulation and the dryness of the atmosphere. Beyond this limit the charge escapes as rapidly as it is developed or imparted. Charges at even the highest potentials mentioned are not danger- ous, unless the quantity of the charge is much larger than is gener- ally the case. 398. The Leyden Jar and Other Condensers. The Ley den jar (Fig. 378) is a device for accumulating and stor- ing a large charge. Its name is derived from the city of Leyden, in the Netherlands, where the principle of its action first became known in 1745. It consists of a glass jar, coated inside and out for about two thirds its height with tin-foil. A brass rod, terminating in a knob at the 496 ELECTROSTATICS top, extends through the cover, and is connected with the inner 'coat of the jar by means of a chain, attached to its lower end. To charge the jar it is held in the hand, and the knob is brought near one terminal of an electrical machine; or it may be placed on the table, and the knob connected with the machine by means of a conductor. In either case the other terminal of the machine should be con- nected to earth by running a chain or wire from it to the table. To discharge the jar, FIG. 378. Leyden jar and one end of a short conductor Discharger. j g touched to j ts Quter CQat and the other end brought near the knob, as shown in the figure. Before the gap is closed a spark passes, dis- charging the jar. The conductor is provided with an in- sulating handle to protect the operator from the danger of a shock. While the inner coat of a Leyden jar is receiving a charge, the outer coat is also receiving one, of opposite sign, al- though its potential remains at zero. The latter is an induced charge, attracted by the charge on the inner coat, and is received from the earth, by conduction through the table or the body of the person holding the jar. This induced charge reacts inductively on the inner one, and by its attraction enables the inner coat to receive a much greater charge from the machine than would otherwise be possible. To prove this we have only to charge and dis- charge the jar while it is standing on a large sheet of vul- canite or glass. With the outer coat thus insulated, it can not become charged, and the jar can be made to yield only a short, weak spark. ELECTROSTATICS 497 The extent to which the electrical capacity of a Leyden jar is increased by the mutual induction of the opposite charges on its inner and outer coats is strikingly shown by the action of the jars of an in- duction machine (Fig. 374). When the machine is operated, the charges accumulate principally in the jars, the positive charge in one, the negative in the other. The outer coats of the jars become oppo- sitely charged by induction, each receiving its charge from the other through metal conducting rods, by which they may be connected at the will of the operator. Under these conditions the machine gives a thick, brilliant spark, at intervals of several seconds; but when the outer coats of the jars are disconnected, the sparks are thin and FIG. 379. Photograph of a Lightning Flash. r faint, and occur much more frequently. The quantity of electricity that is discharged with each spark is very much less in the latter case owing to the reduced capacity of the jars. The loss of capacity is due to the fact that the outer coats of the jars do not become charged, the wooden base of the machine being practically an insulator for such rapid action. 399. Atmospheric Electricity. The sparks obtained from electrical machines and Leyden jars suggested to a number of the early experimenters in electricity that lightning was an electrical discharge between one cloud and another or between a cloud and the earth. Benjamin 498 ELECTROSTATICS Franklin put this theory to an experimental test in 1752. "He sent up a kite during the passing of a storm, and found the wetted string to conduct electricity to the earth, and to yield an abundance of sparks. These he drew from a key tied to the string, a silk ribbon being interposed between his hand and the key for safety. Leyden jars could be charged, and all other electrical effects produced, by the sparks furnished from the clouds. The proof of the identity was complete." Thunder corresponds to the snapping sound produced by an electric spark. The sudden heating of the air along the path of a lightning flash causes it to expand with explo- sive violence, producing sound waves of great intensity. If the flash is short and straight, the sound is a short clap; if it is long and zigzag, the sound produced by its differ- ent parts have unequal distances to travel to the observer and are heard in quick succession as a continuous rattle. The rolling sound of distant thunder is due to various reflections of the sound from clouds, from the ground, and often from neighboring hills. Experiments have shown that the atmosphere is gen- erally electrified even in fair weather. In fair weather the electrification is almost always positive; in stormy weather it is sometimes positive and sometimes negative. The potential increases with the altitude; but differs widely in different localities and with different states of the weather. The rise of potential has been found as great as 600 volts per meter of elevation above the ground. The potential of thunder-clouds, as estimated from the length of light- ning flashes, runs into the hundreds of millions of volts. The aurora or northern light is due to electric discharges in the upper air. (Art. 500.) Various theories have been advanced to account for the ELECTROSTATICS 499 electrification of the atmosphere; but very little is defi- nitely known about it. Evaporation is very probably one of the principal causes. 400. Lightning Conductors. The use of lightning conductors to protect buildings was first suggested by Benjamin Franklin. The usual device consists of one or more iron rods, extending some dis- tance above the highest points of the building and connected by means of large iron or copper conductors with damp earth, or, better, with water. A conductor ending in dry earth is worse than useless, it is dangerous; for dry earth is not a sufficiently good conductor. Each rod is terminated by a gilded copper point. The action of a lightning conductor depends largely upon induc- tion. A charged cloud induces an opposite charge at the surface of the ground under it and on houses, trees, and other objects within this area. The inductive action is strongest upon the highest objects, and causes lightning rods to become highly electrified. Under these conditions a rapid and continuous brush discharge takes place from the sharply pointed tips of the rods. This quiet discharge of oppo- site electrification toward the cloud is often sufficient to prevent light- ning; but, if a stroke does occur, it is received by the rod and the building is not damaged. 401. The Electric Field. The attraction or repulsion between two electric charges varies in amount with the intervening medium. For example, it is one sixth as great through glass as it is through an equal thickness of air. But electric forces act as readily through a vacuum as they do through air; from which it appears that the ether is the one essential medium, as it is in the case of magnetic forces. Magnetic action and electrostatic action are, however, fundamentally different in their nature; for an electric charge neither attracts nor repels a magnetic pole. The space within which an electric charge can be detected is called an electric field ; and a line of electric force is a line in the field along which an electric charge would move, under the attraction or repul- sion of the field. The energy of a charge is stored, not on or in the conductor, but in the electric field, and is due to a state of strain in the ether and other insulating media which occupy the field. This is proved by the occurrence of electric sparks, which are due to the breaking 500 ELECTROSTATICS down of the material structure of the medium. "If a spark passes through a sheet of paper or a pane of glass, a hole is made in it; if the spark is in air, the molecules of its gases are broken into parts. This proves that the medium must have been greatly strained just before the sparks passed; and, if it was strained, it must have possessed potential energy." Although the ether transmits electric force, it does not transmit electricity. A perfect vacuum is a perfect insulator, and a spark in it is impossible. CHAPTER XIII ELECTRODYNAMICS I. INTRODUCTION 402. Effects of the Electric Current. An electric cur- rent can not be seen, but its presence is known by the effects which it produces. Some of the effects of high-potential currents are already familiar, such as the electric spark and the shock experienced when the current passes through any part of the body. Other effects are produced by cur- rents of both high and low potential. These are classed as magnetic effects, heating effects, and chemical effects. The heating effect is well known through its application in electric lighting. It is an obvious fact that the light of an arc or an incandescent lamp comes from a white-hot body. Small incandescent lamps are made which are bril- liantly lighted by the current from a battery of three or four dry cells. Magnetic effects are also familiar, but their nature is less evident. One may ring door bells and use telephones, and yet remain in ignorance of the fact that these useful appliances owe their existence to the magnetic field which surrounds an electric current. The existence of such a field is shown by the deflection of a magnetic needle, when near a wire in which a current is flowing. If the wire is extended parallel to the needle, at a distance of several centimeters above it, and is then brought down close to it (Fig. 380), the needle will be deflected through a greater or less angle, depending upon the strength of the current. 501 5 02 ELECTRODYNAMICS We have here a most important difference between elec- tricity at rest and electricity in motion; when in motion it acts upon magnets, when at rest it does not. The chemical effects of electric currents are less familiar, but of great and growing im- portance in the chemical industries. One of the simplest examples of FIG. 380. Magnetic Action of a Current. . . . electrochemical action is the separation of water into its constituent gases, hydrogen and oxygen, when a current is passed through water con- taining a little sulphuric acid. In the study of electric currents we shall become ac- quainted with these effects in detail, and with many of their more important applications. 403. Sources of Electric Currents. Electric currents for all practical purposes are generated either by electric batteries or by dynamos. Batteries are used where the cur- rent required is comparatively small, as in ringing bells, operating telegraph instruments, etc. Large ," storage" batteries generate electricity in sufficient quantity to run electric automobiles and electric launches; but they must be " charged" at frequent intervals by currents from dyna- mos. Dynamos are thus the only primary or original source of electrical energy on a large scale. It might be supposed that an induction machine would be capable of supplying a considerable current, comparable, at least, with the current from ordinary batteries; but a simple test proves that this is not the case. A small elec- tric bell can be rung by means of a single electric cell of PRIMARY CELLS 503 almost any type; but, when the bell is connected with the knobs of an induction machine, it remains silent, however vigorously the machine may be operated. The relative strength of the currents is more definitely shown by their action on a magnetic needle, when flowing through a wire directly above it (Fig. 380). The current from a cell deflects the needle several degrees perhaps 20 or 30. If we connect the ends of a wire with the knobs of an in- duction machine, the current from the machine flows through it ; but the needle is not deflected when the wire is brought near it. Since the magnetic action is too weak to affect the needle, the current must be very small. II. PRIMARY CELLS 404. General Facts Concerning Electric Cells. There are many forms of electric cells, but they are all alike in certain respects. Every cell has two plates. In most cells one plate is of zinc and the other of copper or 'carbon. Every cell contains a liquid in which the plates are im- mersed, or two liquids, with one of the plates immersed in each. (In the so-called dry cell the liquid is held by a porous solid.) Different liquids are used, a common one being dilute sulphuric acid. To obtain a current from a cell a wire is connected with its plates. For convenience in making the connection, each plate has a binding post at the top. A cell in working condition supplies a current continu- ously, as long as its plates are connected by a conductor. Now we have learned in the study of electrostatics that a difference of potential is a necessary condition for the flow of electricity. Hence we may reasonably infer that the plates of a cell are at different potentials. The difference is very small, however, and can be detected only with 504 ELECTRODYNAMICS a very sensitive apparatus. For this purpose the elec- troscope must be provided with two metal disks, which are covered on their contact surfaces with an insulating coat of shellac, and act as con- densers (Fig. 381). One disk takes the place of the customary knob of the electroscope, and FIG. 381. Demonstrating that the Plates of a Cell are Charged. the other is provided with an insulating handle. To make the test the disks are placed together, and each is connected by a wire with a plate of the cell, as shown in the figure. The wires are then removed and the upper disk lifted off. The charge on the disk of the electro- scope is now free, and is shared with the leaves, causing them to diverge slightly. When this charge is tested in the usual manner, it proves to be negative if it was received from the zinc plate, and positive if received from the copper or the carbon plate of the cell. If stronger charges are desired for the test, they may be obtained from a battery of two or more cells connected in series. When the plates of a cell are joined by a wire, positive electricity flows through the wire from the positively charged copper or carbon plate to the negatively charged zinc, i.e. from positive to negative potential. This is called the direction of the current. There is an equal flow of negative electricity in the opposite direction; but a nega- tive current is equivalent in its effects to an equal positive current, and it is the universal practice to regard the entire current as positive. (The newer theory Art. 512 gives a different account of the process.) PRIMARY CELLS 505 The plates of a cell are often called poles or electrodes (from the Greek electro + hodos, way, i.e. a way for elec- tricity). The copper or carbon is called the positive pole or electrode, and the zinc the negative pole or electrode. 405. Further Comparison of Electric Cells and the In- duction Machine. The flow of water through a pipe increases with an increased difference between the pressures at the inlet and the outlet; so, too, the flow of electricity through a given conductor increases with an increased potential difference between its terminals. This being the case, it might be supposed that an induction machine, which can be charged to a potential difference of 100,000 volts, would send a much larger current through a wire than an electric cell, the plates of which differ in potential by one or two volts at the most. But we have seen that the current from the cell is much the greater. This apparent contradiction is explained when we take account of all the facts. When the knobs of an electrical machine are connected by a wire, no spark can be drawn from the machine, not even the shortest, and no shock is felt when the knobs are touched with the fingers. The positive and negative electricities flow through the wire and neutralize each other as fast as they are developed, and no accumulation of charges is possible. It is like pouring water into a sieve. However industriously this may be done, the sieve remains empty. In like manner the knobs and jars of the machine remain practically at zero potential. We see, then, that if a cell can maintain a potential difference of one volt, or even a small fraction of a volt, between its poles, it is superior to the induction machine as a source of a con- tinuous flow of current. 506 ELECTRODYNAMICS When an induction machine is operated in the usual manner and a spark passes, there is a sudden rush of cur- rent, which lasts perhaps for a millionth of a second. This reduces the potential difference to zero, and it must again become very great before another spark discharge can take place. The time intervals between successive sparks are very great compared with the actual duration of the current; hence the quantity of electricity that passes per second is exceedingly small. A cell, on the contrary, maintains a steady flow of current; from which. we know that the plates must be recharged, by some action within the cell, as rapidly as they are discharged through the wire. The charged plates of a cell possess energy, which is constantly expended in maintaining the current and con- stantly renewed by the action within the cell. Evidently there must be a certain store of energy in the cell, which is available for this work. The source of this energy and something of the manner in which it is liberated may be learned from a study of the earliest and simplest form of electric cell, invented by the noted Italian physicist, Ales- sandro Volta, in 1800. It is named after him the voltaic cell, and consists simply of a zinc and a copper strip in dilute sulphuric acid. To explain the action of the cell, we must begin with the chemical behavior of these materials. 406. Action of Dilute Sulphuric Acid on Zinc and Cop- per. When a strip of zinc is placed in dilute sulphuric acid, it is attacked by the acid and gradually eaten away or dissolved. At the same time small bubbles form in great numbers on the surface of the zinc, to which they adhere until detached by the buoyancy of the liquid. The escape of the bubbles at the surface gives the liquid the appearance of boiling. PRIMARY CELLS 507 These visible effects are due to chemical action. The sulphuric acid molecule consists of two atoms of hydro- gen, one of sulphur, and four of oxygen, and is represented by the formula H 2 S0 4 . In the chemical action the two hydrogen atoms of the acid molecule are replaced by one atom of zinc (Zn), forming a molecule of zinc sulphate (ZnSO 4 ); and the two hydrogen atoms unite to form a hydrogen molecule. The hydrogen molecules gather in the form of bubbles and escape. The zinc sulphate re- mains in solution in the liquid. If the liquid is evaporated, the sulphate remains in the form of a white solid. As the action of the acid on the zinc continues, the tem- perature of the liquid rises. Evidently heat is generated in the process. The acid and the zinc possess a certain amount of chemical energy, which is converted into heat when the two substances unite. This transformation of energy is similar to that which takes place when fuel is burned. The zinc may be compared to coal and the acid to the oxygen of the air, with which the coal unites in burning. When a strip of copper is placed in the acid, no bubbles are formed and the copper does not waste away however long it may remain in the liquid. There is no appreciable chemical action. 407. Electrochemical Action in the Voltaic Cell. When a zinc and a copper strip are in the same vessel of dilute sulphuric acid, but are not in contact, neither strip is af- fected by the presence of the other, and hydrogen bubbles appear only on the zinc. When the strips are connected by a wire, hydrogen bubbles form on both, and a mag- netic needle indicates the presence of an electric current in the wire (Fig. 380). 508 ELECTRODYNAMICS While the current is flowing the acid appears to attack both strips, since bubbles form on both; but the appear- ance deceives. The copper does not waste away however long it may be used. The hydrogen liberated at the copper represents useful consumption of zinc and acid. By this action the strips are charged and the current is maintained in the wire. The hydrogen liberated at the zinc repre- sents useless consumption of zinc and acid, by which the chemical energy of the materials is immediately converted into heat in the liquid. This useless action takes place whether the cell is generating a current or not. The waste- ful action may be greatly reduced by treating the zinc with mercury (Art. 410). The electrical nature of the action in the cell remains to be con- sidered. This is explained by the theory of electrolytic dissociation. According to this theory many substances, which are known in chem- istry as acids, bases, and salts, become more or less dissociated when they are dissolved in water. A dissociated molecule is one that is broken up into two or more electrified parts, called ions. An ion is an atom or a group of atoms having a positive or a negative charge. Since the solution as a whole is not charged, the sum of all the charges on the positive ions must be equal to the sum of all the charges on the negative ions. The sulphuric-acid molecule forms two positive hydrogen ions (H + ,H + ) and one negative ion (SO")- The latter is called a sulphion. Its negative charge is equal to the sum of the positive charges on the two hydrogen ions. When a strip of zinc is placed in the acid, it immediately begins to dissolve by giving off positively charged zinc ions (Zn ++ ) to the liquid. The charge on the zinc ion is twice as great as that on the hydrogen ion. The loss of these positive charges leaves the zinc negatively charged; while the liquid immediately surrounding the zinc plate is positively charged by the presence of the zinc ions. These charges quickly increase to a definite limit; for a zinc ion, on the point of leaving the plate with its positive charge, is retarded by the attraction of the negative charge on the plate and also by the repulsion of the positive charge of the liquid. These retarding PRIMARY CELLS 509 forces soon become great enough to prevent the further escape of the zinc ions into the liquid, unless the charges are carried off in some way as they accumulate. A copper strip, placed in the acid and connected by a wire with the zinc, completes an electric circuit through which the discharge can take place. The positive zinc and hydrogen ions are repelled from the space about the zinc plate, where the potential of the liquid is the highest, and drift toward the copper plate. On arriving at the copper plate, the hydrogen ions give up their positive charges to it, and unite in pairs, forming uncharged hydrogen molecules. These accumulate in the form of bubbles and escape. The positive charge on the copper is conducted through the wire to the zinc plate. This raises the potential of the zinc and enables it to give off more ions. The current is thus maintained as long as the plates are connected, or until the zinc is entirely dissolved or the supply of hydrogen ions is exhausted. 408. Mechanical Illustration of the Action of a Cell. - We have already made use of mechanical analogies in dis- cussing electrical phenomena, and fur- ther helpful ideas may be gained in the same way. The action of a cell in producing and maintaining a difference of potential may be compared to the action of a pump in producing and main- taining a difference of water-level. With the device shown in Fig. 382, water can be forced through the lower pipe from L .to R, by means of the rotary pump, which is driven by the weight W. The back pressure of the water in R increases with the increasing difference of the water-levels in R and L until, finally, it becomes great enough to stop the pump. If now the stop- cock in the upper pipe is opened, water will flow through FIG. 382. Water Analogy of an Electric Cell. 510 ELECTRODYNAMICS it from R to L, the level in R will fall somewhat, thus decreasing the back pressure against the pump, and the pump will start again. Under these conditions there will be a continuous circu- lation of the water, the pump supplying the energy expended in maintaining the flow. In this illustration the tank R represents the copper plate, L the zinc plate, and the action of the pump the chemical action in the cell. The upper pipe, with the stop- cock open, represents the connecting wire between the plates. When the stop-cock is closed, the return flow is cut off; and, as soon as the difference of level in R and L reaches the possible maximum, the pump stops. Similarly, when the plates of a cell are disconnected, their charges accumulate until the potential difference reaches its maxi- mum value for that particular type of cell. All action within the cell then ceases, unless there is wasteful consump- tion of the zinc. The maximum potential difference which a cell is able to produce, i.e. the difference between the potentials of its plates when they are disconnected, is called the electro- motive force of the cell. Potential difference is usually denoted by P.D. and electromotive force by E.M.F. The E.M.F. of the zinc-copper-sulphuric-acid cell is approxi- mately one volt. 409. The Electric Circuit. When the plates of a cell are connected by a wire, the positive electricity flows through the liquid from the zinc to the copper plate, thence through the wire to the zinc plate. Its path is a complete circuit, continuous from any point back to that point again. In general, an electric circuit consists of a series of conductors, forming a closed loop. A circuit is said to PRIMARY CELLS 511 be closed when it is complete, open or broken when there is a gap at any point. Electricity flows with a velocity comparable with that of light, and the current is established in all parts of a circuit practically at the same instant. When we say that a cell " generates an electric current," the expression is to be understood to mean that the action in the cell establishes and maintains a flow of electricity. Before the circuit is closed the electricity is already in existence, in the form of charges carried by the ions in the liquid and charges on the plates. In the same sense we may say that a pump generates a current of water. Just as the same body of water may be caused to flow endlessly round a circuit (Fig. 382), conveying water-power from one point to another without loss of the water itself, so the same electricity may be caused to flow endlessly round a circuit, conveying electrical power, and this power can be used in ringing bells, lighting lamps, running motors, etc., without loss of the electricity itself. Some of the water may be lost through leaky pipes, and some of the electricity may be lost through poor insulation; but neither of these losses is a necessary or useful part of the process. 410. Local Action on the Negative Plate. We have seen that hydrogen is liberated at the zinc plate of a voltaic cell whether the circuit is open or closed, and have learned that this represents a wasteful consumption of the zinc and acid. This action is due to small particles of iron, lead, and carbon, which are present as impurities in com- mercial zinc. Any such particle on the surface of the zinc and in contact with the liquid acts as a positive pole, and forms a minute voltaic cell with the adjacent zinc and liquid. This causes a local or parasitic current at the 5 1 2 ELECTRODYNAMICS spot, which adds nothing to the flow through the wire. With chemically pure zinc this local action, as it is called, does not occur. The zinc is consumed only when the cir- cuit is closed, and the hydrogen is set free only at the posi- tive plate. The same result is obtained, though somewhat imperfectly, with a plate of commercial zinc, when covered with a coating of mercury. The mercury dissolves a por- tion of the zinc, forming a pasty amalgam, which covers the surface and keeps the acid from contact with the impu- rities. The zinc in this condition is said to be amalgamated. Amalgamation prevents a great deal of waste where the liquid of the cell is an acid solution, as in the bichromate cell (Art. 413). With the various types of cells in general use, the materials used in the liquid are such that local action is avoided, and amalgamation is unnecessary. 411. Polarization of the Positive Plate. The simple voltaic cell has another defect, due to the accumulation of hydrogen on the positive plate. In addition to the bubbles, a thin, invisible film of hydrogen spreads over the surface. The result is a very considerable weakening of the current. This may be shown with a battery of one or more cells, which, when first connected in circuit, is just sufficient to operate a telegraph sounder. The current very quickly becomes too weak for the purpose. The power of the battery can be restored by drying the copper plates in a Bunsen flame. A deposit of hydrogen on the positive plate of a cell decreases the current for two reasons. In the first place, hydrogen is a non-conductor, and cuts off the current from the part of the surface that it covers. The cell as a whole thus becomes a poorer conductor of the current, or, in other words, its electrical resistance is increased. In the PRIMARY CELLS 513 second place, the hydrogen on the surface of the copper tends to go into solution again as positive ions, and conse- quently sets up an opposing E.M.F. which retards the ap- proach of other hydrogen ions with their charges. This action is shown by the diminished P.D. between the plates, when measured with a voltmeter (an instrument to be described later). The accumulation of hydrogen on the positive plate, with its attendant effects, is called the polarization of the cell. In some of the common types of cells polarization is diminished by the use of chemicals which yield oxygen to the hydrogen, forming water; in others it is avoided altogether by the use of solutions in which the positive charges are carried by metallic ions in place of hydrogen. No single type of cell is best for all purposes. Some of the most common types are described below. These descriptions should be studied as opportunity is afforded in the class-room and the laboratory for observation and use of the different cells. 412. The Electrical Resistance of a Cell. All sub- stances, even the best conductors, offer greater or less oppo- sition to the flow of electricity through them. The greater this opposition, or electrical resistance, the less will be the current that a given E.M.F. is able to maintain in the cir- cuit. The effect of resistance in diminishing the current is the same whether it is met with in the external part of the circuit or within the cell. The resistance of the plates is negligible; that of the liquid may be considerable. The resistance of the liquid varies with the kind and quantity of the materials in solution. With a given liquid it is reduced by using larger plates, and by shortening the distance between them ; for this provides a wider and shorter path for the current through the liquid. 514 ELECTRODYNAMICS A cell supplies its greatest possible current through an external circuit having the least possible resistance, such as a short copper wire. The current is then proportional to the E.M.F. of the cell and inversely proportional to its resistance. When thus connected the cell is said to be short-circuited. The subject of resistance is considered in detail later. 413. The Chromic Acid or Bichromate Cell. The zinc plate of this cell is attached to a rod, by means of which it is raised from the liquid when the cell is not in use (Fig. 383). The positive pole con- sists of two plates of carbon one on each side of the zinc which are connected to the same bind- ing post at the top. The liquid is dilute sulphuric acid, containing in solution chromic acid or bichro- mate of potassium or of sodium, which acts as a depolarizer. These substances contain oxygen which they give up readily to hydrogen, forming water, chromate Cell *~ ^ e accumu l a tion of hydrogen on the positive plates is thus diminished, but not entirely prevented. Polarization usually diminishes the current by one third or more in a few minutes. The electromotive force of this cell is about two volts, which is considerably higher than that of most other cells. Its resistance is small, for the current has only a very short path in the liquid, and the double carbon pole reduces the resistance further by one half. Owing to its high E.M.F. and low resistance, the bichromate cell is capable of supplying an exceptionally strong current, and on this account is much used in experimental work. The zinc should be kept thor- oughly amalgamated; and it must be raised from the liquid when the cell is not in use, for the amalgam is only partially effective in prevent- ing local action. 