UNIVERSITY OF 6AUFQW MAR 1 2 1923 DIVISIOS PRACTICAL ELECTRICITY Practical Electricity A Laboratory and Lecture Course For First Year Students of Electrical Engineering, based on the Practical Definitions of the Electrical Units BY W. E. AYRTON If F.R.S., ASSOC. M. INST. G.E. Past President of the Institution of Electrical Engineers, Late Professor of Electrical Engineering, Central Technical College REVISED AND LARGELY REWRITTEN BY T. MATHER Wh. Sch., F.R.S., M.I.E.E. Professor of Electrical Engineering, Central Technical College, Imperial College of Science and Technology. South Kensington WITH OVER 300 ILLUSTRATIONS CASSELL AND COMPANY, LTD London, New York, Toronto and Melbourne JS ../, f i First published February 1887. 'Reprinted November 1887, November 1888, January 1890. January and October 1891, October 1892, March and August 1893, October 1894, January 1896. New Edition November 1896. Reprinted December 1897, /w/y 1900, March 1902, /7*6 1903, October 1906 New Edition February 1911. Revised Edition March 1914. Rtprinted December 1916, ^/oy 1919, March 1920. New Edition August 1921. AM RIGHTS RBSERVRD PREFACE TO THE FOURTH EDITION A NEW edition of the book being required, advantage has been taken of the occasion to bring the work up to date. Also to modify, where convenient, the symbols used, in accordance with the list adopted by the International Electrotechnical Commission in 1913. A copy of this list is given in Appendix VII. The sections on Dry Cells have been rewritten, and that dealing with Storage Cells amplified. The addendum to Appendix I, relating to the practical electrical units, has been revised and extended to include more recent work in this subject. I am again indebted to Mr. Maurice Solomon, of the General Electric Company, for valuable information ; to Mr. R. W. Cooper, M.A., Messrs. Benn Brothers, and Edison Accumulators, Limited, for the use of blocks ; and to the India Rubber and Gutta Percha Company. My best thanks are -also due to Dr. Chas. Chree, M.A., F.R.S., for magnetic data, and to Mr. F. E. Smith, F.R.S., O.B.E., for help in connection with absolute measurements of the primary electrical units. The whole world is deeply indebted to Mr. Smith for the masterly way in which he has originated and carried out the researches on electrical standards at the National Physical Laboratory for many years past. His work has placed Britain well ahead of other nations in this branch of precision measurements. T. MATHER. PREFACE TO THE THIRD EDITION MAINLY owing to the numerous public calls on the time of the late Professor Ayrton, and to some extent because of the impaired health resulting from his many and strenuous labours for the advancement of technical education, the second of the two volumes in which it was intended to issue the second edition ol " PRACTICAL ELECTRICITY," was not completed. Shortly before the Professor's much lamented death, it was decided that the book should be re-written and published in a single volume of somewhat larger size, in the joint names of Professor Ayrton and myself. The result is the present work, which deals with the matters treated of in Volume I. of the 1896 edition, as well as those which the second volume was intended to contain. The arrangement of the book follows, in the main, that of the original work, but an attempt has been made to illustrate in greater detail the intimate relations that exist between electri- cal and mechanical quantities, and to show how the practical system of electrical units is founded on the C.G.S. system of mechanical units. It is hoped that this treatment will enable beginners to realise that definite ratios must exist between the electrical and mechanical units of power and energy, instead of regarding these relations as somewhat mysterious. Students should make themselves acquainted with the C.G.S. system of mechanical units before commencing the study of electricity. The experimental proof of Ohm's Law, -which formed a pro- minent feature in previous editions, has been improved by using a zero electro-dynamometer to measure current strengths instead of a calibrated galvanometer. Many new figures and new examples have been added, and new chapters on the potentiometer, the induction of electric currents, and on the magnetisation of iron, are now included. As the measurement of electric energy has become, of late years, a subject of much technical and commercial importance, more space is devoted to electric meters than in former editions. viii PREFACE Some of the meters described are intended for continuous currents, whilst others may be used both for continuous and alternating currents ; but as this work deals with continuous currents only, no reference to the use of meters for alternating currents is made in the text. No attempt has been made to treat the subject of meters completely, for their number and diversity are now so great that such treatment would alone require a whole volume. A knowledge of sizes of wires, and the resistances of copper wires is of considerable utility, so a table of the Legal Standard Wire Gauge is printed as an appendix, giving, in addition to diameters and areas of cross -section, the relations between Length and Resistance, Resistance and Weight, and Weight and Length, both in British and Metric measure. Other tables, very useful in making calculation of windings of instruments and machines, give the number of wires per lineal inch or centimetre, the number of turns of wire per square inch or per square centi- metre of windings, as well as the resistances per cubic inch and per cubic centimetre for the various sizes of copper wire insulated in several different ways. Tables of this kind, based on the thicknesses of insulated coverings adopted by the London Electric Wire Company, have been in use at the Central Technical College since 1890. For the calculation of the present tables I am indebted to my son, W. H. Mather, who has also re-worked many of the examples. The tables have been checked by Mr. S. S. Watkins, A.C.G.I., B.Sc. Although the book is primarily intended for students following a first year's lecture and laboratory course in Electrical Engineer- ing, it deals with most of the subjects required for the intermediate examinations in Electricity and Magnetism in the universities of London and the provinces. I therefore hope it will be useful in assisting students to acquire the knowledge necessary for these examinations, not as a " cram book," but as one that, with the help of laboratory instruments and apparatus such as are described and illustrated, will give them a sound quantitative knowledge of the several phenomena which form the basis of electrical science and industry. The " Short History of the Absolute Unit of Resistance, etc." written by the late Professor Ayrton, has been reprinted and extended by a statement of the work done since 1896, and of the resolutions adopted by the " International Conference on Electrical Units and Standards," held in London in October, 1908. Acknowledgment is due and is hereby tendered to several correspondents for pointing out a few errors and misprints in PREFACE ix previous editions. Notice of similar imperfections in the present work will be greatly appreciated. My thanks are also due to Mrs. Ayrton, who has kindly read the proofs and made several valuable suggestions, to my colleague Prof. G. W. O. Howe, M.Sc., and my assistants, Mr. F. E. Meade, Mr. S. S. Watkins, A.C.G.I., B.Sc., for help in collecting data, etc., to Mr. F. E. Smith, A.R.C.S., of the National Physical Laboratory ; Mr. Maurice Solomon, A. C.G.I., of the Birmingham Carbon Works ; The Electrician Publishing Company ; and to the several firms who have furnished blocks and information about their manufactures, which have been useful in bringing the book up to date. T. MATHER. PREFACE TO THE SECOND EDITION EXACTLY ten years have elapsed since the preface to the first edition of this book was written a decade which has seen a vast development in the applications of electricity to industrial purposes, and the springing up in all parts of the kingdom of Technical Schools and Colleges where much attention is devoted to the study of electrotechnics. Hence, to-day it is far more easy for a student to connect his experimental apparatus with the electric light mains and use a comparatively large current at a pressure of 100 volts, than it was in 1886 to obtain a small current at a much lower pressure from the battery which he had to set up for the purpose. This possibility of carrying out the experiments on a larger scale has led to considerable simpli- fication in certain cases ; for example, in experimentally deter- mining the heat equivalent of electric energy, it is no longer necessary to distract the beginner's attention with a variety of corrections for the loss of heat, &c. After many issues of the book had appeared in its original form, it seemed desirable to bring it up to date ; and since the practice, not unfrequently resorted to by writers, of inserting a number of new patches in an antiquated ground work, would be out of place in a book which had been written to aid electro- technical teaching and not for purposes of profit, a proposition was made to entirely rewrite it. This the publishers accepted ; and, guided by the success which the book had achieved, they generously, and I anticipate wisely, modified the arrangements so as to justify my devoting a large amount of time to the pre- paration of what in reality is an entirely new book, although called by its old title " PRACTICAL ELECTRICITY." The reception of the first edition took me by surprise. I anticipated that the book would be regarded as " faddy," and that the critics, while admitting that perhaps it would do well enough for my own classes, would not recommend its use PREFACE xi for students in general. It did not occur to me that the world was ready for using such a text-book and prepared to adopt the methods of teaching advocated in its pages. To-day, however, the following reasons suggested in the original preface for even elementary students of electricity spending much time in the laboratory would be advanced by many teachers : " One of the great difficulties experienced by people in master- ing the quantitative science of electricity, arises from the fact that we do not number an electrical sense among our other senses, and hence we have no intuitive perception of electrical phenomena. During childhood we did not have years of unconscious experi- menting with electrical forces as we had with the forces connected with the sensations of heaviness and lightness, loudness and softness, heat and cold. Beyond a shock or two taken perhaps from some medical galvanic apparatus, or from a Leyden jar, our senses have never been affected by electrical action, and hence we ought to begin the study of electricity as a child begins its early education. Quite an infant has distinct ideas about hot and cold, although it may not be able to put its ideas into words, and yet many a student of electricity of mature years has but the haziest notions of the exact meaning of high and low poten- tial, the electrical analogues of hot and cold. That it is desirable that students should learn physics, as they learn to ride the bicycle, by experimenting themselves, is now generally admitted, and this is especially true in the case of electricity, since it is by experimenting, and only by experimenting, that a student can obtain such a real grasp of electricity that its laws become, so to say, a part of his nature." " Hence, in the courses of electricity which I arranged at the City and Guilds of London Technical College, Finsbury, and at their Central Technical College, Exhibition Road, for every hour that a student spends at lecture, he spends several in the laboratory.'' When Dr. Hopkinson this year, 1896, in his Inaugural Address as President of the Institution of Electrical Engineers, advocated commencing the study of electricity with the electric current, more than one teacher testified to the value of the method by claiming it as his own, apparently forgetful that when his order of treating the subject was introduced by the author in 1879 there was no precedent for such an innovation. Indeed, when even seven> years later there appeared the first edition of " PRACTICAL ELECTRICITY " it was thought advisable to introduce the method by inserting the following explanatory paragraphs : " Readers who have been accustomed only to the ordinary Xll PREFACE books, commencing with certain chapters on statical electricity, continuing with one or more on magnetism, and ending with some on current electricity, will be surprised at the arrangement of the subjects in this book, and will probably be astonished at what they will condemn, at the first reading, as a total want of order. But so far from the various subjects having been thrown together haphazard, the order in which they have been arranged has been a matter of the most careful consideration, and has been arrived at by following what appears to me to be the natural as distinguished from the scholastic method of studying electricity. I have endeavoured to treat the subject analytically rather than synthetically, because that race of successful experimental philosophers children adopt this method. " For example, it is not by studying geometrical optics, much less physical optics, that an infant gradually learns to appreciate the distance of objects ; and later on it is not by studying a treatise on struts, nor by listening to a course of lectures on structures, that the child finds out that the table has legs, hard legs, round legs. Feeling, looking, trying, in fact a simple course of experimental investigation, gives a child its knowledge ; and this, therefore, I venture to think, is the method we should adopt when commencing the study of electricity." " The subject of current is treated first, because in almost all the industries in which electricity is practically made use of, it is the electric current that is employed ; also, because currents can be compared with one another, and the unit of current (the ampere) denned, without any knowledge of potential difference or resistance. Potential difference is next considered, and resist- ance the last of the three, because the very idea of resistance implies a previous acquaintance with the ideas of current and potential difference, since the resistance of a conductor is the name given to the ratio of the potential difference (measured electrostatically) between its terminals to the current passing through it. And it is Ohm's experimental proof that this ratio was constant for a given conductor under given conditions, together with the num- berless experimental verifications that this conclusion has received, that has led to resistance gradually coming to be considered as a fixed definite property of a given conductor like its weight or length." The international, or Board of Trade, unit of P.D. the volt cannot, however, be defined until the definition of the unit of resistance the ohm has been fixed, because for legal purposes the units of current and resistance have been taken as the primary ones, and Ohm's law has been employed to fix the third or derived PREFACE xin unit viz. that of P.D. Hence, the actual sequence adopted in the present volume is (i) current and the ampere ; (2) the relative measurement of P.Ds. with a zero electrostatic voltmeter ; (3) Ohm's law ; (4) resistance, and the ohm ; (5) the volt, and current-voltmeters. Electric energy and power, with their units the joule and the watt are next treated ; and, lastly, the conception of the E.M.F. in a circuit, and the necessity for the E.M.F. of a good cell being constant, are derived from the laws of energy. If should be obvious that any method of trying to experimen- tally prove Ohm's law with a current-voltmeter, such as may be found in certain text-books, begs the question. If a voltmeter be used it must be of the electrostatic type, and to simplify the definition of one P.D. being twice another this electrostatic voltmeter should be a zero instrument, which, without the need of any independent electrification, would be suitable for measuring P.Ds. no larger than those commonly employed in laboratories for sending currents. Such an instrument I have long felt the need of, and now thanks to the ingenuity of Mr. Mather it is available for use, and will be found described for the first time in pages 163-166* of the present volume. It will be observed that the apparatus required for each experi- ment is mounted complete on a board. This is to enable it to be easily carried backwards and forwards between the laboratory and the lecture-room without disarranging it. At first sight it might appear that the student, finding each set of apparatus joined up quite complete, with current laid on all ready for carrying out the experiment, would be deprived of all incentive to exercise his own ingenuity in overcoming experimental difficul- ties, and, therefore, would fail to acquire habits of self-reliance. For first year students, however, I have found it a good plan to have each set of apparatus complete in position ; firstly, because it is only with some such arrangement that fifty or more students can commence work almost simultaneously, and in the course of two or three hours have all performed some quantitative experi- ment ; secondly, because when the apparatus is so arranged that even beginners can perform several experiments successfully, they acquire faith in the possibility of success, and are less discouraged with the difficulties they subsequently meet with when selecting and arranging the apparatus for conducting some investigation. The practical side of electricity has grown so rapidly that the original single volume has expanded into two. The present volume of the rewritten book is intended to assist students in * Pages 134-137 of third edition. XIV PREFACE acquiring experimentally an exact working knowledge of current, difference of potentials, resistance, energy, and power, with their electric transmission, cells and their cost of working. This subject of the cost of converting chemical energy into electric energy is not, as far as I am aware, to be found in any text-book. Hence, in view of the " booms " in primary batteries, which appear to be periodic, the question of cost has been entered into in considerable detail. The past four years have seen the legalisation in several countries of an international system of electrical units, so that, while the units of length, volume, mass, and money vary from country to country, there is now but one ampere, one ohm and one volt throughout the whole world ; a fact of which electri- cal engineers may feel justly proud. Some thirty pages at the end of the book are, therefore, devoted to " A Short History of the Absolute Unit of Resistance, and of the Electrical Standards of the Board of Trade." In spite of the fact that the present volume contains some 140 pages more than the original book, the subjects of secondary cells, electric quantity, coulombmeters, capacity, &c., have had to be left for a second volume. This has arisen not merely from primary cells, including dry cells and the Clark's standard cell, having been treated somewhat fully, but from the subjects of electric energy and power, the various meters used for measuring these quantities, the efficiency of electric transmission, the ratio of the power received to the maximum power receivable in various cases of transmission, &c., having been entered into at length in consequence of the commercial importance that the electric transmission of energy now possesses. And it may be mentioned that generally where problems of maxima or minima have been considered, attention has been directed to the kind of change that is produced in the value of the quantity under consideration, when the value of the variable is altered from that required to make the quantity a maximum or a minimum. In fact, the aim has been to treat a few subjects fairly thoroughly in a simple manner, and not to prepare a list of short instructions for carrying out a large number of experiments, nor to write a treatise, mainly of value as an electrical dictionary, which should give a little information about everything that can be comprised under the head of electricity, whether it be electric eels, the history of the invention of the telegraph, the aurora, or the earliest forms of fractional machines. In the letterpress, small capitals have been used to represent instruments, parts of apparatus, &c., while large capitals PREFACE xv systematically stand for electric quantities other than resistances, these being throughout designated by small letters in italics. Thus, A, A, a stand respectively for an ammeter, the current in amperes flowing through it, and its resistance in ohms. * In the preface written in 1886 it was mentioned that, with the exception of two or three blocks that had been lent, the 180 figures had been specially drawn for the book, and were not time-honoured representations of historical apparatus. Of these 180 figures only 64, however, have been employed in the present volume, partly because the fresh matter required many new figures to illustrate it, and partly because several of the blocks specially executed for the original book have lost their freshness from the appreciative use of them by other writers. Hence, 183 of the 247 figures contained in the present volume will not be found in the former book, and 163 of these fresh illustrations have been specially drawn for this new edition. A large number of new examples have been added, and any that have been reproduced from the original book have been reworked, either to check the accuracy of the results, or because the so-called legal units referred to have been replaced by those that have now been adopted internationally. My thanks are due to my past and present assistants Dr. Sumpner, Mr. Haycraft, and Mr. Severs for much assistance in the preparation of this book ; to my daughter for compiling a very comprehensive and judiciously arranged index ; and to Messrs. Spiers, Twyman, and other students for carefully exam- ining the proofs. In conclusion, I desire to express to-day even more warmly than in October, 1886, my indebtedness to Mr. Mather for the very earnest, thoughtful, and painstaking way in which for many years he has assisted me in developing the course of instruction for students of electrical technology, of which the present volume represents part of the elementary portion. W. E. AYRTON. October, 1896. * As "International Symbols" have been largely used in the present edition, this rule no longer holds. T. M., 1921. CONTENTS CHAPTER I THE ELECTRIC CURRENT AND ITS MEASUREMENT SECTION FACE 1 . What is meant by an Electric Current, and by its Direc- tion of Flow i 2. Production of an Electric Current : Electric Circuit . 2 3. Cells and Batteries 3 4. Conductors and Insulators 3 5. Properties of an Electric Current 4 6. Current Strength . 7 7. The Strength of an Electric Current ; by which of its Properties shall it be directly measured ? . . 12 8. Definition of the Unit Current ; Ampere . . . 18 9. Electrolysis ; Electrochemical Equivalent . . . 21 10. Definition of Unit Quantity of Electricity : Coulomb . 24 n. Definition of the Direction of the Current : Ions . . 26 12. Objection to the Usual Mode of Constructing Voltameters 30 13. Description of Practical Forms of Sulphuric Acid Volta- meters 31 14. Relative Advantages of Voltameters and Galvanometers 33 15. Measurement of Current by Galvanometers : Tangent Galvanometers . . . . . . . 36 1 6. Meaning of the Relative and the Absolute Calibration of a Galvanometer ....... 39 17. Comparison of Tangent Galvanometer with a Voltameter 40 1 8. Absolute Calibration of Tangent Galvanometer . . 42 19. Calibrating any Galvanometer by Direct Comparison with a Tangent Galvanometer . . . . 43 20. Graphically Recording the Results of an Experiment . 44 21. Practical Value of Drawing Curves to Record Graphically the Results of Experiments .... 47 22. To Construct a Galvanometer Scale from which the Relative Strengths of Currents can be at once ascertained . . . . . 5 CHAPTER II MAGNETIC FIELDS 23. Magnetic Fields . .' 53 24. Lines of Magnetic Force . . . . . .56 25. Strength of Magnetic Poles 59 xviii CONTENTS SECTION JAGH 25^. Hibbert's Magnetic Balance 61 256. Balance for Finding Strength of Pole .... 62 26. Magnetic Moment ....... 63 27. Absolute Measurement of Strength of Magnetic Field and of Magnetic Moment ..... 66 28. Mapping Magnetic Fields . . .68 29. Comparing the Relative Strengths of different Parts of a Magnetic Field by the Vibration Method . . 72 30. Comparing the Relative Strengths of different parts of a Magnetic Field by the Magnetometer Method . 74 300. Difference of Magnetic Potential : Equipotential Sur- face 79 Addendum to Chapter II. : Electric Lines of Force and Elec- trostatics 8 1 CHAPTER III GALVANOMETERS, ELECTRODYNAMOMETERS AND AMMETERS 31. The Tangent Galvanometer 83 32. Adjusting the Coil of a Tangent Galvanometer . . 84 33. Scale for a Tangent Galvanometer . . . 86 34. Tangent Law 88 35. Variation of the Sensibility of a Tangent Galvanometer with the Number of Windings, and with the Diameter of the Coil 90 36. Values in Amperes of the Deflections of a Tangent Gal- vanometer controlled only by the Earth's Magnetism 96 37. Pivot and Fibre Suspensions 99 38. Sine Galvanometer 100 39. Electrodynamometers ....... 102 40. Construction of Galvanometers in which the Angular Deflection is directly proportional to the Current 105 41. Galvanometers of Invariable Sensibility . . . 107 42. Permanent Magnet Ammeters 109 43. Moving Coil Ammeters . . . . . .112 44. Soft Iron Ammeters : Spring and Gravity Control . . 118 45. Hot- Wire Ammeter 124 CHAPTER IV DIFFERENCE OF POTENTIAL AND RESISTANCE 46. Difference of Potentials 126 47. Potential of the Earth arbitrarily called Nought ; Posi- tive and Negative Potentials . . . . 132 48. Measurement of Potential Difference 49. Electrometer .... 50. Ohm's Law ... 51. Resistance 52. Ohm : Unit of Resistance 133 134 138 142 143 CONTENTS xix 53. Resistance Coils and Resistance Boxes . . .145 54- v olt 148 55. Ohm's Law applicable to Complete Circuits : E.M.F. 149 550. Electromagnetic Definition of E.M.F 151 56. Current Method of Comparing P.Ds. and Resistances . 153 57. Reason for using High Resistance Galvanometers for P.D. Measurements, and Low Resistance Galvanometers for Current Measurements . . . . . 154 58. Voltmeter 155 59. Resistances of Ammeters and Current Voltmeters . . 158 60. Ammeters used as Voltmeters 158 61. Moving Coil Voltmeter 160 62. Calibrating a Deflectional Voltmeter . . . .160 63. Voltmeters used as Ammeters 163 64. Gold-Leaf Electroscope . . . . . .166 65. Sensibility of Gold -Leaf Electroscopes . . . .168 CHAPTER V GALVANIC CELLS 66. Chemical Action in a Simple Voltaic Element . . 170 67. Daniell's Use of a Depolariser : Two-Fluid Cell . . 173 68. Local or Prejudicial Action J 75 69. Gravity Daniell's Cells . .... 178 70. Minotto's Cell 71. Resistance of Daniell's Cells 72. Grove's and Bunsen's Cells 73. Potassium Bichromate Cell 74. Storage or Secondary Cell . 75. Leclanche Cells 76. Dry Cells .... 77. Hellesen and Dania Dry Cells 78. G.E.C. and Obach Cells 79. Blue Bell and Columbia Cells 80. Extra-Sec and Inert Cells . 81. Edison-Lalande Cell 179 1 80 183 185 187 189 193 196 197 198 198 199 82. Standard Cells, Clark's and Weston's Cells. . . 200 83. Calculation of the E.M.F. of a Cell from the Energy Liberated by the Chemical Action . . . 206 CHAPTER VI RESISTANCE ! ITS LAWS AND MEASUREMENT 84. Comparing Resistances : Voltmeter and Ammeter Method 210 85. Ohmmeter : Megger ....... 211 86. Simple Substitution Method of Comparing Resistances 214 87. Differential Galvanometer, A Null Method . . . 