Hatt OfoUegc of 3lgrtcult«w Kt dorncU IniaerBtta art^aia. Si. a. Cornell University Library QC 21.C93 A text-book of general physics for colle 3 1924 002 939 613 Cornell University Library The original of tiiis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/cletails/cu31924002939613 A TEXT-BOOK OF General Physics FOR COLLEGE STUDENTS ELECTRICITY, ELECTROMAGNETIC WAVES, AND SOUND BY v.n. )/ J. A. CULLER, Ph.D. PROFESSOR OP PHYSICS, MIAMI UNIVERSITY; AUTHOR OF GENERAL PHYSICS FOR COLLEGES, MECHANICS AND HEAT PHILADELPHIA J. B. LIPPINCOTT COMPANY Copyright, 1914 By J. B. LippiNCOTT Company Printed by J. B, Litpincotl Company The Washington Square Press, Philadelphia, V.S.A, PREFACE Several aims have been kept in mind in the preparation of the following text: (1) To make the descriptions, proofs, state- ments, and illustrations clear to the average student. (2) To emphasize the physical side of physics, point out its applications in the commercial world, and give more than an outline in the development of a topic. (3) To incorporate in the body of the discussions and in their proper place the electronic and electro- magnetic theories now so well established. The discussion of electricity begins with the experimental evidence which led to a belief in the electron, and the ordinary phenomena of electricity are explained in accordance with this theory. Magnetism comes early in the discussion and emphasis is placed on the fact that a magnetic field is the result of an electric charge in motion. The explanation of electrolysis is given in accordance with modem theories, and an unusual amount of space has been given to the very important subject of electromagnetic induction. After a study of electricity the next natural and logical step is to electromagnetic waves. These embrace the whole subject of ether disturbance, including those short waves which produce light. Light is therefore treated as a subdivision of this general heading. Here also is introduced the evidence in favor of the belief that light is an electromagnetic disturbance resulting from rapid vibrations of electric charges, such, for example, as the spec- troscopic effect when a source of light is in a magnetic field, the phenomena of radio-activity, etc. No attempt has been made to produce a compendium of physics but rather a logical development of the live topics which, it seems, should be included in a text-book for college students. We desire to thank the Leeds & Northrup Co. for cuts show- ing standard resistance, shunts, and inductance; the General Electric Co. for Figs. 101, 102, 106, 127, and 128; the Weston Electrical Instrument Co. for Fig. 69; Wm. Gaertner & Co. for Fig. 97; and the Electric Storage Battery Co. for Fig. 55. J. A. Culler. CONTENTS CHAPTER I electricity section page 1. What Electricity Is 1 2. Evidence for the Electron Theory 3 3. Kinds of Electricity 7 4. Quantity of Electricity 7 5. Field of Force 8 6. Potential 10 7. The P.D. Due to a Charged Point 12 8. Equipotential Surfaces 13 9. Electric Induction 14 10. Charging by Induction 15 11. Capacity of a Conductor 16 12. Location of the Charge 17 13. The Field within a Hollow Charged Conductor 17 14. Distribution of the Charge on a Conductor 19 15. The Electric Spark 20 16. The Energy of a Charge 21 17. Lightning 22 18. Condensers 24 19. Capacity of a Spherical Condenser 24 20. Electrometers 27 21. Specific Inductive Capacity 29 22. Residual Charge 29 23. The Electric Machine 30 24. Dimension of Electrostatic Units 33 CHAPTER II magnetism and the electric current 25. What Magnetism Is 36 26. Permeability. ' 38 27. Magnets 39 28. Theory of Magnets 40 29. Poles of a Magnet 41 30. Law of Magnets : 44 31. Unit Magnetic Pole 44 32. Magnetic Lines of Force 45 33. Intensity of a Magnetic Field 46 34. Magnetic Induction 46 V vi CONTENTS. SECTION PAGE 35. Intensity of Magnetization 47 36. Susceptibility 47 37. Permeability 47 38. The B-H Diagram 48 39. Hysteresis 48 40. Effect of Temperature on Magnetism '. 50 41. Magnetic Moment 51 42. The Field Produced by a Magnet 52 43. Action of a Magnetic Needle in Two Fields at Right Angles 64 44. Determination of M and H 55 45. Terrestrial Magnetism 56 CHAPTER III electromagnetic and practical units 46. Unit Strength of Current 61 47. The Ampere and Coulomb 62 48. Comparison of e.m. and e.s. units 62 49. Unit Difference of Potential 63 60. The Volt 63 61. Electromotive Force 63 52. Unit of Resistance. Ohm's Law 64 53. Concrete Standards 65 54. Energy Relations 65 CHAPTER IV conduction of electricity through solutions 55. An Electrolyte 68 56. The Electrolytic Cell 68 57. Action within an Electrolytic Cell. Dissociation 68 58. Transfer of Electricity in Electrolytes 73 59. Faraday's Laws of Electrolysis 75 60. Electrochemical Equivalent 77 61. Charge on an Ion 78 62. The Voltameter 79 63. Uses of Electrolysis 80 CHAPTER V battery cells 64. The Voltaic Cell 82 65. Contact Difference of Potential 83 66. Polarization 86 67. Origin of E.M.F. in an Electric Cell 88 68. Energy Relations 90 CONTENTS. vii section page 69. Other Primary Cells 90 70. Standard Cells 91 71. Storage Cells 93 72. Arrangement of Cells 94 CHAPTER VI galvanometers 73. The Tangent Galvanometer 98 74. The Astatic Galvanometer 100 75. The D'Arsonval Galvanometer : 101 76 The String Galvanometer 105 77. The Ballistic Galvanometer 106 78. Ammeters and Voltmeters 107 79. The Electrodynamometer 108 80. The Wattmeter 109 CHAPTER VII measurement of electrical resistance 81. Relation of E, i, and R 112 82. Specific Resistance 112 83. Temperature Coefficient of Resistance 113 84. Currents in Series and in Parallel. Kirchhoff's Laws... . 113 85. Wheatstone Bridge 115 86. Shunts 120 87. Potentiometer 122 CHAPTER VIII thermoelectricity 88. Heat and Electricity 125 89. The Seebeck Effect 125 90. The Peltier Effect.! 128 91. The Kelvin Effect 130 92. Uses of Thermoelectricity 130 CHAPTER IX condensers 93. Capacity of Condensers in e.m. and Practical Units 132 94. Standard Condensers 132 95. Measurement of Capacities 134 96. Ratio of e.m. to e.s. Unit of Quantity 136 97. Dielectric (Constant 137 viii CONTENTS. CHAPTER X electromagnets section page 98. Solenoids 139 99. Magnetic Flux 140 100. Magnetomotive Force 141 101. Magnetic Reluctance 142 CHAPTER XI electromagnetic induction 102. Induction 144 103. Nature of Electromagnetic Induction 144 104. The Principle of the Dynamo 149 105. The Direct Current Dynamo 153 106. Eddy Currents 155 107. Winding of D.C. Dynamo , 156 108. Polyphase Generators 157 109. The Induction Motor 160 110. Synchronous and D.C. Motors 161 111. Self-induction 162 112. Measurement of Inductance 165 113. Mutual Inductance 168 114. The Induction Coil 168 115. Alternating Current Transformer 169 116. Tesla Transformer 171 117. Effective Value of Alternating Current 172 118. Impedance 173 119. Lag and Lead 174 120. Power of Alternating Current 177 121. Advantage in Use of an Alternating Current 178 122. Rectifiers 179 123. Dimensions of Electromagnetic Units 181 CHAPTER XII * electromagnetic waves 124. Electric Oscillations 185 125. Length of Ether Waves 185 126. Electric Resonance 186 127. Experiments of Hertz 187 128. Electric Nodes and Internodes 189 129. Wireless Telegraphy 190 130. Light 193 131. Source of Light Waves 194 132. Radio Activity 194 CONTENTS. ix SECTION PAGE 133. Velocity of ;8 Particles 196 134. The Ratio of e to »i and the Value of e 198 135. Optics 199 136. Theories of Light 199 137. Wave-front and Huygens' Principle 201 138. Interference of Waves 202 139. Rectilinear Propagation : 203 140. Velocity of Light 206 141. Reflection from a Plane Surface 208 142. Curvature 209 143. Reflection of a Spherical Wave from a Plane Surface 210 144. Reflection from a Curved Surface 211 145. Spherical Aberration. Caustics 216 146. Parabolic Mirror 217 147. Refraction 219 148. Refraction of a Spherical Wave at a Plane Surface 221 149. Relative and Absolute Index of Refraction 222 150. Critical Angle. Total Reflection 223 151. Refraction by a Prism 225 152. Spectrometer 226 153. Refraction of Spherical Waves at a Spherical Surface 229 154. Lenses 230 155. Optical Centre 234 156. Location of Images by Drawings 235 157. Spherical Aberration 236 158. The Spectrum 237 159. Diffraction Grating 238 160. Wave-length of Light 240 161. Dispersive and Resolving Power of Gratings 241 162. Prismatic Spectra 242 163. Kinds of Spectra 243 164. Limits of the Spectrum 246 165. The Spectroscope 247 166. Chromatic Aberration 248 167. Color 249 168. Complementary Colors 250 169. Color Resulting from Absorption 251 170. Polarized Light 251 171. Polarization by Reflection 253 172. Brewster's Law 254 173. Polarization by Refraction 255 174. Polarization by Double Refraction 256 175. The Nicol Prism 258 176. Circular and Elliptic Polarization 258 177. Color of Thin Plates 260 178. Rotation of the Plane of Vibration 261 X CONTENTS. SECTION PAGE 179. The Zeeman Effect 261 180. Artificial Light 264 181. Candle-power 265 182. Intensity of Illumination 266 183. Photometry 267 CHAPTER XIII SOUND 184. Media of Communication 270 185. Nature of Sound Waves , 270 186. Phenomena of Sound Transmission 272 187. Velocity of Sound Transmission 273 188. Effect of Changes in Temperature and Pressure 274 189. Musical Tones 275 190. Pitch 276 191. Intensity 276 192. Resonance 277 193. Resonance in a Tube Open at Both Ends 280 194. Quality 281 195. Nodes in Resonant Air Columns 282 196. Organ Pipes 284 197. Velocity of Sound by Kundt's Method 285 198. Vibration of Strings 286 199. Diatonic Scale 288 200. Construction of a Major Diatonic Scale 289 201. Scale of Even Temperament 291 Appendix 294 GENERAL PHYSICS ELECTRICITY, ELECTROMAGNETIC WAVES, AND SOUND CHAPTER I ELECTRICITY I. What Electricity Is. — Various theories as to the nature of electricity have been proposed from time to time. All have been of service in the advancement of science, and each in its time was probably the best that covild be formulated from the experimental data then at hand. A one-fluid theory was proposed by Benjamin FranHin in 1750. According to this theory electricity was assumed to be a self-repeUant fluid distributed through all matter. AU bodies in a normal state were assumed to contain a definite quantity of this fluid. It was then explained that a body deficient in its normal quantity of fluid would attract a normal body but would repel another body which was also deficient. This involved the assumption that bodies devoid of any electric fluid would repel each other. Thus a body was regarded as positively charged when it con- tained an excess of electric fluid, and negatively charged when it contained less than what naturally belonged to it. Another theory prevalent at that time was the two-fluid theory which assumed that there were two weightless and continuous fluids in every body of matter. One, such as that on glass when rubbed with silk, was called positive. The other, such as that on rosin or hard rubber when rubbed with cat fur or woolen cloth, was called negative. In a normal body these two fluids were assumed to be present in equal quantities, thus neutralizing each other. When a conductor was brought into an electric field, the two fluids present in it would be separated, one being repelled and the other attracted. In case of insulators such as mica, glass, \( ' 1 2 • GENERAL PHYSICS. silk, etc., the fluids were not easily separated, and when bodies were electrified by rubbing one on the other only the surface layers were affected, an excess of positive fluid being found on one and an excess of negative on the other. This theory was probably never intended as an explanation of the nature of electricity, but it has been valuable as a means of describing and investigating electrical phenomena and as such has been extensively used. A modem theory which is offered in place of the two-fluid theory assumes that atoms of matter are composed of positive and negative particles or corpuscles. These may become detached from their atoms and are then free to move along a conductor under the influence of an outside electrical force. Insulators would then be explained as substances in which corpuscles are not easily separated from their atoms. The chief difference be- tween this and the old two-fluid theory consists in the assumption that instead of two continuous fluids there are two kinds of dis- crete particles whose presence and movement cause the electrical phenomena which are observed. A modem theory which seems most plausible because most nearly in accord with experimental results is a modification of the one-fluid theory. Each atom of a substance is here regarded as made up of a number of minute particles called corpuscles or electrons all of which are negative. These are very minute, having a mass of about rsW that of an atom of hydrogen. There is a great deal of experimental evidence tending to prove the existence of these minute negative particles. It is assumed that there is in each normal atom sufficient positive electricity to neutralize the electrons all of which are negative. Positive particles are always found to have nearly the mass of the atom. Consequently it is believed that in conductors there are many roaming electrons which have been detached from atoms and which may at times connect themselves to other atoms. Under the influence of an electric force these are set in motion, causing what is called the electric current. In solid conductors the positive portion of the atom does not move. Whenever a body contains an excess of corpuscles it is said to be charged negatively, i.e., there are more electrons than are needed to neutralize the positive. Whenever a body is deficient ELECTRICITY. 3 in electrons it is said to be charged positively, for there are not enough electrons to neutralize all the positive. It is unfortunate that the terms positive and negative were not reversed in their first application to the two kinds of electricity. Preference will be given to this so-called electron theory in the following discussions. 2. Evidence for the Electron Theory. — ^A theory cannot be regarded as established nor is it accepted by the modem scientific world until experimental tests show that it is not only in accord- ance with actual results but that the line of laboratory tests which the theory suggests gives results which would be expected. A theory is often crude at first and yet may serve as a working hypothesis. Prom time to time it may be modified by experi- mental results tuitil, if false, it is abandoned, or, if true, is estab- lished and generally accepted. It must not be understood, how- ever, that a theory is first formulated and an effort then made to confirm it by experimental results. There is rather first a great mass of facts often isolated and not satisfactorily explained. A theory is then proposed which is in accord with these facts and which gives a rational basis for the explanation of all. The im- portant thing in science is not the theory but the experimental facts, and the true scientist works diligently for the latter without being prejudiced by the former. The electron theory is not the result of a sudden discovery but rather a growth from the accumulated evidence of years of experiment. Until near the end of the 19th century the atom was the smallest particle of which science had any experimental knowledge. Experiments with electrons had been made before that time but they were not recognized as such. Now it is believed that the atom is composed of many perfectly distinct particles. The atom holds its place in our conception of matter just as before, but it is no longer regarded as an indivisible particle nor as the ultimate particle of which all matter is composed. In Fig. 1 is shown a glass bulb with terminals A and C sealed in the walls. If the air is now pumped out until the pressure within is about .01 mm. of mercury and a current of high voltage, such as that from an induction coil or electric machine, is passed in at A and out at C, there will issue from C a stream of particles 4 GENERAL PHYSICS. called cathode rays. These were so called because, according to conventional terms, C is the cathode and A the anode when the current flows in the direction indicated. If the direction is re- versed, cathode rays will stream from A. These rays move in a straight line normal to the surface from which they come. The glass at 5 will fluoresce with a greenish-yellow light. Various kinds of crystals will become luminescent when cathode rays are directed upon them. A light wheel will be vigorously turned when these rays are directed against the paddles on one side of the wheel. A metal disc placed in the path of the rays will effec- tively screen the space beyond. It was at first thought that cathode rays were some kind of ether waves, but this notion has been entirely abandoned. Sir Fig. 1. William Crookes maintained that -cathode rays consisted of nu- merous material particles projected from the cathode. A strong evidence in favor of his claim is that when a magnet M (Fig. 2) is held near the tube as shown, the rays are drawn down between the poles of the magnet or repelled in the opposite direction, de- pending on the position of the magnet; i.e., if one position attracts the rays. Then on turning the magnet so the poles are reversed the rays ,are repelled. This effect is just the same as that of a magnet on a conductor carrying a current of electricity (see Fig. 95). Ether waves are not affected in this manner by a magnetic field. If the cathode stream is in fact composed of discrete particles charged with electricity it would seem at first thought that they woidd be about the size of ordinary atoms and so could not pass out through the walls of the tube any more than atoms of air ELECTRICITY. 5 could pass into the vacuum within. Lenard in 1893 constructed a tube with a small aluminimi window and, by directing cathode rays against the aluminum, showed that the particles which compose these rays pass out and move on through several centi- metres of air before they are stopped, while at the same time no air could enter the tube. This seemed to be a strong objection to the claim that cathode rays were composed of projected par- ticles of matter. But when it is once asstimed that this cathode stream is composed of minute parts of atoms, each bearing about the same relation to the dimensions of the atom as a speck of dust to the dimensions of a room, it is seen that these minute particles could pass through the thin partition of aluminum and on for a distance into the air before they would all be stopped by atoms which might lie in their paths. Fig. 2. The mass of these small particles — electrons — ^have been cal- culated many times from different experimental data. The re- sults always show a mass about tsVtj that of the lightest atom — that of hydrogen. A method of making this calculation is given in § 133. An important property of cathode ray particles is that they carry a negative charge of electricity, i.e., all electrons are nega- tive. This may be shown by use of a tube designed by J. J. Thomson. Here A, Fig. 3, is the anode and C the cathode. A perforation in A permits some of the cathode rays from C to pass into the chamber T and faU upon the wall at some point P. Now by use of a magnet as explained above the rays may be deflected from their course, and instead of falling on P may be made to 6 GENERAL PHYSICS. fall on M which is a conductor connected at £ to an electroscope. The conductor n is surrounded by a metal tube which is connected to the ground and thus shields n from any electrification except that caused by the cathode rays. The effect on the electroscope shows that cathode rays always carry a negative charge. The direction in which the rays are deflected by a magnet also shows that the charge must be negative. Further evidence of the electron is found in radio-active sub- stances such as uranium, radium, etc., which give oflE several kinds of rays as will later be described more fully. Among these are the beta rays, which possess properties almost identical with cathode rays and so are not atoms but minute parts of atoms which are being thrown off in natural changes which are taking place in matter. r z^ Fig. 3. These negative particles called electrons — also, by some, called negative corpuscles — ^are the parts of the atom with which we can deal in experiment, i.e., we get them off by themselves, ob- serve their properties, and calculate their mass and velocity. They appear to be constituents of all kinds of matter. They do not differ when different electrodes and different gases are used in the vacuiun tube. But if electrons are constituent parts of the atom, then, according to our present knowledge of electricity, it is necessary to assume that in each atom there is a quantity of electricity in some form which will just neutralize the negative of the electrons and give us the neutral atom with which we com- monly deal. Rays which carry a positive charge are observed in experiment, but in all cases the particles which compose these rays have nearly the dimensions and mass of the atom itself. It is probable that these positive particles are only atoms from which one or more electrons have been detached. . Other strong evidences of the electron theory have been dis- ELECTRICITY. t4 covered and will be described in connection with other topics. Enough has been said at this point to indicate the line of thought and experiment which has led up to this theory. 3. Kinds of Electricity. — ^Long before the corpuscular nature of electricity was known it had been agreed to call that charge positive which appears on glass when it is rubbed with silk. In some way, not well known, the contact and separation of glass and silk cause a deficiency of electrons on the glass and an equal quantity in excess on the silk. Likewise it had been agreed to call that charge negative which appeared on ebonite or hard rub- ber when these were separated after contact with fur or woolen goods. Here electrons are found in excess on the ebonite while the fur and wool have less than the normal number. If a pith ball is suspended from a silk fibre and touched with a charged rod of ebonite, some electrons will pass from the rod to the ball, thus charging it negatively. The rod will then repel the baU. If a charged glass rod be touched to another pith ball some electrons will pass from the ball to the glass, leaving the ball positively charged. The two pith balls will now attract one another. Hence the law : Like charges repel and unlike charges attract one another. A convenient instrtunent for testing these and other phenomena of this so-caUed static electricity is shown in Fig. 4. A pith ball is attached to the end of a slender fibre of glass. The glass is supported by silk fibres as shown. The ball will swing horizontally in response to a very small force. If, by use of the ebonite rod, the ball is given more than its normal number of electrons it will be charged negatively and can then be used as an electroscope to detect the kind of charge in other bodies. 4. Quantity of Electricity. — The quantity of static electricity is measured by the force which it would exert upon a known Fig. 4. 8 GENERAL PHYSICS. quantity at a given distance from it. If Qi and Qa are two quan- tities at a distance r from one another, then, as experiment shows, the force F between them will vary directly as the product of these quantities and inversely as the square of the distance be- tween them. This is shown in formula by: Fo.^ (1) For tmit quantity r is 1 cm. and Qi is equal to Qi. Then when, in air, Qi is such a quantity that F is 1 dyne, Qi or Q2 is the electro- static unit of quantity. In words, the electrostatic unit of quantity is that quantity which when placed at a distance of 1 cm. in air from an equal quantity will act upon it with a force of 1 dyne. Equation (1) may then be written: F^^ (2) This will give the correct value of F in dynes, provided the experi- ment is made in air. Strictly, however, the definition should specify " in vacuum " instead of " in air," but the difference in the value of F would be very slight, and it is much more conveni- ent to use air as a mediimi between the charges. If, however, some other substance such as a plate of glass, mica, or sulphur is interposed between the charges, the value of F is diminished in nearly aU cases, only a few gases being exceptions. The sub- stance between the charges is called the dielectric, and the ratio of the value of F in air to its value when some other dielectric is used is called the dielectric constant and is usually designated by K. Equation (2) for any dielectric may then be written: F = ^ (3) Values of K for various dielectrics are given in Appendix, Table 10. 5. Field of Force. — ^A field of force is any region in which force may exist. The strength or intensity of an electrical field is the force with which a unit charge, as defined above, would be urged if placed at that point. A unit field is one in which a unit charge is acted on by a force of 1 djme. The direction of a field is the direc- tion in which a positive charge would move if placed in that field. ELECTRICITY. 9 A field is often represented by lines called lines of force. These are lines whose direction at any point is the same as the direction of the field. Each square centimetre of surface at right angles to the direction of the field is regarded as including as many lines of force as there are units of field intensity at that point. This is true of magnetic, gravitational, and electrical fields, though in the last the term tubes of force is frequently used instead of lines of force, and the ntmiber of tubes is numerically equal to the intensity of the field divided by 4ir. By so doing the number of tubes indicates the number of unit charges on the surface of a conductor and also the electrical density on any portion of the surface. Hence we may say that 1 sq. cm. of surface at right angles to the direction of the electrical field will include as many tubes of force as there are units of intensity or field strength at that point divided by 4t. The reason for this will appear from the following: If a unit of electricity is included within a given area on the surface of a positively charged body and lines are drawn from every point of the boundary of this area in the direc- tion of the field of force, these lines will include a tube of force. These tubes are always regarded as passing out from a positive charge and ending on a negative one. The greater the charge the greater the number of tubes of force per square centimetre at any point ia the field. Let a very small spherical conductor B, Fig. 5, be charged with Q imits of positive electricity. Q tubes of force will then radiate from S in all directions. Let .A be a portion of an imagin- ary surface of a large sphere concentric with B and at a distance r cm. The area of the surface of A is iirr^ and it intercepts the Q tubes of force from B. Hence the number of tubes per square centimetre on A is Q/Awr^. Also, since B is very small, its charge may be assumed to be at the centre of the sphere A. The strength of the field at a distance r from the Q units on B is Q/r^ as shown by equation (2) and the definition of strength of field. But r» iirr'' and it has just been shown that Q/47rr' is the number of tubes per square centimetre. Hence the number of tubes of force per square centimetre at right angles to the direction of the field is 10 GENERAL PHYSICS. B equal to the intensity or strength of the field divided by 4jr. Instead of drawing tubes, the field may be represented quite as well by drawing one line along the axis of each unit tube. If two charges, equal but opposite in kind, are placed near each other, the lines or tubes of force would be as shown in Fig. 6, A, but if the charges are the same in kind, the lines would be as in 5. In a tmiform field the tubes are everywhere parallel and each has the same cross section throughout its length. The method of representing a field by lines of force originated with Faraday. He regarded the lines as being tmder tension, i.e., as elastic bands stretched between the two charges, and also, though not like elastic bands, as repelling each other laterally. The direction of the lines and the stress between the two charged bodies would point to such a concep- tion. The two oppositely charged bodies in Fig. 6, A, would accordingly be drawn together while those at B would be pushed apart. The old method of explaining at- traction and repulsion of magnets and electric charges as "action at a dis- tance" is simply a statement of a con- dition which could not be explained. It is not possible for one body to affect another except through some medium of communication between them. This medium is now believed to be the ether and Faraday's lines are not simply a convenient fancy, but represent a strain in the meditmi — the ether — ^in the region of the charged bodies. The bodies will then always move in such a direction as to relieve the strain and reduce the amount of potential energy. 6. Potential. — The potential of a point in an electrical field has the same meaning as potential energy in reference to a point in the earth's gravitational field. The potential of any point above the earth's surface may be defined as the amount of energy required to lift a unit mass from the earth to that point. To lift it to a greater height would require more energy or work and so Fig. 5. ELECTRICITY. 11 the potential energy would -be greater. When the body is raised to a point at the limit of the earth's gravitational field, i.e., to an infinite distance, its potential in reference to the earth is the greatest possible. It is customary, however, to regard the out- most boundaries of a field as having absolute zero, of potential and to define absolute potential of any point in a field as the quantity of work required to bring unit mass — if the field is gravitational — from infimity to that point. In case of gravity the potential of a point defined in this manner is evidently negative, for work is not being done by an external agent against a resisting field but the field itself moves the mass nearer to the earth. V f/'?^'\1 // . In a similar maimer, absolute electrical ■^■^^^i^^^k'j-ii' potential of a point is defined as the work ^i^^^^^^^^^^?""^ required to bring a unit positive charge ' / v\^^:^'J\\^ from an infinite distance to the point. If ^ — ^* the field is one due to a positive charge, ._ -^v^ •, \ i/ /'/V^ the potential of any point in the field is — .'■\\'! )/•, • ///>" positive, for work must be done by an ...~7:r:-j^'' g >&.;:jr ^ external agent against a resisting field and "'' V-'-/^/!\\ /f^""" — •• so potential energy of the unit positive " / / ! \\l\ \ \"^-' charge will be increased when it is moved ' ■■'''"• \ , , , . , . „. . Fig. 6. nearer the charge which is repelhng it. The potential difference (P.D.) between two points is defined as the amount of work required to move unit positive charge from one point to the other. When an electrical P.D. exists between two points on a con- ductor, electricity will flow from the higher to the lower potential. Hence if two oppositely charged bodies, or bodies having the same kind of charge but at different potential, are connected by a conductor, a current will flow from positive to negative or from a point of higher to a point of lower potential. This is the con- ventional direction of current. In fact, if the electron theory is correct, the only thing that moves in a solid conductor is the electron, and its motion is always from a negative to a positive charge or, in general, in such a direction as will relieve any strain caused by a separation of electrons from their atoms. We will see that in case of electroljrtes and ionized gases both positive and negative charges move, but in solids the electron is the only 12 GENERAL PHYSICS. part that appears to move and its motion constitutes what is called an electric current. If, then, + and—. Fig. Q,A, are con- nected by a conducting wire, electrons will pass from — to + and in so doing will carry one end of lines of force with them. Thus the lines are shortened and finally disappear. Ether strain is thus removed as far as these electrons are concerned. 7. The P.D. Due to a Charged Point.— Let A, Fig. 7, be a point charged with Q units of electricity. It is desired to find the P.D. between p\ at a distance n from A, and p2 at, a distance Q Q Ti from A. The strength of the field at pi is — ^ and at p2 is — ^ (see section 5 and equation 2). By definition given above, P.D. is the work required to move unit positive charge from pi to pi, i.e., over a distance ra— n. If the strength of field were uniform between pi and pi, the amount of work could be easily obtained A hjL O Fig. 7. by multiplying the force, i.e., the strength of the field, by the distance r2 — n. This force, however, varies inversely as the square of the distance from A. The average force in such a case is the geometric mean of the forces at pi and pi, i.e., it is the square root of the product of — and — or -^. This average strength of field times the distance gives the work or P.D. sought. Hence V:Q.= ^{n-ri)=Q(\-\\ (4) If the absolute potential, Vi, of the point pi is desired, move pi to an infinite distance from A. Then the P.D. between the two points will be the work required to bring unit positive charge from infinity to pi. But if 7'2=oo, l/r2 = l/oo=0. Hence, from equation (4), P.D.= Fx = g('--0 V - (5) ELECTRICITY. 13 Therefore the potential due to a charged point varies directly as the charge and inversely as the distance from the charge. (See also Appendix 1.) It may be shown that if this charge were placed on the surface of a sphere having A as a centre, the potential at pi would not be changed, i.e., if the sphere were enlarged till its radius is n, the potential at its surface would be the same as at pi when the charge was at A. 8. Equipotential Surfaces. — ^An equipotential surface is one on which no work is performed in moving an electric charge from one point to another, i.e., there is no P.D. between points on the Fig. 8. surface. Such surfaces are spherical shells concentric about a charged spherical conductor. There is a P.D. between the shells, but aU points on the same shell have the same potential. The tubes of force from an electric charge are always at right angles to the equipotential surfaces, otherwise there would be a component of force along the surface and this times the distance between two points would indicate that work must be done in moving a charge from one point to another on the surface. Thus when the Unes of force in any electrical field have been determined, the equipotential surfaces may be drawn everjnsrhere at right angles to these Unes. The simplest representation of such a field 14 GENERAL PHYSICS. is shown in Fig. 8, where the radiating lines are lines of force and the circles are cross sections of equipotential shells around the charge. 9. Electric Induction. — Electric induction is the phenomenon observed when an insulated conductor is brought into an electrical field. Let AB, Fig. 9, represent a brass rod supported on a glass stand. According to the electron theory the uncharged brass contains a sufficient number of electrons to neutralize all the atoms there. Many of the electrons, however, may at any time be de- tached from their atoms and are then free to move under the influence of an electrical force. If a rod of hard rubber R is elec- trified by contact with cat fur, it is Surrounded by a field of force. If then the brass rod is moved into this field the free electrons will be driven toward A, and consequently negative electricity fi O Fig. 9. will be in excess of positive at that end of the rod while, in conse- quence of a deficiency of electrons at B, there will be an excess of positive at that end. The fact of opposite charges at the ends of AB may be readily shown by use of the proof -plane P, which is simply a disc of metal attached to an insulating handle. If P is touched to B, electrons will pass from P to .B and the disc will then be positively electri- fied, as may be shown by bringing it near an electroscope, such as the charged pith ball of Fig. 4. Now neutralize P by touching it to the hand or some body connected to the earth. Then touch it to i4. The disc will receive an excess of electrons and will be negative, as the electroscope will show. On removal of the rod R from the region of AB, electrons in excess at A wiU be distributed throughout the brass rod and the atoms will again be neutral. ELECTRICITY. 15 10. Charging by Induction. — If a conductor, as AB in Fig. 9, while in a negative electrical field, is touched at any point by the finger or by any conductor connected to earth, electrons will escape from it. If then the finger and afterwards the rod R be removed, the whole body AB will show a deficiency of electrons, i.e., will be positively charged. This will be the case no matter at what point AB is touched, for contact of AB with another body increases the region into which electrons may be driven. O m Fig. 10. Then after contact is broken and the field removed, those electrons which were driven to the end A are not sufficient in number to neutralize the positive of the whole body. If the field had been positive electrons would have passed from the finger into AB, which would then have been found to be negatively charged. Such is the method of charging any body by induction. The gold leaf electroscope is usually charged in this manner. Gold is used because leaves of it can be made very thin and light. The leaves are attached to a conducting rod which is carefully insulated 16 GENERAL PHYSICS. from other bodies. Fused sulphur is a good insulator for this purpose. When this insulated rod with gold leaves attached is charged by induction either positively or negatively, the leaves will be driven apart, because like charges repel one another. In the form shown in Fig. 10 a single leaf is used which wiU swing away from a brass strip that is similarly charged. The electro- scope can therefore be used to detect the presence and character of other charges. Suppose the leaves are charged negatively, then if a positively charged body is brought near the top of the electroscope, the leaves will fall toward one another because they have become neutral by the withdrawal of electrons from them. If the positive charge is brought stiU nearer or is made stronger, the leaves will fall completely together and will then diverge again, for so many electrons have been withdrawn that the leaves have become positively charged. In a similar manner the action of the leaves may be explained when they are fiirst charged positively. 11. Capacity of a Conductor. — If an isolated conductor is given a charge Q, its potential is raised a certain amount, i.e., a certain amount of work must be done to bring unit positive charge to the conductor. If the charge is doubled, the strength of the field is also doubled and so likewise the potential. If the conductor ■ is made larger, the same quantity Q will not, however, cause as great a change in potential (see § 6). Capacity, C, may therefore be defined as the charge per unit change of potential, or as the ratio of the charge Q to the potential V. Expressed in formula, Capacity of a conductor varies not only with size and shape but is also directly dependent on the dielectric constant K. Equa- tion (3) shows that the strength of the field varies inversely as K. Hence when K increases, less work will be done in moving unit charge through the field, i.e., the potential at the surface of the conductor is less. Hence when the dielectric is other than air, equation (5) should be written: V=% (7) Substituting this value in (6), C'^'Kr (8) The capacity of a spherical conductor in air is therefore numerically ELECTRICITY. 17 equal to its radius r, but, in another medium, capacity equals radius times dielectric constant. 12. Location of the Charge. — If a metal vessel such as a spheri- cal shell with an opening to the inside, or a cylinder made of wire gauze, is coimected to an electroscope and a charged metal ball suspended from a silk string is lowered into the vessel, a charge like that on the ball appears on the outside of the vessel and an unlike charge on the inside. This is simply a case of electric induction. If the ball is charged negatively — i.e., has an excess of electrons — a number of electrons equal to this excess will be driven to the outside of the vessel and also into the electroscope. Hence there will be a deficiency on the interior walls which the excess of electrons on the ball will just be able to supply. If the ball is now touched to the inner wall and then removed, it is found to be neutral. The leaves of the electroscope still stand apart and are not affected by again touching the ball to the inner walls or by its removal. The excess of electrons driven to the outside remain there. Since electrons repel each other, they will occupy the surface of largest area, which is the outer surface of the vessel. This is Faraday's " ice pail experiment," so called because the vessel he first used for this purpose was a metal ice pail. His explanations, however, were not given in terms of electrons as here. Faraday also noted that there was a divergence of the gold leaves whether the vessel was filled with air, oil, or other nonconducting medium. Hence he gave the name dielectric to media of this character. The fact of a charge appearing only on the outside of a vessel may be shown by charging any hollow metal body and then by use of the proof-plane and electroscope, testing the inner and outer surfaces. 13. The Field within a Hollow Charged Conductor. — ^When a closed hollow conductor is charged with electricity, the interior at all points is devoid of any resultant field intensity. The reason for this in the case of a spherical shell is apparent from a considera- tion of Fig. 11. When this sphere is charged, the electric surface density will be the same at aU points. By density is meant the quantity of electricity on each square centimetre of surface. Let P be any point within the sphere. Let two lines be drawn through P, making very small angles with one another. These lines 2 18 GENERAL PHYSICS. enclose areas which are sections of cones with vertices at P, their bases s and Si being portions of the shell. Then if electric density is d, the quantity of electricity on 5 and Si is respectively sd and S\d. The strength of field at P due to the charge on s is, by equa- tion (2), sd/r^, and that due to the charge on Si is sid/r^. These forces are in opposite directions. In similar cones the areas of their bases are directly as the squares of the altitudes, i.e., s/s\ = r^/n^. Hence ds/dsi=r^/n^, i.e., the quantities of electricity here considered vary directly as the squares of their distances from P Fig. 11. But the forces at P due to these charges vary inversely as the square of the distances. Consequently the forces at P are equal and opposite and so there is no resultant force at that point. Since the whole interior may be regarded as made up of similar sets of cones, and since P is any point within the sphere, what is true of P is true of any point within the sphere. The same is true of an enclosed cylinder or other body. Fara- day made a large cubical enclosure covered with tin-foil. When he entered this with a delicate electroscope, he could not detect any field of force while the outside was highly charged. A similar condition may be shown by inverting a wire gauze vessel over an electroscope and then electrifying the cylinder by contact with the charged ebonite stick. ELECTRICITY. 19 14. Distribution of the Charge on a G)nductor. — ^The charge on an insulated conductor is most dense at points where the curva- ture is greatest. On a sphere electrical density is uniform over the surface, but on other surfaces, such as that of an ellipsoid, density is found to be greatest at points of greatest curvature. An explanation of this may be found in the fact that adjacent portions of any given charge repel one another. Consequently the components a, a, Fig. 12, of forces driving electricity toward P are greater than oi, oi, which tend to drive the charge to Pi. Another method of explaining the increased density at points is that since density is the quantity of electricity per square centimetre, that on the surface of a sphere of radius n, when charged with Qi units, is Qi/4^irr.i'. Let another sphere of radius ra, less than n, be in contact with the first sphere. It will contain a charge Q2 and the density on its surface is Qi/Arri^. Since the spheres are in contact the potential, V, is the same on both. Then from equation (5), Qj = Vn and Qi = Vh. Substituting these values of Qx and 02 in the above expressions for density, \rx di "iwi^ (9) Fig. 12. ri This shows that the densities on the spheres are inversely as their radii. The greater the curvature the greater the density. Now when one portion of a surface has a greater curvature, i.e., shorter radius, than other portions, the former may be considered part of the surface of a small sphere attached to a larger one. Hence the increased density on the portion of greatest curvature. When the ciu-vature becomes very great, as in case of fine 20 GENERAL PHYSICS. points, the strain may be such as to break down the dielectric. A discharge into the air follows, causing the so-called " electric wind," which may be detected by holding a candle flame in front of the point. The force which sets the air in motion would tend to move the point in an opposite direction, as is shown by the " electric whirl." A point discharge in the dark has the appear- ance of a brush and so is called a brush discharge. This is a common phenomenon in nature. On the masts of ships it is known as St. Ehno's fire. 15. The Electric Spark. — Gases are ordinarily poor conductors of electricity. Two bodies oppositely charged are almost perfectly insulated from one another by dry air between them as long as the P.D. is not too great. If two knobs are mounted as shown in Fig. 13 so that the space between them may be adjusted, then when one is charged positively and the other negatively a spark will pass between them whenever the P.D. is suflSciently great -O o- Fig. 13. for that distance. If the knobs are pushed closer together, a smaller P.D. will ca,use a spark. The length of the spark gap depends on several conditions such as atmospheric pressure, the size and character of surface of the knobs, and the presence of ions in the air as explained below. It may be stated roughly that under ordinary atmospheric conditions, with knobs more than 2 cm. in diameter and more than 2 mm. apart, 30,000 volts are required for every centimetre of the air gap. This is the same as 100 electrostatic units of potential. If, however, cathode rays or X-rays are projected into the region between the knobs, the air there becomes a good conductor and the knobs are soon discharged. When air is thus made a conductor it is said to be ionized. An atom from which an electron has been separated exhibits a positive charge, and one which carries an electron in excess of the normal number is negatively charged. Cathode rays. X-rays, a flame, and many other agencies will ionize a gas in this manner. As long as the ionizing agent ELECTRICITY. ' 21 continues, the gas will conduct electricity in proportion to the num- ber of ions present. After the ionizing agent ceases, ions will continue for some time in the gas but those of opposite charge will be gradually drawn together and neutralized; or, in an electro- static field such as that between the knobs of Fig. 13, a positive ion will move with the tubes of force, i.e., toward the negative knob, while the negative ion will move in the opposite direction. Thus the gas is soon cleared of most of its ions and again becomes a nonconductor. Now when the P.D. between the two knobs is gradually increased, the air at first acts as an almost perfect nonconductor, for there are very few ions present in normal air. With increase of P.D. a point is reached where the electric force will of itself ionize the gas and thus cause it to become a good conductor. A discharge then takes place in the form of an electric spark. Succeeding sparks readily follow because of ions produced by former sparks. A gas flame held between the knobs facilitates the passage of a spark because a flame ionizes the air. 16. The Energy of a Charge. — The energy of an electric charge is equal to the amount of work that must be done to produce the charge. Suppose a body A is devoid of any charge. Let unit charge be brought- to it from an infinite distance or from some body such as the earth which, for purpose of reference, may be assumed to have zero potential. Both the charge and the poten- tial of A wiU. thus be raised. Let other tmit charges be brought in succession to A until the total charge is Q and the potential V. Each increase in charge increased the potential and also the work required to bring up the next quantity of electricity. The potential at first was zero and increased uniformly to V. Hence the average potential is V/2. The total work, therefore, is the sum Q of all the unit charges times V/2. If work or energy is represented by W, then W=^QV (10) Substituting in (10) the value of V from (6), W = ^ (ip or by substituting the value of Q from (6), Q \y^ W=~-CV^ [O'- -^ (12) 22 GENERAL PHYSICS. 17. Lightning. — Lightning is an electric spark of great magni- tude which occurs between different portions of a rain cloud, between two clouds, or between a cloud and the earth. The origin of the electric charge in the atmosphere or in a cloud is not yet a settled question. Clouds are formed by condensation of moisture -in air, but this cannot occur unless there are nuclei of some kind on which a droplet may begin to be formed. It has been shown by experiment that it is difficult to cause condensation of moisture in air which has been freed from dust particles, but if this air be ionized in the maimer explained in § 15, the ions serve as nuclei and condensation readily occurs, first on negative ions and later on positive ones. If at any time there is a sufficient number of ions in air which is saturated with moisture, they may serve as centres of condensation and charges of opposite kind may thus be formed. Each droplet is charged to a certain potential but when several of these tmite into a larger drop the potential rapidly rises. Sup- pose n small drops tmite into a large one, then since volumes of spheres vary as the cubes of their radii, the radius of the large drop will be n^ times that of one of the small ones. Since the capacity of a sphere is proportional to the radius (§11), the capac- ity is increased m* times. Also, since potential is equal to charge divided by capacity and there are n small charges tuiited on the large drop, V=—r- = n* n' Hence the potential on the large drop is w' times that on one of the smaU ones, e.g., if 27 small charged drops unite, the potential will be raised nine times. Thus, after the small drops are once charged, the vmion of these may produce a high potential. It does not seem possible, however, to secure by this means a potential sufficiently great to cause a lightning stroke one mile in length. Their lengths vary from a short stroke up to even two miles in length. A potential great enough to cause a spark one mile long would certainly produce an enormous brush discharge — a thing which is not observed. Most lightning strokes pass from one portion of a cloud to another portion. Comparatively few strokes reach the earth. It is possible that the potential in a cloud may not be very great and that a disruptive discharge at one point is ELECTRICITY. 23 followed by a succession of discharges throughout the length of an electrified region. These follow so rapidly that they appear as one stroke. When a large body of electrified raindrops fall toward the earth, a stroke occurs between it and the earth. A stroke of Hghtning may do much damage, particularly where numerous electrical lines lead to electrical plants, telephone centres, etc. To pro-vide against this, various forms of " lightning arresters " are provided and by them lightning is diverted to the ground before it can pass into and destroy expensive machinery. A simple form of such arrester is a conductor from the line wire to the ground with a short air gap at some point in the conductor. The current on the line is therefore not grounded unless, as in case of lightning, the potential should rise to such a point that the current would cross the air gap and pass to the ground. A similar device consists in breaking the ground wire with a plate of mica. This prevents grounding of the ordinary current, but at a certain potential, depending on the thickness of the mica, the dielectric breaks down and the current is grounded. Another device with many advantages is the aluminum cell. This consists of aluminum plates covered with a film of aluminum hydroxide and placed in a suitable electrolyte. One terminal may be con- nected directly to the line and the other to the ground. The cell prevents the passage of a current until the potential rises to a certain critical point which can be fixed in the structure and opera- tion of the cell. For any higher potential, as in case of lightning, the cell permits a free discharge to the ground. In the protection of buildings, lightning rods are serviceable if good conductors of sufficient size and without breaks extend from moist earth, five or six feet below the surface, to sharp points which rise well above the roof and chimneys. These protect in two ways. First, the sharp points slowly discharge electricity to the ground and, by thus relieving the dielectric strain, minimize or prevent a stroke. Second, if a disruptive discharge occurs, the rod may serve to conduct it to the ground. A few terminals over a building, however, even when well grounded, are not a guarantee of protection from lightning. The interior of a metal building would not be injured by lightning. A building with a metal roof connected to ground at three or four points by rods or heavy copper wires will be fairly well protected. 