414. The Leclanche Cell. The poles of this cell are a zinc rod and a block of carbon (Fig. 384). The latter is inclosed in a cylin- drical cup of porous earthenware, and is packed round with small PRIMARY CELLS SIS fragments of carbon and manganese dioxide (Mn0 2 ). The liquid is a solution of ammonium chloride, or sal ammoniac (NEUCl), which FIG. 384. Leclanche Cell. FIG. 385. Dry Cell. dissociates into negative chlorine ions, Cl , and positive ammonium ions, NH4 + . When the circuit is closed, zinc ions displace the ammo- nium, forming zinc chloride (ZnCk), which remains in solution. The NH4 ions move toward the carbon plate, where they break up into ammonia (NHs), which dissolves in the solution, and hydrogen, which combines with part of the oxygen of the MnOo, forming water. There is no local action; hence the zinc is not amalgamated. The E.M.F. of the cell is about 1.5 volts. Its resistance is at least three or four times as great as that of the bichromate cell, and its maximum current is correspondingly less. In a modified form of the Leclanche cell the positive plate is made of a mixture of carbon and man- ganese dioxide, and the porous cup is dispensed with. The action of the manganese dioxide is not rapid enough to prevent the cell from becoming polarized if used constantly. Hence the cell is satisfactory only for intermittent service, as in ringing door bells. It has the merit of not requiring attention for months at a time. 415. The Dry Cell. In this cell (Fig. 385), as in the Leclanche cell, the current is due to the chemical action of ammonium chloride on zinc. The zinc plate forms the containing vessel, and the solu- tion forms a porous paste with plaster of Paris and smaller quantities of other materials. The E.M.F. of the cell is about 1.4 volts. Its resistance is very low; and, on short circuit, it supplies a larger cur- rent than the bichromate cell. ELECTRODYNAMICS The special merits of the dry cell are its convenience and its port- ability. It requires no care, and when exhausted is thrown away. It is extensively used on electric-bell circuits, and supplies the current for pocket electric lamps and for "spark ignition" in the gasoline engines of motor cycles, automobiles, etc. 416. The Gravity Cell. The positive pole of this cell consists of several copper strips fastened together, and is placed at the bottom of the jar (Fig. 386). The zinc is near the top, and commonly hangs suspended from the edge of the jar. The lower portion of the liquid is a saturated solution of copper sul- phate, or bluestone (CuSO 4 ). The upper portion, in which the zinc is suspended, is a weak solution of zinc sulphate. (Very dilute sulphuric acid will serve in setting up the cell.) This is of less specific gravity than the lower solution, and rests upon it without mixing, except by the slow process of diffusion; hence the name "gravity cell." The zinc and copper ions of the solution are positive; the sulphions (S0 4 ) are negative, as in dilute sulphuric acid. When the circuit is closed, copper ions pass out of solution at the copper plate, upon which they are deposited, giving up their positive charges. The nega- tive ions are repelled toward the zinc plate, where an equal number of positive zinc ions are passing into solution, leaving the zinc negatively charged. The solution of copper sulphate is continually renewed from a supply of copper sulphate crystals at the bottom of the cell. Thus the zinc and the copper sulphate are gradually consumed, the amount of zinc sulphate in solution increases, and metallic copper is added to the copper plate. When the cell is not in use it should be kept on closed circuit through a considerable resistance (20 to 30 ohms). The small cur- rent then flowing prevents diffusion of the liquids into each other; otherwise the copper sulphate, coming in contact with the zinc plate, will deposit copper upon it, and the cell will then furnish little or no current. The zinc is not amalgamated, as there is no local action. FIG. 386. Gravity Cell. THE MAGNETIC ACTION OF A CURRENT 517 The E.M.F. of the gravity cell is about 1.08 volts; it is constant, for the deposit of copper on the copper plate does not affect its elec- trical properties in any way. The resistance of the cell is 3 or 4 times as great as that of the Leclanche cell, and from 12 to 20 times that of the bichromate cell. Owing to its low E.M.F. and high resistance, its maximum current is very small. It is especially serviceable in experimental work requiring a constant E.M.F., and is much used for purposes requiring a current all or nearly all of the time, as in telegraphy. 417. The Daniell Cell. The materials in this cell are the same as in the gravity cell, but they are differently arranged (Fig. 387). The two solutions are separated by the walls of a porous cup, which contains the zinc pole and the zinc sulphate. The cup stands in the solution of copper sul- phate, and is nearly surrounded by a sheet of copper, which serves as the positive pole. FIG. 387. Daniell Cell. III. THE MAGNETIC ACTION or A CURRENT 418. Oersted's Experiment. We have seen that an electric current affects a magnetic needle, and must there- fore be surrounded by a magnetic field. This great fact was discovered by the Danish physicist, Hans Christian Oersted, in 1819, nineteen years after Volta's invention of the electric battery. It is related that, at the close of a lecture one day, Oersted held over a magnetic needle a wire carrying a current, and observed, much to his surprise, that the needle set itself at right angles to the wire (Fig. 380). Thus was discovered the fundamental fact of electromagnetism, which, in less than a century, has be- come one of the controlling factors in the industries of the world, through the invention of the telegraph, the tele- phone, the dynamo and electric motor, etc. 5 1 8 ELECTRODYNAMICS Nearly all that follows in our study of electricity has to do in one way or another with the magnetic fields of cur- rents; and we shall therefore need to become very thor- oughly acquainted with them. Magnetic fields have previously been considered in their relation to the poles of magnets; they are now to be studied in their relation to the direction and strength of electric currents, and the shape of conductors. 419. The Magnetic Field of a Current in a Straight Conductor. The magnetic lines of force about a wire can be shown with iron filings, provided the current is quite strong. A battery of three or four bichromate cells, connected in parallel (Art. 448), will generate a sufficient current; but a single cell is as good or better when the con- necting wire is formed into a large coil of some 15 or 20 turns, placed close together. With a given current the field is strengthened in proportion to the number of parallel wires, but in other respects it is the same as if there were only a single wire. Following either plan, let a current be sent through a straight, vertical conductor, which pierces a sheet of cardboard. Iron filings sprinkled on the card- board show that the lines of force are concentric circles about the wire (Fig. 388). The lines of filings are most distinct close to the wire, and at a distance of a few inches none are formed. Evidently the field grows weaker from the wire outward in all directions; but it has the same strength at a given distance on all sides and all along the wire. The magnetic field is in the form of a cylinder, with the wire extending along its axis; and each circular line of force lies in a plane at right angles to the wire. The direction of the lines of force round the wire depends on the direction of the current. If the current is flowing THE MAGNETIC ACTION OF A CURRENT 519 upward, the north pole of a magnetic needle points counter- clockwise round the wire, as shown in the figure; if it is flowing downward, the north pole of the needle points FIG. 388. Direction of Lines of Force about a Wire in which FIG. 389. The Right- the Current is flowing upward. hand Rule. clockwise round. There is, then, a definite and invariable relation between the direction of the current and the direc- tion of the magnetic lines of force. This relation is most serviceably stated in terms of the right-hand rule: Grasp the wire with the right hand, with the extended thumb pointing in the direction of the current; then the fingers will point round the wire in the direction of the lines of force (Fig. 389). With the aid of this rule we can determine the direction of the current in a wire by observing the deflection of a compass needle near it; and, conversely, if we know the direction of the current, the rule gives the direction of the lines of force. 420. The Magnetic Field of a Current in a Circular Coil. The magnetic lines of force about a curved con- ductor are crowded together on the concave side and spread apart on the convex side (Fig. 390). Their direction round 520 ELECTRODYNAMICS any part of the conductor bears the same relation to the direction of the current as in a straight wire. All the lines within a loop extend through it in the same direction, as shown in the figure. The field at the center of a coil is especially important, since this is where the magnetic needle is placed in instru- ments for measuring the strength and the voltage of electric currents. When a strong current is sent through a coil of fifteen or more turns, the lines of force within and about the coil can be shown with iron filings (Fig. 391). At the center the lines are straight, and perpendicular to the plane of the coil. Their direction relative to the direction of the current is given by the right-hand rule for coils: Close the right hand and place it within the coil, with the fingers pointing round in the direction of the cur- rent; then the extended thumb will point in the direction of the lines of force through the coil. FIG. 390. Mag- netic Field about a Coil. FIG. 391. Section of the Field of a Coil. FIG. 392. Magnetic Field of a Helix. 421. The Helix and the Electromagnet. An elongated cylindrical coil of wire is called a helix or solenoid. If it is made of bare wire, it must be wound with an open space between adjacent turns, in order to carry an electric cur- THE MAGNETIC ACTION OF A CURRENT 521 rent properly (Fig. 392). With insulated wire, it may be close-wound in one or more layers of turns; and the insu- lation compels the current to travel round each turn in succession, from one end of the wire to the other. When such a coil is wound round a bar or rod of soft iron, the iron is called a core, and the core and coil together form an electromagnet. We shall see that electromagnets play the leading role in the generation and use of electric currents in daily life; without them electricity would be of very little use indeed. The action of an electromagnet depends, in the first place, on the magnetic field of the helix. This field is similar to that of a flat coil, as described above, the only difference being that it is elongated in the direction of the axis. Within the coil the lines of force extend from end to end in approximately straight lines. Outside the coil the field is like that of a bar magnet. The lines spread out from one end and return to the other, each line forming a closed curve. The helix, when a current is flowing in it, behaves like a magnet. The end from which the lines of force emerge repels the north pole of a needle and attracts the south pole. This is the north end or pole of the helix. The other end is a south pole. If a helix is supported so that it is free to turn in a horizontal plane, while carrying a current, it turns into a north-and-south line, like a compass needle. The direction of the current round a helix determines its polarity, in agreement with the right-hand rule for coils: If the helix is grasped in the right hand so that the fingers point round it in the direc- tion of the current, the extended thumb will point toward the north pole of the helix (Fig. 393). It should be noted that, if two coils are oppositely wound, the current enters at the north end of one" and at the south end of the other. 522 ELECTRODYNAMICS The magnetic effects of a helix are greatly intensified when a soft iron core is inserted. The magnetic field of the coil induces magnetism in the core, with like poles of the core and coil at the same end. If the strength of the cur- rent is sufficient, the core will be magnetized to saturation, and its magnetic strength will FIG. 393. Right-hand Rule. , r be from 1000 to 2000 times as great as that of the helix alone, and many times as great as that of a permanent steel magnet of the same size. By testing the power of the core to hold iron filings or nails, it will be found that the coil is instantly magnetized when the current is started, and instantly demagnetized when the current is stopped. An electro- magnet is thus under the perfect control of the current. It is this property of controllability which gives to electro- magnets their wide field of usefulness. The property next in importance is their great strength. Both of these properties are strikingly shown by the large lifting magnets, which are now widely used in iron-foundries and machine- shops for lifting iron and steel castings, etc. In Fig. 394 is shown one of the smallest of these magnets in the act of lifting an 8oo-lb. load. The magnet itself (the low cylinder just below the hook) is 10 in. in diameter and weighs 75 Ib. Magnets from 50 to 60 in. in diam- eter and capable of lifting from 20,000 to 50,000 Ib. are now in daily use. A lifting magnet is suspended from a crane by means of chains and pulleys, so that it can be raised and lowered and moved from place to place. The closing or opening of a switch, turning the elec- tricity on or off, causes the magnet to pick up or release its load. 422. The Strength of Electromagnets. For an electromagnet to attain its full strength, the current sent through the coil must have a certain strength. A weaker current will only partly magnetize the core. On the other hand, when the core is already magnetized to saturation, a further increase of current has practically no effect. THE MAGNETIC ACTION OF A CURRENT 523 The greater the number of turns in the coil the smaller will be the current required to produce an equal magnetization of the core. For the core is magnetized by the mag- netic field of the current, and the inten- sity of this field is proportional jointly to the strength of the current and the number of turns. Hence if the number of turns is increased, the strength of the current may be decreased, and vice versa. For certain purposes the coils of electromagnets are wound with hundreds or thousands of turns of fine wire. These require only a very small current. For other uses the coil has only a few turns, and a proportionately larger current is necessary. The strength of an electromagnet depends also upon the quality of iron used in the core. The softest iron can be most strongly magnetized and re- quires the least current. It is also the most completely demagnetized when the current is turned off. The shape of an electromagnet affects its strength, as well as its adaptability to particular uses. Other conditions being the same, the shorter the air gap be- tween the poles the greater will be the strength; hence the horseshoe form is generally preferable (Fig. 395). When the two poles act together on the same mass of iron, A , the attraction is much more than twice that of either pole alone; for the iron virtually brings the poles of the magnet together, and each strengthens the other by induction. The coils on the two arms of such a magnet are oppositely wound, as shown in the figure. (Why?) A bar of soft iron extending across between the poles of a magnet, either in contact with them or near them, is called an armature. FIG. 394. Ten-inch Magnet Lifting 800 Pounds. FIG. 395. Horseshoe Electromagnet. 524 ELECTRODYNAMICS 423. The Electric Bell (Fig. 396) is a simple and familiar application of the electromagnet. Bells differ more or less in details of construction, but the essential parts are the same in all. Generally, as a matter of convenience in construction, the electric circuit includes a short path through the metal frame of the bell. In any case the con- nections and insulations must be such that the only path offered the current through the bell is by way of the coils of the electromagnet and across between the free end of a spring^ a, and the end of a screw, c. The spring s, which carries the arma- ture, is so adjusted that it holds the armature away from the magnet, and, at the same time, presses the spring a against the screw. When the circuit is closed by pressing a push button, placed at some convenient point in the circuit, the electromagnet attracts the arma- ture, and the clapper attached to it strikes the bell. At the same time the spring is pulled away from the screw at c, thus breaking the circuit. With the stopping of the current, the magnet releases the arma- ture, which is then pulled back by the spring s. This closes the circuit at c again, and the process is repeated as long as the push button is pressed. A simplified diagram of the bell and the electric circuit is shown in Fig. 397. In this figure the binding post B is connected with the armature, while in the preceding figure it is connected with the screw. It is, of course, immaterial whether the current passes from the armature to the screw or vice versa. In either case' the armature acts as an automatic circuit-breaker, which is the essential thing. FIG. 396. Electric Bell. THE MAGNETIC ACTION OF A CURRENT 525 424. The Electromagnetic Telegraph was the first great industrial triumph of electricity. Its invention followed close upon the experimental researches of Joseph Henry, of Albany, New York, on the electromagnet. The earli- est electromagnets were wound with a single layer of bare copper wire, insulated wire being then unknown. Henry covered his magnet wire with silk, and constructed, coils of many layers of turns (1830). He then began to experiment on the transmission of signals by the action of an electromagnet at a dis- tance; but he soon be- came engrossed in other lines of investigation, and it was left for others to work out the practical application of his impor- tant discoveries. And others there were, both in England and America, who took up the problem with more or less successful results. In the United States the successful inventor was Samuel F. B. Morse, of New York, who constructed his first practical instruments in 1838. The first commercial telegraph line in the United States was built by Morse between Washington and Baltimore in 1844. At the receiving station, in the original Morse system, the message was automatically recorded in dots and dashes Clapper Gong FIG. 397. Diagram of a Bell Circuit. 526 ELECTRODYNAMICS on a strip of paper; but operators soon found that they could read messages by listening to the clicking sounds of the recording instrument. This method was soon adopted, and the receiving instrument was modified into a sounder. 425. The Sounder (Fig. 398) is operated by an electro- magnet, the poles of which point upward. An armature of soft iron is fixed across a lever just above the poles. When a current passes through the coils of the magnet, the armature is attracted down, carrying the lever with it, and a screw near the end of the lever makes a click as it strikes the support be- neath. When the cir- cuit is broken the armature is released. FIG. 398. Telegraph Sounder. . , and a spring throws the lever back against a screw above it. The two clicks of the lever sound differently and are thus easily dis- tinguished from each other. They together constitute a "dot" when one follows immediately after the other, and a "dash" when there is a brief interval between them. The letters of the alphabet, the punctuation marks, and the numbers from zero to nine are represented by certain combinations of dots or dashes, or of dots and dashes together. 426. The Key (Fig. 399) is a device by which the oper- ator makes and breaks the circuit in the act of sending a message. It is fastened to a table by two screws, the one at the left in the figure being insulated from the metal base. FIG. 399. Telegraph Key. THE MAGNETIC ACTION OF A CURRENT 527 One wire of the line is fastened to each screw. There is a small platinum point at the top of the insulated sciew, and another, P, just above it on the under side of the lever. The circuit is closed by the contact of these points, when the lever is depressed. When the key is not in use, the circuit is kept closed by the switch 5, which con- nects the base of the instrument with the insulated post, as shown in the figure. This switch is moved to the right while a message is being sent, leaving the circuit open and under the control of the operator by means of the Jever. On a short line a key and a sounder at each station are the only instruments required, and all are in the same circuit. 427. The Relay. Owing to the high resistance of a long tele- graph line, the current is too weak to operate a sounder. This diffi- culty might be overcome by using a battery of a very great number of cells; but it is more convenient and more economical to make use of an additional instrument, called a relay (Fig. 400). The relay acts on the same principle as the sounder, but the electromagnet is horizontal and the armature lever vertical. The lever is light and delicately balanced, and responds to much smaller forces than the lever of the sounder does. The coils of the magnet are connected with the line circuit, which runs to the distant station. This connection is made at two binding posts, A and B, at which the ends of the magnet wire terminate. A second pair of binding posts, C and D, connects the relay with FIG. 400. Telegraph Relay. 528 ELECTRODYNAMICS a local circuit, containing the sounder and a battery to operate it. This circuit runs from one post to the armature lever, and from the other post to the screw which the upper end of the lever strikes when drawn over by the electromagnet. This contact closes the local circuit. When the line circuit is broken, the lever is pulled back by a spring, and strikes an insulating stop at the end of the opposite screw. The local circuit is then open. Thus when the line circuit is closed or opened by means of the key, the local circuit is at the same instant closed or opened by the action of the relay; and the message is read from the sounder. The relay alone would not serve, as its sounds are scarcely audible. 428. A Complete Telegraph Line. A diagram of a complete telegraph line connecting two cities is shown in Fig. 401. In actual New York Sounder Philadelphia Sounder Re Local Battery Local Battery Earth Earth FIG. "401. Diagram of Telegraph Circuit. practice there are generally many stations on the same circuit, and at each station the line wire connects with a key and the magnet coils of a relay. At the terminal stations the line wire is connected, with the earth by means of metal plates sunk in moist ground. The earth completes the circuit, taking the place of a return wire. There is a line battery * at each terminal station, consisting of many cells in series (Art. 448). Since the circuit remains closed when the line is idle, a non-polarizing cell is required, such as the gravity cell. Small dynamos are now very generally used on long lines instead of batteries. A local battery supplies the current for the sounder at each station. * In diagrams of electric circuits a cell is commonly represented by the symbol 1 1 , the long thin line representing the positive plate and the short thick line the negative plate. A battery is represented by a series of these symbols, one for each cell. THE MAGNETIC ACTION OF A CURRENT 529 When an operator wishes to send a message, he opens the switch of his key and calls the receiving station. The sounders at all the stations deliver the message, but the operator at the station called is the only one who pays attention to it. 429. Later Developments in Telegraphy. By the system of telegraphy described above, only one message at a time can be sent over a wire. More complicated instruments are now in general use by means of which four messages can be sent at one time, two in each direction, over the same wire. This is known as quadruplex telegraphy. Eight operators are employed on each wire, one to send and one to receive each of the four messages. A skilled operator can signal about 35 words per minute; hence by the quadruplex system the capacity of a single wire is about 140 words per minute. Even at this rate many wires are required for the ordinary business of telegraph offices in large cities; and other lines are occupied in transmitting long press despatches for the daily papers. These great and growing demands have led to the invention of various systems of high-speed telegraphy, in which both the sending and the receiving instruments operate automatically. In the Barclay printing-tele- graph system the messages are first punched in the Morse charac- ters on a long tape, by a special form of typewriter. The tape is then fed into the sending machine. The holes in the tape allow elec- trical contact to be made, which sends impulses over the wire just as they are sent by an ordinary operator's key, only much faster. One wire will transmit messages as fast as three or four girls can perfo- rate the tapes. At the receiving end of the wire, an electrically oper- ated typewriter takes the message and prints it in letters instead of dots and dashes. This is done automatically, and no operator is required at the receiving end. Another recent system, invented by two Hungarian electricians, Anton Pollak and Josef Virag, also makes use of a perforated strip in the sending instrument. The receiver records the message on photographic paper, by means of a pencil of light reflected from a tiny mirror. This weightless pencil writes the message in ordinary script at the rate of 800 words per minute, or four times as fast as a person can talk! The Delany telepost system, now in operation in New England and the middle West, is still more rapid, its "ordinary" rate being 1,000 530 ELECTRODYNAMICS words per minute and its maximum 2000 or more. The sending instrument is operated automatically by means of a perforated tape, as in the other automatic systems. At the receiving end the message is recorded in dots and dashes on a chemically prepared tape. 430. Open-circuit Systems for Amateurs. Boys who would like to set up a telegraph line for their own amusement or instruc- FIG. 402. Diagram of an Open-circuit Telegraph Line. tion will find it less expensive to adopt either of the open-circuit systems shown in Figs. 402 and 403. In either case the battery is in action only while a message is being sent, and dry cells may be used. By the method shown in Fig. 402 the batteries at the two stations are connected so as to oppose each other. The keys are left open . . *3oi/acJer Line* FIG. 403. Diagram of an Open-circuit Telegraph Line. when not in use; and the circuit is then closed through the two bat- teries, the sounders, the line wire, and the ground. There is no cur- rent, since the batteries are opposed. Closing the key at either station operates both sounders. (Explain.) The system shown in Fig. 403 requires a two-point switch at each station in addition to the key and sounder. When the line is not in use, the switches are turned so as to close the circuit through the MEASUREMENT OF ELECTRIC CURRENTS 531 sounder at each station, but not through the batteries; hence there is no current. The person who wishes to call throws his switch so that it connects the line with the battery and key at his station. He can then call the other station and send messages in the usual manner. To answer the call, the other person must throw the switch at his station so as to include his own battery and key in the circuit. (Explain in detail.) IV. MEASUREMENT OF ELECTRIC CURRENTS 431. Current Strength. By the strength of an electric current is meant the quantity of electricity which flows past any point of the circuit in one second. This quantity is the same at all points along an .undivided circuit, regard- less of the kind or amount of work which the current may be doing. Thus if a small elec- tric lamp is lighted by the cur- rent from a battery (Fig. 404), electrical energy is lost from the circuit by transformation into heat in the filament of the lamp; but there is just as much current after it has passed through the lamp as before (see Art. 400). This is proved by *u vu FlG - 44- measuring the current with an ammeter (Art. 435) before it reaches the lamp and again after it has passed through. It may also be shown in a very simple way by lighting two or more small lamps in series (i.e. placed one after the other in the circuit), all the lamps being alike (Fig. 405). The number of cells required will be in proportion to the number of lamps; but the sig- nificant fact for our present purpose is that all the lamps are equally lighted, the last of the series as brilliantly as the first. This proves (as nearly as the eye can judge) 532 ELECTRODYNAMICS FIG. 405. that the current continues undiminished through all the lamps. The practical unit of electric current is called the ampere, after the French physicist, Andre Marie Ampere (1775- 1836), who made important discoveries concerning the magnetic action of cur- rents. The ampere is de- fined as the current that would produce a certain magnetic or a certain chem- ical effect. These defini- tions are of the greatest scientific and practical im- portance; but they are of no service to the beginner in electrical science, since they relate to matters with which he is not familiar. A definite or even approximate idea of the ampere and other electrical units can be gained only through personal acquaintance with the effects of currents under known conditions. This acquaintance will come, in some measure, through the experiments of the class-room and the laboratory. Compared with the quantities of electricity present in electro- static phenomena, the quantity carried in one second by a current of one ampere is enormous (Art. 405). For example, a flow of one am- pere for one two-hundred-thousandth of a second would be sufficient to charge an insulated sphere a foot in diameter to a potential of 300,000 volts. Yet a single dry cell, on short circuit, will supply a current of 15 amperes or more. Very little can be gained, however, by comparing plectric cur- rents with electric charges. The most serviceable ideas of current strength are derived from a knowledge of the effects which currents produce; but it should be noted that the voltage of a current is also a determining factor in producing these effects. In short, the power of a current is proportional jointly to its amperage (strength) and its voltage (pressure), just as water-power is proportional jointly to the MEASUREMENT OF ELECTRIC CURRENTS 533 quantity of the flow per second and the pressure or head. The power of a current of one ampere, maintained by an E.M.F. of one volt, is equal to r horse-power, or .74 ft.-lb. per second; but the same 740 current, when maintained by an E.M.F. of 746 volts, transmits one horse-power. (These matters will receive further attention later.) 432. Methods of Measuring Currents. Electric currents can be measured by means of their heating, chemical, or magnetic effects (Art. 402); but the magnetic effect is the only one adapted to general use. Instruments which meas- ure currents by their magnetic effects are called galvanome- ters. These are of various forms; but their action in all cases depends upon the fact that the magnetic field of a current in a given circuit is proportional to the strength of the current. 