216 88. Wheatstone's Bridge : its Principle . . . .218 89. Wheatstone's Bridge : its Use and Simple Method of Constructing 221 xx CONTENTS SECTION PAGE 90. Bridge Key . . . . . . . . 225 91. Use of a Shunt with the Bridge 227 92. Meaning of the Deflection on a Bridge Galvanometer 227 93. Conditions Affecting the Resistance of a Conductor . 228 94. Variation of Resistance with Length s . . . 229 95. Variation of Resistance with Cross-Section . . . 230 96. Variation of Resistance with Material . . .231 97. Resistance of Metals and Alloys per Centimetre Cube and per Inch Cube . . . . . .231 98. Resistance of Metals and Alloys for a Given Length and Weight * . .234 99. Variation of Resistance with Temperature . . 236 100. Conductors of Large Specific Resistance have Small Temperature Coefficients ; : .. . . 239 101. Conductivity and Conductance ..... 242 102. Comparison of Electric and Heat Conductivities . 243 103. Resistance and Conductance of Several Conductors in Series or in Parallel . . . . . . 244 104. Currents in Parallel Conductors 248 105. Kirchhoff's Rules 248 106. Shunts . . . . . . . . 251 107. Multiplying Power of a Shunt . -'. . 252 1 08. Usual Method of Constructing a Shunt Box . . 253 109. Increase of the Main Current produced by Applying a Shunt . . ... ... . . 255 no. Principle of Universal Shunts ..... 259 in. Method of Constructing a Universal Shunt Box, and its Advantages 260 112. Standard Resistance Coils . . . . . . 264 113. Ordinary Forms of Wheatstone Bridge . . .265 114. Portable Forms of Wheatstone Bridge . . . 269 115. Dial and Bar Patterns of Bridge . . . .271 CHAPTER VII ELECTRIC ENERGY AND POWER 1 1 6. Work done by a Current . , . . . .273 117. Electric Unit of Energy : Joule 277 ti8. Heat Produced by a Current . . . . .277 119. Measuring the Heat Equivalent of Electric Energy . 278 120. Power 282 121. Electric Unit of Power : Watt 283 122. Joule's Law . ....... 285 123. Instruments for Measuring Power : Wattmeters . 286 124. Commercial Forms of Wattmeters . . . .288 125. Joule or Energy Meter : Clock Form . . . 290 126. Board of Trade Unit of Energy . . . . 294 127. Energy Meter : Motor Form 296 128. Quantity or Ampere-hour Meters .... 302 129. Electric Transmission ol Energy . . . . . 308 130. Power Developed by a Current Generator . . . 312 CONTENTS xxi 131. Connection between the E.M.F. of a Battery, the P.D. between its Terminals, the Resistance and the Current 313 132. Electromotive Force of any Current Generator . . 314 133. Power Absorbed in the Circuit Exterior to the Genera- tor : Back E.M.F 315 134. Distribution of Power in an Electric Circuit . . 318 135. External Circuit that Receives Maximum Power from a Given Current Generator . . . . . 319 136. Arrangement of n Cells -to give Maximum Power to an External Circuit of Fixed Resistance . . 325 137. Minimum Number of Cells required to give a fixed Amount of Power to a given External Circuit . 329 138. Importance of Low Resistance and High E.M.F for Large Powers -333 139. Modifications Introduced into the Previous Results by Limitation of the Maximum Current a Cell may Produce . . 333 140. Efficiency ......... 335 141. Efficiency of Electric Transmission of Energy . . 339 142. Connection between Electrical Efficiency of Transmis- sion and Ratio of the Power Received to the Maxi- mum Power Receivable 343 143. Economy in Electrical Transmission of Energy : Kelvin's Law 346 CHAPTER VIII QUANTITY AND CAPACITY 144. Electric Quantity and its Measurement . . . 348 145. Ballistic Galvanometer ... . 349 146. Measurement of Quantity by Ballistic Galvanometer 353 147. Correction of Ballistic Galvanometer for Damping . 356 148. Determination of Decrement and Logarithmic Decre- ment 357 149. Constant of a Ballistic Galvanometer . . . 359 150. Comparison of Quantities 361 151. Capacity 3 62 152. Condensers : Mechanical Analogies .... 363 153. Units of Capacity ; Farad ; Microfarad . . 365 154. Variation of the Capacity of a Condenser with the Area of its Coatings and the Distance between them . 366 155. Relation between the Electrostatic Unit of Capacity and the Farad 3 6 7 156. Capacity of Spherical and Plate Air Condensers in Farads 368 157. Capacity of Cylindrical Condensers . . . .369 158. Specific Inductive Capacity 370 159. Dielectric Strength of Insulators .... 372 160. Resistivity of Insulators 373 161. Construction of Condenser of Large Capacity . .374 162. Condensers for Large P.Ds.. Leyden Jars . . . 376 xxii CONTENTS 163. Comparison of Condensers . ..... 379 164. Potential Divider . . . . . . 3 80 165. Combined Capacity of Several Condensers . . . 382 1 66. Charged Condensers are Stores of Electric Energy, not of Electricity " . 384 167. Energy wasted in Charging a Condenser from a Source of Constant P.D. 386 168. Absolute Measurement of Capacity .... 387 169. Measurement of Specific Inductive Capacity, and Resisti- vity of Insulators 388 170. Standard Air Condensers ...... 392 171. Ratio of Electromagnetic and Electrostatic Units of Quantity .... ... 394 172. Use of Condensers for Comparing E.M.Fs. of Cells or other Current Generators . . . . . 397 173. Condenser Method of Measuring the Resistance of a Cell ;...*. . 398 CHAPTER IX POTENTIOMETER MEASUREMENTS 174. Poggendorff's Method of Comparing the E.M.Fs. of Cells or Batteries 400 175. Principle of the Potentiometer 403 176. Calibration of Potentiometer Wire .... 404 177. Industrial Form of Potentiometer .... 406 178. Modern Form of Crompton Potentiometer . . . 408 179. Dial Potentiometer 409 1 80. Calibration of Voltmeter by Potentiometer : Volt (or Ratio) Boxes 411 181. Standard Resistances for Current Measurements . 414 182. Calibration of Ammeters 416 183. Comparison of Resistances by Potentiometer . . 417 184. Measurement of Power 420 185. Advantages and Disadvantages of Potentiometer Measurements . . . . . . . 420 CHAPTER X INDUCED CURRENTS 186. Introductory 4 2 3 187. Direction of Induced Currents due to Magneto-Electric Induction 424 1 88. Lenz's Law : Fleming's Rule 425 189. Relation between Quantity Induced and Resistance of the Circuit 426 190. Determination of Constant of a Ballistic Galvanometer by Earth-Inductor Method .... 429 191. Distribution of Magnetism in a Bar Magnet . . 43 * 192. Flux Density over Cross-Sections and over Surfaces of a Magnet 43 2 193. Mutual Induction r .435 CONTENTS xxiii SECTION PAGE 194. Unit of Mutual Induction: Henry .... 437 195. Self-induction 438 196. Induction Coil , 438 197. Induction of Currents in Parallel Wires . . .441 CHAPTER XI MAGNETISATION OF IRON 198. Lifting Magnets 442 199. Relation between Lifting Force and Current-Turns . 442 200. Lifting Force and Flux Density : 446 201. Magnetic Saturation 450 202. Magnetic Field produced by Current in a Straight Con- ductor 450 203. Magneto-Motive Force . . . . . 454 204. Testing Magnetic Properties of Iron by the Ballistic Method 457 205. Permeability 460 206. Hysteresis of Iron 460 207. Remanent Magnetism : Coercive Force . . . 463 208. Loss of Energy due to Hysteresis. Mechanical Analogy 464 209. The Magnetic Circuit : Reluctance .... 467 Appendix I. Short History of the Absolute Unit of Resistance, and of the Electrical Standards of the Board of Trade . . . . . . . 473 Appendix II. Comparison of C.G.S. and British Systems of Units . 5I4 Appendix III. Relations between the Practical C.G.S. Electro- magnetic and C.G.S. Electrostatic Units . . Appendix IV. Specific Gravities, Specific Resistances, and Specific Conductivities of Mixtures of Pure Sulphuric Acid and Distilled Water . . Appendix V. Showing the Dimensions of Wires according to the British Standard Wire Gauge (S.W.G.) as well as the approximate Relations between Lengths, Resistances, and Weights of Pure Copper Wire at a Temperature of 15 C. . ^5 Appendix VI. Windings Tables (a) Ordinary Cotton Covered (single) . . . ^g (b) Ordinary Cotton Covered (double) . . . CJQ (c) Specially Fine Cotton Covered (single) . c 2O (d) Specially Fine Cotton Covered (double) . c 2 i (e) Silk Covered (single) 2 22 (/) Silk Covered (double) ^ (g) Enamel Insulated - 2 . (h) Enamel Insulated and Cotton Covered (single) 525 () Enamel Insulated and Cotton Covered (double) 526 Appendix VII. Table of Symbols 527 Index * ... 529 PRACTICAL ELECTRICITY CHAPTER I THE ELECTRIC CURRENT AND ITS MEASUREMENT I. What is meant by an Electric Current, and by its Direction of Flow 2. Production of an Electric Current : Electric Circuit 3. Cells and Batteries 4. Conductors and Insulators 5. Properties of an Electric Current 6. Current Strength 7. The. Strength of an Electric Current : by which of its Properties shall it be Directly Measured ? 8. Definition of Unit Current ; Ampere 9. Electrolysis, Electrochemical Equivalent 10. Definition of Unit Quantity of Electricity; Coulomb u. Definition of the Direction of the Current 12. Objection to the Usual Mode of Constructing Volta- meters 13. Description of Practical Forms of Sulphuric Acid Volta- meters 14. Relative Advantages of Voltameters and Galvanometers 15. Measurement of Current by Galvanometers ; Tangent Galvano- meters 16. Meaning of the Relative and the Absolute Calibration of a Galvanometer 17. Comparison of Tangent Galvanometer with a Voltameter 18. Absolute Calibration of Tangent Galvanometer 19. Calibrating any Galvanometer by Direct Comparison with a Tangent Galvanometer 20. Graphically Recording the Results of an Experiment 21. Practical Value of Drawing Curves to Record Results 22. To Construct a Galvanometer Scale from which the Relative Strengths of Currents can be at once ascertained. i. What is meant by an Electric Current, and by its Direction of Flow. In the various industries in which electricity is em- ployed, as in the telegraph, telephone, electric lighting, electro- typing, electroplating, torpedo exploding, electric traction, the electric transmission of power, and in the working of machinery by the aid of electromotors, it is the so-called " electric current " that is made use of. Hence a knowledge of the laws of this electric current, a clear conception of its so-called properties, combined with a practical acquaintance with the modes of measuring it, must be of especial importance for a right under- standing of the working of the apparatus employed in the above- mentioned industries. Indeed, such knowledge is absolutely necessary if the user of electrical apparatus is desirous of em- ploying it to the best. advantage, of being able to correct faults when they occur, as well as of effecting improvements in the appliances themselves. B 2 ;\ PRACTICAL ELECTRICITY It/jstcasiottiaiy to: speak of an electric current as if it had an independent 1 existenorapart from the " conductor " through which it is said to be flowing, just as a current of water is correctly spoken of as something quite distinct from the pipe through which it flows. But in reality we are not sure that this is the case. Modern theory, however, suggests that electricity is atomic in its nature and of two kinds, and that the two kinds pass in opposite directions along the conductor. So the student must not assume that the conventional expression, " The current flows from the copper pole of a galvanic battery to the zinc pole through the external circuit," implies a certain knowledge of, the real direction of flow, any more than the railway expressions, " up train " and " down train," mean that either train is necessarily going to a higher level than the other. In the case of a stream of water flowing along a river bed we are quite sure that there is water in motion, and everyone is agreed as to which way the water is flowing ; a cork or a piece of wood thrown on the water indicates by its motion the direction in which the water is moving. Nor, again, must an electric current be supposed to be like waves of sound travelling along, since, in this latter case, although there is no actual travelling along of matter, still the direction of motion of the wave of sound is perfectly definite. Indeed, a wire along which an electric current is flowing is more like a wire at each end of which a musical instrument is being played, so that the sound is travelling in both directions along the wire at the same time. In short, the statement that an electric current is flowing along a wire is only a short way of expressing the fact that the wire and the space around the wire are in a different state from that in which they are when no electric current is said to be flowing. So that when a body and the space around the body possess certain properties that they do not usually possess, an electric current is said to be flowing through that body. 2. Production of an Electric Current: Electric Circuit. Perhaps the simplest method of producing an electric current is to place a piece of copper and a piece of zinc in simple 1 ceifan4 g ckcuit. a jar of dilute sulphuric acid and join the Fig. i. Simple cell and circuit. PRODUCTION OF AN ELECTRIC CURRENT 3 two plates together by a piece of wire*, thus forming what is called an " electric circuit " (Fig. i). 3. Cells and Batteries. The jar and the dissimilar metals in dilute acid shown in Fig. i constitute what is termed a " galvanic cell," or more shortly a " cell " ; and a number of such cells Fig. ib. Five simple cells Connected in Series. connected together form a " battery." Fig. la shows dia- grammatically a circuit containing one cell, and Figs, ib and ic, a circuit with a battery of five cells in series. The peculiar properties exhibited by an electric circuit, such as described in Section 2, rapidly become less marked, and in a short time practically disappear. If, however, the jar be divided into two parts by a porous partition between the plates, and copper sulphate be substituted for sulphuric acid in the compart- ment containing the copper plate (Fig. 2), the length of time during which the properties are appreciable is very greatly increased. Such an arrangement is called a " constant " cell because the effects it produces are much more constant than those of the cell previously described. Many forms of " constant " cells have been devised and are now used in preference to simple" cells. 4. Conductors and Insulators. When a wire is joined to the two plates c and z (Fig. 2), the wire and the space around it are found to possess properties which they did not previously possess. This fact is expressed by saying that " an electric cur- rent is flowing through the wire," and the wire is spoken of as being a " conductor " of electric current. If the wire be cut and Fig. ic. Diagram of five simple cells Connected in Series. PRACTICAL ELECTRICITY Porous joartitrioti Fig. 2. "Constant" Cell. the cut ends be kept apart in the air, the peculiar properties disappear ; we then say the current " ceases," or is " stopped." As an air space between the cut ends of the wire stops the current, the air is said to be a " non-conductor " of electric current. If the two ends of the cut wire be pressed to- gether, or against a plate of clean metal, or against another piece of wire, or be dipped into mercury, the current again " flows," but if glass or dry wood, silk or cotton, or oil, replace the metal plate, wire, or mercury respectively, the current does not flow. We can therefore say that some materials are conductors and others non-conductors. Non-conductors are called " insulators," and wires covered with non-conducting coatings are spoken of as " insulated wires." 5. Properties of an Electric Current. These properties are : (1) A suspended magnet put in nearly any position near a conductor through which an electric current is flowing will be deflected, showing that a force is exerted on the magnet (Fig. 3). This force is mutual, so that if a magnet be brought near any substance traversed by an electrical current, this sub- stance will generally be acted upon by a force tending to move it (Fig. 4). Also any piece of soft iron put near a conductor carrying a current will become magnetised (Fig. 5). The action in all these cases is just as if the body conveying the current had become magnetic. This is further shown by the fact that any two wires through each of which a current of electricity is passing, act upon each other with a magnetic force in nearly every position in which the wires may be placed relatively to one another (Fig. 6). (2) If the circuit through which an electric current is flowing be partly solid and partly liquid, then the liquid will generally be decomposed into two parts, one part going to one side of the liquid in the pig. 3. Magnet deflected by Conductor Carrying Current PROPERTIES OF AN ELECTRIC CURRENT 5 direction in^which the current may be said to be flowing, and the othei part going to the other side of the liquid in the opposite direction to the flow of the current (Fig. 7) . (3) The body conveying the current becomes more or less heated (Fig. 8). In popular language the current is said : (1) To deflect the magnet, and magnetise the iron. (2) To decompose the liquid. (3) To heat the body through which it is flowing. But as we have no evidence of the current apart from the conductor through which it is said to flow, it is more accurate to speak of a current being said to flow through a con- ductor in which these effects are found to be produced, than to say that the current produces these effects. The latter expression, how- ever, for brevity's sake, is generally adopted ; and, indeed, the heat generated in a wire conveying a current has so many analogies with the heat produced in a pipe by the friction of a stream of water passing through it, that we can frequently assist ourselves by thinking of an electric current as a stream of matter passing through the wire as water would pass through a pipe filled with sponge, or loosely packed with sand. But the analogy, like many other analogies, must not be pressed too far, . ., . Fig. 5. Iron Rod Picking up Nails when a Current Flows, especially as there IS through a Wire Coiled round it. Fig. 4. A piece of Tinsel Coiling itself round a Magnet when a Current Flows through the Tinsel. 6 PRACTICAL ELECTRICITY this very great difference between a current of water flowing in a pipe and a current of electricity in a wire, viz. that in the former case no effects are produced external to the pipe, whereas in the latter the whole space surrounding the wire is affected. For example, if an electric current is flowing through a conductor, a compass needle brought within two or three inches of it is deflected. But suppose not merely is there a current of electricity flowing, but also a steady stream of water Fig. 6. Two Coils Standing on Narrow Bases Falling Down when a Current Flows through them in Opposite Directions. passing through the interior of the conductor, the conductor being in reality a pipe. The water stream, however, is a perfectly steady one, therefore it makes no sound ; and supposing the water has been previously brought to the temperature of the pipe, the presence of the water inside the pipe cannot be detected by the pipe feeling hotter or colder to the touch ; consequently, it would be extremely difficult to detect this stream of water by any test made outside the pipe.* The Magnetic, Chemical, and Heating effects of a current are utilised practically in a number of electrical instruments and processes ; for instance : * A "compo " gas pipe answers very well for this experiment. PROPERTIES OF AN ELECTRIC CURRENT 7 Fig. 7. Tube ABC contains solution of Common Salt with a drop of Hydrochloric Acid, and is coloured red with Litmus. When a current flows Chlorine is liberated in limb A, which bleaches the liquid, while Caustic Soda is formed in limb c, making the liquid dark blue. the effect which always occurs when a current flows. 6. Current Strength. The magnitude of the effects pro- duced in and near an electric circuit can be varied in several ways. As a rule, these are more pronounced when the number or size of cells employed is in- creased, and we say that the * It is desirable to show in opera- tion to students as many as possible of the instruments enumerated under the three heads, Magnetic Property, Chemical Property, and Heating Property, but at this early stage it is only necessary to describe the instru- ments in so far as their operation illustrates the respective property of the current. Magnetic Property. Needle telegraph,* the Morse instru- ment, telephones, electric bells, arc lamps, dynamo machines, electromotors, and, in fact, all instruments using electro- magnets. Chemical Property. Electro- plating, electrotyping, the ex- traction of aluminium and other metals from their ores, the pro- duction of sodium and chlorine from salt, the manufacture of pure copper, the cleansing of the mercury used in separating gold from sand, etc. Heating Property. Elec- tric welding, electric heat- ing and cooking apparatus, electric lamps, contri- vances for lighting gas or oil lamps electrically, fuses for torpedoes, etc. The heating effect of the current is, as we shall see, Fig. 8. Glow Lamp. 8 PRACTICAL ELECTRICITY increased effect is due to a " stronger " current, or to a current of " greater strength." An electric current is thus said to be strong or weak according as the magnitudes of the effects it pro- duces under given conditions are large or small ; in other words we take the amount of the effect produced as a sort of measure of the current strength. As, however, the magnetic, chemical, and heating effects of a current do not all alter in the same propor- tions when the current is changed in any way, it is important to consider which property should be taken as a direct measure of the current. In Section 5 we have stated that the production of heat always accompanies the passage of a current, and it might seem that the amount of heat produced in a given time ought to be taken as a measure of the current. But in addition to the difficulty of measuring the small amounts of heat produced by weak currents the only way we have of measuring the amount of heat given to a body is an indirect one, and consists in measuring its rise of temperature by means of a thermometer. Further, a thermo- meter measures merely rise of temperature, and not the amount of heat, and as the rise of temperature depends on the mass and nature of the material heated, and on the facilities for cooling which exist, as well as on the strength of the current to be measured, it is evident that this method of measurement is neither simple nor convenient. To ascertain which of the properties of a current can be best employed for measuring its strength, an experiment may be made with the following apparatus : A, B, c, D, E, F, G (Fig. 9) are instruments so arranged that the same electric current will be sent through them all by the " bat- tery," b b, on joining the wires P and Q. A is a " sulphuric acid voltameter " consisting of two platinum plates dipping into moderately dilute sulphuric acid in a vessel v, closed by an air-tight stopper s. Through this stopper passes a glass tube, /, open at both ends, with its lower end nearly touching the bottom of v, and graduated at its upper part in fractions of a cubic centi- metre or Cubic inch. B consists of two thin copper plates, p p, partly immersed into a solution of copper sulphate (the blue vitriol of commerce), and is called a "copper voltameter." c is a coil of insulated wire with a magnet, m, suspended so as to turn freely inside the coil, the whole arrangement forming what is called a " galvanoscope." D is an " electromagnet " consisting of a piece of iron of horse-shoe form round the ends of which are coils of covered wire wound in opposite directions. E represents an " electric fan " formed by metal blades mounted on the shaft to PRACTICAL ELECTRICITY of a small electromotor. F is a coil of bare wire immersed in paraffin oil, the temperature of which can be measured by the thermometer T, the arrangement being called a " calorimeter," and G is an electric lamp. Connect the two wires P and Q, and attow the current to pass for a convenient time through these seven pieces of apparatus, then it will be found that : (1) The liquid has risen a distance d^ in the tube t of the vol- tameter A, indicating that the passing of the current through the liquid from one of the platinum plates to the other has caused c l cubic inches of gas to be generated. (2) One of the plates in the copper voltameter has increased in weight by W t grains. (3) The magnetic " needle " m of the galvanoscope c has all the time been kept deflected from its original position through a number of degrees NJ. (4) If at any time during the passage of the current the " armature " a was placed carefully on the ends of the horse-shoe electromagnet D it required a pull of w Ibs., as measured by the spring balance, to pull it off, when the handle at the top of the apparatus was slowly turned. (5) The fan will have made a certain number of revolutions R r (6) The mercury in the thermometer T of the calorimeter F has risen through DJ (7) The lamp G has been emitting light of a certain intensity LI- Next increase the strength of the current passing through the apparatus, A, B, c, D, E, F, G, by increasing the number of cells forming the battery b b, or in any other way, such as will be described later on, and repeat the experiment for the same time as before, then each of the effects previously observed with these instruments will be increased, and instead of the results c v W lf NJ, w lt R lf DJ, Lj, we shall obtain c 2 , W 2 , NJ, w 2 , R 2 , D, L 2 . But it will be found that the new values do not all bear the same ratio to the corresponding old ones. For example, if c 2 is twice c lt then N may be more or less than twice NJ, but will generally be less than twice, while w z , R 2 , D, and L 2 may be found to be much greater than twice w lf R x , DJ, and L x respectively. On the other hand, if the strength of the second current be so chosen as to make D exactly twice DJ, then generally it will be found that w 2 and L 2 are rather more than twice w l and L I} while c 2 and W 2 are less than twice c x and W x respectively. R 2 may be either less than or greater than twice R x . The needle m of the galvanoscope c, Fig. 9, is shown suspended MEASURING EFFECTS OF A CURRENT 11 by a silk fibre, and the needle is deflected arid moves relatively to the coil when the current passes. If the silk be replaced by a fine wire fixed to the needle at its lower end and at its upper end to a torsion head, the needle could be brought back to the position it occupied when no current was flowing, by turning the torsion nead, H Fig. 10, and thus make the relative positions of needle and coil the same. The angle through which the torsion head requires turning to bring the needle back, is a measure of the moment of the couple exerted on the needle by the current- carrying coil. Let these angles be di and d\ in the two experiments de- scribed on page 10 ; we shall then find that if c 2 is twice c lt then d z will be twice d lf and W 2 twice Wj. If then we arbitrarily define the strength of the current as being directly proportional to the gas evolved in a given time in the sulphuric acid voltameter, we must con- clude that if c 2 is exactly double Cj we have doubled the current strength, that the amount of copper deposited is directly pro- portional to the current, and that the couple exerted between a coil and a magnet in fixed relative positions is directly pro- portional to the strength of the current. But, on the other hand, if we prefer to say that strength of current is directly proportional to the angular deflection of the needle m in the galvanoscope c, then we must conclude that, as N is less than twice NJ, we have not quite doubled the strength of the current ; whereas if we prefer to say that current strength shall be regarded as proportional to the force required to detach the armature a of the electromagnet D, or, instead, proportional to the rise of temperature of the liquid in the calorimeter F in a given time, or to the light given out by the lamp G, then we must conclude that the strength of the current has been more than doubled. Which of these is right Fig. 10. Torsion Galvanometer. 