24 GENERAL PHYSICS. 18. Condensers. — ^A condenser is an arrangement by which the capacity of an insulated conductor is increased by the presence of another conductor, the two being separated by a dielectric. The second conductor is usually connected to the earth or to the opposite pole of an electrical generator. To understand how this arrangement increases the capacity of a conductor, let A, Fig. 14, represent a portion of a spherical conductor charged to a certain potential. Its field of force would extend out indefinitely in all directions, as shown in Fig. 8, and its potential is the amount of work required to bring unit positive charge through this field to the conductor. Now let a concentric spherical shell B be placed in the field near A and connected by a wire to earth. The poten- tial of B will then be the same as that of the earth, i.e., zero. Hence no work will be required to bring unit charge from the earth to B. The potential of A is now only the P.D. between A and B, for the only work done in moving unit charge to A is that caused by the P.D. between B and A. The P.D. between A and any point on B is the same as before B was placed in the field, hence the fall of potential at A is the same Pig. 14. r, rrf A ■ ■ • , as that at B. To restore A to its original potential, so that as much work would be done in moving unit charge from S to A as was originally required to move it from infinity or earth to A, the sphere A must receive a greatly in- creased charge. 19. Capacity of a Spherical Condenser.— Let A, Fig. 15, be an insulated spherical conductor surrounded by a spherical shell B which is connected to earth. Let r and n be the respective radii of these spheres and let A have a charge of Q units. The potential on the surface of A is Q/r (see equation 5) . At a distance fi from the centre the potential is Q/n. When B was connected to earth its potential fell to zero, i.e., it lost Q/n units of potential. Since Q/r included the quantity Q/n before B was reduced to zero, the potential V on A must now be r n \r nj \rn ) ELECTRICITY. 25 Since capacity C is defined as the ratio of charge Q to potential V (see equation 6), V*" Q _ rn (14) C-- <^) "■ The two conductors A and B are very close to one another and so no serious error will be introduced by using r^ in place of m. Then if ri— r is represented by d, 4ir2 5 47rd C = -=- d iird (15) Where S is the total surface of the sphere A. The values of C in (14) and (15) are correct when air is the medium between the Fig. is. plates. For other media the values must be multiplied by the dielectric constant K. Thus Aird (16) 26 GENERAL PHYSICS. In case of parallel plate condensers the area of the plates may be regarded as portions of the areas of very large spheres. The ratio of the area of one of the plates to the area of surface of this large sphere is the same as the ratio of the capacity of the plate condenser to that of the large spherical condenser. Hence if Cp stands for the capacity of the plate condenser and S^ for the area of one of the plates, KS„ C„ = liTTd (17) "V where d is the thickness of. J;he dielectric. Plate condensers, instead of being composed of two large plates separated by a dielectric, are built up in the manner shown in Fig. 16, where sheets of tin-foil or o^her thin metal are separated by \thin sheets of mica, paraf5Sned paper, or other dielectric. The alternate sheets of foil are con- nected to A and the others to B. In this way a condenser of large capacity can be made up in compact form. In calculat- ing capacity by use of equa- tion (17), Sp is the area of all the plates on one side, i.e., the area of all the foil connected to A in Fig. 16, or, in other words, Sp is the area of that portion of the sheets of mica or other dielectric which is covered by tin-foil on both sides. The reason for this will appear on a reconsideration of § 18 and Fig. 14. To charge a condenser, let A of Fig. 16 be coimected to one terminal of an electric machine or battery and B to the earth or, better, to the other terminal of the machine. Another common form of condenser is the Leyden jar, which is simply a glass jar covered inside and outside to about three- fourths of its height with tin-foil. The jar with its coverings may then be regarded as a plate condenser in form of a cylinder. Its capacity may be roughly calculated by use of equation (17). Fig. 16. ELECTRICITY. 27 In calculating the capacity of a condenser by use of equation (16) or (17), the quantity obtained is, according to equation (6), the number of electrostatic units of quantity that must be placed on one plate, the other being connected to earth, to cause a change of one erg in the amount of work that must be done in moving unit charge from one plate to the other, i.e., to cause imit change in the P.D. between the plates. Fvirther consideration of condensers is given in later chapters. 20. Electrometers. — To measure the P.D. between two plates which are oppositely charged, as in a condenser, various instru- ments have been devised. In one invented by Lord Kelvin, the plates A and B, Fig. 17, are connected respectively to any two points whose P.D. is sought. A disc cut from the central portion of the upper plate ...-t-t-T TN is suspended from one arm of a bal- /\ ance and may move freely up or down. / \ , ' Sights are provided to determine ..! ■.;. ..ni...,i;ii;i>Mi.iii8 r — ,. when the disc is exactly in the plane 'ii 'i'-;'i'!'riiM'"i"!i' , H! i<^ I oiA. The P.D. between the plates will ^v^ be the same as that between the points to which they are connected. The disc will then be attracted toward the lower plate with a force F, which may be determined by placing weights in the pan at the other end of the balance beam until the disc is returned to its position in the plane of A. The part of the plate A surrotmding the disc is called the guard ring. It insures a uniform field between the disc and an equal portion of B directly beneath. At the edge of the plates tubes of force do not run straight across but curve outward from edge to edge, hence the field there is not uniform. Now let 5 be the area of the disc and d the distance between the plates. This forms an air condenser whose capacity is 5 C = - — : (See equation 15) iira From equation (12) the energy is Eliminating C from these two equations, 2W ^ S V 47rd 28 GENERAL PHYSICS. Since the field is uniform the work which must be done in moving the disc from B to A may be expressed by Fd. This may then be put in place of W above. Hence 2Fd_ S V ~^Trd y=4 STd'F (18) ICnowing S, d, and F it is possible to calculate the number of ergs of work required to carry unit electrostatic charge from one point to the other, i.e., to find the P.D. between the points. Since P.D. is found in this manner and not by comparison with some standard, this instrument is called an absolute electrometer. Another instrument used for a similar purpose but much more sensitive than the one just de- scribed, is called the quadrant electrometer, also an invention of Lord Kelvin. A simplified form now in common use is the Dale- zalek electrometer. The instru- ment consists essentially of a metal drum, Fig. i8, which is divided into quadrants or quar- ters, each quadrant being sup- ported on an insulating pillar and connected by a wire with the one opposite, forming two pairs. Within this drum is suspended a "needle" of light silver paper attached to a stem which carries a small mirror. The suspending fibre is quartz or a fine bronze wire. Now let all the quadrants be momentarily groimded to bring them to the same potential. The needle is then turned so that its posi- tion will be symmetrical in reference to the two pairs of quadrants, i.e., will stand in the line separating the A and B quadrants. The needle is then charged to a high potential. It will not turn toward the A or B quadrants for they are equal in potential. If now the A quadrants are connected to earth and the B quadrants to a body which is charged, say, positively, then the needle, if positive, Fig. 18. ELECTRICITY. 29 will be repelled by the positive quadrants and will turn until its turning moment equals the torsion of the suspending fibre. The deflection of the needle is determined by the movement of a beam of light reflected from the mirror onto a scale or by means of a telescope and scale. If this observation is made for a known P.D. in the pairs of sectors, then, assuming that the deflection of the needle is proportional to P.D., an unknown P.D. may be determined. This instrument is also valuable in other ways, e.g., in observing the rate of electric discharge, the activity of an ionizing agent, or the strength of radium and other like substances. 21. Specific Inductive Capacity. — The capacity of a condenser differs when different substances are interposed between the plates. The value of a substance in this respect is usually indicated by K and is called the dielectric constant. It was named by Faraday the specific inductive capacity and is defined as the ratio of the capacity of any condenser to the capacity of the same condenser when air is used as the dielectric. An approximate value of this constant for various dielectrics is given in the appendix. 22. Residual Charge. — ^When a condenser, such as a Leyden jar, is highly charged, it may be discharged by means of a wire one end of which touches the outer coating and the other is brought near the knob. A bright spark passes and the jar appears to be completely discharged. A little later, however, another discharge, much weaker than the first, may be obtained. This is called a residual charge. Since the foil or metal plates of a condenser are good conductors, it appears that a residual charge must in some manner depend on the slow recovery from a strained condition of the dielectric. The importance of the dielectric in this respect may be strikingly shown by use of a dissectible Leyden jar. Let such a jar be charged, then remove the two coatings. These may be handled or touched together, but no evidence of a charge will be found in them. When the coatings are then replaced, the jar is found to be charged as strongly as before it was dis- sected. If the glass whUe separated from the coatings is touched with the hand over its entire surface, the charge is removed. It appears from this that the metal plates simply serve as distributors of the charge to all parts of a dielectric, the latter being always a 30 GENERAL PHYSICS. nonconductor. The inner and outer walls of the glass jar possess charges of the same sign as that of their respective coatings before removal. Apparently electrons are not easily detached from atoms of a dielectric nor can they make their way through sub- stances of that kind. Hence opposite charges on the coatings cause a strained condition in the glass and, like most mechanical strains, a complete recovery does not at once follow the removal of the stress. Therefore several residual charges, each smaller than the preceding one, may be obtained from a condenser. 23. The Electric Machine. — The fundamental principle of the electric machine may be shown by a simple device called the electrophorus. This, as shown in Fig. 19, consists of a plate R of hard rubber, rosin, or any nonconductor that may be electrified. H + + + + + ■+ + + + + Fig. 19. Upon this is placed a metal disc M. By use of the insulating handle H, M may be Hfted from R. Let R be electrified nega- tively, i.e., have a greater number of electrons than is needed to neutralize the positive there. This may be done by striking the rubber with cat-fur or woolen cloth. When M is now placed on R, the negative charge there acts inductively on the metal plate, driving a number of electrons to the upper side and thus leaving the lower side positive. Let plate M, while still on R, be touched with the finger or any conductor that leads to the earth. Electrons will escape from the plate. If M is then lifted and removed from the influence of R, it is found to be positively charged, and if brought near to some conducting body, electrons will flow to the plate and restore its neutral state. This operation may be repeated indeflnitely, and the charges thus obtained may be stored in an insulated conductor or a con- ELECTRICITY. 31 denser such as a Leyden jar. These repeated operations do not impoverish the charge in R. The electricity produced represents a certain amount of energy. This, however, comes, not from the charge on R, but as a result of work done in separating the two opposite charges when M is removed. The action of i? on Af is only inductive and there is no transference of electric charges from one to the other. An electric machine constructed in accordance with the prin- ciple just described may be made continuous and automatic in its operation. This will be apparent from Fig. 20. A glass plate Fig. 20. which may be rotated on the axis o has a number of small metal discs fastened on its front surface, as a and b. These correspond to the upper plate of the electrophorus. Behind the glass plate is a stationary nonconducting body R, which corresponds to the lower plate of the electrophorus. If R is now charged, say nega- tively, it will act inductively on a, and by turning the glass plate in the direction indicated by the arrow, a is made to touch the metal brush c. This is the same as touching the upper plate of the electrophorus. Under the influence of i? a ntimber of electrons 32 GENERAL PHYSICS. are driven from o through the metal rod co to the ground. After a passes c it also moves from the influence of R. This is the same as lifting the upper plate of the electrophorus. When a, therefore, has moved on to a position b it is deficient in electrons and hence shows a positive charge. If R had been positively charged, elec- trons would have passed from c to a at the moment of contact, and so a negative charge would have appeared at b. Thus after R is once charged it is only necessary to rotate the glass plate. Each metal disc will, then, when it reaches b, carry a charge which may be taken off by a conductor and carried to a condenser or other body which is to be charged. A complete machine which operates in accordance with this principle is shown in Fig. 21. This one is known as the Toepler- Holtz machine. There are two glass plates, one of which, S, is stationary while the other may be rotated. Two pieces of paper, as shown by the dotted lines, are pasted on the stationary plate. These are called inductors. Conducting arms o and o' extend from the inductors over to the front of the revolving plate and end in a brush of tinsel which touches the buttons a, 6, c, etc., as they pass. The operation is the same as that described above, except that here the electrophorus is double and the electric charge on the buttons is given up chiefly to the inductors through the conductors o and o'. If the inductor R is negative, the button b will carry a positive charge to R'. Then R' acts on d just as R acted on a, but with an opposite effect, so that e carries a nega- tive charge to R. Thus as the glass plate is turned the inductors are more and more highly charged, up to a limit depending on several conditions such as size of glass plates, atmospheric pressure, ionization of air, etc. The arms Q and Q' are held in position by insulating supports, one branch terminating in fine points which lie close to the front surface of the revolving plate, while the other branch terminates in the knobs E and E'. Under the influence of the inductors now highly and oppositely charged, electrons are driven out to one knob, E, and withdrawn from the other, E'. This may continue until a spark passes be- tween the knobs. The P.D. between the knobs falls when the spark occurs, but is soon restored by the operation of the machine, so that a rapid succession of sparks may be produced. The con- ducting rod k carries a stream of electrons from the end over the ELECTRICITY. 33 negative inductor to the end over the positive inductor, both inductors acting to produce this effect. Leyden jars may be attached to Q and Q' to accumulate the charge. Sparks will then pass less frequently between E and E' but a greater quantity of electricity will pass when the spark does occur. The Wimshurst machine is, in principle, the same as that just described, but in construction and operation it is difEerent. There are two plates which revolve in opposite directions, each carrying a number of metallic sectors on its outer face. These sectors act alternately as carriers (i.e., like the buttons on the Toepler-Holtz machine) and as inductors. 24, Dimension of Electrostatic Units. — ^There are three sys- tems of electrical units, electrostatic, electromagnetic, and practical. Only the first of these will be considered here. Electrostatic tmits are founded on the definition of unit quan- tity of static electricity. Q is the c.g.s. electrostatic unit when 3 34 GENERAL PHYSICS. it repels an equal charge Q' with a force of 1 dyne, the distance between the charges being 1 cm. in air. From equation (3), and since F is a force, r a distance, and Q is equal to Q', the dimen- sional equation for unit quantity is Q^ = \MLT-->\\U\\K\ or Q = [M^L*r-i2?*] (19) The strength of a field has been defined as the force per unit charge, i.e., the force with which unit charge would be urged if placed in the field. Hence Strength of field = [MLT^^] h- \M^I} 7^» K^\ = \M^L-^T-^K-^\ (20) Unit strength of field, or field intensity, is 1 dyne. Unit P.D. exists between two points when one erg of work must be done in moving unit charge from one point to the other, i.e., potential difference is the work per unit charge. Hence, letting V represent this unit, = \^L^T-^K-^\ (21) Capacity is defined as the ratio of the charge to the difference of potential, hence C = \ls/^L^T-^K>\-¥\l^jyT-''K-^\ = LK (22) The strength of current in this system is the number of electro- static tmits of quantity that pass a cross section of a conductor in unit time. Hence, letting i represent strength of current, = \M>L^T-^K^\ (23) Problems ' ' 1. Two points, 6 cm. apart, are each charged with 20 units of negative electricity. With what force will the charges repel one another? 2. What is the intensity of an electrical field at a point 10 cm. from a charge of 500 electrostatic units? ELECTRICITY. 35 3. If a charge of 25 units is placed at one comer of an equilateral triangle and 70 units at another comer, the sides being 30 cm. long, what will be the intensity at the third comer? 4. A spherical conductor, whose radius is 3 cm. and its potential 130 units, is touched to an uncharged conductor. The potential fell to 20 units. What is the capacity of the second conductor? 5. A sphere whose diameter is 22 cm. is charged with 136 units of static electricity and then connected by a fine wire, whose capacity may be neglected, to an uncharged sphere 10 cm. in diameter. What will then be the potential of each sphere? 6. How much electrical energy is stored in a sphere whose capacity is 25 cm., when the charge is 1000 electrostatic units? 7. What is the P.D. between two charged discs 8 cm. in diameter when a force of 10 g. is needed to hold one 5 cm. from the other? 8. A spherical conductor of radius r is surrounded by a spherical shell of radius i'. If r' is connected to the earth and air is the dielectric, how much is the capacity of the conductor increased? The capacity of the conductor is r while that of the condenser is rr" /r' —r. Subtract and then multiply both numerator and denominator by 47r. Take r'— r as equal to d and 5 equal area of surface. Ans. 1. 16 dynes. 2. 5 dynes. 3. .0946 dyne. 4. 16.5 cm. 5. 8.5 units. 6. 2(10)< ergs. 7. 350 units. 8 -^ CHAPTER II MAGNETISM AND THE ELECTRIC CURRENT 25. What Magnetism Is. — It has already been stated that, according to the electron theory, an electric current is the move- ment of electrons within a conductor. Each electron possesses a negative charge of electricity and, while it is at rest, is surrounded only by an electrostatic field in which like charges are- repelled and unlike charges are attracted. But when an electron is set in motion, it carries its tubes of force with it and sets up another field whose lines of force are at right angles to the electrostatic lines and also to the direction of motion of the electrons. This second field is magnetic. Whenever electrostatic lines of force are moved through the ether, a magnetic field is produced. Magnet- ism, then, is a condition of the ether and is caused by the motion of an electrostatic charge. The existence of such a field may be shown by passing a strong current, i.e., a stream of electrons, along a conductor, as in Fig. 22. Some iron filings sprinkled on the card will become magnets and will arrange themselves in concentric circles about the conductor. Such a magnetic field exists at all points along the conductor while the electrons are moving through it. It may be shown by experiment that an electric charge will, when moving through the ether, set up a magnetic field. Rowland in 1876 made an experimental investigation of this subject. He mounted a gilded ebonite disc in such a manner that it could be rapidly rotated and also charged to a high potential. A delicate magnetic needle suspended above the disc and shielded from all influences except that of a magnetic field was observed to turn to the right or left according to the sign of the electric charge, i.e., the needle turned just as it would if placed near a wire on which a current is passing in one direction or the other. If the conducting wire is bent around in form of a circle, as shown in Fig. 23 where the effect is increased by use of several parallel strands, then the current flowing around the circle causes a strong magnetic field within, as shown by the effect on iron filings. If a magnetic needle is placed within this circle it will promptly take 36 MAGNETISM AND THE ELECTRIC CURRENT. 37 a position parallel to the lines of filings. If the direction of the current is reversed, the needle will turn through 180° and so wUl point in an opposite direction. In Fig. 24 is a conductor wound in form of a slender spiral coil the ends of which are attached, one to a strip of zinc and the other to a strip of copper. These Fig. 22. project into a battery solution, the whole being floated on water as shown. When this floating battery is placed in the field caused by a current flowing around the large coil, the small coil will take a position at right angles to the plane of the large coil, or, if the current is reversed, will turn just as did the magnetic needle in Fig. 23. Fig. 23. Fig. 24. It appears from experiments such as these that the funda- mental thing regarding magnetism is electricity in motion, i.e., the movement of electric charges which here are the electrons. A theory which attempts to explain magnets, such as a magnetic 38 GENERAL PHYSICS. needle or a bar magnet, should therefore be based on the motion of electric charges as a cause. Such a theory has been worked out by P. Langevin. Before considering it, attention should be given to the matter presented in the next two paragraphs. 26. Permeability. — The lines of filings shown in Fig. 23 indi- cate a magnetic field set up in air. If a body of iron is placed in such a field, magnetic lines will crowd toward it and there will be a greater magnetic flux through the iron than through the same region in air. (See Fig. 25.) That property which a body possesses in its relation to the flux of magnetic lines through it is called permeability. A technical definition will be given later. The permeability of air is taken as unity. A body which is more permeable than air is said to be paramagnetic or simply magnetic. Such a body will be attracted by a magnet. A needle formed from such a substance will, when suspended in a magnetic fields Fig. 25. set itself parallel to the lines of force. The most strongly para- magnetic substances known are iron, nickel and cobalt. To these should also be added certain alloys discovered in 1903 by Heusler and known by his name. These are composed of manganese, alumi- num, and copper. Two parts by weight of Mn to 1 part of Al. The Mn is dissolved in molten copper and the Al then added. The metal is then moulded in desired form and is found to be about as permeable as cast iron. Several other magnetic alloys have since been found, all of which contain either manganese or chromium. A substance which is less permeable than air is said to be dia- magnetic. It will, when placed in a magnetic field, move to the weaker parts of the field, and a needle made of a diamagnetic sub- stance will set itself at right angles to lines of force in a magnetic field. Magnetic lines will bend away from a diamagnetic body, and the lines within it are less crowded than they would be in air. Most substances are diamagnetic. The one most strongly so is bismuth. MAGNETISM AND THE ELECTRIC CURRENT. 39 The above classification refers to the action of bodies in air. In another medium the action may be different. A body which is paramagnetic in air will, when immersed in a liquid more para- magnetic than itself, appear diamagnetic. This may be illustrated by filling a slender glass tube with a weak solution (about 50 per cent.) of ferric chloride and suspending it in a magnetic field in air. The tube will tium so as to stand parallel to the magnetic lines. Now place in the field a beaker filled with a concentrated solution of the ferric chloride, and the tube will, when immersed, turn to a position at right angles to the magnetic lines. Iron, nickel, cobalt, and certain alloys such as the Heusler alloys, are intensely affected by the magnetic field and so are usually classed by themselves as ferromagnetic. Other paramagnetic substances such as chromium, cerium, and oxygen, and also all the diamagnetic substances, exhibit only feeble effects. 27. Magnets. — The term magnets is usually applied to a body of iron in such a state that it sets up a magnetic field and attracts other magnetic substances. A valuable iron ore called magnetite, Fe304, is found in various localities.* It is magnetic by nature and was first observed on the .^Egean coast in Magnesia. It was called the Magnesian stone, whence our word magnet. If a long, slender piece of this ore is suspended at its middle point, it will turn lender the influence of the earth's magnetic field and take a stand with its long axis pointing north and south. Thus it was used as a guide to mariners and was called lodestone. The Chinese appear to have used magnets in this manner as early as 1100 B.C. The most powerful and most valuable magnet now in use is the electro-magnet. This consists of a soft-iron core around which insulated wire is wrapped. A magnetic field is produced while the electric ctirrent flows aroimd the core, but ceases when the circuit is broken. A magnetic field of any strength, within certain limits to be described later, may therefore be produced by regulat- ing the strength of the current. This may vary in degree from the feeble field which operates a deUcate relay to powerful electric cranes where masses of iron weighing several tons are lifted by * Magnetite is found chiefly in Norway, Sweden, Finland, the Urals, New York State, Pennsylvania, New Jersey, and Michigan. 40 GENERAL PHYSICS. simple contact with the magnet and may be dropped by breaking the circuit. The electromagnet occupies a very important place in the industrial life of the world. A permanent magnet is one which retains its magnetic property independent of any magnetizing agency. It is made of steel, and when placed in a strong magnetic field, such as that produced by a strong electromagnet, will continue to be a magnet after the field is removed. Such magnets are useful in many ways but can- not be compared with electromagnets in strength and adaptability. 28. Theory of Magnets. — ^A knowledge of the fact that the electron is a constituent part of an atom, that the electrons are in rapid orbital motion within the atom, that each carries a charge or is a charge of electricity, and that the movement of an electrical charge causes a magnetic field, furnishes a basis upon which a fundamental theory of the magnet may be constructed. If all material substances may be regarded as composed of molecules, the molecules in turn being composed of atoms, and the atoms made up of positive and negative electric charges, the negative at least being known and believed to have an orbital motion, then it should be possible to explain the. behavior of different sub- stances when placed in a magnetic field. Suppose in the first place that, in case of certain substances, the structure of the atom is perfectly symmetrical, i.e., that for every electron revolving in one direction there is one revolving in an opposite direction. The simi of their magnetic effects would then be zero. Now when such a substance is brought into a magnetic field, the effect is to accelerate the motion and increase the orbits of electrons moving in one direction and to decrease the orbits of those moving in an opposite direction (see Fig. 61). The effect is to destroy the symmetry and produce a resultant field, which is the same in kind as that of the external magnet. Since likes in magnetism repel one another, a substance of the kind described would be repelled by a magnet. Such substances are diamagnetic. The effect of a magnetic field on a diamagnetic substance is, however, always feeble. If a substance contains electrons which revolve in different orbits, some larger and some smaller, and S3mmaetry is wanting, then the molecule taken as a whole would most probably exhibit resulting magnetism, i.e., the magnetism produced by electrons MAGNETISM AND THE ELECTRIC CURRENT. 41 revolving in one direction would not be neutralized by others, in an opposite direction. Such a molecule may therefore be re- garded as a small magnet and in a magnetic field will be affected accordingly. (See Fig. 24.) Substances of this kind are called paramagnetic or simply magnetic. If the resultant magnetism of the group of atoms within a molecule is large and the molecules influence each other across the space which separates them, the substance is said to be ferro- magnetic, i.e., it is very strongly paramagnetic. In all these cases the first effect of a magnetic field is that described for diamagnetic substances, and so all substances may be said to be diamagnetic, and paramagnetic effects are the result of stronger forces in an opposite direction. Such, briefly, is a statement of the most probable theory of magnetism in the light of our present knowledge of electricity and the constitution of matter. 29. Poles of a Magnet. — ^According to the theory just explained, a magnet, as the term is ordinarily used, is a rod of soft iron, hard steel, or other magnetic substance, in which the planes of the orbits of electrons are parallel and facing in the same direction. Not all the orbits are thus turned but only those which produce a predominating magnetic field. The molecules as a whole are turned to such a position that their resultant fields are parallel and in the same direction. AU paramagnetic and ferromagnetic phenomena are therefore molecular in character. In soft iron this arrangement of molecules is maintained only while the iron is under the influence of an outside field, while in hard steel the arrangement persists after the field is removed. If a bar of steel is tmiformly magnetized and then dipped in iron filings, the fiUngs become magnets and will cling to the ends of the bar, diminishing in quantity along the sides toward the middle where no filings are foimd. It was, therefore, wrongly assumed that there was a greater quantity of magnetism at the ends of a magnet than at other points. The ends were called poles, and magnetism was once regarded as a kind of fluid which could flow from one point to another or could exist in a free state. It is evident from the electron theory, however, that there is as much magnetism at the centre of the bar as at any other point. The greater magnetic effect on filings or other magnetic bodies 42 GENERAL PHYSICS. near the ends of the bar is a resultant effect of the magnetism distributed all along the bar. In Fig. 26 let 5 N represent a perma- nent magnet with the so-called poles at S and N. Let an electron orbit, e, with its magnetic field be placed midway between N and 5 in the field of the hit magnet. Then e will face about so that its lines are parallel to and in the same direction as the lines out- side the bar. But since it is impossible to have one pole of a magnet Fig. 26. without having another of opposite magnetic efiEect and exactly equal in strength, then N is equal to S and each is at the same distance from e. Hence e is under the stress of balanced forces of attraction. If e is placed to the right or left of the centre of the bar, it will still keep its lines parallel to those of the field in which it moves and will be more and more strongly attracted by that end of the bar to which it moves. A pole of a magnet is techni- cally defined as the point of ap- plication of the resultant of all the forces acting on one-half of the magnet and tending to produce ro- tation about the magnetic centre when the magnet is placed at right angles to a uniform magnetic field. Thus, in Fig. 27, n and 5 represent the resultants of a number of parallel lines which increase in length from o to each end where they are longest. If the sum of all the moments at each end is divided by the sum of all the forces, the quotient will be the dis- tance from 0, where, if the sum of all the forced is applied, the moment will be the same as before. The positions of poles thus Fig. 27. MAGNETISM AND THE ELECTRIC CURRENT. 43 defined, in case of a permanent bar magnet, are distant from one another about five-sixths, or less, of the length of' the magnet. The pole which points northward when a magnet can freely rotate in the earth's magnetic field is called the north-seeking pole and the other the south-seeking. Often they are called simply the north and south poles, but since like poles repel one another while unlike poles attract, it seems inconsistent to call that pole north which points to the earth's north magnetic pole. This is avoided by using the terms north-seeking and south-seeking.* Any number of poles, called consequent poles, may exist be- tween the two at the ends. If a straightened piece of clock spring, for example, is touched at several points with a magnet, iron filings will show magnetism at these points. (Fig. 28.) This is the prin- .A^^^^^^^^^Sfcl ■f^^^^B^^^^BP?- Fig. 28. ciple of an instrument called the magnetophone by which sound may be recorded on a steel wire or disc and later reproduced. The steel wire may be wrapped spirally on a cylinder and when the cylinder is rotated successive portions of the wire are moved under an electromagnet which varies in intensity in response to waves of sound as in the transmitter of a telephone. The different por- tions of the wire are therefore magnetized differently and, when the cylinder is turned in the same direction imder a suitable dia- phragm, the same sounds will be produced. It is possible to have a magnet without any poles. A mag- netised iron ring will have all its Unes within the iron. If, however, a portion of the ring is cut away, poles will be apparent at the ends of the gap. * In France the end of a magnetic needle which points northward is called the south pole and the other end the north pole. 44 GENERAL PHYSICS. 30. Law of Magnets. — ^According to a law announced by Coulomb in 1785, the force of attraction or repulsion between two magnetic poles is directly proportional to the product of the pole strengths and inversely proportional to the square of the distance between them. This may be written ^ W1W2 ,„,, Foe ^- 24) where F is the force, mi and mt the strengths of the poles, and r the distance. If, for any system of units that may be adopted, the value of F is always determined in vacuum, then equation (24) may be written F=^ (25) but for different media between the poles the attraction or repul- sion is different. This depends on the permeability of the medium. The more permeable a medium is the less will be the force. Hence equation (25) should be written F=^ (26) where 11 stands for permeability. The value of 11 for vacuum is 1, and for air is 1.000005. Hence the permeability of air may also be taken as 1. 3L Unit Magnetic Pole. — ^As stated in a previous section there are three systems of electrical tmits (§24). One of them is the electromagnetic system which depends on the definition of the unit magnetic pole. It should be remembered, however,that magnetism of one kind cannot, like an electric charge, be set off by itself. It is not possible to have an N pole without an S pole. It is possible, however, in case of a long, thin magnet to have the poles so far apart that the action of one may be considered without reference to the influence of the other. The conception of a single pole is often an advantage in this discussion. From equation (26) we see that it is possible to make mi equal to nii and of such strength that F is one dyne when r is 1 cm. and the medium is air. Such a strength of pole may then be taken MAGNETISM AND THE ELECTRIC CURRENT. 45 as the unit. Hence, unit pole is a pole of such strength that, when placed in air, 1 cm. from an equal and similar pole will repel it with a force of one dyne. 32. Magnetic Lines of Force. — The region about a magnetic pole possesses properties wliich are not found when the magnet is removed. The ether in this region is assumed to be under strain. A ferromagnetic body placed in such a field will become a magnet and will move or tend to move in such a direction as to reduce the strain. This is somewhat analogous to mechanical strains which cause a stress tending to restore a body to an un- strained condition. A magnetic field is any region in which Fig. 29. magnetic effects are produced. Faraday represented such a field by lines or tubes of force. These lines are assumed to pass from the south-seeking (5) to the north-seeking (A^) pole within the magnet but from TV to 5 in the field outside the magnet. Out- side the magnet lines always start at the N pole and end on the S pole. The direction of the lines at any point in a field is the same as that in which a small N pole would be urged if placed at that point. Some idea of these lines in one plane of the field may be obtained by sprinkling iron filings over a paper beneath which a magnet or magnets are placed. Thus in Fig. 29, where four magnets are arranged as shown, the filings become magnets and, clinging together, arrange themselves along the lines of force. 46 GENERAL PHYSICS. Faraday thought of these lines as being under tension. He also assumed a repulsion between lines. Hence there would be attrac- tion between the two upper magnets in the figure and repulsion between the two lower ones. 33. Intensity of a Magnetic Field. — The intensity or strength of any point in a magnetic field is the force with which unit pole would be acted on if placed at that point. If the strength of field is represented by H and the force by F, then F= H as long as the force is only that on the unit pole. If the unit pole is replaced by one of strength m, then F=Hm (27) A unit field is one in which the force on unit pole is 1 dyne. Such a field is usually represented by one line per square centi- metre of surface at right angles to the field. Unit strength of field is called a gauss. The number of lines per square centimetre then represents the strength in gausses. 34. Magnetic Induction. — Magnetic induction is denoted by B and is defined as the total number of lines per square centi- . metre appearing in a body IT-I ~.^ ^^g=5^^5SSS^ ^r' * body and those which would v"( v'<:c:^ ' — :z ->^t)' ~ ■* be in the same space if the -^^^~^-=.^^j==— -<^^^A — » body were removed. Let a Lin I_*^^^r^=-^]^i^r ~ HX bar of iron, Fig. 30, be placed : I_^rtZ!riri_'~_Z~L~_l5 "^ ^ fi^^*i of intensity H. The molecular magnets of Fig. 30. _ , "n • the iron will be turned so that their lines coincide with those of the field. The bar therefore becomes a magnet, and a much greater number of Hnes now pass through the iron from S to N. This additional magnetism has not been created. It was already in the iron and has only been made by the field to have a conmion direction. If the total induc- tion per square centimetre is B and the number of lines in the same region of the field is H, then B — H is the niunber of lines due to the molecular magnets of the iron. MAGNETISM AND THE ELECTRIC CURRENT. 47 If a unit pole is placed at the centre of a sphere 1 cm. in radius, then the strength of the field at any point on the surface of the sphere will be unity. Since there are 4jr square centimetres of surface there must be 4ir lines coming from the unit pole. If there are m unit poles at the centre of the sphere, ivm lines would arise from it. Hence when the pole strength per square centimetre of the face of the iron bar is m, there will be 4irm lines coming out from that area at the north-seeking end, passing through the air to the south-seeking end, and thence through the iron from S to AT. Consequently the total induction B is expressed by B=H+imr^'^'' (28) 35. Intensity of Magnetization. — Intensity of magnetization is denoted by / and may be defined as the number of unit poles per square centimetre induced in a bar of iron or other magnetic sub- stance when placed in a magnetic field. This means the number of unit poles per square centimetre of the face of the magnet. From what has been said in the previous section we may, according to this definition, write 7 = ^ (29) 4:7r Comparing this with equation (28) we see that I=m. Hence we may write 5 = H+47rI (30) 36. Susceptibility. — Magnetic susceptibihty is usually denoted by k and is defined as the ratio of the intensity of magnetization to the strength of the field, H, which produces it. This may then be expressed by the equation ^ k=jj (31) 37. Permeability. — Permeability has already been discussed in § 26. We are now ready to define it in terms of other quantities. It is usually denoted by n and may be defined as the ratio of the total induction, B, to the strength of field H. Hence M= f (32) Combining equations (28), (29), and (30), /. = H-47rtfe (33) 48 GENERAL PHYSICS. 38. The B=H Diagram. — ^When a bar of soft iron or other magnetic substance is placed in a magnetic field that may be varied at will, then as the field, H, is increased step by step, the induction, B, will also be changed. If the values of H and B are determined for each change and are plotted as abscissas and ordi- nates respectively, we obtain a curve like that in Fig. 31. The curve is different for different substances, but, as a rule, three different stages of magnetization may be observed. From o to a the induction increases slowly. From a to 6 a small increase in H will produce a large increase in B. Then the ratio of B to H begins to grow less and the curve finally becomes a straight fine which, if B and H are plotted on the same scale, will be inclined 45° to the axes, i.e., the increase of lines of force in the iron is the same as it would be in air at that place if the iron were removed. Iron in this final state is said to be saturated. A physical interpretation of the B-H curve would probably be that ferromagnetic molecules of an im- magnetized bar of iron are influenced by the magnetic fields of their neigh- bors and bound together loosely in irregular groups which can be broken up only by the application of force. This would explain the curve from o to a. After the orientation of the molecules begins, a slight force would suffice to turn them to a position where their fields would more nearly be in the same direction, and when all were turned, a further increase in the field H could have no effect in increasing the magnetization of the iron. An I-H diagram would be similar to that of Fig. 31, but, since I, from equation (29), is proportional to B—H, then I includes only the lines which are induced as a result of H and not the lines of H. Hence this curve at the point of saturation would be parallel to the abscissa. 39. Hysteresis. — Hysteresis is a terra which means lagging behind. It is observed when iron is subjected to the influence of an alternating magnetic field, for then the induction B lags behind the magnetizing force H. This will be understood by a MAGNETISM AND THE ELECTRIC CURRENT. 49 consideration of the curves in Fig. 32 where the abscissa represents the magnetizing force and the ordinate the resulting induction. When an alternating current flows in a coil of wire, the field in the ' coil is alternately directed one way and then the other. A core of soft iron within the coil will then be subjected to a field which changes its direction each time the current is reversed. Suppose that the iron is completely demagnetized and therefore in a state represented by o in the figture. Now let B., which is zero at o, increase in a positive direction until the iron is saturated. (+ and — here signify only opposite directions of the magnetizing field.) The induction B will in- crease from o to 5 and the B-H curve will be oaC. If now the strength of the current in the coil is decreased, the field H is decreased and may be restored to the state o where it is zero. But the curve, instead of returning along the path Cao, follows a path CD, and although there is now no magnetizing force, yet the induction is still the large quantity oD, called residual magnetism or the remanence. The ratio of oD to the value of Ch at saturation is called retentivity. It is observed that the induction lags behind the magnetizing force and hence the application of the term hysteresis. In order to remove this residual magnetism, i.e., to extend the curve CD to E, it is necessary to reverse the current and thus apply the force of a reversed magnetic field oE. This quantity oE is therefore called the coercive force. In case of hard steel this force is very large and hence the possibility of permanent magnets made of that material. In soft iron and mild steel coercive force is small. If the reversed magnetic field be increased in strength to h', the curve CDE will be extended to F, where the induction will be the same as at C, but with the lines of force in an opposite direction, i.e., the molecular magnets in the iron have been made to turn through 180° while the field changed from h to h'. Now let the negative field decrease in strength from h' to o 4 Fic. 32. 50 GENERAL PHYSICS. and the induction will then be oK. Here again B is lagging behind H and a certain coercive force oj is needed to remove the residual magnetism. A further increase of the positive field to h restores the state C. Thus a cycle has been completed. Any- subsequent cycle of changes in the field from h to h' and back to h will result in a cycle of changes in the induction which, when traced on a'' B-H diagram, will describe a hysteresis loop CDEFJC. At all points of the loop induction lags behind the magnetizing force, which shows that it requires a certain force to cause the molecules of iron to face about. If no such force were required, the lines oE and oJ would, as far as they represent coercive force, become zero, i.e., E and J would coincide with o and instead of a loop there would be a single curved line which would be retraced each time H changed from h to h' or from h' to h. Under this condition the area of the loop would be zero, but since a force has been applied in producing a change, the area of the loop represents the amount of work done in one cycle of operation. This work is lost as far as this cycle of operations is concerned and appears as heat in the iron. It is evident from this that the quality of the iron is an impor- tant consideration in the construction of electric machines where the induced magnetism must rapidly change direction under the influence of a rapidly alternating field as in case of transformers, armatures of dynamos, etc. 40. Effect of Temperature on Magnetism. — Since both heat and magnetism are treated as molecular phenomena, we would expect that a change in temperature would result in a change of the magnetic properties of a magnetic substance. When a bar of soft iron is placed in a weak magnetic field and heated, no sensible increase in permeability is noticed tiU the temperature rises above 600° C. Then there is a rapid rise in permeability which later falls to zero at about 800° C. — i.e., at 800° C. the iron both loses its magnetism and ceases to be magnetic. As the iron cools it regains its magnetic property but at a lower temperature. In stronger magnetic fields the changes are not so abrupt, but in all cases there is a so-called critical temperature for magnets at which they cease to be magnetic. Nickel and cobalt possess this same property, but their critical temperature is lower. MAGNETISM AND THE ELECTRIC CURRENT. 51 It appears that when the molecules are greatly agitated by heat the magnetic field is not able to keep the molecular magnetic fields of the iron in line. 41. Magnetic Moment. — Magnetic moment is defined as the product of the pole strength and the distance between the poles. If M is the magnetic moment and m the pole strength, then M=ml (34) This is the same as the turning moment which a magnet would experience when placed at right angles to a unit magnetic field. If in Fig. 33 the value of the uni- form field, H, is unity — i.e., such that unit pole placed at any point would be acted on by a force of one dyne — ^then m unit poles will be acted on by a force of m dynes in one direction, while an equal force will act on the other pole of the magnet in an opposite direction. This gives a me- chanical couple of which the moment is ml. The magnetic moment, therefore, varies only with the strength of the poles of a magnet and the distance between the poles. If the magnetic moment is known, it is plain that the moment of force tending to turn a magnet placed as in Fig. 33 in a field of strength H is Hml or HM. If we use the expression Fh to represent moment of force we may write Fh=HM t-\,\;\ui (35) If, however, the magnet is inclined at angle 6 to the direction of the field as in Fig. 34, then the component of Hm which is effective in turning the magnet is Hm sin 8 and the moment of the couple is Hml sin 6. Hence a general equation for any posi- tion of the magnet is Fig. 33. Fh = Hml sin 6 = HM sin 6 (36) 62 GENERAL PHYSICS. If the magnet is free to turn it will take a stand parallel to the field just as a compass needle turns to a position where it is parallel to the earth's magnetic field. If then a magnet is suspended as Fig. 34. shown in Fig. 35 and is made to swing as a torsion pendulum, it will execute simple harmonic motion provided 6 is so small that sin may be regarded as equal to 6 measured in radians. With this imder- standing we may write equation (36) as follows: Fh — =HM (37) B Since H and M are constant quantities, 6 must vary directly as Fh. This is a condition of S.H.M. Consequently, from equation 127 of " Mechanics and Heat " we may write 47r2J fy or V HM= O "-^'Vs (38) (39) Hence, if we determine the moment of inertia, I, of a magnet and its period of oscillation P, we can readily calculate the value of HM from equation (39). 42. The Field Produced by a Magnet. — Since it is impossible to have an isolated magnetic pole, the strength of field at any Fig. 35. MAGNETISM AND THE ELECTRIC CURRENT. 53' point near a magnet will be a resultant effect of both poles. For example, let SN, Fig. 36, be a magnet 10 cm. long, and suppose it is desired to know the strength of field at G, a point 10 cm. from the north-seeking pole N. Let the pole strength of the magnet be 50 units, i.e., 50 unit poles. Then since strength of field at G is the force with which unit pole at that point is urged, we have from equation (25), for the field due to the N pole, 50/10^ and for that due to the S pole, 50/20^. But since the poles are opposite in kind, their resultant effect at G would be their difference or .375 gauss.