433. The Tangent Galvanometer. In the tangent galvanometer (Fig. 406) the current to be measured is sent through a vertical, circular coil of insulated wire. A short, magnetic needle is mounted with its center at the center of the coil. A long, non- magnetic pointer is attached at right angles to the needle and turns with it (Fig. 407). Any deflection of the needle is indicated by the pointer as it moves over a dial grad- uated in degrees. FIG. 406. Tangent Gal- T . vanometer. In using a tangent galvanometer, it must be turned so that the plane of the coil is in the mag- netic north-and-south line. This adjustment is made while there is no current in the coil. The coil and the needle are then in the same vertical plane, and the ends of the pointer are at the zero points of the scale on the dial. When the galvanometer is connected in a circuit, the 534 ELECTRODYNAMICS FIG. 407. Compass of Galvanometer. magnetic field of the current in the coil acts on the needle and tends to turn it at right angles to the plane of the coil (Fig. 391). But the deflection of the needle is opposed by the earth's field. Each pole of the needle is thus acted upon by two forces at right angles to each other (Fig. 408), and the needle comes to rest in line with the resultant of these forces. Since the force due to the cur- rent is proportional to the current, it is evident that the angle through which the needle turns (called the deflection) will be greater or less according as the current is stronger or weaker. The current, however, is not proportional to the angle of deflection, but to the tangent of the angle ; hence the name tangent galvanometer. (See Lab. Ex. 63 for a discussion of this relation, and detailed directions for the use of the instrument.) The value of a cur- rent in amperes can be obtained by multiplying the tangent of the angle of deflection by a factor, which is a constant for the same instrument at the same place, or the scale can be graduated to read in amperes directly. This graduation, if correct for one locality, would be incorrect for another, unless the earth's magnetic field happened to be of equal intensity at the two places. (Why?) 434. The D'Arsonval Galvanometer. There are two principal types of galvanometers. In the one the current is sent through a fixed coil, and a magnetic needle is de- FIG. 408. Component and Resultant Forces on Galvanometer Needle. N MEASUREMENT OF ELECTRIC CURRENTS 535 fleeted. The tangent galvanometer is an example. In the other type the magnet is fixed and the coil is deflected. Such instruments are called D' Arson val galva- nometers, after the French scientist who originated this type. They are made in a great variety of forms, adapted to use under different conditions. The general plan of a laboratory or a lecture-table D' Arson val is shown in Fig. 409 and a complete lecture-table instrument in Fig. 410. A coil of fine wire hangs between the poles N and S of a strong permanent magnet. The circuit includes the coil, the slender metal ribbon by which it is suspended, and a similar ribbon, in the form of a spiral, ' . r . ,. FlG - 409- Diagram leading from it beneath. A fixed cyhn- ofD'ArsonvaiGai- drical core of soft iron is mounted within variometer. the coil to strengthen the magnetic field. When no current is passing, the connecting ribbons hold the coil so that its plane is parallel to the line joining the poles of the magnet; but, with a current flowing, the coil tends to set itself with its north side facing the south pole of the magnet. This rotation is opposed by the torsion of the ribbons, and the coil turns through a greater or less angle, depending upon the strength of the current. In some instruments the coil carries a non-magnetic pointer, which moves over a scale (Fig. 410); in others it carries a small mirror, M (Fig. 409), which indicates the deflection by the angle at which it reflects a beam of light or the image of a scale placed at some distance in front of it. One important advantage of the D'Arsonval galvanometer is that it is independent of the earth's field, which is negligible 536 ELECTRODYNAMICS in comparison with the strong field of the magnet; hence the instrument does not need to be turned in any particular direc- tion. The sensitiveness of this galvanometer is increased by de- creasing the size of the supporting ribbon, by increasing the strength of the magnet, or by increasing the number of turns of the coil. With the most sensitive instru- ments a current less than a millionth of an ampere can be detected and measured. 435. Ammeters. A galvano- meter whose scale is graduated to read in amperes is called an am- meter (contracted from ampere- meter). Ammeters for industrial use are usually of the D'Arson- val type, and are so constructed that they can be carried about FIG. 410. D 'Arson val Galvanometer. without danger of injury. The Weston ammeter (Fig. 411) is a common instrument of this character. The magnet is horizontal, its poles being at the narrow side of the case, opposite the scale. The coil is pivoted on fixed bearings and FIG. 411. Weston Ammeter. FIG. 412. Sectional End View of Weston .Ammeter. carries a pointer, B (Fig. 412). The turning effect of the current is opposed by coiled springs, D and D. In Fig. 412 the nearer pole of OHM'S LAW 537 the magnet is represented as partly cut away, to afford a better view of the coil, C, and other interior parts. An ammeter is of necessity a low-resistance instrument; for, if its resistance were an appreciable fraction of the whole resistance of any circuit in which it was placed, it would reduce the current which it was placed there to measure, and would thus fail to serve the in- tended purpose. V. OHM'S LAW 436. Electromotive Force and potential difference are, as a rule, equivalent expressions. They always mean the same kind of quantity; but usage restricts the one to electric cur- rents, while the other may refer either to currents or to electric charges. The E.M.F. of a cell or a battery is the maximum potential difference which it can produce, i.e. the P.D. between the poles of the cell or the battery when the circuit is open (Art. 408, end). When the circuit is closed through a good conductor, the poles are discharged so rapidly that the P.D. between them falls more or less below the E.M.F. of the battery, and may even become practically zero; but the E.M.F. of the battery is not changed unless there is polarization. The volt is the practical unit of E.M.F. or P.D. As in the case of the ampere, the pupil's idea of the volt must be gained through personal acquaintance with the phenom- ena of electric currents. As a starting point, it will be of service to remember the approximate values of the E.M.F. of the different cells used in the laboratory and the class room. The following table is given for reference: ELECTROMOTIVE FORCE OF CELLS (APPROXIMATIONS) VOLTS VOLTS Storage cell 2.2 Dry cell 1.4 Bichromate cell 2.0 Daniell 1.08 Bunsen 1.9 Gravity 98 Grove 1.9 Simple voltaic cell 98 Leclanche" 1.4 Edison-Lalande 7 538 ELECTRODYNAMICS 437. Electrical Resistance. Let a cell be connected with an ammeter or other low-resistance galvanometer and the current measured. Again measure the current from the same cell, .When it is sent through various con- ductors in turn, e.g. a small lamp, a piece of German sil- ver wire, the coils of an electromagnet, etc. It will be found that, with any of these additions to the circuit, the current is reduced more or less, probably in most cases to a small fraction of its original value. If the cell is one that does not polarize, the E.M.F. acting in all these circuits is the same, and the currents are unequal only because the different conductors offer unequal opposition to the flow of the current through them. This opposition is a measureable quajitity, and is called electrical resistance, or, simply, resistance. With a given E.M.F., the resistance of the entire circuit (including the resistance ojLthejDell) is^by definition, inversely propor- tional to the current. For example, if the current is re- duced one half^when^gLConductor is added to the circuit, we know that this^addition has doubled the resistance of the circuit. The unit of resistance is denned as that resistance through which an E.M.F. of one volt will maintain a current of one ampere. This unit is called the ohm, after the German physicist Georg Ohm. The ohm is approximately the resistance of 157 ft. of No. 18 copper wire (diameter = 1.024 mm.) or 249 ft. of No. 16 (diameter = 1.29 mm.). 438. Ohm's Law. The current maintained through a given resistance is directly proportional to the E.M.F. or the P.D. between the terminals of the conductor. This is known as Ohm's law, it having been discovered experi- mentally by Ohm in 1826. OHM'S LAW 539 Ohm's law, when taken together with the definition of resistance, is stated as follows : The current strength in any circuit is directly proportional to the E.M.F. and inversely proportional to the total resistance. This is one of the most general and most important laws of electrical science. It holds under all circumstances for steady currents. (When the E.M.F. and the current are rapidly changing, other factors are involved.) < If C denotes the current measured in amperes, E the E.M.F. measured in volts, and R the resistance of the entire circuit in ohms, then C = | - (Ohm's law) ' . (i) Ohm's law holds for any part of a circuit, as well as for the entire circuit. Thus if P bc denotes the potential differ- ence between the points b and c of a circuit (Fig. 413) and R bc the resistance of the conductor between these points, then it being understood that there is no cell or other source of E.M.F. between the given points. EXAMPLES. The E.M.F. of a battery of six storage cells (Fig. 413) is 12 volts, and its resistance is i ohm. Two lamps, LI and L%, are placed in the circuit, the resistance of the first being 14 ohms and that of the second 9 ohms. Find (i) the current strength, (2) the potential difference, P, between the terminals of each of the lamps, and (3) the loss of potential in the battery. (1) The total resistance of the circuit is i + 14 + 9 = 24 ohms, assuming that the resistance of the connecting wires is negligible; E 12 hence C = = = .5 ampere. (2) Pbc = C X R^ = -5 X 14 = 7 volts. P de = C X #de = -5 X 9 = 4-5 volts. 540 ELECTRODYNAMICS (3) The loss of potential in the battery is the same as it would be in any conductor of one ohm resistance, when carrying a current of .5 ampere, which is .5 X i = -5 volt. The P.D. between the poles of the battery, a and/, is, therefore, 12. - .5 = 11.5 volts; which, of course,' is equal to the fall of potential in the two lamps (7 +4.5 = 11.5). 439. Fall of Potential along a Circuit. In the above example C = -^ = ~^; whence P bc : P de :: R bc : R de ; or, -ft-bc -tvde taking the numerical values, 7 14.5 :: 14 19. That is, the fall of potential in the one lamp is to that in the other as the resistance of the first is to the resistance of the second. This relation is general. The fall of potential in the differ- ent parts of a circuit (except- ing only the part or parts FIG. 413- *,. *. , * within which the source of the E.M.F. is located) is proportional to the resistance of the several parts. In that part of a circuit in which the source of E.M.F. is located, as the part fa in Fig. 413, there is a rise of poten- tial, owing to the chemical action or other source of energy. In a cell this rise of potential takes place abruptly at the surface of the negative plate, as the current passes from the plate to the liquid (Art. 407). Within the liquid there is a fall of potential from the negative to the positive plate, when a current is flowing, as in the case of any other conductor. PROBLEMS 1. How would the current from a given battery be affected by a fourfold increase in the resistance of the circuit? by a tenfold increase? 2. What is the resistance of a no- volt lamp if it is lighted by a current of half an ampere? LAWS OF RESISTANCE 541 3. The E.M.F. of a bichromate cell is 2 volts and its resistance .25 ohm. What current will it supply (a) through an external resistance of .1 ohm? (6) through an external resistance of 12 ohms? 4. The E.M.F. of a dry cell is 1.4 volts and its resistance .1 ohm. (a) What current will it supply through an external resistance of 00.5 ohm? (6) What will then be the P.D. between its poles? 6. The E.M.F. of a gravity cell is .98 volt and its resistance 3.5 ohms, (a) What current will it send through a conductor whose resistance is negli- gible? (6) What current will it send through an external resistance of 3.5 ohms? (c) What will be the P.D. between its poles in the first case? in the second case? VI. LAWS OF RESISTANCE 440. Variation of Resistance with Length and Cross- section. Experiment shows that the resistance of a uni- form conductor of any given material varies directly as its length and inversely as its cross-section. Equal parts of a uniform conductor have equal resistance, and the resistance of the whole conductor is the sum of the resistances of its parts ; hence the law of lengths. Similarly the resistance of a circuit which is made up of any number of different conductors joined in series (i.e. so that the entire current passes through each conductor, as in Figs. 405 and 413) is equal to the sum of the resistances of its parts. Since the cross-section of a circular wire is proportional to the square of its diameter, its resistance varies inversely as the square of its diameter. Thus if we take two wires of the same length and material, the one having a diameter of i mm. and the other a diameter of 2 mm., the resistance of the larger is one fourth as great as that of the smaller, since its cross-section is four times as great. The resist- ance of No. i copper wire, which is about the size of a lead pencil, is approximately .62 ohm per mile of length. 441. Specific Resistance. Let the current in a bat- tery circuit be measured when wires of equal length and 542 ELECTRODYNAMICS cross-section, but of different materials, are included in it in turn. It will be found that the current is strongest through copper, considerably weaker through iron, and still weaker through German silver. Evidently the re- sistance of a conductor depends upon the material, as well as upon the length and cross-section. The less the resistance of a wire of given length and cross- section, the greater is said to be the conductivity of the material of which it is made and the less the specific re- sistance of the material. Conductivity and specific resist- ance are reciprocal quantities. Silver has the greatest conductivity and the least specific resistance of any known substance. Copper is only slightly inferior to silver. The following table gives the specific resistances of several com- mon materials, relative to copper as the standard. The value for copper is arbitrarily taken as unity. The values are to be regarded only as approximations, for the specific resist- ance of different specimens of the same material is found to vary considerably with the purity of the specimen, the process of manufacture, the tempering, etc. SPECIFIC RESISTANCES (RELATIVE TO COPPER) Silver, annealed .... 0.94 German silver, (varying Copper, annealed .... i.oo with the composition) . . 13 to 20. Aluminum 1.7 Manganin 33. Iron, pure 6. Mercury 59. Platinum 7. Carbon, arc and incandes- Iron, telegraph wire 9. cent lamp 2500. Copper wire is almost exclusively used for lighting and power cir- cuits, which must have a very low resistance. Aluminum is the only alternative for this purpose, and is used to some extent. 442. Variation of Resistance with Temperature. The resist- ance of metals and of most other substances increases with a rise of temperature. The rate of increase is nearly the same for all pure LAWS OF RESISTANCE 543 metals, and is such that at 100 C. their specific resistances are about 40% higher than at o. The resistance of alloys, particularly man- ganin and German silver, is much less affected by change of tempera- ture; hence these alloys are used for standard resistance coils (Art. 444). The resistance of carbon, dilute acids, and other conducting solutions decreases with a rise of temperature. The carbon filament of the common incandescent lamp has only about half the resistance when white hot that it has when cold. If a conductor is maintained at a constant temperature, its re- sistance is the same whatever the strength of the current. 443. Laws of Divided Circuits. Electric circuits often have two or more branches between two points, as between A and B (Fig. 414). The branches are said to be connected in parallel, and either of two branches is called a shunt to the other. This is the usual arrange- ment of electric bells and incandescent lamps in circuits (Figs. 415 and 416). FlG> 4I4 * The sum of the currents in all the branches between two points is equal to the current in the undivided part of the circuit. If the branches all have equal resistance, they take equal portions of the current ; if their resistances are unequal, the currents in them are inversely proportional to their resistances. This is proved for a two-branch circuit as follows: Let RI denote the resistance of one branch be- tween A and B (Fig. 414), and Rz the re- sistance of the other ; and let Ci and C z ' denote the currents FIG. 415. Electric Lamps m Parallel. in the respective branches. By Ohm's law the P:D. between A and B is equal to C\R\ and also to C^Rz ; hence CiRi = C 2 Rz, or 544 ELECTRODYNAMICS The resistance between any two points of a circuit is decreased by adding one or more conductors in parallel between the points; for this is equivalent to an increase in the cross-section of the original conductor between the points. In the simple case of n branches, having equal resistance, their combined resistance is one n th of the resist- ance of one of them; for the n branches are together equiva- lent to a single conductor of the same length as one of the branches and of n times the cross-section. A good example is that of incandescent lamps in parallel (Fig. 415). PROBLEMS 1. (a) If the lamps oh a no- volt circuit (Fig. 416) have each a resist- ance of 220 ohms when lighted, what is the joint resistance of 6 of them in parallel ? (b) What current flows in the leads (the main wires C HH> a b c FlG. 416. Electric Bell Circuits. MEASUREMENT OF RESISTANCE 545 of the circuit) when the 6 lamps are turned on? (c) What is the cur- rent when only one lamp is lighted? 2. (a) If on the above circuit two of the lamps are connected in series between the leads, what would be their combined resistance? (b) What would be the P.D. between the terminals of each lamp? (c) What current would flow through them? 3. Describe the bell circuits shown in Fig. 417. In which circuits does only one bell ring when one button is pushed? In which do two bells ring when a single button is pushed? In which does the same bell ring when either of two buttons is pushed? Which circuits require a larger current than circuit a? Which do not? 4. (a) What is the ratio of the cross-sections of aluminum and copper wires having equal resistance per unit length? (b) What is the ratio of the weight per unit length of such wires? (c) Which material has the advantage in the matter of conducting power for a given size? for a given weight? VII. MEASUREMENT OF RESISTANCE AND ELECTROMOTIVE FORCE 444. Standards of Resistance. The standard ohm is so denned that it can be reproduced in any scientific lab- oratory. It is the resistance at o C. of a column of mercury 106.3 cm. long and i sq. cm. in cross-section. The mass of the mercury is 14.4521 g. This standard is used only for comparison in making FIG. 417. Resistance and testing more convenient standards of wire for ordinary use. The latter are known as resist- ance coils. They are made of some alloy having a high specific resistance, usually manganin. The ends of each coil are joined to brass blocks, A and B, B and C (Fig. 417), arranged in rows on the insulating top of a resistance box, with the coils inside (Fig. 418). Adjacent blocks are separated by a gap, which is bridged by means of a brass plug. With all the plugs firmly in place the box resistance is practically zero; but wherever a plug is removed the 546 ELECTRODYNAMICS current can pass only through the coil at that place. This introduces the resistance of the coil into the circuit of which the box forms a part. The amount of this resistance is mark- ed on the top of the box. The total box resistance included in the circuit is the sum of the resistances of the FIG. 418. Resistance Box. are out. The coils of a box make up the series .1, .2, .3, .4, i, 2, 3, 4, 10, 20, 30, and 40 ohms, which is sometimes extended to higher resistances. 445. Measurement of Resistance. There are vari- ous methods of measuring resistance. The practical elec- trician uses some form of a special apparatus known as a Wheatstone bridge (Lab. Ex. 69). The method of substi- tution requires only a resistance box and any low-resist- ance galvanometer. By this method the resistance to be measured R (Fig. 419) is connected in circuit with the galvanometer and a cell of constant E.M.F., and the deflection is read as accurately as possible. The unknown resistance is then removed from the circuit and the resist- ance box put in its place. The box resistance is adjusted to give exactly the same deflection as ^-]L_^f~\ G before. In making this adjustment, the f 'I coils are tried in order from larger to x^,^ / smaller, as weights are tried in weighing. R If the deflection is too great, the resist- Fra 4IQ - ance is too small, and vice versa. (Why?) The unknown resistance is equal to the box resistance when the deflec- MEASUREMENT OF RESISTANCE 547 tions are equal. This follows from Ohm's law. For the equal deflections indicate equal currents; and, with a con- stant E.M.F., the currents will be equal only when the entire resistance of the circuit is the same in both cases. Hence the unknown resistance must be equal to the box resistance which took its place. 446. Measurement of E.M.F. and P.D. The Volt- meter. In dealing with electric charges, their potentials and potential differences are measured by utilizing the elec- trostatic attractions and repulsions of the charges. The greater or less divergence of the leaves of an electroscope can be made to serve this purpose. In dealing with electric currents, high-resistance galvanometers are used. The coil of such an instrument is made of a very long, fine wire, and has a resistance of hundreds or even thou- sands of ohms. Owing to the^, great number of turns in the coil, a very weak current catwes a relatively large deflection. If the scale is graduated in volts, the instru- ment is called a voltmeter. Voltmeters are usually of the D'Arsonval type (fixed magnet and movable coil). An E.M.F. , when measured in volts, is often called voltage. To determine the P.D. between two points of a circuit, e.g. the terminals of the lamp L (Fig. 420), the voltmeter, V, is con- nected as a shunt between FlG " 42o.-Diagram of Connections for Volt- meter and Ammeter. these points. By Ohm's law, the current through the voltmeter is proportional to the P.D. between the points a and b. Hence, with a tangent ^^^ jr.] ^^ b f a */. il fi ii - 548 ELECTRODYNAMICS instrument, the P.D. is proportional to the tangent of the angle of deflection. With instruments of the D' Arson val type, the scale can be graduated, once for all, in volts. To find the E.F.M. of a cell or a battery, the circuit is closed through the voltmeter only (Figures 421, 422, and 423). The necessity for a high resistance in the voltmeter arises from the fact that, being connected as a shunt, it tends to diminish the resistance of the circuit between the points with which it is connected (Art. 443). But any appreciable decrease in this re- sistance would reduce the P.D. be .ween the points, since the fall of potential along a circuit is everywhere proportional to the resistance to be overcome (Art. 439). Hence if the voltmeter had only a mod- erate resistance, it would lower the P.D. which it was intended to measure. On the other hand, if it has a very high resistance, it does not appreciably affect the resistance or the P.D. between the points with which it is connected. 447. Measurement of Resistance with an Ammeter and a Voltmeter. If R denotes the resistance of a conductor (in Fig. 420, the lamp, L), and P the P.D. between its ends when a current, C, is flowing through it, then, by p p Ohm's law, C = ^ or R = -~ (Equation 2). Hence, if the P.D. is measured with a voltmeter, V, and the cur- rent with an ammeter, A t the resistance of the conductor p is given by the quotient pr This is the simplest and quickest method of measuring resistance. To find the resistance of a cell by this method, its E.M.F. is measured with the voltmeter, and the current is meas- ured with the cell short-circuited through the ammeter (the resistance of the ammeter being negligible). MEASUREMENT OF RESISTANCE 549 448. Arrangement of Cells in a Battery. A battery of two or more cells will, in general, supply a larger current in a given circuit than a single cell of the same kind; but the current obtained from a given number of cells in a given circuit is largely determined by the manner in which the cells are joined together. They may be joined in series, or in parallel, or in groups with both series and parallel connections. By applying Ohm's law we can readily deter- mine which arrangement is best in any given case. For this purpose it is necessary to distinguish between the resistance of the battery and the resistance of the external part of the circuit. The former is often called the internal resistance and the latter the external. Throughout the present discussion, E denotes the E.M.F. and R { the resist- ance of a single cell, and R e the external resistance. Ohm's law, when applied to a circuit in which there is only one cell, then takes the form - C = "r> ET* (4) -Ki + Ke Cells are said to be connected in series when the entire current passes through each in succession, as with other conductors in series (Fig. 421). The cells are so joined that the E.M.F.'s of all act in the same direction round the circuit, i.e. the ^p If- positive pole of the first cell is con- nected with the negative pole of the , , . FIG. 421. Cells in Series. second, the positive pole of the second with the negative pole of the third, etc. The E.M.F. of the battery is the sum of the E.M.F.'s of the cells, and its resistance is the sum of the resistances of the cells, as in the case of other resistances in series. With a battery of n like cells in series the E.M.F. is nE and the resistance nR r Ohm's law, when applied to a circuit in which the 550 ELECTRODYNAMICS current is maintained by such a battery, takes for them nE = ^TT^e (5) Cells are said to be connected in parallel when the circuit is divided at the battery, with a cell in each branch (Fig. 422). The negative plates of all the cells are joined to the negative terminal of the battery, N, and the positive plates to the positive terminal, P. The branches may all start from one point and meet at one point; but it serves p N FIG. 422. Cells in Parallel. the same purpose and is more convenient in making con- nections to place them one after the other, as shown in the figure. This is exactly like the connection of lamps in parallel between two leads (Fig. 415). All the negative plates are at the same potential, since there is no source of E.M.F. between them, and all the positive plates are at the same potential, for the same reason. The E.M.F. between the negative and the positive plates of the battery is that of one cell only, or E. If there are n like cells in parallel, the resistance of each being R b the resistance of r>- the battery is according to the law for conductors n in parallel. Hence for this case Ohm's law takes the E form C = - (6) + , n 449. When to Connect Cells in Series and when in Par- allel. When cells are joined in series the E.M.F. of the MEASUREMENT OF RESISTANCE 551 battery varies as the number of cells; but the resistance of the battery increases in the same ratio. Hence, if the bat- tery resistance is practically the whole resistance of the circuit (as when the external circuit is a short copper wire), the current from any number of cells in series is not appre- ciably larger than a single cell would supply; i. e., con- _f '77 TT> sidering R e negligible, C = - ^ = =-. But if the external nK\ KI resistance is relatively large, an increase in the battery resistance has little effect on the result and the current increases nearly in the same ratio as the E.M.F., when cells are added in series. A telegraph battery is a good example (Fig. 401). By joining cells in parallel the battery resistance is decreased in proportion to the number of cells. If the external resistance is very small, this decreases the total resistance of the circuit in nearly the same ratio, and there is a proportionate increase in the current. But if the exter- nal resistance is large, a decrease in the battery resistance has little effect on the total resistance, j^\ '| ~\\ and a single cell will furnish practically as large a current as any number of cells in parallel. FIG. 423. a, Series-parallel, and b, Parallel- Hence, in general, series Grouping ' cells should be joined in series when the external resistance is relatively high, and in parallel when the external resist- ance is low. 450. Mixed Series and Parallel Grouping. With a medium ex- ternal resistance, the largest current from a given number of cells is 552 ELECTRODYNAMICS sometimes obtained by series-parallel grouping. The result is the same whether the cells of each group are joined in series, and the groups in parallel (Fig. 423, a), or the cells of each group in parallel and the groups in series (Fig. 423, b). The formula for either ar- rangement shown in the figure is ^ + R C 2 With a given number of cells and a given external resistance, the largest current is obtained when the cells are so connected that the resistance of the battery is as nearly as possible equal to the external resistance. PROBLEMS " 1. What is the combined resistance of three incandescent lamps in par- allel, the resistance of each lamp being 200 ohms? 2. The fall of potential through a coil of wire is 1.5 volts when a current of -.2 ampere is flowing. What is the resistance of the coil? 3. What E.M.F. will maintain a current of 1.5 amperes through a re- sistance of 80 ohms? 4. If the E.M.F. of a chromic acid cell is 2 volts, and its resistance .3 ohm, what current will it supply through an external resistance of .1 ohm? 5. What would be the current through the same external resistance from a battery of 4 such cells (a) in parallel? (6) in series? 6. What current would be supplied by a battery of 12 Leclanch6 cells, each having an E.M.F. of 1.4 volts and a resistance of i ohm, through an external resistance of 1.5 ohms (a) with the cells . connected in series? (6) in parallel? (c) in three groups of four each, the cells of each group being in series, and the groups connected in parallel? Draw a diagram for (c). 7. Show that, when the external resistance of a circuit is negligible in comparison with the resistance of a cell, the current is porportional to the number of cells connected in parallel, but a single cell furnishes as large a current as any number of cells connected in series. 8. Show that, when the battery resistance is negligible in comparison with the external resistance, the current is porportional to the number of cells connected in series, but a single cell furnishes as large a current as any number of cells in parallel. ELECTRICAL ENERGY 553 VIII. ELECTRICAL ENERGY. HEATING EFFECTS OF ELECTRIC CURRENTS 451. Electrical Energy. A battery or a dynamo may be compared to a pump which raises water from a lower to a higher level. The work done by the pump, or the energy imparted to the water, would be measured by the product of the weight of water raised and the height to which it is raised. Similarly the function of a battery or a dynamo is to raise electricity from a lower to a higher potential, in doing which it imparts energy to the current; and the energy is measured by the product of the whole quantity of electricity supplied during the time that the current is flowing and the potential or E.M.F. at which it is supplied. (Compare with the energy of an electric charge, Art. 396). With a constant current, the quantity of electricity sup- plied by a generator (battery or dynamo) in a given time is equal to the product of the current strength and the time ; just as the quantity of water delivered by a pump would be measured by the product of the number of pounds or gal- lons per second and the number of seconds. The unit quantity of electricity is the quantity which passes any point of a circuit in one second when the current strength is one ampere. This quantity is called an ampere-second. (A larger unit of quantity is the ampere-hour.) A current of C amperes, flowing for t seconds, transports Ct ampere- seconds of electricity past every point of the circuit. As stated above, the electrical energy generated in a given time is measured by the product of the E.M.F. of the generator and the whole quantity of electricity sup- plied; i.e. - Electrical energy = volts X ampere-seconds. 554 ELECTRODYNAMICS The unit of electrical energy is the energy imparted to a unit quantity of electricity when its potential is raised one volt; or, a current of one ampere generated at an E.M.F. of one volt conveys a unit quantity of electrical energy in one second. This unit is called the joule, after the English physicist of that name (Arts. 242 and 454). Hence, if E denotes the E.M.F. of the generator in volts, C the num- ber of amperes, and / the number of seconds, the quantity of electrical energy generated in that time is ECt joules; or, briefly - Electrical energy = ECt joules. (7) The equivalent of one joule in mechanical energy is .74 ft.-lb., approximately. 452. Energy Expended by an Electric Current. The energy imparted to an electric current by the generator is expended (changed into other forms of energy) in the circuit. It is transformed (i) into heat in overcoming the resistance of the circuit, as in the electric lamp, (2) into chemical energy in producing chemical change, as in charging a storage battery, and (3) into mechanical energy in doing mechanical work, as in running electric motors. The work, of whatever kind, done by a current in the different parts of a circuit is everywhere proportional to the fall of potential. Thus if the P.D. between any two points, a and b, of the circuit is denoted by P ab , the work done by the current between these points in / seconds is P ab O joules, or, briefly, Work done between a and b = P ab Ct joules. (8) 453. Electrical Power. Industrial Units of Electrical Power and Energy. The power of an agent, as defined in the study of Mechanics (Art. 136), is its rate of doing work. ELECTRICAL ENERGY 555 The power of a battery or a dynamo is the rate at which it generates electrical energy, and is measured by the energy generated in one second, that is Electrical power = EC joules per second. (9) One joule of work per second is a unit of power, and is called a watt, after James Watt, the inventor of the modern steam engine. Since this unit is very small, a unit 1000 times as large, called the kilowatt, is adopted for industrial use. Dynamos and motors are very generally rated in kilowatts. A watt is equal to yjg f a horse-power, or to .74 foot-pound per second. A kilowatt is equal to "We " horse-power, or f horse-power, very nearly. In terms of these units, Electrical power = EC watts, (10) EC kilowatts, (n) C horse-power. (12) The power expended in any part of an electric circuit is proportional jointly to the fall of potential in that part and to the strength of the current, or Power expended between a and b = P ab C watts. (13) For example, if a no- volt lamp takes a current of .5 ampere, the power consumed in lighting it is 110 X .5 = 55 watts, or -2 = horse-power, nearly. 