12 PRACTICAL ELECTRICITY and which wrong ? So long as no one of the effects varies we may be safe in concluding that the strength of the current is constant, but if the different effects to which we have been re- ferring vary from one time to another, then which of them shall we take to represent by the magnitude of its variation the change that has taken place in the current strength ? In the case of measuring the velocity of a stream of water, or the number of gallons of water per minute discharged by a river, no two experimenters could differ much. One of them, by the employ- ment of better constructed measuring instruments, or it may be from having greater experience in making such measurements, might get answers slightly different from, and more accurate than, those obtained by the other experimenter. But they could not have such totally different conceptions of what should be meant by the velocity of the water in a particular part of the channel, or of the total discharge in gallons per minute, that the results obtained by one observer were, apart from all mere errors of experiments, twice as great as those obtained by the other. And this is because they would be dealing with the actual flow of a material substance water. The flow of an electric current, however, being merely a con- ventional method of expressing the fact that a conductor has acquired certain properties that it does not usually possess, there is no question of right or wrong, but only one of convenience, in selecting whichever we choose of the so-called properties of the current as the one we arbitrarily decide to employ as the measure of the current strength. 7. The Strength of an Electric Current : by which of its Pro- perties shall it be Directly Measured? To assist us in deciding whether the amount of the magnetic action, or of the chemical action, or the amount of heat produced in a given time, shall be arbitrarily taken as that magnitude to which the current strength shall be defined as being directly proportional, we ob- serve that of the seven pieces of apparatus A, B, c, D, E, F, G employed in the previous experiment, A and B are the only two which give results that steadily increase in the same proportion when the current is increased ; but if c were replaced by a zero- torsion instrument (Fig. 10) the results obtained with a third piece of apparatus would increase in the same proportion as in A and B. Consequently, while on the one hand, our estimate of the relative strength of two currents would be quite different according as we selected the angular deflection of the magnet m (Fig. 9), or the force of detachment of the armature a to be the direct measure of the current strength ; on the other hand we DEFINING STRENGTH OF A CURRENT 13 should arrive at practically the same estimate whether we chose to say that the current was directly proportional to the rate of production of gas in the sulphuric acid voltameter A, or to the rate of deposition of copper in the copper voltameter B, or to the couple exerted between a coil and a permanent magnet in fixed relative positions. But in addition to this agree- ment between the relative amounts of different chemical actions pro- duced by two currents there is another equally important fact, viz., that the rate at which a -par- ticular chemical effect is produced by a current is practically in* dependent oj the size and shape of the apparatus. Thus, suppose we have two sulphuric acid volta- meters, the platinum plates being of totally different shapes and sizes (Fig. n) ; two copper voltameters also of different shapes and sizes (Fig. 12), the copper plates, for example, being much larger, and, either much nearer together, or much farther apart 1 in the one than in the other ; also two galvanoscopes (Fig. 13) , which may look very much like one another, but the bobbin of the instrument to the right is wound with a few turns of thick wire, while that of the other galvanoscope to the left is wound with many turns of fine wire ; two electromagnets (Fig. 14), which differ from one another in the same sort of way as do the galvanoscopes, and two calorimeters (Fig. 15), the two instruments in each case being selected so as to be distinctly different in size and form. Then, if an experiment be made with each pair of apparatus, a certain current being sent through both sulphuric acid volta- meters for a certain time, and a current, which may or may not be of the same strength as the former, through both the copper Fig. ii. Two Sulphuric Acid Voltameters having Platinum Plates of Different Sizes and at Different Distances Apart. 14 PRACTICAL ELECTRICITY voltameters, etc., the following results will be observed : In the two sulphuric acid voltameters quantities of gas equal in volume at the same pressure and temperature, and, therefore, possessing Fig. 12. Two Copper Voltameters having plates of Different Sizes and at Different Distances Apart. the same mass, will be developed in the same time, in spite of the platinum plates being of a very different size and at a very different distance apart in the two voltameters.* Similarly, in spite of the difference in size and form in the two copper volta- meters, the increase in weight of the plate of the one will be practically the same as the increase in weight of the corresponding Fig. 13. Galvanoscope to the Left Wound with Many Turns of Fine Wire ; Galvanoscope to the Right with a Few Turns of Thick Wire. plate of the other, unless the current be so strong that the deposited copper falls to the bottom instead of forming an * Equality of pressure may be obtained by using for the voltameters two vessels of the same size as well as two tubes of the same bore, and filling the vessels with the same quantity of dilute sulphuric acid of the same specific gravity. In that case, if the level of the liquid in the two tubes be the same to start with, the liquids will be found to rise at exactly the same rate in them on the same current being sent through the two voltameters. DEFINING STRENGTH OF A CURRENT 15 adherent deposit. But in the case of the^two galvanoscopes, the two electromagnets, and the two calorimeters, although the current passing through the two apparatus in any one pair is the same, the effects depend on the shape, on the size, and on very many 'details in the arrangement, etc. Hence, to specify the strength of a current by the magnitude of the deflection of the needle of a galvanoscope, it would be necessary to state the exact mode of constructing each part of the galvanoscope in great detail, as well as the exact position of the instrument relatively to neighbouring magnetic pieces of iron. Whereas, to specify the strength of a current by the amount of gas pro- Fig. 14. Electromagnet to the Left Wound with Many Turns of Fine Wire; Electromagnet to the Right with a Few Turns of Thick Wire. duced in a given time in a sulphuric acid voltameter, or by the amount of copper deposited in a given time on one of the plates of a copper voltameter, neither the shape nor size of the plates, nor the distance between them, need be taken into account within wide limits. In both the voltameters it is chemical decomposition that takes place in the former, this decomposition being the splitting up of the liquid into gases ; in the latter, the splitting up of the copper sulphate, and the deposit of copper on one of the copper plates, together with a loss of an equal weight of the metal of the other copper plate to give back to the solution the amount of copper taken out of it. In c and D (Fig. 9) the effects produced are both magnetic, but we have found that N does not bear to NI the same ratio that w+ bears to w ; whereas in the case of the 16 PRACTICAL ELECTRICITY voltameters we always find that c 2 bears to c almost exactly the same ratio that W 2 bears to W t . Consequently, so far as we have seen at present, the amount of chemical action produced in a given time by a current appears to be a more direct measure of its strength than the magnitude of some of the magnetic effects produced, and is also proportional to the magnetic effect between a coil and magnet in fixed relative positions. Fig. 15. Thermometer to the Left Surrounded with Many Turns of Fine Wire ; Thermometer to the Right with a Few Turns of Thick Wire. Let us examine this point still further. In Fig. 9 all the ap- paratus is joined up " in series " that is to say, the current passing through any one instrument passes through every other. But in Fig. 16 C 2 and C 3 are two sulphuric acid voltameters " in parallel," and not in series, with one another. For the current which comes along the wire W 1 and passes through sul- phuric acid voltameter Cj divides into two portions, one of which passes through C 2 and the other through C 3 ; the two portions then recombine and flow away together by the wire W 2 . Also from the DEFINING STRENGTH OF A CURRENT 17 construction of the apparatus it will be seen that the rise of liquid in the tube T 1 measures the production of gas in the voltameter C 1} while the rise of liquid in the tube T 2 measures the sum of the quantities of gas produced in the voltameters C 2 and C 3 together. Now, experiment shows that, if precautions similar to those referred to in the note on page 14 for using the apparatus in Fig. ii be taken, the liquid rises at exactly the same rate in the tube T X that it does in the tube T 2 . Consequently the rate of production of gas in c t is equal to the sum of the rates of pro- duction in C 2 and C 3 together. Further, whether c lf C 2 , and C 3 be all sulphuric acid volta- meters, or all copper volta- meters, or all silver voltameters, or, indeed, all voltameters of the same character, it will be found that, no matter what be the shapes or sizes of the different voltameters, and no matter what be the areas of the platinum, copper, or silver plates immersed in the respec- tive liquids, or the distances apart of the plates, the amount of chemical action produced in a given time in Cj is almost exactly equal to the sum of the amounts of chemical action pro- duced in c and together. The plates, in any one of the voltameters, c lf may be large or small, near together or far apart may be, in fact, moved about while the chemical action is going on. The current may be strong and the chemical action take place rapidly, or it may be weak and the action proceed slowly, and it may be varied while the action is progressing ; but the same general result still remains true. Measure the amount of chemical action that has taken place in C 2 and m C 3 , add the two together, and it Vill be found to be practically equal to the action that has taken place in c t in the same time. Now, when a river divides in consequence of the existence of Fig. 16. Voltameters c, and c, in Parallel with One Another, but in Series with Voltameter c,. i8 PRACTICAL ELECTRICITY an island in mid-stream, we know that the number of gallons of water flowing per minute on the two sides of the island must together equal the total number of gallons per minute flowing in the main stream, simply because the water which does not go past one side of the island must go past the other ; and similarly, if we are to look upon a current of electricity in the same way as a current of water, we must expect that, when it divides into two parts, the sum of these parts must always be equal to the whole, whether the current which divides is a large one or a small one. The experiment just described shows that, if we say that a current is directly proportional to the rate at which chemical action is produced in a voltameter, this statement will always be true, whatever be the current in the main circuit ; but it will not generally be true if we take any of the other effects occurring in the instruments indicated in Fig. 9 (page 9) as a direct measure of a current. Thus, in Fig. 16, if c lf C 2 , C 3 represent galvano- scopes, such as c in Fig. 9, the deflection of the first will not generally be equal to the sum of the deflections of the other two ; and even if this were the case for one current in the main circuit, it would not be the case for any other. Nor will any simple relation be found to connect the deflection of the first instrument with those of the other two, unless elaborate precautions be taken in the construction of the apparatus. 8. Definition of the Unit Current ; Ampere. We may therefore define the strength of a current as being proportional to the amount of chemical decomposition it can produce in a given time ; and an unvarying current which, when passed through a solution of nitrate of silver in water, deposits silver*' at the rate 0/0*00111800 of a gramme per second, is taken as the unit of current and called one " international ampere," or, more shortly, one ampere. The reason why the number crooiiiSoo is chosen is as follows: An ampere was originally defined as " one-tenthf of a C.G.S. (centi- metre gramme second) unit of current." Now the C.G.S. unit of current is defined as that current which, flowing through a conductor of i centimetre length placed along the circumference of a circle of I centimetre radius, exerts a force of i dyne% on unit magnetic pole at the centre.^ * Silver is used because it gives a heavier deposit than other metals, and does not oxidise readily. The deposit can therefore be weighed with greater accuracy. f The ampere was taken as ^ of a C.G.S. unit because at the time the " practical " system was adopted, the C.G.S. unit was considered too large tor practical purposes. J A dyne is a force approximately equal to ^| T of the weight of a gramme in London. Another way of expressing this, and one more easily realised ex- perimentally, is that current which, flowing through a circular conductor of i centimetre radius, exerts a force of 2-0- (6-283) dynes, on unit pole at the centre ; for the circumference of a circle of unit radius is 2?r. THE UNIT CURRENT; THE AMPERE 19 To understand this definition we must know what is meant by " unit magnetic pole." When a bar magnet is dipped into iron filings, the filings adhere to the magnet, especially towards its ends, and if the magnet is long and thin, they are attracted only near the ends. The magnetism thus appears to be concentrated towards the ends of the magnet, and these ends are called " poles." If a pole of one magnet be brought near one of the poles of another magnet, a force is exerted between them. Suppose two long thin magnets exactly alike, and placed so that one pole of one is near one pole of the other magnet, and that the other two poles are as far apart as possible, then the force between the magnets will be due almost entirely to the mutual action of the adjacent poles. If the magnets be such that the poles exert a force of one dyne on each other when at a distance of one centimetre apart, they are said to be " unit poles," or " poles of unit strength." A unit pole is therefore one that exerts a force of one dyne on an equal pole, when the two poles are one centimetre apart, and a current which exerts a force of one-tenth of a dyne on such a pole, under the conditions stated in the definition of the c.G.s. unit (page 18) is found by experiment to deposit silver from a solution of silver nitrate at the rate of 1-118 milligrammes per second. We may here add that the direction of the force between two poles is found by experiment to be in the line joining the poles, and that of the force exerted by a current in a short conductor on a magnetic pole is at right angles to the plane containing the conductor and the pole. When the distance between two magnetic poles is changed, experiment shows (see Sect. 250) that the magnitude of the force between them varies inversely as the square of the distance, i.e., if the distance be doubled, the force is reduced to one-quarter of its previous value. A similar law between force and distance exists in the case of a short conductor carrying a current and a magnetic pole. 5 Fig. 17. Silver Voltameter for Measuring Currents of about One Ampere. The metal deposited by the current does not adhere well to the plate of a voltameter or " electrolytic cell" if the action pro- ceeds too rapidly ; also errors will arise in the estimation of a current by the electrolytic method, unless certain precautions be carefully attended to. Thus, when measuring a current of about one ampere with a silver voltameter, it is advisable to adopt the following arrangement : The " cathode," (sometimes spelt " kathode") or plate on which the silver is deposited, should take the form of a light bowl K (Fig. 17), not less than 20 PRACTICAL ELECTRICITY 10 centimetres* in diameter, and from 4 to 5 centimetres in depth, and made of platinum, so that it may be easily cleaned with nitric acid. The " anode," or plate from which the silver is electrically removed, should be a disc of pure silver, A, of about 30 square centimetres in area, and from 2 to 3 millimetres thick. Riveted to the anode is a strip of pure silver s, and by means of the metal clamp c and nut N the anode is supported centrally within the cathode bowl with its upper surface just below the level of the liquid. The liquid usually employed is a neutral solution of pure silver nitrate, containing about 15 parts by weight of the salt to 85 parts of distilled water. Fig. 1 8. Desiccator used with the Silver Voltameter. Electric contact is made between the wire W 1 and the bowl by means of three metal pins p, on which the bowl rests ; and the wire W 2 is electrically joined to the anode disc by the strip s being held fast in the rnetal clamp c, to which the wire W 2 is attached. In addition to the surface of the anode plate being turned into silver nitrate by the passage of the current, there is a tendency for small bits of silver to become detached and to fall into the bowl, thus making its weight too great. To prevent this, the anode may be wrapped round with pure filter paper, secured at the back with sealing-wax as shown at A, Fig. 17; * One metre is 39-370 inches, therefore 10 centimetres correspond with a little less than 4 inches. One square metre is 1,550 square inches therefore 30 square centimetres is a little less than 4! square inches. ELECTROCHEMICAL EQUIVALENTS 21 When making an observation, the current should be allowed to pass for about half an hour, and be maintained as constant as possible. A full description of the method of making a measure- ment is given in Appendix I., page 490. To obtain a uniform adherent deposit of silver, it is desirable that the cathode should possess about 30 square centimetres of surface for every ampere passing. Hence, if a large current of several hundred amperes had to be measured by means of a silver voltameter, the apparatus would necessarily be large and costly. In the voltametric measurements of large currents, therefore, it is usual to replace the platinum bowl and the silver disc by copper plates, and the solution of silver nitrate by one of acidulated copper sulphate. The chief reasons for using the rate of deposition of silver in the practical definition of the " international ampere " are (i) currents can be measured with precision by the silver volta- meter in any civilised country at moderate cost, the quantities to be determined, viz., mass and time being susceptible of very accurate measurement ; (2) the amount of deposit is indepen- dent of the value of gravity, of temperature, of humidity, and of atmospheric pressure to a very high degree. Although the primary definition of the ampere is based on the magnetic property of electric currents and the C.G.S. system of mechanical units, its precise realisation necessitates the con- struction of very accurate and costly instruments, a knowledge of the strength of pole or moment of a magnet, and also of the acceleration of gravity at the spot where the experiment is carried out. Another method of determining the ampere is based on the forces which exist between coils carrying currents, and a very exa'ct measurement has been carried out by one of the authors and Mr. F. E. Smith, of the N. P. L., using an apparatus designed in 1898-99 at the Central Technical College (now the City and Guilds (Engineering) College). Experiments of this nature have of necessity to be made to realise the ampere as based on the C.G.S. unit of current strength, and the results are usually expressed in terms of the amount of silver deposited per second. 9. Electrolysis ; Electrochemical Equivalent. If a number of voltameters containing, for example, solutions of silver nitrate, copper sulphate, zinc sulphate, etc., respectively, be placed in series, and a current be sent through them for a certain time, the weights of the metals deposited on the cathodes of the respective voltameters, or the weights of the other constituents of the respective salts set free at the anodes, are very approxi- mately proportional to the chemical equivalents. Thus since 22 PRACTICAL ELECTRICITY the atomic weights of silver, copper, and zinc are respectively 107-88, 63-57, an d 65-37,* and, since silver is monatomic while copper and zinc are diatomic, it follows that, as an ampere is the current that deposits 0-001118 gramme of silver per second, the weights of copper and zinc that will be deposited per second per ampere are respectively about X ^ X 0-001118, or 0-00032.94 gramme, f 2 107** and - X 5_3Z x 0-001118, or 0-0003387 gramme. The first quantitative experiments on " electrolysis" the name given to electric decomposition, were carried out by Faraday in 1833, and although he found that the proportion of the weights of different substances liberated by a given current flowing for a certain time differed sometimes by as much as 2 per cent, from the ratio of their chemical equivalents, he attributed this to inaccuracy in his experiments. He, therefore, concluded that the " electrochemical equivalents " of substances were directly proportional to their chemical equivalents. Among the many investigations that have been conducted for comparing the rates of deposit of copper and silver the most com- plete is probably that carried out by Prof. T. Gray. He found that the amount of copper deposited per second per ampere varied slightly with the size of the cathode and the temperature of the copper sulphate bath. If, however, the anode and cathode plates have each an area of about 50 square centimetres per ampere passing, and if the solution in the bath be formed by dissolving pure copper sulphate in distilled water until the density becomes 1-18, and afterwards adding about i per cent, of sulphuric acid, the weight of copper deposited per second per ampere is very approximately 0-0003286 gramme, and is but little affected by temperature. The difference between 0-0003286 and the theoretical value 0-0003294 arises mainly from the fact that copper plates lose weight when immersed in acidulated solu- tion of copper sulphate. To allow for this the experimental value has been used in the calculations which follow. It will be observed that the weight of silver deposited per second per ampere in a silver voltameter is nearly four times as great as the weight of copper deposited in a copper voltameter. * These are " International Atomic Weights " (1909), based on that of oxygen being 16-00 ; the atomic weight of hydrogen on this basis being I -008. \ See pp. 22 and 25 for experimental value 0-0003286. ELECTROCHEMICAL EQUIVALENTS 23 This reason would alone render the silver voltameter much to be preferred for the measurement of small currents, but for large currents the cost of silver is excessive> so copper is employed. A current of one ampere, when passed through a solution of dilute sulphuric acid, decomposes about 0-00009334 gramme of the liquid per second, The acid in the voltameter may be con- veniently diluted with water until the specific gravity of the mixture is about ii, which corresponds with a mixture of about 15 per cent, by weight of pure sulphuric acid at 15 C. The volume of mixed gas (oxygen and hydrogen) that is produced per second by the decomposition, corresponding with a current of one ampere, is about 0-1734 cubic centimetre, when the temperature at which the gas is evolved is o Centigrade, and the atmospheric pressure that of 76 centimetres of mercury. When the temperature is T Centigrade, and the height of the barometer h centimetres, the volume of gas evolved by one ampere in one second is approximately 0-1734 X 76 X (273+7) ,. i^_j_ -- v /J L cubic centimetres. AX 273 Example i. How many amperes would deposit 5 grammes of copper in half an hour, the current being supposed constant ? As 0-0003286 gramme is deposited in I second by I ampere, 5 grammes are deposited in i second by - - amperes. 0-0003286 Hence 5 grammes are deposited in 30 X 60 seconds by amperes. 0-0003286x30x60 Answer. About 8-453 amperes. Example 2. How many grammes of copper would be deposited by a steady current of 40 amperes acting for 5 hours ? i ampere acting for i second deposits 0-0003286 gramme, therefore 40 amperes acting for 60 X 60 X 5 seconds deposit 0-0003286 X 40 X 60 X 60 X 5 grammes. Answer. About 236-6 grammes. Example 3. How many amperes would deposit 9 grammes of copper in 2\ hours, the current being constant ? Answer. About 3-043 amperes. Example 4. How many grammes of copper would be deposited by a steady current of 1-5 amperes acting for 16 seconds ? Answer. About 0-007886 gramme. 24 PRACTICAL ELECTRICITY Example 5. How many grammes of dilute sulphuric acid would be decomposed by a steady current of 12 amperes acting for one hour ? Answer. -About 4*032 grammes. Example 6.- A current is passed through two voltameters in succession, one having silver plates and the other copper. After the current has ceased a deposit of 2-03 grammes of silver is found in the former voltameter ; how much copper has been deposited in the latter ? Answer. 0*597 gramme. Example 7. If the mixed gas produced in a sulphuric acid voltameter beat 20 C., and the barometer stand at 77-5 centi- metres, what volume of gas would be produced in half a minute by a steady current of 18 amperes ? i ampere in i second produces about Q-I734X76X (273+20) cubic centimetres of 77-5X273 therefore 18 amperes in 30 seconds produce about 0-1734 X 76 X 293 X 18 X 30 cubic centimetres of gas. 77-5X273 Answer. About 98-5 cubic centimetres of gas. Example 8. If the temperature of the mixed gas in a sulphuric acid voltameter be 19 -5 C., and the height of the barometer 75 centimetres, what current would produce 50 cubic centimetres of mixed gas in one minute ? Answer. About 4-43 amperes. Example 9. A silver voltameter and a copper voltameter are arranged like C 2 , C 3 , in Fig. 16, so that the main current divides between them. A steady current of 3 amperes is kept flowing in the main circuit for one hour, and it is then found that the deposit of copper in the copper voltameter is 0-4 gramme. What is the deposit of silver in the other voltameter ? Answer. About 10-71 grammes. io. Definition of Unit Quantity of Electricity : Coulomb. In the preceding section we have seen that the amount of chemical decomposition is proportional to the strength of the current, and to the time the current flows ; it is therefore proportional to the product of the current strength /* and the time t. A similar rule holds in the case of the flow of water or gas, the amount carried depending on the current and the time, current being considered as the velocity of flow multiplied by the area of the channel or pipe. The product of current (of water or gas) * The letter / has been adopted internationally as the symbol for current strength. THE UNIT QUANTITY; THE COULOMB 25 and time is called the " quantity " of liquid or gas, and may be expressed in gallons or cubic feet or other convenient units. In the same way the product of electric current and time is called " electric quantity," and when the current is one ampere and the time one second, the quantity conveyed is called " one coulomb." We may therefore define a coulomb as the quantity of electricity conveyed by a current of one ampere flowing for one second. For any other values of / and t we have coulombs = amperes X time in seconds, = It. There is another unit of quantity in commercial use, viz. the " ampere hour "* and as I hour is 3,600 seconds, i ampere hour =3, 600 coulombs. As the amount of chemical decomposition in a voltameter is proportional to the current and the time, it is proportional to the quantity of electricity which passes through the voltameter, and we may express the electrochemical equivalents of substances as so many grammes, or milligrammes, per coulomb . For example we have : Electrochemical equivalent of silver = 1-118 mgs. per coulomb. copper = 0-3286 zinc = 0-3387 water = 0-09334 hydrogen = 0-01044 oxygen = 0-08290 Voltameters are. in reality coulomb -meters, as the amount of chemical decomposition depends on the number of coulombs of electricity passed through them. Special forms of voltameters are frequently employed by Electric Lighting Companies as house meters, to register the quantity of electricity the " con- sumer " has allowed to pass through his lamps. (See Sect. 128.) Example 10. How many coulombs pass through the volta- meters mentioned in example I ? state also the quantity in am- pere hours, 1st method : Quantity in coulombs = current in amperes X time in seconds. = 8-453 X number of seconds in half an hour. = 8-453x30x60. Answer = 15,215 (approximately). An ampere hour is the quantity of electricity conveyed by a current of one ampere flowing for one hour. 26 PRACTICAL ELECTRICITY Quantity in ampere hours = am peresx hours. = 8-453 Xj. Answer = 4-226 (approximately). 