- u w-t ilnji ^ "^ LaX^ ^ Fig. 36. It is desirable to know the field strength, F, due to a magnet, in terms of the magnetic moment of that magnet and the distance from the centre of the magnet to the point of the field being con- sidered. Let this distance be r, Fig. 36, the length of the magnet /, and the poles +m and —m. Then the strength of field at G would be, as shown above, — -^ due to the N pole, and — — ^ due to the S pole. The resultant field at G is therefore the alge- braic sum of these quantities. Hence F=- — ^t; (40) (--t) But ml is the magnetic moment, hence we may write (-4) If the distance r is many times greater (20 times or more) than the length of the magnet, no appreciable error will result from the omission of ZV^- Hence we may express the strength of field by F=-;r (42) 54 GENERAL PHYSICS. 43. Action of a Magnetic Needle in Two Fields at Right Angles. — It has already been shown how it is possible to find the strength of the field due to a magnet. Let NS, Fig. 37, be the magnet and let a short magnetic needle, suspended so that it may freely turn, be placed in this field at a distance r from the centre of the magnet. The effect of the magnet alone would be to turn the needle to a position parallel to its lines of force F. Let another field, such as the horizontal component of the earth's magnetism, act with an intensity H at the same point but in a direction at right angles to F. The effect of H alone would be to turn the needle parallel to H. The needle will therefore occupy Fig. 37. a position of equilibriimi between the two forces. Let this posi- tion be such that the needle makes an angle 6 with the field H, and let n be the pole strength of the needle whose length is /'. Then the turning moment of the if -field is Hnl' sin 6, while that of the F-field is Fnl' cos 0. Since the needle is in equilibrium imder these two moments, we have the equation Hnl' sin B=Fnl' cos or F=Htan.e (43) Substituting the value of F from equation (42), 2M „ —r=Hta.x).e H~ 2 (44) MAGNETISM AND THE ELECTRIC CURRENT. 55 44. Determination of M and H. — The value of HM may be found from equation (38) by the use of the vibration magnet- ometer shown in Fig. 35. The value of the moment of inertia I can be calculated from the dimensions of the magnet and the period of vibration, P, can be observed. Also the ratio oiMtoH can be found from equation (44) by use of a deflection magnet- ometer shown in Fig. 38. Here a small magnet fastened to the back of the mirror serves as the magnetic needle, and the magnet used in Fig. 35 is placed so that its axis is at right angles to the length of the needle and at a distance r, as shown in diagram Fig. 37. The mirror turns with the needle so that by use of a tele- scope and scale the value of d can be observed. If equations (44) and (38) be multiplied together we get the equation M^= (45) in which all the terms on the right can be determined. Hence the magnetic moment, M, can be foimd. The division of (38) by (44) gives H^=^^^ (46) PV3 tan e from which the intensity of a field, as if in Fig. 37, may be determined. It is also observed from equation (38) that M and I are constant quantities for the same magnet, con- sequently different fields may be compared by the relation i.e., the intensities of two fields vary inversely as the squares of the periods of vibration of the magnet in those fields. This and equation (44) will give the total intensity only when the Hues of force of the field are parallel to the plane of vibration of the ■ magnet. Otherwise the magnet will be influenced only by that component of the field to which it is parallel. The lines of the earth's magnetic field, for example, are not horizontal except at o 56 GENERAL PHYSICS. points near the equator. Hence, since the plane in which the magnet vibrates is horizontal, the methods described above will give, in this case, only the horizontal component of the earth's magnetic intensity. 45. Terrestrial Magnetism. — The earth is surrounded by a magnetic field of moderate intensity. The lines of force in a general way run north and south and lead in one direction to a north magnetic pole at a point northwest of Hudson Bay, about 97° W. longitude and 70° N. latitude. In the other direction the lines lead to a south magnetic pole south of Australia, about 148° E. longitude and 74° S. latitude. These poles, however, are not fixed but are slowly shiftiag. If a magnetic needle is mounted so that it is free to swing in all directions it will take a position parallel to the earth's lines of force. At the equator, or near it, the needle will be horizontal. As it is moved northward the north- seeking end will point more and more downward, until at the north magnetic pole it will stand in a vertical position. At the south pole the needle will again be vertical but with south-seeking end downward. The angle which the needle makes with a horizontal plane at any point on the earth's surface is called the inclina- tion or dip. This angle has been determined for many points on the earth. Lines drawn through points of equal dip are called isoclinic lines. These, though irregular, correspond in a general way to parallels of latitude. When a magnetic needle is moimted so that it can swing only in a horizontal plane, as in the case of compasses used in naviga- tion and survejmig, it is evident that, except where there is no dip, only a component of the field intensity is effective in directing the needle. In Fig. 39 let R be the direction of the magnetic field. It is also the direction of a dipping needle. Then 6 is the dip or inclination. Let the line R represent the total intensity Fig. 39. MAGNETISM AND THE ELECTRIC CURRENT. 57 of the field, then Y and H are its vertical and horizontal com- ponents. The value of H is of most importance and one method of finding it has been described in previous paragraphs, where it is shown that by use of the vibration and deflection magnetometers aU the values for equation (46) may be found. When H has been found the total intensity R may be calculated if the dip, B, is meas- ured, for by Fig. 39, i?=— ^ (47) cos 6 For example, if H is .2 gauss — i.e., the horizontal iatensity of the field is such as will exert a force of .2 dyne on unit pole — and if the dip at that point is 70°, then the total intensity R is about .58 gauss. It is also observed that V may be found from F=i?tan9 (48) If now we confine our attention to the position of a needle in a horizontal plane we find that it does not in general point in the Pig. 40. direction of a geographical meridian but makes an angle with it. This angle is called the declination or variation and fines which pass through points of equal declination are called isogonic lines. These correspond in a general way to geographical meridians but, as shown in Fig. 41, are very irregular lines. 58 GENERAL PHYSICS. A line drawn through all points of zero declination is called an agonic line. The agonic line in the United States passes through Michigan, Ohio, Kentucky, Tennessee, North Carolina, and South Carolina. At points east of the agonic line a magnetic needle points west of north, and, west of the line, east of north. On the map, Fig. 40, the heavy chain lines indicate the dip at points through which they pass. The broken lines show the number of minutes of change in dip each year. One of these is marked 0', i.e., there is no change in dip at points on this line. Below this line the dip is increasing Fig. 41. a certain number of minutes each year as marked at the end of the line. In New England and on the Pacific coast dip is decreasing as the map shows. This map is taken from one prepared by the Coast and Geodetic Survey and includes all observations up to 1907. The map shown in Fig. 41 is taken from the same source and includes observations on declination up to the year 1910. The heavy chain lines are the isogonics. The one marked 0° is the agonic. The broken lines indicate the annual rate of change in declination. One of these marked 0' is a line on which there is no annual change. All broken lines west of this are moving east- ward and those on the east side are moving westward at a rate per year indicated by the number of minutes at the end of the line. MAGNETISM AND THE ELECTRIC CURRENT. 59 Declination, iijclination, and magnetic intensity are called the magnetic elements. These are now being detennined at many- points in all civilized countries. Accurate observations have been made in England since the year 1540. It is expected that when sufficient data covering a long period of time are collected some law in regard to magnetic changes in the earth's field wiU be estab- lished and also that some plausible theory of the origin of the earth's magnetism may be evolved. One thing that is clearly shown is that the magnetic poles and all the magnetic elements are changing. The most important of these is the so-called secular change which it is presumed will complete a cycle in about 470 years. If a magnetic needle were pivoted at the centre in such a manner that it is free to indicate both inclination and declination at the same time, each pole will very slowly describe an irregular curved line which, it is thought, will form a closed curve. Such a curve may be plotted, as far as the data go, from observations of inclination and decUnation that have been made from time to time. From observations such as these it appears that the secular change in England should com- plete a cycle in about the year 2020 a.d. In addition to the secular change there are also cycles of change each day and each year, though these are sUght. The magnetic field is also modified by the influence of the moon, by conditions which give rise to sun spots, and by magnetic storms. Problems 1. If the strength of a magnetic pole, regarded as isolated from its oppo- site pole, is 147 units, at what distance in air will the strength of the field be 3 gausses? 2. The pole strength of a magnet is 350 units. The distance between poles is 10 cm. What is the strength of field at a point 10 cm. distant from the N pole and in line with the axis of the magnet? 3. What is the force exerted on a pole whose strength is 3 units when placed at a point 3 cm. directly above the 5 pole of a horizontal magnet 4 cm. long, the pole strength of the latter being 50 units? 4. A rod of soft iron placed in a magnetic field acquires a pole strength of 200 units. How many lines of force are thus made to pass through the rod? 5. If the distance between the poles of a magnet is 4.2 cm. and the mag- netic moment is 50 c.g.s. units, what is the pole strength? 60 GENERAL PHYSICS. 6. The magnetic moment is 300 c.g.s. units and the magnet is placed in a uniform field whose strength is 12 gausses. What will be the moment of the couple? 7. If the strength of the earth's magnetic field is .2 gauss and a needle whose magnetic moment is 300 c.g.s. units is placed so that it is inclined 30° to the field, what is the moment of the couple tending to turn the needle? 8. What is the permeability of a sample of wrought iron when a field strength of 2 gausses causes an induction of 8000 gausses? 9. If an induction of 8000 lines per square centimetre results from placing an iron bar in a field of 12 gausses, what is the intensity of magnetization? 10. If the period of vibration in a magnetometer at a place where the earth's field is .19 gauss is 15.12 seconds, what is the strength of field at another place where the period of the same instrument is 16 seconds? 11. What is the total strength of the earth's field where the horizontal component is .2 gauss and the dip is 70°? Ans. 1. 7 cm. 2. 2.725 gausses. 3. 12.39 dynes. 4. 2513. 5. 11.9 units. 6. 3600 dyne centimetres. 7. 3 dyne centimetres. 8. 4000. 9. 635.6 unit poles. 10. .17 gauss. 11. .585 gauss. CHAPTER III ELECTROMAGNETIC AND PRACTICAL UNITS 46. Unit Strength of Current. — In 1820 Oersted of Copenhagen made the discovery that when a current of electricity flows in a conductor, a magnetic field is set up at right angles to the direction of the current. He showed that when a magnetic needle is placed in this field it tends to take a position at right angles to the con- ductor, the N pole of the needle being driven in one direction along the lines of force and the 5 pole in the opposite direction. If now this conductor is bent in form of a circular loop, Fig. 42, the magnetic fields which surround the con- ductor at all points will create a magnetic field at the centre of the circle, the intensity of which will depend on the strength of the current. This principle has been used in determin- ing a unit electromagnetic strength of cur- rent, the definition of which is as foUows: The electromagnetic unit of current is that cur- rent which when flowing through an arc 1 cm. long, of a circle 1 cm. in radius, will act on p^^ ^^ unit pole at the centre with a force of 1 dyne. The loop shown in Fig. 42 is filled with lines of force, as has been shown in Figs. 22 and 23 for one plane in the field. A con- vention for finding the direction of lines of force is as follows: Grasp the conductor with the right hand so that the thumb mil point in the direction the current is flowing. The fingers will then point in the direction of the lines of force of the magnetic field. If then a positive pole is placed at o in Fig. 42 it will be moved along the lines of force in a direction at right angles to the plane of the coil. A south-seeking pole would be urged in an opposite direction. According to the definition for unit electromagnetic (e.m.) strength of current the strength of field at the centre of unit circle wotild be 2ir dynes, for there are 2rr cm. in the complete circle. For n loops of wire in unit circle the strength of field at o would be 2im. If in addition the strength of current be made i instead 61 62 GENERAL PHYSICS. of one unit as defined, the strength of magnetic field becomes 2jrm. If, finally, the radius of the circle is made r cm. instead of 1 cm., then, since the length of a conductor looped in form of a circle varies directly as r and the magnetic force from any point varies inversely as r', we have, letting F represent the magnetic intensity at 0, j, ^jU-^d jj y-'^,^} .i.k>-' F — -r- = ^r (49) /'/ ^*-'' «'''^%; -^ 1^"-^ r^ r This gives the strength of field, i.e., the number of gausses, or dynes of force on unit pole, at o when i is taken in e.m. imits. 47. The Ampere and Coulomb. — For commercial purposes the e.m. unit as defined above is inconveniently large. So a practical unit called the ampere has been adopted. It is one-tenth (10~') of the e.m. unit. The g.m._.uniJLjof (j'wgwfo'<;j>^^oldectricity is the quantity per second which passes any cross section of a conductor in which the strength of the current is one e.m. unit. When the strength of current is one ampere, the quantity which passes per second is called one coulmnbj The coulomb is then the practical unit of quantity and is one-tenth (10~0 of the e.m. unit of quantity. By strength of current is meant the quantity rate at which elec- tricity is transferred on a conductor, while quantity of electricity, measured in coulombs, is independent of the time in which it may be made to flow from one point to another. By analogy the strength of a current as measured in amperes may be compared to the number of gallons of water per second flowing through a pipe. We may speak of the strength of the stream in terms of the number of gallons per second. But the quantity of electricity as measured in coulombs would correspond to the number of gallons of water in a vessel, no matter how or at what rate it had been placed there. The strength of current is one ampere when one coulomb per second passes any cross section of a conductor. 48. Comparison of e.m. and e.s. Units. — In the electrostatic system, as has been shown, the unit quantity of electricity was first defined and then unit strength of current, *', would be the ntunber, Q, of e.s. units of quantity per second which pass a cross section of the conductor. This is expressed by »=f (50) ELECTROMAGNETIC AND PRACTICAL UNITS. 63 This equation also expresses the proper relation of the quantities in the e.m. system, but there it was i that was first defined, while in the e.s. system Q is defined as the fundamental unit. Probably a more fundamental unit of current would be the number of electrons which pass any point per second or a strength which would cause a repulsion or attraction of 1 dyne between two conductors at a distance of 1 cm. But the systems which we have were established at a time when it was not so well known as now that aU magnetism is electromagnetic, and that when a charge of static electricity is set in motion it will create a magnetic field in the same manner as does the current from a battery. If a given quantity of electricity is measured first in e.s. units, and then in e.m. units, in a manner which will be indicated later, it will be found that the former is, in round numbers, 3(10)'" times larger than the latter. This means that the e.m. unit of quantity is 3(10)"* times as large as the e.s. unit. 49. Unit Difference of Potential. — ^The P.D. between two points has already been defined (§6) as the number of ergs of work that must be performed in moving a unit quantity of elec- tricity from one point to the other. This is the definition no matter what system of units is used, but since the e.m. unit of quantity is 3(10)'° times as large as the e.s. unit, 3(10)'° times as many ergs of work must be done on the e.m. unit quantity as on the e.s. unit for the same actual difference of potential. Hence the P.D. indicated by 1 erg of work in the e.m. system is a very small quantity. In the e.m. system two points have unit potential difference when 1 erg of work is done in moving unit e.m. quantity from one point to the other. The unit P.D. in the e.m. system is therefore i(10)~'° part of the corresponding unit in the e.s. system. 50. The Volt. — Since the e.m. unit P.D. is inconveniently small, (10)' of them have been taken as a practical unit called the volt. The volt then may be defined as the P.D. between two points when (10)' ergs of work must be done to transfer unit e.m. quantity of electricity from one point to the other. The volt is therefore wr of the e.s. unit P.D., for i(10)~'° times (10)' =4(10):''. 51. Electromotive Force. — ^When potential difference is re- garded as a cause of the flow of electricity it is usually called 64 GENERAL PHYSICS. electromotive force. Thus in Fig. 43, if a is charged positively and b negatively, there is a P.D. between a and b which is the E.M.F. causing a flow of current when the points are connected by a conductor. If we follow a convention and say that a current flows from a to b, there will be a drop of potential at each succes- sive point from a to b, the sum of which is equal to the P.D. between a and b. If two points, as c and d, are joined by a con- ductor, a current will flow on dec for the potential at d is higher than at c. This P.D. may be called the E.M.F. of the current dec. The greatest P.D. which a battery is capable of creating is called the E.M.F. of that battery; for example, the P.D. between the terminals of a cell on open circuit. The same may be said of other electric generators. How a battery produces a P.D. that acts as an E.M.F. will be described in a later section. b Saa/v. Fig. 43. Electromotive force, like potential difference, is measured in volts. 52. Unit of Resistance. Ohm's Law. — The German physicist, G. S. Ohm, in 1827 announced a law, known by his name, that for any given conductor the strength of the electric current is directly proportional to the electromotive force. Thus if * is strength of current and E is E.M.F., then Ohm's law may be stated by ^ = i? (51) where Risa. constant quantity which has been called the resistance. If, however, the conductor is changed in dimensions, state, or kind of material, the resistance will change and the E which was formerly used will give a different value for i. Equation (49) is therefore better written i = I (52) The absolute e.m. unit of resistance is the resistance of a conductor when a P.D. of pne e.m. unit maintained at its terminals causes a current of one e.m. unit. ELECTROMAGNETIC AND PRACTICAL UNITS. 65 The practical unit of resistance is called the ohm. An ohm is the resistance of a conductor in which a P.D. of one volt will cause a current of one ampere. Since a volt equals (10)' absolute e.m. units of potential and the ampere equals (10)~* absolute units of current, then, from equation (51), the ohm equals (10)'^(10)-i = (10)» absolute units of resistance. 53. G)ncrete Standards. — The c.g.s. electromagnetic tmits as defined above are the fundamental tmits of the e.m. system. But although these units are real they are not practical in actual application for comparison, measurement, or legal action. Hence certain concrete units, made to conform as nearly as possible to the fundamental ones, have been prepared. This is much like the fundamental definition of the metre as one-ten-miUionth of the earth's quadrant, but the practical metre is a platinum- iridium bar which was made to agree as nearly as possible with the fundamental definition. The ohm, ampere, and volt have been defined in terms of con- crete standards and are called international units. The international ohm is the resistance offered to an unvarying current by a column of mercury 106.3 cm. long, of constant cross- sectional area, and at the temperature of melting ice, the mass of the mercury being 14.4521 grams. The area of cross section of such a column is practically one square miUimetre. The international ampere is the unvarying electric current which when passed through a solution of nitrate of silver in water, in accord- ance with standard specifications, deposits silver at the rate of .001118 g. per second. The international volt is that electromotive force which will pro- duce a current of one international ampere in a conductor whose resistance is one international ohm. International units as defined by these concrete standards agree so closely with the fundamental ones that for all practical measure- ments no distinction need be made. 54. Energy Relations. — Since a volt is such a P.D. between two points that (10)* ergs of work must be done to transfer 1 e.m. unit of quantity of electricity from one point to the other, then to transfer one coulomb (10~^ e.m. unit) would require 10' times 10"* = 10' ergs. This is one joule of energy. If the number of 5 66 GENERAL PHYSICS. coulombs is represented by Q and the number of volts by V, then Energy in joules = VQ (53) Now the strength of a current in amperes is the number of coulombs which are transferred per second. Hence if amperes are repre- sented by i and / is the time in seconds, Q=it as in equation (50). Hence equation (53) may be written Energy in joules = Vit (54) Also from equation (52), where E is the same as V, V = Ri. Substituting this value of V in (54), Energy in joules =i'^Ri (55) "A current of electricity consists of electrons moving from the negative to the positive terminal of a conductor, i.e., the current is a negative one, according to the electron theory, and the direc- tion in which electrons move is opposite to that in which an electric current has been assumed to move. While electrons thread their way between molecules of a conductor, they are impeded in their movement, particularly by poor conductors, and their energy appears as heat. At least this is one conception we may have of how an electric current heats a conductor. The speed of an electron may not be great, possibly about 1 cm. per second when the P.D. is one volt, but the speed with which an impulse travels along a line of electrons, setting them all in motion, is enormously great — about 3(10)"' cm. or 185,000 miles per second. When work is thus done in moving electricity against the resistance of a conductor and the only resistance is that due to the nature of the conductor itself, then all the energy expended appears in the conductor as heat. If the conductor is immersed in water in a calorimeter, the nimiber of calories of heat may be measured by simply multiplying the mass of water plus the water equivalent of the calorimeter by the rise of temperature. But the mechanical equivalent of one gram-calorie is 4.187 joules, hence, from equations (54) and (55), For i or R may be found when the other quantities are known. Also the number of calories of heat may be calculated when the strength of current in amperes, the resistance of the conductor in ohms, and the time are known. ELECTROMAGNETIC AND PRACTICAL UNITS. 67 Thus if both sides of equation (55) are divided by /, the mechanical equivalent of heat (see p. 246, " Mechanics and Heat "), Energy in joules _ i^Rt number of calories = 7—07 ~ -24*'^' (^6) A similar equation could be deduced from (52), and it is plain that these relations furnish an excellent means of determining the mechanical equivalent of heat if the electric quantities involved can be accurately measured. Now Vit is the total energy of a current where V is volts, i.e., 10* ergs; * is amperes, i.e., 10"' e.m. unit; and t is time in seconds. If i is 1 second, then Vit is 10' ergs or one joule. One joule per second is taken as the unit rate of electrical work and is called the watt. The watt is the unit of electrical power and may be found bymultiplying volts byamperes. One kilowatt is 1000 watts. One kilowatt-hour is the quantity of work done in one hour when the power is one kilowatt. There are 746 watts or .746 kilowatts in one horse-power. Problems 1. What power is required to maintain a current of .5 ampere at a pressure of 110 volts? 2. What will it cost to maintain 50 lamps, the resistance of each being 220 ohms, the current required being .5 ampere, when the rate is 10 cents i per K.W. hour? t-V^i, rf "^ H i>0 : 7, ^ Tr^o 3^ «>. r 5 ^^^0 ■^-- 3. If 2.7(10)' electrostatic units of quantity of electricity pass through a conductor per second, what is the strength of current as measured in amperes? 4. How much work is done in moving 25 coulombs of electricity through the space between two points whose P.D. is 2 e.s. units? 5. What will be the strength of field at the centre of a coil of wire, the diameter being 30 cm., the number of turns 20, and the strength of current 2 e.m. units? » 6. How much heat will be developed by the passage of 500 coidombs of electricity through a wire whose resistance is 25 ohms? Ans. 1. 55 watts. 2. 27.5c. per hr. 3. 0.9 ampere. 4. 15,000 joules. 5. 16.75 gausses, i)^-^^ • 6. 2985.43 calories. ^ CHAPTER IV CONDUCTION OF ELECTRICITY THROUGH SOLUTIONS 55. An Electrolyte. — ^An electrolyte is a solution which serves as a conductor of electricity. Electrolytes of greatest importance are aqueous solutions of strong acids, bases, and salts. Pure water is a nonconductor of electricity and so likewise is pure sulphuric acid, but a solution of sulphuric acid in water is a good conductor. We shall see, however, that the method by which electricity is conveyed through an electrolyte is very different from that in a metallic conductor. In case of electrolytes, as we have seen, osmotic pressure, lowering of freezing point, and elevation of boiling point are all abnormal. (See p. 220, " Mechanics and Heat.") 56. The Electrolytic Cell. — ^Faraday in 1832 made a close study of the effects of passing a current through solutions — a process which he called electrolysis. The solution E, Fig. 44, is the elec- trolyte which we will say for our present purpose is copper sulphate (CUSO4). Plates A and C, extending down into the electrolyte, are called electrodes, A being the anode * (the way up) and C the cathode (the way down) when the current flows in the conventional direction, as indicated by the arrow heads. A battery B creates a potential difference between A and C. The effects observed in this particular case are a deposit of pure metallic copper on the cathode and a passing of copper from the anode into the solution. 57. Action within an Electrolytic Cell. Dissociation. — ^Various attempts have been made to construct a theory which would con- sistently explain experimental facts of electrolysis. In 1805 Grotthus advanced the theory that molecules of a dissolved sub- stance are composed of positive and negative atoms and when these are placed in an electrostatic field, as between the electrodes in Fig. 44, they are turned so that their positive sides are toward * The terms anode and cathode originated with Faraday. He placed the electrolytic cell so that the current passing through it would be parallel and in the same direction as the earth currents which were assumed to flow from east to west or in the direction of the sun's apparent motion. Hence the plate of the cell on the side where the sun comes up was called the anode, and on the side where the sun goes down, the cathode. 68 ELECTRICITY THROUGH SOLUTIONS. 69 the cathode and the negative sides toward the anode. Then when the P.D. becomes sufficiently great, the positive and nega- tive atoms were supposed to be forced apart and caused to move by steps — i.e., from molecule to molecule — toward that electrode by which they are attracted. Thus it was supposed that molecules continued as such in a solution and were broken up only by the action of an electric force. Faraday likewise regarded the electric current as the immediate cause of the breaking up of molecules in a solution. The parts into which molecules are separated possess a charge of electricity, either positive or negative. Faraday called these ions (from the A>^*- ,H .^ «i Pig. 44. Greek word to ?o). The ions which , „. move toward the cathode are called oa tk-^-j "•'--■- ^ the cations, and those moving in the opposite direction, anions. Thus the ions of hydrochloric acid "& + - + ': ^, are H and CI ; sodium chloride, Na "5 t/^ --H + "i and CI; sulphuric acid, Hj ando + - ' SO4; copper sulphate, Cu and SO4. An ion is an atom or a group of atoms possessing a charge of ' q' AAt^~- electricity. An objection to these theories was brought forward by Clausius in 1857. His objection was based on the fact that very little or none of the electric energy of the current is expended in the separation of molecules into ions as the Grotthus theory would require. In a cell like that shown in Fig. 44 where the electrodes are copper and the elec- trolyte CUSO4, the current flows as soon as there is a P.D. be- tween the electrodes and is proportional to the P.D. as Ohm's law requires for metallic conductors. It is true that when the substance deposited by the cation is different from that of the cathode there is a counter E.M.F. due to the tendency of the substance to go back into solution and therefore a certain critical P.D. of the electrodes, different for different substances, is then necessary before electrolysis will begin, but when once started the current is proportional to E.M.F. whereas, according to the 70 GENERAL PHYSICS. older theory, the current should at once greatly increase as soon as the molecules have been broken up into ions. Clausius, therefore, proposed a modification known as the dissociation theory. This has already been alluded to under Osmotic Pressure. The theory assumes that in an electrolyte the ions of molecules are dissociated whenever a solution is formed. No cur- rent is needed to break up the molecules. In fact the presence of ions with positive and negative charges is a necessary condition of an electroljrte, and without these a solution is not a conductor of electricity. A solution of sugar in water is a nonconductor of electricity because the molecules of sugar are not separated into positive and negative ions. It is not necessary to assume that all the molecules of an elec- trolj^e are ionized. In a strong sblution, as will be shown later, the ionization is only partial, though in a very dilute solution all the molecules may be separated into ions. Free ions may not remain so, but in strong solutions are constantly uniting with others of opposite sign and then separating. At any given time, however, there are a number of free ions and a number of neutral molecules where ions have united, except in very dilute solutions where all ions may be assumed to continue free. The effect, then, of a difference of potential between electrodes immersed in an electro- Ijrte is not to cause ionization but to direct the motion of ions already in the solution. Attempts have been made to explain the cause of dissociation. Clausius was of the opinion that it resulted from violent impacts against water molecules. Another possible cause may also be given. It has been shown (equation 3) that the force between two electric charges depends on the medium in which the charges are placed — i.e., the force varies inversely as the dielectric constant of the medium. Now for water, which is by far the most important solvent, the value of if is 80 as compared to air. It is possible that this weakening of the bonds which hold together the ions of a molecule may, in certain substances, render effective the impacts of water molecules and result in dissociation. This theory did not find a ready acceptance until the publica- tion in 1887 of two remarkable papers,* one by Van't Hoff on the * See Modem Theory of Solutions: Harper's Scientific Memoirs. Pub. Harper & Bros. ELECTRICITY THROUGH SOLUTIONS. 71 laws of osmotic pressure of dilute solutions and the other by Arrhenius giving an explanation of abnormal osmotic presstire as due to dissociation. The same number of molecules of any sub- stance in a given volume of solution should give the same osmotic pressure. To- obtain the same nimiber of molecules a normal solution is made up, i.e., one containing a gram-molecule of the substance to a litre of solution.* A gram-molecule is a number of grams equal to the molecular weight. For sodiimi chloride, one gram-molecule is Na(23.05)+Cl(35.45)=58.5g. In like manner one gram-molecule of grape sugar is 342.2 g. If such a quantity of these substances be dissolved, each in say two litres of water, there would be an equal number of molecules per cubic centimetre in each solution. It would be expected, then, that the osmotic pressure would be the same in each case, but experiment shows that in dilute solutions the NaCl solution exerts a presstue twice as great as that of the sugar solution. The nattu^al explanation would be that when NaCl is dissolved in water the molecules are dissociated into ions, each of which acts as a physical molecule and affects the osmotic pressure as much as would one undissoci- ated chemical molecule of sugar. A solution of NaCl is an elec- trolsrte. All electrolytes produce an osmotic pressure abnormally great, but never more than twice as great in case of substances which cannot break up into more than two ions. In addition to this evidence in favor of dissociation, Raotilt has shown by an extensive series of experiments that the lowering of the freezing point and elevation of boiling point of electrolytes are abnormally great. This, as above, would be explained as a result of dissociation, each ion then acting as a physical molecule and virtually increasing the concentration of the solution. Again, if the theory of dissociation as explained above is correct, the electric conductivity of a dilute electrolsrte should be greater than one containing the same member of molecules in a concentrated solution, for in the former case the ions are so far separated that most of them (in very dilute solutions all of them) remain separated from their neighbors and act as carriers of electric charges; while in more concentrated solutions a part of * In case of a substance like CuSOj, a gram-molecule per liter is usually called a molar solution. One-half of this amount per litre of solution is then a normal solution, for Cu carries two positive charges, i.e., its valence is 2. 72 GENERAL PHYSICS. them, though not always the same ones, would be united with others of opposite sign, thus forming neutral molecules. Many experiments have been performed to test this point. The best way to do this is to determine what is called the molecular conduc- tivity of a solution, i.e., the conductivity with different dilutions but with the same number of molecules of the substance in solu- tion. This may be done by use of an electrolytic cell like that shown in Fig. 45, where two platinum plates, a and b, are set at a fixed distance apart. Then some normal solution is introduced — enough to cover the plates. A current can then pass from A to C" only by passing through the electroljrte from a to b. The resistance of that part of the circuit between a and b is then determined by use of a Wheatstone bridge as described in § 85. The reciprocal of this resist- <^: .A^ u Fig. 45. Electron Curre Pig. 46. ^lil^ ance is the conductivity; e.g., if the resistance of any conductor is three times as great as some standard of resistance, its con- ductivity would be one-third as great. Now remove one-half of the electrolyte and put in its place an equal quantity of pure water. This is dilution 2. Find the conductivity as above and multiply by 2, since one-half the molecules have been removed and we desire the conductivity when the number of molecules is the same for all dilutions. Hence, multiplying by 2 gives the same result for this dilution as would have been obtained if none of the electrolyte had been removed and a volume of water equal ELECTRICITY THROUGH SOLUTIONS. 73 to the total volume of the electrolyte has been added. Again remove one-half of the electrolyte and add an equal volume of water. This is dilution 4. Find the conductivity and multiply by 4. The next operation would give dilution 8. Thus the process may be continued until the electroljrte is very dilute. Details of this experiment may be found in laboratory manuals. An experiment of this character shows that, as the dilution increases, molecular conductivity also increases and approaches a certain fixed value. The physical interpretation of this would be that as the dilution increased a greater and greater number of molecules were permanently ionized. At very great dilutions the percentage of ionization may be regarded as 100. 58. Transfer of Electricity in Electrolytes. — The laws of electrol- ysis, as far as they apply to deposition of metals, the evolution of gases, or the principle of dissociation, are definite and as well estab- lished as are most of the laws of physics. But when we attempt to get a distinct physical concept of just what takes place between the electrodes and the electrolyte there is a great deal of uncertainty and lack of agreement among scientists. It may be profitable, however, to consider one possible way of looking at the matter. For a first case let the electrolyte E, Fig. 46, be a solution of hydrochloric acid in water, the electrodes being strips of platinum. An electron current of electricity flows from the negative pole of the battery B to the electrode C, and from A to the positive pole. These electrons, however, do not pass through the electro- lyte. C is still called the cathode and A the anode according to + estabUshed conventions. The ions in this electrolyte are H and — + CI. The H ions wiU steadily move toward the cathode C which + is negatively charged, i.e., has an excess of electrons. When H reaches C, electrons will pass to H, making it negative, when it will at once combine with a hydrogen ion which has not yet reached C and wiU form a neutral hydrogen molecule H2 which, with others, rises as a bubble through 'the electrolyte. On the other side the CI ions move to A, where electrons pass from CI to A, leaving the chlorine ion positive, when it at once combines with one of the CI ions which has not yet parted with its excess 74 GENERAL PHYSICS. of electrons, thus forming chlorine gas which is absorbed in the water of the electrolyte. According to this explanation, each of + one-half of the H ions took two electrons from C, and the same nimiber of CI ions gave two electrons to A. Now let the electrolyte be a solution of copper sulphate, CUSO4, and the electrodes, strips of copper. The ions are then ++ — ++ Cu and SO4. The Cu ions move up to the cathode and four elec- _ trons pass from C, making Cu negative, when it combines with a positive Cu, forming a molecule of copper which is deposited on the cathode. On the other side the SO4 unites with an atom of the copper anode and brings it into solution as fresh CUSO4. This permits four electrons to escape from A to the positive pole of the battery. This is the principle used in electroplating. The solution maintains its strength, but the mass of the anode grows less. If platinum electrodes are substituted for copper ones, copper will be deposited on the cathode as before, but at the anode the ++ SO4 displaces O of a water molecule and combines with Hj, forming H2SO4. The O then gives up four electrons to the anode, thus becoming positive, and then unites with a negative O, forming a neutral molecule of oxygen gas. These collect and escape as bubbles of oxygen. ^ In this case the solution soon becomes weak in copper ions, for copper is deposited on the cathode and is not replenished at the anode. In the electrolysis of water a cell like that shown in Fig. 47, called a water voltameter, may be used. Since pure water con- tains very few, if any, ions, it cannot be electrolyzed directly. ++ By the addition of a little H2SO4 the water will swarm with H2 and SO4 ions and, by use of them, molecules of water, H2O, may be separated into their constituent parts. In this cell the electrodes are platinum. The cell is filled with water acidulated with HaS04. When a battery having sufficient E.M.F., at least 1.5 volts, is con- ++ nected as shown, A becomes the anode and C the cathode. The Hj ELECTRICITY THROUGH SOLUTIONS. 75 ions then move up to C, take four electrons, and combine ++ with H2 near by, thus forming two neutral molecules of hydrogen gas, H2. At the anode the SO4 ion displaces the oxygen of a water ++ molecule, then combines with the H2 thus set free, forming a mole- cule of H2SO4. The O which has been displaced then parts with ++ four electrons to the anode, becoming O when it unites with an ++ O, thus forming a molecule of oxygen, O2. But since for each Hj ion two molecules of hydrogen are formed, then for the same trans- ference of electrons the volume of hydro- gen gas evolved should be twice as great as that of oxygen. Experiment shows such a result for the volume of water displaced at H is twice as great as that at 0. In electrolytic actions such as those described above, the results which are ob- tained and their quantitative relation to the current employed are simple matters of experimental verification, but a de- scription of just what occturs within an electrolytic cell can be supported only by claim of consistency with known laws of electricity and chemistry. It will be observed that the word elec- trolysis, meaning to set loose by means of electricity, no longer retains its etymological meaning, and, likewise, the terms cathode and anode should be defined not in refer- ence to the direction in which a current flows but rather in reference to the direction in which positive or negative ions move. 59. Faraday's Laws of Electrolysis. — ^Faraday in 1832 an- nounced two laws which embody the main results of his investi- gation of the relation between electric currents and the products of electrolysis. It has never been necessary to change his state- ment of these laws. They are : Fig. 47. 76 GENERAL PHYSICS. 1. The mass of any given substance produced by electrolysis is proportional to the quantity of electricity which flows in the electric circuit. 2. The masses of different substances produced by equal quantities of electricity are proportional to the chemical equivalents of those substances. The first law states a result which wotild be expected from the nature of action in an electrolytic cell as described in the preceding paragraph, for a current flows through the circuit, outside the cell, only as a result of ionic charges which are taken from the current at the cathode and an equal number of others which are added to the current at the anode. In other words the battery circuit is open at the electrol3^ic cell, and a current is kept moving by taking electrons from the negative terminal and adding an equal number of other electrons to the positive terminal. Then, since all the ions of any given substance are alike in their ability to give or take electrons, the quantity of substance deposited or evolved at either the anode or cathode must be proportional to the current. In reference to the second law a few terms may need explana- tion. The chemical equivalent of any substance is its atomic weight divided by its valence. The valence of H is 1 ; that of O is 2. This means that it requires two atoms of hydrogen to com- bine with one of oxygen in forming a molecule of water, H20. Each atom of matter always carries a definite quantity of elec- tricity. The quantity on a hydrogen atom was called by Hehn- holtz an atom of electricity. The number of atoms of electricity on an atom of matter determines the valence, i.e., an atom of matter may have an excess or a deficiency of electrons, but the number either way is always an exact multiple of the atom of electricity on hydrogen. Hence the valence of an atom or radical is sometimes defined as the number of hydrogen atoms which would need to combine with it to form a neutral molecule. For example, in H2SO4 we say that the valence of the radical SO4 is 2 because it takes two atoms of H to unite with SO4 and form a molecule of sulphuric acid. Then in CuSO^ the valence of Cu must be 2, for it takes the place of Hj. A given quantity of electricity will therefore produce in elec- trolysis twice as many atoms of H as of O because the valence of O is 2, i.e., there need be only half as many atoms of O to carry ELECTRICITY THROUGH SOLUTIONS. 77 the same quantity of electricity. Now the atomic weight of H is 1, and of O, 16. Also the masses of equal volumes of different gases under the same conditions are to each other as their atomic weights. But since the volume of oxygen is one-half that of hydro- gen, then 1 g. of hydrogen is chemically equivalent to 8 g. of oxy- gen. If quantities of hydrogen and oxygen in the proportion of 1 to 8 by weight be united by combustion, none of either gas would be left over, for they are chemically equivalent. Likewise if we are considering the deposit of copper from CUSO4, since the valence of Cu is 2 and its atomic weight is 63.6, the chemical equivalent is 31.8. In case of silver from AgNOs, the valence of Ag is 1 and its atomic weight 107.9, hence its chemical equivalent is 107.9. Therefore the same quantity of electricity that will cause a deposit of 31.8 g. of Cu from CUSO4 will cause a deposit of 107.9 g. of Ag from AgNOs. The mass deposited in any case by a given quantity of electricity varies directly as the atomic weight and inversely as the electric charge (the valence) which the ion carries. 60. Electrochemical Equivalent. — The mass of any substance resulting from the electrolytic action of one coulomb of electricity is called the electrochemical equivalent of that substance. The advantage of a knowledge of this quantity wiU appear from the following: Faraday's laws may be expressed in form of an equation by a macq— (57) V where m is the mass, q the quantity of electricity, a the atomic weight, and v the valence. By this it is possible to compare the masses of different substances deposited when q, a, and v are known, but we could not find m for any one given substance. If, however, we can find the value of m for any substance on the assumption that q, a, and v are unity, then this value, say k, when multiplied by qa/v would give m for any substance. Hence we may write m = qk- (58) V The value of k may be found once for all by experiment with any desired electrols^e. The most exact results are obtained from a deposit of silver by use of a cell Uke that in Fig. 48, where the Fig. 48. 78 GENERAL PHYSICS. anode — a plate of silver — ^is suspended in a solution of silver nitrate in a platinum cup which is the cathode. By noting the number of amperes, the time, and the increased weight of the cup, the number of grams of silver per ampere-second, or coulomb, is readily calculated. This is found to be .001118 g. per coulomb. ^ Since the atomic weight of silver is 107.9, its valence 1, and q for this value of m is 1 coulomb, equation (58) becomes .001118 = 107.9jfe or /fe = 1.036(10)-6 The quantity k— is by definition the elec- trochemical equivalent, for it is the quan- t tity which when multiplied by the number of coulombs gives the mass deposited. For hydrogen the atomic weight and valence are both unity, hence, for one coulomb, equation (58) becomes, for this gas, Mi =ife = 1.036(10)"' The value of k is therefore the electrochemical equivalent of hydrogen. If one coulomb will cause the evolution of 1.036(10)"* g. of hydrogen (see Fig. 47), for 1 g. of hydrogen it would require 1.036(10)-=^^^^^ '=°"^°°^^' From the second law the mass of substance resulting from any given quantity of electricity is proportional to the chemical equiva- lent. Hence the 96,525 coulombs which produced 1 g. of hydrogen would, in a proper electrolyte, produce 107.9 g. of Ag, 31.8 g. of Gu, 8 g. of O, or a mass equal to the chemical equivalent of any substance capable of electrolysis. 61. Charge on an Ion. — If the total quantity of electricity carried by ions were known, and also the number of ions concerned, it would be a simple matter to calculate the charge on each ion. In case of hydrogen whose density under standard conditions is 8.95(10)"* g. per c.c, the volume of one 1 g. is 11,173 c.c. Some of the best determinations of the nvunber of molecules in 1 c.c. ELECTRICITY THROUGH SOLUTIONS. 79 of hydrogen under these conditions are 2.7(10)""- The number of atoms wovild be 5.4(10)". Hence the total number of atoms in Ig. is 11173 X 5.4(10)" =60334(10)" Each of these atoms was an ion in the electrolytic cell and the sum of their charges was 96,525 coulombs per gram of the gas. Hence the charge on each ion, was 96525 ■ = 1.6(10)"'' coulombs 60334(10)" This, then, is what has been called one atom of electricity — ^the electron. In e.m. imits it would be 1.6(10)"^" since 1 coulomb equals (10)"' e.m. unit of quantity. In e.s. units it would be 4.8(10)~" since the e.m. unit is 3(10)'° times as large as the e.s. imit. According to this the fundamental unit quantity of electricity is 4.8(10)"*" of the electrostatic unit, and this is the charge on all atoms of matter whose valence is 1. If th^ valence is 2, the atom carries twice this charge. Assuming this result to be correct, it is possible to calculate the number of molecules per cubic centimetre of any substance deposited by electrolysis. An ion of Cu carries two imit charges or 3.2(10)~" coulombs. It would then require, to carry 96,525 coulombs, 96525 3-:^(ior^=^«^«^(^«)"^°^^ These would form 15,082(10)" molecules. The mass of copper deposited by 96,525 coulombs is 31.8 g. Hence the number of molecules per gram is 15082(10)" ,„,„„„ , , — — =4.74(10)^' molecules per gram ol.o =4.03(10)^^ molecules per c.c. 62. The Voltameter. — ^A voltameter, more properly called a coulometer, is an instrument used to measure the quantity of electricity by the electrolj^ic effects produced. Any of the electro- lytic cells which have been described may be used as coulometers. 80 GENERAL PHYSICS. The silver cell, Fig. 48, is the standard cell. The copper coulom- eter was used in former times to determine the quantity of direct current used in electric lighting. The change in weight of the cathode and the electrochemical equivalent are the only two factors needed to calculate the quantity of current used. 63. Uses of Electrolysis. — The deposit of a thin layer of one metal over the surface of another is a very common practice in an art called electroplating. Gold, silver, copper, nickel, zinc, tin, and platinimi are some of the most common metals used for this purpose. It is necessary in all cases that the electrolyte be a solution of some salt of the metal to be deposited, e.g., sulphate or cyanide of copper, chloride of gold, etc., and the anode of the plating vat should be the substance which is to be transferred to the cathode. Extensive use is also made of this principle in a process called electrotyping, by which an exact reproduction of a page of type is made in copper and preserved for future printing. Ordinary commercial copper contains many impurities. If this is placed as the anode in a copper sulphate solution, an electric current will, when proper precautions are observed, transfer only pure copper to the cathode. This operation is carried out on a large scale and the product is called electrolytic copper. About 1 kg. of copper may thus be refined by 250 watt-hours of electrical energy, though the amount differs widely for different conditions and different kinds of copper. Electrolysis often has an injurious effect on metal pipes laid in the ground. Where the current of an electric circuit is grounded, electricity will flow back to the dynamo, or other point of different potential, on the metal pipes rather than through the ground. The pipe is often surrounded by moist earth containing a solution of electrolytic salts. The pipe then acts as an anode at the point where the current leaves it, and this is where the metal of the pipe wastes away. Note. — In this and the following chapter chemical symbols are, for con- venience, kept in the same form after ionization as in the neutral molecule, the number of plus and minus signs indicating the number of atoms of elec- tricity to be taken into account whatever the valence of the substance may be. Some authors write 2H with a plus sign over the H to indicate two ions of H. Others write two plus marks even when they use the form 2H. ELECTRICITY THROUGH SOLUTIONS. 