746 14 The industrial units of electrical energy are the watt-hour and the kilowatt-hour. The watt-hour is the energy expended in one hour at the rate of one watt, or one joule per second; hence it is equal to 3600 joules. The kilowatt-hour is equal to 1000 watt-hours. The cost of electrical energy to consumers for light and other uses is generally between 7 cents and 10 cents per kilowatt-hour. An instrument for measuring the power of a current is called a watt-meter. A watt-hour-meter records on a set of dials the number of watt-hours of energy consumed. In this instrument the current 556 ELECTRODYNAMICS drives a small motor, so designed that its speed is proportional jointly to the number of volts and the number of amperes. The motor drives a train of gear-wheels; and the wheels move the hands of the dials, as in the gas meter. / 454. Heat Generated in a Conductor. Joule's Law. - When a current of several amperes is sent through a piece of fine German silver or platinum wire, the wire becomes white hot, and may even melt; but the remainder of the circuit, if it consists of larger copper wire, is not appre- ciably warmed. Electrical energy is converted into heat in overcoming the resistance of the circuit. If no other work is done by the current, all its energy is thus transformed. Since all conductors have resistance, all are heated more or less when carrying a current ; but with good conductors, having a size adapted to the strength of the current, the amount of heating is slight and generally passes unnoticed. Since the fall of potential in the different parts of a cir- cuit is proportional to the resistances of the parts (Art. 439), and since the loss of electrical energy is proportional to the fall of potential (Formula 13), it follows that the heat generated in the different parts of a circuit (in all of which parts the same current is flowing) is proportional to their resistances. The German silver or platinum wire in the experiment described above becomes very hot, while the copper connecting wires remain cool, because its resist- ance for a given length is many times greater than that of the copper wires. Electric lighting furnishes another example of the same conditions. The circuit wires are of copper, large enough to carry the current without noticeable heating; but the lamp filaments have a high resistance. Hence the fall of potential and the conversion of electrical energy into heat take place almost wholly in the lamps. ELECTRICAL ENERGY 557 Since the energy converted into heat in a conductor is equal to P.D. X Ct (Formula 8), and since P.D. = CR, R being the resist- ance of the conductor (Ohm's law), it fol- lows that the heat generated is equal to C?Rt joules. In order to express this quantity of heat in terms of the custom- ary heat unit, the calorie, it is necessary to know the equivalent of the joule in calories. This is found by experiment to be .24 calorie; hence Heat generated = .z^C-Ri calories. (14) This is known as Joule's law, it having FIG. 424. been experimentally determined by him in 1841. The experiment consists in passing a known current for a measured length of time through a coil of known resistance, placed in a calorimeter contain- ing water (Fig. 424). The electrical energy converted into heat in the calorimeter is equal to C 2 Rt joules. The heat generated is com- puted in calories from the mass of the water, the mass and specific heat of the vessel, and the rise of temperature. 455. Incandescent Lamps. In an incandescent lamp (Fig. 425), the current passes through a slender filament or wire which, owing to its high resistance, is heated white hot. The ends of the fila- ment are attached to short platinum wires, which pass through the glass and connect with the metal casing, d, and plug, g, at the base of the lamp. When FIG. 425. Diagram of the lamp is screwed into the socket, g is Ele " brou S nt into contact with h; the circuit is closed at c by turning the key x. The air is exhausted from the bulb to prevent combustion of the filament when heated. 558 ELECTRODYNAMICS In the earliest type of incandescent lamp, which is still the most common, the filament is of carbon. These lamps are made for both no- volt and 2 20- volt circuits, and are of various candle powers, the i6-candle lamp being in most general use. The i6-candle, no- volt lamp has a resistance of about 220 ohms when lighted, and hence takes a cur- rent of about half an ampere. The power consumed is no X .5 = 55 watts. The i6-candle, 220- volt lamp also consumes 55 watts, and hence takes a current of .25 am- peres (55 -z- 220 = .25). By Ohm's law the resistance of the heated filament is 220 -f- .25 = 880 ohms. To secure this high resistance, the filament is made longer and of smaller size than that of the no- volt lamp. The newer types of incandescent lamps have metallic filaments. None of the common metals or alloys would serve for this purpose, since their melting points are all too low. The best results have been ob- tained with tungsten and tantalum. As the spe- cific resistance of these materials is much less than that of carbon, the filaments must be long and very slender even for iio-volt circuits. The advantage of the metallic-filament lamps is their high efficiency, or large candle power for the number of watts consumed. In this res P ect tne tungsten lamp (Fig. 426) ranks first, requiring only 1.25 watts per candle power, while the ordinary carbon-filament lamp requires 3.5 watts per candle power (55 ^- 16 = 3.5). 466. The Electric Arc and the Arc Lamp. When the rounded ends of two pieces of carbon loosely touch each other and a current of several amperes passes between them, the ends quickly become white hot. The heating is due to the relatively high resistance where the surfaces ELECTRICAL ENERGY 559 make imperfect contact with each other. If the source of the current is a dynamo or a storage battery, capable of maintaining a P. D. of 40 or 50 volts between the carbons when they are separated slightly, the current will be conducted across the gap by the heated air and carbon vapor. The luminous track of the current through the air is known as the electric arc (Fig. 427). The greater part of the light comes from the carbon points, especially from the depression or crater, which forms at the end of the positive carbon. The carbon rods of an arc lamp burn FlG ' 427 ' ~ Electric Arc ' away more or less rapidly; and, if they were held in a fixed position, the arc would increase in length and finally go out, when the resistance of the gap became too great for the cur- rent to pass. Arc lamps are therefore provided with auto- matic devices of various sorts, by means of which the upper carbon is permitted to drop and touch the lower one, for the purpose of starting the arc, and is thereafter lowered a little from time to time as the carbons burn away. This "feed- ing" mechanism, as it is called, is controlled by the current which lights the lamp, through the action of electromag- nets. The regulating device shown in Fig. 428 consists of two magnet coils, Ci and Ci, with their movable iron cores, n and n, a lever, /, and a clutch, c. C\ is a low-resistance coil, in series with the carbons; C 2 is a high-resistance coil, connected as a shunt to the carbons. When there is no current flowing in the circuit, the clutch releases the metal rod to which the upper carbon is fastened, and the rod drops till the carbons meet. When the current is turned on, prac- tically the whole of it passes by way of the carbons and the coil C\. The magnetic pull of this coil draws the core n up, thus lifting the 560 ELECTRODYNAMICS right end of the lever /. This motion is communicated to the clutch, which grips the rod and raises it slightly; the rod lifts the upper carbon, and the arc is established. As the carbons burn away, the resist- ance of the arc increases, the current across the arc and through the coil Ci decreases, and an increasing fraction of the current flows through the shunt coil C%. This goes on until the mag- netic pull of C 2 is sufficient to raise the core n. As n rises, it momentarily releases the clutch, and the rod falls a short distance, bringing the carbons closer together. The electric arc requires a P. D. of 45 to 50 volts. The current varies with the diameter of the carbons. Street lamps of the usual size take about 10 am- peres. These give a light of about 1000 candle power in the direction of greatest intensity. The average for all direc- tions in space about the lamp (called the "mean spherical" candle power), lies between 300 and 500 candle power. The arc lamp, therefore, takes approximately one watt per candle power. There are several types of arc lamps. The enclosed arc receives its name from the fact that it is completely inclosed in a small globe. Although this is not air tight, it practically excludes fresh air; and the carbons last much longer than they do with an open arc. In the flam- ing arc lamp the carbons have a central core, filled with lime and other minerals which change the arc to an intensely brilliant flame. The flaming arc has the highest efficiency of any artificial source of light, requiring only .6 watt per mean spherical candle power. There are various other forms of electric lamps, and the number is increas- ing from year to year. FIG. 428. Diagram of Arc Lamp. ELECTRICAL ENERGY 561 457. Electric Forging and Smelting. The temperature of the electric arc is estimated at 3500 to 3800 C., and is the highest that can be produced by any known means. At this temperature the most refractory substances are melted and vaporized. The heat of the arc is utilized in the reduction of ores and in the manufacture of nu- merous chemical products, e.g. calcium carbide, carborundum, and graphite. Such processes are conducted on a large scale at Niagara Falls and other places, where water-power is abundant and can be cheaply converted into electrical power. The materials to be treated are placed in a huge furnace, and a current of hundreds or thousands of amperes is passed through the mass between large carbon elec- trodes. In recent years the electric current has largely superseded the forge fire as the source of heat for welding, brazing, shaping, and tem- pering metals. The heat is generated within the metal by sending through it a very large current at a low voltage. In welding, the two pieces of metal are heated in this manner while pressed firmly together. When they have become soft, they are squeezed together slightly, the current is shut off, and the weld is complete. 458. Cooking and Heating by Electricity. Heat is obtained from electricity for cooking and other uses in the home by sending the current through coils of suitable resistance. The electric flat-iron is perhaps the most familiar example. The resistance coil is within the hollow body of the iron, just above the bottom plate (Fig. 429). A layer of asbestos, placed over the coil, largely prevents the conduction of heat upward. The heating coil of cooking utensils is either within the vessel or directly underneath it. Electric cooking and heating appliances are of ten connected to lamp sockets; but it should be borne in mind that they take a much greater current than a lamp does. Flat- irons, chafing-dishes, coffee urns, etc., take from 250 to 500 watts, or as much as 5 to ance Ribbon. 10 carbon-filament lamps; and the larger ap- pliances, such as stoves, ovens, and broilers, take from 500 to 1800 watts. Now the wire of lamp circuits is not large enough to carry more than a few amperes without overheating, which destroys the 562 ELECTRODYNAMICS insulation and may set the house on fire. Hence in making such use of lamp circuits one must be careful to avoid an " overload " of current. The wiring for an " electric kitchen " is of much larger size than that regularly used for lighting circuits. The electric current is a cleanly, safe, and efficient source of heat in the home. The one obstacle in the way of -its extensive use for this purpose is the expense, which, so far as the general heating of dwellings is concerned, is prohibitive. The output of electrical energy from a modern power station is, on the average, not more than 10% of the energy of the fuel consumed in producing it. And this small fraction is obtained at great expense; so that, in the end, a given amount of heat from the electric current costs the consumer probably fifty times as much as an equal amount obtained directly from fuel. On the other hand, the heat of the current can be put where it is wanted and generated only as long as it is wanted; hence very little is wasted, and for light cooking, ironing, and similar uses, the cost is not excessive. 459. Safety Fuses. In the use of electricity for light- ing and power purposes, there is always the possibility that the mains or leads may be accidentally connected by a conductor of low resistance, or short-circuited. If this happens, the current instantly increases perhaps a hundred or a thousandfold; for the resistance of the circuit is then only a few ohms at the most. With such a current the wires quickly become hot, and are likely to set fire to woodwork and other inflammable materials near them before the trouble is discovered and remedied. As a pro- tection against the danger of a short-circuit and the lesser danger of an overload in ordinary use, safety fuses are placed at suitable points in all lighting and power circuits. Fuse wire is made of lead and tin, or other alloy which melts at a low temperature. It is sometimes used as a bare wire; but in reg- ular practice it is in the form of an inclosed fuse (Fig. 430). This consists of a fuse wire inclosed in a fiber tube, which is filled in with some non-conducting, infusible material. The wire is joined to brass c ELECTRICAL ENERGY 563 caps which cover the ends of the tube; and the caps connect with the circuit through the spring clips by which they are held in place on the fuse block. The latter is of por- celain to prevent the possibility of fire when a fuse " blows out." Fuses are of different sizes, vary- ing with the number of amperes which they are designed to carry. The fuses in any circuit should have a capacity just above the maximum a b which the circuit should carry. Then, FIG. 430. Fuses, a, Single Fuse; if for any reason the current exceeds b > the Same i n Position; c, * Fuse Block for three Circuits, this maximum, the fuse wire will in- stantly melt, breaking the circuit, and no further damage will be done. PROBLEMS 1. What is the power of a battery that is able to maintain a current of 4 amperes through a resistance of 6 ohms? 2. (a) Assuming equal efficiency, how does the current taken by a 50- candle lamp compare with that taken by a i6-candle lamp for the same voltage? (6) How does the resistance of the 5o-candle lamp compare with that of the other? 3. An electric oven takes 1500 watts and a chafing-dish 500 watts on a 1 10- volt circuit. What is the resistance of the coil in each? 4. What is the relation between the resistance of a coil or a lamp and the heat generated in it per second on a constant- potential circuit? 6. What is the relation between the resistance of a coil and the heat generated in it per second on a constant-current circuit? 6. What power will be required to light 225 tungsten lamps of 32 candle power, the efficiency being 1.25 watts per candle power? 7. If in lighting the above lamps 10% of the power generated by the dynamo is lost in the distributing wires, what must be the power of the dynamo in kilowatts? 8. If the above dynamo is run by a turbine water-wheel having an effi- ciency of 80%, what water-power will be required to run the wheel? Express the result in horse-power. 9. At 9 cents per kilowatt hour, what will it cost to heat to the boiling point 4 liters (a little more than i gal.) of water, taken at 20 C., assum- ing that 75% of the heat generated goes to the water? 564 ELECTRODYNAMICS IX. ELECTROMAGNETIC INDUCTION 460. Historical Note. The generation of electric cur- rents by magnetic action, as in the dynamo, is termed electromagnetic induction, or current induction. The phenomena of current induction were first observed and studied by Joseph Henry in America and Michael Fara- day in England. Knowing that an electric current affects a magnet, Faraday reasoned that a magnet should react upon a current. Working from this hypothesis, he tried again and again during a period of several years to dis- cover such an effect, and at last, in 1831, his efforts were rewarded with success he had discovered electromagnetic induction. The importance of this discovery could not have been foreseen at the time. Faraday himself was so far from suspecting it that he says in a letter, "I am busy just now again on electromagnetism, and I think I have got hold of a good thing, but can't say. It may be a weed instead of a fish that, after all my labor, I may at last pull up." He had indeed got hold of a good thing. Years afterward Tyndall wrote: "I can not help thinking that this discovery is the greatest experimental result ever obtained. It is the Mont Blanc of Faraday's own achieve- ments. He always worked at great elevations, but higher than this he never attained." While great honor is due to Faraday for this most useful contribution to science, Henry's name will also be remembered. His discoveries in part preceded Faraday's, but he did not publish them till later. 461. Current Induction by a Magnet. Induced cur- rents can be generated in various ways, but these are only different ways of bringing about the one essential condition, namely, a change in the magnetic field within a closed circuit. This change may be either an increase or a decrease in ELECTROMAGNETIC INDUCTION 565 the strength of the field, or a change in the direction of the lines of force relative to the circuit. The induced currents obtained in experiments are usu- ally very weak; and, in order to increase the induction, a long wire in the form of a coil of several hundred turns is taken for the circuit (Fig. 431). This coil is joined with a sensitive galvanometer, which serves to show the presence of a current and to determine its direction. With such a circuit and a strong magnet, the laws of current induc- tion can be readily determined. Let the magnet be thrust quickly into the coil, and, after a few seconds, quickly removed. The galvanometer indicates a momentary current in one direction while the mag- net is being inserted, and in the opposite direction while it is being withdrawn; but there is no current while the mag- net remains at rest within the coil. The field of the magnet induces a current not by 'its presence merely, but by its motion relative to the coil, as the magnet is moved. As stated above, this is FIG. 431. Current in- .. . . . . ... duction by Magnet. only one of several devices by which the strength of the magnetic field within a circuit may be changed. 462. Lenz's Law. In the above experiment the direc- tion of the current through the galvanometer is indicated by the direction of the deflection. Starting from the gal- vanometer, we can trace the direction of the current round the coil; and we can then find the polarity of the coil by applying the right-hand rule. Proceeding thus, we find that the nearer (upper) end of the coil is N while the N 566 ELECTRODYNAMICS FIG. 432. Direction of In- duced Current. pole of the magnet is being inserted, and is S while the N pole is being withdrawn (Fig. 432). Inserting the S pole of the magnet makes the nearer end of the coil S; re- moving it makes this end N. Thus while either pole of the magnet is being inserted, the motion is opposed by the repul- sion of the nearer (like) pole of the coil; and while either pole is being withdrawn the motion is opposed by the attraction of the nearer (unlike) pole of the coil. Hence, in general, " The induced current is in such a direction as to oppose by its electromagnetic action the motion of the magnet or the coil (see Art. 463) which produces the induction." This is known as Lenz's law. To move the magnet against the opposing magnetic forces requires an expenditure of mechanical energy, which becomes the electrical energy of the induced current. The amount of energy thus transformed in the experiment is exceedingly small; but the process is of the same nature as that which takes place in all dynamos, and in the largest is capable of generating electrical energy at the rate of several thousand horse-power. The principle involved is therefore of the greatest importance. Lenz's law is only a special case under the general prin- ciple of the conservation of energy. If the direction of the induced current were such that its magnetic action aided the motion which produces the current, a dynamo would run of itself when once started, and generate elec- trical energy out of nothing; or, in other words, it would be a " perpetual-motion machine." ELECTROMAGNETIC INDUCTION 567 463. Current Induction by a Current. Results sim- ilar to the above are obtained when a long, slender coil, in which a current is flowing, takes the place of the magnet in the experiment. This coil is called the primary coil, and its current the primary or inducing current. The larger coil is known as the secondary coil, and the current induced in it is often called the secondary current. The primary current is supplied by a battery. If the primary coil is small enough to go inside the sec- ondary, the results obtained when it is inserted or with- drawn are the same as in the corresponding case with the magnet. Thus if the lower end of the primary coil is N, thrusting it in induces a current which makes the upper end of the secondary coil N. Hence in this case the direc- tion of the induced current is opposite to that of the pri- mary current, and it is therefore called an inverse induced current. A direct induced current is one whose direc- tion round the coil is the same as that of the primary current. Experiment shows that the induced current is inverse when either pole of the primary coil is inserted, and direct when it is withdrawn. It will be useful to remember that, to an observer looking in the direction of the lines of force of the inducing magnetic field, an inverse induced current flows counter-clockwise round the coil, and a direct induced current clockwise. If the primary circuit is closed or broken while the pri- mary coil remains at rest within the secondary, the induc- tion is the same as when the primary coil is inserted or removed with the current flowing; for this is only another way of changing the magnetic field within the secondary. The induced currents are in all cases much stronger when the primary coil contains a soft iron core; for the iron greatly increases the strength of the magnetic field. 568 ELECTRODYNAMICS A review of all the cases considered will show that the direction of the induced current is given by the following general law: An increase in the strength of the magnetic field within a closed circuit induces an inverse current, and a decrease in the strength of the field induces a direct current. These directions are such that the magnetic action of the induced current is always in agreement with Lenz's law. 464. Magnitude of the Induced E.M.F. An induced current is due to an induced E.M.F., and, in a given cir- cuit, is proportional to it. The E.M.F. induced in a given circuit is proportional to the rate of increase or decrease of the magnetic field within the circuit. This can be shown qualitatively by varying the speed with which the primary coil or the magnet is thrust into the secondary. With the primary coil and an iron core, the deflection of a sensitive galvanometer is very large when the motion is rapid; but it becomes less and less indefinitely as the coil is moved more and more slowly. With a given pair of coils and a given primary current, the greatest possible E.M.F. is in- duced by breaking the primary circuit while the primary coil and the iron core are at rest within the secondary; for the magnetic field within the secondary is very strong to begin with, and breaking the circuit removes it in the quick- est possible way. The induced E.M.F. at "break" is so great, even with coils of ordinary size, that a distinct shock is received from the secondary when its terminals are touched with the fingers. Other conditions remaining the same, the induced E.M.F. is proportional to the number of turns in the secondary coil. For any change in the magnetic field induces an E.M.F. in each turn, just as if the other turns were not present. Primary coils are commonly made with a relatively small ELECTROMAGNETIC INDUCTION 569 number of turns (from 100 to 200), and secondary coils with many thousand turns. With such coils, the induced E.M.F. is relatively high, as indicated by the fact that shocks may be obtained from them. If the primary cur- rent is sent through the larger coil and the smaller one is used as the secondary, the induced E.M.F. will be low. 465. Self-induction. Joseph Henry seems to have been the first to observe that a brilliant spark occurs when a circuit containing the coil of a large electromagnet is broken. If one end of such a cir- cuit is joined to a file and the free end of the other wire is drawn over the file (Fig. 433), the circuit is rapidly closed and broken, producing a shower of brilliant sparks. Without the electromagnet in the cir- cuit, the sparks are very feeble. The effect of the coil is due to induction. When the circuit through the coil is broken, the core instantly loses its magnetism, and a direct E.M.F. is induced in the coil, just as if a strong magnet were with- drawn from it. This induced E.M.F. may be hundreds of times greater than that of the battery which supplies the current. Its effect is to prolong the current after the break by driving it across the gap, thus producing the spark. With even a small coil, such as that of a small elec- tric bell, the in- duced E.M.F. at break is great enough to give a shock, if the bare wires are held in the moistened fin- gers when the cir- cuit is broken; and with a large coil the shock may be painfully strong, although the primary current is sup- plied by a single cell. Whenever the strength of the current in a coil is changing, its magnetic field is also changing; and the changing field reacts induct- FIG. 433. Self-induction. 570 ELECTRODYNAMICS ively on the coil, just as it does on a secondary coil when one is pres- ent. If the current is increasing, as at the instant when the circuit is closed, the induced E.M.F. is inverse and opposes the primary or battery E.M.F. Its effect is to retard the growth of the current, which therefore requires a fraction of a second to gain its full value. When the circuit is broken the current falls to zero very suddenly; and, since the induced E.M.F. is proportional to the rate of change of the magnetic field, it may be thousands of times higher at " break " than at " make." The inductive action of a changing current on itself is called self- induction, and the current due to self-induction at break is called the extra current. Self-induction in a coil increases with the number of turns, and is enormously increased by the magnetic action of a soft iron core. A coil with a core, placed in a circuit for the purpose of producing a spark at break, is called a spark coil. Spark coils are used in battery circuits for lighting gas jets, and for igniting the explosive mixture in some gas and gasoline engines. 466. The Induction Coil. The induction or Ruhm- korff coil (Fig. 434) is an instrument for generating induced /? G FIG. 434. Induction Coil. currents of very high potential. A simplified diagram of the essential parts is shown in Fig. 435. These are an iron core, AB, a primary coil consisting of one or two layers of turns of large insulated wire, a secondary coil of very ELECTROMAGNETIC INDUCTION 571 fine wire, well insulated and often many miles in length, an automatic make-and-break device or current inter- rupter, CD, which is included in the primary circuit, and a condenser, E. There is generally also a device, called a switch, for reversing the current through the primary coil without changing the battery connections. When a battery current is sent through the primary coil, it magnetizes the iron core, and the core attracts the iron block, C, which is supported near the end of the core upon a spring. This spring is the movable part of the interrupter, and the primary current passes between it and the point of a screw, D, against which it rests. By the attraction of the magnetized core the spring is drawn away from the point, breaking the circuit. The core instantly loses its mag- netism, and the spring flies back again, closing the circuit. (This action is the same as in the electric bell.) The pri- mary circuit is thus closed and broken many times every second, causing alter- nately an inverse and a direct induced E.M.F. in the secondary coil. The ends l 1 } e' of the secondary coil are FlG - 43S '~ Diag c r f 1 of an Induction connected with the binding posts, R and G, and may be extended, by means of rods or wires attached to the posts, until the gap, H, is made as small as desired. When this gap is not too great, a spark passes between the terminals with every interruption of the primary current. The maximum length of the spark depends upon the induced E.M.F., and increases with the number of turns in the secondary coil and with the rate of change of the magnetic field. Since this change is much more abrupt 572 ELECTRODYNAMICS at break than at make (Art. 465), the spark passes only at break. The purpose of the condenser (Art. 398) is to prevent or at least diminish the spark in the primary circuit at the interrupter, and thus to increase the abruptness of the break. This it does by serving as a temporary reservoir into which the extra current flows, instead of jumping across the gap. The induced E.M.F. of a coil may be roughly estimated at 25,000 volts per centimeter of the longest spark that it will give. To pro- duce a spark only a few centimeters in length requires many thou- sands of turns in the secondary. The sparking distance of the largest coils runs from two to three and a half feet, and the E.M.F. is from 2,000,000 to 3,000,000 volts or even higher. For large coils the inter- rupter is a separate mechanism, different from that described above. The induction coil has many important uses. The so-called physi- cian's battery is a small induction coil, operated by a battery current. The handles which the patient holds are the terminals of the second- ary. Large induction coils are used in operating X-ray tubes (Art. 504) and in wireless telegraphy (Art. 499) . A simple form of induc- tion coil, without an interrupter, is used in the telephone (Art. 483) ; and another form, known as the transformer, is an indispensable factor in the transmission of electrical power over long distances (Art. 479). The induction coil is generally preferred to the primary or spark coil for igniting the charge in gas and gasoline engines, the prin- cipal advantage being that the secondary spark will jump across a gap between the fixed terminals of a "spark plug," whereas the pri- mary spark requires a make-and-break device within the cylinder of the engine. 