2nd method : Massdeposited=numberof coulombs X electrochemical equivalent. /. 5=number of coulombs X 0-0003286. .'. Number of coulombs = 5-^-0-0003286. = 15,215 (approximately). Ampere hours = number of coulombs -=-3, 600. _ 15,215 3,600 = 4-226 (approximately). Example n. Express the quantities of electricity used in examples 2 to 8 in coulombs and ampere hours. Answers to Example n. Coulombs. Ampere hours. No. 2 720,000 20O No. 3 . . . . . 27,389 7-608 No. 4 24 0-006 No. 5 43,200 '12 No. 6 1,816 0-504 No. 7 540 0-15 No. 8 . . . . . . > 266 0-0772. ii. Definition of the Direction of the Current : Ions. The next thing to define is the direction of the current, which, as already explained, can only be done in a conventional way. In the case of a sulphuric acid voltameter, we have hitherto only spoken of the total quantity of gas given off at both platinum plates, but if these gases be collected in separate tubes, as can very conveniently be done in the Hoffmann's voltameter (Fig. 19), then it is found that at one of the plates P oxygen gas o is given off and at the other plate hydrogen H is liberated, and the current is said to travel through the liquid towards the plate at which the hydrogen is given off, or, in other words, the current flows through the liquid with the hydrogen. Hence in the Hoffmann's voltameter, shown in Fig. 19, the current would be said to flow through the liquid in the short horizontal tube, from right to left. The gases are evolved exactly in the proportions in which they have to be combined together to form water viz., two (or more accurately 2-002 at 15 C.) volumes of hydrogen and one of oxygen.* So that the electrolytic action effected by sending a * That the gases are hydrogen and oxygen can be proved by the fact that on turning the stop-cocks s, s, the one gas H when lighted will burn with a pale blue flame, and the other o will ignite a glowing piece of wood. DEFINING DIRECTION OF A CURRENT 27 current from one platinum plate to andther in dilute sulphuric acid is exactly the same as if the water had simply been decom- posed. If an acid, a silver, a copper, and a zinc voltameter be all joined together, so that the same current passes through them, then it will be found that the hydrogen in the first, the silver in the second, the copper in the third, and the zinc in the fourth, all travel in the same direction in the circuit ; so that if through the liquid in an acid voltameter the current be said to go in the direction in which the hydrogen travels, then through the liquids in a silver, a copper, and a zinc voltameter, it must be said to go in the direction in which the silver, the copper, and the zinc travel. Or generally the current in a voltameter may be said to travel with the metal from the anode towards the cathode, hydrogen behaving in this respect, and, as is well known, in other respects, like a metal. The components into which an " electrolyte " is decomposed by the passage of a current are called " ions," and the ion which travels with the current is called the " electropositive ion," while the one which travels against the current is called the " electronegative ion" Other names for these ions are cation and anion, meaning the ions which appear at the cathode and anode respectively. With the definition given above of the direction of a current, we find that if a com- pass needle, n s (Fig. 20), be pivoted so as to turn in a plane at right angles to the plane of the paper, and a current flow along any wire, A B c D, which is in the plane of the paper, then the north-seeking end* of the compass needle will * The " north-seeking " end oi a magnet is the one that points towards the geographical north. The simple expression " north " end is confusing, since in England it refers generally to the end of a magnet that points to the north, while in France it refers to the end that points to the south, the French using that definition because that end is attracted by the earth's magnetism situated in the southern hemisphere, and the unlike ends attract one another. Calling the ends of magnets " red " and " blue " is equally confusing, as some people use one of these two colours, and others the other colour, to indicate the. same end. As, however, the north- seeking end of a magnet is usually marked by instrument makers with a scratch or a cut, it would probably be best to call the " north-seeking " 28 PRACTICAL ELECTRICITY come towards the observer if the current flow round the wire in the direction indicated by the continuous arrow that is, counter- clockwise ; whereas the south-seeking end of the needle will come towards the observer if the current flow in the direction of the dotted arrow that is, clockwise. Similarly, if A B (Fig. 21) be any bit of wire in the plane oi the paper, the north-seeking end of the needle (n, say) will come towards the observer if the current flow along this bit of wire, A B, in such a direction that A B may be regarded as forming part of a counter-clockwise circuit round the needle. Therefore, in the upper three of the illustrations of Fig. 22, the end n will come towards the Flg- 20 * observer, while in the lower three it will be the end s that will come out towards the observer. Or, again, if a wire conveying a current be coiled round a piece of iron shown end-on to an observer, then the end of the iron nearest him will act as the north-seeking end of a magnet when the current appears to the observer to flow round the wire in a counter- clockwise direction. If the observer now look at the other end of the bar, he will of course see the south-seeking end, and in his new position the current will now appear to him to flow round the wire in the same direction as that in which the hands of a clock go (or clockwise). ^ The relative magnetic polarity of the iron bar and the direction of the current, as indicated by the arrows, are shown in Fig. 23. Flg * 2I- The magnetic polarity of the end of an iron bar round which a current is flowing does not depend on whether the current is flowing from the left to the right-hand end of the bar, as in the first of Fig. 23, or from the right to the left-hand end, as in the last of Fig. 23 ; but merely on the direction the current flows round the bar. Now, in spite of the difference of the winding of the wire on the first and last of Fig. 23, it will be found that in both cases, if the bar be looked at end-on from the right, the and "south-seeking" ends of a magnet the "marked end" and "unmarked end" respectively. In this work where the words "north end" or "north pole" are used they are to be understood to mean "north-seeking. DIRECTION OF A CURRENT 29 Fig. 22. current is clock- wise, whereas if the bar be looked at end-on from the left the cur- rent is counter- clockwise. Perhaps the simplest method for remembering the connection between the magnetic polarity of an iron bar and the direction in which a current circulates round it is that, if a current circulates round the bar in the direction in which the handle of a corkscrew (Fig. 24) is turned when the corkscrew is screwed down or up, the point of the screw will move towards the north-seeking magnetic end of the iron bar. Example 12. A compass needle is supported under a telegraph- wire running north and south. How will the needle deflect if a strong current flow through the wire towards the south ? Answer. The north-seeking end of the needle will turn towards the east. Example 13. A flat vertical conductor is fastened against a wall, and in front is suspended a mag- netic needle pivoted so as to turn on a vertical plane par- allel to the wall. The north - seeking end of the needle is weighted so that the needle stands ver- tically when no current is flowing. Which way must a current flow in the conductor to make the upper end of the needle point to the right ? A nswer. Down- wards. Example 14. Fig. 23. Draw an arrow on 3 o PRACTICAL ELECTRICITY the movable card of a compass, so that when the compass is placed above a horizontal conductor conveying a strong current the arrow will indicate the direction of the current. Answer. 12. Objection to the Usual Mode of Constructing Voltameters. The sulphuric acid voltameters, as usually pictured in books, which are still the only forms obtainable at some shops, are extremely unsuitable for practi- cal use, as it is troublesome, after the tubes in which the gas is collected are full of gas, to fill them with liquid again for a new experiment.* The apparatus shown in Fig. 19, page 27, is very convenient when it is required to collect the oxygen and hydrogen separately, but it has the incon- venience that, the platinum plates being small and far apart, it requires the employment of several galvanic cells to make the gas come off quickly. For, although the quantity of gas pro- duced in a given time by the same current is practically in- dependent of the shape and size of the plates, the ease with which this current can be generated depends very materially on the size of the plates and their dis- tance apart, and if we wish to * The improved forms of volta- meters described in Section 13 have been adopted by many instrument- makers since the first appearance of this book. FORMS OF ACID VOLTAMETERS produce chemical decomposition quickly, we ought to have the plates large and very near together, and the liquid employed ought to contain something like 33 per cent, of strong sulphuric acid by weight, the mixture having a specific gravity of about 1-25 at 15 C.* Such a mixture conducts electricity more readily than solutions of other strengths. 13. Description of Practical Forms of Sulphuric Acid Volta- meters. In Fig. 25 is shown a very convenient form of voltameter, designed by Prof. Ayrton, consisting of a glass vessel closed at the top with an indiarubber stopper I and containing moderately dilute sulphuric acid. The two platinum plates p are held to- gether by indiarubber bands, but prevented from touching one another by small pieces of glass tubing put between the plates at the top and bottom, or to save the expense of thick platinum plates, two pieces of thin plat- inum foil may be used, stuck at the bottom with bicycle or other suitable cement, to a piece of glass tube, the weight of which causes the two pieces of foil to hang vertically, and therefore at the same distance apart all the way down. Wires coated with gutta-percha to prevent their being corroded by acid being spilt over them, or better still, platinum wires go from the plates, one to the " key " K (which is raised up above the general level of the apparatus to prevent its being corroded by drops of acid), and the other wire to one of the terminal binding screws seen in the figure. On pressing down K, the current produced by a generator attached by wires to the two binding screws, seen at the right-hand side of the figure, is allowed to pass through the apparatus. The gas which is generated is unable to escape from the vessel when the pinch-cock c is closed, and accordingly forces the liquid up the graduated tube t. This tube passes air-tight through the indiarubber stopper I, reaches nearly to the bottom of the vessel, and termin- ates at the upper end in a thistle funnel F, so that if the current is by accident kept on for a longer time than is necessary to cause See Appendix IV, Fig. 25. Ayrton's Form of Sulphuric Acid Voltameter. PRACTICAL ELECTRICITY the liquid to rise to the top of the graduated tube, the liquid collects in the funnel instead of spilling over. This tube is also sloped so that the rise of liquid in the tube may increase the pressure of the gas in the upper part of the voltameter as little as possible.* The second tube might be simply terminated with a piece of indiarubber tubing closed with a spring pinch-cock, c, on opening which the gas is allowed to escape, and the liquid runs back out of the tube /. If this is done suddenly, however, there is a tendency for small particles of the liquid to be jerked out of the lower tube. To prevent these particles being thrown on to the stand of the apparatus, the tube is carried up, and its end bent over into the thistle funnel F. Instead of observing the distance the liquid travels up the graduated tube t (Fig. 25) in a given time, we may notice the time it takes to travel from a certain fixed mark at one end of the tube to another fixed mark at the other. In other words, instead of measuring the volume of gas produced in a given time, we may measure the time taken to produce a given volume. And since for different currents the times taken for the same volume of gas to be produced must be inversely as the volumes of gas produced in the same time, we can deduce the current by employing a tube which has not been subdivided into equal volumes, but only has two marks on it. With this method of ( using a voltameter to measure ^currents there is no necessity for the tube to be long, since it can be Fie. 26. Mather's Form of Sulphuric conveniently expanded into a bulb B (Fig. 26), and great sensibility can be combined with compactness by the bore of the tube being made small at the places where the reference marks m 2 and m l above and below the bulb, are made. The wires pass through the indiarubber stopper inside glass tubes to ensure that all the current passes through the liquid. The spring pinch-cock should not be left squeezing the indiarubber tube of the voltameter (Figs. 25 and 26) when the instrument is out of use, for continued pressure on the sides of the * If the vessel be full of liquid so that there is no gas between the top of the liquid and the indiarubber stopper I at the commencement of the experiment, the error arising from the compression of the gas produced by the rise of liquid in the tube t may be neglected. SULPHURIC ACID VOLTAMETERS 33 tube causes it to acquire a permanent set and prevents it from regaining its circular form when the pinch-cock is removed. Another form of voltameter, devised by J. A. McMichael, Esq., is shown in Fig. z6a. Connection with the platinum plates is made through wires sealed into glass tubes containing mercury, which are seen projecting just above the top of the rubber stopper. The measuring tube on the right is graduated in ampere-minutes, and by passing a current for a period of one minute through the voltameter its strength can be read off directly in amperes. 14. Relative Advantages of Voltameters and Galvanometers. One great advantage that voltameters possess over galvano- meters is that a given current produces the same rate of chemical decomposition at any place on the earth's surface, this rate being quite independent of the force of gravity, or of the earth's mag- netism, both of which differ in intensity at different places. The indications of galvanometers, and of most other forms of current measurers, are influenced by gravitational or magnetic forces, and so do not possess the same immunity from local conditions as voltameters. For these reasons the electrochemical definition of the ampere is now employed for international purposes. The disadvantages of employing a volta- meter for the practical measurement of currents are (i), that it requires a strong current to produce any visible decomposition in a reasonable time ; and (2), that a measure- ment of time is necessary. Even the current of one ampere, which is about six times that used in an ordinary 8-candle incandescent lamp, would require nearly three hours to decompose one gramme of dilute sulphuric acid, whereas the weak currents used in telegraphy, and, still more, the far weaker currents used in testing the insulating char- acter of specimens of gutta-percha, india- rubber, etc., might pass for many days through a sulphuric acid voltameter without causing any noticeable amount of chemical decomposition. Indeed, not to mention the enormous waste of time, and the difficulty of keeping the current strength which it was desired to measure constant all this time, the leakage of the gas Fig. 2 6. McMichaei's Form , . , , of Acid Voltameter grad- Which WOuld take place at all parts OI uated in ampere-minutes, the apparatus that were not hermetically 1 a " 34 PRACTICAL ELECTRICITY sealed,* would render such a mode of testing quite futile. Hence, although the voltametric method is a fairly direct way of measuring a current strength, and is one of the most accurate ways of determining the strength of current; exceeding a few tenths of an ampere, that can be kept constant for half an hour or so, still the very fact that the amount of chemical decomposition produced in a given time by a certain current is independent of the shape or size of the instru- ment, makes it very difficult to increase its sensibility Con- sequently some other apparatus must be employed for practically measuring small currents, and the law of the apparatus that is, the connection between the real strength of the current and the effect produced in the apparatus must either be experimentally ascertained by direct comparison with a voltameter, or an instrument constructed so that the current can be calculated from its dimensions and the C.G.S. unit of current strength denned in Section 8. When the law of the apparatus has been found, it is said to be " calibrated." But if we are going to compare together the indications of two instruments produced by various currents, the second in- strument cannot be much more sensitive than the first; what advantage, therefore, can arise from employing an instrument as unsensitive as a voltameter ? This leads us to the fact that it is very much more difficult to increase the sensitiveness of volta- meters than of " galvanometers."^ We might increase the magnitude of the indications of a voltameter, such as that shown in Fig. 25, by using a tube / of very small bore, or by putting several such voltameters in series, and collecting the gases given off by each into one vessel ; but we cannot by either of these means succeed in constructing a voltameter which possesses anything like the sensibility that can be very easily given to a galvanometer. The indications of any measuring instrument may be increased in three distinct ways. As an illustration, let us consider an ordinary spring-balance, like the one attached to the apparatus D in Fig. 9, page 9. We may, in the first place, use a microscope, or we may fit the balance with a wheel and pinion, or employ * A glass vessel is said to be hermetically sealed when any opening that previously existed in it has been closed by heating the glass round the opening until it becomes soft and sticky, and pressing the edges together. f While a " galvanoscope " is the name given to an instrument used for ascertaining whether a current is flowing, or merely which of two currents is the stronger, a " galvanometer " is the name given to an instrument by means of which the relative strengths of currents can be compared. Any galvanoscope when so calibrated becomes a more or less sensitive galvano- meter. VOLTAMETERS AND GALVANOMETERS 35 some other magnifying arrangement to render the extension of the spring more apparent; or the electromagnet may be so constructed, either by employing more iron or by putting more convolutions of wire round its limbs, so that the pull on the " keeper " or armature a (Fig. 9), caused by passing a given current round the coils of wire, is increased ; or, lastly, we may use a weak spring in the balance, so that, for a given pull on the keeper, the movement of the index may be large. Each of these three methods can be applied with great success to galvanometers. In the first place, the sensitiveness may be increased by using a long pointer, and the pointer may be made light, and therefore easily moved, by forming it of a very fine glass tube, or of a narrow strip of some light substance like alu- minium. But the best of all methods, and therefore the one employed with very sensitive galvanometers, consists in using a ray of light several feet long, but, of course, quite weightless, reflected from a small mirror attached to the needle, thus making what is called a " reflecting galvanometer." The sensi- bility of a galvanometer can also be made large by winding the bobbins with very many turns of very fine wire (see Sect. 35) ; also by placing the bobbins very near the suspended needle. Friction can be diminished by suspending the little magnet with a thin fibre of untwisted silk. And lastly, by employing a very weak " controlling magnet " or by putting it at some distance from the galvanometer, the " torque"* required to turn the needle can be reduced to a very small amount, and therefore a consider- able deflection can be produced by an extremely weak current. And so successful have been the various attempts to increase the indications of galvanometers that it is now possible to measure accurately an electric current which is so small that it would have to flow for a million years through a voltameter before it produced as much chemical action as a current of one ampere could produce in a single hour. Now, experiment shows that a galvanometer of a particular shape and size, and with a definite magnetic needle, acted on by a definite controlling force, produced, say, by the earth's magnetism, or by some fixed permanent magnet, has a perfectly definite law connecting the magnitude of the deflection with the strength of the current producing it, although the absolute value of the current in amperes necessary to produce any particular deflection can be increased, or diminished, by using thick wire and few turns, * Torque is the tendency that any system of forces has to cause a body to turn, so that torque bears the same relation to turning that a force has to motion in a straight line. 36 PRACTICAL ELECTRICITY or fine wire and more turns, to make a coil of the same dimensions. If, for example, with a particular gauge of wire employed to fill up the bobbin it requires 2f times as many amperes to produce a deflection of 40 as it requires to produce a deflection of 20, then if a much finer gauge of wire be employed to fill the bobbin there will still be required 2| times as many amperes to produce a deflection of 40 as are required to produce a deflection of 20. But in the second case y^^ of an ampere may be all that is required to produce the 20 deflection, whereas five amperes may be required to produce the same deflection in the first. The law of the instrument remains the same, although its sensibility has been increased 5,000 times by using finer wire to wind on the bobbin. Thus, while we may take advantage of the absolute character of the amount of chemical action to furnish us with our " standard current meter," we can avail ourselves of the variation that can easily be made in the deflection of a galvanometer needle cor- responding with the same current, to furnish us with instruments of greater and greater degrees of delicacy. 15. Measurement of Current by Galvanometers : Tangent Galvanometers. As galvanometers are capable of being used over such a wide range of current strength, it is advisable to study them somewhat closely at the present stage. A very useful form of instrument is shown in Fig. 27, page 37, consisting of a small magnetic needle n s suspended by a fibre of unspun silk at the centre of a comparatively large, circular coil of wire. A thin glass pointer, pp, attached to the needle, moves over a graduated scale, which is fixed to a disc of looking-glass to avoid " parallax " in reading the deflections. When no current passes through the coil, the magnet n s behaves like a compass needle and sets itself in a direction nearly north and south, and if turned in any other direction by the finger or other means, will promptly return to the north and south position when freed. The force (or rather torque) which is operative in causing this return to the zero position, is called the " controlling force," and is usually due to the earth's magnetism ; in such cases the needle is said to be controlled by the earth's magnetic field. The space in the vicinity of any magnet where a force would be exerted on a magnetic pole if such were present, is spoken of as the magnetic field of the magnet, and the field is said to be strong or weak according as the force exerted on a unit pole is large or small. In fact, the strength of a magnetic field at any point in space, is, by defini- tion, measured by the force in dynes exerted on a unit magnetic pole placed at that point. The sense of the field is taken as that in TANGENT GALVANOMETERS 37 which a north-seeking pole placed at trie point would tend to move, and may be found practically by a small compass needle with its centre at the point considered. Over considerable distances the earth's field (where undisturbed by masses of iron, or other magnets, or by electric currents), is uniform in strength and direction, and urges the north-seeking pole of a magnet northwards and its south-seeking pole southwards. G B Fig. 27. Tangent Galvanometer ; the smaller diagrams A, B and C show various modes of supporting the Fibre. With A the needle can be moved sideways by sliding the roller R in the spring clips s, s, and can be raised or lowered by turning this roller. With B the pin p is held by a single screw s instead of between two brass plates, as shown in the complete galvanometer. With C the pin P is held by a set screw s in a support made with a ball top B. This fits in the hole h in the plate i and forms a ball and socket, so that the needle can be accurately centred. The ball and socket joint is clamped to the semicircular support A with the screw s. In most places on the earth's surface, the direction of the earth's magnetic field is neither horizontal nor vertical, its true direction being shown by a magnetic instrument called the " dipping needle." The angle between a horizontal line in the magnetic meridian* at any place, and the position taken up by the axis of a dipping needle, is called the " angle of dip " at that place. For * The vertical plane in which a freely suspended magnet sets itself at any place, is called the magnetic meridian at that place. 38 PRACTICAL ELECTRICITY galvanometric purposes it is customary to consider the earth's magnetic force resolved into two components, the horizontal component and the vertical component respectively, and the former is frequently employed as the controlling field in galvano- meters.