81 Problems 1. A current flowing for one hour increases the weight of the cathode of a copper voltameter 2.9646 g. What was the strength of current? 2. How many coulombs of electricity will produce 50 c.c. of oxygen at 22° C. and under a barometric pressure of 73 cm.? 3. If a certain current produces 3 g. of copper in a copper voltameter, how much nickel would be deposited by the same current? Make a propor- tion according to the second law. 4. If 46 is the atomic weight of a certain substance and its valence is 2, how much electricity will be required to deposit 11.5 g. of it? 5. How many molecules in 1 c.c. of gold? 6. How long a time must a current of .5 ampere flow to cause a deposit of .4 g. on the cathode of a silver voltameter? 7. What would be the increased cost per kg. of dectrolytic copper where a current of 20 amperes at a pressure of 2 volts is maintained at a cost of 10 cents per kilowatt hour? 8. A surface of 10 sq. cm. is plated with gold. A current of 1.3 amperes flows for 30 minutes. How thick will the gold plate be? Ans. 1. 2.5 amperes.- 2. 769.6 coulombs. 3. 2.77 g. 4. 48,262.5 coulombs. 5. 2.95(10)^. 6. 11 min. 55.6 sec. 7. 16.44c. per kg. or 7.5c. per lb. 8. .08257 mm. CHAPTER V BATTERY CELLS 64. The Voltaic Cell. — ^From the time of the earliest experi- ments in electricity to the beginning of the 19th century, static electricity, so called, was the only kind known. Galvani (1737-1798), professor of physiology at Bologna, investigated the relation of electricity to animal life. When a current from his electric machine was passed through the body of a dead frog the muscles would contract. He also noted that when the frogs were hung on brass hooks attached to an iron railing, their legs would be drawn up by muscular contraction whenever they were caused to swing against the iron. Galvani's explanation was that the electricity was in the body of the animal and, when a conductor cormected the outside of the frog's leg to a nerve which led to the inside, a discharge much like that of a Leyden jar took place, thus producing a union of the negative charge on the outer surface with the positive charge within. The metal, as he claimed, only acted as a conductor of these charges. Volta (1745-1827), professor of natural philosophy at Padua, claimed that Galvani's contention was not correct and that the electricity resulted from the contact of dissimilar metals, the nerve and muscles of the frog acting only as conductors. To prove his claim Volta devised a number of experiments and showed that when dissinoilar metals — copper and zinc for example — were placed in contact, the one with the other, one became positive and the other negative, as was shown by a delicate electroscope. This claim is discussed in the next paragraph. He also invented the voltaic pile, which may consist of alternate strips of copper and zinc, paper moist- ened with acidtdated water being placed between each successive pair of plates — i.e. , there would be copper, paper, zinc, copper, paper, zinc, etc., piled up to any height. Such a pile may be made to give a very vigorous current of electricity. From this it was an easy step to the voltaic cell as we now know it, consisting of dilute sulphuric acid in which are placed the elements zinc and copper. A wire connecting the zinc and copper plates conducts the electrons 82 BATTERY CELLS. 83 from zinc to copper. This is the only current that actually flows, though according to a custom we speak of a positive current as flowing in the wire from copper to zinc. The former is a negative current and the latter, which is only assumed, is called a positive current. 65. Contact Difference of Potential. — ^Volta claimed that it was an inherent property of metals that when different kinds were placed in contact there would result a difference of potential between them, and he named a series of metals any one of which would be positive in reference to any other one lower in the series. Experiment seems to confirm the truth, of this claim and a differ- It o o o H,SQ, Zn Cw ft 4 Currerft. \fM9J,t4>^ Fig. 49. ence of potential does exist, but later experiments have shown that when the metals are surrotmded by gases other than air, a series entirely different from that given by Volta and others is obtained. This seems to show that the effects observed are a result of the relation of the metals to the surrounding gases rather than an inherent property of the metals themselves. This matter, however, is not yet fuUy settled. Between certain metals and an electrols^te there is a definite contact difference of potential which can be measured in several different ways. In the voltaic cell, Fig. 49, it is found that the contact difference of potential between zinc and the electrolyte 84 GENERAL PHYSICS. is .62 volt and that between copper and the electrolsrte is .46 volt, the former being lower and the latter higher than the electro- lyte. Hence the total difference between the zinc and copper in the cell is 1.08 volts. This furnishes an E.M.F. which causes a current to flow on the conducting wire. One method of measuring the contact difference of potential is by use of Lippmann's capillary electrometer. The principle of this instrument is that when a current of electricity is passed through a point of contact between mercury and dilute sulphuric acid there will be a change of surface tension at that point which will cause a movement of mercury one way or the other, depending Pig. 50. Fig. 51. on the direction of the current. If, for example, a drop of mercury, d (Fig. 50,A), is introduced into a glass tube, the remainder of the tube being filled with dilute sulphuric acid, each end of the mercury drop will be positive in reference to the electrolyte with which it is in contact, and as a result of this the surface tension will be weakened at both ends. If now a positive current be passed from the platinum electrode a through the liquids to b, the effect will be to decrease or neutralize the contact difference of potential on the left end of the mercury and to stUl further increase the difference at the right-hand end. Thus the surface tension on the left may increase to its natural strength while that on the right will decrease. The mercury will therefore move along the BATTERY CELLS. 85 tube in the direction of the positive current. Lippmann's appa- ratus consists of a glass tube T drawn to a fine point (Fig. 50, B). It is partly filled with mercury and the lower end is inserted in an electrolyte. There will be a contact of mercury and electro- Ijrte at c and the point c will be lower on account of the contact, for the surface tension of the merciu-y has thereby been weakened. Now let the meniscus of the merctiry at c be observed by a fixed microscope and let a positive current pass from the electrolsrte up through c. This will tend to neutralize the contact difference of potential at c, the surface tension will increase, and the mercury meniscus will rise to a wider portion of the tube where the pressure due to surface tension upward is balanced by the pressm-e of the column of mercury above. By raising the tube 5 the height of mercury in T may be increased tiU the meniscus is again brought to its position as first seen through the microscope. As the poten- tial of the current is increased a point will be reached where the height of mercury in T necessary to bring the meniscus back to c will be a maximum. This occurs when the appHed difference of potential is just equal and opposite to the contact difference, for any further increase in the potential of the current would cause a difference at c opposite in sign. This would again weaken surface tension and the meniscus would fall to a narrower part of the capillary. Hence the known E.M.F. which must be applied to produce maximum surface tension, as measured by the height of merciury in T necessary to bring the meniscus back to c, is also the contact difEerence of potential between the mercury and the electrol3rte. Suppose, now, it is desired to know the contact difference of potential between zinc and dilute sulphuric acid. The P.D. at the terminals E, Fig. 51, is the potential difference between the mercury and zinc. This can readily be measured by use of a potentiometer and is evidently the difference between e' and e when these are both considered in reference to the same electro- lyte. For example, E is found to be 1.48 volts and e' is found by the capillary electrometer to be .86 volt. Then E'^e'-e 1.48 = . 86 -e .-. e=-.62 86 GENERAL PHYSICS. Thus the potential of the mercury is .86 volt higher than the electrolj^e, and the electrolyte is .62 volt higher than the zinc. In the voltaic ceU, Fig. 49, copper is .46 volt above the electro- lyte and the electrolyte .62 volt above the zinc. Hence the P.D. between the copper and zinc is 1.08 volts. 66. Polarization. — The passage of electricity through a voltaic cell or any electric cell, is accomplished in the same manner as in an electrolytic cell. The difference between the two cells is chiefly the fact that an electric cell produces its own electromotive force. Polarization is the condition of a cell when a counter electromotive force is set up whereby the flow of current is checked or stopped. As already suggested in the previous chapter, when a small difference of potential is maintained between plati- num electrodes in a solution of sulphttric acid, a current will begin to flow but will soon cease. A counter electromotive force has been set up equal and opposite to that between the electrodes. If the P.D. of the electrodes be increased — in this particular case it must be more than about 1.7 volts — the current will again flow notMrithstanding the counter E.M.F. which still opposes it. The existence of the counter E.M.F. may be shown experimentally by the arrangement shown in Fig. 52 where, when the switch 5 is thrown to a, the battery B will cause a positive current to flow as indicated by arrow heads on the battery circuit. The electrolytic cell E, having platintun electrodes and an H2SO4 solution, will be polarized. If now the switch s is thrown to b, the battery will be cut out and the cell E will for a short time send out a current in an opposite direction, as will be shown by the galvanometer G. Thus, as is usually the case in electrolytic action, the passage of a current may cause a change in the character of the surface of one of the electrodes or a change in the electrolyte adjacent to the electrodes, either of which wiU require the expenditiire of external energy in the process of electrol3rtic action. This occurs when the electrodes are of a different material from that which is produced by electrolytic action. In the same way most electric cells will soon polarize if the current which they produce is allowed to flow through a conductor from one electrode to the other. The voltaic cell. Fig. 49, is one of this kind, and its current soon becomes weak. BATTERY CELLS. 87 When, on the other hand, the substance deposited by electro- lytic action is the same as that of the electrode, there is no polariza- tion. This is the case when the electrodes are bathed in a solu- tion of their own salts, e.g., copper electrodes in a solution of copper sulphate. Here the passage of electricity consists simply of charges on atoms of copper going into solution at the anode and other charged atoms being deposited from the solution upon the cathode. Metals going into solution always carry a positive charge, i.e., are always deficient in electrons. For every positive charge which goes into solution in this case, an equal positive charge goes out and no counter electromotive force is developed. This suggests the possibility of a nonpolarizing cell and such is realized in the DanieU cell where, in one form, a zinc plate, the anode, is bathed in a solution of zinc sulphate, ZnSOi, and a copper plate, the cathode, in a solution of copper sulphate, CuSOi, the two solutions being kept apart B i ■> : > 4 — Zn - ZnSO^ " CUSO4 =_ L_C — u 1 . — T Fig. 52. Fig. 53. by a difference in density as shown in Pig. 53. Here zinc goes into solution at the anode, and copper is ddposited on the cathode. This causes a difference of potential between the electrodes, for each atom of zinc that goes into solution carries a positive charge and so the zinc anode is left with an excess of electrons, i.e., is negatively charged. Also, the positively charged ions of copper on reaching the cathode take up electrons which pass over the conducting wire from anode to cathode. Thus a negative current passes over the conductor from zinc to copper, or we may regard it as a positive current from the copper to the zinc. No current as such passes through 88 GENERAL PHYSICS. the electrolyte except in a sense already explained for an electro- lytic cell. Ohm's law and the heat effects are, however, the same in electrolytes as in ordinary conductors. The style of Daniell cell shown in Fig. 53 is called a gravity cell. Its voltage is about 1.09 and it maintains a very constant ciurent. When not in use the circuit should be closed and enough current allowed to flow to prevent a diffusion of CuSOi up to the zinc. Otherwise a muddy precipitate will form on the zinc and interfere with the action of the cell. In another form of the Daniell cell the electroljrtes are separated by an unglazed earthenware cup. The zinc may be placed in the cup in a solution of H2SO4 or ZnS04. • The outer vessel is filled with a solution of CuSOi in which is placed the copper electrode. 67. Origin of E.M.F. in an Electric Cell. — ^According to a theory developed by Nemst a metal in a solution exerts a certain pressure called solution pressure as a result of which it tends to go into solution. On the other hand the solution also exerts a pressure called its osmotic pressure as a result of which it tends to deposit the material of its ions on the metal, i.e., to go out of solution. When these two pressures are in equilibrium there will be no action of either kind. When an atom of metal goes into solution it is positively charged, and the metal which it leaves will therefore be negative. When a metallic ion goes out of solu- tion it receives a negative charge from the plate to which it attaches itself. When electrodes and electrolj^es are such that both opera- tions are possible, a conducting wire will convey the excess of electrons from the electrode which possesses large solution pres- sure to another electrode to which ions are passing from a solution. This constitutes an electric current as produced by a primary cell. These changes may be illustrated by reference to the cell shown in Fig. 53, where a zinc plate is immersed in a solution of ZnS04 and the copper plate in a solution of CUSO4. The solution pressure of the zinc plate is greater than that of the copper plate. The zinc plate begins to send zinc ions into the ZnS04 solution and copper begins to go out of the CUSO4 solution to the copper plate, but, while the circuit is broken at s, an inappreciably small change of this kind will make both the ZnS04 solution and the copper plate positive, thus setting up a positive electrostatic St BATTERY CELLS. 89 field which opposes any further change. As soon as the circuit is closed at s, electrons will pass on the wire from zinc to copper, thus diminishing the electrostatic stress. Copper ions will then pass out of solution to the copper plate, zinc ions will go over to the SO4 radicals of the copper sulphate solution, and the zinc plate will then be free to send more positive ions into solutions. Copper ions go out of solution more readily than zinc ions, so that copper only is deposited on the cathode as long as there is copper in solution. In this cell crystals of CuSO* are kept in the CuSOi solution, and these will be dissolved as they are needed. According to the above explanation of a Daniell cell if the ZnS04 solution is made more dilute or the CUSO4 solution more concentrated, or both, the E.M.F. of the cell should increase, for the osmotic pressure of the zinc solution is thus decreased while that of the copper solution is increased. Experiment shows this to be a fact. There is also, as might be expected, a contact differ- ence of potential at the interface between the two electrolytes, though this is ordinarily small. In fact, a primary ceU may have electrodes of the same metal each immersed in a separate solution, the solutions being in contact, or both metals and electrolsrtes may be the same, but the latter of different concentration. In aU primary cells the general principles of operation are the same as described above. In the simple voltaic cell zinc goes into solution and hydrogen ions go out. In the DanieU cell, when H2SO4 is used in place of ZnS04, zinc ions go into solution and ++ ++ ++ displace H2 of the H2SO4 solution. The H2 and Cu are then free .++ ++ to go to the cathode but, since Cu goes out more readily, the H2 will remain to balance the SO4 radical from which the copper has been separated. Hence the collection of hydrogen on the cathode and the consequent polarization which occurs in the voltaic cell is avoided in the Daniell cell. It has been assumed in the descriptions given above that the zinc is pure or has been amalgamated with mercury. Then no zinc will go into solution unless an equivalent quantity of some other substance carrying a positive charge of electricity can go out. This condition is supplied when a current can flow through a conductor from anode to cathode. If, however, the zinc contains 90 GENERAL PHYSICS. small particles of carbon or iron and is not amalgamated, a local action is set up between these foreign substances and the zinc, whereby zinc may enter an H2SO4 solution forming ZnS04 and setting hydrogen free without any flow of current over the con- ducting wire of the cell. 68. Energy Relations. — In an electric cell the quantity of energy which is set free by the substance which goes into solution is always greater than that absorbed by the substance which goes out of solution, hence the operation of the cell results in a decrease of potential energy, i.e.^ in a decrease of the total amount of available energy. This is in accord with the operation of any isolated system in nature. In a cell where the total decrease of potential energy appears in the electric current and where the quantity of heat, if the energy had appeared as heat instead of an electric current, is known, it is possible to form an equation from which the E.M.F. of a cell may be calculated. The difference between the heat of formation of ZnS04 and that of CUSO4 for one chemical equivalent of each metal is 25,065 calories. This, expressed in heat units, is the quantity of energy which may be used in producing electricity in a Daniell cell when that quantity of zinc goes into solution and an equivalent quantity of copper goes out. The quantity of elec- tricity involved in this change is, as shown in the preceding chapter, 96,525 coulombs. This quantity times the electromotive force, E, gives the quantity of work in watt-seconds. One watt-second is one coulomb per second under a pressure of one volt. (See § 54.) Hence (10)" 1 watt-second = ^g,,^,^.^ = -2388 calorie .-.96525 EX.2388 = 25065 and £ = 1.087 volts This method of calculating the voltage of a cell can be used only when the temperature coefficient is very small, i.e., when the change in E.M.F. due to a change in temperature is a negligible quantity. 69. Other Primary Cells. — Numerous eflEorts have been made to construct voltaic cells which would not polarize. Polarization BAtTERY CELLS. 91 is due to a change in the surface of the cathode resulting from a deposit of hydrogen upon it. Anything that will prevent this is a depolarizer. In the Grove cell the electrodes are platinum and zinc. The platinum is immersed in strong nitric acid, HNO3, in a porous cup. The cup in turn is immersed in a solution of H2SO4 in which the zinc is placed. Zinc goes into solution and hydrogen ions move toward the platinum electrode where it combines with HNO3, forming water and nitrogen dioxide according to the chemical equation 3H2 + 2HNO3 = 4H2O + N2O2 in which change electrons are taken from the platintmi electrode and N2O2 escapes into the air where it combines with another molecule of oxygen, forming the noxious red peroxide or nitrogen tetroxide. The advantage of this cell is high E.M.F., about 1.9 volts, and freedom from polarization. The Bunsen cell is the same as Grove's except that a plate of carbon is used in place of platinum. ' The Bichromate cell is the same as Bunsen's except that a solution of bichromate of potash is used in place of HNO3 and the porous cup is not used. This is, then, a single-fluid cell with a solution of H2SO4 and K2Cr207. When the former unites with the latter, chromium trioxide, Cr03, is formed, and this is a strong oxidizing agent which takes up hydrogen ions and prevents polari- zation. The E.M.P. of this cell is 2 volts or more. The Leclanche cell is made up of zinc and carbon as electrodes, the latter being surrotmded by manganese dioxide as a depolarizer. The electrolyte is a solution of ammonium chloride (sal ammoniac) . This cell, in some form, is in common use for open circuit work, but it soon polarizes on closed circuit. Its voltage is about 1.5. The advantage of its use is that it needs no attention for a long time and will recuperate during a period of rest. One very con- venient form of this cell is the so-called dry cell in which the electrolyte is a moist paste composed of ammoniimi chloride, zinc chloride, zinc oxide, and plaster of Paris. The composition of the paste is varied in different makes of this cell. 70. Standard Cells. — It would be an advantage to have a cell which under specified conditions would always give the same 92 GENERAL PHYSICS. electromotive force. After its E.M.P. is once determined with accuracy it is possible, by comparison, to determine the E.M.F. of any other cell or the P.D. between two given points on a con- ductor. One cell that has been selected for this purpose is the Clark cell. As shown in Fig. 54 it consists of an H-shaped glass vessel in which the electrodes are ptare mercury on one side, m, and an amalgam coptainiag about 15 per cent, of zinc on the other, a. On the mercury is placed a paste, p, of merctu'ous sulphate. The cell is then filled with a sat- urated solution of zinc sulphate, s, and is kept saturated by crys- tals of ZnS04 which are added. Connection is made with the Fig. 54. Fig. SS. electrodes by platinum wires which are sealed into the glass. The E.M.F. is 1.4325 volts at 15° C, but there is considerable change in this value when the temperature changes. For any temperature t, E.M.F. = 1.4325-.00119(*-15) -.000007(i-15)2 Another standard which is often preferred because of its small variation with change of temperature is the Weston standard cell. In it cadmitmi amalgam and cadmium sulphate are sub- stituted for zinc amalgam and zinc sulphate of the Clark ceU. In other respects they are alike. This ceU is also made in the form shown in Fig. 54 where m is pure mercury, a is an amalgam of cadmium and mercury, ^ is a paste of mercurous sulphate, c is cadmium sulphate in the form of crystals, and 5 is a saturated BATTERY CELLS. 93 solution of cadmium sulphate. A cell of this kind constracted and used in accordance with standard directions will remain constant for a number of years. The international E.M.F. of the Weston cell is 1.0183 volts at 20° C, and, for any other temperature, t, E.M.F. =1.0183-.0000406(«-20) -.00000095(^-20)2 Standard cells such as the Clark and Weston can be used only in what is called the zero method, i.e., by balancing its E.M.F. against another which is equal and opposite to it. This is explained in the next chapter. If a current is allowed to flow, the cell will soon polarize and become useless. 71. Storage Cells. — ^A storage cell, also called a secondary cell or accumulator, is one in which the passage of a current from an outside source will effect such changes that the cell wiU afterwards reverse its action and give out a current. The cell shown in Fig. 52 may be considered as a storage cell, for part of the energy ex- pended was rettumed. This, however, would be a very poor and inefiBcient kind of storage cell. The Daniell cell may in a sense also be considered a storage cell, for if a current is passed through it in a direction opposite to that which it would give out, copper goes into solution and zinc is deposited on the zinc plate. Thus potential energy is stored up and will be returned whenever the cell operates in the ordinary manner. Such a cell is said to be reversible. A common form of storage cell consists of lead grids filled with a paste of lead sulphate. These are the electrodes. The electro- Ij^e is a solution of sulphuric acid. When a diSerence of potential is maintained at the electrodes, i.e., when, as we say, a current is passed through this cell, hydrogen ions from the H2SO4 solution go to the cathode where they act on the PbS04 according to the equation H2 + PbSOi =H2S04 + Pb Thus this plate, the negative one, is reduced to metallic lead in a very spongy condition. This plate may be recognized by its gray color. On the other hand, SO4 ions go to the anode and convert it into peroxide of lead, PbOz, according to the reaction SOi+PbSOi f2H20 = 2H2S04+Pb02 The Pb02 remains in the grid and forms the positive plate which 94 GENERAL PHYSICS. may be recognized by its dark red color. In both reactions H2SO4 is added to the solution. In the uncharged cell the density of the solution should be about 1.17 g. per cubic centimetre, but in the process of charging this will increase to 1.21 g. per cubic centimetre. When fully charged, any further current will cause hydrogen to be given off at the cathode and oxygen at the anode just as described for the electrolysis of water. During discharge the reactions described above take place in a reverse direction and the electrodes are changed again to lead sulphate. The E.M.F. of this cell is very constant. On full charge the voltage rises to 2.5 volts but falls at once to 2.2 volts and during discharge remains steadily at 2 volts until nearly exhausted. The voltage then falls rapidly but should not be allowed to go below 1.8 volts or exceed the normal discharge rate, for an over-discharge will cause the plates to buckle. In the recent Edison storage cell the electrodes are nickel hydrate, positive, and iron oxide, negative. The electrolj^e is a 21 per cent, solution of caustic potash and the container is made of nickel-plated sheet steel. The chemical changes which take place are complex and not yet well understood. The E.M.F. of this cell is 1.2 and its efficiency about 60 per cent., but it can be stored with about twice as much energy as the same weight of lead cell. The efficiency of a new lead battery is about 80 per cent., i.e., it will return 80 per cent, of the energy put into it. 72. Arrangement of Cells. — ^When a current flows from an electric cell, the electrolyte itself is regarded as forming part of the circuit, and its resistance, called internal resistance, may be represented by r. The remainder of the circuit is called external resistance and may be represented by /?< Then by Ohm's law the strength of current, i, is where E is the E.M.F. Two or more cells connected together constitute a battery. A battery may be made up of cells joined in series — i.e., with the negative electrode of the first coimected to the positive of the second, negative of the second to the positive of the third, etc., the negative of the last being connected through the external circuit to the positive of the first. BATTERY CELLS. 95 Cells may also be joined in parallel, i.e., one terminal of the external circuit is joined to all the positive electrodes and the other to all the negative electrodes. Another arrangement is the series-parallel which is a combina- tion of the two above. Here there are several groups of cells, those in each group being joined in series and the groups then joined in parallel. (Fig. 56.) When cells are in series, the E.M.F. is multiplied as many times as there are cells and the same is true of the internal resist- ance, but when in parallel the internal resistance is divided as Ser/'es -©-e-<5)-€>-€>-G-€M=HGH8)-©-e— 1 Fbrol/ef €>-©-€H!>-©-^l 1 i-©-G-€>-e-i Series' Parallel Fig. 56. many times as there are cells, the E.M.F. remaining the same as for a single cell. A general equation for any possible arrangement would then be pE _ pE t = - pr +R ^+R (60) where p is the number of cells in series in any group, n the total number of cells, and n/p the number of groups. li p=n, then •-;^ <«) i.e., there is but one group with all the cells in series. li p = l, then . E -+R n i.e., all the cells are arranged in parallel. (62) 96 GENERAL PHYSICS. For a maximum value of i the best arrangement is one where the internal resistance of the entire battery is, as nearly as possible, equal to the resistance of the external circuit. This will appear from a consideration of equation (60) which may be written E n p (63) If the terms of the denominator of the second member of this equation are multiplied together we have rR n which is a constant quantity, for none of these quantities, r, R, or n, change, whatever the arrangement of the battery may be. But whenever the product of two variables is a constant, their sum is least when they are equal. Consequently i, the strength of current, will be maximum when Pl_R n p or i^= R (64) The first term in (64) is the internal resistance of the battery and the second is the external resistance. Hence when p is made such that these terms in equation (60) are equal, or as nearly so aa possible, i will be maximum. For example, if jf?=3 ohms and r=2 ohms and we have 24 cells, the best arrangement to securg the greatest strength of current is in 4 groups with 6 cells in series in each group, for then ph 36X2 24 ■=3 = i? Problems 1. What arrangement of 20 cells will give maximum current when for each cell the E.M.F. = 1.09 volts and internal resistance = 2 ohms, the ex- ternal resistance being 2.5 ohms? 2. The 50 cells of a battery are joined 10 in series, with 5 groups in parallel. The resistance of the entire battery is 4 ohms. What is the resistance of each cell? BATTERY CELLS. 97 3. If the resistance of a lamp is 440 ohms and it requires .25 ampere to operate it, how many storage cells, 2 volts each, will be required, assuming that the internal resistance of the cells may be neglected? 4. The E.M.P. of a cell is known to be 2 volts. A current of .475 ampere is observed to flow through an external circuit of 4 ohms. What is the internal resistance? 5. Six lead storage cells, the E.M.F. of each being 2 volts and internal resistance of each .15 volt, are joined in series. The external circuit contains 3 electrolytic cells in series, each offering a counter electromotive force of 1.5 volts and having a resistance of 2.15 ohms. The resistance of the connecting wires is 1.1 ohms. What strength of current will flow? Ans. 1. 5 cells in series. 2. 2 ohms. 3. 55 cells. 4. .21 ohm. 5. .887 ampere. CHAPTER VI GALVANOMETERS 73. The Tangent Galvanometer. — ^A tangent galvanometer is one where the strength of an electric ctirrent flowing through a coil is proportional to the tangent of the angle of deflection of a magnetic needle at the centre of the coil. This instrument is based on the deflnition of the e.m. unit of current as described in § 47. It is there shown that the strength, F, of the field at the centre of a coil of n turns is F= 2imi where r is the mean radius of the coil and i is the strength of current in e.m. units. If such a coil be set up with its plane parallel to the earth's field, the field of the coil will be at right angles to that of the earth. Let the strength of the former be denoted by F and of the latter by H. Let be, Fig. 57, be a magnetic needle of pole strength m, and length / placed at the centre of the coil. Then the moment of the couple tending to turn the needle to a position parallel to the field of the coil is Fm times ad. But ad is equal to be, or /, times the cosine of 0. Thence the moment of the couple is 2irm , ml cos d r Fig. 57. mt • , There is at the same time a moment tending to turn the needle to a position parallel to the earth's field, and the moment of this couple is Hm(ab+cd). But ab+cd is equal to I sin 0. Hence the moment is Hml sin d 98 GALVANOMETERS. 99 Since the needle is in equilibrium, the two moments are equal and we have the equation Hml sin tf = ml cos (65) r or H tan = r or t = -^Htane (66) The strength, i, of a current may by this method be found in e.m. units at any place where H is known. Since i varies as the tangent of the angle of deflection, the instrument is called a tangent galvanometer. The quantity contains only terms whose value is fixed in r the construction of the instrument and so is called the galvanom- eter constant, usually denoted by G. Hence (66) may be written rr t=-— tane (67) G Since an ampere is one-tenth of an e.m. tmit, there would be ten times as many amperes in any given strength of current. Hence (67) may be written i = 10— tan d amperes (68) G It is assumed that the field in which the needle is placed is uniform, and to secure this condition as nearly as possible the length of the needle. Fig. 58, should not be greater than about one-twentieth the diameter of the coil. The tangent galvanometer is a fundamental instrument, for it is bviilt up on the definition of the e.m. unit of current. It was by its use that the definition of the ampere given in § 53 was deter- mined, for by connecting an electrols^tic ceU in series with the galvanometer, the relation between strength of current and the quantity of metal deposited on the cathode could be found. This method, in turn, may then be used to find H, the value of i being calculated from the deposit of silver, copper, or other substance from the electroljdje. 100 GENERAL PHYSICS. 74. The Astatic Qalvanometer. — The coil of a tangent gal- vanometer must have a^large diameter in comparison with length of the needle, consequently the field at its centre is not of great intensity and the needle does not respond to very small changes in current. A very sensitive instrument of this kind was devised by Lord Kelvin and is known as the Kelvin galvanometer. The coils contain many turns of fine wire and are close to the needles Fig. 58. at their centres as shown in Fig. 59. The magnetic needles are attached to a very light frame suspended by a delicate fibre. The poles of the needles in one coil point in a direction opposite to those in the other. This is an arrangement known as an astatic system, for if the magnetic moments are exactly equal, the ten- dency of the earth's field will be to turn the upper needles in one direction and the lower in the other. As a result they will not GALVANOMETERS. 101 take a stand in the earth's field, i.e., they will be astatic. A con- trol magnet may be placed above or below the instrument to bring the needles back to a zero position after they have been disturbed. The needle is therefore free to respond to very weak fields produced by the coils, and since the wire in the two coils is wound in opposite directions, the turning moment will be in the same direction for both sets of needles. A small mirror at- tached to the frame makes it possible, by means of a telescope and scale, to read the deflections of the magnets. The frame is usually sus- pended by a fine fibre of quartz, and the whole suspended system weighs but a small fraction of a gram. This instrument is valuable for the detec- tion of small currents and the direction in which they flow, but it is not a tangent galvanometer, for the needles do not turn through a uniform field. It may, however, be used to measure strength of current if the relation of current to deflection is determined by previous experiment. 75. The D'Arsonval Galvanometer. — The galvanometers described in the two preceding sections are of the mov- ing magnet tjrpe, and it is evident they would be easily disturbed by any ex- ternal magnetic field or by any mass of iron near by. Another type of instrtunent is the moving coil galvanometer in which the magnets are stationary and the coil tixms. These may be made just as delicate as the others and are practically unaffected by outside influences. A common form of such an instrument is known as the D'Arsonval galvanometer. This consists of a strong permanent magnet formed so that its poles are near each other and in this gap is suspended a coil of wire. When a current passes from e to /, Fig. 60, the coil tends to turn to a position such that its plane will be perpendicular to the field of the magnets and its magnetic field within the coil will be parallel and in the same direc- tion as the field of the magnets. Fig. 69. 102 GENERAL PHYSICS. The same principle applies to the turning of the coil in a gal- vanometer as we shall see later ia the explanation of an electric motor. In Fig. 61 let ba and dc be conductors carrying electricity N P Fig. 60. through the magnetic field between N and 5. The motor rule is as follows: Extend the thumb, the first finger, and the second finger of the left hand so that each will be at right angles to the other two, then if the first finger ^ ^ points in the direction of the lines of force, and the second finger in the direction the pos- itive current flows, the thumb will indicate the direction the condtictor will move across the magnetic field. Appljring this rule to Fig. 61, we find that the conductor ba when canying a current will move across the field in the direction m while cd will move in the direction n. This is just the condition shown in Fig. 60, and when a current flows through the coil in the di- rection there indicated, the coil will turn in the direction shown by m and n. In the definition of the e.m. unit of current (§ 46), the current, under conditions there given, will act on unit pole with a force of Pig. 61. GALVANOMETERS. 103 1 djme. But by the law of action and reaction we can as well say that the e.m. unit of current is of such strength that each centi- metre of its length, when at right angles to the direction of a tinit magnetic field in which it is placed, is acted on by a force of 1 djme. Consequently if the strength of the field, Fig. 60, is H, and the length of the side ab is I, then when a current i flows, the force urging one strand of ab in the direction m is f=ilH (69) An equal force is acting at the other side of the coil in the direction K. The moment of this couple is fk=iklH (70) where h is the distance between the two sides. The top and bottom of the coil are parallel to the field of the magnets and so do not affect the turning moment. If the coil is turned so that all its sides are at right angles to the field the turning moment would be zero and the only effect of the field on the current in the coil would be a force tending to push out or draw in all foiw sides. If there are n turns of wire in the coil, (70) would become Fh=ihlnH (71) Now, hi is the area enclosed by one turn of the coil, and hln is the siun of all such areas. If this is denoted by A, Fh=iAH (72) This equation gives the turning moment no matter what the shape of the coil may be. When the coil begins to turn it is resisted by the torsion of the suspension fibre and so will come to rest in a position of equilibrium between these two forces. The restoring force exerted by the twisted fibre is its moment of torsion times the angle through which it is twisted. It has been shown (see " Mechanics and Heat," p. 117) that moriient of torsion is the ratio of moment of force, Fh, to the angular displacement, d, which is produced, i.e., it is the moment of force which will cause an angular displacement of 1 radian. The moment of force for 6 radians would then be Fh — e e 104 GENERAL PHYSICS. When the coil has turned through radians, the component of its area parallel to the magnetic field is A cos 0, hence Fh iAH cos = — e e Ph If is very small — i.e., if the coil is made to turn only sUghtly from its position in Fig. 60 — cos d may be taken as unity without appreciable error. Then (73) becomes 7-JH ™ Under these conditions the second member of (74) is called the constant of the galvanometer, for all its terms are constant. Hence, if the ratio of i to 6 is once determined, the strength of another current is the angular displacement produced times this constant. Attached at the top of the coil is a mirror, m, so that by use of a telescope and scale set at a known distance, the angle through which the coil turns when a known current flows through it is readily found. Then, knowing i and 9, the ratio of the former to the latter is the constant, k, which is sought. One convenience in the use of this instrument is that it is practically dead-beat, i.e., the coil promptly turns to its position of equilibrium, or returns to zero, without frequent oscillations before it comes to rest. This is the result of electromagnetic damping which occurs whenever a closed conducting circuit, such as the copper frame seen in front of the coil, Fig. 62, is moved across magnetic lines of force. The suspension wires of this instrument are usually steel wires or phosphor-bronze ribbons whose moment of torsion is very small. Within the coil is a soft iron core supported from the frame. The core serves to concentrate the lines of force and make a more intense field in the region of the coil. The sensitiveness of a galvanometer is often designated by what is called the figure of merit. This is the strength of current in amperes which will cause a deflection of 1 mm. on a scale 1 m. GALVANOMETERS. 105 distant from the coil. If m, Fig. 63, is the mirror attached to coil and sr a scale graduated in millimetres and 1 m. from m, then a current which will ttim m so that a beam of light from/, 1 mm. from o, will on reflection from m return to a telescope at o, is the figtire of merit. This is jggo of the constant, k, as defined above, for the angle turned through by a reflected beam is twice that through which the mirror turns. Hence, when the mirror turns through one radian, the reflected beam will turn two radians, say or. But mo is 1000 mm., hence or is 2000 mm., or 2000 times of. The term sensitiveness also often means the number of megohms — i.e. 10' ohms — ^which must be connected in series with the galvanometer to reduce the deflection to 1 mm. on a scale 1 m. distant when the electric pressure is 1 volt. This is the reciprocal of the figure of merit, for by Ohm's law (equation 52), since £ = 1 and i is the same as for figure of merit, then R = 1/i. For example, suppose k for a certain galvanometer is 9 (10)"'. Then the figure of merit is 4.5(10)"', and the sensibility is 220 megohms. 76. The String Galvanometer. — ^An instrument of extreme sen- sitiveness has been invented by Einthoven and is known as the string galvanometer. The fundamental principle involved is the same as that described above for the D'Arsonval, but in this case a single conducting fibre is stretched across the field as shown at ab, Fig. 64. The magnets are pierced so that a bright light may be focused on the fibre by the lenses in the tube through N, and this in turn is focused on a screen by the projection micro- 106 GENERAL PHYSICS. scope through 5. The fibre ab will move across the magnetic field in response to the slightest current which passes through it, and this movement is magnified on the screen. The " string " is made of platinum or quartz coated with silver and is so fine that it can be seen by the naked eye only vhen brilliantly lighted on a black background, the diameter being only 2 or 3/*. A current as small as about (10)"'^ amperes may be detected. One use of such an instrument, as described by L. F. Baker of Johns Hopkins University, is in making electrocardiagrams, i.e., photographic records of the beating of the hiunan heart. The point where a b Pig. 64. muscle is stimulated to activity is found to be negatively electri- fied, and hence a current will flow on a conductor connecting that point with another point of the body at the moment of excitation. It is sufficient to have the patient grasp the electrodes leading to a and 6 and then, by moving a photographic film before the eye-piece of the microscope, a record of the action of the heart is recorded. 77. The Ballistic Galvanometer. — In the discharge of a con- denser or other charged body it is often desirable to know the quantity of electricity, Q, which passes over a conductor dtiring the very short time required for the discharge. This may be done by use of a galvanometer, preferably the D'Arsonval, in which the coil is made to have a long period and damping is reduced to a minimum. Under these conditions the current has practically all passed before the coil begins to swing. A momentary impulse is given and the swing, or throw, follows. Hence the use of the term ballistic. Any delicate galvanometer which is not " dead GALVANOMETERS. 107 beat " may be used in this manner. If, for example, the copper frame in Fig. 62 be removed, the instrument becomes a ballistic galvanometer. After the throw is observed, the coil may be readily brought to rest by pressing a key which closes its circuit and thus, for the time, introduces electromagnetic damping. The relation between quantity, Q, and the throw n may be approximately expressed by Q=kn (75) where fe is a constant which may be determined by passing a known quantity of electricity through the coil and observing the throw n as measured in scale divisions. Fig. 65. 78. Ammeters and Voltmeters. — ^For commercial purposes and also for use in the laboratory it is desirable to have instruments which are so constructed and caUbrated that either voltage or amperage will be indicated by a pointer which moves over a graduated scale. The principle of these, for direct current, is the same as that of a D'Arsonval galvanometer, but the coil, instead of being suspended, is pivoted on jeweled bearings and is returned to the zero point by coiled springs which also serve to conduct the current to or from the movable coU. A long pointer attached to the coil moves over the scale (Fig. 65). In the ammeter a low resistance conductor is in series with the main circuit, and only a small fraction of the current is shunted through the coil. In the voltmeter a large resistance is put in series with the coil 108 GENERAL PHYSICS. and the main circuit so that only a very small current flows. The P.D. is here desired and not strength of current. It would be better if no current whatever passed through the voltmeter, but in instruments of this kind it is always necessary to take ofiE enough current between two points of different potential to operate the coil. The greater the P.D. the greater the current, and hence the greater the deflection of the needle. For very high voltage, electrostatic voltmeters may be used. These do not require any current. The principle here involved is the same as that of the quadrant electrometer. Fig. 18. As Fig. 66. shown in Fig. 66, conductors leading from points of different poten- tial are connected, one to A which is fixed, and the other to B which is movable and carries a pointer. A and B will thus be charged electrostatically, one positively and the other negatively, and so B will turn and move the pointer over a scale which may be graduated so as to read in volts. 79. The Electrodynamometer. — It has been shown (§ 25) that when a current flows through two adjacent coils, they will turn so as to make their magnetic fields coincide in position and direc- tion as nearly as possible. This principle has been applied in the GALVANOMETERS. 109 construction of a d3mamometer for measurement of the strength of a current. One form of this instrument is shown in Fig. 67. The central coil is fixed, while the other, consisting of a single heavy wire, carries a pointer which reaches up to a graduated circle at the top of the instrument. The coils are in series so that whatever current passes through one will also pass through the other. A spiral spring connects the movable coil with a torsion head. When a current flows there will be a force tending to make the coils parallel. The number of scale divisions through which the torsion head must be turned to keep the coils at right angles is a measure of the strength of current, but since the coils are in series an increase of current in one means an equal increase in the other, so that their mutual effect produces a force which varies as the square of the current. Hence, if S is the number of scale divisions moved over by the pointer at- tached to the torsion head. i= k y/e (76) Fig. 67. where & is a constant which | may be found by observing 6 when a known current, i, flows through the instrument. This instrument can be used to measure either direct or alternating currents, for when a current is reversed in one coil it is reversed in the other at the same time. 80. The Wattmeter.— The watt (§ 54) is the unit of electrical power. The power is 1 watt when 1 joule of work is done per second. The two factors whose product determines electrical power are strength of current and electrical pressure. The product of the number of volts by the number of amperes gives the number of watts. If, for example, it is desired to know the number of watts needed to operate an incandescent lamp, a voltmeter may be connected, as shown in Fig. 68, so as to show the P.D. at the 110 GENERAL PHYSICS. terminal of the lamp, and an ammeter in series with the main current will show the strength of current. The product of the reading of these two instruments gives the number of watts. If now an instrument is constructed on the same principle as the electrodynamometer, one coil, called the pressure coil, having many turns of fine wire — i.e., having large resistance Uke a volt- meter — ^and the other, the current coil, having few turns of coarse wire; and if these coils are connected, not in series, but as in Figj 68, the instrument becomes a wattmeter. Such coils may be ar- Fig. 68. ranged as shown in Fig. 69. Then to determine the wattage of a lamp it is only necessary to connect the opposite sides of the lamp to the pressure coil while the current coil forms part of the main circuit. Problems 1. The coil of a tangent galvanometer, 14 cm. in diameter, contains two turns of wire and is set parallel to the earth's field. What is the strength of current that will deflect the needle 22.4° at a point where the intensity of the earth's field is .2 gauss? 2. A current whose strength is 5(10)-' amperes causes, in a D'Arsonval galvanometer, a deflection of 10 cm. on a scale 1 m. distant. What is the sensibility? 3. If the torsion head of an electro-dynamometer must be turned through 16 scale divisions to keep the coils at right angles when a current of 10 amperes flows, what is the strength of current for 49 scale divisions? 4. If 4 lamps are in parallel and each requires 40 watts for their operation, what is the strength of current when the pressure is 110 volts? 6. If 330 watts are required to heat an electric iron on a 110-volt circuit, what will be the cost at 10 cents per K.W. hour, and how many calories of heat will be produced in one hour? Fig. GALVANOMETERS. Ill 6. The coil of a tangent galvanometer is 46 cm. in diameter and contains 150 turns of wire. It is joined in series with a CuSO< voltameter. What is the strength of the earth's magnetic field if .0275 g. of copper is deposited on the cathode in 30 minutes, the needle of the galvanometer being deflected 45° during that time? Ans. 1. .451 ampere. 2. 200 megohms. 3. 17.5 amperes. 4. 1.45 amperes. 5. 3.3 cents per hr. 283,735.3 calories. 6. .19 gauss. CHAPTER VII MEASUREMENT OF ELECTRICAL RESISTANCE 81. Relation of E, i, and R. — By Ohm's law, stated in § 52, the ratio of the electromotive force E to the strength of current i is a constant quantity R for a given conductor at a constant temperature. Hence the value of E between any two points on a conductor is, from equation (52) E=iR (77) If R becomes large, E must also be large to produce a current of the same strength, and if E remains constant, i may be reduced to any desired strength by increasing R. 82. Specific Resistance. — It is found that the resistance of a conductor varies directly as the length, inversely as the area of cross section, and is different for different materials at the same _ temperature. This may be expressed by R=k\ (78) A where k is the specific resistance or resistivity of any given material. If the length /is 1 cm. and the area of cross section A is 1 sq. cm., then R=k, i.e., k is the resistance of a 1-cm. cube of the material. This value of k for a number of different materials is given in the appendix. When the value of k is once found, the resistance of any length and area of cross section may readily be calculated. The resistance of a mil-foot is also frequently used as the specific resistance of a conductor. A mil-foot is a wire 1 foot long and one-thousandth of an inch in diameter. The area of cross section of this unit is called one circular mil. Then, since the areas of circles are to each other as the squares of their diameters, the square of the diameter of any wire will give the area of cross section in circular mils provided the diameter is taken in thou- sandths of an inch. This is the method used in commercial and construction work, and equation (78) is then written R=k „ , , CM. 112 MEASUREMENT OF RESISTANCE. 