467. Current Induction in the Dynamo. An induced E.M.F. can be generated either by the motion of a magnetic field within a stationary coil, or by the motion of a coil in a stationary field. The latter method is employed in most forms of dynamos. The small dynamo shown in Fig. 436 is an example. The current is generated in a set of coils, A, which are wound on an iron core to increase ELECTROMAGNETIC INDUCTION 573 FIG. 436. Small Bi-polar Dynamo. N, S, Poles of Field Magnet; C, Coil of Field Magnet; A, Armature; P, Driving Pulley. the induction. The coils and core together are called the armature. The armature is nearly surrounded by the curved poles, N and 5, of an electromagnet. This is termed the field mag- net, since it produces the magnetic field in which the armature turns. The coil , C, of the field magnet is called the field coil. It is connected as a shunt to the external circuit (in this dynamo), and takes a small part of the cur- rent generated in the armature. The armature is driven at high speed by a belt running over a pulley, P. The rota- tion of the armature coils in the field of the electromagnet causes the induction. The induction in each coil of the armature runs through a complete cycle of changes during each revolution. The nature of this cycle can be determined with the aid of Fig. 437, which represents a single loop of an armature coil, AB, turning in the direction of the curved arrow, between the vertical poles, N and 5, of the field magnet. The direc- tion of the lines of force of the field is from N to S, or from left to right in the figure. We shall suppose that the coil starts from the horizontal position and makes one complete turn, in four stages of 90 each. At the start the plane of the coil is parallel to the lines of force, and hence none of them extend through it. As the coil turns into the vertical position, it takes in a constantly increas- ing number of the lines. The effect is the same as if the N pole of a magnet were thrust into the coil from the left 574 ELECTRODYNAMICS JT FIG. 437. Simple Alternator. side; hence the direction of the induced E.M.F. is counter- clockwise, as indicated by the arrow-heads (Art. 463). As the coil turns from the vertical to the horizontal position through the second 90, the portion of the field extending through it d e - creases to zero, and the induction is the same as if the N pole of a magnet were withdrawn from the coil. Hence the direction of the current is clockwise. As the coil turns from the horizontal to the vertical position again, through the third 90, the cross-sec- tion of the field included within it again increases to a maxi- mum, and the current flows counter-clockwise, as at first. But since the opposite side of the coil now faces toward the observer (the sides A and B being interchanged), the direc- tion of the current with respect to the coil itself is really the same as it was during the second 90 of the revolution. In other words, the current reverses its direction in the coil as the coil passes the vertical (or the position at right angles to the lines of force), but not when it passes the horizontal (or the position parallel to the lines of force) . Hence a second reversal of the current takes place as the coil begins the fourth quarter of the revolution, i.e. when the coil passes the vertical with side A below. Thus a continuous rotation of the coil induces an alternating current in it, the reversal of the current taking place twice during each revolution, as ELECTROMAGNETIC INDUCTION 575 the coil passes through the position at right angles to the lines of force. (If the poles of the field magnet were hori- zontal, and the lines of force of the field vertical, as in Fig. 436, we should have to substitute vertical for horizontal and horizontal for vertical throughout the above discussion, for the induction depends upon the relative position and motion of the coil and the lines of force.) 468. The Dynamo Rule. The direction of the induced E.M.F. in any part of a circuit, as the sides of a rotating coil, can be readily determined by the dynamo rule, which is as follows: Extend the thumb and the first and second fingers of the right hand at right angles each to each; then turn the hand so that the first finger points in the direction of the lines of force of the inducing field, and the thumb in the direction in which the wire is moving across the lines of force. The second finger will then point along the wire in the direction of the induced E.M.F. (Fig. 438). Thus for the side A of the coil in Fig. 437 the forefinger points toward the right (from the ./V" pole to the 5" pole of the mag- *net), and the thumb upward. The second finger then points FlG " 438. -Right-hand Rule for Deter- mining Direction of Induced Current, in the direction of the arrow- head, or away from the observer. For the side B the forefinger points to the right and the thumb downward. The second finger then points in the direction of the arrow-head, or toward the observer. According to the rule, the induced E.M.F. in either side of the coil continues in one direction as long as the wire continues in one direc- tion (upward or downward) across the lines of force, and reverses at the instant when the wire starts across the lines in the opposite di- rection. Hence, as already stated, the current reverses its direction in the coil as the coil passes through the position at right angles to the lines of force. The induction in the wire at the ends of the coil is across the wire, and not in the direction of its length; hence it has no effect on the current. 576 ELECTRODYNAMICS 469. Arbitrary Use of the Term "Line of Force." Variation of the Induced E.M.F. in a Rotating Coil. A line of force, according to our previous use of the term (Art. 375), is simply a mathematical line indicating the direction of a resultant force. It has no real existence. In this sense a line of force passes through every point in a magnetic field; hence their number is umlimited or infinite. But it is custom- ary to speak of lines of force as if they really existed, and were pres- ent in limited numbers throughout a magnetic field, in proportion to the intensity. It is arbitrarily assumed that, in a field of unit intensity, there is one line of force per square centimeter of the cross- section perpendicular to the lines, that in a field of twice this inten- sity there are two lines per square centimeter, etc. This is only a mathematical fiction ; but it serves a useful purpose as a basis for stat- ing electromagnetic relations in simple and definite terms. In the first place, the number of lines of force per sq. cm. of the plane perpendicular to the lines becomes the measure of the intensity of the field (Fig. 439) ; and hence the induction in any circuit is pro- portional to the rate of change in the number of lines of force passing through it. The E.M.F. induced in any part of a circuit, as a certain length of wire, is proportional to the number of lines of force which it crosses or "cuts" in a second. An armature coil, turning at a FIG. 439. Strength of Magnetic uniform rate, cuts the lines of force Field is Measured by Number o f the field most rapidly when its of Lines of Force per Square Centimeter of Cross-section. sides are moving at right angles to the lines, e.g. when the coil AB (Fig. 437) is passing through the horizontal position. The induced E.M.F. is then at a maximum. As the coil turns through 90 from this position, its sides cut the lines of force more and more obliquely, and hence at a diminishing rate; and the induced E.M.F. decreases in proportion. As the coil passes the perpendicular to the lines of force, its sides are moving parallel to the lines and are cutting none; hence the induced E.M.F. is then zero. In one revolution, starting with the coil perpendicular to the lines, the induced E.M.F. rises from zero to a maximum, decreases to zero, rises to a maximum in the opposite direction, and again decreases to zero. ELECTROMAGNETIC INDUCTION 577 470. Devices for Leading the Current through an Ex- ternal Circuit. The alternating current generated in an armature coil may be led off through an external circuit either as an alternating current or as a direct current, depending upon the mechanism by which the coil is FIG. 440. Side and End View of Single Loop and Commutator. connected with the terminals of the dynamo. In the alternating-current dynamo, or alternator, this mechanism consists of two copper collecting rings, c and d (Fig. 437), mounted on the armature shaft. They are insulated from the shaft and from each other, and an end of the coil is connected with each. The rings are connected with the FIG. 441. Direct-current Dynamo with Single- loop Armature. external circuit, HG, by means of stationary terminals or brushes, e and /, consisting of copper strips or blocks of carbon. As the rings turn with the shaft, the brushes make a sliding contact with them. The current is of the same character in the external circuit as in the coil. 578 ELECTRODYNAMICS In the direct-current dynamo a device known as a com- mutator takes the place of the collecting rings. For a single-coil armature this consists of a copper ring split in halves, c and d (Figs. 440 and 441), with the parts insu- lated from each other. Fig. 440 shows both a side and an end view of the coil and the commutator. In the other figure are end views, showing the position of the brushes, e and /. The curved arrow indicates the direction of rota- FIG. 442. Direct-current Dynamo with Ring Armature. tion, and the other arrows indicate the direction of the current. The small cross within a small circle represents the tail of a receding arrow, and indicates that the current in that side of the coil flows from the observer. The dot within a small circle represents the head of an approaching arrow. The brushes must be set on opposite sides of the commu- tator, in such positions that each changes contact from one ELECTROMAGNETIC INDUCTION 579 commutator segment to the other at the instant when the current changes its direction in the coil. Thus each seg- ment is in contact with brush/ while it is positive, and with brush e while it is negative. The current in the external circuit is therefore always in one direction ; but, if generated in a single coil, as in the illustration, it is pulsating or inter- mittent, rising to a maximum and falling to zero twice during each revolution of the coil. 471. The Direct-Current Dynamo. The E.M.F. in- duced in the armature of- a dynamo is proportional jointly to the total number of turns in all the coils, to -.--. the strength of the field, ~-j-\~- and to the rate of rota- ::i: : _ tion. The number of -_-_-_"_ turns varies with the Irrrl- size of the machine, and :_ : 5 with the desired voltage. For a Small Current at FlG - 443- Lines of Force through a Ring . Armature. high voltage the arma- ture is wound with many turns of small wire; for a large current at a low voltage it is wound with fewer turns of large wire. In any case the winding is evenly distributed round the coil, in order that the induction may be con- stant. This produces a steady direct current, instead of a pulsating one. Armatures are of two principal types, known as ring armatures and drum armatures. In the former the iron core is a ring (Fig. 442) ; in the latter it is a cylinder (Figs. 436, 444, and 445). The winding of an eight-coil ring armature is plainly shown in Fig. 442. The coils are all wound in the same direction and are joined 580 ELECTRODYNAMICS FIG. 444. Winding of a Four- coil Drum Armature. in series. Each junction between coils is connected with one of the commutator segments, a, b, c, etc. The lines of force of the field crowd into the iron ring, but do not penetrate the space within it (Fig. 443) ; hence they are cut only by the outer side of the coils, and there is no induction on the inside. By applying the dynamo rule (Art. 468) it will be found that the direction of the induced E.M.F. is as shown by the arrow-heads. The current thus flows toward the positive brush BI in both the right and the left halves of the armature. As each coil crosses from the left to the right side above or from the right to the left side below, the in- duction in it falls to zero and reverses. Hence the induced E.M.F. is always in the right direction; and, as there are always three active coils on each side, it is practically constant. The coils of a drum armature surround the entire core (Fig. 444), and induction takes place on both sides, as explained in connection with Fig. 437. The coils are joined in series and connected with the armature segments, as in the ring armature. The plan of a four-coil armature is shown in Fig. 444. The arma- tures of commercial dynamos usually have from thirty to one hundred coils. The dynamos thus far considered are bipolar, i.e. they have one field magnet with two poles. Four-pole dynamos are more common (Figs. 445 and 446). In these the poles are alternately N and 5, and the current reverses in each conductor as it passes from one pole to the next, or four times in-each revolution. FIG. 445. Parts of a Four-pole, Direct-current Dynamo or Motor. ELECTROMAGNETIC INDUCTION If an armature core were made of a single piece of iron, currents would be induced in it as in the coils. Such currents are worse than useless, for energy is wasted in generating them, and, be- sides, this energy is trans- formed into heat in the core. If this were permitted, the armature would become so hot as to injure or destroy the insulation. Armatures are therefore built up of thin disks of sheet iron, in- sulated from one another. These disks or lamina extend at right angles to the induced E.M.F.; hence the current tends to pass from one to the other, but is prevented by the insulation. Cores of this description are called laminated cores. FIG. 446. Dynamo of Fig. 445 Assembled. 472. Winding of the Field Coils. A direct-current dynamo supplies the current by which its field magnets are excited. In one type of machine the field coils consist of many turns of small wire, connected as a shunt to the FIG. 447. Shunt- wound Dynamo. FIG. 448. Series- wound Dynamo. FIG. 449. Com- pou nd - wound Dynamo. external circuit (Fig. 447). These are called shunt-wound dynamos. In the series-wound dynamo (Fig. 448) the 582 ELECTRODYNAMICS field magnets are wound with a few turns of large wire connected in series with the external circuit, and the entire current flows through them. In the compound- wound dynamo (Fig. 449) each field magnet has both a series and a shunt coil. Each variety of winding has certain advantages depending on con- ditions of use. Stated briefly, a shunt-dynamo, when provided with a suitable regulating device, is a constant-potential machine. It supplies a varying current at a constant potential on a circuit of varying resistance. A series dynamo, also provided with a regulating device, is a constant-current machine. It supplies a constant current at a potential which varies with the resistance of the circuit. Such machines are used for lighting arc lamps in series. Compound dyna- mos are used on constant-potential circuits where the current is very fluctuating, as in incandescent electric lighting and electric street- car service. In starting a dynamo, the current is at first very weak, since the field magnets retain very little magnetism; but this small current flowing through the coils of the magnets strengthens them, producing a stronger current. This mutual action continues until the magnets gain their full strength. 473. Transformation of Energy in the Dynamo. The energy of the current generated by a dynamo is derived from the mechanical energy expended in driving the arma- ture. A part of the energy supplied to a dynamo is lost in overcoming frictional resistance, and there is a further loss in the coils of the armature and the field magnets, owing to their electrical resistance. These losses, taken together, vary from about 15% in the smaller machines to 5% in the larger sizes. Hence the efficiency of a dynamo, as a device for converting mechanical energy into avail- able electrical energy, is from 85 to 95%. This means that the power required to run the armature of a dynamo at a given speed is from seven to twenty times greater while ELECTROMAGNETIC INDUCTION 583 the machine is generating a current than it is when the circuit is open. If the student will open and close the circuit of a small hand-power dynamo while he is running it, he will learn by personal experience that the armature carries a "load" while generating a current. The sound of the machine tells the story to all who are within hearing. With the circuit open, the moving parts emit a light, chattering sound; but on closed circuit the sound is deep and labored. The added resistance to the rotation of the armature when it is generating a current is due to opposing magnetic forces, developed according to Lenz's law (Art. 462). The in- duced current, by its magnetic action, opposes the motion which produces it. How this comes about can be under- stood by referring to Fig. 437. With the current in the direction indicated, the N side of the coil faces the N pole of the field magnet, toward which it is turning. The motion of the coil is therefore opposed by the repulsion of the N pole and also by the attraction of the 5 pole of the mag- net. In fact the magnetic forces tend to turn the coil the other way about. These opposing forces are further increased by the magnetized core of the armature. This can be better shown from Fig. 442. The current in the armature coils constantly magnetizes the iron core in right and left halves, with the south poles of the two semicir- cular magnets together at the top and their north poles together at the bottom. These poles are each constantly turning toward (but never reaching) the field pole which repels it and away from the pole which attracts it. 474. The Direct-Current Motor. The energy trans- formation in a dynamo is reversible. When a current is passed through the armature and field coils of a dynamo ELECTRODYNAMICS the armature revolves and is capable of driving machinery. The dynamo then becomes an electric motor, and converts electrical into mechanical energy. "Manufacturers sell their standard direct-current dynamos to be used either as generators or motors. It is only when the machines are built to be used for some special purpose that they can not be conveniently interchanged in their action." As shown above, the magnetic forces acting on the arma- ture of a dynamo tend to turn it in the direction opposite to that in which it is driven. Hence if a dynamo is sup- G M FIG. 450. Relation of Dynamo to Motor. plied with a current which flows through its field and arma- ture coils in thte same direction as the current which the dynamo itself generates, the direction of rotation will be reversed ; but if the current supplied flows in the same direc- tion through the armature coils and in the opposite direc- tion through the field coils, the rotation will be in the same direction. The latter case is shown in Fig. 450, in which G represents a dynamo supplying a current to a motor, M. Both are shunt-wound machines of identical construction. The current flows in the same direction in the two arma- tures; and this direction is such as to maintain a double 5 t ELECTROMAGNETIC INDUCTION 585 pole at the top of the core and a double N pole at the bot- tom. In the dynamo these poles are driven in opposition to the magnetic attractions and repulsions of the field poles; in the motor the armature is turned by the attractions and repulsions of the field poles. Since, in the figure, the N field pole of the motor is in the position of the 5 field pole of the dynamo, and vice versa, the armatures turn in the same direction. It should be noted that the current leaves the dynamo armature by the positive brush and the motor armature by the negative brush. 475. Transformation of Energy in the Motor. An E.M.F. is induced in the armature coils of a motor, when it is running, for the coils cut the lines of force of the field just as they do in the dynamo. By applying the dynamo rule to the motor diagram of Fig. 450 it will be found that this induced E.M.F. opposes the current which runs the motor. Hence it is called a counter or back E.M.F. This action can be shown with any small motor, driven by a bat- tery current (Fig. 451). An ammeter, A, placed in the circuit will show that the current is much smaller when the motor is running than it is when the armature is held at rest. When the motor is run at different speeds by varying the friction at the pulley, the cur- rent decreases as the speed increases. At the same time a voltmeter, V, will show that the fall of potential in the armature increases as the speed increases. When the armature is at rest, the fall of potential in it is due simply to the resistance of the coils. The added fall of potential when the motor is running is really the in- duced E.M.F., which is working against the current. Hence the current is reduced, as FlG shown by the ammeter. It is as if a smaller battery were placed in the circuit at this point, with its E.M.F. opposed to that of the principal battery. 586 ELECTRODYNAMICS If the armature of a motor is not permitted to turn, the energy expended in it is all converted into heat in over- coming the resistance of the coils, as in any other conductor; but when a motor is running, the greater part of the elec- trical energy (generally from 85 to 95% of it) is expended in overcoming the counter E.M.F. in the armature. It is this part of the energy which is transformed into mechan- ical energy by the motor in doing work; and the rate at which the work is done is measured in watts by the product of the current and the counter E.M.F., in agreement with equation 10, page 555. v 476. Starting a Motor. Motors for industrial use are operated on constant-potential circuits. These may be no- volt or 2 20- volt lighting circuits, or separate power circuits, usually at 500 volts. The latter is the usual voltage for street-car service. The armature resistance of motors is very small, usually only a fraction of an ohm ; hence if the full voltage of the circuit were applied to a motor in start- ing it, there would be a suddden rush of current, amounting to several hundred amperes, and the armature would be ruined by overheating. Hence the current is turned on gradually through a starting box, which contains a number of resistance coils in series with each other and with the armature. As the motor gains speed, these resistance coils are cut out, one after the other, by moving the lever arm of the starting box. Finally all of the box resistance is cut out, leaving the armature connected directly to the circuit; for the back E.M.F. is sufficient to prevent an excessive current when the motor is running at full speed. The use of a starting box may be observed on any electric street car. The entire regulating mechanism is contained in a large, up- right iron box, and is called a controller. In addition to the resist- ance coils of an ordinary starting box, the controller contains a set of switches by which the two motors of the car can be joined either in series or in parallel, and by which also the field coils of each motor can be joined either in series or in parallel. Occasionally a motorman will cut out the starting resistances too quickly. The motors are protected against this mischance by an automatic circuit breaker ELECTROMAGNETIC INDUCTION 587 which breaks the circuit by the action of an electromagnet when the current is greater than it should be. This device is placed under the roof of the car, near the motorman. Its action causes a loud noise and a spark, and is sometimes the occasion of needless alarm to pas- sengers unacquainted with such matters. 477. The Alternating-current Dynamo. For reasons presented in the following articles, it is possible to transmit electrical energy for power purposes over distances of 100 to 200 mi., by means of alternating currents, while with direct currents only short distances are practicable. The alternating-current dynamo, or alternator, is, therefore, of very great industrial importance. The study of alter- nating currents and alternating current machinery is very extensive and can only be touched upon in an elementary course in general physics. A large alternator (Figs. 452 and 453) has many field poles, always an even number. They are alternately N and S, the coils of all the N poles being wound in one direction and those of the 5 poles in the opposite direction. The exciting cur- rent is direct, and may be supplied from a direct-cur- rent winding on the armature of the alternator itself, or by a separate ma- chine. The arma- ture is wound with as many coils as the number of field poles, and alter- nate coils are oppo- sitely wound, as shown in the fig-^ ure. As the arma- ture turns, the induced E.M.F. is in the same direction round all the coils which are passing N field poles, and in the opposite direction round all the coils FIG. 452. Diagram of Multipolar Alternator. 588 ELECTRODYNAMICS which are passing 5 field poles. But, owing to the opposite winding of three two sets of coils, the induced E.M.F.'s are all in the same direction through the armature circuit (i.e. through the armature from K 2 to Ki, or from Ki to X 2 ). The ends of this circuit are joined to collecting rings, as shown; and the current passes off to the external circuit by one and returns by the other. The current reverses in all the coils at the same instant, as they turn from one field pole to the next; and the current reverses in the external circuit with each reversal in the armature. With an eight-pole machine driven at the rate of 15 revolutions per second, the num- ber of alternations per second will be 8 X 15 = 120. Since the current runs through a complete series of changes, or one cycle, during the interval between one reversal and the second one follow- ing, there will be, in the present instance, 60 cycles or current waves per second. Alternating currents for electric lighting are usually 6o-cycle cur- rents. For general power purposes the frequency is sometimes as low as 25 cycles per second. Fig. 453 represents a modern alter- nator, direct-connected to the engine which runs it. The term "direct-connected" means that the armature of the dynamo is mounted on the shaft of the engine. Another method is to connect the dynamo and engine by means of a belt. A generating unit includes both the dynamo and its engine or other source of power. The combination of a dynamo and a turbine water-wheel is termed a hydro-electric unit. Generating units of 5000 to 10,000 horse power are common in modern electrical power stations. 478. Transmission of Electrical Energy. In transmitting elec- trical energy over a line for use at a distance, a certain percentage of it is lost as heat, owing to the resistance of the conductor. This loss limits the distance over which it is practicable to operate power lines. The conditions which determine the loss in transmission are disclosed by a comparison of the numerical examples presented in the table below. In this table EI denotes the E.M.F. generated, E 2 the E.M.F. at the end of the line, C the strength of the current, and R the resistance of the line. The power generated is EiC watts and the power delivered for use EzC watts. The fall of potential in the line is E\~E^ volts, and the power lost in transmission (EiE 2 )C or CzR watts. ELECTROMAGNETIC INDUCTION 589 Given: i 2 3 4 E. M. F. delivered (E 2 ), 500 volts 500 volts 1000 volts 5000 volts Current (C). 10 amperes 20 amperes 10 amperes 2 amperes Resistance of line (R), 25 ohms 25 ohms 25 ohms 25 ohms Then: Power delivered (E 2 Q, 5000 watts 10,000 watts 10,000 watts 10,000 volts Power lost (C 2 R), 2500 watts 1 0,000 watts 2500 watts 100 watts Power generated (]C), 7500 watts 20,000 watts 1 2, 500 watts 10,100 watts Fraction of power lost, 33-3% 50% 20% i % (nearly) E. M. F. at the start (,), 750 volts 1000 volts 1250 volts 5050 volts It will be seen from the formulas and the numerical examples that a greater amount of power can be delivered over a given line by increas- ing either the current or the voltage at which it is delivered. The loss in watts is determined by the current and the resistance of the line, being equal to C 2 ft, and is independent of the voltage (examples i and 3 in the table). Thus if the power generated is increased ten- fold by a tenfold increase in the potential, the current remaining the same, the percentage loss in transmission is reduced nine tenths. In recent years rapid progress has been made in the utilization of water-power through the agency of high-potential currents, generated FIG. 453. Alternator Direct-connected to Steam Engine. in electrical power stations and transmitted to distant points for use in manufacturing, mining, and transportation. The voltage at 5QO ELECTRODYNAMICS which such lines operate determines the distance to which the power can be transmitted without prohibitive losses. Higher and still higher voltages are employed from year to year. Fifteen years or so ago the limit was 30,000 volts. At the present time various lines are operating at 60,000 to 100,000 volts. The most difficult problem to be solved in this development has been to provide efficient insula- tion. An insulator for a high potential line is made of the best porcelain, and must have a widely ex- tended surface thoroughly protected from moisture. The 60,000- volt insulator shown in Fig. 454 is 14 in. in diameter, 12 in. high, weighs 26 lb., and costs about $45. Such an insulator is required at each point of support of the line wire. One of the largest transmission lines in the FIG . 454. Porcelain Insu- WO rld is the Niagara-Syracuse-Auburn line, lator for 6o,ooo-volt Circuit. 1-1, , v which transmits 30,000 horse-power over a distance of 163 mi. The line in parts is designed to carry 60,000 horse-power. The Colgate plant, Yuba River, California, connects by way of Oakland and Mission San Jose to a line 222 mi. in length. This plant has a capacity of 15,000 horse-power, and there are over 100 sub-stations on 1375 mi. of circuit on the system. At McCalls Ferry on the Susquehanna River a dam and power station have recently been constructed at a cost of nearly $10,000,000. The power house is equipped with ten twin-turbine wheels, each with a capacity of 13,500 horse-power. Similar examples of power development on a large scale, in the United States and other countries, could be named by the score; but those mentioned will serve to convey some idea of the tremendous importance which modern electrical science has given to nature's perennial source of energy, run- ning water. 479. The Transformer. The currents generated by the dyna- mos of long-distance power lines do not leave the power-house. They are alternating currents, generated at a pressure of 2000 to 6000 volts, and are sent -through the primary of a huge induction coil of special design, called a transformer. A high-potential current is induced in the secondary coil of the transformer; and this is the current which is transmitted over the line to sub-stations, or transformer ELECTROMAGNETIC INDUCTION 591 houses, located near points where the energy is to be used. At a sub-station the line current passes through the high-potential coil of a transformer, inducing a current at a relatively low voltage in the secondary coil. The essential parts of a transformer are an iron core, and two coils having an unequal number of turns. A core in the form of a closed loop (a ring or a rectangle) is more efficient than a straight one, since it carries all the mag- netic lines of force through both coils (Fig. 455). An alternating current sent through either coil magnetizes the core first in one direction, then in the other; FIG. 455. Diagram of Trans- and these reversals of magnetism induce an alternating E.M.F. of the same frequency in the other coil. If this coil is on a closed circuit, an alternating current will be gener- ated in it. Disregarding a small percentage of loss in the transfor- mation, the E.M.F.'s of the primary and induced currents are in direct proportion to the number of turns in the two coils. If the primary current is sent through the coil of fewer turns, the transformation will be from lower to higher potential. This is the action of the step-up trans- former used at generating stations. Transformation from higher to lower potential is effected by sending the primary current through the coil having the greater number of turns. This is the action of the step-down transformer used at sub-stations. A first-class commercial trans- former (Fig. 456) will thus transfer FIG. 456. Commercial Transformer , o . ,, . . . , Removed from its Case. from 93 to 98% of the electrical energy from one circuit to another completely insulated from it. If Ei and Ci denote the primary E.M.F. and current respectively, and E 2 and C 2 the induced E.M.F. 59 2 ELECTRODYNAMICS and current, then the pawer of the one is Eid and the power of the other E 2 C 2 . Disregarding the small loss in transformation, EiCi = E 2 C 2 or Ei:E 2 ::C 2 :Ci; i.e. a transformer changes the E.M.F. and the current strength in reciprocal proportion. Thus if the E.M.F. is increased ten-fold, the induced current is one- tenth as great as the primary current, and vice versa. The core of a commercial transformer is rectangular, and is built up of thin plates of soft steel, like the armature core of a dynamo. The coil on each side contains both primary and secondary windings. Low Pressure Mains Alternator imps Fig. 457. Electric Light Circuit. the one surrounding the other but insulated from it. The winding of fewer turns is made of the larger wire, since it carries the larger current. Transformers are inclosed in iron cases for protection. In cities where alternating currents are used for house lighting, the current is generated at a relatively high pressure, generally either 1 100 or 2200 volts. This current is distributed over high-pressure mains to convenient points, where step-down transformers are lo- cated (Figs. 457 and 458) ; and the secondary coils of the transformers supply the current for the lamp circuits. 