* The magnitude of the earth's horizontal component is different at different places on the earth's surface, and also changes daily and alters from year to year. At any one place, however, the daily and yearly changes are comparatively small. For the year 1918 its mean value in London was approximately 0-1846, the average yearly change and daily variations amounting to about i and 2 parts in 1000 respectively, so that for many pur- poses we may take the horizontal component of the earth's field in undisturbed areas as approximately constant. If, therefore, an instrument such as that shown in Fig. 27 be placed in an undisturbed area, then when a current passes through the coil the needle will be influenced by two magnetic fields, one due to the earth's horizontal field and the other due to the current, the field of which is at right angles to the plane of the coil (see Fig. 37). The needle will be deflected, and take up a position of equilibrium along the direction of the resultant field, the position of which can be found by the parallelogram of forces Let NQ (Fig. 27^) represent in magnitude and direction the earth's controlling field, and NP the field at n (Fig. 27), due to the current in the coil, then the diagonal NR of the parallelogram NQRP represents in magnitude and direction the resultant of the two fields, and the magnet n s will set itself in this direction. The current causes the magnet to move through the angle QNR, from its zero position, and this is called the angle of deflection. If the lines NQ and NP are perpendicular, i.e., if the needle n s lies in the plane of the coil when there is no current passing, then we have Np = QR = NQ tan QNR Now NP represents the deflecting field, which is, by definition, proportional to the strength of the current, and may be written N p = k /, where k is a constant and / the current in amperes, and N Q represents the horizontal component of the earth's magnetic field (usually denoted by H). We may therefore write the equation as k I == H tan d, where d = angle QNR. or 7 - tan d, (i) k * In some cases, permanent magnets are used to produce controlling fields stronger or weaker than those due to the earth alone. RELATIVE AND ABSOLUTE CALIBRATION 39 and since H and k are constants, we sefr that the current is pro- portional to the tangent of the deflection in the instrument used as described. For this reason galvanometers having large plane coils and small needles are called Tangent Galvanometers. From what has been said above it will be seen that the magnetic needle n s will set itself along the direction of the resultant NR, whether the needle be a strong magnet or a weak one ; in other words, the deflection of a galvanometer needle controlled by a con- stant magnetic field is independent of the strength of the deflected magnet. This is true of any form of galvanometer having a moving needle, magnetically controlled. It is, however, desirable to use strongly magnetised needles, for by doing so the forces operative are in- creased, thus diminishing any error that may be introduced by torsion of the suspending 'fibre, or by friction at the pivots ; and the motions of the pointer are quickened. 16. Meaning of the Relative and the Absolute Calibration of a Galvanometer. Two distinct things are required to be known with reference to a particular galvanometer first, the law connecting Fig. 27* the various deflections with the relative strength of the currents required to produce them ; secondly, the absolute values of the currents that is, the number of amperes required for the same purpose or, what is sufficient if the first has been ascertained, the number of amperes required to produce some one deflection. The first is sometimes called the " relative calibration," the second the " absolute calibration "of the galvanometer. A galvanometer with its bobbin wound with thick wire may be compared directly with a voltameter, and the relative calibration of the galvanometer determined ; then if the same space on the bobbin be wound with any other gauge of wire the relative calibra- tion of the galvanometer will be the same, and therefore known, provided that nothing but the winding has been altered. Or if a galvanometer wound with thick wire be compared with a voltameter, and its absolute calibration determined, and if, further, the law of change of sensibility with gauge of wire has also been ascertained experimentally 'then the absolute calibration of the same galvanometer, when wound with any gauge of wire, filling the same space, will be known without further experiments, provided that only the winding has been changed. 40 PRACTICAL ELECTRICITY 17. Comparison of Tangent Galvanometer with a Voltameter. The apparatus shown in Fig. 28 can be used to show by experi- ment that the tangent of the deflection of a galvanometer of the form shown, is proportional to the rate at which chemical decomposition occurs in a voltameter in the same circuit ; and in this way we may prove that the electromagnetic definition of current strength is relatively consistent with the chemical de- finition. A voltameter, v, is connected up with the galvanometer, G, and a " set of resistances," each consisting of a coil of wire with its ends connected with two successive terminals, t lt t 2 , etc. These coils are wound on bobbins, and are placed underneath the base board to which the whole of the apparatus is fixed, and by means of which it can be bodily carried from place to Fig. 28. Comparison of Tangent Galvanometer with a Voltameter. place ; (from the laboratory to the lecture-room, for example, for demonstration to a class). The magnitude of the current is altered by joining the wire, w, to the various terminals, t lf t 2 , t 3 , etc., on the base board. T, x are the main terminals, or binding screws, to which the wires coming from the current generator are attached. It may be noticed that in the particular experiment shown in Fig. 28 it is quite unnecessary to know the length or gauge of the wire that has been wound on the various bobbins ; nor is it at all necessary that all the coils should be made of the same length or thickness of wire, since, whatever resistance be inserted in the circuit, the current that passes through the voltameter is the same as the current that passes through the galvanometer, so that the variation in strength of the current is known from the voltameter observations, and not from the length of wire that has been introduced into the circuit. Indeed, the resistances in this experiment may be dispensed with altogether when there is any GALVANOMETER AND VOLTAMETER 41 other easy mode of altering the current Strength by using, for example, different numbers of " cells " or a different kind of battery to produce the current, but in practice this result is generally most easily attained by the use of a set of resistance coils. The comparison might be performed by observing for a number of different currents, the rise of the liquid in the graduated tube of a voltameter such as that shown in Fig. 25, in a given time, and the corresponding steady deflec- tion of the needle, or of the pointer, of the galvanometer. But more accurate observations can be made if, instead of observing the different lengths of the tube through which the liquid rises in a given time corresponding with the different currents used, the times be noted during which the liquid rises through a given volume viz., that between the two marks m lt m 2 , of the volta- meter tube (Figs. 26, 28). A calculation can then be made of the rate at which gas is evolved by the current, and from this the strength of the current in amperes can be found. Thus, let v be the volume, in cubic centimetres, of the bulb B (Figs. 26, * 28) between the marks m lt w 2 , and suppose the strength of the current be such that it takes t seconds for the liquid to rise from the mark m l to the mark m 2 , then the number of cubic centimetres of gas generated by the current in every second is v. cubic centimetres. t An actual experiment carried out by first-year students at the City Guilds College, gave the results tabulated below, the volume of the bulb being 9^6 cubic centimetres : TABLE I. Observed quantities. Deduced from observed quantities. Time in seconds (t) to fill bulb. Steady de- flection (d) of galvanometer in degrees. Cubic cms of gas per second. Tangent of deflection. Ratio, cubic cm tan d. v 0-187* (A mpere) . 573 260-5 177 122 *3 15 30-3 40-4 50-9 61 0-0168 0-0369 0-0542 0-0787 0-116 0-268 0-584 0-851 I-230 1-804 0-0625 0-0631 0-0637 0-0640 0-0641 0-090 0-197 0-290 0'42I 0*619 From column 5 it will be seen that the rate of production of gas (i.e., the relative current strength) bears a nearly constant ratio 42 PRACTICAL ELECTRICITY to the tangent of deflection of the galvanometer, the greatest variation from the mean being ij per cent., and this is within the possible error of experiment. It is therefore evident that the voltameter and tangent galvanometer are in substantial agree- ment with each other. Columns 2 and 3 of the table give a relative calibration of the galvanometer. As the numbers in column 3 are proportional to -, (being equal to ,) it is evident that column 2 and a column giving the values of would also constitute a relative calibration of the instrument, and for this purpose the volume of the bulb need not be known. 18. Absolute Calibration of Tangent Galvanometer. The above experiment also enables the relation between the current strength in amperes, and the deflection of the galvanometer (or the tangent of its deflection) to be determined, and in this way gives the absolute calibration of the instrument. For example, one ampere liberates in a sulphuric acid voltameter 0-1734 cubic centimetre of gas per second, at standard temperature and pressure, and if we suppose the actual temperature to be 18 C. and the height of the barometer 750 millimetres, it follows from the expression given in Section 9 that the volume liberated per ampere second under these conditions is 4- 18 0-1734 XX-= 0-I8 7 . Hence the current is - amperes. 0-187^ Calculating out the values of - for each observation in the 0-187* table, we obtain the strength of current in amperes used in each experiment. These are tabulated in column 6. Columns 2 and 6, therefore, give the absolute calibration of the galvanometer. Another important quantity relating to the particular galvano- meter as used in the experiment, can be deduced from the above TT table, viz., the value of in equation (i), Sect. 15. This quantity fv is often called the constant of the tangent galvanometer, under the conditions in which it was used, and is such that the current in amperes producing any deflection equals the constant of the instrument multiplied by the tangent of the deflection. The same thing may CALIBRATION OF GALVANOMETERS 43 be expressed by saying that the " constant " of a tangent gal- vanometer is equal to the current in amperes which will produce a deflection of 45, since tan 45 =i. In the case under considera- tion the constant is 0-351, so that the instrument could now be used to measure strength of currents in amperes ; the con- trolling field H remaining constant. 19. Calibrating any Galvanometer by Direct Comparison with a Tangent Galvanometer. Knowing the law connecting the deflections of a tangent galvanometer and the currents producing them, viz., current :: tan of deflection, we may now use this instrument as a " standard " with which to compare other gal- vanometers, and thus determine the relation between deflections and current strength in cases where the construction of the instrument makes it impossible to predict the relative calibration. To do this the instrument to be calibrated is connected up in the same circuit as the tangent galvanometer, so that the same current passes through both, and simultaneous readings of the two instruments taken. The current is then changed to another value and a second pair of readings observed ; this is done for several other different currents, such as will give deflections distributed over the whole scale of the instrument to be calibrated. Apparatus for making an experiment of this kind is shown in Fig. z8a, where a rough and portable galvanometer, D (sometimes called a "detector"), is coupled in circuit with a tangent gal- vanometer, G. The current is varied by sliding two zinc rods in or out of a V-shaped tube containing a solution of zinc sulphate. Table II. gives the results obtained in such a calibration, and constitutes a " relative calibration table " for the detector D. TABLE II. Deflection of galvano- meter to be calibrated. Deflection of Standard G al vanometer . Tangent of deflection of Standard Galvano- meter (or relative current strength). o-o degrees 3-8 10-0 15-0 20-5 36-5 50-0 65-0 79' O'O degrees 21-3 30-1 387 42-3 52-2 55-8 61-6 697 O-OO o-39 0-58 0-8o O'QI 1-29 1-47 1-85 270 44 PRACTICAL ELECTRICITY The method of calibrating a galvanometer described above is suitable for use when the standard instrument and the one to be calibrated are approximately equal in sensitiveness. Cases, however, frequently occur where one galvanometer gives a large deflection for a current which produces only a small deflection on the other one. When this happens part of the current may be diverted from the coil of the more sensitive instrument by con- necting a wire from one of its terminals to the other, and the length of this wire may be adjusted so that the two galvanometers give about equal deflections. The galvanometer whose terminals are so connected by wire external to the instrument is said to be " shunted," the external wire being called " a shunt" If the shunt be arranged so as to produce no magnetic effect on the Fig. z8a. Calibrating a Detector by Comparison with a Tangent Galvanometer. needle of the instrument, and this is quite easy to do, the relative calibration is not affected by the shunt, although the absolute cali- bration may be much altered by it. For example, if the un- shunted galvanometer requires I amperes to produce a deflection, d, the shunted instrument will require a larger current, /', to give the same deflection, and the ratio of 7 ' to / will be larger the shorter the length of wire (of given size and material), used as the shunt. 20. Graphically Recording the Results of an Experiment. The results of the experiment given in the above table are best recorded graphically by points on a sheet of squared paper.* * Prior to the commencement of the courses at the Finsbury Technical College, in 1879, squared paper was practically used in England only for the recording of results of original experiments. And as these results, rather than the training of the experimenter, were the most important part of the investigation, the paper was very accurately divided, and sold at a high price totally out of the reach of students. It became, therefore, USE OF SQUARED PAPER 45 This has oeen done in Fig. 29, where the points a, b, c, d, e, f, and g represent the numbers in the first and last columns of the table. In plotting a " calibration curve " the distances of the points a, b, c, etc., from the line o Y (Fig. 29) measured parallel to o x necessary to have squared paper specially made cheap, and at the same time sufficiently accurately divided for students' purposes ; and such paper, machine-ruled, can now be obtained at less than a farthing per sheet, or at about one-thirtieth of the cost of the older squared paper. 46 PRACTICAL ELECTRICITY should be taken to represent the deflections of the galvanometer calibrated, and the distances of the same points from o x, measured parallel to o Y, the corresponding values of current strength. It may be asked how distances along a line can represent the angular deflections of a galvanometer, or the strengths of currents producing such deflections. What is meant is this : the line o x is subdivided into a number of equal divisions by the ruling of the squared paper ; one, or any convenient number, of these subdivisions is taken arbitrarily to stand for i, then any de- flection, say of d, is represented by d times the number of divisions arbitrarily chosen to stand for i. Similarly one, or any convenient number of divisions along o Y is taken arbitrarily to stand for one unit of current strength, and n times this number of divisions will represent a current strength of n times the unit chosen. In Fig. 29 one division along o X stands for i deflection of the galvanometer calibrated, and 10 divisions along o Y is taken to represent unit current strength, the current which produces a deflection of 45 on the tangent galvanometer being taken as unity. Curves representing other quantities may be drawn in the same way. For example, the height of the baro- meter from hour to hour, the variation of the price of some commodity from day to day, or the depth of water from point to point along some section of a river, can be readily shown on squared paper, and generally a kind of picture illustrating how one thing varies relative to another may be given by such curves. In selecting the scale to which a curve is to be plotted, that is, determining what number of divisions along o X or along o Y should be taken to stand for i deflection, or for unit current strength respectively, we should be guided by the consideration that the resulting curve should represent the experimental num- bers quite as correctly as, or preferably, rather more correctly than, the accuracy to which the readings of the instruments were taken. For example, if the deflections of the galvanometer to be calibrated can be read, say, to J of i degree, the scale along o x should be such that the length representing J can be seen on the squared paper quite easily, in order that there may be no difficulty in reading the curve to J. Similar considerations will show what scale along o Y should be chosen. Care should be taken to select convenient scales so that the numbers may be plotted easily, and when plotted, be easily read. Squared paper is usually divided decimally, that is, the cardinal lines are spaced 10 divisions apart ; it is therefore desirable to choose the scale, where possible, so that one division represents one unit of the quantity to be plotted, or 10 units, or CALIBRATION CURVES 47 a tenth of a unit, or some decimal multiple or sub-multiple of 10. This cannot always be done, and scales of two or five divisions per unit, or two or five units per division, or decimal multiples, or sub-multiples of these may then be used without much inconvenience ; and the aim in choosing scales should be to make the curves as easy to read as possible, consistent with their being read with the necessary precision. The larger the scales to which a curve is plotted the more accurately it can be read, but in many cases the use of a very large scale is objectionable, for where this enables the points to be plotted with a far greater accuracy than was attained in making the observations, the large scale merely magnifies the errors of observation, and points which, if plotted to a reasonable scale, would lie in proximity to a curve, appear to be dotted about in an irregular manner like the stars of a constellation. Having chosen suitable scales and plotted a sufficient number of points from the experimental results, a curve, as regular as possible, should be drawn through the points. This can be done by bending an elastic strip of wood so as to pass as nearly as possible through all the points plotted and using this as a ruler to draw the curve. Unless the experiment has been performed with great accuracy to attain which requires care and practice it must not be expected that a curve so drawn will pass through all the points. Some of them, such as b, Fig. 29, are sure to be a little too low, meaning that either the deflection of the standard galvanometer had been read too low, or that of the galvanometer calibrated had been read too high. Other points, such as e, will be too high on the paper, owing to errors of reading in the opposite directions, or it may happen that the observations corresponding to b and e have been correctly made, and mistakes made in the plotting. To avoid errors of the latter kind all plottings should be checked,. 21. Practical Value of Drawing Curves to Record Graphically the Results of Experiments. It may be asked, But is it not possible that the points b and e, although not on the curve, may be quite correct ? The answer is, No, because experience makes us quite sure that the connection between the deflection of the galvanometer G and the current strength must be a con- tinuous one, and, therefore, that the points correctly representing the true connection must all lie on an elastic curve, or on such a curve as can be obtained by bending a thin piece of wood or steel, and, consequently, that if no mistake has been made in plottmg the points b and e, some mistake must have been made in taking the observations. But what is even more important, we are also 48 PRACTICAL ELECTRICITY sure that the points b' and e' on the curve, obtained by drawing lines through b and e respectively parallel to OY, give far more accurately the relative strengths of the currents producing respectively the two deflections in question, than the currents obtained directly from the experiment itself. Drawing the curve, then, corrects the results obtained by the experiment. But it does something more than that it gives, by what is called " inter- polation" the results that would have been obtained from inter- mediate experiments correctly made ; that is to say, it tells us what would be the relative strengths of the currents that would pro- duce deflections intermediate between the deflections that were actually observed. For example, suppose it be required to know the strength of current which will produce a deflection of 41 J, for which deflection no experiment has been made, compared with that which will produce a deflection of, say 28 J, for which deflection also no experiment has been made, then all that is necessary is to draw a line parallel to OY*, through the point A in ox corresponding with 41^, similarly to draw a line parallel to. OY, through the point B in ox, corresponding with 28J, and read off the lengths of the lines between ox and the points p and Q, where they cut the curve, then the strength of the current which produces the deflection 41^ on this particular galvano- meter bears to the strength of the current that produces the de- flection 28J the ratio of the length AP to the length BQ. If the curve is an absolute and not merely a relative calibration curve, then the scale on which it is drawn will be known and therefore the number of amperes corresponding with either AP or BQ. The method of plotting the results of experiment: on squared paper, and drawing a curve through them to graphically record the result, has a third important use in that, just as a map gives a better idea of the shape of a country than pages of description, a curve enables us to see at a glance the general character of the result obtained. For example, suppose that the results obtained in some particu- lar calibration of a galvanometer are : Deflection. Relative Strength of Current. 10 24 I7-3 4i-5 22-8 547 29-5 . . . . . . . . 70-8 37-4 97 * Or imagine such a line drawn. USE OF SQUARED PAPER 49 no exact notion of the law of the galvanometer can be obtained by a glance at these figures ; but if they be plotted on squared paper a straight line passing through zero (Fig. 290) is obtained, and from this we see at once that this particular galvanometer has, in some way or other, been so constructed that the angular deflection of the needle is directly proportional to the strength of the current. In the great majority of cases the angular deflection of the needle of a galvanometer is not proportional to the current strength, and a calibration curve is then needed to show the connection between them. After a little experience the eye becomes accus- tomed to the peculi- arities of curves, and a glance at the calibration curve is then sufficient to convey much infor- mation about the instrument to which it refers. It is always difficult for anyone to grasp the meaning of a table of figures, even if it be as simple as that just given, but the curve which repre- sents them is much more readily under- stood, and its chief characteristics can also be more easily remembered. The curves are rendered more expressive if they are always plotted so that the horizontal distances, or " abscisses," represent the values of the thing easily observed for example, the angular deflections of the needle of a galvanometer, the hours of the day, the days of the week, etc. ; and the vertical distances, or " ordinates," represent the values of the variable quantity which it is desired to record for example, the relative strengths of the current producing these observed deflections, the heights of the barometer, the price in pounds of some commodity, etc. current. g CO CO C 3 C / / . 7 s 60 / / / / r S 20 10 ; / / 2j 10 20 30 40 Galvanometer Deflection,. Fig. 2911. 50 PRACTICAL ELECTRICITY It might at first sight appear to be a matter of indifference which of the two quantities was plotted horizontally ; so also the north on a map might be at the bottom or at either of the two sides. But, just as convention has led to maps being always drawn with the north at the top and with the east t the right hand, so by common agreement the values of the previously unknown quantity are plotted vertically, and the values of the quantity which is assumed to vary regularly in a known manner are plotted horizontally. Hence in graphically recording the tem- perature at different hours of the day, temperature is plotted verti- cally and time horizontally ; or in drawing a curve to indicate the depth of the Atlantic at different points between England and America, depth is plotted verti- cally, and distance measured, from either England or America, along the surface of the sea Fig. 30. Protractor used in subdividing u~_',,~,,.4.~Ti. r , a Galvanometer Scale. horizontally. 22. To Construct a Galvano- meter Scale from which the Relative Strengths of Currents can be at once Ascertained. Galvanoscopes, and even cheap galvanometers, are frequently constructed with scales divided simply into degrees, so that it is generally impossible by the mere inspection of the deflections produced by different currents to determine the exact relative strengths of these currents. If a calibration curve has been drawn from the results of previous tests, the relative strengths of any currents can, of course, be ascertained by using the curve to interpret the meaning of the galvanometer deflections. Constant reference, however, to a calibration curve or to a table of values, leads to much waste of time, and therefore, when a galvanometer is to be used under the same general conditions, it is better to construct a scale with the graduation marks so drawn that the relative strengths of currents are directly proportional to the deflections they produce as measured by the numbers on this specially constructed scale. Such a scale may be made as follows : Ascertain from the calibration curve (Fig. 29) the angles in degrees measured along o X, which correspond with currents whose relative strengths are, say, o-i, 0-2, 0-3, 0-4, etc., and make a table of them as follows : DIRECT READING SCALE Relative current strength, o-o o-i 0-2 0-3 .. 