113 where I is the length of a wire in feet, CM. is the area of cross sec- tion in circular mills, and k is the resistance of a mil-foot in ohms. 83. Temperature Coefficient of Resistance. — ^AU pure metals and most alloys increase in resistance as the temperature rises. The change in resistance per degree rise in temperature per ohm of the resistance at 0° C. is called the temperature coefficient of resistance. If this is expressed by k, then fe=^ (79) and when the value of k is once found for any substance, the resistance, Rt, for any given temperature is found by Rt = RaO-+kt) (80) The value of k for all pure metals is about .004 ohm. (See appendix.) The value of k for platinum is very constant through a wide range of temperature and so this, with other properties, makes platinum a proper metal for use in resistance thermometers. (See p. 199 of " Mechanics and Heat.") For alloys this coefficient is much lower than for pxure metals, in some cases even zero or negative. For example, manganinis an alloy usually composed of 12 parts nickel, 84 parts copper, and 4 parts manganese. Its temperature coefiBcient is practically negligible, being about .000001 ohm. This metal is valuable in the construction of measuring instruments not only because k is so small, but also because the thermo-electric current generated when a point of contact of dissimilar metals is heated is' also small. Substances other than metals, notably carbon, decrease in resistance when the temperature increases. The resistance of a carbon filament lamp when hot is about one-half of the cold resistance. 84. Currents in Series and in Parallel. Kirchhoff's Laws. — ' The battery in Fig. 70 is the seat of an E.M.F. which will cause a current to flow through the circuit abcdh. The conductors ah and he are joined in series, i.e., the entire current flows through the first and then passes on through the second. Between d and e the conductors are in parallel. Part of the current will flow on one branch and part on the other. 8 114 GENERAL PHYSICS. Before stating the laws of resistance in these two kinds of circuits it may be an advantage to state certain principles implied in Ohm's law but usually stated in what is known as KirchhofiE laws. 1. The sum of all the currents meeting at a point is zero. The currents flowing toward a point are positive, and away from the point, negative. The first law then simply states that no electricity accumulates at any point in a circuit. The strength of current flowing to c, Pig. 70, is equal to that flowing away from it. The sum of the currents on the two branches leaving d is the same as the current flowing to that point. l|^^ Fig. 70. 2. In any closed circuit, however complex, the sum of fhe products obtained by multiplying the resistance of each part of any one path by the strength of current flowing through that part is equal to the sum of the E.M.F.'s in the circuit. Thus, in Fig. 70, if a current ii flows from a around the cir- cuit, part flowing on each branch of the divided circuit, then iiRi+iiRi+iiRs+iir = E.M.F. in the circuit, r being the internal resistance of the battery or other soiurce of E.M.F. The same result is obtained if the other branch of the divided circuit is followed and isRi is used in place of i^Rs. If we consider only the path dgefd, which does not contain a source of E.M.F., the sum MEASUREMENT OF RESISTANCE. 115 of iiRs and isRt, taken with the proper signs, is zero. In general, the sum of the iR's of a closed circuit which does not contain an E.M.F. is zero, provided that in following the circuit completely around, the iR's are regarded as positive in the direction the current flows, but negative when opposed to the current. If we consider only the portion of the circuit from o to c in Fig. 70 and denote its total resistance by R^, then R,ii = Riii-\- Riii or R. = Ri+Ri (81) In general the total resistance of conductors in series is the sum of the individual resistances. In case of that portion of the circuit from d to e, if Rp is the resistance of the two branches, then, by Ohm's law, equation (52), and since the potential difference E between d and e is the same for either branch, E . E . E t2+i3=—; *2= -^S *3= ^ tip Ki Ki Hence E__E_E^ Rp Rs Rt Kp Ki J\i In general the reciprocal of the total resistance of any number of conductors in parallel is equal to the sum of the reciprocals of the resistance of the individual conductors, whatever the number of branches in parallel may be. In case there are but two conduc- tors in parallel, from (82) „ XV3XV4 i.e., the total resistance of two conductors in parallel is equal to the product of the individual resistances divided by their sum. 85. Wheatstone Bridge. — The best method of determining the resistance of a conductor is by means of a Wheatstone bridge. This, as shown in Fig. 71, consists of a divided circuit. Part of the current from the battery flows through ABC and a part through ADC. A conductor with a galvanometer in series is 116 GENERAL PHYSICS. connected between B and D. When these two points do not differ in potential no current will flow through the galvanometer. A coil of unknown resistance X is inserted in the arm AB; a resist- ance box R is inserted ia AD, and by inserting or removing plugs the resistance, R, in this arm may be varied at will; the arms a and b each contain known resistances which may be plugged in or out of the circuit. Let X, R, a, and 6 represent the resistances inserted in the four branches of the bridge. R is so adjusted that the galvanometer shows no deflection. The bridge is then said to be balanced. Now, according to Kirchhoff's second law, the sum of the products of resistance by current in all parts of the closed circuit ABDA is, since it contains no source of E.M.F., equal to zero.j Hence iiX-itR=0 (84) (85) Likewise in the closed circuit BCDB, iia—itb=0 Dividing (84) by (85), X R a b or x-l. (86) MEASUREMENT OF RESISTANCE. 117 Since the resistance of a, b, and R are known, that of X is readily fotind, and if a is equal to b, X is equal to R. If b is made 100 ohms while o is 1 ohm, then X-^ ^"100 For example, if 245 ohms were introduced in i? to balance the bridge, the resistance of X is 2.45 ohms. According to KirchhofiE's first law it is evident that when the bridge is balanced the current which flows through AB will con- tinue with the same strength through BC, and likewise for the other side of the bridge. The connections of the galvanometer and battery should ordinarily be as shown in Fig. 71. Maxwell's rule for this is that of the two resistances, that of the battery and that of the galvanom- eter, the greater should connect the junctture of the arms having the two largest resistances to that of the other two arms. The resistance of the galvanometer wUl nearly always be greater than that of the battery and the resistance of R and b, Fig. 71, will in most cases be greater than that of X and a. Hence the greatest deflection will be obtained when the galvanometer is coimected as shown. The slide wire form of the Wheatstone bridge does not differ in principle from the form already explained. As shown in Fig. 72 the ratio coils of Fig. 71 are here replaced by a straight resist- ance wire. A balance is approximately seciired by adjusting the resistance in R while c makes contact with the middle of the wire. Then by sliding c one way or the other along the wire an exact balance is secured. The resistances of a and b are assumed to be in proportion to their lengths, and so, as already shown, R~ b a or X=—R b This form of bridge is convenient but is not so accurate in actual practice as a form to be described later. It is difficult to secure a wire which is exactly tmiform, and therefore the assump- tion that resistance is proportional to length involves a probable 118 GENERAL PHYSICS. error. Errors also are made in observing the point of contact on the bridge wire and in finding the exact point of balance. Thermo- electric currents are also likely to be set up as a result of heating the points of contact by the hand or otherwise. These errors may be partially eliminated by exchanging X and R and combining the observations made before and after the exchange. Thus, let the bridge wire, Fig. 72, be 1000 mm. long. Suppose that when X and R are placed as shown in the Fig. 72. figtire an error e is included in reading the position of c. Let a\ be the distance from A to c in the first observation. Then X_ R ai+e 1000 -ai-e (87) Now let X and R be exchanged. Adjust c for a second balance, assuming that the same error has been made. Let the distance from A to c ia this case be 02. Then R 1000 -oa-g Adding numerators and denominators of (87) and (88), X_ 1000+(ai-a2) R~ 1000 -(01-02) (88) (89) Thus e is eliminated and the only observations necessary are the lengths ai and 02 of the bridge wire. The best form of bridge is one where the rheostat, the ratio coils, and the bridge proper are all conveniently arranged in a box, commonly known as the Post Office Box, from its use in the Postal Service of England. The external appearance of such a box is shown in Fig. 73 and the cormections, as shown in Fig. 74, MEASUREMENT OF RESISTANCE. 119 are the same as in Fig. 71 but the battery and galvanometer wires are made to pass through keys on the top of the box. By pressing these at any time, the galvanometer will indicate whether or not the bridge is balanced. By removing plugs from their sockets Fig. 73. between the brass blocks a certain number of ohms of resistance, as marked on the lid of the box, is thrown into that circuit. Sup- pose that the bridge is balanced when the 1-ohm plug on the right and the 100-ohm plug on the left are removed from the ratio !,QQQP.QaQOl-=. Hence the ratio of the e.m. to the e.s. unit of quantity is very nearly 3(10)1°. ^his, in centimetres, is the velocity of light, a fart which has been significant in the de- velopment of the electro- — ^'"— " magnetic theory of light. 97. Dielectric Con= stant. — The dielectric con- stant or specific inductive capacity has been defined (§ 21) as the ratio of the capacity of a condenser when any given dielectric is used to the capacity when air is the dielectric. Hence, by use of the appa- ratus shown in Fig. 89 it is possible to introduce various dielectrics between the plates and then, by charging with the same battery and discharging through a ballistic galvanometer, to compare capacities. These would be in proportion to the throws of the galvanometer if the plates are separated by the same distance in each case. The charge in a condenser depends in some measure on the time of charge. As explained in § 22 a certain quantity of elec- tricity is absorbed by the dielectric, and this increases with the time of charging. The total charge is therefore not given back Fig. 89. 138 GENERAL PHYSICS. on the first discharge, but will appear on subsequent discharges. Mica is comparatively free from this objection and so is used as the dielectric in standard condensers. Problems 1. What quantity of electricity will a condenser of .5 microfarad capacity hold when the P.D. is 4 volts? 2. What is the capacity of a condenser that holds .0033 of a coulomb when the pressure is 110 volts? 3. How long a time will be required for a current of 1.5 amperes imder a pressure of f 10 volts to charge a condenser of 30 microfarads? 4. A cable two miles long is charged by a battery and then discharged through a galvanometer, causing a deflection of 120 divisions on a scale. A condenser of .5 microfarad charged by the same battery causes a deflection of 150 divisions. What is the capacity of the cable? 5. Three condensers having capacities .5, .2, and .2 microfarad, respec- tively, are connected in series. What charge will be needed to store them tmder a^pressure of 120 volts? 6. Three condensers are in parallel. Their respective capacities are .5, .2, and .1 microfarad. What quantity of electricity measured in e.s. units will charge them at a pressure of 20 volts? Ans. 1. 2(10)-« coulombs. 2. 30 microfarads. 3. .0022 sec. ,4. .2 microfarad per mile. 5. 10(10)-* coulombs. 6. 4.8(10)' e.s. units. CHAPTER X ELECTROMAGNETS 98. Solenoids. — ^A solenoid is a continuous conductor wound in form of a helix. From principles already explained in § 25 we see that, when a current of electricity flows on the conductor, a magnetic field is produced within the solenoid which then exhibits the properties of an ordinary bar magnet as shown in Fig. 24. The direction of the field within the solenoid depends on the direction in which the current flows. A conventional rule for this is as follows: Grasp the conductor at any point with the right hand so that the thumb points in the direction the positive current flows, and the fingers will then encircle the conductor in the direction of the lines of force, which direction within the solenoid is the direction of the field there. The end from which these lines emerge is the north-seeking pole of the solenoid. a Fig. 90. The strength of the field within a solenoid may be calculated, as shown in Appendix 2, by use of the equation //= -jT- gausses (107) where n is the number of turns per centimetre and i is the strength of current in amperes. The strength H may be made large by increasing n and *. The best method of magnetizing a bar of steel or a magnetic needle is by placing it in such a strong field. When a core of soft iron is placed in the solenoid, the number of magnetic lines is enormously increased. This arrangement constitutes what is called^an electromagnet. The extensive use of the electromagnet results from the fact that the iron core is a magnet while a current flows in the solenoid, but ceases to be a magnet when the electric circuit is broken. Hence its use in operating sounders of telegraph instruments, ringing bells, producing magnetic fields of dynamos, 139 140 GENERAL PHYSICS. shifting heavy masses of iron by means of the electric crane, and numerous other uses of this character. If the core of an electromagnet is short, the number of lines of force induced in it will not be so great as when longer and in a longer solenoid, for the poles of the magnet exert a force within the coil opposite to that of the coil. When the poles are farther apart their effect in this respect is not so great. If the helix of the solenoid is bent in form of a circle as shown in Fig. 91, the lines of force form a closed ' circuit and there are no poles. The lines of force are then found only in the space within the helix, as iron filings show. The field is ji.,Q 91 nearly uniform and an iron ring in this field will be magnetized equally in all parts. 99. Magnetic Flux. — The total number of lines of magnetic force passing through any given area is called the magnetic flux through that area. The letter H has been used to represent strength of magnetic field, and strength of field is defined as the number of lines per square centimetre. Hence, if (f> stands for magnetic flux, then for a given area. A, across a uniform field, such as in a long solenoid or a ring solenoid, = HA (108) or if, as explained in § 34, the field contains a paramagnetic sub- stance, the total induction per square centimetre is B, which includes H, then =BA (109) The number of lines per square centimetre is the flux density. In air this is H, but in other material it is B and the relation of B and H is B = nH where n is the permeability of the substance. (See § 37.) Hence by calculating H from equation (107) and observing B, the permeability of any substance may be found, or if ai is known, the induction B is readily calculated. In this manner the various values of H which were plotted as abscissae in Figs. 31 and 32 were found, the resulting induction being plotted as ordinates. ELECTROMAGNETS. 141 One turn of the wire of a solenoid canying one ampere is called one ampere-turn. In equation (107) it is seen that the mag- netizing force H, or the flux density in air, varies directly as ni, i.e., as the ntmiber of ampere-turns per centimetre. Thus, in one centi- metre, a single turn of wire in which five amperes flow will produce the same value for H as five turns in which one ampere flows. 100. Magnetomotive Force. — ^Just as in electricity we speak of an electromotive force as that which causes a current to flow, so in magnetism, although nothing actually flows, yet a force is required to set up that condition known as a magnetic circuit. This is called the magnetomotive force. Electromotive force is defined as the number of ergs of work performed in carrying unit charge around the electric circuit, and in a similar manner the magnetomotive force is defined as the number of ergs bf work required to carry unit pole around a magnetic circuit. The former is often called electric pressure and is measured in volts, whUe the latter is called magnetic pressure and is sometimes expressed in gilberts. In case of a magnet the magnetomotive force (m.m.f.) is the work required to move a unit N pole from the S pole to the N pole. In a ring solenoid such as in Fig. 91, it is the work done in moving unit pole once around the circuit in opposition to the field within the helix. If equation (107) is multiplied by the length of the solenoid, L, then ■TT^ ^■^Lni i7L=^^ (110) and since n is the number of turns per centimetre, Ln is the total number of turns and may be represented by N. Then ^irNi HL=^^ (111) Now H is the intensity of the magnetizing force — i.e., the force with which unit pole would be iKged — and L is the distance, hence HL is the m.m.f. of the circuit. Then iirNi m.m.f. = — — - =1.257 Ni (112) i.e., the m.m.f. is equal to the total number of ampere-turns times 1.257. 142 GENERAL PHYSICS. 101. Magnetic Reluctance. — Ohm's law as expressed by equa- tion (52) simply states that the strength of an electric current is directly proportional to that which causes it and inversely pro- portional to that which resists it. This law is, then, a general principle which has been shown to be rigidly applicable to the electric current. Likewise, in the magnetic circuit we have magnetic flux, 4>, which corresponds in a sense to i of the electric circuit, m.m.f. corresponds to e.m.f., and magnetic reluctance Rm corresponds to electric resistance R. Hence we may speak of Ohm's law of the magnetic circuit and ex- press it by 4,= m.m.f. Rm (113) Reluctance may be found when the dimensions and spe- cific reluctance of a given sub- stance are known, by use of an equation similar to (78), thus Rm = K- (114) where L is length, A is area of cross section, and K is the specific reluctance, i.e., the re- luctance of one cubic centi- metre. But since reluctance is the reciprocal of permeability just as resistance is the recip- rocal of conductivity, (114) may be written L Fig. 92. ■Rm = flA (115) Reluctances in series and in parallel are calculated as in resist- ances, and in shunted magnetic circuits the flux is inversely pro- portional to the reluctances of the branches. If a current of electricity is passed around the iron core D, Fig. 92, lines of force will be set up through C and A, but if a gap is made in the iron ELECTROMAGNETS. 143 circuit at P, the reluctance of the upper branch will be increased and there will be an increased flux through C. The apparatus here shown is a transformer which raises the voltage of an alternating current from 110 to 20,000 volts and the strength of the induced current is regulated in the manner de- scribed above. Problems 1. What is the intensity of the magnetic field within a solenoid when the current strength is .2 e.m. unit and there are 6 turns per centimetre? 2. A solenoid 50 cm. long and having 300 turns is bent in form of a ring and filled with iron for which /i=450. What flux density will be caused by a current of 5 amperes? 3. What is the reluctance of a rod of iron 2 cm. in diameter and 100 cm. long, when ^ = 1000? 4. If a solenoid 40 cm. long has four turns per centimetre, what will be the m.m.J. when the current is 2 amperes? 5. If a magnetizing force of 5 gausses is needed to cause an induction of 9000 gausses in a certain specimen of iron, how many ampere-tums are needed to set up 100,000 lines in a ring of this metal 200 cm. long and 40 cm.' in cross section? 6. An iron rod 3 m. long and 75 cm.' in cross section is bent in form of a circle but with an air gap of 3 cm. between the ends. If /'=800, what is the reluctance of the entire circuit? Ans. 1. 15.08 gausses. 2. 16969 gausses. 3. .032 unit. 4. 402.24 units. 5. 220.98. 6. .045 unit. CHAPTER XI ELECTROMAGNETIC INDUCTION 102. Induction. — Electric induction, in a general way, refers to the effects produced on bodies in an electric or magnetic field. Electrostatic induction has already been defined as the appearance of positive and negative charges in a conductor when placed in an electrostatic field, and, likewise, magnetic induction is the phenomena observed when a magnetic substance is placed in a magnetic field. In 1819 Oersted discovered the fact that a current of electricity would influence a magnetic needle or any form of magnet. Not only would the current affect a magnet but would cause such substances as iron to become magnets so that strong electromagnets could be made. A current of electricity therefore produces a magnetic field about its path. If therefore a current produces magnetism, it seemed very probable that in some manner a mag- netic field could be made to produce electricity. An investigation of this subject by Faraday in England and Henry in America resulted in the discovery of electromagnetic induction. Faraday published his results in 1831 while Henry, although his discovery antedated that of Faraday, delayed publication till 1832. 103. Nature of Electromagnetic Induction. — ^As long as a cur- rent flows on a conductor, a magnetic field is maintained in the region of that conductor, but, on the other hand, the existence of a steady magnetic field in the region of a conductor will not cause any flow of electricity. This may be illustrated by a water analogy. Suppose numerous slender elastic rods are fastened at one end and that the free ends project into a stream of running water. As long as the water flows the rods will be bent and their strained condition may be taken to represent the magnetic field which accompanies the flow of an electric current. If, however, the water is not in motion and the rods are all bent in the same direc- tion by some outside force, then, although the water may be momentarily set in motion, the fact that the rods are kept in that strained condition will not cause a current to flow. Similarly we would not expect that a strained condition of the ether would 144 ELECTROMAGNETIC INDUCTION. 145 cause a steady flow of an electric current, for such a condition would be in direct conflict with all principles of conservation of energy. Faraday and Henry discovered a variety of ways by which a current may be induced in a conductor, the principle common to all being that any change in the number of magnetic lines enclosed by a conducting circuit will induce a current in that circuit. If two coils, A and B of Fig. 93, are placed near one another, the terminals of A being connected to a source of electricity, and the coil B being closed through a galvanometer, then whenever the circuit of A is made or broken, the galvanometer will show a momentary current in B. Either starting or stopping the current in A causes a change in the number of magnetic lines passing through B. If an iron core is placed within the coils, the induction will be greater, for there will be a greater change - -^ in the number of magnetic lines 3 Fig. 93. Fig. 94. when the current is started or stopped. A steady flow of current in A will not cause any current in B, for then the magnetic lines enclosed by B will be neither increasing nor diminishing, but any change in the relative position of the two coils will induce a current in B. If A is replaced by a bar magnet, whenever a pole of the magnet is advanced toward B or withdrawn, induction follows. The numerous kinds of changes by which a current is induced in a closed conducting circuit may all be explained in accordance with the general principle stated above. The direction of the induced current is always such as to oppose the motion which produces it. This is known as Lena's law and is illustrated in Fig. 94, where a north-seeking pole of a magnet moved toward a coil induces a current in such a direction as to produce a north-seeking pole at that end of the coil, and thus oppose the approach of the magnet. The withdrawal of the magnet produces 10 146 GENERAL PHYSICS. a current in the opposite direction, thus making that end of the coil a south-seeking pole and again resisting the motion which causes the induction. Similar phenomena may be shown in case of the two coils in Fig. 93. Any change which causes induction requires, according to Lenz's law, that force must be exerted through a distance, i.e., that work must be done. When a con- ductor is moved across lines of force the direction of the induced current may be conveniently found by what is known as the dynamo rule, viz., extend the thumb, the first, and the second fingers of the right hand so that each is at right angles to the | | i|l|l '|i ||, fnM 'l|l l |i|l l !ll iU l l l i l|l ill |l|llll| . i|| l lllll / mmifmmm TOffl N Fig. 95. Fig. 96. other two, then if the first finger points in the direction of the magnetic lines (from N to S) and the thumb in the direction the conductor is moved, the second finger will indicate the direction in which the cturent flows. Thus in Fig. 95 a current will flow from A to S if the conductor AB is moved in the direction of the arrow. This rule applies only to the conventional current. For the electron or negative current the left hand must be used. The electromotive force of the induced current is proportional to the rate at which a conductor cuts lines of force, or to the rate of change of flux included in a closed circuit. The conductor may be only a straight rod or wire, and if it is moved across lines of force a difference of potential will be produced at the two ends. If this ELECTROMAGNETIC INDUCTION. 147 conductor forms part of a closed circuit, a current of electricity will flow. To show that, in accordance with principles already stated, the e.m.f. is proportional to the change of flux, suppose that a uniform magnetic field exists between the poles N and S, Fig. 96. Let two conducting rods mo and np be placed in this field and connected at mn by a wire. A sliding rod ab laid on the two rods then closes the circuit abnm. Suppose now that this circuit contained an e.m.f. which would cause a current to flow around in the direction bamnb, then, by the motor rule, the sHde would be urged toward mn with a certain force F. But if the circuit does not contain an e.m.f. and a force F is exerted to move the slider from the position ab to a'b', then by the dynamo rule a current will flow around the closed circuit in the direction bamn. Let the intensity of the magnetic field be H gausses, the strength of current i electromagnetic units, and the length of the sUder icm. The current here is that which results from moving the slider across the lines of force. Then, by the definition of the e.m. unit of current, the force applied in moving the slider is F=iHl dynes (116) Let this force be applied through a distance d, then the work, W, done is W = Fd=i Hid ergs (117) This is the amount of energy expended in moving the slider across the field and thus producing the current i. The current should receive all the energy expended in producing it. It has already been shown that the energy of a current is equal to the product of its strength i, the electromotive force E, and the time t during which it flows. If, then, t is the time occupied in moving the slider from ab to a'b', the energy of the current produced may be repre- sented by W=iEt ergs (118) We therefore have from (117) and (118) iHld=iEt or E=Hld (119) t 148 GENERAL PHYSICS. Now H is the number of lines per square centimetre and Id is the area abb'a'. Hence Hid is the change of flux through the closed circuit. This divided by the time gives the change per unit of time, i.e., the rate of change. Hence the e.m.f. is equal to the rate of change of flux, or the rate at which lines of force are cut. From equation (119) we see that the e.m.f. in e.m. units is unity when one line of force is cut per second. Since the volt is 10' times this unit, the e.m.f. is 1 volt when a conductor cuts lines at the rate of 100,000,000 per second, or when that is the rate of change of flux. It should be noted in Fig. 96 that the movement of the slider causes a current which tends to move the slider in an opposite direction. This is in accordance with Lenz's law, and therefore force must be used in moving the slider and energy must be expended in producing the current. An interesting illustration of in- duction by cutting lines of force is found in the use of the earth in- ductor. This, as shown in Fig. 97, consists of a coil of insulated wire having several hundred turns motmted so that it may be rapidly turned through 180° in the earth's magnetic field. As shown in Fig. 39, the earth s field may be regarded as having a horizontal and a vertical com- ponent, each of which may be found separately. Let the in- ductor be set so that it will when rotated cut only the horizontal component. Let a be the area in square centimetres of each turn of the coil and H the number of lines per square centimetre in the field. If there are n turns in the coil, the total area is na. The coil is first set so that its plane is at right angles to the field. When it turns through 90° the total change of flux is Han. The same change of flux will occur in the next quarter revolution. The total change is therefore 2 Han. If the half revolution is accompUshed in time t the e.m.f. induced is £=?^ (120) ELECTROMAGNETIC INDUCTION. 149 for this expresses the rate of change of flux. Therefore by Ohm's law, equation (50), . 2Han Rt (121) and this is the strength of current, as measured in electromagnetic units, that would flow through any circuit connected to the ter- minals of the coil, R being the resistance of the entire circuit including the wire on the coil. Since the quantity of electricity is Q=it, then ^ 2 Han ,,„„, 0=—^ (122) and -ff=F^ (123) 2an where H is the strength of the horizontal component of the earth's field in gausses. The vertical component may be found in the same manner by placing the inductor so that it wiU cut lines only in that direction. To find Q the terminals of the inductor coil may be connected to a ballistic galvanometer. The commutator of the inductor is so made that the circuit is broken the instant the coil completes its turn through 180°. The throw of the galvanometer may then be taken as proportional to the quantity Q of electricity sent through it. A standard condenser may then be made to cause the same throw, and from a knowledge of the capacity C of the condenser and the e.mj. used in charging it, all reduced to e.m. imits, the value of Q is found by equation (6). 104. The Principle of the Dynamo. — ^A djmamo is a device for producing electric currents according to the principles of elec- tromagnetic induction. In any dynamo, then, conductors are made to cut Unes of force either by moving the conductors across the Unes or by moving the magnets so that the lines will cross the conductors. Let the coil dbdc, with terminals at e and /, Fig. 98, be rotated in a clockwise direction in a magnetic field. The ends bd and ac will not cut any lines and so we need consider only ab and cd. Since ab is moving downward across lines of force, and cd is moving upward, the direction of the induced e.m.f. will, 150 GENERAL PHYSICS. according to the (i3maino rale, be from 6 to a and from c to d — i.e., it would, if e and/ were joined by a conductor, cause a current to flow around the loop. It is evident, however, that, although the speed of rotation is uniform, the rate of cutting lines is different at different points in the rotation. Let the plane of the coil be in such a position that it makes an angle d with a perpendicular to the field. Let the velocity of ab be indicated by v and represented in Fig. 98 by the vector ay. This vector may be resolved into the components xy perpendicular to the field and ax parallel to the field. But xy = ay sin d—v sin (124) Pig. 98. Hence the velocity at right angles to the field is, for any position of the coil, V sin 0. If I is the length of ab and H is the strength of the field, then for both ab and cd, e.m.f. =2 Hlv sin (125) When the plane of the coU is parallel to the field, 6 =90° and sin is unity, and lines are then being cut at the maximum rate. Equa- tion (125) then becomes identical with (119), for v is the distance moved directly across lines of force in imit of time. When the plane of the coil is at right angles to the field, 9 becomes zero and so the e.m.f. is zero, i.e., no lines are being cut when the coil is in that position. During the first quarter of a rotation the e.m.f. increases, for both ab and cd are moving into positions where they cut lines at a maximum rate. During the second quarter the e.m.f. decreases ELECTROMAGNETIC INDUCTION. 151 to zero. At the beginning of the third quarter the direction of the e.m.f. is reversed in the coil, as an application of the djrnamo rule will show, and wiU increase to a maximum in that direction, returning again to zero at the end of the last quarter. These changes when plotted give a sine curve as shown in Fig. 99, where the abscissas are the successive angles made by the coil with a plane perpendicular to the field and the ordinates are the corre- sponding values of the e.m.f. as calculated by use of equation (125). If now a metal ring is attached to each terminal e and / of Fig. 98 and made concentric with the axis of the coil but insulated from it, then the circuit of the coU may be closed by an external conductor terminating in brushes that rest on the rings. The e.m.f. induced by rotating the coil will now cause a current of electricity to flow in the circuit in. one direction during the first half of the rotation and in an opposite direction during the second half, i.e., the current through the coil and the external circuit will be alternating, changing direction twice in each rotation of the coil. Fig. 99. A complete set of changes is called a cycle. The curve shown in Fig. 99 is for two cycles. In the arrangement shown in Fig. 98 there would be as many cycles as there are rotations of the coil. Sixty rotations per second or 3600 R.P.M. would give 60 cycles per second. The number of cycles per second is called the fre- quency of the alternating current. A very common frequency in use for both power and light is sixty, and where the current is used for power only, twenty-five cycles per second is in common use. To secure a frequency of 60 it would be necessary to rotate the coil or armature of Fig. 98 very rapidly. This would be ob- jectionable for mechanical reasons. The same frequency may be obtained at a lower rate of rotation by using more poles to pro- duce the field through which the coils of the armature are made to pass. This is shown in Fig. 100 where the electromagnets 152 GENERAL PHYSICS. producing the field are so wound that they are alternately N and S poles when a direct current is made to flow through them. The armature is a drum of soft iron with a slotted surface. In these slots coils of wire are wrapped in manner shown in Fig. 100, B, there being as many coils as there are poles in the field. The coils are a continuous conductor wrapped back and forth on the sur- face of the drum, the direction of the wrapping being reversed in each successive coil so that, when one is passing an N pole of the field and the next an S pole, the directions of the induced e.m.f. will not oppose each other but will produce a current in the same Pig. 100. direction throughout the wire of the armature coils. The direction of the current in the armature will be reversed each time the coils pass opposite poles and so there will be as many alternations as there are poles and as many cycles as there are sets of N and 5 poles. For example, if there are eight poles there would be four cycles in each rotation of the armature and sixty cycles per second would be produced by 900 R.P.M. In the machines just described the field magnets are stationary and the coils into which the current is induced are moved rapidly Pig. 101. ELECTROMAGNETIC INDUCTION. 153 in the magnetic field. All that is required, however, for induction is that there be relative motion of coils and field. The coils may be stationary and the field magnets in motion. It is found to be of considerable advantage both in construction and operation to build alternating generators in this manner. In this case the armature coils are placed in the slots of a heavy iron frame as shown in Fig. 101. This part of the machine is called the stator. The rotor is shown in Fig. 102 and consists of the field coils mounted on an axis. This is placed within the stator and magnetized by a direct cturent from a separate djmamo called the exciter. This current passes around the poles in such a direction as to make them alternately N and S poles. When the rotor is turned we have, then, the same relative motion of magnets and armature as in the previous case. The induced current flows directly from Fig. loa. the stator and the coils of the armature can be more effectively insulated and are not so liable to mechanical injury as when they constitute the moving part. 105. The Direct Current Dynamo. — In all dynamos the current induced in the armature coils is an alternating one. This current, however, may be made unidirectional on the line leading out from the dynamo. This is done by use of a commutator, the principle of which is shown in Fig. 103. Here a copper or brass ring is split into equal parts and insulated. This is mounted on the shaft of a rotating armature. Let the terminals e and / of Fig. 98 be attached one to each segment of the commutator. Then, as the coU rotates, the brushes a and b, Fig. 103, will slide from one segment to the other each 154 GENERAL PHYSICS. time the current is reversed, and so the current which passes out on the line will always be in the same direction, i.e., the current will be direct, usually indicated by D.C. while the alternating current is indicated in writing by A.C. Although the commutator just described produces a direct current, it by no means produces a constant current, i.e., a current having the same intensity at all times. A plot such as that of Fig. 97 would show that through 180° the current rises from zero to maximum and then back to zero, but during the next 180° this is repeated, producing a curve like that in Fig. 104. The two most common kinds of armatures are called the ring armature and the drum arma- ture. The former is often known as Pacinotti's ring or the Gramme ring. It consists of a bundle of wires or laminated iron upon which coils of insulated wire are wound as shown in Fig. 105. There are as many segments of the com- FiG. 103. Fig. 104. mutator as there are coils on the ring, and the coils are con- nected to each other so that they form a continuous circuit. Because of the permeability of the iron ring the lines of force from AT' to S pass through the . material of the ring and there are practically none in the space within the ring. When, there- fore, this armature is rotated, only that part of the coils on the outside of the ring will cut lines of force. Let the rotation be in the direction indicated by the arrow. Then, applying the Ajaaxao rule, it will be seen that a current will flow through the coils on both the right and left sides of the ring toward the brush a and from the brush b. This causes a difference of potential between a and b and consequently a flow of electricity on any cir- cuit terminating in these brushes. Fig. 105. Fig. 106. ELECTROMAGNETIC INDUCTION. 155 Coils at the upper and lower points of the ring are not cutting lines of force, while those coils 90° from these points are cutting lines at the maximum rate. Intermediate coils are approaching or receding from these maximum points. Since there are many coils on the ring, one scarcely has passed the point of maximum cutting before the next has come into that position. The B while H is de- creasing and G is in- creasing to a sufficiently high potential to send a current from A' \xi B t® ^S^i'^st the counter e.m.f. of the battery. The principle of this action of the reactance coil has been explained in the paragraph on self-induction. When the current from A flowed through the coil E, it stored the coil with a quantity of energy in form of a magnetic field. When the current ceased the energy of this field was converted into current flowing in the same di- rection. Thus the current is maintained from A to B while G is rising to sufficient voltage to send current from A' to B. Then the same operation takes place from that side and through reactance F. No current flows directly from the lines through E and F, for these coils are at that instant discharging in an opposite direction. By this arrangement the entire wave form is used and a con- tinuous current is sent through the battery or other load. ♦0 2 Fig. 128. ELECTROMAGNETIC INDUCTION. 181 123. Dimensions of Electromagnetic Units. — The electro- static system of units, as we have seen, is based on the definition of unit quantity of electricity. The electromagnetic system, how- ever, is based on the definition of unit magnetic pole. The force exerted between two poles is expressed by mm' In definition of unit pole m and m' are equal, hence their product is the square of either one. Then substituting the dimensions of force for F, distance for r, and retaining n in the equation. The strength of a magnetic field at any point is measured by the force which would be exerted on a vmit pole at that point, hence, if a pole m is urged by a force F, the strength of field H is m and substituting dimensions for F and m Unit current strength is that current, unit length of which at tmit distance will produce unit field, then for a length 2nr — i.e., circumference of a circle — at a distance r from the centre, and a strength of current i, the strength of field at the centre of the circle is r Hence, since r is length and the dimensions of H are given above. Since quantity of electricity is equal to it we have only to multiply the dimensions of ihy[T] to get 182 GENERAL PHYSICS. Potential difference between two points is defined as the work required to move unit charge from one point to the other. Work is the product of a force by a distance, hence if V is potential or e.m.f. V = [MLT-^] [L] ^ [li^LJ'ir^ = [M''L*T-^n!'] Resistance, R, is by Ohm's law equal to e.m.f. divided by current, hence i? = [L7^V] Capacity, C, is equal to quantity divided by e.m.f., hence Thus from the definition of any unit in a system the dimensions of that imit may be written. Both the e.m. and the e.s. units are absolute units based on the centimetre, gram, and second as units of length, mass, and time. The value of a few of the more common practical units in terms of e.m. and e.s. units is given in the table below. e.m.f. Current . . . Quantity. . Resistance Inductance Capacity. . Work or energy Power Practical unit. Volt Ampere. Coulomb Ohm {Henry Millihenry f Farad l Microfarad 1 Volt -Coulomb = 1 joule Ampere —Volt = 1 watt 10» 10-' 10-» 10» 10' 10« 10-" 10-« 108X10-1 = 10' ergs 10-iX10« = 10' ergs per sec. 510-« 3(10)» 3(10)« 1 (10)-" 4(10)-» 4(10)-» 9(10)" 9(10)5 J10->X3(10)» = 10' ergs 3(10)»XJ(10)-' 10' ergs per sec. Sym- bol. Derivation of number of e.s. units. 10»-!-3(10)w 10-»X3(10)" 10-' X3(10)" R = -L s £^1000 '^ V C-!-(10)« If a given quantity of electricity is measured in e.s. units and also in e.m. units, it has been shown in § 48 that the ratio of the number of e.s. units to the number of e.m. units is very nearly SCIO)!". ELECTROMAGNETIC INDUCTION. 183 If, now, we take the ratio of the dimensions of quantity in the two systems, regarding ju and k as unity in air, we have L*M* ~ T Distance per unit of time is velocity v and this would suggest the equation The velocity of light is 3(10)1" cm./^^^ If we equate the dimensions of quantity in the two systems and include ii and k we have . L 1 ^ Vkn = v Experimental determinations of the value of this ratio also give very nearly 3(10)"". Theoretical considerations of such relations as these led Max- well in 1873 to suggest that light and all other ether radiations are electromagnetic in character. Many experiments since that time strongly confirm Maxwell's theory. This subject is more fully discussed in the next chapter. Problems 1. A straight wire kept in a horizontal position in an east and west direc- tion is 15 m. long. It is let fall from a height of 50 m. at a place where the horizontal intensity of the earth's magnetism is .2 dyne. What is the maxi- mum e.m.f. generated in the wire and which end of the wire has the higher potential? 2. A rectangular coil of wire 20X50 cm. contains ten turns and makes 400 revolutions per second on its longer axis, in a magnetic field perpendicular to the axis. The strength of the field is 10,000 lines per square centimetre. Find the maximum e.m.f., the average e.m.f., and the factor which multiplied by the former will give the latter. 3. If the inductance in a circuit is 70 millihenrys, what is the consequent reactance of an alternating current of 60 cycles? 4. If the virtual value of an alternating current is 17.675 amperes, what is its actual value at phase 210°? 184 GENERAL PHYSICS. 5. An alternating current whose maximum value is 40 amperes lags 25° behind the voltage. What is the value of the current at the instant when the voltage is in the 80° phase? 6. If a virtual current of 25 amperes lags 40° behind a virtual e.tn.f. of 110 volts,-what is the power? 7. An electric line having a resistance of 20 ohms carries a current of 50 amperes under a pressure from the dynamo of 2000 volts. How much power will be saved for useful work at the end of the line by raismg the voltage to 60,000 volts? 8. The coil of an earth inductor is 20 cm. in diameter and contains 100 turns of wire. The terminals of the coil are attached to a ballistic galvanometer and the resistance of the entire circuit is 100 ohms. When the coil is placed so that its axis is vertical and turned rapidly through 180°, the galvanometer deflection as read on the scale is 20 cm. It is then fotmd that a condenser of 3 microfarads capacity, charged by a battery of 2 volts, will cause the same deflection. What is the strength of the horizontal component of the earth's magnetism? Ans. 1. 9.35 millivolts. The east end. 2. 2513.28 volts. 1600 volts. .636. 3. 26.39 ohms. 4. 12.5 amperes. 5. 32.76 amperes. 6. 2106.5 watts. 7. 49,920 watts. 8. .19 dyne. CHAPTER XII ELECTROMAGNETIC WAVES 124. Electric Oscillations. — It has already been shown that an electric charge is surrounded by an electrostatic field and that points in this field may differ in potential by a certain number of volts. When the charge is set in motion its field is carried with it and also there is set up a magnetic field at right angles to the direction of the motion. The strength of this magnetic field depends on the rate of flow of electric charges, i.e., on the amperage. If the direction in which the charge moves is suddenly reversed, the direction of motion of the electrostatic field is reversed and also the direction of the magnetic lines of force. If these reversals are repeated regularly and at short intervals, the ether surrounding the charge is subjected to alternate strains in one direction and then in the other, thus causing ether waves which may move out in aU directions from the point of disturbance. The length of such waves is determined by the rapidity of the alternations or, as here called, the oscillations of the current. The greater the nimiber of oscillations the shorter the wave, just as the greater the number of vibrations of a tuning fork the higher the pitch. 125. Length of Ether Waves. — Since an electromagnetic pulse moves with a velocity of about 3(10) "• cm. per second, if the num- ber of oscillations are comparatively few the wave will be of enormous length. The ordinary alternating current of 60 cycles per second causes waves several thousands of miles long. It is possible to construct generators having many poles and then rotate the armature with such speed as to produce a frequency of 10,000 or 15,000 cycles per second. The length of the wave is thereby proportionately reduced. The best method of producing oscillations of great frequency is by the discharge of a condenser. The natvu-e of an oscillatory discharge has already been described in § 116. The circuit through which the discharge takes place, however, may be such as to pre- vent oscillations and the discharge is then simply a diminishing 185 186 GENERAL PHYSICS. one. This is analogous to the vibrations of an elastic rod. If such a rod is clamped at one end, then if the free end is pulled to one side and released, vibrations will continue for some time. This, however, will be the case only when the rod meets with little resistance as in air. If the rod is immersed in some viscous fluid, when released from its strained condition it will slowly return to a position of rest without vibration. Lord Kelvin in 1853 published a mathematical consideration of this subject and showed that the kind of discharge which will occur depends on the resistance R, the inductance L, and the capacity C. If i2<2^ the discharge will be oscillatory and the frequency n is given by «=^V^ LC § (1^^> If the second quantity under the radical is so small in comparison with the first that it may be neglected, the equation becomes "=^Vzj LC (!*«> and the period P, which is the reciprocal of the nimiber of vibra- tions per second, is P = 2irVLC (149) Equation (149) is often called the fundamental equation of wire- less telegraphy for L and C are very important factors in its operation. If L is given in henrys and C in farads, P is the period in seconds. A wave-length X is equal to the product of period by the velocity V with which the wave travels. Hence \=Pv=2irvVLC (150) 126. Electric Resonance. — ^When an electric circuit is such that oscillations in it will produce waves of a certain length, then oscillations may be produced in this circuit by waves of the same ELECTROMAGNETIC WAVES. 187 length from another oscillator. This is the principle of electric resonance and is analogous to a similar phenomenon in sound where one tuning fork will respond to another of the same pitch. Illustrating this principle an experiment was devised by Lodge, the apparatus for which is shown in Fig. 129. Two Leyden jars of about equal capacity are used. One, A, is continuously charged by means of an induction coil or influence machine so that sparks will pass across the short air gap at c. The other jar is provided with conducting rods connecting with the inner and outer coatings and extending out as shown. If the two jars be placed a short distance apart with the conducting frames parallel, whenever a spark passes at c there will be a corresponding surging of electricity in the neighboring conductor provided it is " tuned " for reson- QO Fig. 129. ance. This timing may be effected by moving the shder s until the proper length of circuit is secured. To make the resonance apparent, a wire clip n is placed over the edge of the jar so as to touch the inner coating but leave a short spark gap between it and the outer coating. Some of the electricity wiU then leap across this short gap and will be seen as a bright spark whenever a spark cxicurs at c. 127. Experiments of Hertz. — ^As early as 1862 Maxwell showed from theoretical considerations that an oscillatory discharge of a condenser causes ether waves which travel out with finite velocity and that this velocity may be expressed by 1 v = as already shown in § 123. V kyL 188 GENERAL PHYSICS. ©■ -0=0 and consequently i is equal to r. 142. Curvature. — The curvature at any point is the rate of departure of a curved line from a tangent to the curve at that point. Curvature is conveniently expressed in terms of the angle formed by the tangents drawn through the extremities of unit length of arc, as 0', Fig. 147. It is seen that 9' is equal to d, the angle at the centre of the circle. If 5 is length of unit arc, R the radius of the circle, and is measured in radians, then s^ Rd e 1 - = - (164) The first term of this equation is, by definition, the curvature. Hence if C stands for curvature, C'^ (165) 210 GENERAL PHYSICS. Curvature may therefore be expressed as the reciprocal of the radius. If an angle larger than B be taken, the arc s will be increased in the same proportion and 6/s will still equal 1/R, i.e., the curvature does not change on this account. If R is made longer, say oc, twice oa, then the arc s' is twice as long as s, hence the curvature of s' is one-half as great as that of s. In gen- eral, then, ctu'vature varies inversely as the Fia. 148. radiusandfromourdefinitionisequaltol/i?. In the particular case shown in the figure where 5 is assumed to be of unit length, say 1 cm., equation (164) may be written but whatever the angle or length of arc, it is seen from the reason- ing above that angle _ 1 arc R The sagitta of an arc may also be taken as a measure of curva- ture. In Fig. 148 the sagitta is the line s drawn from the middle of the chord mn to the vertex of the arc. It is readily seen from the construction of the figure that 2r-s where a is one-half the length of the chord. If is small, as it should be in the following discussions (see Fig. 158), 5 in the denominator may be neglected in comparison with 2r, hence The sagitta is therefore inversely proportional to the radius of curvature, i.e., directly proportional to curvature. 