480. The Magneto-Telephone. The method of using a modern telephone is a very simple matter indeed. The receiver is placed to the ear, a number is spoken into the transmitter, and in a moment two persons, perhaps many miles apart, are talking to each other as if they were in the same room. It is almost as simple as pushing a button to " turn on " an electric light. But simplicity of use in either case is the net result of a very complex and wonder- ful application of scientific principles. A first glance at the intricate mechanism of a complete telephone system, ELECTROMAGNETIC INDUCTION 593 including the subscriber's telephone and the central exchange, where any one of several thousand subscribers can be connected in less than ten seconds with any other, gives the impression that no one but an expert could make anything out of it all. But a close inspec- tion of any single detail will show that it is only an ap- plication of some one or more of the principles already familiar to the stu- dent. T book will permit us to con- sider only the main points. The simplest possible electric telephone line consists of two receivers, permanently joined by wires (Fig. 459). Each receiver serves also as a transmitter. This is the original telephone line, invented by Alexander Graham Bell in 1876. The working parts of each instrument are a permanent magnet, M, a coil of fine wire, C, and a disk of thin sheet iron, D. The disk is supported all round its edge, and is free to vibrate like the head of a drum or the tym- panum of the ear. When the speaker's mouth is close to the disk at either end of the line, the disk is forced to vi- brate in* unison with r *f t fV/o FIG. 458. Transformer on Electric Light Pole. H, H, high potential wires; L, L, low potential wires; T, trans- former. the sound waves which beat upon it. Being of soft iron, it is magnetized more or less as it approaches the pole of the magnet or recedes from it in vibrating. This varies the strength of the magnetic FIG. 459. Diagram of Original Telephone Line. 594 ELECTRODYNAMICS field within the coil of wire, and induces a current in it, first in one direction, then in the other, in rapid succession. This current, flowing through the coil of the other tele- phone, alternately increases and decreases the strength of its magnet. When the magnet is strengthened, it draws the disk more strongly; when it is weakened, the disk springs back. The disk of the receiving telephone thus repeats the movements imparted by the sound waves to the disk of the transmitting telephone; and, by its vibra- tion, it reproduces the sound with remarkable accuracy. A telephone line of this sort does not require a battery. It works successfully over short distances; but over a long line the current is too weak to reproduce intelligible speech. The Bell telephone has continued in use as a receiver; but as a transmitter it soon gave place to a device based on an entirely different principle. The parts of a modern receiver are shown in Fig. 460. The mag- net is in the form of an elongated U, in order that both poles may act on the disk. A short, flat bar of soft iron is fastened to each pole, and about it is wound a coil of fine wire. These iron cores are more sensitive to a varying current in the coils than permanently magnetized steel would be. The coils are joined to the line circuit through a flexible conducting cord, which carries strands of small wire. The working parts are inclosed in a hard rubber case, the cap of which, when screwed on, holds the disk in position. The telephone re- FIG. 460. Modern Telephone ceiver is one of the most sensitive Receiver, Dissected. . , , T ,. instruments ever invented. In ordi- nary use it takes only one ten-thousandth of an ampere, and a cur- rent one thousand times smaller than this produces audible sound. ELECTROMAGNETIC INDUCTION 595 481. The Microphone. The principle of the telephone transmitter is beautifully illustrated by the simple micro- phone, from which it was developed. This instru- ment, as its name implies, reproduces faint sounds with increased intensity. Its action depends upon the fact that the electrical resistance of a loose contact FIG. 461. Microphone in between two conductors varies with the pressure. A contact of carbon with carbon gives the best results. A common form of microphone is shown in Fig. 461. The pointed ends of a carbon rod, C, rest loosely in cavities in carbon supports, A and B. These supports are fixed to a small sounding board, and are joined in series with a Bell receiver and a battery of one or two cells. Vibrations of the sounding board are transmitted to the carbon rods, causing a rapid change of pressure at their points of con- tact. A slight increase of pressure enlarges the area of contact and decreases the resistance. This permits a larger current to flow. When the pressure is lessened, the resistance increases and the current is reduced. The fluctuating current varies the strength of the receiver mag- net, causing the disk to vibrate in unison with the sounding board but with a greater amplitude. An inaudible rub- bing or tapping of the sounding board with the finger causes the receiver to emit a loud, rattling sound, and a watch lying on the board is heard very distinctly. The microphone was invented by David E. Hughes, an English physicist, in 1878. Edison made a similar discov- ery of the action of loose carbon contacts in the same year; 596 ELECTRODYNAMICS and it was not long before various inventors had devised practical telephone transmitters based on this principle. 482. The Granular-carbon Transmitter. Modern transmitters are of the granular-carbon type. The details of construction differ in different makes; but the general form shown in Fig. 462 is typical. M is the mouthpiece, D the vibrating diaphragm. The latter is generally of aluminum, and is held in position by springs not shown in the figure. C is a small metal cup, covered by a mica diaphragm, M'. This flexible cover is attached to the principal diaphragm, Z>, by a short screw, S, and vibrates with it. The battery circuit connects with two carbon disks, E and E'. E is attached to the bottom of the cup and is stationary; E' vi- brates with 'the mica diaphragm. The space between the disks is loosely filled with small carbon granules, which serve to conduct the current between E and E 1 . These granules are subjected to a varying pressure, due to the vibration of E'; and as there are many points of loose contact, the variation of the resistance is large. The corresponding fluctuations of the battery current are there- fore much greater than the feeble currents generated in the Bell receiver, when used as a transmitter. 483. A Complete Telephone Line. The simplest tele- phone line, complete in itself, is one which is used only for communication between two points. This requires at each end of the line a transmitter, a receiver, an electric call-bell, a battery, and switching devices for making the necessary connections. This apparatus, with the excep- tion of the battery, is all assembled in the battery-call telephone (Fig. 463). The term "battery-call" signifies that the battery supplies the current for ringing the bell as well as for talking. (This is possible only on short lines. On a long line the bell requires a more powerful source of ELECTROMAGNETIC INDUCTION 597 current, and a small hand-power dynamo or magneto- generator is used for this purpose.) Fig. 464 is a diagram of the connections in a battery-call tele- phone. When the receiver is on the hook, its weight pulls the hook down, bringing, it into electrical contact with a terminal at a. This connects the bell with the line, for the purpose of receiving signals. A signal is sent by press- ing the button, B, which brings the spring, K, into contact with the terminal, g. This con- nects the battery with the line, and rings the bell at the other station. When the receiver is taken from the hook, the hook is pushed up by a spring. This disconnects the bell at a, closes the local battery circuit at d, and connects the receiver with the line at h. The F l G 463. Battery-call battery circuit includes the transmitter and the primary winding, P, of a small induction coil. The secondary winding, S, is included in the line circuit, in series with the receiver. Line or Ground FIG. 464. Diagram of a Battery-call Telephone. The induction coil serves as a miniature step-up transformer. It may be dispensed with on short lines; but it is an advantage if not a necessity on long lines, owing to the greater resistance to be overcome. 598 ELECTRODYNAMICS 484. The Telephone Exchange. The individual subscribers' lines of a telephone system or exchange all terminate in a switchboard at the central station. This switchboard is so contrived that the oper- ators can connect any line with any other by means of conducting cords. At each end of a cord is a plug, provided with metal terminals. When the plug is inserted in a small hole, about the size of a lead pen- cil, its terminals are brought in contact with two springs, which form the terminals of the subscriber's line. On a large switchboard as many as 5000 to 10,000 such terminals are within the reach of a single operator. The details of a modern switchboard and the auxiliary apparatus necessary for its operation are numerous and complicated. A stor- age battery at the central station supplies the " talking current" for all lines of the system, and a dynamo supplies an alternating current for ringing the bells. A subscriber calls " central " simply by remov- ing the receiver of his telephone from the hook. The current which then flows over his line operates a relay in the central station. This closes a local circuit through a tiny electric lamp, mounted in the switchboard beside the terminal of the subscriber's line. The oper- ator connects her telephone with the line indicated by the lamp, learns what number is wanted, rings the bell on that line, then con- nects it with the line of the calling subscriber. The lamp is auto- matically cut out when connection is made with the line; and when the receiver of either telephone is hung up, another lamp lights, as a signal for the operator to disconnect. In an automatic exchange no operators are required, as all connec- tions are made by automatic devices, operated by electromagnets. 485. Danger from Electric Currents. Electricity has come to be such an important factor in daily life that every one should know its real dangers, and should not be troubled with imaginary ones. Telephone, telegraph, and incandescent lighting currents are not at all dangerous. At the worst a lighting current at no volts will give an unpleasant shock, and a current at 220 volts a severe "jolt." The exposed metal parts of sockets and lamps are insulated from the circuit; hence in ordinary use there is no opportunity to come in con- tact with the current. The circuits for street-car lines and for general power purposes are commonly operated at 500 volts. Contact with bare wires at this voltage is distinctly dangerous, especially in the ELECTROMAGNETIC INDUCTION 599 case of alternating currents, but seldom fatal. Such currents, how- ever, will kill a horse. The distributing mains which run to the trans- formers on incandescent lighting systems, the circuits of street arc lamps, and long-distance transmission lines in general carry currents at 1000 volts or higher. Contact with such wires means death, as a rule; and linemen who have occasion to climb the poles on which the wires are strung are now and then victims of this mischance. It sometimes happens that a wire carrying a high-tension current breaks and falls to the ground, or comes in contact with telephone or incandescent light wires. Such an accident is a serious menace to life and property; but, fortunately, it is a very rare occurrence. Every electric circuit should be regarded with suspicion, unless its character is known. A wire carrying a deadly current differs in no wise in appearance from one which is perfectly harmless. When current is taken from a lamp socket for ironing or cooking, the circuit should never be broken by turning the key of the socket. A special plug switch, provided for this purpose, should always be used. The parts of a lamp switch are designed to break only a small current, such as is taken by a lamp. A larger current is very likely to form an arc and burn out the switch. If in any emergency it is necessary to handle a "live wire" at a dangerous or disagreeable voltage, it should be remembered that insulation protects. A few thicknesses of dry cloth between the wire and the hand renders 500 volts harmless. Again, the current that will pass through the body depends upon the resistance of the cir- cuit of which the body forms a part. If only one wire of the circuit is touched, the current passes through the body to the ground. If any fairly good insulator is interposed in this path, as when the person is standing on a dry board, the current is very small compared with what it would be if he were standing on the ground, especially when the ground is damp. Not the slightest shock is felt on touching one wire of a 2 20- volt circuit, provided no other part of the body is in contact with a better conductor than wood; but if both wires are touched at the same time, the shock is rather severe, for the body then receives the full voltage. Knowing the danger of high-potential currenfs, it is often a matter of surprise to students that an electrostatic machine or an induc- tion coil which works at 30,000 volts, or even higher, can be treated as a plaything without fear of serious consequences. The reason is 600 ELECTRODYNAMICS that these machines develop a high potential only on open circuit. Touch the knobs of an electrostatic machine and the potential in- stantly falls practically to zero (Art. 405). Touch the terminals of a small induction coil and the potential falls to a very moderate and harmless value. On the other hand, the voltage of a lighting or power current is maintained on closed circuit, by the action of a powerful dynamo. PROBLEMS 1. If a dynamo is run by a motor, could the current generated by the dynamo be used to run the motor? 2. In what respects is a D'Arsonval galvanometer like a motor? In what respects does it differ? 3. How would you reverse the direction of rotation of a motor? 4. The lamps of a street car are lighted by the 5oo-volt current taken from the trolley wire. How must loo-volt lamps be connected for this purpose? 5. A dynamo generates a current of 50 amperes, at a pressure of 500 volts, on a line whose resistance is 2 ohms. Find (a) the power generated, (b) the power lost in the line, (c) the power delivered, (d) What per cent of the power is lost in the line? 6. What per cent of the power would be lost in the above line if the dynamo generated 25 amperes at a pressure of 1000 volts? 7. A dynamo supplies current for lighting 5000 no- volt lamps, each taking .5 ampere. Allowing 10% loss in the wires and the transformers, what power in kilowatts must the dynamo generate? 8. The current taken from a trolley wire returns to the power house through the rails. Why is it impossible to get a shock from the rails? 9. If the primary coil of a transformer has 800 turns and the second- ary 200 turns, what voltage will be induced in the secondary by a primary voltage of 220? X. CHEMICAL EFFECTS OF THE ELECTRIC CURRENT 486. Electrolys^. If energy is given out in any phys- ical or chemical process, an equal amount of energy is taken in when the process is reversed. Thus, as we have seen, a gram of steam gives out 537 calories of heat in condensing CHEMICAL EFFECTS OF CURRENTS 601 at 1 00, and 537 calories must be supplied to a gram of water at 100 to vaporize it. Similarly water gives out heat in freezing and absorbs an equal amount of heat in melting. Energy is put into the spring of a watch in wind- ing it, and is paid out again as the spring unwinds itself. The solar energy utilized by a plant in separating the car- bon from the carbon dioxide of the air is recovered as heat when, in decaying or burning, the carbon of the plant unites with the oxygen of the air again (Art. 244). Since, under suitable conditions, various chemical changes produce electrical energy, as in electric cells of all kinds, it is reasonable to suppose that, by an expenditure of electrical energy, these changes could be reversed. In the simple cell, for example, the current is generated by the action of dilute sulphuric acid on zinc, the product, zinc sulphate (ZnSO 4 ), remaining in solution (Art. 407). Reversing the process, metallic zinc is obtained from a solution of zinc sulphate by means of an electric current, as in the follow- ing experiment. A bent tube (Fig. 465) is partly filled with the solution. A narrow strip of platinum, soldered to a wire, is placed in the liquid in one arm of the tube and connected FIG. 4 6 5 . Electrolytic with the positive pole of a battery of three dry cells in series; and an iron or a copper wire is in- serted in the other arm of the tube, and connected with the negative pole of the battery. While the current is flowing, bubbles continue to rise from the platinum terminal; and, after a few seconds, it will be found that the iron or copper wire is covered with a layer of zinc. The current in pass- ing through the solution decomposes the zinc sulphate, and carries the zinc with it to the negative terminal. The other 602 ELECTRODYNAMICS product of the decomposition, consisting of sulphur and oxy- gen in combination (SO 4 ), goes in the opposite direction to the positive terminal. Here it unites with hydrogen from the water, forming sulphuric acid and setting oxygen free. This oxygen forms the bubbles that rise from the strip of platinum. The presence of the sulphuric acid can be shown by adding a few drops of blue litmus solution to the con- tents of the tube before the current is passed; for the acid changes the color of the liquid from blue to red about the platinum terminal. The above experiment is an example of the chemical decomposition of a compound by means of an electric cur- rent sent through a solution of the substance. The process is called electrolysis (electro-analysis), the substance de- composed is termed an electrolyte, and the vessel in which the process is carried out an electrolytic cell. The terminal by which the current enters the solution is called the positive electrode or anode; the one by which it leaves the liquid is the negative electrode or cathode. (These terms are from the Greek, meaning, ode, way or path; an, up; cat, down; electrode, a way for electricity; anode, the way up or against the current ; cathode, the way down or with the current.) If an anode of zinc is substituted for the platinum in the experi- ment, the sulphur-oxygen product of the decomposition (864) unites with it, forming more zinc sulphate. The anode thus loses as much zinc as the cathode gains, and the strength of the solution remains constant. Similar results are obtained with copper electrodes and a solution of copper sulphate. The anode wastes away and an equal weight of copper is deposited upon the cathode. Water can be elec- trolyzed in a cell containing dilute sulphuric acid and platinum elec- trodes. The acid takes part in the chemical changes; but only the water is consumed, hydrogen appearing at the cathode and oxygen at the anode. CHEMICAL EFFECTS OF CURRENTS 603 Cofhod* FIG. 466. "Migration" of the Ions in an Electrolytic Cell. 487. Ionic Theory of Electrolysis. In a solution of zinc sul- phate a certain percentage of the molecules are dissociated, forming positive zinc ions (Zn ++ ) and negative sulphions (SC>4~ ~) (Art. 407). When the electrodes of a battery circuit are placed in the solution (Fig. 466), the negatively charged cathode attracts the positive and repels the negative ions, while the positively charged anode does just the opposite. The result is a slow drift of the positive and negative ions in opposite directions through the solution. With a battery E.M.F. of three or more volts, the zinc ions, on arriving at the catKode, give up their positive charges to it, and adhere to the surface, form- ing a layer of metallic zinc. With a platinum anode, the sulphions, on coming in con- tact with it, give up their negative charges, and immediately decom- pose water molecules, forming sulphuric acid and setting oxygen free. When a zinc anode is used, the sulphion acts on it, and new zinc ions are formed as rapidly as they are deposited from the solution at the cathode. A current of electricity in solid conductors and in molten metals passes through matter; in solutions of acids, bases, and salts (electro- lytes) the electricity is transported by matter, in the form of positive and negative charges of moving ions. All ions of one kind carry equal charges; and the charge on any kind of ion is either equal to the charge on a hydrogen ion or is an exact multiple of it The charge of the hydrogen ion is thus the natural unit of electricity. It is denoted by a single plus sign in ionic symbols, as in H + for the hydrogen ion and Na + for the sodium ion. Zinc and copper ions carry double charges; hence their symbols are Zn ++ and Cu ++ . The unit negative charge is represented by a single minus sign, as in the symbol Cl ~ for the chlorine ion and OH~ for the hydroxide ion. The sulphion (SO4~~) carries a double negative charge. The fol- lowing are further examples of common electrolytes. 604 ELECTRODYNAMICS Substance Ions formed in solution Sulphuric acid, H 2 SO 4 ^ H+ + H+ + SO 4 ~ ~ Zinc sulphate, ZnSO 4 ^ Zn f+ + SO 4 ~ Copper sulphate, CuSO 4 ^ Cu+ + + SO 4 ~ Hydrochloric acid, HC1 ^ H + + Cl~ Sodium chloride, NaCl ^ Na+ + Cl~ Zinc chloride, ZnCl 2 ^ Zn ++ + Cl~ + Cl' Silver nitrate, AgNO 3 Ag + + NO 3 ~ Sodium hydroxide, NaO ^ Na+ + OH~ Aluminum oxide, A1 2 O3 ^ 2A1 + + + -f 36" ' In every case, the sum of the negative charges is equal to the sum of the positive charges. This is shown experimentally by the fact that the solution as a whole has no charge, as would be the case if there were an excess of either positive or negative electricity within it. 488. Laws of Electrolysis. Since in an electrolyte the current is carried only as charges of the ions and all like ions carry equal charges, the amount of any one substance liberated by electrolysis is pro- portional to the quantity of electricity (ampere-hours] which passes through the electrolyte. Thus it is found by experiment that a current of one ampere, flowing for one hour through a solution of any silver salt, e.g. silver nitrate or silver cyanide, always deposits 4.025 g. of silver. Similarly a current of one ampere liberates in one hour .0376 g. of hydrogen from any acid solution, 1.203 g- of zmc from any electrolyte whose positive ions are zinc- ions, 2.444 g. of gold from a solution of a gold compound, etc. Having determined the quantity of any substance which a given current deposits from solution in a given time, the problem can be reversed and the strength of an electric current determined by find- ing the weight of the substance that the current deposits in a certain time. Since mass and time can be measured with very great accu- racy, the method serves as the basis for the practical definition of the ampere. "The international ampere, as thus defined, is the steady current which deposits silver at the rate of .ooiuSg. per second (4.025 g. per hour) from a solution of silver nitrate in water, the solu- tion being of a given fixed strength to insure regular action." The quantities of different substances liberated from their solutions by the passage of equal quantities of electricity are chemically equivalent to CHEMICAL EFFECTS OF CURRENTS 605 one another. The term "chemically equivalent quantities" means that there is just enough of the one to displace the other completely in a chemical change. Thus 4.025 g. of silver will displace .0376 g. of hy- drogen from nitric acid (HNOs) in forming silver nitrate (AgNOs). The two statements above in italics are known as Faraday's laws of electrolysis. They express a quantitative relation between elec- tricity and matter, which is explained in part by the ionic theory of solutions (Art. 487). This is supplemented by the new theory con- cerning the nature of electricity and its relation to matter (Art. 512). 489. Industrial Applications of Electrolysis. The industrial applications of electrolysis cover a wide range, and are increasing in number and importance from year to year. Electrolytic processes are of two general types. In electroplating and electro typing the current deposits a thin film of metal gold, silver, nickel, or copper upon a prepared surface. In electrometallurgy the current per- forms the work of chemical decomposition by which cer- tain of the metals are obtained from the ores and minerals in which they occur. Electrotyping is a process of reproducing pages of type and wood- cuts and other illustrations by an electroplating of copper. The matter to be printed is first set up in common type, and a mold of this is made in wax by pressing it hard upon the type. The impressed side of the mold is very thinly covered with powdered graphite to make it a conductor of electricity. The mold is then suspended as the cathode in an acid solution of copper sulphate; the anode is a plate of copper. When the current is passed, copper is dissolved from the anode and deposited as a thin sheet upon the mold, forming an exact copy of the original. This sheet is removed from the mold, "backed up" by a filling of type metal to give it strength, and mounted on a wooden block. It is then ready for use in printing. Most books, are now printed from electrotype plates, which, as a rule, are preserved for many years, and may be used again and again in printing new editions. In electroplating, the article to be plated is carefully cleaned and made the cathode in a solution of some salt of the metal to be depos- 6o6 ELECTRODYNAMICS ited. The anode is a plate of the same metal, and, by dissolving, maintains the strength of the solution (Fig. 467). In silver plating the solution is a complex cyanide of potassium and sil- ver; in gold plating, a complex cyanide of potassium and gold; and in nickel plating, a double sulphate of nickel and ammonia. The last is com- FIG. 467. Electroplating. monly known as nickel salts, and can be bought in the market. The details in electroplating differ with the different metals, and they must always be attended to with great care, in order to secure a smooth and coherent deposit. The pupil who wishes to try electroplating on a small scale should first acquaint himself with the details by reading up on the subject. The use of the electric current in the treatment of ores and min- erals and in refining metals is termed electrometallurgy. The larg- est industry of this character is the refining of copper. The process is similar to that descibed under electrotyping. " The crude copper produced by the ordinary smelting processes is cast into heavy plates which are used as anodes in depositing vats. The solution in these vats is copper sulphate with a little sulphuric acid. The cathodes at first are thin sheets of pure copper, but they grow by deposition into thick plates of copper, .which may be worked into bars or drawn into wires as desired." The current is supplied by large dynamos. "Other metals, such as gold, silver, and lead, are extracted from their ores and purified by electricity, though the older processes are still used. All the aluminum, magnesium, and sodium of commerce are now manufactured by passing an electric current through their fused compounds." 490. The Secondary or Storage Cell. When a current is sent through an electrolytic cell containing dilute sulphuric acid and lead CHEMICAL EFFECTS OF CURRENTS 607 electrodes, the electrolysis of the liquid liberates oxygen at the anode and hydrogen at the cathode. The hydrogen gathers in small bubbles and escapes. The oxygen, or a part of it, combines with the anode, forming a brown layer of lead peroxide (PbO 2 ) upon its surface. When the anode is in this condition the cell is said to be charged, and is itself capable of generating an electric current. This can be shown by disconnecting the electrodes from the charging battery and con- necting them with a galvanometer. It will further be found that the direction of the current generated by the cell is opposite to that of the charging current; hence the positive plate of the cell is the one that receives the deposit of peroxide. This experiment illustrates the principle of the secondary or storage cell. If the electrodes are several inches square, the current from the cell will probably be sufficient to ring an electric bell or run a small motor; but only for a moment. While the cell is generating a current, hydrogen ions of the acid go to the positive plate, and unite with oxygen from the peroxide, reducing it to monoxide (PbO). The monoxide reacts with the acid, forming lead sulphate (PbSO4), and the latter remains as an insoluble deposit on the surface. At the same time the sulphions go to the negative plate, with which they combine, forming a layer of lead sulphate upon it. When the plates have thus been brought to the same condition, the cell is exhausted. It can be charged again by means of a current, as before. It should be noted that the charging current does work of chemical decom- position within the cell and stores chemical potential energy, not electricity. The essential difference between the storage cell and ordinary or primary cells is that in the former the chemical actions are reversible, and hence the materials of the cell can be used over and over indefinitely. By repeatedly charging and discharging the cell used in the above experiment, the chemical action extends to an increasing depth below the surface of the lead plates, and the cell becomes capable of receiving a greater charge. When the cell is charged, the layer of active material on the negative plate is pure lead in a spongy condi- tion, and that on the positive plate is a porous crust of lead peroxide. In the manufacture of storage cells the active material is generally formed from a paste made of one or more oxides of lead mixed with dilute sulphuric acid. This paste is firmly imbedded in the openings of a lead grid which forms the body of the plate (Fig. 468). Cells of 6o8 ELECTRODYNAMICS FIG. 468. Storage Cell. large capacity have several positive plates joined together, alternat- ing with negative plates, also joined together, as shown in the figure. The solution is one fourth (by weight) of pure sulphuric acid and three fourths distilled water. The E.M.F. of the cell, when fully charged, is 2.2 volts. It slowly falls during the discharge to 1.8 volts, and from that point on the drop is rapid, with more or less permanent damage to the cell. Hence the regular practice is to discharge only to 1.8 volts. Storage bat- teries are charged from direct-current dynamos, or, if from alternating current circuits, the current is first rectified, i.e., changed to direct, by sending it through a special device for that purpose. A good storage battery gives out in useful service about 80% of the energy expended in charging it. The storage battery is principally used as an auxiliary source of power in electric lighting and power stations. The battery is brought into service to supplement the output of the dynamos during those hours of the day or night when the demand for power is greatest, as during the early night hours on lighting systems; and is charged dur- ing those hours when the engines and dynamos would otherwise be idle or working on light load. A single cell of a battery for such pur- poses, complete with plates and acid, weighs from 200 to 7000 lb., according to the size and number of the plates. Storage batteries are also used for running electric launches and automobiles. Their great weight is a hindrance to their more general adoption for such uses, and inventors have long sought to perfect a type of cell that would store a much greater amount of energy for a given weight. The new nickel-iron storage battery of Thomas A. Edison is a great advance in this direction, as it reduces the weight one half. CHAPTER XIV RADIATIONS. THE ELECTRICAL NATURE OF MATTER 491. Introduction. We have now compassed the field of elementary physics. Much has been omitted that might well receive attention, if time were available for it; but, at least, no department of the subject has been slighted. The present chapter is in the nature of a sequel, and is also, in a sense, an introduction, inviting the student to more advanced fields of study. Viewing the subject in retrospect, it is evident that physics depends from first to last upon the physical prop- erties of matter, using the term matter in the broadest sense to include the ether. Thus we have the mechanics of solids, liquids, and gases, the mechanics of sound and sounding bodies, and molecular physics including heat, all of which depend upon the physical properties of ordi- nary matter; while the remaining branches, light, magnet- ism, and electricity, involve primarily the properties of the ether. Between ordinary matter and the ether we find an impassable barrier neither can be converted into the other; yet, under certain conditions, they exert a mutual action by which the energy of either can be transferred to the other, as in the emission and absorption of radiant energy, the transformations of electrical energy, etc. The primary facts of the material universe are summed up in the terms matter, ether, electricity, and energy. These are the actors in the drama of nature. 