0-4 .. ., etc etc. Inflection in degrees, o-o 0-7 i'3 2-5 4-0 etc etc. Then, by means of a protractor (Fig. 30), set off these angles on a blank scale, making each 5th mark longer than the inter- mediate ones, and numbering each 5th (say) as indicated in Fig. 31, which shows the resulting scale. As the scale is intended merely as a relative one, the relative current strengths may be multiplied by 10, or by any other convenient number, to avoid decimals on the scale. If more than 30 divisions were wanted on the scale the spaces could be further subdivided by eye, or 3i. Dkect Reading Scale. the relative current strengths in Fig. 29 could be multiplied by a number which would make the current strength corresponding to the highest point of the scale to be used, equal to the number expressing the number of divisions desired. If this multiplier was not a convenient whole number it would be advisable to plot a new curve, using as current strengths the original values 52 PRACTICAL ELECTRICITY multiplied by the number in question, and then make a new table for use in marking off the scale. It will be noticed that the divisions on the scale (Fig. 31) are crowded together at the beginning and end, and spaced farther apart in the intermediate portions, and on referring to the curve (Fig. 29) we see that the crowded portions of the scale correspond with the steep parts of the curve, and the longer divisions with the part having the least slope. The shape of the calibration curve therefore enables us to see what would be the character of a direct reading scale constructed from it. Example 15. From the calibration curve shown in Fig. 29 find the relative values of the currents required to produce deflections of 5, 30 and 70 respectively, taking the current which gives a deflection of 5 as unity. "Answer. i : 2-39 : 4-5. Example 16. It is required to use currents 2, 3 and 4 times as large as that which produces a deflection of 8 on the instrument whose calibration curve is given in Fig. 29. What deflections of the galvanometer correspond with these currents ? Answer. 35, 63, 75-5. CHAPTER II MAGNETIC FIELDS 23. Magnetic Fields: Magnetometer 24. Lines of Magnetic Force 25. Strength of Magnetic Poles 250. Hibbert's Magnetic Balance 256. Balance for Finding Strength of Pole 26. Magnetic Moment 27. Absolute Measurement of Magnetic Field and of Magnetic Moment 28. Mapping Magnetic Fields 29. Comparing the Relative Strengths of different parts of a Magnetic Field by the Vibration Method 30. Comparing the Relative Strengths of different parts of a Magnetic Field by the Magnetometer Method, 300, Difference of Magnetic Potential ; Equipotential Surface. Addendum to Chapter II. : Electric Lines of Force and Electrostatics, 23. Magnetic Fields. As the action of most galvanometers depends on the magnetic fields produced by the electric current passing through their coils, it is desirable to study them in some detail. Two of the characteristics of magnetic fields at any selected point are direction and strength. We have already mentioned (see Section 15) that the direction of a field is determined by that direction in which a small compass needle sets itself when placed at the point in question, and also that the strength of a field at a point is measured by the number of dynes exerted on a unit magnetic pole placed at the point. Although it is possible to measure the strength of a magnetic field in the way just stated, this is by no means the most convenient way. Nearly all measurements are most easily made by comparing the thing to be measured with another thing of the same kind, this latter being taken as the unit or standard.. For example, in measuring the length of a room we compare its length with that of a foot rule or something equivalent to this. The foot rule is thus taken as the unit of length, and the ratio of the length of the room to that of the foot rule, is the number expressing the length of the room in feet. To be of real use, the foot rule in the above example must be of fixed length, it should not be made of material easily stretched or compressed, or such as is greatly affected by atmospheric temperature, pressure, or humidity. In other words, its length 53 54 PRACTICAL ELECTRICITY must be constant. Similarly, in the case of magnetic fields, if we have, or can produce, a field whose strength is constant, we may take this field as our standard field and measure other fields by comparing them with this standard field. As previously stated in Section 15, the earth's magnetic field, at a given point in an undisturbed area, is very nearly con- stant, and for many measurements its horizontal component is taken as a standard field, the strengths of other magnetic fields being measured by comparison with it. Probably the simplest way of making this comparison is to arrange that the direction of the field to be measured is horizontal, and at right angles to the earth's horizontal component, as was done in the case of the tangent galvanometer (Sect. 15). The field to be measured is thus superimposed on the earth's field at right angles to the latter, and will deflect a small magnetic needle, placed at the point at which the field is to be measured, from the magnetic meridian through an angle depending on the strength of the super- imposed field. If, therefore, we measure the deflection of the small magnet which is caused by the superimposed field, we can find its strength by the triangle of forces, for, as magnetic fields are measured by the mechanical forces they exert on a unit magnetic pole, all propositions relating to the composition of forces are applicable to magnetic fields. Let the line o H (Fig. 32) represent in direction and magnitude the strength of the earth's hori- zontal component (H), and o F, that of the field (F) to be measured, the angle H o F being a right angle, then the diagonal o R will repre- sent the resultant of o H and o F, and from the above figure we 32. have o F = H R, 1 . = o H,tan a, % or F = H tan a. (2) In words we may say that when the superimposed field is horizontal and at right angles to the earth's horizontal component, the strength of the superimposed field is equal to the strength of the earth's horizontal component multiplied by the tangent of the deflection produced by the superimposed field. It is not essential that the directions of the two fields to be compared be at right angles to each other, but this relative direction gives a simple formula, (2) above, and nearly maximum, MEASURING MAGNETIC FIELDS 55 deflection. If the directions be inclined at any angle obtuse (Figs. 33 and 330), we have acute or Sin HOR sin HRO sin HOR (in both figs.) sin ROF sm a or F = sin 08 - a)' H sin a sn where HOR = a, - )' (3) The method of measuring magnetic fields just described above, is called the magnetometer method, and is of great convenience and utility. In fact, in all galvanometers which have moving needles controlled by a constant magnetic field and deflected by the current to be measured, this method is made use of. Any freely suspended magnetic needle arranged so that its deflection can be observed may be used as a magnetometer. A simple application of it is illustrated in Fig. 34, in which the strength of field atji point along the axis of a magnet is being measured. Fig. 35 shows an actual magnetometer in which the magnet m is provided with a long pointer moving over a large graduated circular scale, and a straight scale 140 centimetres long, on which the distances of the ends of magnets from the needle can be read off. By observing the deflections produced by a given magnet M placed at several distances from the needle, and plotting a curve between distances and tangents of the corresponding deflection, the manner in which the strength of 56 PRACTICAL ELECTRICITY field of the magnet varies with the distance along its axis can be readily shown. A reflecting magnetometer, by means of which more accurate observations can be made, is shown in Fig. 350. The small N Fig. 34. Principle of the Magnetometer. magnetic needle, to which a mirror is attached, is suspended in the support A and its deflections measured on the scale c. A stand B supports M so that its axis produced passes through the centre of the suspended needle. 24. Lines of Magnetic Force. If a bar magnet be placed below a sheet of glass and fine iron filings are sifted over the plate, and the plate tapped, the filings will arrange themselves in a pattern such as shown in Fig. 36. On inspection it will be seen that the Fig. 35- Magnetometer with Pointer. filings set themselves in lines, and the direction of one of these lines passing through a given point is approximately that in which a very small compass needle would come to rest if placed with its centre at the point considered. The lines, therefore, show the direction of the magnetic force at various points in the field of the magnet, and are consequently called " lines of magnetic force," or more shortly, " lines of force." They are the lines along which the magnetic force acts. LINES OF MAGNETIC FORGE 57 Since a compass needle, or a magnetic pole, placed at any point in the vicinity of the magnet producing the field, shown in Fig. 36, would experience a force, a line of force may be said to pass through this point, and as the number of such points is Fig- 35. Reflecting Magnetometer. infinite we might say that in a given magnetic field there exists an infinite number of lines of force. It is therefore impossible to draw a diagram representing all the lines of force of a magnet, and even if such a diagram could be drawn it would tell us merely the direction of the field at any and every point, but nothing about the strength of the field at various points. A convenient conven- tion, however, exists, according to which only a few of the lines of Fig. 36. Lines of Force with a Bar Magnet. force are drawn, and tfre proximity of the lines in the vicinity of any point indicates the strength of field at that point. For example, the lines may be drawn so that the number passing through an area of one square centimetre, normal to the lines of force at any point, is numerically equal to the force which unit magnetic pole would experience if placed at that point ; in this case the density of lines (number per square centimetre) in the vicinity of a given point would express tie strength of the field at that 58 PRACTICAL ELECTRICITY point. Diagrams drawn according to this convention have a quantitative, as well as a qualitative meaning, and are much more instructive than charts showing direction only. Fig. 37 is a diagram of the magnetic field produced by a current passing through a circle of round wire, c c c, made on this plan, whilst Fig. 38 shows a diagram such as would be obtained by the iron filings method. The former gives more information of value than the latter. From Fig. 37 it will be seen that the lines are very close together in proximity to the wire, especially near the Fig* 37 Lines of Force due to a Current in a Circular Coil (to scale). inner circumference, whilst near the centre of the circle they are farther apart and approximately parallel to the axis of the coil. This shows that the field is strongest near the wire, and very nearly uniform in the immediate vicinity of the centre. It will also be noticed that all the lines of force which are completed in the diagram form closed curves. This is an important property of ah* lines of magnetic force. In a uniform magnetic field the force exerted on a given mag- netic pole is the same at every point of the field ; . and when such a field is represented by lines drawn in the way described, the lines are parallel to each other, and at equal distances apart. If we consider the magnetic field produced by an isolated LINES OF FORGE 59 pole of unit strength,* we know that the strength of field at a distance of i centimetre from the pole is unity (for at this distance the force exerted on another unit pole is I dyne), and is the same at every point on a spherical surface of I centimetre radius concentric with the pole. Such a field would be represented by drawing radial lines from the pole, equally distributed all round the pole, the number being such that the density of lines over the surface of the sphere is one line per square centimetre of area. As the surface of a sphere of unit radius is 4 TT, we can say that the number of lines of force emanating from unit magnetic pole is 4 TT, or 12-56 approximately. 25. Strength of Magnetic Poles. We have already defined a magnetic pole of unit strength (see Section 8) as one * Although an isolated pole cannot be obtained, the pole of a very long, thin, straight magnet is an approximation thereto. 6o PRACTICAL ELECTRICITY that acts on an equal pole at a distance of one centimetre from it with a force of one dyne. If one of the unit poles be re- placed by another magnetic pole which exerted a force of two dynes on the unit pole, such pole would be said to be of strength 2. Similarly, if the other unit pole were now replaced by a pole of strength 2 the force would again be doubled. We therefore see that the force exerted between two poles at unit distance apart is equal to the product of the strengths of the two poles. This may be expressed thus : the force between two magnetic poles of strengths m and m' respectively, when placed at a distance of one centimetre apart m m' dynes. If the distance between two given poles be altered, experiment shows that the force between them varies inversely as the square of the distance (the law of inverse squares), so we may write the complete law of force between magnetic poles as : force between two magnetic poles of strength m and m' respectively, when placed at a distance d centimetres apart mm' = jr d y nes - (4)- This law is exactly the same as the Newtonian Law of Gravitation. The same law also holds for the force between two quantities or charges of electricity (see page 81), and for this reason a magnetic pole of strength m is sometimes said to possess, or have a charge of, m units of magnetism. By assuming m' in the last equation to be unity, we see that the force exerted on unit pole by a pole of strength m placed at d centimetres away is m -p dynes, and as the force on unit pole measures the strength of the magnetic field it follows that an isolated pole of strength m pro- m/ duces a field of strength at a point d centimetres from the pole. W T hen two poles acting on each other are of the same kind, say both north-seeking, or both south-seeking, the force is found to be one of repulsion, whilst if they are dissimilar poles they attract each other. The two kinds of magnetism, therefore, have opposite properties and may be spoken of as -f or magnetism respectively, north-seeking magnetism being con- sidered -f. A north-seeking pole of strength m acts on a north- seeking pole of m' strength with a force LAWS OF MAGNETIC FORGE -\-m m f m m r ~W ; and that between two south-seeking poles is 61 whilst a north-seeking pole m and a south-seeking pole m' exert a force of m (m)' . m m' ~^~ ~^r- A positive sign for the force, therefore, is associated with a repulsion, and a negative sign indicates an attraction. 39- Hibbert's Magnetic Balance. 250. Hibbert's Magnetic Balance. The law of inverse squares may be proved in a simple manner by means of the Hibbert's Magnetic Balance, shown in Fig. 39. In this instrument a long, thin, magnetised steel rod s n is suspended so that it can swing in a vertical plane, and balanced so as to rest horizontally when all other magnets are far removed from it. Another magnet n f s' is fixed horizontally to a slider s, capable of being moved up and down, its position being indicated by the vertical scale shown. When ri is placed aSove n a force of repulsion is exerted between them, and the end n is depressed. By placing a small weight on the lower magnet between o and s, and adjusting it, the hori- zontal position of s n can be restored. When this has been done approximately, the needle n' s' is moved in the direction of its own axis into the position which produces maximum force between n and n f . The weight is again adjusted until s n is level, and the horizontal distance h of the weight from o 62 PRACTICAL ELECTRICITY (which can be read off on the horizontal scale) is proportional to the force exerted between n and n'. By bringing n' s' lower down the distance between n and n' is decreased, the force between them is increased, and to obtain balance the weight has to be moved farther from the centre, say to A r Calling d and d l the distances between n and n r in the two cases described above, the following relation will be found to hold, viz. : hd 2 = h^j 2 (approximately). h d* 2 " -* The results of an actual experiment carried out in the way described above, are given in the following table : Values of d. Values of h. Values of hd * 9 cms. 8-5 cms. 688 10 6-86 686 II 57 690 12 4'8 692 13 M 4'i 693 14 M 3-55 694 From this table it will be seen that the law stated is very nearly true. An exact agreement between the products h d 2 is not to be expected, since the force exerted between s and s' have not been taken into account, and the distances h and d cannot be measured very accurately. 256. Balance for Finding Strength of Pole. The strength of magnetic poles of long, thin magnets can be determined by a balance such as is shown in Fig. 40. It is convenient to use three magnets of similar shape for this purpose. Call them I., II., and III. respectively, and the strengths of these poles m lt m 2 , and m B . Place I. on the pan hanging from the beam, and counterpoise it when the other magnets are far removed ; then put II. vertically above or below I. with similar poles adjacent to each other. Balance the instrument by moving the rider R and call the equivalent weight placed at the end of the beam c. Replace II. by III. and let the weight be denoted by b. Next replace I. by II. and call the weight a. Thus c = 2 m m 2 d 2 approximately,* b = 2 m m 3 d 2 , a = 2 m 2 m 3 d 2 , * The equation is only approximately correct, because all forces except those between adjacent poles have been neglected, STRENGTH OF MAGNETIC POLES from which we get 63 or Similarly m 2 c b = 4 m^ m 2 m 3 -f- d*, d fab NIC From the above it will be seen that if the strength of pole of one magnet (say mj is already known, then a single observation will enable m 2 to be found. Fig. 40. Balance for Finding Strength of Magnets. 26. Magnetic Moment. The two poles of any magnet are found to possess equal quantities of magnetism, but of opposite sign ; and if the magnet be short or be shaped so as to bring the two poles very near together, the magnetic field produced at a distant point by one pole is very nearly equal and opposite to that produced by the other pole, and so the resultant field at the dis- tant point is very small, in spite of the fact that the strength of each pole may be considerable. 6 4 PRACTICAL ELECTRICITY On the other hand, a magnet, whose poles are of the same strength as those of the magnet considered above, but are situated far apart, say near the ends of a fairly long, straight bar, would produce at a distant point a magnetic field of strength much greater than that given by the short magnet. The effect produced by a magnet depends, therefore, not only on the strength of its poles but also on the distance of those poles apart : in fact the effect is proportional to the strength of the poles, and also proportional to the distance between the poles. The product of the strength of the poles of a magnet and the distance between the poles is consequently an important magnitude, and has been called the moment of the magnet, or more shortly, its magnetic moment. Magnetic moment may be defined by the equation Mm I, where m is the strength of each pole and / the distance between the poles. A fairly direct way of measuring the moment of a magnet in mechanical units, is to suspend it by a torsion wire in a uniform magnetic field of known strength, such as that produced by the earth in an undisturbed area, and observe the torque* required to maintain the magnet in a position perpendicular to the meridian (Fig. 41). Calling the torque T and the strength of field H, we have . T = M H, T and M - . H If the known field be of unit strength, the value of H above is i, and the equation reduces to M = T, from which we see that the moment of a magnet is measured by the torque it Fig. 41. Torsion Apparatus for Measuring Magnetic Moment. * The "constant" of the torsion wire, i.e. the torque required to produce a twist of unit angle in the wire (i radian), can be found by well-known mechanical methods. One of these is the vibration method, in which a non-magnetic bar or disc of known moment of inertia K, is suspended from the wire, and the time, T seconds, of torsional vibration determined. Calling c the "constant," the relation T -- enables c to be calculated, for c = K. MEASURING MAGNETIC MOMENT 65 exerts when placed perpendicular to the Jines of force in a field of unit strength. Another method of measuring magnetic moment is by magneto- meter, as indicated in Figs. 34, 42, and 43. D Fig. 42. Measuring Magnetic Moment by Magnetometer. Let a small magnet n s (Figs. 42 and 43) be placed at p, distance D from the middle of the magnet N s, whose moment is to be determined, and suppose the strength of the field at p, when N s is taken away, is H, and its direction x Y perpendicular to P Q. Also let / be the distance between the poles of N s. The pole near N produces a field at P in the direction P Q of strength 2 , (see section 25), and the pole near s produces a field at p in the direction of p Q' (opposite to p Q) of strength = The deflecting field acting on n s is the difference of these, viz. : F = m m 2 D ml 2D M since M = m I. Fig. 43. Measuring Mag- netic Moment by Mag- netometer. 66 PRACTICAL ELECTRICITY Now a deflecting field F acting on a needle controlled by a field H at right angles to F will produce a deflection a, where F = H tan a, (see section 23). 2 D M therefore or M = = /i tan a, tan a, (/ \ 2 ^ j is small compared with D 2 the expression reduces to M HD* tan a (approx.). (5) If the magnet N s be placed with its centre on the line x Y (Fig. 42), and perpendicular to this line, as shown in Fig. 43, and the deflection produced be a', then M = H \D* + (-} \ z tan. a',* V2/ / which reduces to M = H D 3 tan a', (approx.). when ( - ) is small compared with D 2 . (6) 27. Absolute Measurement of Strength of Magnetic Field and of Magnetic Moment. As will be seen from Sections 15 and 26, the measurement of currents and of magnetic moments there described depends on the knowledge of the strength of the magnetic field con- trolling the needle. It is, therefore, desirable to know how such strengths may be measured. The usual way of doing this is, in outline, as follows : Suspend a bar magnet of moment M (as yet unknown) by a torsionless fibre and determine its periodic time of vibration, T, when placed in the magnetic field H, to be measured. w Fig. 45. * The student should work this out as an exercise on the principles involved. MEASURING MAGNETIC MOMENT 67 When the magnet is at an angle a to the magnetic meridian XY (Fig. 44), the moment of the forces tending to bring it back to the meridian is m H I sin a, / being the distance between the poles of the magnet. Similarly, in the case of a simple pendulum (Fig. 45), the restoring moment, when the angular displacement from the vertical is a, is W l^ sin a. The law of control is there- fore of the same form in the two cases, and the time of vibration of the suspended magnet can be deduced by analogy from that of the simple pendulum. In the latter case, as is well known (for small oscillations), T l = 27c A/ , where g is the acceleration of gravity. s This may be written (by multiplying numerator and denomina- tor under the root by where u represents the mass of the pendulum. It will be noticed that u l-f is the moment of inertia K of the pendulum, and ug its weight W, hence The period of small oscillations of a simple pendulum is therefore equal to 2 it multiplied by the square root of the ratio of its moment of inertia to the controlling moment which would exist if the dis- placement from the vertical were go 9 . Applying this rule to the suspended magnet we get ~lK K being the moment of inertia of the magnet, which can be calculated from its mass, size and shape. This experiment enables the product M H to be determined, for from the last equation ':.: ;. (7) By using the same bar magnet to deflect a needle n s, as in M Fig. 43 (say), we can find the ratio , for M = H Z) 3 tan a' (approx.) [See (6) section 26] ; i.e. ~=D* tan a' (approx.). ti 68 PRACTICAL ELECTRICITY Dividing M H by we get , (approx.). (8) from which equation H may be calculated. The same experiment also enables the magnetic moment M to be found, for MHx = H Z) 3 tan a'. (9) This experiment determines both the moment of a magnet and the strength of a magnetic field in absolute measure, and is of great importance in magnetic measurements. (For details of the. conditions and corrections necessary for obtaining very accurate values, a work dealing with terrestrial magnetism may be consulted, such as Stewart and Gee's Practical Physics.) 28. Mapping Magnetic Fields. In Figs. 36 and 38 we have seen that the lines of force of a coil carrying a current, or of a magnet, can be shown by iron filings. There are various easy ways of fixing these curves marked out by the iron filings, and so enabling a record to be kept of the " lines of force," from which we can at once see the position in which a little compass needle will place itself when put anywhere in the magnetic fieM. One of the simplest is to use waxed paper instead of the glass of section 24 ; then, after the filings have been lightly scattered and the paper gently tapped in order to assist the filings in taking up their proper positions, to warm the paper with the flame of a Bunsen gas-burner moved quickly over it. The wax is thus melted, and the filings stick to it when it becomes cool and hard again. Figs. 36, 46, 47, and 48 show the lines of force obtained with a straight magnet, with two straight magnets placed end on with poles of the same name near one another, and with two horse-shoe magnets. The horse-shoe magnets have fitted to them curved pole pieces of soft iron, and with the second one (Fig. 48) there is in addition a cylinder of soft iron placed between the pole pieces to render the lines of force more or less radial, a result of great value in certain cases (see Moving Coil Ammeters, Sec. 43). The direction of the lines of force may also be traced out by using a small compass needle, n s (Fig. 49) ; for at any particular spot where this little compass may be put the needle places itself MAPPING MAGNETIC FIELDS 69 so that its axis is a tangent to the line of force at that spot. A sheet of paper having been placed on the horizontal table, and fixed by means of the spring clips, the little compass is placed at some particular spot, and as soon as the needle has come to rest a point is marked with a pencil close to each end of the needle, n, The compass is then re- moved and these two points joined with a straight line ; next, the compass is placed a little farther on, so that the n end of the needle is close to the point formerly occupied by its s end. A second short line is now drawn joining points 2 and 3, and thus by draw- ing a number of such adjacent short lines we have a line of force marked out by a large number of its chords. The compass method of tracing out lines of force is, of course, a much more lengthy one than that of using iron filings, but it gives far more ac- curate results, since the friction resisting the com- pass needle taking up the AUn nn Fig. 46. Lines of Force with Two Bar Magnets ; Like AISO, Un- Poles near one another. right position is very ^ small compared with that ^ between the filings and ^ \ ^S||8 ,, , . ,, *&z&-K. the paper on which they are scattered, less the filings be scat- tered extremely sparsely, the magnetism induced in them sensibly disturbs the magnetic field, so that they indicate, not the magnetic field due to the coil alone, but the magnetic field due to the coil as disturbed by the presence of a large number of little magnets. PRACTICAL ELECTRICITY 47. Horse-Shoe Magnet with Curved Iron Pole Pieces, Fig. 48 Horse-Shoe Magnet with Carved Iron Pole Pieces ; the Magnet has also an Iron Cylinder between the poles. MAPPING MAGNETIC FIELDS 7* Further, it is important to remember, when mapping out a field due to a magnet, or to a coil carrying a current, and especially when the delicate compass method is employed, that the result can only be correct when no other magnet is near enough to pro- duce a disturbance. Close to the magnet, or coil, under test the disturbance will be small, unless the disturbing cause be very near or very powerful ; but at some distance from the magnet, or coil, under test the force which is being examined is itself so small that its direction and magnitude may be seriously altered, unless care be taken to eliminate all disturbing magnetic actions such as Fig. 49. Mapping Out the Lines of Force with a Compass Needle. that of the earth, etc. To test whether this condition is fulfilled remove the magnet whose field is to be examined away to some distance, or stop the current passing through the coil, if it be the magnetic field due to a coil that is being investigated, and examine whether the compass needle, when placed anywhere in the area under examination, shows no tendency to .place itself in one position more than another that is, shows that it is not acted on by any directive force. In order to arrive at this state of things it is clear that the earth's magnetic force, which is present everywhere, and the magnetic action set up by any iron pipes, rails, etc., in the neighbour- hood, must be neutralised by magnets or currents judiciously disposed. In obtaining the lines of force seen in Fig. 49, no precaution was taken to neutralise the disturbing action of the earth's field, the 72 PRACTICAL ELECTRICITY direction of which is shown by the arrow. Hence the lines of force in the further parts of the figure are twisted somewhat in a northerly direction, while in the nearer portion they are bent southwards, the effect of which is clearly seen at the left-hand lower corner. % If the main disturbance be that due to a uniform magnetic field such as is produced by the earth, a very convenient method of neutralising it over an area of two or three square feet consists in placing a uniform sheet of copper just on or under the area in Fig. 50. Arrangement for Neutralising a Uniform Magnetic Field. question and sending a current through it in such a direction and of such a strength as to set up a uniform magnetic field exactly equal and opposite to the disturbing one. To avoid the use of a strong current, which would be necessary if we desired to employ a large current sheet, a set of strips Sj, s 2 , etc., of copper (Fig. 50) may be joined up in series, the whole current passing through them all in succession.