143. Reflection of a Spherical Wave from a Plane Surface.— Let 0, Fig. 149, be a point from which light waves emanate, and let mm' be a reflecting surface. When a wave-front reaches the mirror, as it will first at d, this point becomes a centre of disturb- ELECTROMAGNETIC WAVES. 211 ance and sends out a spherical wave in the opposite direction. So also for points in the wave to either side of d when they are inci- dent on the mirror. Let a and b be two elementary areas on an advancing wave. The total surface of the spherical wave may be regarded as com- posed of such areas, each of which is a plane. Now when a or 6 or any other of the infinite number of these small planes reaches the mirror there will be reflection in the manner described for Fig. 146 where it is shown that when a, for example, is reflected it will retain the same inclination to the mirror as it had before incidence but will be oppositely directed — i.e., the end of a which first reaches the mirror will after reflection be farthest from it. Consequently when all these small plane waves have been reflected, the spherical wave-front will be reversed and the ciu-ve sds', which wotild have advanced to c if the mirror had not been there, now becomes ses'. »n Fia. 150. The reflected wave-front will therefore have the same curva- ture as the incident wave, and an observer looking into the mirror will receive spherical waves as if emanating from o', for this is the centre of the reflected curves. What is seen at o' is called a virtual image of o, for the waves appear to have originated from o' rather than from o. It is also evident, from the fact that the curvatures of the mcident and reflected waves are the same, that the image is as far back of a plane mirror as an object is in front of it. 144. Reflection from a Curved Surface. — Consider first a con- cave reflector and let it be a spherical mirror with radius R — ^the distance mc in Fig. 150. The ctirvature of the mirror is then 1/R. The point c is the centre of curvature, and the line mn drawn through c to the mirror is called the principal axis. 212 GENERAL PHYSICS. Let the line ab represent a plane wave perpendicular to this axis and approaching the mirror. The ends a and b will first reach the mirror and will be reflected while the middle portion of the wave-front will be reflected last. This will give to the wave-front a curved form having its centre, say, at /. This point / is called the principal focus of the mirror. We may therefore define a principal focus as a point on the principal axis to which a plane wave, perpendicular to this axis, will converge after re- flection, or, since rays are perpendicular to a wave-front, as the point at which rays parallel to the principal axis meet after re- flection. Let the distance from w to / be denoted by F. Then at the instant the whole of ab has been reflected and is ready to start back, the curvature of the wave-front is 1/F. The concave mirror will therefore change the wave-front from one whose curvature is zero to one whose curvature is 1/F. This is the change in curva- ture which the mirror is capable of impressing on a plane wave- front and is sometimes called the focal power of the mirror. Let the incident wave ab, Fig. 151, be spherical with a centre at o. It is plain that when a and b reach the mirror the central part of the wave is nearer m than it would be if the wave were plane, consequently the curvatiu-e produced by reflection will be less than 1/F by the amount of curvature which the wave possessed before incidence. If represents the distance from m to o, then the curvature of the incident wave is 1/0, and if after reflection the centre of the wave is at i and we denote by I the distance from m to i, the curvature of the reflected wave is l/I, Hence 1 1 1_ F 0~ I If the source, o, of light waves is at any point between c and an infinite distance from the concave mirror, a real image i will be formed between / and c, for when o is at an infinite distance » will be at /, as shown in Fig. 151 ; and when o is at c the curvature of the incident wave is the same as that of the mirror, and so i will also be at c. ELECTROMAGNETIC WAVES. 213 If o is at any point between/ and c, the curvature of the inci- dent waves will be greater than that of the mirror, and the middle portion of the wave will be incident first. The distance of the ends a and h, however, from the mirror is not so great as it would be if / were the source of the waves, and the curvature impressed by the mirror is 1/F less the curvature which the incident wave had. Hence equation (166) can, as before, be used to find the position of the image when that of the object is known, or the position of the object when the distance of the image from the mirror is known, the focal distance being given in both cases. When the object, o, is between/ and c. Fig. 152, a real image, i, will be at some point beyond c, for when o is at c the image will be there also, and when o is at / the wave will be made plane by reflection and so i will be at an infinite distance. Fig. 151. Fic. 152. If is between / and m, the curvature of the incident wave is greater than that from /, consequently the ends a and b must move so far before they are reflected that the point m in the wave gets ahead of other portions of the reflected wave-front. The change in curvature is sufficient not only to make the wave plane but to actually reverse it, so that the image * is virtual and appears on the opposite side of the mirror. Equation (166) will under these conditions give a negative answer as it should. Thus, sup- pose the focal length of the mirror is 12 cm. and the source of light waves is 3 cm. from the concave side of the mirror, then 1 + 1 = 1- 3 7 12 .-.7= -4 This means that the image of the point o is virtual and is located 4 cm. back of the mirror. 214 GENERAL PHYSICS. It may be shown by equation (166) that the principal focus / is half way between the mirror and its centre of curvature c. Let R be the distance from m to c — i.e., the radius of the sphere of which the mirror is a part — then 1/R is the curvature of the mirror. Now when the source of light is placed at c, both and I are equal to R. Hence for this condition l + i = l R R F •■•-! Another way of stating the same fact is that since F~ R the curvature impressed by a concave mirror on a plane wave is twice the curvature of the mirror. Fig. 153. Fig. 154. When a plane wave is incident on a convex mirror, the middle portion of the wave is reflected first and the ends a and b, Fig. 154, are then at a distance from the mirror equal to the sagitta of the arc in case the wave were incident on the concave side, conse- quently the change in curvature will be the same as if the plane wave were incident on the concave side, i.e., twice the curvature of the mirror. Hence the image will be at / as before, i.e., the principal focus, /, is again midway between c and m. If, now, a spherical wave originating in o is incident at w on a convex mirror. Fig. 155, points a and b will have to move forward not only to the position of the plane wave, but, in addition, what- ever movement the plane wave went through before all portions ELECTROMAGNETIC WAVES. 215 of it were reflected. The change of curvature in a plane wave is from zero to 1/F, hence, using for distances the same letters as above, i + i = i F-0 I The image in a convex mirror will therefore always be virtual and will be located between / and m. In cases described above where the image is real, o and i are called conjugate points or conjugate foci, for if the soturce of light is at either one the image will be at the other. To make a drawing which will show the position, size, and nature of an image in a spherical mirror it is more convenient to apply the principles of geometrical optics and use rays instead Fia. 155. Fio. 156. of wave-fronts. Thus in Fig. 156 let oo' be a luminous object, every point of which is a source of light waves. It will be suflScient in this case to locate the image of the two ends of oo'. To locate the image of o, draw two rays, op parallel to the principal axis and on perpendicular to the mirror. After reflection op will pass through / and on will return on its incident path. The two will meet at i, hence * is the image of o. In the same manner draw two rays from o'. These wiU meet at *'. Hence *' is the image of o'. The image of any other point of oo' may be located in a similar manner. All will be found to be at some point between i and »'. The image in this particular case is thus found to be real, inverted, and smaller than the object, and since points midway on oo' are farther from the mirror than the extremities, the line W will be slightly convex toward the mirror. 216 GENERAL PHYSICS. In the same manner images may be foimd when oo' is between c and /, / and m, or on the convex side of the mirror. 145. Spherical Aberration. Caustics. — In previous discussions it was assumed that a spherical wave-front would still be spherical after reflection from a spherical mirror. This, however, is not the case, for the farther the point of inci- , dence, -p. Fig. 156, is from the vertex, m of the mirror, the more nearly will / ap- proach m. Let a ray parallel to the prin- cipal axis be incident at p. Fig. 157. The j,jg J j7 reflected ray will cross the principal axis at some point /. Let cp, the radius of curvature of the mirror, be designated by R, cj by a, and jp by h. Angle = angle r = angle » hc=a cos hp=b COST .'. R=2acos0 and a = - (168) 2 cos « Equation (168) shows that as 6 becomes greater its cosine becomes less and therefore a will be greater, i.e., f will be farther from the centre c. But if S is so small that its cosine may be regarded as unity, R For practical purposes, therefore, we say that all the rays from a point are focused at a common point, provided only those rays which are close to the principal axis are admitted to the mirror. Hence only about 15 or 20 degrees of the aperture of such a mirror should be used. If a large number of rays be drawn to a spherical mirror of large aperture, the reflected rays will intersect as shown in Fig. 158, and a curve drawn from / tangent to all these rays is called a caustic curve, foa and fo'b, the cusp of which is at the principal focus /. This effect is greatest when the incident wave is plane, ELECTROMAGNETIC WAVES. 217 as in Pig. 157, and decreases as the source of light approaches the mirror. When c is the source, the reflected wave-front is spherical. A caustic curve may be plainly observed by letting sunKght or lamplight fall on the concave surface of a cup or glass partly filled with milk or turbid water. 146. Parabolic Mirror. — ^A concave mirror whose surface is a paraboloid of revolution will bring plane waves to an exact focus — i.e., without spherical aberration — and, conversely, if the source of light is at the focus, the reflected wave will be plane. . ■ / / a. Jr * / \ \ ^ Fio. 158. Fia. 159. Let the curve aoh. Pig. 159, be a section of the paraboloid of which/ is the focus and cd the directrix. Any point on a parabola is equally distant from the focus and the directrix, as af=ac, bf=bd, etc. Let a point of light be placed at /. If the mirror had not been there a spherical wave-front would at a certain instant have reached mpn, but the ray fa was reflected to ae, fb to bh, and so for other rays. At the instant the wave would have reached m it will now reach e, ae being made equal to am. Likewise hh is made equal to bn and so for other rays. Hence ce=fm and dh =fn But fm =fn .'.ce=dh 218 GENERAL PHYSICS. The line eh, which is the position of the wave-front after reflection, is therefore parallel to cd, and so the reflected wave- front is plane. It is evident, then, that when a plane wave is incident on such a mirror it wiU be made to converge to / without aberration. Mirrors of this kind are extensively used as reflectors of head- lights and search-lights, for the light of an arc lamp placed at the focus will be projected as a powerful beam. A valuable use of paraboloid reflectors is found in objectives of reflecting telescopes. Such mirrors may be made of polished speculum metal composed of an alloy of copper and tin, or of silvered glass. A very perfect mirror of this kind may be formed by rotating a pan of pure mercury. If the rotation can be kept steady and uniform the surface of the mercury will be depressed at the centre and raised at the sides of the pan, thus forming a paraboloid as long as the rotation is continued. Problems 1. Two plane mirrors are placed at an angle of 60° to one another with an object midway between them. By drawing rays find the position of th« images. 2. If a plane wave is focused at a point 10 cm. from a concave mirror, what must be the position of a luminous point that its image may be 12 cm. from the same mirror? 3. Find the position of a pair of conjugate foci of a concave mirror whose radius of curvature is 40 cm. 4. By use of rays make a drawing to show the position and nature of the image of a candle flame in a convex mirror. 5. A ray of light parallel to the principal axis is incident at a point 60° from the vertex of a concave mirror whose radius of curvature is 50 cm. How far from the principal focus will the reflected ray intersect the principal axis? 6. An object 2 cm. long is placed 10 cm. in front of a concave mirror whose radius of curvature is 12 cm. What will be the length of the image? 7. A beam of light is reflected from a plane mirror to a scale 2 m. distant from the mirror. If the mirror is turned through .2 radian, how many centi- metres will the spot of light move on the scale? Ans. 1. Five images. 2. 60 cm. 3. e.g., 25 and 100. 4. Erect and virtual. 5. 25 cm. 6. 3 cm. 7. 80 cm. ELECTROMAGNETIC WAVES. 219 147. Refraction. — ^When waves of light pass from one medium to another of different density, part of the light is, as a rule, reflected and another part enters the second medium where it may be at once absorbed, or, if the medium is transparent, is transmitted with a change in velocity and direction. The change in direction which occurs when hght passes from one medium to another is called refraction. Let AB be a plane wave-front incident on a plate of glass G, Fig. 160. The greater part of the light will enter the glass and pass on through it. \ Let / be the time y A required for the wave ^ ^ to pass from A to C v ^ and V\ the velocity in ^ air or other rare me- dium. When one end of the wave is at A the other end B is just entering the glass. The point B then be- comes a centre of dis- turbance and sends out waves with veloc- ity V2 in the glass. ^"■^*°- . When A has arrived at C, B will have moved a certain distance BD, less than A C, for the velocity is less in a denser medium. Hence AC vit _ vi BD Vit Vi As the successive points in AB reach the glass each becomes a centre of disturbance. When A has arrived at u, for example, e is at and, as shown above, uC \ \> ^ c \ X- ^ '\ - ^^ ^ i-V G and om sC_ fn Vl Vl Hence BD -.om :im=BC -.oC -.pC 220 GENERAL PHYSICS. Consequently a line from C to D is tangent to all the spherical waves, and so is the position of the wave-front in the glass. Let normals to the surface be drawn at C and B. Then * is called the angle of incidence and r the angle of refraction. The angle i is equal to the angle ABC and r to BCD. Hence AC=BC&mi and BD=BCsmr Hence AC BD sm» smr = n (169) where n is called the index of refraction, defined as the ratio of the sine of the angle of incidence to the sine of the angle of refrac- tion. Refraction, therefore, is caused by a change in the velocity of the wave-front when passing from one medium to another of different optical density. Fio. 162. If the wave-front is parallel to the interface separating the two media then it is evident from Fig. 159 that, although the velocity is different in the second medium, the direction is not changed. When the sides of a second medium are parallel, as in case of plate glass G, Fig. 161, the direction of an incident wave will be deflected toward the normal on entering the glass, but, on emerging from the glass to air, will be deflected an equal amount from the normal so that the direction is the same as before incidence. This is apparent from Fig. 161, for the wave traverses a distance BE or CF in glass in the same time as it moves over .A C in air. When the wave at E is just emerging into air, the point C must still ELECTROMAGNETIC WAVES. 221 traverse a distance CF in glass. Hence EH will be equal in length to AC and HF is parallel to BA. Consequently ab lies in the same direction as cd. There is, however, a lateral displacement bf, the amount of which depends onthe refractive power of the medium and the distance between the two sides. This may be calculated when the angle of incidence and the index of refraction are known, for 6/= 6c sin (t— r) and be = cosr Let the thickness of the glass, bm, be denoted by d, then sin {i—r) bf=d- cos r When i and /x are given, r can be found from equation (169). This phenomenon may be observed by holding a , pencil back of a piece of heavy glass. The part seen obliquely through the glass will appear to be out of line with that above the edge of the glass. 148. Refraction of a Spherical Wave at a Plane Surface. — Let a vessel W, Fig. 162, be filled with a transparent Uquid and let AB be the surface separating the hquid from the air above. Let c be the origin of spherical Ught waves. In a certain time a wave would have moved through the Hquid to a with AaB as the wave-front, but at o it began to emerge from the liquid into a less dense medium and hence to move faster, so that in the same time that the wave would have advanced from o to o in water it moved from o to 6 in air. Let the velocity in air be vi and in water vz, then Likewise where y, is the index of refraction from air to water or other medium denser than air. Ob oa - ^ 111 = M .ob = M ■ oa o'V = M • o'a! 222 GENERAL PHYSICS. The arc AbB is not spherical but when the arc is short, as that used in vision, it may be regarded as spherical. Since the curvature is increased, the centre of the wave will now be at a point c' instead of c, and an observer looking down into the water will see c', an image of c. Since the sagittas of the arcs are inversely proportional to the radii of curvature (see § 142), £^ = iA = ^==^ (170) oa c'A V2 When an observer looks vertically down into a transparent medium both A and B will be very close to o, for the area of cross section of the pencil of light that can enter the pupil of the eye is very small. The arcs and their sagittse will consequently be very short and may be disregarded in comparison with the radii. Hence under these conditions we may put cA =co and c'A =c'o. Then ob CO Vi /^'^^\ oa CO Vi Since the index of refraction from air to water is f , CO 4 . , 3 . . C O = —CO 4 i.e., the vertical depth of water appears to be only three-fourths of what it actually is. These principles may be applied in finding the relative velocity of light in air and other media, i.e., the index of refraction from air to other media. Suppose a microscope is focused on c. Fig. 162, when the vessel is empty. Then let the vessel be filled with a liquid to a height d above c. The microscope must then be raised a distance cc' to bring it into focus again. Hence from equation (171) CO d Vi c'o d—cc' Vi 149. Relative and Absolute Index of Refraction. — The ratio of the velocity of light in vacuum to the velocity in another sub- ELECTROMAGNETIC WAVES. 223 stance is called the absolute index of refraction. The ratio of the velocity of light in one substance to that in another is called the relative index. The absolute index for air under standard conditions is 1.00029, i.e., the velocity of light in vacuum is only 1.00029 times as great as in air. Hence the term index of refraction of a substance ordi- narily refers to the relative index from air to a substance. The absolute index may be foimd by multiplying this relative index by the above number. The relative index for any two substances is found by taking the ratio of the relative indices of these substances to a third substance, usually air. For example, it is known that the index for air to glass is -g-, and for air to water is -g-. Hence, expressing the index as a ratio of velocities, velocity in air 3 velocity in glass 2 velocity in air 4 and — ; — : — ; = — velocity m water 3 Dividing the former by the latter, velocity in water 9 velocity in glass 8 Hence we may write 9 8 M,. = - Hence the relative index of any meditim in reference to a second medium may be found by dividing the iadex of the second medium in reference to air or a vacuum by that of the first medium. 150. Critical Angle. Total Reflection. — ^A critical angle of re- fraction is an angle of such a value that the refracted ray is parallel to the interface which separates the two media. Let o. Fig. 163, be a point source of light waves in a dense medium such as water or glass. The disturbance propagated along the ray oa will emerge into air without bending. Other rays such as of will be refracted more and more from the normal 224 GENERAL PHYSICS. as the angle r becomes greater. At a certain angle C called the critical angle, the ray oc is refracted to ch, i.e., the angle of refrac- tion is 90". (The angle made by a ray with the normal will, as a rule, be called r in the denser medium and i in the rarer medium whatever may be the direction of the ray.) Whenever i is 90°, r is the critical angle. When a portion of the wave, such as moves along the ray oe, reaches the surface, the deflecting effect resulting from an attempt to pass into a rarer medium turns that part of the wave over and it turns back into the first medium according to the law of reflection. Since none of the light passes into the second meditmi, this is called total reflection. The critical angle of any medium in its relation to air may readily be calculated when the index of refraction, n, is known. Thus, sm I -. — =/i smr sin* = sin 90° = ! r = C— critical angle . . sm C=— M The index for water, for example, is 1.33, hence sm C=-—— 1.33 and C=48''36' (172) ELECTROMAGNETIC WAVES. 225 The greater the index of refraction of any substance is, the smaller the critical angle; e.g., the index for diamond is about 2.47, hence its critical angle is about 24° 26'. The brilliancy of a diamond is therefore due to the fact that a considerable portion of the Ught which enters it is totally reflected. Many of the best optical instruments, which require that light be reflected at any point in its path, are provided with total re- flecting prisms instead of mirrors. Let ABC, Fig. 164, be a cross section of a prism, right angled at A and the side AB equal to AC. A ray of hght entering the prism at right angles to AB will pass straight through to c, mak- ing an angle of incidence »=45°. Fio. 164. Fio. 165. If the prism is made of either crown glass or flint glass, whose critical angles are 43° 2' and 37° 34' respectively, the light will be totally reflected at c and wiU pass out at right angles to A C. 151. Refraction by a Prism. — ^An optical prism is usually made of glass or some substance denser than air. Since the sides are not parallel, a beam of light entering the prism on one side will emerge from the other side in a changed direction, the re- fraction being toward the base of the prism. Let a beam abed be passed through the prism ABC, Fig. 165. The emergent beam will deviate from the path of the incident beam by the angle D. The exterior angle D is equal to 0-|- <^, but e=H—ri and =ii—rt .'. D =i\—r\-\rit—rt =*i+ig-(n+rz) Let the angle at the vertex of the prism be denoted by A. 226 GENERAL PHYSICS. Then A is the supplement of the angle hfc. n+rg is supplementary to the same angle. Hence A=ri+r2 and D=ii+i2-A (173) Both theory and experiment show that the value of D is least when ii =it. Under this condition D = 2i-A or t = —^ (174) Also ri=r2 .•.r=| (175) Hence the index of refraction of the material of the prism is . D+A , . sm — - — sm* 2 ,,_„^ A* = -T— = 7- (176) sm r .A sm- By measuring the angle A and the angle D when it is minimtmi, the value of m may readily be calculated by equation (176). 152. Spectrometer. — One important use of a spectrometer is the measurement of A and D of equation (176). The instnunent consists essentially of a collimator C, Fig. 166, a telescope T, and a graduated circle on which the positions of C and T may be read. The collimator is provided with an adjustable slit ^ for the admis- sion of waves of light and a lens I which makes the wave-front plane, i.e., makes the rays parallel. An image of the slit is seen in the telescope. A prism-holder h is located in the centre of the circle. The telescope may be tiomed to any position around the circle. The details of manipulation may be found in laboratory manuals. • One method of finding the angle of a prism is to turn the tele- scope to a position Ti, Fig. 167, where it is perpendicular to the side AB, and note the scale reading. Then the telescope is turned to a position T^, perpendicular to the side AC, or the table carry- ing the prism is turned so as to bring the side A C perpendicular ELECTROMAGNETIC WAVES. 227 to the telescope in position Ti. The number of degrees between these two positions is evidently the angle which is the supple- ment of A. Hence Fio. 166. In this method the collimator is not used, but to aid in setting the telescope exactly in positions Ti and T^ it is provided with a Gauss eye-piece. This, as shown in Figs. 166 and 168, is a short n X r" Tia. 167. e FiQ. 168. tube with an opening o in one side. Light admitted at o falls on a plate of glass set at an angle of 45°. The light is thus directed upon the spider lines x and on through the telescope to the face of the prism where it is reflected. An observer at e may then see 228 GENERAL PHYSICS. the spider lines directly and also their image. When they exactly coincide, the telescope must be at right angled to the reflecting surface. In a second method of measuring A, use is made of both collimator and telescope. Let C be a beam of parallel rays from the collimator falling upon the edge of the prism as shown (Fig. 169). Part of the rays fall on the face AB and are reflected to Ti while other rays fall on face AD and are reflected to Tt. Fia. 169. Fia. 170. If the telescope be turned to position Ti an image of the slit of the collimator will be seen. The telescope is then turned to the position Ti where the image of the slit will again be seen. The number of degrees through which the telescope is moved between these two positions is twice the angle A. This will be apparent if we consider that ai=bi ai = ci A Also A = bi+ci = A bi+Ci = A bi+ci+bi+Ci The spectrometer is also used in finding D, the angle of mini- mum deviation. A source of monochromatic light such as a sodium flame is ELECTROMAGNETIC WAVES. 229 placed in front of the slit of the collimator and a beam C, Fig. 170, directed against one side of the prism. Refraction will occur, as shown in Fig. 165, and the beam will emerge from the prism in the direction Ti where it is received in the telescope. By turning the prism table and following the image of the slit with the tele- scope, a position will be found where, although the rotation of the prism is continued, the motion of the image will be reversed. It is this position where the image stops that is sought, for then the incident and emergent beams of light make equal angles with their respective faces of the prism, and any increase or decrease of the angle of incidence will cause a greater deviation of the direction of T"j from the direction of C. Let 7\ be the position of the tele- scope when the cross hairs are centred on the slit at the moment the image is ready to reverse its direction. If then the prism is removed and the telescope is turned to position Tt, the slit is observed directly in line with the collimator where the deviation is zero. The angular distance between T\ and Tz is therefore the angle of minimum deviation. 153. Refraction of Spherical Waves at a Spherical Surface. — The change of curvature which a medium with curved face may impress on a wave-front depends not only on the curvature of the face on which the Ught is incident but also on the refractive index of the medium. o /[Y Or m ia ''d \\ *= f b ^ikf +++*-- Fig. 171. Fio. 172. Let a body G, Fig. 171, denser than air, say glass, have a curved surface smn. Let ah be a plane wave-front, entering the glass. The wave would in a certain time in air reach the position of the dotted line a'b', but when any part of ab enters the glass it moves more slowly and falls behind other portions which are still in air. In the same time that the wave would move from m to o' in air it moves from w to o in glass, and since index of refrac- tion, fi, is the ratio of the velocities of light in the two media under consideration, n .mo is called the air equivalent of mo. 230 GENERAL PHYSICS. After the wave enters the glass its centre of curvature is /, called the principal focus. Let F be the distance from / to o, then tt is the curvature-producing power of glass in the form shown in the figure. Any increase in the density or curvature of the glass increases the curvature-producing power. If a spherical wave is refracted at a convex surface, the inci- dent wave ab. Fig. 172, must first lose its curvature yr, and what then remains of the power of the glass to produce a curvature -= t is used in producing the curve a'o'b' with a curvature -=-, where O, F, and I are the respective distances of o, /, and i to their wave- fronts at the surface of the glass, the first before refraction and the others just after. A convex lens always reverses or tends to reverse' a spherical wave-front which emanates from a point, as o. Hence the equa- tion for this kind of a surface is l_i = l F 0~ I 1,1 1 or — = — 1 F If the surface were concave instead of convex it is plain from what is said above that refraction tends to increase the curvature which the incident wave already has. Hence the equation for this condition is i + i = i F I 1 1 1 or = — I F 154. Lenses. — ^There are two general classes of lenses, convex and concave. The former always tend to produce a converging wave-front and the latter a diverging one. The double-convex lens Xi, Fig. 173, the plano-convex Xi, and the concavo-convex Xs are the common forms of convex lenses. The double-concave Vi, the plano-concave Vi, and the ELECTROMAGNETIC WAVES. 231 convexo-concave F3 are the common forms of concave lenses. The form X3 is both concave and convex, but the convex side has the greater curvature, and so the sum of the effects is that of a converging lens. Likewise the sum of the effects of V3 is that of a diverging lens. To deduce a general lens formula involving index of refraction, curvature, and conjugate fod, let a double-convex lens shown in Fig. 174, one face of which has a 1 , , , 1 and the other, be curvature _ .-^xv^ ^^^ ^u^v,*, „ , placed in the path of spherical waves originating at o. After the waves pass through the lens they converge to i, so that o and i are conjugate foci. While the wave ab is passing ^ ^.^^^^ through the lens from w to m, a por- ^"^ ~~^^ /^a tion of the wave at a moves through the air a distance ap+pa'. The arc pdp' is described with radius op, and ^'^' '••- -^ """^ y pvp' with radius ip. Hence ap+pa' = md +vn =sm+sd+vs-\-5n The air equivalent of mn is mn . n, i.e., this quantity is numer- ically equal to ap+pa'. Hence sd+vs+sm-i-sn = ii{sn+sm) (177) All these distances are measured from the common chord, pp', of the several arcs and so are the sagittas f,o. 173. of these arcs. Now, putting the reciprocals of the radii in place of the sagittse as a measure of curvature, and using 0, as before, for the object distance and I for the image distance, we have ^+7+i;+i:=''(i;+i;) ^ + 7=(M-l)(^+^) (178) 232 GENERAL PHYSICS. This is a general equation for a double-convex lens within the limits of error involved in assuming that curvature is proportional to sagittae of arcs. (See § 142.) For a thin lens the equation may be regarded as correct, for then the arcs are a short portion of the entire circumference of the circle. It is also assumed that the wave-front after refraction is spherical, an assumption which will introduce no sensible error provided only the middle portion of the lens is used. Fig. 174. If the object o, Fig. 174, is at an infinite distance from the lens, then 0, the radius of curvature of the wave ab, is equal to 00 and the curvature is zero. , Under this condition equation (178) becomes i'^-^a^) (179) But the point where an image is formed when the incident rays are parallel to the principal axis is called the principal focus of the lens, and the distance of this point from the lens is called the focal length. Calling this distance F, F~ I and, from equation 179, When the radii of curvature of the faces of a lens, and the index of refraction, are known, focal length can be calculated by equation (180). ELECTROMAGNETIC WAVES. 233 The general equation for lenses may therefore be written 1 + 1 = i (181) O^ I F where is the object distance, I the image distance, and F the focal distance. As the object approaches the lens, the image moves farther away, and so there mtist be a point where they are at equal dis- tance from the lens. Hence if =1, equation (181) shows that each is 2F distant from the lens. When the object is at the principal focus, equation (181) shows that i + i = l F^ I F 1 . . — = zero and I = 00 i.e., the diverging wave-front will be made plane. If the object is between the principal focus and the lens, the wave after passing the lens will still be divergent, but less so. Hence / will be negative, i.e., the image will be virtual and will appear on the same side as the object. If the lens is plano-convex, Xt, Fig. 173, the curvature of one face is zero and (178) becomes If the lens is concavo-convex, X3, Pig. 173, then (178) becomes for each face tends to neutralize the bending effect of the other, their resultant effect being the difference. In case of concave lenses it is evident that all plane or divergent waves will be made more divergent, so that both the focal and 234 GENERAL PHYSICS. the image distances are negative, i.e., both the image and the principal focus are virtual. Hence, for this condition, equation (178) must be written h-1-^-^ii-i) , .. ^. . . . , (184) 1-1 = i (185) 155. Optical Centre. — The optical centre of a lens is that point through which, if rays are drawn, they will not be changed in direction. In double-convex and double-concave lenses having equal cur- vatiu-e on the two sides, the optical centre is at the centre of the lens. This point, for any spherical lens, may be found by drawing parallel radii as cm and c'm'. Fig. 175. These are perpendicular to the surface of the lens, and tangents at m and m' are parallel. Hence a ray of light drawn through mm' will not be bent from its course. Fia. 175. Fig. 176. Lines joining points corresponding to m and m' for any pair of parallel radii will all pass through n, the optical centre. In a similar manner the point may be found for other lenses. In plano- convex and plano-concave_ forms the optical centre is at the ver- tex of the curved side, for there is the only point where a tangent . is parallel to the other side. In the concavo-convex and convexo- concave forms the optical centre is outside the lens as at m. Fig. 176. Any ray passing through the lens and the point n may within certain limits be regarded as a straight line. In all such cases there will be lateral displacement (see Fig. 161), and it is assumed that when a ray enters the lens at, say, m, it will be so slightly bent that it will practically pass through a parallel tangent at m'. ELECTROMAGNETIC WAVES. 235 156. Location of Images by Drawings, — ^When the focal length and optical centre of a thin lens are known, the position, size, and character of an image may be shown by drawing rays in accordance with the principles given in previous paragraphs. Each point of an object is a source of spherical waves, and each point of an image is formed by waves which came or appeared to come from a corresponding point of the object. Thus, in Fig. 177, A, oo' is the object and an infinite number of rays go out from each point, but it will be sufiicient to consider only the waves from the ends o and o' and thus fix the ends of the image. 0" \/ i It is also a convenience to select two rays from o and two from o' whose direction we know after they emerge from the lens. Rays from each point drawn parallel to the principal axis will after refraction pass through the principal focus /, and rays through the optical centre n will not be deflected. Hence the two rays from o will meet at i and so will all other rays from o. This last statement may be regarded as correct only within certain limits. (See § 157.) Likewise the waves from o' meet at i' and so the image is at «'. The lines oo' and ii' may be regarded as the bases of triangles having equal angles at n, and therefore the size of object and image bear the same ratio as their distance from the lens. 236 GENERAL PHYSICS. If the object is nearer to the lens than the principal focal length, Fig. 177, B, the rays after passing the lens will still be divergent, though less so than the incident rays. Consequently, to an observer at E, the waves will appear to originate at i, i', and intermediate points. The image is virtual, erect, and enlarged. A convex lens used in this manner is commonly known as a magnifying glass or simple microscope. In concave lenses. Fig. 177, C, rays will pass straight through the optical centre, and those parallel to the principal axis will, after re- fraction, if extended backwards, pass through the virtual focus /. If a lens is thick, then in place of an optical centre there are two points called the principal points, n and n'. Fig. 178, such that a ray from o to n will emerge from the lens as though coming from «', and n'i will be parallel to on. The location of the principal points is at the intersection of the principal planes ac and bd with the principal axis. If a ray from o, parallel to the principal axis, is incident at a, it will emerge as if from the opposite point h on the other plane and will pass through the principal focus /. a m o c \ !^ ^ \} s i' i b V ^^ Fig. 178. Fig. 179. The position of these planes is different for lenses of different shape and material. For a double-convex lens of crown glass they are about one-third of the thickness of the lens from each side. Focal distance as well as object and image distances are measured from these planes, and equation (181) will then express the relation of these distances as for thin lenses. In thin lenses it is assumed that the principal planes coincide. 157. Spherical Aberration. — In the previous discussions it has been assumed that a wave-front is spherical after it has passed through a spherical lens. That this is not strictly true is shown as follows: Let ab. Fig. 179, be a plane wave-front entering a double- convex lens. When the wave has just passed through, it has traversed the centre of the glass a distance mn, while in air it ELECTROMAGNETIC WAVES. 237 would have passed over a distance ac=mn . /i. Then, drawing a circle through c and n, with its centre on the principal axis, this is the form the wave would have if it were spherical. But acq, the distance the wave travels through the point o of the lens, is also practically aU in air and so is equal to ac. Hence q will fall within the circle and the wave wiU converge to some point as i' instead of i. This effect is slight when only the central portion of the lens is used, and we may then assume that the refracted waves are spherical and meet at a common point. Problems 1. A block of glass with parallel sides is 3 cm. thick. Its index of refrac- tion is 1.6. A wave of light is incident at 40°. What will be the lateral dis- placement of the wave when it emerges from the opposite side? 2. What is the focal length of a double-convex lens made of glass having an index of refraction 1.5, the radii of curvature of the faces being 20 cm. and 40 cm.? 3. Where is the principal focus of a double-convex lens, the two sides having equal curvature and the index of refraction being 1.5? 4. What is the critical angle in passing from a certain liquid to air if the index of refraction of the liquid is 1.65? 5. If an observer is at the bottom of a pond of clear water 3 m. deep, what is the radius of the circle at the surface of the water through which he can see the sky? 6. If the angle of a prism is 60° and the index of refraction is 1.5, what is the angle of minimum deviation? 7. A convex lens placed 17 cm. from a candle flame forms an image on a screen. When the lens is moved 65 cm. nearer the screen another image is formed. What is the focal length of the lens? 8. A luminous disc 2 cm. in diameter is placed 15 cm. from a convex lens whose focal length is 10 cm. What is the size of the image? Ans. 1. 9.18 mm. 2. 26.6 cm. 3. At centre of curvature. 4. 37° 18'. 5. 340 cm. 6. 13° 44'. 7. 14 cm. 8. 4 cm. in diameter. 158. The Spectrum. — Spectrum is a name applied to a band of various colors which appear on a screen, or may be seen directly by the eye, when white light or other light of a composite charac- ter is so resolved that waves of the same length are grouped to- gether and the groups separated from each other. 238 GENERAL PHYSICS. There are two important methods by which spectra are usually produced: (1) By use of a diffraction grating, and (2) by passing light through a prism. That produced by the former method is called a diffraction spectrum and the other a prismatic spectrum. 159. Diffraction Grating. — ^A simple form of grating consists of a plane piece of glass on which parallel lines are ruled with a diamond. The number of lines may be several thousand per centimetre. The space between the lines is clear glass and so will transmit light, but the grooves cut by the diamond are vir- tually opaque, for the light incident on them is diffused. The principle of the grating will be tmder- stood from the follow- ing consideration. Let AB, Fig. 180, represent a cross sec- tion of a grating where all parts are greatly magnified. The short spaces c, d, e, f, and g are the ends of slits through which light passes, but the spaces between the slits are opaque. Let ab be a wave-front of yellow light or any light of one wave-length. When ab is incident on the grating each slit will become a centre of dis- turbance and the wave-front uv, parallel to ab, will continue to the right of the grating and, passing through the lens L, will be focused at i. The only effect of the grating for this wave-front is a decrease in the quantity of light. It is to be noted, however, that there are other lines, as Am and An, along which waves are in the same phase of vibration. Hence these lines are also wave-fronts. When Am passes through the lens it is f ocused'at p, and An at q. Similar focal points, or images of the shts, will be found at p' and q' and still others both above q and below q'. So now, in addition to the ordinary image at i, we have images of the first, second, etc., orders on either side of i. The reason for this is that the opaque parts of the grating prevent FiQ. 180. ELECTROMAGNETIC WAVES. 239 that destructive interference of waves which would occur at all points except * if the grating were removed. (See Fig. 143.) Let the angle which the wave-front Am makes with the grating be denoted by 6. Also let the distance between two slits, as A to c, c to d, etc., be denoted by s. This distance s is called the grating space or grating constant. From c to Am is one wave-length, X. Hence X =5 sine (186) This is for an image of the first order, i.e., at p. Let An make an angle with the grating, then since the dis- tance from c to An is 2X, 2X=5sin (187) This is for an image of the second order, i.e., at q. Then if 6 is the angle which any wave-front makes with the grating and n is the order of the image on the screen, n\=s sine (188) This is a general equation and shows that 8 varies with X. Now white light may be regarded as the resultant of many different wave-lengths. Hence when such light passes through the grating, 8 wiU have a different value for each wave-length, as shown by equation (188). Instead of a line image we wiU there- fore have a band containing in succession all the colors into which white light is capable of being resolved. The different colors are the sensations produced by different wave-lengths, so that for each wave-length the line or plane Am, Fig. 180, wiU have a differ- ent inclination to the grating. The longest light waves are red and the shortest are violet, consequently the red at p will be farthest from i and next to it will be orange, then yellow, green, blue, and finally violet nearest to i. A similar band of colors win appear at p'. These are diffraction spectra. The ones at p and p' are spectra of the first order. Other spectra will appear at q and q' and at other points beyond, but wiU be less brilliant on account of the overlapping of images there. In spectra of the first order each color, not only those named above but all intermediate shades, occupies a separate position on the screen. Hence this is called a pure spectrum. 240 GENERAL PHYSICS. As long as is small we may, in accordance with eqtiation (188), say that dispersion — i.e., B — is proportional to wave-length. For this reason a diffraction spectrum is taken as the standard and is called the normal spectrum. 160. Wave=Iength of Light. — Principles discussed in the pre- vious paragraph suggest an excellent method of finding the wave- length of hght in any desired part of the spectrum. The method of procedure will be plain if Fig. 181 is compared with Fig. 180. In both, AB is the grating and Am is the wave-front producing a spectrum of the first order. But, in Fig. 181, L is the lens of the eye and CD is the retina. The light whose wave-length is sought is placed at X and admitted through a narrow slit close to the light so that the image may be more exactly located. When FiQ. 181. this light passes through the grating an image is formed on the retina at p and p', and the eye locates the images at i and i'. The angle between AB and Am is equal to the angle made by pi or p'i' with the axis x. The distances from X to the lens and from X to i are measured. Call these distances x and d, respectively. Then d=^tan d From this is found and, by a simple substitution of its value in equation (186) with the value of the grating space 5 which is given by the makers, the value of X is found. In a similar manner any image in spectra of the second or third order may be located and the value of X computed. Another simple method of measuring B is to mount the grating on the spectrometer (Fig. 166) and note the angle through which ELECTROMAGNETIC WAVES. 241 the telescope must be turned from a position in line with the collimator to a position where a color of the first or other order may be seen. 161. Dispersive and Resolving Power of Gratings. — By dis- persive power is meant the difference in the value of 6 for the various colors of the spectra. The greater the dispersive power the longer the spectrum will be. It may be readily inferred from Fig. 180 and also from equation (188) that since n\ sm 6= — • 5 any decrease in s, the grating space, will increase sin 6. If 6 is small it may be used in place of its sine, and then the dispersion 6 is inversely proportional to the grating space, s. The closer the lines of a grating are to each other, i.e., the greater the number of lines per centimetre, the greater the dis- persion. The equation also shows that dispersion varies directly with the order of the spectrum. In the second order the spectrum is twice as long as in the first. The resolving power of a grating is its ability to cause images having different wave-lengths to stand out sharply defined. This power is proportional, not to the number of Unes per centimetre, but to the total number of lines on the grating. It has been shown that when an image of any particular color is formed at p or q, Fig. 180, the space between p and q or p and * is dark because of destructive interference. The closer this region of interference approaches an image the sharper the image will be. If a wave-front is such as to make an angle with the grating only very sUghtly greater or less than Am makes, an image would be formed just above or below p, i.e., the image at p would not be distinct. But if there are a sufBcient number of grating spaces there wiU be one, many spaces from c, whose distance from the wave-front differs by one-half wave-length from the distance of c to the same wave-front. Hence there wiU be complete inter- ference of the waves from these two grating spaces. If, then, there are as many grating spaces below the one whose waves are 242 GENERAL PHYSICS. a half wave-length behind as there are from it up to c, all the waves from one-half of the grating that would form an image on either side of p are destroyed by waves from the other half, for each grating space in one-half has a corresponding one in the other, differing by a half wave-length in their distance from the wave-front. A grating made by Professor Michelson is nine inches long and is ruled with 114,300 lines. The gratings described above are called transmission gratings, for light passes through them. Other gratings having lines ruled on polished metal are called reflection gratings. The principles involved in the use of the latter are the same as for the former. 162. Prismatic Spectra. — ^The deviation of a beam of light when passed through a prism depends on the angle of the prism, the index of refraction, the wave-length, and the angle of inci- dence. The shorter the wave-length the greater as a rule will be the deviation. Hence, when light of a composite character, such as white light, is passed through a prism, it is resolved into groups of similar wave-lengths, and if these are made to fall on a screen there will appear, for white light, all shades of color from red to violet, the deviation being least for red and most for violet. The angle between rays of violet and red waves is the disper- sion for these two wave-lengths. It appears, then, that since the shorter waves are most re- fracted, they are the ones whose velocity is most retarded in pass- ing through a denser medium. When light from the sun or other source of light is admitted to a darkened chamber and passed through a prism, a spectrum is formed consisting of a succession of images in different colors. These images, however, overlap, and there is not a distinct sepa- ration of the various wave-lengths. Much better results are obtained by first passing the light through a narrow slit and then, after dispersion by a prism, interpose a lens which will form on the screen a succession of images of the slit in the various colors into which the prism has resolved the light. It is obvious that index of refraction for any given substance should be given in terms of some specified wave-length or for some designated position or line in the spectrum, for refraction is different for different colors. The D line in the yellow part of ELECTROMAGNETIC WAVES. 243 the spectrum is usually employed in determining what is called the mean index of refraction. It is found, however, that prismatic dispersion is not propor- tional to refraction or to wave-length. Long waves as a rule are but Uttle dispersed, and short waves are abnormally spread out by a prism. Also, for prisms of different material, the relative dispersion in various parts of their spectra do not agree. Dis- persion of this kind is called irrational as contrasted with normal dispersion produced by a grating. 