609 6io RADIATIONS Elementary physics is mainly concerned with the scenes and incidents of this drama (phenomena and processes), rather than with the personality (real nature) of the actors themselves. But the scientist is not content to stop here. He seeks to know what matter and ether and electricity are, and what the invisible mechanism is by which they act upon one another. If matter is composed of atoms, what is the atom and how do atoms differ? Are there in fact two kinds of electricity, positive and negative, or does positive denote an excess and negative a deficiency of one and the same thing? Whichever may be true, the question still remains, What is electricity, and what is its relation to ordinary matter and to the ether? Science has not arrived at the full and final answer to any of these questions. Nevertheless a wonderful advance in this direction has been made in recent years, and it is with this advance that the present chapter mainly deals. I. SPECTRA AND SPECTRUM ANALYSIS 492. Continuous and Bright-line Spectra. One very important source of information about matter is the character of the light which it emits when heated to incandescence in the gaseous state. The light from all incandescent solids and liquids gives a continuous spec- trum (Art. 354), showing the presence of all wave lengths between the longest and the shortest. Hence there is nothing in the spectra of incandescent solids and liquids which serves to distinguish one substance from another. With luminous vapors and gases, however, the case is very different. Their spectra consist of bright, colored lines (isolated images of the slit), separated by black spaces, and no two of these spectra are alike. Thus the spectrum of sodium vapor is a single yellow line (or a very close SPECTRA AND SPECTRUM ANALYSIS 611 double, with wide dispersion); that of lithium consists of two lines in the red. Hydrogen gives a red, a blue, and a violet line; phosphorus, three lines in the green; strontium, several lines in the red; calcium, many lines extending through the red, orange, yellow, and green; and so on. Every element, when in the gaseous state, gives a characteristic bright-line spectrum by which the element can be identified. The great importance of this fact in physics, chemistry, and astronomy has led to the invention of various forms of spectroscopes and spectrometers, by means of which the radiation from any body can be dispersed into a very pure spectrum and the positions of its different lines accurately determined. A spectrometer may be designed either for direct observation and measurement, or for taking photo- graphs of the spectra. The plan of a simple prism spectro- scope is shown in Fig. 469. The essential parts are the collimator, the prism, and the telescope. The collimator FIG. 469. Section Diagram of Spectroscope. consists of a metal tube having a narrow, vertical slit, S, at one end, and an. achromatic lens at the other. The rays from any point of the slit emerge from the lens parallel, the effect being the same as if the slit were a distant lumi- nous object. The light to be analyzed passes through the collimator, is dispersed by the prism, and is brought to a focus by the telescope. The observer views the spectrum through the telescope as he would view a distant object. 612 RADIATIONS It is best to use a spectroscope in a dark room. If the room is not darkened, a black screen should be placed a short distance beyond the collimator to shut out diffused sunlight, unless it is the spectrum of sunlight that is under observation. When a common gas burner is placed before the slit, the spectrum is continuous, for the light of the flame comes from incandescent particles of solid carbon. (This solid carbon is deposited in the form of soot when the flame plays upon the surface of cold porcelain or metal.) The non-luminous Bunsen flame gives no spectrum, or, at the most, only a very faint one. A strip of tin or a wire heated to incandescence in the Bunsen flame gives a continuous spectrum. Spectra of metals which are easily volatilized can be studied by holding in the Bunsen flame a bit of asbestos which has been dipped in a solution of a salt of the metal. A convenient holder is made by fusing an end of a short piece of platinum wire in the end of a piece of glass tubing. The free end of the wire is wrapped round the asbestos (Fig. 470). In this way we may observe the bright-line spectra of sodium, potassium, calcium, strontium, barium, ^== ==_ \& a etc. (A holder should be provided for each solution, otherwise the asbestos will contain traces of the different metals and their spectra will appear together.) The line spectra of the metals which vaporize only at very high temperatures are obtained from the electric arc formed between rods of the metal whose spectrum is required; or, instead of the arc, the sparks from an induction coil will serve. The spectrum of a gas, e.g. oxygen, hydrogen, nitrogen, etc., is obtained by passing the discharge from an induction coil through the highly rarefied gas in a vacuum tube (Art. 500). 493. What the Bright-light Spectrum Teaches. It will be recalled that the different colors of the spectrum are the optical effects of different wave lengths of the light (Art. 355), and that different wave lengths are due to dif- ferent rates of vibration, according to the formula v = In (Art. 276). It follows that a bright line in a line spectrum is formed by ether waves of one definite length, and that SPECTRA AND SPECTRUM ANALYSIS 613 the source of these waves is a body having a fixed rate of vibration. It follows further that the atom of any ele- ment in the gaseous state is the source of as many differ- ent fixed rates of vibration as the number of bright lines in the spectrum; and this number varies from a dozen or less for several of the elements to many thousand for iron and uranium. Let us mark well the meaning of the last statement. Every atom has a definite number of natural rates of vibra- tion, just as a piano or an organ has. The comparison is not far-fetched. On the contrary, it fails, if anything, to do justice to the atom; for the range of the piano is only eighty-eight notes, or different wave lengths, while that of the iron or uranium atom is several thousand. Minute as an atom is, the idea that it is a simple, structureless body must evidently be set aside as wholly untenable. Judged by their spectra, atoms must be very complex bodies indeed, differing widely among themselves in this respect, accord- ing to their kind; but on the average they are seemingly quite as complex as musical instruments. Other facts which are presently to be considered will offer some sug- gestions as to what this complex structure may be. The continuous spectrum of a substance in the solid or the liquid state is presumably due to forced vibrations of the molecules, resulting from their frequent collisions with one another. These forced vibrations, owing to their irreg- ular character, give rise to light waves of all lengths. In gases and vapors, where collisions are relatively infrequent, there is a preponderance of free vibrations in the various natural rates of the molecules or their constituent atoms. 494. Absorption Spectra. There is still a third class of spectra, due to selective absorption by the medium 614 RADIATIONS through which the light passes. We have seen that col- ored glass and colored liquids produce such spectra (Art. 360), and that, as a rule, the absorption includes broad regions of the spectrum, sometimes at either end, sometimes in the central portion, and sometimes in two or more places, with bright areas between. A spectrum of this character is called the absorption spectrum of the body which pro- duces the absorption. The absorption spectra of gases and vapors are of special interest and importance, for to this class belong the spectra of the sun and stars. The general conditions necessary for producing the absorption spec- trum of a vapor in class-room or laboratory are, first, a source of white light at a very high temperature; and, second, a flame at a lower tem- perature, in which the substance is vaporized, so placed that the light from the source passes through it before reaching the prism. The electric arc is best as the source, since it is the hottest obtain- able; but the oxyhydrogen lime-light will serve. With the arc light the vapor may be produced in a Bunsen flame; but with the lime- light a flame of lower temperature will be necessary, such as that of an alcohol lamp. The spectrum can be projected upon a screen with a lantern, or viewed through a spectroscope. In the latter case the alcohol lamp or Bunsen burner is placed near the slit, and the lime or arc light a short distance behind it. The chloride or nitrate of the metal whose spectrum is desired is vaporized in the flame, e.g. a little table salt rubbed into the wick will give the vapor of sodium. With the Bunsen burner a piece of asbestos, dipped into a solution of the salt, is held in the flame or wrapped round the top of the burner. If sunlight is used as the source, the vapor will, in general, merely intensify some of the lines already present.* The absorption spectrum of a gas or vapor is always a dark-line spectrum, differing from the complete spectrum of white light only in the fact that it is crossed by one or * The teacher will find detailed directions for these and other projection ex- periments in light in the admirable little book on Light, by Lewis Wright, published by the Macmillan Company. SPECTRA AND SPECTRUM ANALYSIS 615 more narrow dark lines. Moreover, these dark lines are the same in number and occupy precisely the same positions as the lines in tJte bright-line spectrum of the substance. This is admirably shown when the emission and absorption spectra of a gas are produced side by side. The bright lines of the one join accurately with the dark lines of the other (Fig. 471). The meaning of this is that a gas or vapor absorbs light of the same wave lengths that it emits when heated to incandescence. Gaseous absorption is evidently FIG. 471. Comparison of Solar Spectrum with that of Iron. a case of sympathetic vibration. The atoms respond to (absorb) the vibrations which agree with their own natural periods, just as a tuning fork responds to the vibrations of another fork of the same pitch (Art. 287). It follows from the above that a substance in the gaseous state can be identified by means of its absorption spectrum just as certainly as by its emission spectrum. It is to this fact that the spectroscope owes its great importance in astronomical research. 495. The Solar Spectrum. It will be remembered that, when a fairly pure solar spectrum is thrown upon a screen, it is crossed by sev- eral dark lines (Art. 354). Viewed through a good spectroscope, the lines are more sharply defined and there are many more of them. The greater the excellence and dispersive power of the spectroscope the more numerous are the lines. Several thousand are shown in an enlarged photograph of the solar spectrum, over 42 ft. in length, taken by the late Professor Rowland in 1888. The most prominent of the lines are designated by the letters from A to H (Fig. 472). 6i6 RADIATIONS 760| A ! -a 6871 J2 I 656 527 486 431 D E T___ Bed (center 690) > Green( < 530) >"Blue( The lines of the solar spectrum were first carefully studied by Fraunhofer, a noted German optician, in 1814, and they have since been called the Fraunhofer lines. Their meaning, however, remained a mystery until the epoch-making work of the German phys- \OrangeC 600) icist, Kirchhoff, who developed the science }Yellow( 580) of spectrum analysis, and applied it to the heavenly bodies, during the years 1858- 1862. According to the principles enunciated by Kirchhoff and briefly outlined in the preceding pages, the Fraunhofer lines are tt 470) due to absorption in some gaseous medium between the sun's surface and the earth. Now the only gaseous media which inter- vene are the atmospheres of the sun and the earth; and, after making due allowance f-Violet( 4ltft for absorption in the earth's atmosphere, the balance must be attributed to the sun itself. Considering the intense heat of the Fl ?'- 472 TJ; r Q U , nh fer sun > it is certain that the elements known Lines of the Solar Spec- trum. The numbers are upon the earth, if they exist in the sun at all, must be present as vapors in the solar atmosphere. The Fraunhofer lines are un- impeachable witnesses to the truth of this conclusion. Over two thousand of these lines are found to coincide in position with the bright lines in the gas spectrum of iron (Fig. 471), proving that iron exists as a vapor in the sun's atmosphere. The D line of the spectrum coincides with the yellow line of sodium. The lines C and F are due to hydrogen. Similar identifi- cations have been made for about half of the known terrestrial elements, including iron, nickel, cobalt, carbon, calcium, magnesium, sodium, and hydrogen. Wonderful indeed is the fact that an element in the far-distant sun is thus able to reveal its existence to us, just as certainly as if we had a sample for examination in the chemical laboratory. The story of the stars is also written in their light and revealed in the dark lines of their spectra after the same fashion, although the ether waves which carry the message may have been traveling for hundreds of "fcfc the wave lengths, ex- pressed in millionths of a millimeter. ELECTRIC OSCILLATIONS AND WAVES 617 years. Nor is this all. The spectrum of a star shows whether the star is moving toward or from the earth and at what rate. For motion toward the earth has the effect of shortening all light waves, since more of them reach the earth in a given time than would be the case if the distance were constant. The result is a slight shifting of all the dark lines toward the violet end of the spectrum; and, from the amount of the displacement, the velocity of the star can be computed. Motion from the earth has the contrary effect, and all the lines are displaced toward the red end of the spectrum. In this way it has been found that the brilliant star Arcturus is rushing toward us at the rate of nearly 60 mi. per second. II. ELECTRIC OSCILLATIONS AND WAVES. ELEC- TRO-MAGNETIC THEORY OF LIGHT 496. Electric Oscillations. The discharge of a Leyden jar produces what appears to be a single spark; but, when viewed in a rapidly revolving mirror, it is found to consist of a series of sparks, often a dozen or more. If a concave mirror is arranged to reflect the light and focus it upon a photographic plate, the impressions due to the individual sparks of a series are drawn out into a line, owing to the * FIG. 473- Photograph of Oscillating Electric Sparks. slight angle through which the mirror turns in the brief intervals between their occurrence (Fig. 473). Under ordinary conditions these intervals are less than the millionth part of a second, as shown by computation based on the known rate of rotation of the mirror, the distance between it and the photographic plate, and the distance between the spark images on the plate. The meaning of the series of sparks is that the discharge 6i8 RADIATIONS of a Leyden jar is oscillatory. The current surges back and forth between the inner and outer coats of the jar, gradu- ally dying away, just as a pendulum or a spring executes a series of vibrations with diminishing amplitude before coming to rest. The period of the oscillations is deter- mined by the capacity of the jar and the resistance and self-induction of the discharging circuit. The smaller these factors are the shorter will be the period. When two small metal spheres are substituted for the jar, and the discharge takes place by a short, straight path between them, the period may be less than one hundred-millionth of a second. 497. Electromagnetic Waves. Electrical Resonance. - It can be shown in various ways that electric oscillations produce waves in the ether, which radiate from the center of disturbance with the velocity of light (300,000,000 meters per second). The existence of such waves is demon- strated in the following experiment, due to the English physicist, Sir Oliver Lodge. Two Leyden jars, A and B (Fig. 474), of equal capacity are con- nected with discharge circuits, each consisting of a wire rectangle. A 's circuit has a spark gap, S, between two metal balls. B's circuit is without gap between the coats of the jar, and its size is adjustable by means of the sliding wire M . A strip of tin-foil extends from the inner coat of B to FIG. 474. Apparatus for Showing Electri- w ith;n I mm. of the outer cal Resonance. m, . ., coat at e. The two circuits are placed parallel to each other, and the coats of A are connected with the terminals of an induction coil or with the opposite sides of an electrostatic machine. When A discharges, a spark occurs ELECTRIC OSCILLATIONS AND WAVES 619 at e, provided the areas included within the two rectangles are equal or nearly so; but there is no response at e if the area of B's circuit is made considerably larger or smaller than A's. This is explained as follows. Each circuit has a natural period of oscillation, which is determined by the capacity of the jar and the self-induction of the circuit. (The resistance of the circuit is negligible.) The capacities are equal, and the self-induction increases with the area of the rectangle. Hence with equal areas the periods are equal, and B 's circuit responds to an oscillatory discharge in A , just as a tuning fork responds to another of the same pitch. The phenomenon is termed electrical resonance, and the two circuits are said to be in tune with each other. The induced oscillations are set up through the medium of the ether, which transmits the impulses in the form of elec- tromagnetic waves. When the circuits are not in tune, the induced oscillations are relatively weak, too weak to produce a spark at e if the difference between the periods is considerable. 498. The Electromagnetic Theory of Light. As early as 1867 one of England's greatest physicists, James Clerk Maxwell, advanced the theory that light is an electromag- netic rather than a simple mechanical disturbance of the ether, and that it ought to be possible to produce waves of this character by electrical means. Twenty-one years elapsed before his theory was experimentally verified by the German physicist, Heinrich Hertz. Hertz demon- strated not only that such waves were produced by an oscillatory spark discharge, but also that they possess all the properties of light waves, being reflected, refracted, etc., according to the same laws. The only essential difference is in the length of the waves. For the longest 620 RADIATIONS visible waves (extreme red), this is only .000076 cm.; while the ordinary length of electric waves is several meters, and the shortest yet produced were .4 cm. Since the velocity of all ether waves is the same, their frequencies are inversely proportional to their wave lengths (by the formula v = In). If the frequency of an electrical oscillation is 100,000,000 per second, it will send out electric waves of the length - - = 3 m. Conversely the 100,000,000 frequency of the longest visible waves is > or, .00000076 very nearly 400,000,000,000,000 vibrations per second. Oscillations at such a stupendous rate must of necessity be on an inconceivably small scale. As we shall presently see, there are substantial reasons for believing that the individual source of light waves is a vibrating particle of electricity, or a group of such particles, within the atom. 499. Wireless Telegraphy. Since 1895 many systems of wire- less telegraphy have been devised for transmitting signals through space by means of electric waves. The sending mechanism is de- signed to produce electric oscillations which are under the control of the operator. For long-distance working, the apparatus must be very powerful, since the waves spread out in all directions, and grow rapidly weaker as they travel. The receiving apparatus must be as sensitive as possible, for the same reason. So well have these requirements been met that messages have been sent over distances exceeding 3000 mi. The details of construction and operation of any fully developed system of wireless telegraphy are numerous and complicated, and the principles involved are largely beyond the scope of this book. It is, however, a comparatively simple matter to demonstrate the main fact, viz., that intelligible signals can be transmitted by means of ether waves. This is shown in the following experiment. The transmitter or oscillator is a small induction coil, with a plate of sheet metal and a discharge ball connected to each of the secondary ELECTRIC OSCILLATIONS AND WAVES 621 terminals (Fig. 475). The plates are charged by the coil, and they discharge with an oscillatory spark across the gap. The current is controlled by means of a telegraph key in the primary circuit. One essential part of the receiving apparatus is some device that is very sensitive to electric waves. The co- herer, C (Fig. 476), was first used for this purpose in in- FlG - 475- -Hertz Oscillator for Transmitting dustrial wireless systems, and will serve for our experiment. It consists of a glass tube of small bore, containing a small quantity of metal filings (silver, nickel, or iron) and two wire electrodes. The filings, lying loosely between the electrodes, ordinarily offer a very great resistance to the passage of a current; but the loose mass suddenly becomes a good conductor under the action of electric waves. Apparently the waves cause the filings to FIG. 476. Receiving Apparatus for Wireless c j m g together, thus reducing Telegraphy. ^ resistance at their points of contact. Jarring the tube restores the filings to their original con- dition of high resistance. A diagram of the complete receiving apparatus is shown in Fig. 476. The coherer, C, connects two metal plates, PP, of the same size as those of the transmitter. The coherer is also included in cir- cuit with a battery cell B and the magnet coils of a relay R. This relay controls a second battery circuit, connected with an electric bell. The bell is so placed that the clapper strikes the coherer on the back stroke, thus acting as a decoherer. When the key of the transmitter is depressed, sparks pass between the knobs of the oscillator, and electric waves are sent out. These, falling upon the plates of the receiver, set up oscillations of the same frequency between them. This breaks down the resistance of the coherer and permits the passage of a battery current through the 622 RADIATIONS relay. The relay closes the bell circuit; the bell rings; and the clapper, on the return, strikes the coherer. The apparatus is then in condition to receive another signal. In commercial wireless systems the waves are sent out and received by means of an aerial wire or wires, carried up to a height of 100 to 200 ft. on a mast. For sending, this wire is connected with one terminal of the spark gap. The other terminal of the spark gap is connected with the ground. For receiving, the coherer or other sensitive detector of electric waves takes the place of the spark gap. There are various forms of detectors in connection with which the signals are received through a high-resistance telephone receiver. III. ELECTRIC CONDUCTION THROUGH GASES, CATHODE AND R6NTGEN RAYS 500. Electric Discharge in Rarefied Gases. Geissler Tubes. The study of the electric discharge in rarefied gases has led to results of very great theoretical and practi- cal importance ; and the beauty and novelty of the phenom- ena never fail to arouse the liveliest interest. To study the effect of different pressures we may use a glass tube, 20 cm. or more in length, with a side connection for exhaust- ing the air, and platinum electrodes, A and C, sealed into the ends (Fig. 477). The tube is connected with the ter- ^ or *G, by the concave cathode, C. The disk becomes red hot in a few minutes. FIG. 480. Shadow Cast by Mica Cross in Cathode Rays. 602. Nature of the Cathode Rays. Electrons. The nature of the cathode rays has been determined from an exhaustive study of their properties. It is found that: 626 RADIATIONS 1. They are deflected by a magnetic field. 2. They are deflected by an electrostatic field. 3. They impart a negative charge to an insulated con- ductor upon which they fall inside the tube. 4. They are stopped by the glass walls of the vacuum tube, but they can pass through a very thin plate of aluminum. The first three of these properties demonstrate that the cathode radiation does not consist of ether waves (as some at first supposed), but of particles projected from the sur- face of the cathode. A moving electric charge is virtually an electric current; and it sets up a magnetic field along its path, just as an electric current in a wire does. And since a wire in which a current is flowing tends to move sideways across the lines of force in a magnetic field (as in the motor and the D' Arson val galvanometer), it follows that a charged particle, shooting across lines of force in a magnetic field, must be acted upon by a sideward force which tends to deflect it from a straight path. This con- clusion tallies exactly with the observed fact (Fig. 481). Similarly it is found that the cathode rays are deflected when they pass through an electrostatic field between two plates, one of which is posi- FIG. 481. Deflection of Cathode Rays in tively and the Other a Magnetic Field. negatively charged. These experiments, and others which are too elaborate to be considered here, afford the necessary data from which it is possible to compute the mass and velocity of the cath- ode-ray particles, and the magnitude of the charge which they carry. It turns out that the mass of a particle is about of the mass of a hydrogen atom, which is the light- ELECTRIC CONDUCTION THROUGH GASES 627 est atom of all the elements. The hydrogen atom thus gives place to the cathode-ray particle as the smallest thing known in the universe. The charge of a particle is equal to the smallest electrical charge carried by ions in electrolysis (Art. 487). The velocity of the particles varies somewhat, but the average is roughly one tenth of the veloc- ity of light, or from 15,000 to 20,000 mi. per second, or about 60,000 times the velocity of a rifle ball! Experiment further shows that the mass, charge, and velocity of the cathode particle are all independent of the nature of the gas in the vacuum tube or the kind of metal used for the electrode. Hence it is believed that the cathode particle is one and the same thing in all cases. It is called an electron, or, sometimes, a negative corpuscle or negative ion. The most remarkable property of the electron is yet to be mentioned. A moving electrical charge possesses inertia or mass, which appears to be identical in kind with the mass of ordinary matter; for the moving charge has both momentum and kinetic energy. Prof. J. J. Thomson, of Cambridge, England, has shown mathematically that the observed behavior of electrons can be accounted for on the assumption that the whole of the mass of an electron is due to the charge. On this view an electron is the dis- embodied, indivisible, natural unit or atom of negative elec- tricity pure electricity and nothing else. 503. Positive Ions. Canal Rays. When the cathode of a Crookes tube is perforated with many small holes, it is observed that, in addition to the cathode rays, which are emitted from one side, there are rays proceeding in the opposite direction, apparently through the holes in the cathode. These are called canal rays. They can be deflected by a magnetic or an electric field, and the direction of the deflection shows that they carry a positive charge. They are, 628 RADIATIONS in fact, positively charged particles or ions, moving much more slowly than the electrons. Unlike the electron, the mass of the positive ion varies with the material of the electrodes and with the kind of gas in the discharge tube. Apparently the mass is the same as that of the atoms of the elements which happen to be present in the tube. The charge of the positive ion is believed to be equal to that of the electron (but opposite in sign). Taking all the facts together, they present a very strong argument in support of the view that the positive ion of a gas is simply an atom which has lost an electron. In an electrolyte (Art. 487), according to this view the positive ions are atoms which have lost one, two three, or four electrons, according as their charge is one, two, three, or four times that of the hydrogen ion; and the negative ions are atoms or groups of atoms which have, for the time being, appropri- ated one or more extra electrons, stolen from the positive ions. 504. Rbntgen or X-rays. Where cathode rays strike the walls of a Crookes tube or any solid within it, they excite a form of invisible radiation which is said to consist of Rontgen or X-rays. Rontgen is the name of the German physicist who discovered the rays in 18*95. They were called by him X-rays, because their nature was unknown. Rontgen rays proceed in straight lines from their source. They are not reflected or refracted; hence can not be of the nature of light waves. They do not carry an electric charge, and are not deflected by a magnetic or an electric field. They pass through glass, and also through substances which are opaque to light, such as wood, thin sheets of metal, and animal tissues. They affect a photographic plate, and excite fluorescence (light from a cold body) when they fall on certain substances. It is believed that they are single, disconnected pulses in the ether, traveling with the velocity of light, and that each pulse is due to the sudden stopping of a cathode particle when it strikes a solid. ELECTRIC CONDUCTION THROUGH GASES 629 In the treatment of diseased tissues of the body by means of X-rays and in X-ray photography, it is an advantage to have r ^s-^ ^W4%. the source of the rays as small as possible. The focus tube (Fig. 482) is designed with this end in view. The ./#SSSS^^ ? cathode, C, is concave, in order to concentrate its rays upon the center of the plati- num disk, D. The X-rays are generated at this point. X-ray photographs are shadow pictures (Fig. 483). The photo- graphic plate is kept in a plate holder or wrapped in paper to protect it from light, and the object to be photographed is placed against it. The parts or structure of the object will be recorded in the picture just in so far as the different parts are unequally transparent to the rays. Since flesh is quite transparent and bones are rather opaque, FIG. 482. X-ray Tube. FIG 483. X-ray Photograph of a Broken Arm. a photograph of any part of the body shows the bony structure very clearly. X-ray photography is thus an invaluable aid to the surgeon in determining the character of the injury in cases of fractured or broken bones, as well as in locating foreign bodies, such as bullets, needles, etc., for the metals are also less transparent than the flesh. 630 RADIATIONS The same information can be obtained directly by sight, with the aid of a fluoroscope (Fig. 485). This is a darkened box, shaped at one end so as to fit closely round the eyes of the observer, and closed at the other end with a fluorescent screen. This screen is covered with some substance (usually barium platino- cyanide) which becomes luminous under the influence of X-rays. If the hand is placed against the screen Fig. 484. Fluoroscope. , ., , , , , . , , , , while the latter is exposed to the rays, the shadow of the hand will appear upon the screen, the flesh show- ing rather light and the bones dark. IV. RADIOACTIVITY. ELECTRICAL THEORY OF MATTER 505. Discovery of Radioactivity. We now come to another epoch-making discovery, which opened up a hitherto unknown and unsuspected field of investigation of surpassing interest. In 1896 the distinguished French physicist, M. Henri Becquerel, found that the element uranium and all its compounds emit rays which are able to pass through black paper and affect a photographic plate. These rays are given out continually and without the aid of any outside agency. Uranium and certain other ele- ments, which were later found to possess the same property, are said to be radioactive, and the property itself is termed radioactivity. 506. Use of the Electroscope in the Study of Radioactivity. We must turn aside for a moment to learn something of the use of the electroscope in these investigations. In a dry atmosphere, under normal conditions, a well insulated electroscope. (Fig. 367) retains its charge for several hours. Now it is found that dry air can be ren- dered conducting in several ways; and, when this happens within an electroscope, the leaves are discharged more or less rapidly, accord- ing to the degree of conductivity imparted. ELECTRICAL THEORY OF MATTER 631 This action takes place when a beam of rays from an X-ray tube falls upon a charged electroscope, as is shown by the fact that the leaves immediately begin to fall together. The action of the rays is indirect; for when they pass through air in a separate vessel and this air is afterward introduced into the electroscope, the leaves are dis- charged. The theory is that the X-rays, in passing through a gas, create such an atomic disturbance that here and there an atom loses an electron (negative ion), and itself becomes a positive ion. The process is called ionization. The gold leaves are discharged in ion- ized air by attracting to themselves the ions of opposite sign, the negative ions if the charge on the leaves is positive, and vice versa. The rays emitted by all the radioactive substances ionize the air through which they pass, and so have the power to discharge an electroscope. As a means of detecting the presence of minute quan- tities of a radioactive element, the electroscope as far surpasses the spectroscope as the latter does the most sensitive balance, the ratio of sensitiveness in either case being in the neighborhood of 100,000 to one. " The quantity of any radioactive substance which can be detected is to the corresponding amount of the other elements, which can be detected only by the ordinary methods of chemical analysis, as a second is to a thousand years." /. /. Thomson. 