* 29. Comparing the Relative Strengths of Different Parts of a Magnetic Field by the Vibration Method. Not merely does the position of rest of a pivoted compass needle show the direction of the tangent to the line of force at the particular point, but the square of the number of vibrations made by the needle in a given time, when Set swinging, gives a measure of the strength of the magnetic field at that point. This follows from the formula (7) , Sect. 27, viz. : M H = ~j^~ ; which may be written * When the disturbance is due to the earth's field alone, the current must flow from west to east beneath the paper, and if the sheet is laid on the table beneath the paper, or at any rate is not more than an inch or two below it, the current strength must be about 073 ampere per inch width of sheet. STRENGTH OF MAGNETIC FIELD 73 _ M T* or ti = where n is the number of vibrations in unit time. For a given needle, which is not put into so powerful a field that its strength is altered, the quantities K and M are fixed ; consequently such a needle may be used to measure the relative strengths of different parts of a field. If the magnetic field which is to be explored be a some- what strong one, it will be difficult to time accurately the rapid vibrations of an ordinary compass needle. It is better, therefore, to increase its moment of inertia by adding mass to its two ends, which can be conveniently done by selecting two leaden shot of about equal size, making a cut in each, and slipping one over the point of the needle at each end. The needle is then balanced by moving one or other of the shot nearer to, or farther from the centre of the needle, and the shot can be secured in position by slightly squeezing them with a pair of pliers. A compass needle with weighted ends the whole, however, much enlarged is seen in Fig. 51. When such a weighted needle is used to explore the field produced by a current flowing round a large circular coil, like that seen in Figs. 38 and 49, it is found that at all points distant from the centre of the coil by not more than about one- tenth of its radius the number of vibrations per minute made by the needle is practically the same, And, since the map- ping of the lines of force shows that within this little region round the centre of the coil the lines of force are straight and all perpendicular to the plane of the coil, we see that within this region the magnetic field due to the current flowing round the coil is a nearly uniform one. Consequently if a needle not longer than about one- tenth of the diameter of the coil be suspended at the centre of the coil, and if the controlling force be that due to the earth or to a distant magnet, the needle will be acted on by two nearly uniform Fig. 51. Weighted Compass Needle for Measuring the Strength of a Magnetic Field (about two-and-a-half times full size). 74 PRACTICAL ELECTRICITY magnetic fields, and, from what has been already said, it will place itself along the resultant of these two fields. 30. Comparing the Relative Strengths of Different Parts of a Magnetic Field by the Magnetometer Method. The magneto- s meter may in several cases be conveniently used for finding the relative strengths, or the absolute strengths, of different parts of a magnetic field, more especially when such field is due to a coil or magnet which can be readily moved. In Fig. 52 is shown an application of this method for finding how the strength of field varies in the plane of a coil carrying a current ; and also for STRENGTH OF MAGNETIC FIELD 75 determining how the strength of field varies at different points along the axis of such a coil ; a current of constant strength flowing through the coil in each case. The magnetometer, con- sisting of a short needle, to which a long pointer is attached, is suspended from a rod r, by means of a silk fibre, and is contained in a sector shaped box, g g, having a scale at the bottom, on which the deflections of the needle may be read. The box g g is raised above the base, so that the level of the needle is at the same height as the centre of the coil c c. A board, B B, carrying the coil is provided with pins, which enable the coil to slide either along a groove e in the base parallel to the axis of the coil, or along another groove e' at right angles to the former, which ensures that the centre of the needle remains in the plane of the coil. A galvanoscope, G, is connected through a key, K, with the coil c c, and also with a wire w w of variable length ; by these means the current passing from a battery through the coil c c may be maintained constant. Before making an experiment, the coil c c is placed so that the magnetic needle is at the centre of the coil, and the whole base turned to bring the plane of the coil into the magnetic meridian. When in this position the pointer on the needle should read zero if no current be flowing. On passing a current through the circuit the needle will be deflected, and the magnitude of this deflection can be adjusted to a convenient value by altering the length of wire, w w, included in the circuit. Maintaining the current 10 y 06 O 4 6 12 16 20 24 28 32 Distances of centre of needle from plane of coil in centimetres Fig. 53. Variation of Strength of Field along Axis of Coil. constant at this value by aid of G, the deflection d produced by the current when the needle is at the centre of the coil should be observed. The strength of field at the centre will then be #tan d, where H is the strength of the earth's field at 7 6 PRACTICAL ELECTRICITY the point occupied by the needle ; (Formula (2) Sect. 23). On moving c c a few centimetres to the right in a direction perpendicular to its own plane, the deflection will be found to diminish, and by making several observations with the coil at different distances from the needle, the relation between the strength of field at points along the axis of a coil carrying a current, and the distance of the points from the plane of the coil can be readily determined. The figures given in the follow- ing table show the results obtained in an actual experiment. They are plotted in Fig. 53. Axial distance of centre of needle from plane of coil. Deflection of Needle. Tangent of Deflection. Htan d. O 40-4 0-851 0-154 4 393 0-818 0-147 8 34*0 0-674 0-122 12 29-0 o-554 0-0996 16 21-8 0-400 0-072O 20 16-7 0-300 0-O54O 24 12-9 0-229 0-04I2 28 9-3 0-164 O-O295 32 7'5 0-132 0-0238 In the above experiment the coil c c was a circular one of 20 centimetres radius, the current employed 4-9 amperes (approxi- mately), and the value of H, the earth's horizontal component, 0-18 dynes per unit pole. It is possible to calculate from the definition of current strength and the law of inverse squares, the relation between strength of field and distance along the axis of a circular coil, which was found experimentally above. . Let OB (Fig. 54) represent the axis of a single turn coil c c', whose plane is supposed to be perpendicular to the paper, and P a point on the axis at a distance x from o. Consider a short length, /, of the coil at c perpendicular to the paper, and let a current of strength / amperes be flowing. The force exerted on a unit pole at P due to the current / in this short conductor of length / would, by definition, be/= cp2 , and its direction PQ at right angles to CP. Similarly, a length / at c', the opposite end of a diameter, would act on unit pole at P with equal force (since FIELD ALONG AXIS OF CIRCULAR COIL 77 c'p = CP), and the resultant of these two forces, / and /', would be given by p R both in magnitude and direction. But PR = 2 / cos a, _2_ Il_ r_ ~ IO CP 2 CP* _ 7,1 Ir ~ 10 CP 3 ' so that the force exerted on unit pole by the current in two short lengths / of conductor at opposite ends of a diameter of the circle is directed along the axis of the circle, and of magnitude multiplied by the sum of the lengths of the short conductors. This is true of any pair of short conductors at opposite ends of a Q, Fig. 54. Geometrical Construction for Finding the Strength of Field at a point on the Axis of a Circular Coil. diameter, and as the whole circle can be supposed to be divided up into such pairs, the force exerted by the current in the whole circle will be got by writing 2 TT r, the length of conductor form- ing the circle, instead of 2 / in the expression above. Hence the total force F is given by P = 2 TT r ~ 10 CP 3 2 TT 10 do) 7 8 PRACTICAL ELECTRICITY This may also be written : 10 r where p is the angle CPO (Fig. 54), the angle subtended at p by the radius co. Working out the values of F for the distances % used in the experiments recorded in Fig. 53, we get the values of F to be 0-154, 0-148, 0-123, '097> 0-073, 0-055, 0-040, 0-030, 0-023 respectively, which agree practically within the errors of observation with the V* 4 J25 ioo 1-5 5-0 25 O 25 5-o 7-5 10-0 " Distances from centre of coil (centimetres) Fig. 55. Variation of Strength of Field in Plane of CoiL 12-5 corresponding values of H tan d given in the table above. At the centre of the coil we have x = o, and the formula reduces to _._ 2 TC / 10 r (n) The same apparatus can, by moving the coil along the groove e', be used to find how the magnetic force varies along a diameter of the coil. Fig. 55 shows the results obtained in this way. From the curve it will be seen that the force is practically uni- form near the centre of the coil. It is possible to determine from first principles the relation between the force and the dis- tance from centre of the coil, but the calculation is not sufficiently simple to be given here. An approximation which may be used near the centre is : where b is the distance of the point considered from the centre, DIFFERENCE OF MAGNETIC POTENTIAL 79 300. Difference of Magnetic Potential : Equipotential Surface. As a magnetic pole is acted on by a mechanical force when situated in a magnetic field, mechanical work will, in general, be done when the pole is moved from one point of the field to another, and the value in ergs of the work done in moving a unit pole from one point to another point is called the difference of magnetic potential between the two points. If no work be done in moving Fig. 5 5 a. Lines of Force and Equipotential Surfaces (dotted) due to Circular Current the pole from one point to the other the two points are said to be at the same magnetic potential, or to lie on an equipotential surface. It will be evident that if the pole be moved at right angles to the lines of force no work will be done, so any surface which is everywhere perpendicular to the lines of force of a mag- netic field will be an equipotential surface. Such surfaces can be drawn to represent a magnetic field quantitatively in a manner analogous to that employed with lines of force, Section 24. For 8o PRACTICAL ELECTRICITY example, if the surfaces be supposed drawn through points on a line of force whose distances apart are such that one erg of work would be done in moving unit pole from one point to the next, then the series of surfaces would indicate quantitatively the nature of the magnetic field, for the direction of the field at any point would be normal to the equipotential surface passing through that point, and the proximity of the surfaces drawn as above des- cribed would show the strength of the field, this strength being expressed by the reciprocal of the distance apart of adjacent equipotential surface in the neighbourhood of the point con- sidered. Fig. 55 shows the lines of force and also the sections of equipotential surfaces, in dotted lines, due to an electric current flowing in a circular coil. It will be noticed that where the lines of force are nearest together the equipotential surfaces are nearest to each other, so both systems of lines indicate the character of the magnetic field. When the first of the points considered is supposed to be at an infinite distance from the magnet or coil producing the field, the work done in bringing unit pole from the first point to the second is called the potential of the second point. Every point in a magnetic field may therefore be said to have a potential, and the difference of magnitude of this quantity for any two points considered is the difference of potential between the two points, for no matter what path be taken in moving unit pole from one point to the other, exactly the same amount of work must be done against the magnetic forces. An exact analogy exists in the work done against gravity in moving a given mass from a point at one level to a point at another level ; this is quite independent of the path traversed. If the given mass be unit mass, the work done will be a measure of the difference of gravitational potential between the two points. No work is done against gravity by moving a mass from one point to another on the same contour line, (on a map,) because all such points are at the same gravitational potential, for contour lines represent the intersections of level surfaces with the earth's surface. These contours usually differ in height above Ordnance Datum by definite amounts, say I ft., 10 ft., or 100 ft. according to the nature and extent of country shown on the map ; they are nearest together where the slope of the land is steepest, and far apart in parts nearly level. Their closeness therefore, indicates the gradient of the land, or the gravitational potential gradient, as it may be called, in the same way as the closeness of the equipotential surfaces in the map of magnetic field show the magnetic potential gradient. ELECTROSTATICS 81 ADDENDUM TO CHAPTER II. Electric Lines of Force and Electrostatics. When two dissimilar sub- stances, such as silk and glass, or ebonite and cat's fur, are rubbed together and separated, they possess the property of attracting light bodies, such as pith balls, pieces of paper, etc., and of attracting each other ; the new condition of the glass and silk is described by saying they are electrified, or have electric charges on them. A conducting body supported by an insulator, say silk, ebonite, sealing wax, etc., if touched against the rubbed glass and taken away, exhibits similar properties to those shown by the glass, and the body is said to have been electrified by contact with the glass, and to possess an electric charge. Similarly, an insulated conductor touched against the silk would become electri- fied, and if placed near the body which had touched the glass the two would attract each other. But if both the conductors were electrified by contact with the glass, they would repel each other ; they would also repel if both were electrified by touching the rubbed silk. These phenomena are usually regarded as showing that two kinds of electricity exist (called respectively vitreous and resinous, or positive and negative) and that bodies charged with the same kind repel each other, whereas bodies charged with opposite kinds attract each other. It is also found that exactly equal amounts of the two opposite kinds are produced whenever electrification occurs, just as equal amounts of magnetism of opposite kinds always exist in a magnet. The phenomena exhibited by electrically charged bodies are very similar to those possessed by the poles of magnets, and conceptions of lines of electric force, electric fields, electric moments, etc., analogous to the corresponding magnetic quantities have been developed. The system of measurement of these electrostatic quantities is exactly like that used for magnetism. For example, unit charge, or unit quantity* (in electrostatic measure) is defined to be such that the force of repulsion between two unit charges at unit distance is unity, viz. : i dyne, in the C.G.S. system ; similarly, the force exerted between two quantities q and q' at distance d is equal to $-4 , an expression identical in form with the one for magnetic forces in formula (4) Section 25. The strength of an electric field is measured by the force in dynes exerted on unit charge placed in the field, just as the strength of a magnetic field is measured by the force in dynes on unit pole. The direction of the field is taken to be that of the force exerted on a vitreous (or positive) charge. Diagrams representing the directions of the electric forces in the vicinity of electrified bodies can be drawn just as magnetic fields are represented, and equipotential services everywhere at right angles to the lines of force, delineated. If a single charged sphere be placed in a large enclosure the lines of electric force in the vicinity of the sphere will be radial (this follows from considerations of symmetry), and the equipotential surfaces concen- tric spheres. Further, the force exerted on a small charge at any external * This unit of quantity is, of course, quite different from the coulomb defined, in Section 10. In fact, i coulomb is found, by experiment, to be approximately equal to 3,000,000,000 i.e. (3 x io 9 ) electrostatic units, and i electromagnetic unit of quantity (io coulombs) is, therefore, equal to 3 x io 10 electrostatic units (Section 171). It is interesting to notice that 3 x io 10 expresses the velocity of light in centimetres per second, and electromagnetic theory indicates that the ratio of the two units of quantity should be numerically equal to the velocity of light, A proof of this is beyond the scope of an elementary work. 82 PRACTICAL ELECTRICITY point by the charged sphere can be shown to be the same as if the whole charge was concentrated at the centre of the sphere, for in this latter case, assuming the spherical surface removed, the directions of the lines of force must also be radial, and the equipotential surfaces concentric spheres. Consequently the force on a unit charge at a distance d from the centre of a sphere charged with quantity q will be q x i q -V '' jr. ' and the distance between adjacent equipotential surfaces must be such that the product of this force into the distance is unity.* Calling this dis- tance d' t we must have T' x * = ' * = d i~ From this we see that as d increases d' increases in a duplicate ratio, so that the equipotential surfaces become further apart as the distance from the sphere increases. If we suppose this sphere to be surrounded by a hollow concentric sphere of conducting material, the lines of electric force will still be radial, f The " difference of potential " between the spheres will, by definition, be equal to the work done on unit charge when moved from one surface to the other. By summation (or by integration, or by plotting a curve between force and distance from the expression / = |- a , and finding its area) this work can be shown to be q [ --- ), where r l and r t are the radii of the inner and Vl ^2' outer spheres respectively. If r t be infinite, i.e., the inner sphere be alone in space, the above expression becomes - ; this result is generally r\ expressed by the statement that the electric potential of a sphere in space is equal to the charge on it, divided by its radius. To the ratio . - the name capacity is given, so we see potential difference that the capacity of a sphere of radius r^ surrounded by another concentric one of radius r t is given by the expression - , and for a sphere r t fj isolated in space the capacity is r lt i.e., equal to the radius of the sphere. An arrangement of two concentric conducting spheres insulated from each other, or in fact any two conducting surfaces adjacent to each other, is termed a condenser. If we consider two adjacent surfaces which are in definite relative positions, the capacity of the arrangement will be constant, if the insulating medium (or dielectric as it is termed) between the surfaces remains unaltered. The quantity of electricity on each sur- face will therefore be proportional to the P.D. { between them, and as the force between quantities in fixed relative positions depends on the product of these quantities, we see that the force exerted between two surfaces of a condenser is proportional to the square of the potential difference between them. This fact provides us with a method of measuring potential differences (see Section 48). Experiment proves that the capacities of condensers depend on the insu- lating substance between the surfaces, and the ratio in which the capacity of a given condenser is changed by substituting any substance for air is called the specific inductive capacity of the substance. * More accurately expressed by saying that the line integral of the force from one surface to the other is unity. f For the symmetry is not disturbed by its presence. J P.D. is an abbreviation for potential difference. CHAPTER III GALVANOMETERS, ELECTRODYNAMOMETERS, AND AMMETERS 31. The Tangent Galvanometer 32. Adjusting the coil of a Tangent Galvanometer 33. Tangent Scale 34. Tangent Law 35. Variation of Sensibility of a Tangent Galvanometer with Number and Size of Turns 36. Value in Amperes of the deflections of a Tangent Galvano- meter controlled by the Earth's Field 37. Pivot and Fibre Suspen- sions 38. Sine Galvanometer ; Sine Law 39. Electrodynamo- meters 40. Construction of Proportional Galvanometers 41. Galvanometers of Invariable Sensibility 42. Permanent Magnet Ammeters 43. Moving Coil Ammeters ; Single Pivot Galvanometer 44. Soft Iron Ammeters, Spring and Gravity Control 45. Hot Wire Ammeters. 31. The Tangent Galvanometer. In the previous chapters we have seen how currents may be measured by comparing the strengths of the magnetic fields they produce with another mag- netic field of constant strength, and are now in a position to deal with current -measuring instruments more fully. As mentioned in Section 15, the relation between the angular deflections of a magnetic needle and the currents which produce them becomes very simple in the case where the two magnetic fields are uniform and at right angles to each other, the tangent of the deflection is proportional to the current passing round the coil of the galvano- meter. This law holds for an instrument when the following four conditions are fulfilled : (1) The needle is controlled by a uniform magnetic field of constant strength. (2) The diameter of the coil is large compared with the length of the needle. (3) The needle is suspended sufficiently near the centre of the coil for the field which is produced by the current passing round the coil to be a uniform one in the neighbourhood of the needle. (4) 'The axis of the needle is parallel to the plane of the coil when no current is passing. When these four conditions are all fulfilled the calibration curve of the galvanometer, when tested by comparison with a 83 8 4 PRACTICAL ELECTRICITY voltameter, as described in Section 17, will be found to be of the shape shown in Fig. 56 ; and if any three points, p, Q, R, be taken on this curve, it will be found that the lengths A p, B Q, c R, parallel to o Y, bear to one another the ratios of the tangents of the angles represented by o A, o B, and o c respectively. Such a galvanometer (seen in detail in Fig. 27) is, there- fore, called a " tangent galvanometer," and it may be henceforth used without refer- ence to any volta- meter for the com- parison of current strengths, as they will be simply pro- portional to the tangents of the angles through which the magnetic needle is deflected. 