163. Kinds of Spectra. — Spectra are usually classified as: (1) bright-line, (2) continuous, and (3) absorption or reversed spectra. An incandescent gas wUl ordinarily give a bright-line spectra. Atom^ of a gas are widely separated from each other and move in long, free paths without collision. The electrons which are regarded as the source of light waves may then produce a long train of waves of a definite wave-length. These will cause a single bright Hne of a certain color without the other colors of the ordinary spectrum. Within, the same atom electrons may have different periods of vibration, and so there may be several trains of waves of different wave-length each producing a bright line in the spectrum. Light from sodium vapor, for example, as when soditim chloride is bxtmed in a Bunsen flame, will produce a bright yellow line without any other colors. Cadmium vapor gives red, green, and blue lines. Vapor of mercury gives lines of yellow, green, blue, and violet. Other spectra of this kind are obtained by heating salts of various substances to a state of incandescent vapor. If a vapor or gas is put under grea.t pressure the lines become broader and tend to unite in continuous spectra. Incandescent solids and liquids will produce continuous spec- tra, i.e., there is no break in the band of color, however pure the spectra may be. White light from such a source is therefore resolved into every possible wave-length from red to violet. A third form of spectrum of very great importance and utility is the absorption spectra. When sunlight is admitted through a narrow slit and is dispersed by a prism or grating, numerous dark hnes parallel to the slit are observed at intervals throughout the spectrum. Fraunhofer in about the year 1815 counted more than 700 of these lines and defimtely mapped out about 350 of them, 244 GENERAL PHYSICS. E b r JB HK and hence they are known as the Fraunhofer lines. He designated the more important lines by letters, as shown in Fig. 182, where A lies in the extreme red, B and C in the lighter red, D in the yellow, E in green, F and G in the blue, and H at the limit of violet. These lines are of great importance for purpose of reference as in designating wave-length for any particular point of the spec- trum. Thus we say the wave-length for F is 4861 AngstrSm units. An Angstrom unit is lO"*" metres, i.e., 10"'' mm. or 10"* microns. The D line is in fact two fine lines very close to- gether called Z?i and Di, whose wave-lengths are 5896 and 6890 Angstrom tmits. The index of refraction is difEerent foreachchangeof wave-length and these values are given for the various lines of the spectrum. By use of Fraunhofer lines it is possible to determine a number of elements in the sun, for the dark lines are the region where waves that would fall on that part of the spectrum have been absorbed by some substance through which the light passed. Orange YellAW ^am Fia. 182. BIw ViKlct 1 ^!w RMiH#iL*fcipffe 'i^wk\\mr-i- S" tili. JiSi''.riucJlt^^ ■mm { 1 V '|! i 1 axBSBKm.mie.^mmKEa'm BBIKDEIllKS^^sSlltl^P^SIHiSB^IW^^ Fia. 183. The sun is surrounded by an envelope consisting of gases and the vapors of many substances. Light from the hotter portions of the sun must pass out through these vapors, and those waves are absorbed which the vapors themselves would give out if they were highly incandescent. The bright line spectrum of the vapor of iron, for example, is shown in part in the middle band of the photograph. Fig. 183. On either side is a spectrum of sunlight. The exact position of the numerous lines of iron in Une with dark lines of the solar spectrum leaves little doubt that there is iron in the sun. In a similar manner it is shown that many other substances are constituents of the sun. ELECTROMAGNETIC WAVES. 245 The sun is rotating on its axis, and if a spectrum is formed of light coming from that limb which is moving toward the earth, all the lines except A will be shifted toward the violet. This shifting is what would be expected according to Doppler's prin- ciple that when a body from which a train of waves emanates is in motion, wave-length wiU be shorter in the direction of that motion. This is often observed in case of sounding bodies in motion and is just as true of luminous bodies in motion. Hence the shifting of spectral lines to a slightly higher position than they would have if the source of light were stationary. Light from the opposite limb of the sun causes a shift in the opposite direction. Since the A line is not thus affected it is concluded that the earth's atmosphere absorbs those waves that would produce a bright line at that point. There appears to be no limit to the number of Fratmhofer lines, for every improvement in the methods of dispersing light beings more lines in view. The ratio of the difference in deviation of rays that fall on the C and F lines to the deviation of those that fall on the D line is called the dispersive power and may be written Bf-Dc where D stands for deviation and subscripts F, C, and D are lines of the spectrum. This ratio may be expressed in terms of refractive indices as follows: From equation (173) D=ii+i2 — A We also have sin * t'l ii sin r n Ti within limits where the ratio of angles may be regarded as equal to the ratio of their sines. Hence and ii = jurj 246 GENERAL PHYSICS. Substituting these values of ii and ii in the equation above and keeping in mind that ri+r2 = A, we have for the deviation of the three rays under consideration Do = (ixc-l)A Do={hd-1)A Substituting these values in the expression for dispersive power P (m>-1)A-(mc-1)A P = - (mi,-1)A or p=f^E-J^ (189) 164. Limits of the Spectrum. — ^That portion of dispersed waves which appears in various colors is called the visible spectrum. There are, however, in the radiations from such a source as the sun or arc lamp, many waves which are longer than the longest red waves. This is known as the infra red region of the spectrum. The length of wave in the A line is .7604^1. The wave-length of the longest infra red wave yet found is 300^. Designating this interval in octaves as in music, the infra red region extends about 8| octaves below the A line. These long waves may be detected by the heat effects, as when a thermopile or bolometer is moved along the region of the spectrum. (See p. 237, " Mechanics and Heat.") The wave-length can then be computed, for in a normal spectrum wave-length is directly proportional to deviation. The visible spectrum is only a little more than one octave in length — i.e., from about A to H — ^these lines being near the ex- treme limits of red and violet respectively. This interval in wave- lengths is .7604ix to .3968;u. Above the violet is a region called ultra violet. The shortest wave-length that has been measured in this part of the spectrum is .Ijtt. Ultra violet waves, therefore, extend nearly two octaves above the H line. Short waves may be detected by their actinic efiEect — i.e., by the chemical effect which they produce — as when they fall on a photographic plate. Such a plate is affected in a similar manner by Rontgen rays, or X-rays, and these are believed to be very short ether waves ELECTROMAGNETIC WAVES. 247 produced by the impact of electrons on a metal target placed in the path of cathode rays. (See Fig. 1.) X-rays, however, can- not be reflected, refracted, or diffracted, and hence no way is known by which their wave-length can be determined. Both X-rayB and ultra violet waves will cause fluorescence in a substance such as barium platiao-cyanide. 165. The Spectroscope. — ^A spectroscope is an instrument by which spectra may be conveniently and accurately studied. One common form is shown in Fig. 184. The principle of construction is the same as in the spectrometer (Fig. 166), with the addition lio. 184. of a third tube S, which contains a finely divided transparent scale. Light from L passes through the scale and is reflected from a side of the prism into the telescope. Thus an observer may locate the various lines of the spectrum on the scale. A substance in a state of incandescent vapor wiU. give a bright- line spectrum characteristic of that substance. When these Unes and their position have been learned, it is often possible to make a chemical analysis of an unknown substance by vaporizing some of it in a Bunsen flame placed in front of the slit of the collimator. An observer looking through the telescope identifies characteristic 248 GENERAL PHYSICS. spectra of known substances, or light from the unknown vapor may be admitted directly through one-half of the slit and that from a known substance may be admitted at the same time by use of a total reflecting prism placed over the other half. The two spectra are then seen side by side. The known substance is present in the imknown if its lines run straight across both spectra. The third tube is not needed. By another method a photographic camera is put in place of the telescope. With one-half of the plate covered a photograph is made of the spectrum from one source of light, then after shifting the cover to the other half a photograph of the other source is made. A permanent record is thus secured. 166. Chromatic Aberration. — ^When white light is refracted by a lens as in Fig. 177 dispersion will occur as in a prism. The violet which is refracted most will be focused at a point nearer the lens, the red at a point farthest away, and the other wave- lengths at intermediate points. The image will therefore be indis- tinct and surrounded by a fringe of colors. This phenomenon is called chromatic aberration. This defect of lenses may be corrected by use of achromatic lenses. Such lenses are possible as a result of irrational dispersion already explained. If two prisms of the same kind of glass and having the same angle are placed in a reversed position as in Fig. 185, then, although white Kght is dispersed in passing through the first prism, the rays are reunited by the second prism, so that the emergent beam is the same in character and direction as the incident beam. The effect is Fig. 185. ^^^ Same as for a plate of glass with parallel sides. (See Fig. 161.) If, however, the angles and kind of glass are different in the two prisms it is possible to have deviation without dispersion. Such a combination is achromatic. To construct prisms for this purpose which will be achromatic for any desired wave-lengths it is necessary to choose such angles and material of such indices of refraction that dispersion in the region to be achromatized shall be equal. It has been shown in the derivation of equation (189) that deviation is expressed hy A (ju — 1). The difference of devia- ELECTROMAGNETIC WAVES. 249 tion between, say, the F and H lines, would be the dispersion in that region and wovild be expressed by If then two prisms of different angles, one made of flint glass and the other of crown glass, for example, be selected so that A{n^— /Xj.) of one is equal to that of the other and the prisms are placed as in Fig. 185, the dispersion of one between F and H will be cor- rected by that of the other and there wiU still remain a certain amount of deviation. In the same manner two lenses of different curvature and material may be combined to produce an achromatic lens. A plano-concave flint glass is usually combined with a double- convex crown glass and the curvatures so selected that for optical instruments the lens will be achromatic for the brilliant colors — i.e., in the region of the yellow — but for photography the upper regions of the spectrum are thus corrected. If as a result of irrational dispersion it is possible to have deviation without dispersion, it should also be possible to have dispersion without deviation. This latter result may be obtained by choosing two prisms of such values for A and n that the devia- tion j4 (ju — 1) of one is equal to that of the other. This principle is employed in the construction of direct-vision spectroscopes, i.e., spectroscopes through which the observer looks directly toward the source of light. 167. Color. — As already suggested, the different colors which we perceive are sensations caused by different wave-lengths of light. Consequently this topic belongs to physiology rather than to physics. According to the Young-Helmholtz theory of color sensations there are in the retina of the eye three sets of nerve terminals, one of which is particularly sensitive to the waves that produce a sensation called red, another to waves called green, and a third to violet waves. The sensitiveness of these terminals to the various wave-lengths of the spectrum is shown by the length of the ordinates of the curves in Fig. 186, where R, G, and V are points of maximum sensitiveness. It is found, however, by examination of cases of color-blindness, that each set of terminals is sensitive in some 250 GENERAL PHYSICS. degree to all wave-lengths, so that each curve extends over nearly the whole spectrum, reaching a maximum at R, G, and V. A superposition of red, green, and violet of the proper shade and intensity will produce the sensation of white or gray, and by a proper combination of these three primary-color sensations any color of the spectrum may be produced. It is supposed that the nerve terminals respond or vibrate in sjrmpathy with waves of a certain length, and for any combina- tion of waves the sensation is that of the resultant disturbance, but it is not known with certainty just how light afEects the nerve terminals. Another assumption is that the retina may contain substances which are acted on chemically by the light waves. A person who is color-blind is defective in one of the three sets of terminals. For one who is red-blind the entire curve R, Fig. 186, is wanting. To him a combination of green and violet will be white in the region of the ordinate Wi.' For one who is green-blind the curve G is wanting and a green will be gray, for Wi is a common ordinate of the R and V curves in that region. 168. Complementaty Colors. — ^Any two colors which, falling upon the retina at the same time, produce the sensation of white are complementary. Blue and yellow, for example, or red and blue-green are complementary. A good way to mix colors is by the rotation of discs on which colors are arranged in different proportions. A ntimber of discs of cardboard, each of different color, may be slit along a radial line, and any two or more may then be slipped together so that any desired portion of each is exposed. When these are rapidly rotated each color will, on ac- count of the persistence of the image, be impressed on the retina at the same time. Red and green will give yellow; red and blue, purple. Any desired shade of color may be obtained by mixing the various colors in the proper proportion. If one looks intently at a blue card held against a white back- ground for a short time, when the card is suddenly removed a ELECTROMAGNETIC WAVES. 251 yellow image will appear in the region which was covered by the blue. If the card is red, the after image is green. The color of the image is complementary to that of the card. An explanation offered for this is that after looking at blue for a time those nerve terminals which are particularly sensitive to that color become fatigued. Hence, when only white is in view, that part of the retina which was before covered by blue will be most sensitive to that color, yellow, which with blue would produce white, i.e., after the terminals sensitive to blue are wakened by fatigue the eye is for a short time blue-blind and the red and green terminals will, as the diagram shows, produce a sensation of yellow. 169. Color Resulting from Absorption. — In the previous para- graph we have discussed what may be called addition of colors. The color of transmitted light is a result of the subtraction of colors. The waves which pass through a transparent body are those which have not been absorbed or subtracted by the body. Red glass, so called, is glass that absorbs all waves except red. Blue glass absorbs all waves except blue, and so on. A common experiment illustrating this fact consists in passing a beam of light through a solution of copper sulphate. An exanaination with the spectroscope wUl then show that the longer waves of the spec- trum, the red and yellow, have been completely absorbed. The light is then passed on through a solution of bichromate of potash and all shorter wave-lengths, blue and violet, will be absorbed. The only waves which pass through both solutions are the ones which produce a sensation of green. The mixing of colors and the mixing of pigments are, therefore, entirely different processes. By the former process blue and yellow give white; by the latter, green. When white light falls on any mixture of pigments certain waves are as a rule absorbed and the others reflected. A white paint reflects all waves, while a black paint absorbs all. The color of a paint is determined by the waves which are reflected and is not something inherent in the paint. A white body in red light is red and a red body in blue light is black. 170. Polarized Light. — The discussion at the beginning of this chapter would lead us to infer that the disturbance of the ether on the passage of an electromagnetic wave would be in a direction at right angles to the direction of propagation, for when an elec- 252 GENERAL PHYSICS. tron vibrates it sets up a magnetic field only at right angles to the direction of its motion. In way of analogy we may think of a cord of indefinite length, one end of which is fastened to the prong of a tuning fork or other vibrating body. Waves pass out on the cord, but any point on the cord moves back and forth in a direction at right angles to the direction the wave is moving. In each atom of a luminous body are a number of electrons, each of which may be vibrating in a different plane, or any one electron may rapidly change its plane of vibration. As a conse- quence we find that the ether disturbances along a beam of light take place in every conceivable plane, the plane in all cases in- cluding the line of propagation of the wave. In Fig. 187 is a representation of a wave moving in a direction perpendicular to the paper with the numerous planes of vibration of the ether. These planes may all be resolved in two directions xx' and yy' . If by any means the vibrations in one of these two planes a are removed or destroyed, the light is said to be plane polar- ized, Iqx the vibration of the ether is in one plane only. If only a part, less than half, of the planes are removed, the light may be said to be par- tially polarized. Ordinary light, in this sense, is polarized in every direction. Huygens in 1610 discovered that when a beam of light passes through a crystal of Iceland spar it emerges from the crystal as two beams, except when the direction of the light is parallel to the axis of the spar. He also found that if another crystal is held in the path of one of these two beams another division of the light took place when the second crystal was held in one position, but not when either crystal was turned 90° on the line of light as axis. Because of this experiment Huygens is usually given credit for the discovery of polarization, though he could not explain the phenomenon. We now know that the Kght was polarized by double refraction, as will be shown later, and the emergent beam FiQ. 188. ELECTROMAGNETIC WAVES. 253 which exhibited peculiar properties contained vibrations in but one plane instead of the synmaetrical arrangement shown in Fig. 187. Early in the 19th century Malus discovered that when light is reflected from glass at a certain angle it exhibits properties like that observed by Huygens in case of double refraction, i.e., by simple reflection at a certain angle the light was polarized. Experimental evidence of this kind, described in succeeding paragraphs, leads to the belief that the vibrations of the medium through which light travels are not longitudinal as in case of sound but transverse to the line of propagation. 171. Polarization by Reflection. — In Fig. 188 let w be a mirror of plane glass set so that the angle which it makes with the ver- tical line 6c will be about 33°, i.e., <#> = 33°. Then let a beam of light ah fall on m, making the angle of incidence, 8, about 57°. The ray he will then be plane polarized, i.e., will contain vibra- tions in but one plane. How this has come about may be imderstood from Fig. j* 189, where ab is an ordinary beam of light '^ I ^^ \ />» with directions of vibration as shown in ' / V^ , , , Fig. 187, but here resolved into two planes o*'*'**/'^ at right angles. The lines and dots on ab ^^^ jgg represent 189, A, as seen edgewise. Now part of the beam ab will be reflected and part will be refracted and pass on to /. If be is perpendicular to bf, the former wiU be com- pletely polarized and the latter partially so, as will be described later. If be, then, is to contain both planes of vibration, one of them would be in the direction of the propagation of the wave, and that is just what we have claimed cannot be. In the other plane, however, the vibrations continue to be transverse to the direction of the light. Now going back to Fig. 188 the beam be is made to fall on a mirror m' which is backed with black varnish and is set parallel to m. The eye of an observer at d will receive the beam of polarized light and no difference will be observed between it and ordinary light, but when m' is rotated through 90° on cb as axis, no light will be reflected from m'. In place of the beam seen at d there will now be darkness. Just as m removed the transverse vibra- tions in one plane, w' now removes those in the other plane at 254 GENERAL PHYSICS. right angles to the first. A further rotation of m' through another 90° will again bring the light into view, as it should if our theory is correct. At the end of the next 90° there will again be darkness. In an arrangement of this kind m is called the polarizer and m' the analyzer. To determine whether light from an unknown source is plane polarized or not, a mirror such as m' is placed in a proper relation to the beam and rotated in the manner just described. Fig. 190. A very convenient instrument for experimental work in polari- zation is the Norrenberg polariscope shown in Fig. 190. The upper mirror is supported on a collar which may be turned on a graduated circle. 172. Brewster's Law. — The angle of incidence at which polari- zation of reflected light is maximum is called the polarizing angle. A very simple law discovered by Brewster states that when the ELECTROMAGNETIC WAVES. 255 angle between the reflected and refracted rays is 90° the reflected ray is completely polarized. From the construction of Fig. 191 it is seen that r+90°+/ = 180° .-. r = 90°-/=90-i sin i sin i sin i . ,, __, "^ = ^= • mno •x = ^^=tant (190) sm r sm(90 — t) cos t Hence another statement of Brewster's law would be that the index of refraction of a substance is equal to the tangent of the polarizing angle, or *=tan-V (191) The polarizing angle will therefore be different for substances with different refractive indices. Ordinary glass has an index of about 1.55, and this is approximately the tangent of 57°. This is the reason the angle of incidence was made 57° in the experiment described above. Also, since refractive index is different for dif- ferent colors of the spectrum, if white light is polarized by reflection, not all the colors are shut _. , , , . , , , . . , Fia. 191. oil by the analyzer, for when the polanzing angle is correct for red, say, it is not quite correct for violet. Hence a slightly colored spot of light will be seen even when the analyzer crosses the polarizer unless monochromatic light is used. 173. Polarization by Refraction. — ^As shown in Fig. 189 the ray be contains only vibrations at right angles to the plane of polari- zation — i.e., the plane which includes ab, bn, and be — ^but the refracted ray bf includes all vibrations of ab which were in the plane of polarization and also those transverse vibrations which were not reflected with be. The light in bf is therefore only par- tially polarized. If, now, several plates of glass be placed parallel to m in the path of bf, each will reflect out a portion of the vibra- tions Uke those in be, and so at each reflection bf wUl become more nearly plane polarized. A bundle of about 12 or 15 thin glass plates wiU completely polarize the refracted beam. There is no advantage in using a greater number of plates, for, after all the vibrations perpendicular to the plane of incidence ^ w ^ 256 GENERAL PHYSICS. have been reflected, no further reflection will occur and there is a disadvantage resulting from absorption of light by the glass. If such a bundle of glass, seen to the right in Fig. 190, be used as analyzer in place of the mirror of the Norrenberg polariscope, it will be found that, since the light is now already plane polarized by the lower mirror, when light is transmitted with maximtim brightness, as may be observed by looking vertically downward through the bundle of glass, reflected light is a minimum. At a point 90° from this position reflection will be a maximum and trans- mission a minimum. 174. Polarization by Double Refraction. — Certain crystals, notably crystals of calcium carbonate called calcite or Iceland spar, possess a structure such that when a beam of light is incident normally to one side a part of the beam will pass straight through, just as it would go through glass, while another portion of the beam is deflected to one side. Two separate and parallel beams emerge from the crystal, both of which are plane polarized. Fia. 193. The natural shape of a crystal of Iceland spar is shown in Fig. 192. The solid angles at a and b are obtuse, each being in- cluded by three obtuse plane angles of 101° 53'. The shape of the crystal is a rhombohedron. A line ab making equal angles with the three faces of the solid obtuse angles at a and b is called the optic axis. When the sides of the rhombohedron are equal, as in the figure, a6 is a continuous line. A beam of light passing along the optic axis or any line parallel to this axis does not suffer double refraction. The reason for this is that in all directions at right angles to any point on the optic axis the structure of the material is alike, and so the transverse ELECTROMAGNETIC WAVES. 257 vibrations are equally affected in all directions. This is not the case when light is passed through the crystal in other directions. A plane passed down through ad, Fig. 192, so as to include the optic axis, is called the principal plane and is shown in Fig. 193. Let a beam of light from s. Fig. 193, pass through the spar in the direction indicated. It will be divided into two beams, one, so, called the ordinary ray, has its plane of vibration perpendicular to the principal plane; while the other, se, called extraordinary ray, has its plane of vibration in the principal plane. Both rays are plane polarized, but in direction at right angles to each other. This can be easily shown by placing a crystal of Iceland spar on the naiddle support of the polariscope. Fig. 190, over a pin hole through which light is reflected from white paper placed below, to the analyzer above. While the analyzer is tiumed, the two white images of the pin hole, one caused by the ordinary and the other by the extraordinary ray, will alternately appear and disappear, each reaching maximum brightness twice in a complete revolution. This shows clearly the fact and the direction of the polarization. If a crystal of this kind is placed over a dot on a sheet of white paper it wiU be noted that when the crystal is rotated the ordinary ray is stationary and the extraordinary ray turns around it, but the two images and the solid obtuse angle a are always in line. It will also be noted that image of the dot as seen by the ordinary ray is nearer the top of the crystal than is the other image. This shows that the ordinary ray suffers a greater retarda- tion in passing through the crystal (see § 148). Considerations such as those described above lead to the belief that when light falls on a crystal, as shown in Fig. 193, the point of incidence becomes a source of ether waves which attempt to spread out in all directions, but the velocity of propagation is greater for those transverse vibrations which are parallel to the optic axis. Hence the direction of the ray containing vibrations of this kind only will be that of the resultant of velocities in all directions, the greatest component being in a direction at right angles to the optic axis. The extraordinary ray will, therefore, be deflected toward that path in which waves travel with the greatest facility. In case of the ordinary ray the transverse vibrations are per- pendicular to the principal plane and hence to the optic axis. 17 258 GENERAL PHYSICS. The retardation will then be the same in all directions. The waves arising from the point of incidence will be spherical, and the medium is virtually isotropic for this ray. Many other crystals such as quartz, tourmaline, naica, and sugar possess the property of double refraction, but Iceland spar can be most easily found in large transparent crystals. 175.- The Nicol Prism. — ^The most effective way of obtaining plane polarized light is by use of a doubly refracting crystal as explained above. Both of the rays emerging from such a crystal are completely polarized, but they are close together unless a very thick crystal is used, and their planes of vibration are at right angles. Hence it is desirable to get rid of one of the rays and transmit the other. Such an arrangement was devised by Nicol. As shown in Fig. 194 he cut a crystal of Iceland spar along a e plane mn, this plane being parallel to the upper edge of the crystal. The two faces were then polished and "» again cemented together ,with Canada balsam. The ray of light is incident obliquely and the ordinary, being most retarded, will be refracted most. The index of refraction of balsam is less than that of calcite for the ordinary ray but greater for the extraordinary. The plane mn is therefore so chosen that at o the ordinary ray is totally reflected to the black coating on the crystal where it is ab- sorbed. The extraordinary ray passes on through the crystal and is plane polarized. 176. Circular and Elliptic Polarization. — In plane polarization, as we have seen, the motion of ether particles as a wave passes is back and forth in a straight line transverse to the line of propa- gation of the wave. The motion of the ether particle may, how- ever, be circular or elliptical, and the light is then said to be circularly or elliptically polarized. As, has already been indicated, when light has been doubly refracted two images are seen, and if these are viewed by another crystal in the proper position four images will appear. Each of the two rays of plane polarized light has been resolved into two, one ray in each set being an ordinary and the other an extra- ordinary ray. This kind of resolution will occur and will give ELECTROMAGNETIC WAVES. 259 maximum brightness to the images when the plane vibrations of the rays from the first crystal are at an angle of 45° to the optic axis of the second crystal. If, then, a thin piece of mica is placed on the middle shelf of the Norrenberg polariscope and a plane polarized ray passed up through it, the ray will, when the mica is turned to the proper position, be doubly refracted, for mica possesses the same property as calcite in that respect. The single plane of vibration of light incident on the mica is resolved into two rectangular components. The ordinary and extraordinary will not, since the mica is very thin, be appreciably separated on emergence, and their fields of disturbance in the ether will overlap. Also, since the ordinary ray is more retarded than the other in passing through the mica, it will emerge with a change of phase unless the retarda- tion happens to be one complete wave-length. Let the mica plate be of such thickness that the ordinary ray lags a quarter wave-length, X/4, or some odd multiple of X/4. Then if A, Fig. 195, represents a plane vibration before entering the mica, B will represent the condi- tion after resolution and retardation. But if the amplitudes of o and e are equal and o differs X/4 in phase from e, then, just as in compounding any two simple haxmonic motions, the resultant motion is a \^ circle. This kind of disturbance in ether is called / circular polarization. If the diEEerence of phase is not X/4 or the amplitudes are unequal, the resultant motion will be in the orbit of an eUipse, producing elliptic polarization. A plate of mica or other substance which will cause the ordinary ray to lag X/4 is called a quarter- p,g jgg wave plate. Mica is commonly used because it can be split into very thin plates. If the plate is thick the two rays will be so far separated that there will be no overlapping of fields of disturbance on emergence. Double refraction is the most common method of producing polarization of this kind, but there are other methods of pro- ducing similar results. Plane polarized light reflected from a polished pole of a magnet is found to be elliptically polarized. This is known as the Hall effect. o 260 GENERAL PHYSICS. One method of detecting circularly polarized light is by passing the light through a quarter-wave plate. This wiU change the circular back to plane polarization which can then be readily detected with an analyzer. Elliptical polarization can usually be detected by the fact that in that case the image seen in the analyzer will be alternately bright and dim as the analyzer is rotated. The same effect, however, is produced by partially polarized light and when in doubt the quarter-wave plate may be used as above. 177. Color of Thin Plates. — ^When plane polarized light is passed through a thin plate of mica or other substance having similar optic properties a brilliant play of colors will be seen when the plate is viewed through an analyzer. This can be very readily observed by use of the Norrenberg polariscope or simply by holding the mica in light reflected from the surface of a polished table top and viewing it through a pile of glass plates. (See § 173.) A rotation of either the mica or the analyzer will cause colors to appear and disappear. If in one position there are bright colors a rotation of 45° will cause them to disappear. The next 45° will again show colors which are the complements of those first seen. In the next 45° the colors again disappear, and at 180° from the first position the first colors will again be in view. In explanation of these effects suppose the thickness of the mica is such that the ordinary ray lags X/2 behind the extra- ordinary for red light. When the two rays emerge from the mica one will have been changed in phase by 180°. The plane of vibra- tion will therefore be turned through 90°. But with the same thickness of plate the lag for violet is a whole, wave-length or 360°, and the plane of vibration of emergent violet is the same as it was on entering the mica. The two planes of vibration are then at right angles. Hence, when the Nicol is placed so as to extinguish red, violet wiU be transmitted, and when the Nicol is turned through 90°, red will be seen and violet will be cut out. For other colors of the spectrum the mica plate will, as has been shown, produce elliptic polarization. For colors near the red, as orange and yellow, the long axis of the ellipse will be nearly parallel to the plane of vibration of the red, and similarly for colors near the violet. Hence those colors which have wave- lengths approximating red will be mainly transmitted through ELECTROMAGNETIC WAVES. 261 the Nicol with red and other colors with the violet. As the two sets of waves together produce white, any color seen in one posi- tion of the Nicol will be replaced by its complementary color when the Nicol is turned through 90°. 178. Rotation of the Plane of Vibration. — If the Nicol on the Norrenberg polariscope is turned so that a beam of plane polarized light sent up through it is extinguished, and a crystal of quartz having its upper and lower faces normal to the optic axis is inter- posed in the path of light below the Nicol, the beam of light will reappear. The Nicol must then be turned through a certain angle to again extinguish the light, i.e., the quartz turned the plane of vibration through a certain angle though the light remained plane polarized. The experiment indicated above should be made with mono- chromatic light such as a sodium flame. The amount of rotation is different for different wave-lengths, being about three times as great for violet as for red. It also varies with the thickness of the plate. For a plate of quartz 1 mm. in thickness the rotation for red light is about 18°. If white light is used the field of view will always be colored, for in any given position of the Nicol but one color is cut out. In respect to this property it is found that quartz crystals are of two kinds : one rotates the plane of vibration to the right and is called right-handed or dextrogyrate quartz, while the other rotates the plane to the left and is called left-handed or levogyrate quartz. A number of crystals and solutions possess the property of rotary polarization. The most important direct application of polarized light is in the use of a saccharimeter to determine the character and strength of sugar by the direction and number of degrees through which the plane of vibration of plane polarized light is turned while passing through a fixed length of solution. Sugar, like quartz, because of a difference in crystalline structure, is foimd to be left-handed and right-handed, the former being called dextrose and the latter levulose. 179. The Zeeman Effect. — It was long believed that an effect of some kind should be observed in a spectrum when the source of light is placed in a strong magnetic field. After the establish- ment of the electron theory it appeared evident that if the elec- tron is an electric charge and its vibration is the cause of those 262 GENERAL PHYSICS. electromagnetic waves which we call Ught, some changes would be noticed in the spectral lines when the electrons are made to revolve or vibrate in a magnetic field. Many experiments were tried and the problem seemed to be one of making suflSciently powerful instruments to make the effect observable. In 1896 Dr. Zeeman, of the University of Amsterdam, was able to obtain results which strongly confirm the electron theory. Some of his results show great complexity and are not easy to explain, but we wiU describe a simple typical case as, for example, when blue-green cadmium Ught is used. Zeeman found that when the light is viewed in the direction of the lines of force, through a hole in the pole pieces, this cadmium line was resolved into two lines, a doublet, one on each side of its ordi- nary position in the spec- trum. This shows that there has been an in- crease in vibration fre- quency for that line which has moved toward the blue, and a decrease for the other line. He also observed that both lines were circularly po- larized, but in opposite directions. To explain this we assume that electrons are moving in orbits in every conceivable direction. All these orbits may be resolved in three planes as shown in Fig. 196, A. Now let the electrons, moving in these orbits, be placed in a strong magnetic field. Consider first only the orbit abed placed at right angles to the field and viewed from E, Fig. 196, B. Electrons are moving in both directions around the orbit. One moving in the direction abed will, as may be shown by the /. Fia. 196. ELECTROMAGNETIC WAVES. 263 motor rule, be subject to a force tending to drive it from the centre. (See Fig. 61.) It will therefore move in a larger orbit and will make fewer revolutions per second. The waves sent out by this motion of the electron are therefore longer, and the resulting line as seen in the spectroscope is shifted toward the red end of the spectrum. Waves are sent out only in a direction at right angles to the motion of the electron, and since the plane of the orbit is per- pendicular to the Une of observation, the light received at E will be circularly polarized. This accounts for the lower line of the doublet. An electron moving around the orbit in the opposite direction, adcb, will be acted on by a force tending to draw it to the centre. Hence its motion will be accelerated and it will make a greater number of revolutions per second. The waves sent to E will then be shorter, and the line seen in the spectroscope will be shifted toward the region of the violet and the light will be cir- cularly polarized in a direction opposite to that of the other line, for the orbits are in an opposite direction. In appljring the conventional left-hand rule or motor rule it appears that the lower Une as seen from E should be circularly polarized clockwise and the upper line counter clockwise. An exam- ination, however, shows that the opposite is true. Hence the electric current is negative, i.e., electrons are negatively charged. Now let the spectroscope be placed at E' and the orbits viewed in a direction at right angles to the magnetic field. No waves are sent out in the direction of the motion of an electron, hence only those components of the orbits at right angles to the line of vision will send waves to E'. The effect will be the same as the vibration of an electron between a and c on a diameter of the orbit. The larger orbit wiU therefore produce plane" polarized light shifted toward the red in the spectnmi. The smaller orbit will also produce plane polarized light, shifted toward the blue. The plane of vibration in both cases is perpendicular to the mag- netic field. What is actually observed when the light is viewed at right angles to the field is a triplet — i.e., three lines — ^the middle one being in its ordinary position and plane polarized at right angles to the other two. The two outside lines have the position and 264 GENERAL PHYSICS. polarization they should have according to the explanation just given. The middle line of the triplet may be explained by con- sidering the effect of the other two orbits shown in Fig. 196, A, which for sake of simplicity were left out of the lower figure. By the amount of separation of the lines in the spectrum Dr. Zeeman was able to calculate the ratio of the charge to the mass, e/m, of an electron and obtained results comparable with those for beta rays and cathode rays. (See § 134.) 180. Artificial Light. — ^Any light produced by the devices of man is termed artificial as contrasted with natural light such as that from the sun. Most methods of producing artificial light are very inefficient. Optical efficiency is the ratio of light energy emitted to the quan- tity of energy expended in producing that light and is usually designated in watts per candle-power (W.P.C.). Thus a carbon lamp at a temperature of about 1850° C. requires 3.5 W.P.C. ; a tantalum lamp at 2000° C, 2 W.P.C; a tungsten lamp at 2100° C, 1.25 W.P.C; and at 2300° C, 1 W.P.C. The best of these lamps, however, have a low efiiciency, about 2 per cent. Methods of producing artificial light consist chiefly in heating bodies to such a temperature that the waves sent out will come within the limits of the visible spectrum. Such waves, as shown in § 164, cover less than one-tenth of the entire spectrum, visible and invisible. Hence, even at the highest temperatures, a great deal of the radiant energy consists of waves too long to produce light. As long as we rely on a process of heating to produce light we will have optical inefficiency. It is found, however, that as temperature is raised optical efiBciency increases, for the point of maximum energy radiation is thus shifted to the shorter wave- lengths. (See Displacement Law, p. 240, " Mechanics and Heat.") The value of the tungsten, gem, tantalvim, Nemst, and other lamps consists in the fact that a high temperature can be used without fusion or disintegration. But even with the best lights of this kind, efficiency is low. The ordinary carbon arc lamp gives out light from the incan- descent carbon terminals. The arc itself is not luminous. Hence this also is a heat lamp; although the temperature is high, the efficiency is not over about 10 per cent. ELECTROMAGNETIC WAVES. 265 A great improvement has been effected by making the arc luminous. This may be done by impregnating the carbons with certain calcitmi salts which are vaporized at the hot terminals and render the carbon arc highly luminous. Such are called flame arcs. In the so-called luminous arcs carbon is not used. One elec- trode is simply a good conductor which does not bum away, but the other is a mixture or compound of oxides of iron and titaniimi. The arc is highly luminous and the luminous particles are at the same time the carriers of the electric current. In both lamps just described efficiency is high, between 30 and 40 per cent., and the effect is not due to high temperatiu-e. 181. Candle-power. — The unit of light giving power of any source is called a candle, from the former use of the candle for that purpose. By the efforts of the Bureau of Standards at Washington the principal nations of the world have agreed to adopt as a unit of candle-power one international candle. The unit is fixed by an agreement to definite relations which shall exist between it and the various concrete standards in practical use by different nations. The standard commonly used in Ger- many, for example, is the Hefner lamp, which bums amyl acetate and when the flame is turned to a height of 4 cm. gives one Hefner unit. The Carcel lamp bums colza oil and gives one Carcel unit when regulated according to certain specifications. This lamp is in common use in France. Another standard is the Vemon- Harcourt pentane lamp in which pentane is burned in a specified manner. The relation of these standards to the international unit is 1 international unit = 1 American candle. = 1 pentane candle. = 1.11 Hefner units. = .104 Carcel unit. The Hefner lamp is extensively used as a flame standard and, as shown above, the value of the Hefner unit is nine-tenths of the international imit. It is difiBcult to maintain flame standards, for they vary with humidity, barometric pressure, etc. Probably the best practical standard is an incandescent lamp the filament of which has been 266 GENERAL PHYSICS. well seasoned by passing a current through it. Such a lamp when standardized and used without over-voltage will remain constant for a long time. 182. Intensity of Illumination. — ^The intensity of illimiination of a surface upon which light falls is the quantity of light per unit area of that surface. Let L, Fig. 197, be a source of light of one candle-power in the centre of a sphere of radius r. Light wiU pass out in form of cones having their vertices at L. Let A be the area of the base of such a cone,. The solid angle B is then and if is unity fl = -— (see Appendix 2) r A=-r* (192) (193) i.e., the area of the base of the cone as measured on the surface of the sphere is equal to the square of the radius, and since the surface of the sphere is 4irr* there are ^ir unit cones of light. vt U- Q ^\ i Cs ii c. V, c, I I I I I I 0: FlQ. 204. It is seen from the discussion just given that the length of air column must be at least one-fourth of the wave-length of the note to which it resounds. By a process of reasoning similar to that above it may be shown that there will be resonance, though not so intense, when the tube A, Fig. 204, is | of the wave-length, for the condensation Ci will then pass down and back and start a rarefaction down with r2. Meantime the rarefaction n, one-half wave-length behind Ci, returns in time to send a condensation back with Cj, and so on. In a similar manner there will be resonance again when the tube is |, \, etc., of the length of a wave. Making use of this method of finding the length of a wave and knowing the frequency of the fork which produces the waves, it is possible to calculate velocity of sound from y=«X (200') where F= velocity, «= frequency, and X= wave-length. 280 GENERAL PHYSICS. A glass tube, Fig. 205, about 3 cm. in diameter and one metre or more in length, is partly filled with water which may be raised or lowered by use of a side tube or pump. A tuning fork making, say, 512 vibrations per second is set in vibration and the water raised to a point, as a, when a loud resonance will be heard. The distance from a to the top of the air column is then X/4. Now let the water be lowered until a second resonance occurs, say at b. Then the distance from b to the top of the air column is 3X/4. It is found that the effective length of the air column extends a short distance above the mouth of the tube to a point where freedom of the air is such that conden- sations and rarefactions are reflected back into the tube as if from a rarer medium. Hence a certain quantity, x, must be added to the actual length of the resonant tube to obtain the length of the reso- nant air column. The value of x is approximately Fio 205 *^^ radius of the tube but varies with the material and size of the tube. Since the value of x is uncertain, it is better to eliminate it. Let Li be the length of the tube as measured from a to the top, Li as measured from b. Then and Li+x = — - 4 Subtracting. L2 — Li = — and )y = 2(L2-Li) (201) The value of X and n may then be substituted in equation (200') 193. Resonance in a Tube Open at Both Ends. — By applica- tion of the principles explained in § 192 it is seen that when the condensation Ci, Fig. 206, arrives at the open end e of the tube it causes a rarefaction to return to 0. This rarefaction on emerging from the tube at o starts a condensation back into the tube and if the length of the tube is one-half the length of the wave, con- densation Ci will join it, thus producing an increased ampUtude of vibration. Likewise n will enter A, retiun a condensation at e. SOUND. 281 and on emerging from o will send back a rarefaction which is joined by r2. Thus the first resonant length for a tube open at both ends is X/2. If the length of the tube is made 2X/2, ci will return to o in time to send a condensation back with C3 and n with rs. If the length of the tube is 3X/2, Ci will return in time to unite with Ci and n with u. Thus, while for any given train of waves the lengths of reso- nant air colimms in a tube closed at one end are X/4, 3X/4, 5X/4, 7X/4, etc., the lengths in tubes open at both ends are X/2, 2X/2, 3X/2, 4X/2, etc. ^ If the length of the air column is kept constant and the length of the sound waves changed, it is evident there will be a series 2of different wave-lengths for which there will be resonance, for the ratio of wave-length to tube length will be the same, for example, I whether the wave-length is kept constant and the tube made i twice as long or the tube kept constant and the wave-length made e* ft Cs r. c, r, c I I I I I I EX^' Fig. 206. one-half its former length. Hence the series of wave-lengths for which there will be resonance in a closed tube will be in the ratio 1, -3-, -g-, -7-, etc., and the corresponding ratio of the ntimber of waves is 1, 3, 5, 7, etc. The first note, i.e., the one having the longest wave-length, is called the fundamental note. The others are overtones. Likewise for an open tube of constant length there will be resonance when wave-lengths bear the ratio 1, -g", t, T, T> ^^^-i or, in terms of frequency, 1, 2, 3, 4, 5, etc. Thus in closed tubes the frequencies for overtones are odd multiples of the frequency for the fundamental, while in open tubes the multiples are both even and odd. 194. Quality. — Of the three principal properties of musical tones, pitch and intensity have already been discussed. Quality has been deferred imtU after a discussion of resonance. According to the investigations of Hebxiholtz, quality depends on the combination of fundamentals and overtones and the relative phase and intensity of the various waves which enter into this combination. 282 GENERAL PHYSICS. To analyze a complex tone such as that produced by striking the key of a piano, Helmholtz used a series of resonators like those shown in Fig. 207. Each of these is tuned to resonance for a certain vibration frequency. By placing these in succession to the ear while a tone is being sounded, the fundamentals and overtones may be detected by the resonance which they produce. Instead of placing the resonators to the ear they may be con- nected in succession to a recording phonograph, and a permanent record thus made of the fundamental and overtones in any given tone. , Fig. 207. Helmholtz then prepared a number of tuning forks whose fre- quencies were the same as those of the components of the tone he had analyzed. When these were sounded together, each with proper amplitude, the original tone was reproduced. Thus tones may be synthesized as well as analyzed, and quality, or timber as it is sometimes called, is that property of a tone by which it is possible to distinguish the tone produced by one in- stnmient from that produced by another, although the pitch and intensity are the same in both. 195. Nodes in Resonant Air G)lumns. — ^A node is a point in a medium where particles in the path of a wave are quiescent. When the condensation ci arrives at the end e of an open pipe whose length is one-half the length of a wave, the rarefaction has arrived at o. Ci will therefore send back a rarefaction from e to meet n at the centre of the pipe. The particles in the line n will thus be subjected to two equal and opposite forces and will have no amplitude. Each rarefaction will return to their respective SOUND. 