507. Other Radioactive Elements. The discovery of Professor Becquerel brought other workers into the field, and an extended search was made for other radioactive substances. In 1898 thorium and its compounds were added to the number. Thorium is a constituent of the Welsbach gas mantle. A piece of such a mantle, pressed out flat on a photographic plate and left in the dark for a week or more, takes its own photograph. Among the substances examined was the mineral pitch- blende, from which uranium is principally obtained. Madame Curie, of Paris, found that the activity of this ore, as shown by the rate of discharge of an electroscope, was three or four times as great as that of pure uranium. It was evident that pitchblende must contain some other sub- stance of much greater radiating power than uranium. 632 RADIATIONS To extract this substance from the ore proved to be a most difficult task, partly because it was present only in exces- sively minute quantities; but, from a ton of the ore, Mme. Curie finally obtained a few milligrams of the new element, which she called radium (1898). The ray-emitting power of this wonderful element is about 1,800,000 times as great as that of an equal quantity of uranium. From the same ore Mme. Curie extracted another highly radio- active substance, which she named polonium; and M. Debierne found still another, which he called actinium. A ton of pitchblende contains about .17 gram of radium and ^oVu as much polonium. Some idea of the labor involved in extracting these rare elements may be gained from the fact that the price of radium bromide (the form in which radium is usually obtained) is quoted at $75,000 per gram. The value of radium is not alone due to its use in scientific research. Its rays have been found to be a cure for certain disfigurements and affections of the skin, accomplishing in many cases what medicine and surgery can not. 508. The Three Types of Rays. The rays emitted by the radioactive elements are invisible, and can be made evi- dent and investigated only by their effects. Experiment has shown that the rays are of three kinds, and they are designated by the first three letters of the Greek alphabet, a (alpha), ft (beta), and y (gamma). The alpha rays are slightly deviated by a very intense magnetic or electrostatic field, and the direction of the deflection shows that they carry a positive charge; they are very efficient ionizers, as shown by the rapid discharge of an electroscope; they excite fluorescence, (see spinthariscope, below), but their photographic action is slight; they are completely stopped by a sheet of ordinary writing paper or a sheet of aluminum .05 mm. thick; they are emitted by all the radio- active elements named above. The a rays consist of positively charged particles (ions), shot out from the active substance within ELECTRICAL THEORY OF MATTER 633 average jveiocity-jof about TJ^OQQ mi. per. second The charge of an a particle is apparently equal to that of the hydrogen ion and its mass twice as great, or about 3400 times the mass of an electron. The fluorescent action of the a particles is beau- tifully shown by the Spinthariscope (Fig. 485), a device due to Sir William Crookes. "It is formed of a short brass tube, with a screen coated with crystal- line zinc sulphide at one end and an observing lens at the other. A small pointed brass needle is fixed a Fig. 485. Spin- few millimeters above the screen, and on the side of thariscope. S, f fl u or esce n t this nearest the screen a very minute quantity of screen . ^ ra _ radium is deposited by moistening it with a solution dium on point of a radium salt. On evaporation an invisible film of needle - of the salt remains." Viewed through the magnifying lens, the screen presents a perfectly dark background, upon which is seen a multitude of brilliant points of greenish-white light. These flash out and disap- pear in rapid succession, like the light of a thousand fireflies on a dark night. Each flash is due to the impact of an a particle, and is given out by the particular crystalline fragment which happens to be struck. " In these scintillations we have possibly the only direct evidence of the action of one individual atom known to science. The marvel is most impressive when it is remembered that the effect is produced and maintained incessantly by a quantity of radium too small to be vis- ible, and that there does not appear to be the slightest loss of activity with the lapse of time." C. W. Ra/ety. The beta rays are negatively charged particles, for they are strongly deflected in a magnetic or electric field, and the deflection is opposite to that of the a rays. They appear to be identical in character with the cathode rays, i.e. the ft particles are electrons. Their velocities vary from 40,000 to 170,0x30 mi. per second. Their photographic action is strong, their ionizing action relatively weak; they excite fluorescence. Owing to their high velocity they have considerable penetrating power; they pass readily through several millimeters of aluminum, and a thickness of i cm. of lead is required to stop all of them. They are emitted by uranium, radium, thorium, and actin- ium, but not by polonium. The gamma rays are always found in association with the ft rays. They have all the properties of X-rays, and appear to be of the same 634 RADIATIONS nature, i.e. to consist of pulses in the ether, traveling with the velocity of light. It is believed that they are produced by the expulsion of the ft particles from the active substance. They have great pene- trating power. Professor Rutherford states that the y radiation from 30 mg. of radium could be detected by the electroscope through 30 cm. of solid iron. 509. Nature of Radioactivity. According to all chem- ical tests, uranium, thorium, and radium are elements; they can not by any known agency be decomposed into unlike constituents. In forming compounds with other elements they exhibit no unusual properties. But their radioactivity is something wholly different from chemical activities in general. It takes place spontaneously, per- petually, and can neither be hastened nor retarded by any agency known to man. It is not affected in the slightest degree by the cold of liquid air or the most intense heat. It is an inherent and unalterable property of the atom itself, persisting unchanged in all chemical combinations of the radioactive elements with other forms of matter. The student who is familiar with elementary chemistry will readily understand that radioactivity involves changes within the atom itself, changes which are absolutely foreign to those which take place in chemical reactions. With the exception of the gain or loss of a few electrons (electric charges), by which atoms become ions and vice versa, the atom is supposed to retain its identity in all chemical changes whatever. The loss of a particle of atomic size (the a particle) from an atom of uranium, thorium, or radium means that the element itself is changing into some- thing else, a thing which the ancient alchemists sought to accomplish (the transmutation of the elements), but which the modern science of chemistry found to be impos- sible. It is impossible by human agency, so far as is known ; ELECTRICAL THEORY OF MATTER 635 but the facts of radioactivity find no other interpretation than that atomic disintegration is constantly going on in the radioactive elements, as a spontaneous process. 610. Disintegration Products. Origin of Radium. An enor- mous amount of experimental work has been done to determine the nature of the transformation products of the radioactive elements, and whole volumes have been written on the subject. In such work the main dependence must be placed on the evidence afforded by the electroscope; for in no instance have these products been obtained in a weighable amount except from the ores, in which they have been accumulating during geological ages. Radium continually evolves and gives off a gas in exceedingly minute quantities. This gas is known as the radium emanation. It is intensely radioactive and short-lived. It gives rise to a series of disintegration products (eight at least), which are solids and most of which are radioctive. What is supposed to happen is this: The radium atom loses an a particle and becomes an atom of the emana- tion; the latter loses an a particle and becomes an atom of the solid deposit from the emanation; and so on to the end. In some of the changes /Sand y rays are also given out. The last radioactive product of the series is believed to be polonium, which is always found asso- ciated with radium in pitchblende. It is conjectured that polonium changes into lead. The a particles have not been identified with any known element. Their mass seems to be intermediate between the mass of the hydrogen atom and that of helium (one of the rare inert gases). They may possibly be atoms of the latter element; for helium is found occluded in the radioactive minerals. Radium itself appears to be a disintegration product of uranium; it is always found in fixed proportion to the amount of uranium in uranium minerals. It is estimated that half of any mass of radium will disintegrate in the course of 2000 years. It is confidently ex- pected that the atomic shower in a spinthariscope will continue dur- ing that length of time, with a loss of only one half of its intensity. Thorium and actinium give rise to disintegration products, which form a definite series in each case, as with radium. 611. Theory of Atomic Disintegration. The disintegration of an atom is attended by the liberation of energy. The amount of 636 RADIATIONS this energy is no less astounding than the facts already presented. Owing to the high velocities of the a and /8 particles, their kinetic energy is relatively very great, and in the aggregate is seemingly out of all proportion to the mass of the radioactive substance. This energy is directly manifested as heat in any compact mass of radium; for the a particles shot off by atoms within the mass are stopped by collisions before they can escape, and their kinetic energy is thus converted into heat. Now it is found that radium maintains itself at a higher tempera- ture than its surroundings, and is constantly giving out heat at the rate of 100 calories per gram of pure radium per hour. Computa- tion shows that the total heat given out by a gram of radium during its "life" amounts to something like 3,000,000,000 calories, which is enough to raise the temperature of 33 tons of water from the freez- ing to the boiling point. If this energy could all be liberated in a single explosion, it would be capable of blowing a 2o,ooo-ton battle- ship 230 ft. into the air. Inconceivable as it may be, we are forced to the conclusion that this energy is stored within the radium atoms. The atom is pic- tured as a system of particles in rapid orbital motion. Their equilib- rium is not static, but dynamic, like that of the solar system. When, from whatever cause, the atomic system becomes unstable, a particle breaks away from its fellows and pursues an independent course, with the velocity it chanced to have at the instant. 612. The Electron Theory of the Atom. Theories concerning the constitution of .the atom are still in the formative state, but some of the main points appear to be established. Foremost among these is the hypothesis that the electron is a ronsti'tiient of all atoms; or, rather, that-the-atomJs.. largely-made up of electrons, their mim- bejrjmjijyTaiigemen t varying jsvith Jthe_diff erent. _dejnenls . These electrons are in rapid oscillatory or orbital motion, which in stable atoms (as of most elements) persists through geological ages, perhaps for all time, like the revolution of the planets round the sun. Heat imparts motion to the molecule as a whole, and pre- ELECTRICAL THEORY OF MATTER 637 sumably also to its constituent atoms and to the electron systems within the atoms. An oscillating group of elec- trons (negative electric charges) seems to answer all the requirements of a source of ether waves, as proposed in the electromagnetic theory of light (Art. 498). The charac- teristic spectra of the different elements (Art. 493) are accounted for on the supposition that different kinds of atoms have different numbers of electrons, differently arranged; for the natural oscillation period or periods of a system of electrons would certainly vary with their num- ber and grouping. One of the greatest difficulties encountered in formulating a con- ception of the atom is in reference to the positive electricity. It is doubtful whether this has ever been isolated from the atom as the elec- tron has. The positive charge of the atom has been compared by some to the sun, and the electrons to the planets revolving about it. Professor J. J. Thomson, who is the leading authority in such matters, pictures "a sphere of uniform positive electrification" with the electrons revolving inside it. // is rather startling to note that, according to either of these views, matter is nothing but electricity. According to the electron theory, positive electricity, being insep- arable from the atom, is not free to move in solid conductors; and an electric current in such conductors consists simply of a stream of electrons, moving in the direction opposite to what is universally termed "the direction of the current." Whatever may prove to be the truth, it is hardly probable that the present conventional terminology relative to the electric current will ever be changed. APPENDIX Table I. Metric Units Deci- means tenth i decimeter (dm.) = .1 meter (m.) i decigram (dg.) = .1 gram (g.) Centi- means hundredth i centimeter (cm.) = .01 meter i centigram (eg.) = .01 gram Milli- means thousandth i millimeter (mm.) = .001 meter = .1 cm. i milligram (mg.) = .001 gram Kilo- means thousand i kilometer (km.) = 1000 meters i kilogram (kg.) = 1000 grams The area of a square is the second power (square} of the length of one side, (i sq. ft. = i2 2 or 144 sq. in.) i square centimeter (sq. cm. or cm. 2 ) = 100 sq. mm. i square decimeter (sq. dm. or dm. 2 ) = 100 sq. cm. i square meter (sq. m. or m. 2 ) = 100 sq. dm. = 10,000 sq. cm. The volume of a cube is the third power (cube) of the length of one side, (i cu. ft. = i2 3 or 1728 cu. in.) i cubic centimeter (cu. cm. or cm. 3 ) = 1000 cu. mm. i cubic decimeter (cu. dm. or dm. 3 ) = 1000 cu. cm. i cubic meter (cu. m. or m. 3 ) = 1000 cu. dm. = 1,000,000 cu. cm. Table II. Equivalents Metric to English English to Metric i cm. = .3937 in. i in. = 2.540 cm. i m. = 39.37 in. i ft. = 30.48 cm. = 3.281 ft. i yd. = .9144 m. i km. = .6214 mile i mile = 1.6093 km. 638 APPENDIX 639 i cm. 2 = .i55osq. in. i sq. in. = 6.452 cm. 2 im. 2 = 1.196 sq. yd. i sq. ft. = 929.0 cm. 2 = 10.764 sq. ft. i sq. yd. = .8361 m. 2 i cm. 3 = .06103 cu. in. i cu. in. = 16.387 cm. 3 i dm. 3 = 1.0567 qt. (liquid) I CU. ft. = 28,315 cm. 3 i cu. m. 3 = 1.308 cu. yd. i cu. yd. = .7645 m. 3 = 35-3I7 cu. ft. i qt.' = .9463dm. 3 (liters) i gal. = 3.785 liters i gram = .0353 oz. I OZ. = 28.35 g- i kg. = 2.2046 Ib. ilb. = 453-6 g. Table III. Mensuration Rules ratio of the circumference of a circle to its diameter = 3.1416 Circumference of a circle (radius r) = 2 irr Area of a circle = trr 2 Surface of a sphere = 4 irr 2 Volume of a sphere = t irr 3 Lateral surface of a right cylinder (altitude h and radius of base r) = 2 irrh Volume of a right cylinder = irr 2 /* Table IV. Densities (in grams per ccm.) Solids (Except Mercury) Aluminum, cast . 2.58 Iron, bar . 7-8 Antimony, cast . 6.72 Iron, cast 7.2 to 7.3 Beeswax . .96 Ivory 1.9 Bismuth, cast 9.8 Lead . . - ... j 11.3 to 11.4 Brass .... 8-5 Marble . . . 2.72 Copper . 8.8 to 8.9 Mercury, at o C. 13-596 Cork . . . : .14 to .24 Platinum . 21.5 Galena 7.58 Quartz 2:65 German silver 8-5 Silver . . . . . 10.4 "to 10.5 Glass, crown . 2-5 Steel .... 7.8 to 7.9 Glass, flint . . 3 to 3.5 Sulphur, native . 2.03 Gold .... iQ-3 Tin .... 7-3 Ice. .017 Zinc, cast 7.1 640 APPENDIX Liquids and Solutions Alcohol (95%) . Blood .... Carbon disulphide Chloroform . Copper sulphate so- lution . Ether .... Glycerine . Hydrochloric acid Mercury at o C. .82 i.o6 1.29 1.16 .736 1.27 1.22 I3-596 Milk . . . Nitric acid Oil of turpentine Olive oil 1.03 i-5 .87 -915 Salt solution (NaCl), saturated . . 1.205 Sulphuric acid (15%) i.io Sulphuric acid . 1.8 Water (4 C.) . i.ooo Water, sea 1.026 Gases at o C. and 76 cm. Pressure Air 001293 Carbon dioxide . .001977 Carbon monoxide .001250 Hydrogen Nitrogen Oxygen .0000896 .001257 .001429 INDEX THE NUMBERS REFER TO PAGES Aberration, chromatic, 443 spherical, 387, 414 Absolute temperature, 245 Absolute units of force, 126 Absorption, of radiation, 231, 235, 237 selective, 237 Acceleration, 109-119 due to gravity, no, 115-119 Achromatic lens, 444 Action and reaction, 14, 129-131, 144 Adhesion, 200 Aeroplanes, 191-193 Air, buoyancy of, 59 composition of, 7, 266 density of, 43 water vapor in, 266-272 Air pump, 61 Amalgamating zinc, 512 Ammeter, 536 Ampere, 532, 605 Angle, critical, 401 of deviation, 390 of incidence and reflection, 374 of refraction, 390 refracting, of prisms, 398 sine of, 394 visual, 419-420 Anode, 602 Antinode, 348 Arch, 98 Archimedes, principle of, 37 Armature, of dynamos and motors, p 573 Artesian wells, 31 Atmosphere, heating of, 237 height of, 57-58 Atmospheric electricity, 497-499 Atmospheric pressure, 43-51 refraction, 402-405 Atom, 198 nature of, 613, 636 Atomic disintegration, 635 Attraction, electrostatic, 478 magnetic, 457-458 molecular, 199-201 of gravitation, 141-143 Audibility, limits of, 328 Aurora borealis, 58, 624 Axis, of lens, 407, 409 of mirror, 379 Balance, 19 Balloon, 66, 191-193 Barometer, 47-51 Battery, see Cells Beam, of light, 364 Beats, 331-333 Bell, electric, 524 Bellows, 63 Bichromate cell, 514 Biograph, 435 Boiling, 273-275 Boiling point, 223, 274 Boyle's law, 55 Buoyancy, center of, 89 of air, 59 of liquids, 36-38 Caloric theory, 218 Calorie, 248 Calorimetry, 248-251 Canal rays, 627 Capillarity, 211-214 Capstan, 169 Cathode, 602 641 642 INDEX Cathode rays, 624-627 Cells, electric, 503-517 electro-motive force of, 538 in battery, 549~552 storage, 607 Center of buoyancy 89 of curvature, 379 of gravity, 84-86 Centrifugal force, 139 Centripetal force, 135-139 Charge, electrostatic, 579, 589 Charles, law of, 244 Chemical changes, 4 effects of electric current, 600-608 energy, 180, 289 Chromatic aberration, 443 scale, 328 Circuit, electric, 510 divided or shunt, 543 Clouds, 271 Coal, energy of, 289-290 Cohesion, 199-201 Cold storage, 280 , Color, 441, 447-455 by interference, 454 vision, theory of, 453 Commutator, 578 Compass, 476 Compressibility, of gases, 8, 204 of liquids and solids, 8, 208, 215 Compression pump, 62 Condensation, of gases, 279 of water vapor, 271 Condenser, electric, 495 Conduction of heat, 225-227 Conductivity, electric, 542 Conservation, of energy, 181, 288, 290 Convection, of heat, 227-229 Couple, 82, 90 Critical angle, 401 temperature, 279 Crook es tubes, 624 Current, electric, chemical effects of, 601-608 danger from, 598 heating effects of, 556-563 induced, 564-571 magnetic effects of, 517-531 measurement of, 53 J -537 Current, nature of, 637 sources of, 502 unit of, 532, 605 Curvilinear motion, 135-140 Dalton's laws, 264 Daniell cell, 517 D' Arson val galvanometer, 535 Declination, magnetic, 472 Density, 20 and pressure of gases, 57 Derrick, 77, 170 Deviation, angle of, 390 Dew, 271 Dew-point, 267 Diffusion, of gases, 202 of light, 373 of liquids, 207 Dipping needle, 474 Discord, 326, 332, 341 Dispersion, of light, 437-442 Distillation, 275 Diving bell, 63 Divisibility, of matter, 197 Dynamics, 104-193 definition of, 104 of fluids, 183-193 of solids, 104-150 Dynamos, 573-588 Dyne, 126 Ear, 356 Earth, effect of rotation on its shape, 144-145 effect of rotation on weight, 145 magnetic field of, 472-476 revolution and rotation of, 19, 141-142 Echoes, 320 Eclipses, 365 Edison, 371, 595, 608 Efficiency, of machines, 161 Elasticity, 93 Electric, arc, 559 battery, 549~552 bell, 524 cells, 503-51 7 circuit, 510 conductors and non-conductors, 479 Electric cooking and heating, 561 current, see Current discharge, in rarefied gases, 622- 627- energy, 533, 553-556, 588 forging and smelting, 561 fuses, 562 light, 558-561 measurements, 531-553 motors, 583-587 oscillations, 617 potential, see Potential power, 533, 555 resistance, see Resistance spark, 494 telegraph, 525-531 transmission of power, 589-592 Electricity, current, 501-608 atmospheric, 497-499 nature of, Chap. XIV Electrification, 478-485 Electrodes, 505, 602 Electrolysis, 600-605 Electrolytic dissociation, 508, 603 Electro-magnet, 520-523 Electro-magnetic field, 517-520 induction, 564-571 theory of light, 619 waves, 618 Electro-metallurgy, 606 Electro-motive force, 537 measurement of, 547 of cells, 538 unit of, 537 Electrons, 625, 633, 636 Electrophorus, 487 Electroplating, 605 Electroscope, 481 Electrostatics, 477-500 Electrostatic attraction and repul- sion, 478 capacity, 497 charge, 479, 489 condenser, 495 field, 499 induction, 482 machines, 486-488 potential, 490-494 Electrotyping, 605 INDEX 643 Energy, availability of, 182 Energy, conservation of , 181, 288, 290 dissipation of, 182 forms of, chemical, 180, 289 electrical, 533, 553~556 kinetic, 151-157 mechanical, 177 molecular kinetic (heat), 206, 218-220 molecular potential (" latent heat"), 258-259, 276-278 muscular,. 180 of light, 361, 443 of rotation, 178 of sound, 315-317 potential, 177, 180 radiant, 230-239, 290 solar, 289-291 sources of, 160, 289 transformation of, by combustion, 289 by compression and expansion of gases, 278 by dynamos, 582 by electrical resistance, 556 by friction, 179 by fusion and solidification, 258 by motors, 585 by radiation and absorption, 230, 235-237 by steam engine, 293, 297 by vaporization and condensa- tion. 264, 276 transmission of, by electric current, 511, 588-592 by machines, 159-162, 166, i6P 172 units of, electrical, 556 mechanical, 154 thermal, 248 Engine, steam, 292-299 compound, 295 condensing, 294 efficiency of, 297 steam turbine, 302-304 gas, 299-302 gasoline, 302 Equilibrium, of concurrent forces, 69-76 644 INDEX Equilibrium, of floating bodies, 89 of parallel forces, 78-79 of two forces, 69 stable, unstable, and neutral, 84-90 Ether, luminiferous, 233, 360, 471, 618 Evaporation, 260-267 cooling by, 264, 281 Expansion, by heat, 206, 208, 215, 240-245 cooling by, 278 Extension, 16 Eye, 416-425 care of, 422 defects of, 420 Falling bodies, no, 115-118 Faraday, 564, 605 Far sight, 421 Field, electro-magnetic, 517-520 electrostatic, 499 magnetic, 466-471 Field magnet, 573 Floating bodies, buoyancy upon, 38 equilibrium of, 89 Fluids, characteristics of, 7-9 dynamics of, 183-193 Flywheel, 178 Focal length, of lens, 407 of mirrors, 382 Foci, of lenses, 407-410 of mirrors, 379-384 Fog, 271 Foot-pound, 154 Force, 9-12 buoyant, 36-38, 59 centrifugal, 139 centripetal, 135-139 electro- motive, 537 elements of, 70 graphic representation of, 70 moments of, 80-83 resultant, 13 units of, 1 8, 126 Forces, balanced, 12, 68-79 composition of, 73 concurrent, 69-76 molecular, 199-201, 204 Forces, parallel, 78-79 parallelogram of, 74 resolution of, 75 unbalanced, 12, 120-149 Force pump, 64 Franklin, 498 Fraunhofer lines, 440 Freezing, 253-255 Freezing mixtures, 259-260 Freezing point, 223, 256 Friction, 10 heating effects of, 179, 218-220 uses of, 131, 174-175 Frost, 271 Fulcrum, 163 Fundamental tone, 336-342, 351-354 Fusion, 252-259 change of volume during, 255 heat of, 257-259 Galilean telescope, 432 Galileo, 48, 131, 433 Galvanometers, 533-537 Gas engines, 299-302 Gases, characteristics of, 7-9 compressibility of, 8, 204 cooled by expansion, 278 diffusion of, 202 distinguished from vapors, 9, 262- 264, 279 effect of pressure on volume and density of, 55-57 effect of temperature on volume of, 206, 244 kinetic theory of, 207 liquefaction of, 279 molecular properties of, 202-207 pressure of, 52-57 statics of, 43-67 Gasoline engines, 302 Gay-Lussac, law of, 244, 246 Geissler tubes, 623 Glaciers, flow of, 257 Gram, mass and weight, 18 Gram-centimeter, 154 Gravitation, 141-145 law of, 143 Gravity, 84, 145 acceleration due to, no, 115-118 INDEX 645 Gravity cell, 516 center of, 84-88 pressure due to, 24-32 specific, 39-41 Guericke, 44, 6 1 Hail, 272 Harmony, 326, 341 Hearing, 356-358 Heat, 179, 206, 218-304 conduction of, 225-227 convection of, 227-229 expansion due to, 206, 208, 215, 240-245 kinetic theory of, 206, 218-220 mechanical equivalent of, 287 of fusion, 257 of vaporization, 276, sources of, 289 specific, 248-251 unit of, 248 Heat engines, 292-304 Heating of buildings, 283-287 Helmholtz, 291, 356 Hooke's law, 94 Horse-power, 157 Humidity, 268 Hydraulic press, 34 Hygrometer, 270 Ice, 253-258 manufacture of, 280 Illumination, intensity of, 368 artificial, 402 Images, by lens, 409-414 by plane mirrors, 375-378 by small opening, 367 by spherical mirrors, 382-388 real, 380, 383, 409, 412 virtual, 385, 410, 413 Incandescent lamp, 557-558 Inclined plane, 76, 112, 173 Index of refraction, 395 Induced currents, 564-571 Induction, earth's, 474 electro-magnetic, 564-571 electrostatic, 482 magnetic, 460 self, 569 Induction coil, 570-572 Inertia, 9-11, 120, 125 Insulators, electric, 479, 590 Interference, of light, 454 of sound, 329-332 Ions, 508, 603-604, 627, 630 Iridescence, 455 Joule, 220, 288 Joule's equivalent, 287 law, 556 Kilogram-meter, 154 Kinetic energy, 151-157 Kinetic theory, of gases, 202-207 of heat, 206, 218-220 Kinetics, see Dynamics Lamp, arc, 559 incandescent, 557-558 Lantern, optical, 435 Law, Boyle's, 55-57 Dalton's 264 Hooke's, 94 Joule's, 556 of Charles, 244 Ohm's, 539 Pascal's, 33 Laws of motion, Newton's, 120-132 Laws of nature, 28 Leclanche cell, 514 Lenses, achromatic, 443-445 concave, 413 convex, 405-413 Lever, 163 Leyden jar, 495 Lifting pump, 63 Light, 359-456 dispersion of, 437-442 intensity of, 369-371 interference of, 454 propagation of, 362-363, 366 reflection of, 372-375^ refraction of, 389-398 theory of, 359-361, 619 velocity of, 362 wave length of, 366, 441 Lightning, 497 rod, 499 6 4 6 INDEX Lines of force, 466-471, 576 Lintel, 97 Liquids, characteristics of, 7-9 diffusion of, 207 dynamics of, 183-193 molecular properties of, 207-214 statics of, 22-42 Liter, 16 Local action in voltaic cell, 511 Locomotive engine, 296 Lodestone, 457 Loudness of sound, 314 Machines, 159-175 efficiency of, 161 electrical, 486-488 mechanical advantage of, 161 Magdeburg hemispheres, 44 Magnetic declination, 472 effects of a current, 517-531 field, 466-471 inclination or dip, 473 induction, 460 lines of force, 466 meridian, 472 needle, 458 permeability, 462 poles, 458 substances, 460 Magnetism, 457-476 terrestrial, 472-476 Magnetization, permanent and tem- porary, 461 . theory of, 462-465 Magnifying glass, 426 power, 426 Major triad, 326 Manometers, 53-55 Mass, center of, 84 definition of, 17 measurement of, by weight, 19 units of, 1 8 Matter, divisibility of, 197 properties of, 7-9, 194-217 states of, 7-9, 200, 252 structure of, 196-202 Measurement, 15-20 Mechanical advantage, 161 equivalent of heat, 287 Mechanics, definition of, 5, 22 Melting points, 254 effect of pressure on, 256 Meter, 16 Microphone, 595 Microscope, compound, 427 simple, 426 Mirage, 404 Mirrors, parabolic, 387 plane, 376-378 spherical, 379~3S& Molecular forces, 199-201, 204 motion, 202-209, 215 structure of matter, 194-217 Molecule, 197 Moment of force, 80-83 Momentum, 127 Moon, revolution of, 143 Motion, 104-120 accelerated, 108 curvilinear, 135-140 laws of r 120-132 of falling bodies, 115-118 of pendulum, 146-149 of projectiles, 116-118 on an inclined plane, 111-114 uniform, 105 wave, 309 Motor, electric, 583-587 Musical instruments, 344, 350-354 intervals, 324-328 scales, 326-328 sounds, 313, 324 Newton, 132, 142 Newton's law of cooling, 236 laws of motion, 120-132 Nodes, 337, 347 Noise, 324 Octave, 325 Ohm, definition of, 538 \*S Ohm's law, 539 , Opera glass, 432 Optical instruments, 425-436 Organ pipes, 35-3S3 Overtones, 336-341, 35i~354 Parallelogram of forces, 74 INDEX 647 Pascal's law, 33 Pendulum, 146-149 Penumbra, 365 Permeability, magnetic, 462 Phenomena, natural, 5 Phonograph, 344 Photometry, 369-371 Physical changes, 4 Pinhole image, 367 Pitch, of musical sounds, 307, 322, 324-329 Plasticity, 94 Polarization, in voltaic cell, 512 Poles, magnetic, 458 Porosity, 214 Potential, electric, 490 fall of, 540 Potential energy, 177, 180 Pound, weight and mass, 18 Poundal, 127 Power, 157 electric transmission of, 589-592 Pressure, atmospheric, 43-51 of gases, 52-57 of liquids, 24-36 of moving fluids, 183-187 of vapors, 262-264, 274 Pressure gauges, 53-55 Prism, 398 Prism binocular, 433 Projectiles, 116-118 Properties of matter, 194-217 Pulleys, 170-173 Pump, air, 61 compression, 62 force, 64 steam, 65 suction, 63 Quality of sound, 322, 338-341 Radiant energy, 230-238, 361 emission of, 231, 235 reflection of, 236 selective absorption of, 237 transmission of, 230-234 visible and invisible, 232 Radioactivity, 630-637 Radiometer, 235 Radium, 631, 634 Rain, 272 Rainbow, 445 Ray, of light, 364 Reaction and action, 14, 129-131, 144 Reflection, of light, 372-375 of sound, 320 total, 399-401 Refraction, 389-398 atmospheric, 402-405 index of, 395 laws of, 393 relation to velocity, 390-395 Relay, telegraph, 527 Resistance, 161 of the air, 123, 189 of water, 189 Resistance, electrical, 538 laws of, 541-545 measurement of, 546, 548 specific, 542-543 unit of, 538 Resistance coils, 546 Resolution, of a force, 75 of a velocity, 106 Resonance, 346-350 Resultant force, 13, 74, 79 velocity, 106 Rumford, Count, 218, 369 Scales, musical, 326-328 Screw, 174 Screw propeller, 190 Selective absorption, 237 Self-induction, 569 Shadows, 364 Short sight, 420 Shunt circuit, 543 Sine of an angle, 394 Siphon, 65 Sky, color of, 450 light of, 373 Snow, 272 Soap bubbles, 210 Solenoid, 520 Solidification, 252-258 Solids, characteristics of, 7, 8 dynamics of, 108-183 6 4 8 INDEX Solids, molecular properties of, 214- 217 statics of, 68-103 Solution, heat of, 259 Sonometer, 333 Sound, 305-358 intensity of, 314-3*7 interference of, 329-332 loudness of, 314 media, 308 origin of, 306 pitch of, 307, 322, 324-329 properties of, 322-342 quality of, 322, 338-341 reflection of, 320 transmission of, 308-314 velocity of, 317-319 waves, 311, 329 Sounder, telegraph, 526 Speaking tubes, 317 Specific gravity, 39-41 heat, 248^251 resistance, 542 Spectrum, 438-440 invisible, 442 solar, 615 Spectra and spectrum analysis, 610- 616 Speed, 105 Spherical aberration, 387, 414 Stability, 88-90 Stars, distance of, 362 twinkling of, 403 Statics of gases, Chap. IV of liquids, Chap. Ill of solids, Chap. V Steam, heating by, 286 pressure of, 275 saturated, 262 superheated, 262, 298 Steam engines, 292-299 Stereoscope, 424 Stresses and strains, 92-102 Strings, vibration of, 334-341 Sun, energy of, 289-291 Surface tension, 210-212 Suspension cable, 102 Sympathetic vibrations, 342, 345- 350 Tangent galvanometer, 533 Telegraph, 525-531 Telegraphy, wireless, 620 Telephone, 592-598 acoustic, 317 exchange, 597 Telescopes, 429-434 Temperature, 220-224 absolute, 245 ~~ Tenacity, 94, 217 Terrestrial magnetism, 472-476 Theory, definition of, 195 of electricity, 630-637 of gases, 202-207 of heat, 206, 218-220 of light, 359-361, 619 of magnetic action, 470 of magnetization, 462-465 of the structure of matter, 194-217 Thermometers, 222-224, 242 Thrust, 30 Thunder, 497 Time, unit of, 19 Tone, 324 fundamental, 336-342 Torricelli, 49 Total reflection, 399-401^ Transference of energy, see Energy Transformation of energy, see Energy Transformer, 590-592 Truss, loo-ioi Tuning fork, vibration of, 307 Turbine engines, 302-304 Umbra, 365 Units, fundamental, 20 of acceleration, no of current strength, 532, 605 of electrical power, 555 of electrical resistance, 538 ofE.M.F., 537 of extension, 16 of fluid pressure, 29, 55 of force, 1 8, 126-127 of heat, 248 of mass, 1 8 of mechanical power, 157 of time, 19 of velocity, 105 INDEX 649 Units of work and energy, 154 Vacuum, 47 Vapor, atmospheric, 266-272 pressure of, 262-264 Vaporization, 260-282 - heat of, 276 Vapors, 9, 262-264, 279 Velocity, 105-108 graphic representation of, 105 resolution of, 106 uniform, 105 variable, 108-119 Velocities, composition of, 105 Ventilation, 283-287 Vibration, forced and sympathetic, 342-358 of air columns, 346-354 of bells, 342 of pendulum, 147 of strings, 334~34i Vision, binocular, 423-425 Vocal cords, 355 Voice, 354 Volt, 537 Voltaic cell, 507 Voltmeter, 547 Water, compressibility of, 8, 208 density of, 20 electrolysis of, 502 evaporation of, 265-267 expansion of, 243 Water wheels, 185-188 Watt, 555 < Wave motion, 309 Waves, of light, 359~36i of sound, 311, 329 of water, 310 Weather, prediction of, 50 Wedge, 174 Weighing, 19 Weight, 11-12, 16 Welding, 200 electric, 561 Wheel and axle, 167 Wind instruments, 350-354 Windlass, 168, 170 Winds, 229 Work and energy, 150-157 units of, 154 X-rays, 628 Zero, absolute, 245 UNIVEESITY OF CALIFOKNIA LIBEAEY, BERKELEY THIS BOOK IS DUE ON THE LAST DATE STAMPED BELOW Books not returned on time are subject to a fine of 50c per volume after the third day overdue, increasing to $1.00 per volume after the sixth day. 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