283 ends of the tube and then return to w as condensations. Here, again, the particles at n will not be moved. An open pipe when giving its fundamental note will always have a node at its centre and an antinode at each end. c. ^ fe. Fig. 208. For the first overtone of an open pipe, the waves are one-half as long as for the fundamental. Hence the distance oe. Fig. 209, is now the length of one wave. When the first condensation ci arrives at e, n will be half-way down the pipe, ci sends back a rarefaction which collides with n at a point one-fourth the length of the pipe from e. Here, then, is a node n^. But n is reflected I I f I I I I H i e Fia. 209. back from wa as a rarefaction and collides with C2 at the middle of the pipe. Since both pulses displace particles in the same direc- tion, there will be maximum disturbance at this point, ca now starts a rarefaction from the middle of the pipe toward o but collides with ra at mi, one-fourth of the length of the pipe from o. r, FiQ. 210. In a similar manner the three nodes of the second overtone or the four nodes of the third may be traced out. A closed pipe is one-f oiuth the wave-length of the fundamental tone, hence ci, Fig. 210, passes down to e and returns as a conden- sation to o by the time n reaches that point. Hence there will be maximum disturbance at o and the only node in the pipe is at the stopped end e. 284 GENERAL PHYSICS. For the first overtone of closed pipes the waves must be one« third as long as for the ftmdamental, hence the length of the pipe is 3X/4. When ci, Fig. 211, has moved to e, n will have moved into the pipe to a position one-third of the distance from o to e. Then C\ will return as a condensation and collide with ri at a point one-fourth of a wave-length from e. A condensation will return from this point and move toward o, meeting cj at a point one-fourth wave-length from o. Here will be a node Mi, the other being at the closed end. '^ "a <=s 15 i, £i, etc., Ca, T>i, Ei, etc. Each note in an octave is also designated by a name as in line (2)! In line (3) is given the ratio of the frequency of vibration of each note to the frequency for the fundamental or keynote C. C may have any frequency but, whatever it is, -g- of it will give the frequency for D, -j- of it will be the frequency for E, and so on through the scale. If the letters are taken to represent fre- quencies, C : 2? = 8 : 9, C : F = 3 : 4, C : 5 = 8 : 15, and so on for any notes in the scale. In Une (4) is given the ratio of each note to the one preceding. These are called intervals. There are three kinds of intervals in the scale, -j, -g-, and jj , the first two being whole tones and the last a half tone. 200. Construction of a Major Diatonic Scctle. — ^When two trains of waves of different wave-length are moving together as shown in Fig. 217 they move with the same velocity, and there are certain regions in the trains where the distiurbance will be a maximum or a minimtmi. Ci in each train is a condensation, 290 GENERAL PHYSICS. hence the displacement of air particles there will be maximum, and the same effect is produced at all points where there is coin- cidence of the c's or the r's in the two trains. At ce and n, how- ever, the vibrations are opposite in phase, i.e., a condensation and a rarefaction are united, and the result will be silence. If two continuous trains of waves of this kind enter the ear, the sound will be alternately loud, then soft, then loud, and so on. Intermittent sounds produced in this manner are called beats. The number of beats per second produced by any two tones is plainly equal to the difference in their vibration frequencies. When, for example, c' and d' are soimded together, the number of beats is 288-256=32. Now the ear is so constructed that when the number of beats is very small or very large, their effect in not unpleasant, but for intermediate numbers, about 30 or 40 in the neighborhood of middle c, the effect is very discordant. -All discord is caused by beats. tc^crcrcrcTC. \ i I l' I'l'l'l ','■', ', I — ■ -' — J ■ — — » Fig. 217. It is necessary, therefore, ia selecting notes for a musical scale, to avoid, as far as possible, those frequencies which when combined would produce discord as a result of beats. It is found that the ratio of the vibration frequencies of two tones is most pleasing when it may be expressed by the smallest numbers. Thus C : c = 1 : 2, the octave, is the most harmonious ratio. The next is C : G = 2 : 3, the fifth; C : F = 3 : 4, the fourth; C : £ = 4 : 5, the major third; E : G = 5 : 6, the minor third. A combination of three tones whose frequencies are in the ratio 4:5:6 produces a pleasing effect. The three tones so related form a major triad. The diatonic scale may be regarded as made up of three such triads. The first embraces the notes C, E, and G. These must have frequencies in the ratio 4 : 5 : 6. Hence, if C = 4, E=-^ and G=-2- oi that number. Whatever the value of Cmay be, E is -j- and G is-g-of it. If C = l, then E=-^, SOUND. 291 and C=-|- as given in the scale on page 289. This is called the triad of the tonic because the first note in it is the tonic or key- note of the scale. The second triad includes the notes G, B, and d. These must also be in the ratio 4 : 5 : 6, or 1 : | : |. The frequency of these three notes in reference to C is, therefore, G=2, S = 4 of 2 ="r> and d=\ of | =|. This is called the triad of the dominant, for G is called the dominant in the scale. In this triad d falls in the octave above. To complete the eight notes of which C is the tonic, Z)=| is used instead of d=|, the frequency for d being. twice that for D. The third triad includes the remaining notes of the scale, viz., F, A, and c. The ratio of frequencies must again be 4 : 5 : 6 or 1 : 1 : f , i.e., "=c and ^=A. Since c is one octave above C, the ratio of their frequencies is C : c = 1 : 2. In reference to C, there- fore,c=2. ^=c = 2, hence F=|. ■v = ^, hence | of | = A=|. This is called the triad of the subdominant, for F is called the subdominant of the scale. This completes the frequency ratios, in reference to the key-note, for the eight notes of the major diatonic scale. A minor triad is composed of three notes having frequencies in the ratio 10 : 12 : 15. By using three of these a minor scale may be constructed in exactly the same manner as that given above for the major scale. 201. Scale of Even Temperament. — The scale described in the preceding paragraph is what is called a scale of just temperament. It serves the purposes of music very well as long as the key-note is C, but it is often desirable to change the key-note for the pur- pose of producing certain musical effects or to accommodate the pitch to the range of singers' voices. For the latter purpose the change in pitch might be a whole octave and then C of that octave could still be the key-note, but such a large change is seldom desired. For the human voice and instruments, such as the violin, that can be adjusted at will, the scale of just temperament can be used with any key-note. But in instruments with fixed key-boards such as pianos and organs, where each wire or pipe always has the same vibration frequency, the frequencies of several notes in the scale will not be in accordance with the requirements of just tern- 292 GENERAL PHYSICS. perament. This may be shown by a comparison of two scales as given below, the first being in the key of C and the second in the key of D. The numbers are the absolute frequencies in the middle octave for physicists' pitch. do re mi fa sol la si do re 1 1 i 1 1 1 ¥ 2 * Key of C 256 288 320 341.3 384 426.6 480 512 576 do re mi fa sol la si do 1 1 i f 1 1 ¥ 2 Key of Z> . 288 32 360 384 432 480 540 576 It is seen that instead of the 320 vibrations which the piano would give if it had only the major diatonic scale, there should be 324 if D is the key-note. Also, instead of 341.3 the nimiber should be 360, instead of 426.6 it should be 432, and instead of 512 it should be 540. For the key of D, then, provision would have to be made for four extra notes. By comparing in this man- ner the frequencies for scales with other key-notes, it is found that there would need to be about 50 notes in each octave. This difficulty could be partly remedied by introducing sharps and flats within the interval of the whole tones. The interval from C to C# is f| and from D\> to D is also ff. If all the whole tones are broken up in this manner, we would have the so-called harmonic scale of 18 notes, C C# Db D D# Eb E 1 IS li I H S S.etc. The intervals in this scale are much shorter, and the frequencies required for the extra notes when diflferent keys are used wiU very nearly agree with one of these 18 notes. But the scale is long and the agreement not as close as it should be in many cases. It has therefore been agreed that the scale shall consist of 12 notes with equal intervals throughout. This is called the scale of even temperament. _ The frequency for any note in this scale mtdtiplied by \/ 2 or 1.059 will give the frequency for the next note. A comparison of the frequencies as given below shows that the intervals do not seriously differ from those of the true scale. C D EP G A BC Just 236 28S 320 341.3 384 '426.7 480 512 Even 256 271.3 287.4 304.8 322.7 341.7 362.2 383.3 406.6 430.7 456.5 483.5 612 SOUND. 293 The numbers between the letters in the even scale are used for both the sharp of the letter below and the flat of the letter above. They correspond to the black keys of the piano. Problems 1. Compare the velocity of sound in air at —17° C. and at 16° C. 2. A bomb dropped from an air-ship explodes on striking the ground and the sound is heard in the'ship 10 seconds after the bomb has started to fall. Temperature of air = 10° C. How high was the ship? 3. What must be the tension of a string 1 m. long and weighing 4 mg. that it may make 256 complete vibrations per second? (Use grams, centi- metres, and dynes.) 4. A closed organ pipe 1 m. in length gives the correct pitch at 15° C. Will the pitch be raised or lowered when the temperature rises to 35° C? 5. If the frequency of c" is 512 and an observer moves toward the source of sound at the rate of 41.37 m. per second in air at 0° C, what tone will he hear? 6. A metal rod 1 m. long vibrates longitudinally and produces nodes and antinodes in a Kundt's tube. If the distance between the nodes in air at 20° C. is 9 cm., what is Hie velocity of sound in the metal rod? 7. What is the ratio of the radii of the e and g strings on a violin, all other conditions being the same? 8. What is the interval between F and c on the major scale? Between C and Tf on the even-tempered scale? Ans. 1. 16 : 17. 2. 384-t-m. 3. 1.7 kg. 4. Raised 3 vibrations per sec. 5. d". 6. 3811.2. 8. (1) i (2) i72r APPENDIX 1. Prove that the absolute potential at a distance n from a charged point is — It is desired to find the work required to bring unit charge from infinity to the point n (see Fig. 7). The intensity of the Q\ field at n in air is^, i.e., intensity or strength varies inversely as the square of the distance. If F is the force at any point, the work done in moving unit positive charge a distance dr is Fdr. Substituting the value of F=-^ and letting W stand for work, J'' I .W = Q /, r-^dr -Q -<-^) u 2. Prove that the intensity of the magnetic field within a solenoid is H = ——- gausses Let a continuous wire be wrapped so as to form the hollow cylinder shown in Fig. 218. This constitutes a solenoid. A cur- rent flowing on the conductor must pass many times around the enclosed space. 294 APPENDIX. 295 Let o be a point within the solenoid and A an elementary ring having a width I and circumference 2irr'. If there are n turns of wire per centimetre in the solenoid, there are nl turns on the ring. Each short arc of the ring may be regarded as an arc of another circle having its centre at o. Hence a unit pole at o will be urged in a direction at right angles to the plane of the drcle around o, e.g., the small arc ab wiU urge the unit pole in the direction oj. A similar arc on the opposite side of the ring will urge the unit pole along oj'. The sum of all these ele- mentary portions is IW for each turn, and 2-Kr'nl for nl turns. The sum of all the forces on o, according to the definition of unit e.m. current, is, therefore, IWlni where i is strength of current and r is the radius of an imaginary circle having its centre at o. A FiQ. 219. Fig. 218. The resultant of aU the forces due to each elementary part of the ring is oF, parallel to the axis of the solenoid and making an angle 6 with of or of. Hence the total force exerted by the elementary ring on unit pole at o is ^ . 2ir/Z F=tn — r^cos 6 Now- 2irr'Z cos 6 is the solid angle at o included between lines drawn from both edges of the elementary ring to o. This will be plain from Fig. 219, where abed is a small portion of the ring of Fig. 218. If o is the centre of a unit sphere, say 1 cm. in radius, there will be 4:ir square centimetres on the surface, each of which may be regarded as subtending unit solid angle at o. There would then be 47r solid angles about o. But at any distance r, the area of the 296 GENERAL PHYSICS. base of a pyramid varies directly as the square of r. Hence if the area cdmn is divided by r* we get the area s on the unit sphere. The area cdmn = abed cos 6, hence the solid angle at o, Fig. 219, is abed cos The area of the ring, Fig. 218, is made up of such areas as abed and the total number of them is 2irr7. Hence the solid angle sub- tended by the ring is 2irr'l — r-cos The point o is surrounded by portions of the solenoid in every direction except at the ends and, if the solenoid is long as compared with its diameter, o may be regarded as completely surrounded. Then by applying the same argument as above, the entire volume of the unit sphere at o will be filled with solid angles except where subtended by the small open space at the ends, and this is negligible under the conditions given. Since there are 47r solid angles at o, F = iTrni gausses if i is in e.m. units, and F = gausses if i is in amperes. 3. The square root of the average of all the squares of the current or e.m.f. at each instant during a period gives the effective current or e.m.f., called also the virtual current or virtual e.m.f. Since the square root of the average of the squares of a series of numbers is larger than the average of those numbers, the virtual current is larger than the average current. To calculate virtual e.m.f. of an alternating current we have, from the sine curve, y=ran.0 Mean square of y = ~zzi APPENDIX. 297 Substituting the value of y=f sin^, and integrating the de- nominator, /2ir Sir Mean square oi y=^ [ sin* dd6 „ . . 1 cos 2d ^ , . . « But sin* 0= — — • Substitutmg this value of sin' 0, Mean square of y= — / — — 2ae «/o ^1*"^ sin2g 2t L 2 ~ 4 10 r* -— a-— o 2ir i! 2 r -»/mean square of y = — t= = .707r Let i, be the virtual strength of current. The maximum ordinate in a sine curve is r = maximum current 4i. Hence . 4. _„_. tv=—7- = -707t,n V2 Likewise for virtual e.m.f. E.= ~=.707E„ V2 4. Prove that if a current i with a frequency m is flowing through a circuit having inductance L, the counter e.m.f. is 27m Li. If the current is expressed in virtual amperes and the inductance in henrys, the e.m.f. will be in virtual volts. Inductance, L, is for unit current, hence the self-induction for current i is Li. Since the self-induced e.m.f. is proportional to the rate of change of Li, e.m.f. = Lf^ (1) at 298 GENERAL PHYSICS. But i=im sin 2imt (2*) di and - — = 2tm4, cos 2imt (3) at di Substituting this value of -r- in (1) and keeping in mind that virtual values of sine and cosine are equal, e.m.f. = 2TmLim sin 2Tmt (4) Then substituting in (4) the value of i as found in (2), e.m.f. =2TmLi 5. Prove that the velocity of a wave in an elastic medium is v= v: It may be assumed, to begin with, that velocity varies directly as the elasticity E which determines the rapidity of transmission from particle to particle and inversely as the density which has a retarding influence. F Elasticity is a force per unit area, i.e., — • This, expressed in \MLT-^\ ftmdamental units, is — rY^ — Density is mass per unit volume or \ML-^]. L But velocity F= — > hence V= f-l = ^ ^"^ ' LrJ i wL-^ Y Now if the exponents a and b are each made to equal |, the r Hence, by choosing such units that no constant need be introduced, *See p. 32, " Mechanics and Heat." APPENDIX. 299 6, Magnifying Power 1. Simple Microscope. — The principle of the simple microscope has already been illustrated in Fig. 177, B. The magnifying power of such an instrtiment is the ratio of the linear dimensions of an image to that of an object. Thus, in Fig. 220, oo' is an object placed less than the focal distance from the lens. An eye at E will then see a virtual image at «'. The magnifying power is then the ratio of it' to oo'. These lines bear the same ratio as their distances from the lens, so if is the object distance and I the image distance, 0~ oo' The general equation for lenses is O'^ I ~ F Fia. 220. but since the image in a simple microscope is always virtual, 1 1 1 I F I or 1 = I F . I "0 ~ h^ Since the distance I for distinct vision is 25 cm. , or 10 inches, we may write Fia. 221. -r = — + 1 =magmfjHing power. C r 2. Compound Microscope. — In the compound microscope the objective Li, Fig. 221, forms a real image of oo' at ii'. The length of the image is as many times greater than the object as the dis- tance Ii is times Oi. Hence the magnification of oo' by the lens Li is Ox 300 GENERAL PHYSICS. The image ii' falls within the focal length of L2, hence the magnification of Lj is the same as that given above for the simple microscope. The magnifying power of both lenses is therefore Ji/25. t(f+0 3. Astronomical Telescope. — In an astronomical refracting telescope the object glass is of large diameter and long focus. Let the diameter of the object 00', Fig. 222, be denoted by Di and its distance by O. Let the angle which rays from o and o' make at lens Li be B. The object is far distant and so the wave- front from it may be regarded as plane. The distance from Ci to the image W is therefore the focal length, Fj, of Li. Fi<3. 222. If the object is viewed directly by the eye without any lens, its diameter, 0, measured in radians, is The ratio of Pj to Pi is equal to the ratio of Fi to 0. Hence A O and D2 = - Now if the image D2 is viewed through an eye-piece L2 placed at its focal distance, Ft, from ii', the apparent angular dimensions, 0, of the image will be F, APPENDIX. 301 Substituting the value of Dt found above, we have for the apparent size of the image OFt But— is the diameter of the object as seen without the tele- scope, hence this has been multiplied by -=r by means of the tele- rt Fi scope. The magnifsdng power is therefore -^' or the ratio of the focal length of objective to eye-piece. 7. Diopters Opticians and spectacle makers usually express the power of a lens in diopters. A diopter is the reciprocal of focal length in metres. If the focal length is 1 m., the power is 1 diopter. If the focal length is 50 cm., 2 diopters; if 4 m., .25 diopter, etc. One metre is 39.37 inches, but 40 inches is usually taken as equal to 1 m. for this purpose. A lens having a 5-iach focus would then have a power of 8 diopters. If an eye cannot clearly see objects at less than 2 m. distant, the focal length in the eye is too long and an image, if it could be formed, would be back of the retina. To remedy this defect a convex lens must be used. If distinct vision at 25 cm. is desired the curvature of the refracted waves must be increased from j^ to ^. The increase is therefore y^o ^^^ so the necessary power of the glasses is 3.5 diopters. 8. The Interferometer An interferometer is any device by which waves in two beams of light are made to interfere and produce fringes or bands which are alternately dark and hght if monochromatic light is used, or colored bands in case of white hght. A very simple arrangement for showing interference bands is shown in Fig. 223. Two pieces of plate glass, A and B, about 8 cm. long and 1 cm. wide, are placed in contact at a and separated very slightly at b. Thus a very thin wedge of air is included be- tween the two plates. When light falls on the top of A it is partly 802 GENERAL PHYSICS. transmitted through both plates and partly reflected from each of the four surfaces, but that reflected from the bottom of A and the top of B will produce interference. That portion of a beam reflected from the top of B must, after reflection, lag behind the portion reflected from the bottom of A by a distance 2d where d is the thickness of the wedge at that point. These two trains of waves will therefore differ in phase, and if this difference amounts to half a wave-length there is destructive interference or darkness. — 1 -T — -r~ ~T -r-rT" : : : • : : B ^ ^, 1 1 1 1 FiQ. 223. kK h But the conditions of reflection are different in these two cases, for in A the ray is reflected in glass against air while on B the reflection is in air against glass. The effect of a change of medium is just the same as in case of sound waves (§ 190), and the differ- ence of phase is therefore increased by — on this accoimt. Hence the total retardation or difference of phase is 2. + I When this retardation amounts to an even number of half wave-lengths, the band seen at that part of the wedge is bright. When it equals an odd number of half wave-lengths, the bands are dark. At a the two plates are in contact and d = 0. Hence 2d+ ■-■ = -jr and there is a black band at this poiat. At 5i, d=—. Hence 2d+yr='K. Here, then, is a bright band, for the two sets of waves X X 3X are in step. At wi, d=—. Hence 2d-|-— - = — , and again there is APPENDIX. 303 a dark line. In like manner it can be shown that there is a dark band when d = —,—,— , etc., for the corresponding retardation Ji a Ji . 5X 7X 9X , If the dark lines are numbered in order from a to 6, 0, 1, 2, 3, 4, etc., it is seen that twice any one of these numbers, plus 1, gives the number of half wave-lengths of retardation. Hence (2«+l)| = 2d + |- 2n\ „, •'• -;;- = 2d 2 ' and X= — n 9. The Michelson Interferometer A form of interferometer invented by Professor Michelson can be conveniently used in making exact physical measurements in a variety of experiments. The apparatus in a simple and useful form consists of four glass plates whose faces are optically plane. Mi and M2, Fig. 224, are silvered on the front surface. Mi is fixed in position except that its plane may be finely adjusted by means of thumb-screws. M2 may be moved back and forth by turning a smaU craiik or a worm gear. A and C are polished glass plates with parallel sides. Rays of light from some source x are made parallel by a lens L, pass through plate A, and are in part reflected from the surface ab to M2, the other part passing on to Mi. The rays in both cases are reflected back to the surface ab and pass thence to the eye at E. The purpose of the plate C is to make the two paths optically equal. Plates A and C are first made in one piece and then cut in two and made exactly equal in thickness. The rays reflected to M2 pass three times through A, and those reflected from Ml pass once through A and twice through C before reaching the eye at E. Plate C is therefore called the compensator. By careful adjustment the mirror M^ and the virtual image of Ml may be made exactly parallel, and when the difference in their optical paths is one-half wave-length, there will be destructive 304 GENERAL PHYSICS. interference and a series of dark and bright bands will be observed. These will move across the field of view when Mz is moved for- ward or backward. These bands or fringes are most readily found when the light is monochromatic, as when a sodium flame is used. A merciuy vapor lamp is also excellent for this purpose. Fio. 224. The number of waves of light in a given length may be counted by focusing a micrometer microscope on a given mark of a scale placed on the carrier of mirror Mz, and then counting the ntimber of dark bands that cross the field while M2 is moved, say, 2 or 3 tenths of a millimetre. The number for 1 mm. is about 3000, but will be more or less for different wave-lengths. When white light is used the mirrors Mi and M2 must be exactly the same distance from ab. The fringes are then colored except the central band, which is dark. A great deal of patience and care is necessary at first in attempt- ing to operate an instrument of this kind, for wave-lengths are very short and a very slight change in the adjustment will produce a marked change in results. This is particularly true for white light. APPENDIX. 305 In a book entitled " Light Waves and Their Uses," by Professor Michelson, the student will find a description of many uses to which an interferometer may be put, and also a full account of the methods employed in measuring the standard metre in terms of light waves. 10. Specific Inductive Capacity, K Air, standard pressure 1 Beeswax 1.86 Flint glass, light 6.85 Flint glass, heavy 10 Glass, common 3 Hard rubber 2-3 Hydrogen 9997 India rubber, pure 2.2-2.5 Mica 4r-8 Paraffin, solid 2-2.3 Petroleum 2-3 Porcelain 4.38 Resin 2.5 Shellac 2.7-3.6 Sulphur 2.5-4 Turpentine 2.2 11. Electrochemical Equivalents Substance. Aluminum. Antimony. . Arsenic Bismuth... Bromine.. . Chlorine... Copper Gold Helium Hydrogen . . Iron Lead Lithium Magnesium Manganese. Mercury... Nickel Nitrogen. . . Oxygen Platinum... Potassium.. Radium Silver Sodium Sulphur Tin Uranium.. . Zinc ~20 Atomic weight. Valence. 27.1 120.2 75 208.5 79.96 35.45 63.6 197.2 4 1.008 55.9 206.9 7.03 24.36 65 200 58.7 14.04 16 194.8 39.15 225 107.93 23.05 32.06 119 238.5 65.4 3 3 3 3 1 1 1 2 3 1 2 3 2 1 2 2 4 1 2 3 2 2 1 1 1 2 2 2 2 Grams per coulomb. .000936 .000415 .000259 .0007199 .0008283 .0003672 .0006588 .0003294 .0006809 .00001036 .0002895 .000193 .0010716 .0000728 .0001262 .0002849 .0001424 .0020717 .0003040 .0000485 .0000829 .0010098 .0004055 .0011180 .0002387 .000166 .0006163 .0012353 .0003387 306 GENERAL PHYSICS. 12. Specific Resistance and Temperature Coefficient Metal. Resistance per c.c. Resistance per mil-foot at 20° C. Temperature coefficient. 3(10)-' 1.584(10)-' 1.619(10)-' 20(10)-' 10.5(10)-' 42(10)-' 94(10)-' 8.9(10)-' 1.5(10)-' 17.4 10.4 10.65 114r-270 90 250-450 ' 58 ' ' 9.53 .00435 Copper, annealed .0042 Copper, hard .00025 Iron, annealed .005 .00001 .00075 Platinum .00366 Silver .00377 .005 13. Thermoelectric Power in Microvolts with Respect to Lead Temp. =20° C. Metal. Microvolts. Metal. Microvolts. -89 -22 -12 - 4.2 1.0 1.2 Silver 3 Cobalt Zinc 3.7 German silver Copper, pure Iron 3.8 17.5 Antimony Tellurium 24 Tin 503 Gold Selenium 807 When a jimction of any two of these metals is heated a current will flow from the higher to the lower metal in the series. The thermoelectric power of any two metals is the difference of their numbers as given in the table. 14. Wave=lengths of Fraunhofer Lines Letter. Line due to Wave-length in microns. Letter. Line due to Wave-length in microns. A Sun Sun H Na Na He .7604 .718478 .6870 .6563 .5896 .5890 .587598 El 6i F G H K Fe Mg H Fe Ca Ca .5270 .518379 B .48616 C .4308 Di D D, .39686 .3934 APPENDIX. 307 15. Indices of Refraction Description. Indices. Red (C). Yellow (D). Blue (P). \riolet (H). Air 1.000293 1.3318 1.6336 1.5419 1.5509 1.65446 1.48474 1.5254 1.5568 1.5783 1.6795 1.000294 1.3336 1.6433 1.5442 1.5534 1.65846 1.48654 1.5280 1.5604 1.5822 1.6858 1.000296 1.3377 1.6688 1.5496 1.5589 1.6679 1.4908 1.5343 1.5690 1.5929 1.7019 1.0003 Water 1.3442 Carbon disulphide 1.7175 Quartz Ordinary ray Extraordinary ray Ordinary ray Extraordinary ray Light 1.5581 Quartz. . . . 1.5677 Iceland spar Iceland spar 1.6833 1.4978 1.5443 Crown glass Dense 1 5836 Flint glass Light 1 6098 Flint, glflsf!, Dense 1 7306 16. Velocity of Sound Substance. Meters per second. Substance. Meters per second. Air(0°C.) 331 1260 5100 3500 3652 262 3560 5000 to 6000 3950 1269 Alcohol Iron. . . . 5000 Aluminum Marble 3810 Brass Oak wood (along the fibre) Oxygen. 3850 Brick 317 Carbon dioxide Pine wood (jjong the fibre) 3320 2690 Glass Silver. 2610 Granite Slate 4510 308 APPENDIX. 17. Natural Sines and Cosines SINE Deg. -(y iW 20' 30' 4^ 50' 60' Deg. 0.00000 0.00291 0.00582 0.00873 0.01164 0.01454 0.01745 89 1. 0.01745 0.02036 0.02327 0.02618 0.02908 0.03199 0.03490 88 2 0.03490 0.03781 0.04071 0.04362 0.04653 0.04943 0.05234 87 S 0.05234 0.05524 0.05814 0.06105 0.06395 0.06685 0.06976 86 4 0.06976 0.07266 0.07556 0.07846 0.08136 0.08426 0.08716 85 5 0.08716 0.09005 0.09295 0.09585 0.09874 0.10164 0.10453 84 6 0.10453 0.10742 0.11031 0.11320 0.11609 0.11898 0.12187 83 7 0.12187 0.12476 0.12764 0.13053 0.13341 0.13629 0.13917 82 8 0.13917 0.14205 0.14493 0.14781 0.15069 0.15356 0.15643 81 9 0.15643 0.15931 0.16218 0.16505 0.16792 0.17078 0.17365 80 10 0.17365 0.17651 0.17937 0.18224 0.18509 0.18795 0.19081 79 11 0.19081 0.19366 0.19652 0.19937 0.20222 0.20507 0.20791 78 12 0.20791 0.21076 0.21360 0.21644 0.21928 0.22212 0.22495 77 13 0.22495 0.22778 0.23062 0.23345 0.23627 0.23910 0.24192 76 14 0.24192 0.24474 0.24756 0.25038 0.25320 0.25601 0.25882 75 15 0.25882 0.26163 0.26443 0.26724 0.27004 0.27284 0.27564 74 16 0.27564 0.27843 0.28123 0.28402 0.28680 0.28959 0.29237 73 17 0.29237 0.29515 0.29793 0.30071 0.30348 0.30625 0.30902 72 18 0.30902 0.31178 0.31454 0.31730 0.32006 0.32282 0.32557 71 19 0.32557 0.32832 0.33106 0.33381 0.33655 0.33929 0.34202 70 20 0.34202 0.34475 0.34748 0.35021 0.35293 0.35565 0.35837 69 21 0.35837 0.36108 0.36379 0.36650 0.36921 0.37191 0.37461 68 22 0.37461 0.37730 0.37999 0.38268 0.38537 0.38805 0.39073 67 23 0.39073 0.39341 0.39608 0.39875 0.40142 0.40408 0.40674 66 24 0.40674 0.40939 0.41204 0.41469 0.41734 0.41998 0.42262 65 25 0.42262 0.42525 0.42788 0.43051 0.43313 0.43575 0.43837 64 26 0.43837 0.44098 0.44359 0.44620 0.44880 0.45140 0.45399 63 27 0.45399 0.45658 0.45917 0.46175 0.46433 0.46690 0.46947 62 28 0.46947 0.47204 0.47460 0.47716 0.47971 0.48226 0.48481 61 29 0.48481 0.48735 0.48989 0.49242 0.49495 0.49748 0:50000 60 30 0.50000 0.50252 0.50503 0.50754 0.51004 0.51254 0.51504 59 31 0.51504 0.51753 0.52002 0.52250 0.52498 0.52745 0.52992 58 32 0.52992 0.53238 0.53484 0.53730 0.53975 0.54220 0.54464 57 33 0.54464 0.54708 0.64951 0.55194 0.55436 0.55678 0.55919 56 34 0.55919 0.56160 0.56401 0.56641 0.56880 0.57119 0.57358 55 35 0.57358 0.57596 0.57833 0.58070 0.58307 0.58543 0.58779 54 36 0.58779 0.59014 0.59248 0.59482 0.59716 0.59949 0.60182 53 37 0.60182 0.60414 0.60645 0.60876 0.61107 0.61337 0.61566 52 38 0.61566 0.61795 0.62024 0.62251 0.62479 0.62706 0.62932 51 39 0.62932 0.63158 0.63383 0.63608 0.63832 0.64056 0.64279 50 40 0.64279 0.64501 0.64723 0.64945 0.65166 0.65386 0.65606 49 41 0.65606 0.65825 0.66044 0.66262 0.66480 0.66697 0.66913 48 42 0.66913 0.67129 0.67344 0.67559 0.67773 0.67987 0.68200 47 43 0.68200 0.68412 0.68624 0.68835 0.69046 0.69256 0.69466 46 44 0.69466 0.69675 0.69883 0.70091 0.70298 0.70505 0.70711 45 60' sc 40' 30' 20' 10' 0' COSINE APPENDIX. 309 Natural Sines and Cosines — Continued COSINE Deg. W iw 20" 30' 40' SO- 60' Deg. 1.00000 1.00000 0.99998 0.99996 0.99993 0.99989 0.99985 89 1 0.99985 0.99979 0.99973 0.99966 0.99958 0.99949 0.99939 88 2 0.99939 0.99929 0.99917 0.99905 0.99892 0.99878 0.99863 87 3 0.99863 0.99847 0.99831 0.99813 0.99795 0.99776 0.99756 86 4 0.99756 0.99736 0.99714 0.99692 0.99668 0.99644 0.99619 85 5 0.99619 0.99594 0.99567 0.99540 0.99511 0.99482 0.99452 84 6 0.99452 0,99421 0.99390 0.99357 0.99324 0.99290 0.99255 83 7 0.99255 0.99219 0.99182 0.99144 0.99106 0.99067 0.99027 82 8 0.99027 0.98986 0.98944 0.98902 0.98858 0.98814 0.98769 81 9 0.98769 0.98723 0.98676 0.98629 0.98580 0.98531 0.98481 80 10 0.98481 0.98430 0.98378 0.98325 0.98272 0.98218 0.98163 79 11 0.98163 0.98107 0.98050 0.97992 0.97934 0.97875 0.97815 78 12 0.97815 0.97754 0.97692 0.97630 0.97566 0.97502 0.97437 77 13 0.97437 0.97371 0.97304 0.97237 0.97169 0.97100 0.97030 76 14 0.97030 0.96959 0.96887 0.96815 0.96742 0.96667 0.96593 75 15 0.96593 0.96517 0.96440 0.96363 0.96285 0.96206 0.96126 74 16 0.96126 0.96046 0.95964 0.95882 0.95799 0.95715 0.95630 73 17 0.95630 0.95545 0.95459 0.95372 0.95284 0.95195 0.95106 72 18 0.95106 0.95015 0.94924 0.94832 0.94740 0.94646 0.94552 71 19 0.94552 0.94457 0.94361 0.94264 0.94167 0.94068 0.93969 70 20 0.93969 0.93869 0.93769 0.93667 0.93565 0.93462 0.93358 69 21 0.93358 0.93253 0.93148 0.93042 0.92935 0.92827 0.92718 68 22 0.92718 0.92609 0.92499 0.92388 0.92276 0.92164 0.92050 67 23 0.92050 0.91936 0.91822 0.91706 0.91590 0.91472 0.91355 66 24 0.91355 0.91236 0.91116 0.90996 0.90875 0.90753 0.90631 65 25 0.90631 0.90507 0.90383 0.90259 0.90133 0.90007 0.89879 64 26 0.89879 0.89752 0.89623 0.89493 0.89363 0.89232 0.89101 63 27 0.89101 0.88968 0.88835 0.88701 0.88566 0.88431 0.88295 62 28 0.88295 0.88158 0.88020 0.87882 0.87743 0.87603 0.87462 61 29 0.87462 0.87321 0.87178 0.87036 0.86892 0.86748 0.86603 60 30 0.86603 0.86457 0.86310 0.86163 0.86015 0.85866 0.85717 59 31 0.85717 0.85567 0.85416 0.85264 0.85112 0.84959 0.84805 58 32 0.84805 0.84650 0.84495 0.84339 0.84182 0.84025 0.83867 57 33 0.83867 0.83708 0.8.3549 0.83389 0.83228 0.83066 0.82904 56 34 0.82904 0.82741 0.82577 0.82413 0.82248 0.82082 0.81915 55 35 0.81915 0.81748 0.81580 0.81412 0.81242 0.81072 0.80902 54 36 0.80902 0.80730 0.80558 0.80386 0.80212 0.80038 0.79864 53 37 0.79864 0.79688 0.79512 0.79335 0.79158 0.78980 0.78801 52 38 0.78801 0.78622 0.78442 0.78261 0.78079 0.77897 0.77715 51 39 0.77715 0.77531 0.77347 0.77162 0.76977 0.76791 0.76604 50 40 0.76604 0.76417 0.76229 0.76041 0.75851 0.75661 0.75471 49 41 0.75471 0.75280 0.75088 0.74896 0.74703 0.74509 0.74814 48 42 0.74314 0.74120 0.73924 0.73728 0.73531 0.73333 0.73135 47 43 0.73135 0.72937 0.72737 0.72537 0.72337 0.72136 0.71934 46 44 0.71934 0.71732 0.71529 0.71325 0.71121 0.70916 0.70711 45 dC 50' 40' 30' 20' IC 0' SINE 310 APPENDIX. IS. Natural Tangents and Cotangents TANGENT " Deg. C IC 20' 30* 40' 50' 60' Deg. 0.00000 0.00291 0.00582 0.00873 0.01164 0.01455 0.01746 89 1 0.01746 0.02036 0.02328 0.02619 0.02910 0.03201 0.03492 88 2 0.03492 0.03783 0.04075 0.04366 0.04658 0.04949 0.05241 87 3 0.05241 0.05533 0.05824 0.06116 0.06408 0.06700 0.06993 86 4 0.06993 0.07285 0.07578 0.07870 0.08163 0.08456 0.08749 85 5 0.08749 0.09042 0.09335 0.09629 0.09923 0.10216 0.10510 84 6 0.10510 0.10805 0.11099 0.11394 0.11688 0.11983 0.12278 83 7 0.12278 0.12574 0.12869 0.13165 0.13461 0.13758 0.14054 82 8 0.14054 0.14351 0.14648 0.14945 0.15243 0.15540 0.15838 81 9 0.15838 0.16137 0.16435 0.16734 0.17033 0.17333 0.17633 80 10 0.17633 0.17933 0.18233 0.18534 0.18835 0.19136 0.19438 79 11 0.19438 0.19740 0.20042 0.20345 0.20648 0.20952 0.21256 78 12 0.21256 0.21560 0.21864 0.22169 0.22475 0.22781 0.23087 77 13 0.23087 0.23393 0.23700 0.24008 0.24316 0.24624 0.24933 76 14 0.24933 0.25242 0.25552 0.25862 0.26172 0.26483 0.26795 75 15 0.26795 0.27107 0.27419 0.27732 0.28046 0.28360 0.28675 74 16 0.28675 0.28990 0.29305 0.29621 0.29938 0.30255 0.30573 73 17 0.30573 0.30891 0.31210 0.31530 0.31850 0.32171 0.32492 72 18 0.32492 0.32814 0.33136 0.33460 0.33783 0.34108 0.34433 71 19 0.34433 0.34758 0.35085 0.35412 0.35740 0.36068 0.36397 70 20 0.36397 0.36727 0.37057 0.37388 0.37720 0.38053 0.38386 69 21 0.38386 0.38721 0.39055 0.39391 0.39727 0.40065 0.40403 68 22 0.40403 0.40741 0.41081 0.41421 0.41763 0.42105 0.42447 67 23 0.42447 0.42791 0.43136 0.43481 0.43828 0.44175 0.44523 66 24 0.44523 0.44872 0.45222 0.45573 0.45924 0.46277 0.46631 65 25 0.46631 0.46985 0.47341 0.47698 0.48055 0.48414 0.48773 64 26 0.48773 0.49134 0.49495 0.49858 0.50222 0.50587 0.50953 63 27 0.50953 0.51320 0.51688 0.52057 0.52427 0.52798 0.53171 62 28 0.53171 0.53545 0.53920 0.54296 0.54673 0.55051 0.55431 61 29 0.55431 0.55812 0.56194 0.56577 0.56962 0.57348 0.57735 60 30 0.57735 0.58124 0.58513 0.58905 0.59297 0.59691 0.60086 59 31 0.60086 0.60483 0.60881 0.61280 0.61681 0.62083 0.62487 58 32 0.62487 0.62892 0.63299 0.63707 0.64117 0.64528 0.64941 57 33 0.64941 0.65355 0.65771 0.66189 0.66608 0.67028 0.67451 56 34 0.67451 0.67875 0.68301 0.68728 0.69157 0.69588 0.70021 55 35 0.70021 0.70455 0.70891 0.71329 0.71769 0.72211 0.72654 54 36 0.72654 0.73100 0.73547 0.73996 0.74447 0.74900 0.75355 53 37 0.75355 0.75812 0.76272 0.76733 0.77196 0.77661 0.78129 52 38 0.78129 0.78598 0.79070 0.79544 0.80020 0.80498 0.80978 51 39 0.80978 0.81461 0.81946 0.82434 0.82923 0.83415 0.83910 50 40 0.83910 0.84407 0.84906 0.85408 0.85912 0.86419 0.86929 49 41 0.86929 0.87441 0.87955 0.88473 0.88992 0.89515 0.90040 48 42 0.90040 0.90569 0.91099 0.91633 0.92170 0.92709 0.93252 47 43 0.93252 0.93797 0.94345 0.94896 0.95451 0.96008 0.96569 46 44 0.96569 0.97133 0.97700 0.98270 0.98843 0.99420 1.00000 45 60' SC 40' 30' 20* 10' 0' COTANGENT APPENDIX. 311 Natural Tangents and Cotangents — Continued COTANGENT Deg. C IC 20' 30' 40' 80' 60' Deg. Infini 343.77371 171.88540 114.58865 85.93979 68.75009 57.28996 89 1 57.28996 49.10388 42.96408 38.18846 34.36777 31.24158 28.63625 88 2 28.63625 26.43160 24.54176 22.90377 21.47040 20.20555 19.08114 87 3 19.08114 18.07498 17.16934 16.34986 15.60478 14.92442 14.30067 86 4 14.30067 13.72674 13.19688 12.70621 12.25051 11.82617 11.43005 85 5 11.43005 11.05943 10.71191 10.38540 10.07803 9.78817 9.51436 84 6 9.51436 9.25530 9.00983 8.77689 8.55555 8.34496 8.14435 83 7 8.14435 7.95302 7.77035 7.59575 7.42871 7.26873 7.11537 82 8 7.11537 6.96823 6.82694 6.69116 6.56055 6.43484 6.31375 81 9 6.31375 6.19703 6.08444 5.97576 5.87080 5.76937 5.67128 80 10 5.67128 5.57638 5.48451 5.39552 5.30928 5.22566 5.14455 79 11 5.14455 5.06584 4.98940 4.91516 4.84300 4.77286 4.70463 78 12 4.70463 4.63825 4.57363 4.51071 4.44942 4.38969 4.33148 77 13 4.33148 4.27471 4.21933 4.16530 4.11256 4.06107 4.01078 76 14 4.01078 3.96165 3.91364 3.86671 3.82083 3.77595 3.73205 75 15 3.73205 3.68909 3.64705 3.60588 3.56577 3.52609 3.48741 74 16 3.48741 3.44951 3.41236 3.37594 3.34023 3.30521 3.27085 73 17 3.27085 3.23714 3.20406 3.17159 3.13972 3.10842 3.07768 72 18 3.07768 3.04749 3.01783 2.98869 2.96004 2.93189 2.90421 71 19 2.90421 2.87700 2.85023 2.82391 2.79802 2.77254 2.74748 70 20 2.74748 2.72281 2.69853 2.67462 2.65109 2.62791 2.60509 69 21 2.60509 2.58261 2.56046 2.53865 2.51715 2.49597 2.47509 68 22 2.47509 2.45451 2.43422 2.41421 2.39449 2.37504 2.35585 67 23 2.35585 2.33693 2.31826 2.29984 2.28167 2.26374 2.24604 66 24 2.24604 2.22857 2.21132 2.19430 2.17749 2.16090 2.14451 65 25 2.14451 2.12832 2.11233 2.09654 2.08094 2.06553 2.05030 64 26 2.05030 2.03526 2.02039 2.00569 1.99116 1.97680 1.96261 63 27 1.96261 1.94858 1.93470 1.92098 1.90741 1.89400 1.88073 62 28 1.88073 1.86760 1.85462 1.84177 1.82906 1.81649 1.80405 61 29 1.80405 1.79174 1.77955 1.76749 1.75556 1.74375 1.73205 60 30 1.73205 1.72047 1.70901 1.69766 1.68643 1.67530 1.66428 59 31 1.66428 1.65337 1.64256 1.63185 1.62125 1.61074 1.60033 58 32 1.60033 1.59002 1.57981 1.56969 1.55966 1.54972 1.53987 57 33 1.53987 1.53010 1.52043 1.50184 1.50133 1.49190 1.48256 56 34 1.48256 1.47330 1.46411 1.45501 1.44598 1.43703 1.42815 55 35 1.42815 1.41934 1.41061 1.40195 1.39336 1.38484 1.37638 54 36 1.37638 1.36800 1.35968 1.35142 1.34323 1.33511 1.32704 53 37 1.32704 1.31904 1.31110 1.30323 1.29541 1.28764 1.27994 52 38 1.27994 1.27230 1.26471 1.25717 1.24969 , 1.24227 1.23490 51 39 1.23490 1.22758 1.22031 1.21310 1.20593 1.19882 1.19175 50 40 1.19175 1.18474 1.17777 1.17085 1.16398 1.15715 1.15037 49 41 1.15037 1.14363 1.13694 1.13029 1.12369 1.11713 1.11061 48 42 1.11061 1.10414 1.09770 1.09131 1.08496 1.07864 1.07237 47 43 1.07237 1.06613 1.05994 1.05378 1.04766 1.04158 1.03553 46 44 1.03553 1.02952 1.02355 1.01761 1.01170 1.00583 1.00000 45 60' sc 40' 30' 20' 10' c TANGENT INDEX Aberration, 216, 236, 248 Absolute electrometer, 27 index, 222 potential, 11 Achromatic lens, 248 Acoustics, 272 Actinic efiEect, 246 Action at a distance, 10 After image, 250 Agonic line, 58 Alpha rays, 196 Alternating current, 151, 153 advantage in use of, 178 Ammeter, 107 Ampere, 62, 65 turn, 141 Amplitude, 27l' Analysis, by spectroscope, 248 Analyzer, 254 Angle of prism, 227 AngstrSm unit, 244 Anions, 69 Anode, 68, 75 Arc lamp, 264 Artificial light, 264 Atomic weights, 305 Atom of electricity, 76, 79 Atoms, number of, 79 Ayrton shunt, 121 B Balanced bridge, 116 Ballistic galvanometer, 106 Battery, E.M.F. of, 64, 88, 90 kinds of, 91 Beam, 202 Beats, 290 Becquerel, 194 Beta rays, 196 velocity of, 196 B-H diagram, 48 Bichromate cell, 91 Brewster's law, 254 Bunsen's cell, 91 photometer, 267 Cadmium cell, 92 vapor, 243 Calcite, 256 Candle-power, 265 Capacity of conductor, 16 of condenser, 134 Capillary electrometer, 84 Carbon, 113 arc, 264 lamp, 264 Carcel lamp, 265 Cathode, 68, 75 rays, 4, 197 Cations, 69 Caustics, 216 Cells, arrangement of, 94 kinds, 91 Charge on electron, 199 on ion, 78 on points, 19 on raindrops, 22 Charging by induction, 15 Chemical equivalent, 76 Choke coil, 170 Chromatic aberration, 248 Circular mil, 112 polarization, 258 how detected, 260 Clark cell, 92 Clausius, 69 Coefi5cient of self-induction, 163 Coercive force, 49 Coherer, 191 Collimator, 226 313 S14 INDEX. Color, 249 absorption of, 251 blindness, 250 of thin plates, 260 Commutator, 153 Complementary colors, 250 Complex tone, 275 Compound microscope, 299 Concave lenses, 231 mirrors, 210 Concrete standards, 65 Condensations, reflection of, 278 Condensers, 24, 132, 134 capacity of, 24, 132 of induction coil, 169 standard, 132 time to charge, 137 Conductivity, 72 molecular, 72 Conductor in magnetic field, 102 Conductors in parallel, 113 in series, 113 Conjugate foci, 215 Consequent poles, 43 Consonance, 290 Constant of dielectric, 137 of galvanometer, 104 Contact P.D., 83 Continuous spectrum, 243 Convex lenses, 231 mirrors, 214 Corpuscular theory, 199 Coulomb, 62 Coulometer, 79 Counter E.M.F., 86 Critical angle, 223 Crookes, 4 Crystal detector, 192 Curie, 195 Current, alternating, 153 direct, 154 eddy, 155 e.m., 61 energy of, 65 Foucault, 155 heat effects, 66 virtual, 172, 296 Curvature, 209 Cycle, 151 D Damping, electromagnetic, 104, 107, 156 Daniell cell, 87 D'Arsonval galvanometer, 101 constant of, 104 D.C. dynamo, 153 motor, 161 Dead beat, 104, 156 Declination, 57 Delta connection, 160 Detector, 190 Dextrose, 261 Diamagnetic, 38 Diatonic scale, 288 construction of, 289 Dielectric, 17 constant of, 137 Difference of potential, 11 Diffraction grating, 238 Dimensions e.m. units, 181 e.s. units, 33 Diopters, 301 Dip, 56 Dipping needle, 56 Direct current, 154 Direction of field, 8 Direct vision spectroscope, 249 Disc dynamo, 155 Dispersion, irrational, 243 Dispersive power, 241, 245 Dissociation, 68 cause of, 70 Distribution of charge, 19 Dolezalek electrometer, 28 Dominant, 291 Doppler's principle, 245 Drum armature, 154 Dry cell, 91 Dynamo, 149 D.C, 153 rule, 146 Dynamometer, 109 INDEX. 315 Earth inductor, 148 Echo, 272 Eddy current, 155 EflEective units, 172, 296 EfiBciency, optical, 264 Electric induction, 14 machine, 30 nodes, 189 oscillations, 185, 192 resonance, 186 spark, 20 whirl, 20 Electricity, 1 kinds of, 7 Electrocardiagrams, 106 Electrochemical equivalent, 77 table of, 305 Electrodynamometer, 108 Electrolysis, 73 laws of, 75 of water, 75 uses of, 80 Electrolyte, 68, 73 Electrolytic cell, 68 copper, 80 detector, 191 Electromagnet, 39, 139 energy of, 164 Electromagnetic damping, 104, 107, 156 induction, 144 waves, 185 Electromagnetism, laws of, 145 water analogy, 144 Electrometers, absolute, 27 capillary, 84 • quadrant, 28 Electron, charge on, 199, 264 current, 66 evidence of, 3, 196, 261 magnetic field of, 252 mass of, 199 theory of, 2, 3 Electrophorus, 30 Electroplating, 80 Electroscope, 15 Electrostatic voltmeter, 108 induction, 14, 144 Electrotyping, 80 e.ffi. units, 181 e.m. and e.s. units, 62, 136 E.M.F., 63 counter, 297 of battery, 64, 88, 89 virtual, 296 Elliptic polarization, 258 how detected, 260 Energy of battery, 90 of charge, 21 of current, 66, 171 of magnetic field, 164, 169 Equation of lenses, 231 Equipotential surfaces, 13 Ether strain, 10 luminiferous, 199 waves, 185, 270 Even temperament, 291 Exchange of R and X, 118 Exciter, 153 Extra current, 169 Extraordinary ray, 256 Farad, 132 Faraday, 17, 18, 68, 75, 144 Ferromagnetic, 39 Field at right angles, 54 direction of, 8 in solenoid, 139, 295 magnetic, 45, 52, 148 of force, 8 strength of, 46 unit of, 8 within a conductor, 17 Figure of merit, 104 Flame arc, 265 Floating magnet, 37 Flux, 140 density, 141 Focal length, 212 Foot-candle, 266 Foucault current, 155 316 INDEX. Franklin, 1 Fraunhofer lines, 244 table of, 306 Frequency, 151 Fresnel, 200 Fundamental, 275 Galvani, 82 Galvanic cell, 82 Galvanometer, astatic, 100 ballistic, 106 . D'Arsonval, 101 string, 105 tangent, 98 Gamma rays, 196 Gauss, 46 eye-piece, 227 Geometrical optics, 199 Gilbert, 141 Gram-molecule, 71 Gramme ring, 154 Grating, 238 Grotthus, 68 Grove cell, 91 H Hall effect, 259 Heat and electricity, 125 from current, 66 Hefner lamp, 265 HeUx, 139 Henry, 144, 164 Hertz, 187 Heusler alloys, 38 Horizontal component, 54 Horse-power, 67 Huygens' principle, 200 Hysteresis, 48 loop, 50 I Iceland spar, 253, 256 Ice-pail experiment, 17 I-H diagram, 48 Illumination, 266 Image, 211, 212, 235 Impedance, 173, 176 coil, 170 Incandescent lamp, 264 as standard, 266 Inclination, 56 Index of refraction, 220, 222, 226 table of, 307 Inductance, 163 Induction, charging by, 15 coil, 168 electromagnetic, 144 electrostatic, 14, 144 magnetic, 46 motor, 160 Influence machine, 32 Infra red, 246 Intensity, of illumination, 266 of magnetization, 47 of sound, 276 Interference, 202, 205 Interferometer, 301 International candle, 265 pitch, 289 Intervals, 289, 290 Inversion, 126 Ionization of air, 20, 21, 194 Ions, .69 charge on, 78 Irrational dispersion, 243, 248 Isoclinic lines, 56 Isogonic lines, 57 Just temperament, 291 K Kelvin effect, 130 galvanometer, 101 Key-note, 290 Kilowatt, 67 hour, 67 Kirchhoff's laws, 113 Kundt's tube, 285 INDEX. 317 Lag, 175 Laminated core, 156, 169 Langevin, 38 La Place, 274 Lateral displacement, 221 Law of attraction, 7 of Brewster, 254 of incidence and reflection, 209 of inverse squares, 266 of magnets, 44 Laws of electric charges, 7 of electrolysis, 75 of electromagnetic induction, 145 of field in coil, 61 of strings, 286 Lead, 175 Leclanchfi cell, 91 Lenard rays, 5 Lens equation, 231 Lenses, 230 achromatic, 248 thick, 236 Lenz's law, 145 Light, 193 definition of, 201 theories of, 199 velocity of 206 Lightning, 22 arresters, 23 rods, 23 Light waves, 194 Lines of force, 9, 45, 148 Local action, 90 Location bf charge, 17 Lodestone, 39 Longitudinal vibrations, 271 Lumen, 266 Luminous arc, 265 Lummer-Brodhun photometer, 267 Lux, 266 M M and H, 55 Magnesian stone, 39 Magnetic detector, 192 elements, 59 Magnetic flux, 140 induction, 46, 144 moment, 51 saturation, 48 Magnetism, 36 temperature efiEects, 50 terrestrial, 56, 148 Magnetite, 39 Magnetization, 47 intensity of, 47 Magnetometer, deflection, 55 vibration, 52 Magnetomotive force, 141 Magnetophone, 43 Magnets, 39 field of, 52 law of, 44 niqkel and cobalt, 38 poles of, 41 theory of, 40 Magnifying power, 299 Major triad, 290 Manganin, 113 Maxwell's theory, 183, 187, 271 rule, 117 Measurement of inductance, 165 Mechanical equivalent of heat, 67 Media, 270 change of, 278 Mercury arc, 179 vapor, 243 Metre-candle, 266 Michelson, 206 interferometer, 303 Microfarad, 132 Microscope, simple, 299 compound, 299 Middle c, 289 Mil-foot, 112 Millihenry, 164 Minimum deviation, 226, 228 Mixing colors, 250 pigments, 251 Molar solutions, 71 Molecular conductivity, 72 Molecules, number of, 79 Moment, magnetic, 51 318 INDEX. Motor, D.C., 161 induction, 160 rule, 102 synchronous, 161 Multipolar dynamo, 152 Musical tones, 275 Mutual induction, 168 N Negative charges, 7 Neutral point, 126 Newton, 199, 273 Nicol prism, 258 Nodes, 282, 288 Noise, 275 Nonpolarizing cell, 87 Normal cell, 92 solution, 70 spectrum, 239 O Octave, 289 Oersted, 144 Ohn, 65 Ohn's law, 64 of magnetic circuit, 142 0, 7, and F, 212, 215, 230, 232 Optical centre, 234 eflBciency, 264 Optics, 199 Ordinary ray, 256 Organ pipes, 284 Oscillatory discharge, 194 Overtone, 275 Pacinotti's ring, 154 Parabolic mirror, 217 Parallel arrangement, 95, 113, 133, 142, 156 Paramagnetic, 38 P.D., 11 P.D. due to charged point, 12, 294 P.D. in e.m. and e.s. systems, 63 Peltier effect, 128 reversibility of, 129 Pencil, 202 Pentane lamp, 265 Permanent magnet, 40 Permeability, 38, 47 and reluctance, 142 Phase, 157 Photometers, 267 Photometric standards, 265 Physical optics, 199 Physicist's pitch, 289 Pitch, 276, 289 Pitchblende, 195 Polariscope, 254 Polarization, 86 by reflection, 253 by refraction, 256 circular, 258 elliptic, 258 Polarized light, 251 angle of, 255 partial, 252 plane, 252 Polarizer, 254 Poles, consequent, 43 definition of, 42 location of, 42 of earth, 56 of electromagnet, 139 of magnets, 41 of solenoid, 139 unit, 44 Polonium, 195 Polyphase generator, 157 Positive charges, 7 Post oflSce bridge, 118 Potential, 10, 294 absolute, 11 Potentiometer, 122 PoweC factor, 178 of A.C., 177 Practical units, 182 Primary cells, 90 Principal focus, 212, 214, 216 Prismatic spectra, 242 Prisms, 225, 227 Nicol, 258 Pure spectra, 239 tone, 275 INDEX. 319 Quality of tone, 281 Quantity of electricity, 7 unit of, S Quarter-wave plate, 259 Quartz, dextrogyrate, 261 levogyrate, 261 Radio activity, 194 communication, 190 Radium, 195 Rarefaction, 270 reflection of, 278 Ratio e.m. to e.s. units, 136 e to m, 198, 264 Ray, 201 Reactance, 177 Rectifier, 179 Rectilinear propagation, 203 Red blind, 250 Reflection, 208 grating, 242 of sound, 272 Refraction, 219 by prism, 225 by lens, 229 Relative index, 222 Reluctance, 142 Remanence, 49 Residual charge, 29 magnetism, 49 Resistance, 64 of alloys, 113 of carbon, 113 specific, 112, 306 temperature coefficient of, 113 Resolving power, 241 Resonance, 186, 189 in closed tubes, 279 in open tubes, 280 Resonators, 282 Resultant field, 157 Retentivity, 49 Right-angled prism, 225 Ring armature, 154 Rotary transformer, 179 Rotating field, 161 magnet, 153 plane of polarization, 261 Rotor, 153, 161 Saccharimeter, 261 Sagitta, 210 Saturation, magnetic, 48 Scale, diatonic, 288 even tempered, 291 harmonic, 292 just tempered, 291 Sechometer, 166 Secular change, 59 Seebeck effect, 125 Self-induction, 162 Sensitive flame, 205 Sensitiveness, 105 Series, 99, 113, 133, 142, 156 parallel, 95 Shunt, 120 Ayrton, 121 box, 121 Simple microscope, 236, 299 Sine curve, 173 Single-phase generator, 157 motor, 161 Siren, 276 Slide wire bridge, 117 Solar spectrum, 243 Solenoid, 139, 295 in circle, 140 SoUd angle, 266, 295 Sound, 270, 272, 279 intensity of, 277 velocity of, 272, 286, 307 waves, 205, 277 Specific inductive capacity, 29 table of, 305 resistance, 112 table of, 306 reluctance, 142 Spectrometer, 226 Spectroscope, 247 direct vision, 249 320 INDEX. Spectrum, 237, 239, 243 analysis by, 248 bright line, 243 continuous, 243 limits of, 246 of iron, 244 reversed, 243 Speculum metal, 218 Spherical aberration, 216 Squirrel cage armature, 161 Standard cells, 91, 123 condensers, 132 lamp, 266 of candle-power, 265 of induction, 166 Stationary waves, 288 Stator, 163, 161 St. Elmo's fire, 20 Step-up transformer, 171 Storage cells, 93 Edison, 94; Strength of field, 8, 46, 55 of current, 61, 62 String galvanometer, 105 Strings, vibration of, 286 law of, 289 Subdominant, 291 Susceptibility, 47 Synchronism, 162 Synchronous motor, 161 Table of units, 182 Tangent galvanometer, 98 Tantalum, 264 Telescope, 300 Temperature coefficient of resistance, 113 table of, 306 effect on light, 264 on velocity of sound, 274 of inversion, 126 Terrestrial magnetism, 56, 148 Tesla transformer, 171 Theory of electricity, 1, 2 of light, 199 of magnets, 40 Thermocouple, 126 Thermoelectricity, 125 uses of, 130 Thermoelectric power, 127 table of, 306 Thick lenses, 236 Thomson, 5 Three-phase current, 159 Three-wire system, 158 Timber, 281 Toepler-Holtz machine, 32 Tone, 275 quality of, 281 Tonic, 290 Total reflection, 223 Transformer, 142, 169 Transmission grating, 242 Transverse vibrations of light, 253 Tubes of force, 9 Tungsten, 264 Two-phase generator, 158 U Ultra violet,, 246 Unit current, 61 field, 8 '! fundamental, 75 of resistance, 64 P.D., 63 pole, 44 quantity of electricity, 8 virtual, 172 Units, table of, 182 international, 65 Universal shunt, 121 Uranium, 195 Valence, 76 table of, 305 Van't HoflE, 70 Variable standard of self-induction, 166 Variation, 57 Velocity of beta rays, 196 of light, 206, 208 of sound, 27E, 298, 307 INDEX. 321 Vibration of strings, 286 Virtual units, 172, 296 image, 211 Visible spectrum, 246 Volt, 63, 65, 148 Volta, 82 Voltage of cell, 85 Voltaic cell, 82 Voltameter, 78, 79 Voltmeter, 107 electrostatic, 108 W Watt, 67 Wattless, 177 Wattmeter, 109 Watts, P.C, 264 Wave-front, 201 Wave-length of light, 240, 306 of sound, 279 Wave-lengths compared, 193 Wave theory, 200 Waves of light, 194 of sound, 271 in elastic medium, 298 stationary, 288 Weston cell, 92 Wheatstone bridge, 115 P.O. form, 118 slide wire form, 117 Wimshurst machine, 33 Wireless telegraphy, 190 law regarding, 190 X-rays, 246 Y Y-connection, 160 Young, 200 Young-Helmholtz theory, 249 Z Zeeman eSect, 261