32. Adjusting the Coil of a Tangent Galvanometer. We have next to con- sider how we can adjust the coil of a galvanometer so as to be sure that its mean plane is parallel to the axis of the needle when no current is passing. Owing to the coil having a certain breadth, it is sometimes impossible to see the needle when looking down on to the coil ; indeed, it is for this reason that the long light pointer attached to the needle is placed at right angles to the needle. It would not be right to assume that because the instrument has been so turned that the pointer points to the zero on the scale, therefore the plane of the coil is parallel to the magnetic axis of the needle, for even if the scale has been attached to the instrument so that the line of zeros is at right angles to the plane of the coil, it does not follow that the pointer itself is at right angles to the needle. The two may even have been placed at right angles to one another by the maker, and yet the pointer may have been bent subsequently, so that they are not at right angles when used ; or no experiment may have been made by the maker to test this, as he is aware that the user will probably make a test and adjust the pointer for himself. Fig. 56. Calibration Curve of a Tangent Galvanometer. TANGENT GALVANOMETER 85 The test for parallelism of the axis oHhe needle with the mean plane of the coil may most simply be made as follows : Turn the instrument until the pointer points to o, send any convenient current through it, and observe the deflection, then reverse the direction of the current without altering its strength,* and observe the deflection on the other side of the scale. If these deflections are exactly equal, then the plane of the coil is parallel to the axis of the needle when the pointer points to o, and the instrument is properly adjusted. But if one deflection is, say, 47 to the left, and the other, say, 44 to the right, the pointer is not at right angles to the magnetic axis of the needle, supposing, of course, that the scale has been so fixed that the line of zeros is exactly at right angles to the plane of the coil. Next, turn the instrument a little about its centre in the direction opposite to that in which the needle moved when the greater deflection was obtained. The pointer will now, of course, not point to zero ; let it stand at i to the left. Again send a current, first in one direction, obtaining a reading, say, 46 to the left, and in the other direction, when it gives a reading of, say, 45 to the right. Now remembering that the pointer started from i to the left, the true deflections of the needle are respectively, 46 1, or 45 to the left, and 45+ 1, or 46 to the right. Hence, the fault is now on the other side, or the left deflection is smaller than the right, and we have, consequently, turned the instrument too much. Turn, therefore, the coil round a very little in the opposite direction, so that when no current is passing through the instrument the pointer stands at, say, J to the left, and send as before reverse currents of equal strength, obtaining readings, 454 to the left and 44 J to the right, which, corrected for the initial zero error, correspond with equal deflec- tions of 45 to either side. The instrument will now be correct when it is so placed that for no current the pointer stands at J left, and it can be so used, but not, however, with the tangent scale described in the next section. To enable us to employ the side of the dial graduated * This may be done by causing the current to pass through any galvanoscope, the law of which may be quite unknown ; and taking care that the deflection of the needle of this galvanoscope after the current has been reversed is the same in direction and in amount as it was before the current through the galvanometer was reversed, for if we leave the current through the galvanoscope unchanged in direction when its direc- tion through the galvanometer is reversed in the experiment, it will not be necessary to know that the coil and needle of this auxiliary galvanoscope are symmetrical, or that the strength of a current producing a deflection to the right is the same as that of a current producing the same deflection to the left. 86 PRACTICAL ELECTRICITY in tangents, as well as to avoid having to remember the J left error, do not alter the position of the instrument, but bend the pointer until it points to o for the same position of the instrument in which it previously pointed to J left. The instrument will now behave as a correct tangent galvanometer when the pointer stands at o for no current. 33. Scale for a Tangent Galvanometer. The scales of tangent galvanometers are frequently simply divided into degrees, and references have constantly to be made to a table of tangents to enable the galvanometer to be used. A better plan is to divide the scale, not into equal divisions, but into divisions the lengths of which become smaller and smaller as we depart from the zero or undeflected position of the needle, in such a way that the number of divisions in any arc is proportional, but not necessarily equal, to the tangent of the angle corresponding with that arc. Or the scale may, as shown in Fig. 57, be divided into degrees on one side, and on the tangent principle on the other. Such a tangent scale can be most easily constructed in the fol- lowing way : Draw a tangent FAF (Fig. 58) to a circle, and starting from the point of con- tact A of this tangent line with the circle, mark off A B, B c, c D, etc., in both directions all equal to one another. Then join the centre o of the circle with the points B, c, D, etc., by straight lines cutting the circle in the points i, 2, 3, etc. ; then the numbers I, 2, 3, 4, etc., will be respectively proportional to the tangents of the angles AOI, AO2, A03, etc. For tan AOI = ; o A' Fig. 57. Scale for a Tangent Galvanometer, tan AO2 tan A03 A C _2 A B O A* O A* AD 3 A B ; = ; and so on. o A o A TANGENT GALVANOMETER 87 Beginners are apt to think that, because the divisions on such a tangent scale are very much crowded together in the higher part of the scale, the value of a current can be more accurately ascertained by taking a reading on the degree side, and then finding the value of the tangent in a table of tangents, than by reading it off on the tangent scale. But this seemingly greater accuracy is quite delusive, since what has to be ascer- tained in either case is the tan- gent of the angle, not merely the angle, and al- though on the degree side of the Scale the Fi S- 5 8 - Constructing a Scale for a Tangent Galvanometer. angle can be read much more accurately than can be its tangent, or a number proportional to its tangent, on the other side, this only indicates .that the error of a tenth of a degree in a large angle, although a much smaller proportional error than a tenth of a degree in a smaller angle, produces a far greater proportional error in the tangent. For example, if 2O-i be read instead of 20, the error made in reading the angle is 5^, whereas if 85-! be read instead of 85, the error is only B Jo, or less than a quarter of the preceding error. But the tangents are in the first case 0-3659, and 0-3640, the error in the tangent, therefore, is gif^y, or about 1^2, whereas the tangents in the second case are n-66 and 11-43, so that the proportional error is if|g, or about ^, which is nearly four times as great as before. Hence in this case, when the proportional angular error is diminished to one quarter, the corresponding proportional error in the tangents is increased four times. The crowding together of the divisions on the tangent scale at the higher readings is, therefore, a correct indication of the inaccuracy likely to occur in taking readings in that part of the scale. It can be shown that if one current strength has to be measured by a tangent galvanometer, the result, other things being the same, will be most accurate when the deflection produced is 45 ;* or if two currents are to be measured, their ratio will be * The student should prove this as an exercise. PRACTICAL ELECTRICITY most accurate when the deflections they produce are as nearly as possible at equal distances on the two sides of 45. We may here recall attention to the fact that the deflec- tion produced by a given current passing through a tangent galvanometer is not altered by varying the strength of the mag- netic needle of the galvanometer, or by varying its length, provided that the needle is not made so long as to render the tangent law untrue for the particular galvanometer. For altering the strength of the needle alters the deflecting and the controlling forces in exactly the same proportion, so that the direction of the resultant of these two forces remains unchanged. So, also, altering the length of the needle does not change the direction of the resultant force. Hence the advantages gained by using a strongly magnetised needle are, first, that it moves more quickly to the deflected position when a current is sent through the galvanometer, and re- turns more quickly to the zero when the current is stopped ; secondly, that the friction at the pivot on which the needle turns, or the torsion of the silk fibre supporting the needle, introduces less error in a measurement. 34. Tangent Law. The conditions under which the tangent law is true, may be stated most generally thus : If any body N N' (Fig. 59), turning about an axis at o, be acted on by two forces whose directions lie in a plane at right angles to this axis and intersect at a point N, the tangent of the angle made by N o with one of the forces P, will be proportional to the magnitude of the other force Q when : (1) The controlling force, P, is constant in magnitude, but not necessarily in direction. (2) The deflecting force, Q, acts at right anghs to the controlling force. In the tangent galvanometer these conditions, as already explained, are necessarily satisfied by the construction of the apparatus without any adjustment being necessary when the deflecting force is varied in magnitude. So in the apparatus seen in Fig. 60, where both the controlling and the deflecting force are produced by weights, the above conditions will also be auto- matically fulfilled for any position of the rod N N', if N N' be short, and if the pulley p be placed far away from N N' in such a position that the thread k k is horizontal. Any weight w' put into the WHEN THE TANGENT LAW IS TRUE 89 scale pan plus the weight of scale pan, will therefore be propor- tional to the tangent of the angle which N N' makes with the direction of the controlling force. This tangent is proportional to the length z R if the scale s s be horizontal and initially adjusted so that its zero line z coincides with the pointer attached to N N' when the only force acting on N N' is that due to w, the controlling force. For the required tangent is the ratio of z R to o z, ando z is, of course, a constant. With the apparatus illustrated in Fig. 61, which is a more accurate, but at the same time a more expensive one than that shown in Fig. 60, the pulley p is comparatively near the rod N N'. Fig. 60. Simple Mechanical Apparatus for Testing the Tangent Law. Hence an adjustment is necessary to keep the thread k k always horizontal, that is, at right angles to the direction of the control- ling force. This adjustment is made by turning the tangent screw T, and the simplest way of insuring that the pulley p has been raised or lowered sufficiently to keep the thread k k horizontal, when the rod N N' is deflected, is to commence the experiment by turning the levelling screw s, until the level L shows that the bar b b is perfectly horizontal ; then, after putting each of the different weights w' into the scale pan, to turn the screw T until th$ thread k k is seen to be parallel to one of the edges of bar b b. As N N' is not symmetrical above and below the axis o in the ; apparatus shown in Fig:. 61, and, therefore, is not self-balanced, 90 PRACTICAL ELECTRICITY we must, before any measurements are commenced, screw the counterpoise weight c in or out, until the rod remains balanced in any position when the controlling and the deflecting forces are both naught. These forces are easily made naught by resting the weight w on the block of wood B, and by taking the thread k k off the pulley p and resting the scale pan on the base -board of the instrument. The scale s s is adjusted as before, so that when the controlling weight w alone acts on N N', the zero line of the scale coincides with the position taken up by the pointer, only this adjustment can now be made very accurately by using as the pointer the wire stretched along the centre of the moving Fig. 6r. Improved Mechanical Apparatus for Testing the Tangent Law. arm, and ensuring coincidence by observing when the image of this wire seen in the mirror which is attached to the scale, coincides with the zero line. The controlling and deflecting weights may of course be inter- changed, in which case the rod N N' will remain horizontal instead of vertical, when the controlling force alone acts on it, and the tangent of the angle, which is proportional to the magnitude of the deflecting force, will be measured on a vertical scale. 35. Variation of the Sensibility of a Tangent Galvanometer with the Number of Windings, and with the Diameter of the Coil. A tangent galvanometer, whose bobbin contains only one turn of wire, is not suitable for measuring very weak currents, as it is not sufficiently sensitive. In order to obtain a delicate tangent galvanometer, the bobbin must be wound with many turns of fine wire, and the greater the number of turns employed, TANGENT GALVANOMETER SENSIBILITY 91 the smaller will be the current needed to produce a given deflection on the instrument. The exact way in which the sensibility of a tangent galvanometer is dependent on the number of windings may be experimentally tested by means of the apparatus shown in Fig. 62, and this may also be used to ascertain the variation in sensibility pro- duced by changing the diameter of the bobbin on which the wire is wound. The main portion of this apparatus has already been de- scribed (Section 30), but in Fig. 62 a smaller coil c c is shown, whose mean diameter is exactly half that of the lar- ger one. This small coil is mounted on a board b b, and, when placed in position on B B, is concen- tric and coplanar with the larger. On the larger bobbin c c are wound two distinct coils of in- sulated wire, both of the same mean dia- meter, oneconsisting of twelve convolu- tions and having its ends attached to two of the binding screws, i, 2, the other of four con- volutions and hav- ing its ends attached to the other two binding screws, 3, 4. If the binding screw 2 at the end of the first coil be joined by a piece of wire, as shown in the figure, to the binding screw 3 attached to the beginning of the second, the current will go 12+4, or sixteen times round the bobbin; whereas 92 PRACTICAL ELECTRICITY if the wire connect the end of the first coil, 2, with the end of the second, 4, and the current enter and finally leave the bobbin by the two binding screws i, 3, attached re- spectively to the beginnings of the two coils, then the current will go twelve times round the bobbin in one direction and four times in the other, or practically 12 4, or eight times round the bobbin. Now, experiment shows that if a current of constant strength* be passed successively first four, then eight, then twelve, then sixteen times round the bobbin, and if this is kept in a fixed position during the experiment, the tangents of the corresponding deflections produced will be as four to eight, to twelve, to sixteen, that is, simply proportional to the number of times the current passes round the bobbin. This proves that the sensibility of a tangent galvanometer is proportional to the number of turns of wire used on its bobbin. We may next proceed to investigate the effect of the size of the bobbin by experiments made on the small coil c c. The diameter of this coil is only one half that of c c, and there are four convolu- tions of wire wound upon it. When experiments are made it is found that, if the two bobbins c c and c c are placed so as to be in the same plane, and so as to have their centres coincident with that of the suspended magnet, the tangent of the deflection produced by any current flowing round the smaller one is twice as great as the tangent of the deflection produced by the same current flowing four times round the larger bobbin ; and also, if the same current pass four times round the smaller bobbin in one direction, and eight times round the larger in the opposite direction, that no deflection is produced whatever the current may be. From the above observations we learn that the tangent of the deflection produced by a current, that is, the sensibility of the instrument, is directly proportional to the number of convolutions of wire, and inversely proportional to the diameter of the coil. In order, therefore, to get a sensitive instrument we should use coils of small diameter, and wound with many turns of wire, and it might be imagined that a tangent galvanometer intended for the measurement of very weak currents should be made in this way. As a matter of fact, however, the coils of good tangent galvanometers are always large in diameter compared with the length of the suspended needle ; and the number of turns of wire used in winding is always limited by the consideration that the depth and width of the channel in which the wire is wound must not exceed a certain fraction of the diameter of the coil. These * The current may be kept constant, as described in Section 30. TANGENT GALVANOMETER SENSIBILITY 93 restrictions are only imposed in order to ensure the fulfilment of the tangent law, and need not be considered when there is no necessity for the tangent of the galvanometer deflection to be strictly proportional to the current. An instrument which is to be used as a tangent galvanometer must, however, be so constructed that all the conditions mentioned in Section 31, earlier, as necessary to ensure the fulfilment of the tangent law, are complied with. Now when the needle in the box g g, Fig. 62, is deflected, its poles move away from the coil c c, and the force exerted by the current in this coil is less, after the needle has moved, than before. The tangent law will not hold good unless the change produced in this way is small enough to be neglected. In order to test this point, the apparatus shown in Fig. 62 is arranged so that each of the coils c c, c c, can be moved either in its own plane or perpendicular to its plane, as described in Section 30. Experiments such as those described in Section 30 show that as the bobbin is moved the deflection alters, and that the change pro- duced for the same amount of motion is proportionately greater for the small bobbin c c than for the large one c c. For example, when the coil was moved parallel to itself, and so that its axis passes through the centre of the needle, we found that the tangent of the deflection of the needle for a given current was propor- tional to r* where r represents the mean radius of the coil and x the distance from the mean plane of the coil to the centre of the needle. Now it is clear from this formula that for a given change in x there will be a greater change in the value of this fraction the smaller r is. It thus becomes apparent that any error due to want of proper centring of the needle of a tangent galvanometer, or to the actual movement of its poles when it is deflected, must prove far more serious when the bobbins are small than when they are large ; and for this reason instruments in which the tangent law is to be accurately relied upon are constructed with large bobbins. Example 17. A tangent galvanometer wound with 50 con- volutions of wire gives a deflection of 45 when a current of 0-05 ampere passes. What current flowing in a coil of 2 turns of the same diameter would produce the same deflection ? Answer: 1-25 amperes. Example 18. In the above example find the current required 94 PRACTICAL ELECTRICITY to produce a deflection of 25, (a), with the 50 turn coil, (b), with the 2 turn coil. (a) Let / be the current in amperes required. 7 tan 2 Then J. llV-^-Ll ' - 0-05 tan 45 /. / = 0-0233 ampere. ta " * (6, - 1-25 tan 45 .*. / = 0-583 ampere. Example 19. A current of o-i ampere passes through a coil of 20 turns the mean diameter of which is 12 inches. What must be the size of a coplanar concentric coil of 5 convolutions carrying 0-25 ampere, which would produce an equal magnetic force at its centre ? Let d be the diameter required: Then, since the magnetic force is :: to the current and to the number of convolutions, and inversely as the diameter, we have o-i x 20 0-25 x 5 12 d . d _ 0*25x5x12 O'l X2O = 7-5 inches. Example 20. A tangent galvanometer is made with two coils of equal diameter, the first consisting of 500 convolutions of wire, the second of one convolution. If a current of 0-25 ampere Sent through the first cause a deflection of 45, what current sent through the second in the opposite direction, while the same current was still flowing through the first, would cause the de- flection to become one of 10 ? Let / be the unknown number of amperes : 500 x 0-25 ix/ _ tan 10 Then 500 x 0-25 tan 45 Answer. 103 amperes. Example 21. A tangent galvanometer with its needle sup- ported independently of the coil (as in Fig. 62) gives a certain deflection for a current of / amperes, when the needle is at the centre of the coil. Through what distances must the coil be moved along its own axis if a current of io/ amperes is to give the same deflection ? (Radius of coil, io centimetres.) TANGENT GALVANOMETER EXAMPLES 95 Let the distance required be x centimetres ; the question then requires that the strength of field at a distance x along the axis of the coil when a current 10 / is passing, must be equal to that at the centre of the coil when a current / flows through it. Making use of the formula deduced in Section 30, viz. : 2 . F = -- , we have 10 V ), for For a current of 10 / the force at a distance x along the axis Force at centre, (x = o), for current I = - |- Equating the two we get lOf' or 10 r = y+*; squaring both sides and extracting cube root or x* = r * = r 19-1 cms. From this we see that by moving the coil so that the needle is 19-1 centimetres from its centre (a distance nearly equal to the diameter of the coil), the sensibility of the instrument is reduced to ye of its former value. Example 22. From the curve given by the graph, Fig. 53, find the distances along the axis of the coil at which a magnetic needle must be placed so that the sensibility of the galvanometer so formed may be reduced in the proportions \, , and J respec- tively, the sensibility with the needle at the centre of the coil being considered unity. Find also these distances by calculation. Answers. From curve, 15-2, 24-8, 34 (approx.). By calculation, 15-3, 247, 34-6. o6 PRACTICAL ELECTRICITY 36. Values in Amperes of the Deflections of a Tangent Galvano- meter controlled only by the Earth's Magnetism. The sensibility of any galvanometer depends not merely on the coil, but also on the strength of the controlling field. If this controlling field be altered by bringing up a magnet, then even if the magnet be so placed that the position of rest of the needle for no current be unchanged, still the force, and therefore the current required to turn the needle through a given angle will be altered. For let the controlling force N P be increased to N P' (Fig. 63) so that the zero position of the needle is the same, but the needle is held in that position with a greater force, then in order that the angle P'NR' may remain of the same value as before, the deflecting force P R must be increased to P'R', that is, in the same pro- portion as the controlling force. If the current has the same value as before, so that P' R" is equal to P R, then the angular deflection of the needle instead of being P N R' will be reduced to P' N R". Even if the controlling field be merely that due to the earth, this will alter from place to place, and from year to year ; so that a tangent galvano- meter requiring a current equal to I ampere to produce a deflection of 45 degrees in some particular town, will generally need a somewhat different current to produce the same deflection if moved to another town, and even if kept in the same position the absolute calibration will be found to gradually alter with time. When the needle of a tangent galvanometer is supported in such a way that it turns in a hori- zontal plane, and when the controlling force is entirely produced by the " horizontal component of the earth's magnetic force" the following formula connects the current I in amperes, passing through the coil, with the deflection d in degrees, the radius r of the coil in centimetres, and the number of convolutions n of wire on the bobbin T . / =- -tan d, (12) 2-x n where H is the strength of the horizontal component of the earth's magnetic field at the place where the galvanometer is situated. This follows from the formulse given in Sections 15 and 30, combined with the fact just proved, that the sensibility of a tangent galvanometer is proportional to the number of convolutions of wire on its bobbin. INTENSITY OF EARTH'S FIELD 97 The quantity - in the above expression for /, is constant for a given time and place in an undisturbed area, and may be written as k lf the formula then becoming I = tan d. (13) In Table III. is given the average values of H at Greenwich Kew, Valencia, Stonyhurst and Eskdalemuir, for the years 1914 to 1919, and also the values of k l when r is measured in centi- metres and in inches respectively. From 1910 to 1913 the mean values of H at these stations remained practically unaltered, ard those for 1918 and 1919 are equal except at Stonvliurst. where a new magnet was set up in the interval. TABLE III. Value of ki, when r is in Pl___ Vpnr Volno nf H r Jace. i enr. value 01 /7. Centimetres. Inches. 1914 0*1852 0-2948 0*7489 1915 0*1851 0*2946 0*7484 Greenwich 1916 0*1849 0*2943 o*7474 1917 0-1848 0*2942 07471 1918 0*1846 0*2939 0*7463 1919 0-1846 0-2939 07463 1914 0-1849 0*2Q43 0*7474 i9 T 5 0-1846 0-2939 07463 Kew 1916 0-1846 0*2939 0*7463 1917 o 1841 0-2935 o*7454 1018 0-1843 0-2934 ' 07451 1919 0*1842 0-2932 0-7448 1914 0-1789 0*2847 0*7233 *9 r 5 0-1787 0*2844 07224 Valencia 1916 1917 0-1787 0-1786 0-2844 0*2843 07224 0*7221 1918 0*1784 o -2840 07213 1919 0-178; 0-2840 0*7213 1914 0-1735 0*2762 070x3 1915 0-173* 0*2761 0-7009 Stonyhurst 19.6 1917 0*1734 0-1734 0*2760 0*2760 07003 07009 1918 01733 0-2758 0*7004 1919 0*1729 0-2752 0-6991 1914 0*1680 0*2674 0*6792 i9 T 5 0*1679 0*2673 0-6789 Eskdalemuir 1916 1917 0*1676 0*1673 0-2668 0-2663 0*6776 0-6764 7918 0-1671 0*2661 0-6756. 1919 0*1671 0*2661 0-6756 When the controlling force acting on the needle of a tangent galvanometer is due to the presence of a distant magnet, placed H 98 PRACTICAL ELECTRICITY so that the needle is parallel to the plane of the coil when no cur- rent passes, the preceding formula holds true, but the constant, k lt must be determined experimentally. If the value of k^ for the earth's field alone be accurately known for the particular place and the particular time, then the value of k l for any other controlling field may be ascertained by employing the principle described in Section 29. Remove all magnets, pieces of iron, etc., so that the needle of the tangent galvano- meter is acted on by the earth's field alone, and count the number of oscillations, n lf say, that the needle makes in any convenient interval of time. Replace the controlling magnet, or magnets, as desired, and again count the number of oscillations, n 2 , say, that the needle makes in the same time, then the k l for the earth's field alone must be multiplied by n^/n^ to obtain the k l to be used in the preceding formula for the particular com- bination of controlling magnets in question. Example 23. How many amperes would deflect the needle of a tangent galvanometer 60 in the year 1914, the controlling force being the horizontal component of the earth's magnetism at Greenwich, and the galvanometer having a coil 5 inches in radius, wound with six convolutions of wire ? The number of amperes is ?3 ^ X Answer. 1-079 ampere. Example 24. Through what angle would 0-598 ampere deflect the needle of a tangent galvanometer with a bobbin 7 inches in radius, wound with five convolutions of wire, in the year 1918, the controlling force being the horizontal component of the earth's magnetism at Kew ? .,tan ^ 0-7451 x 7 = 0-5731, d = 2924 Answer. 29^4. Example 25. If the horizontal component of the earth's magnetism in 1914 at Stonyhurst be the controlling force in a tangent galvanometer, the bobbin of which is n inches in dia- meter, how many convolutions of wire must be wound on in order that a current of 0-964 ampere may give a deflection of 45 ? Answer. 4 convolutions. PIVOT AND FIBRE SUSPENSIONS 99 Example 26. If the horizontal component of the earth's magnetism in 1917 at Kew be the controlling force in a tangent galvanometer, the bobbin of which is wound with eight con- volutions of wire, what must be the radius of the coil in order that a current of 0-384 ampere may give a deflection of 50 ? Answer. 3-45 inches. Example 27. About how many times the horizontal compo- nent of the earth's magnetism must the controlling force be in a tangent galvanometer, having a coil 5 inches in radius wound with six convolutions of wire, in order that a current of 20 amperes may cause a deflection of 45 ? Answer. About 32 times. Example 28. The needle of a tangent galvanometer when acted on by the earth's field alone makes one oscillation in 1-3 second, whereas, when the controlling magnet is placed in position, it makes one oscillation in 0-433 second. If the coil be 15 centimetres in radius, and be wound with twenty turns of wire, what current will produce a deflection of 30 in 1918 at Greenwich ? Answer. 1-14 ampere. Example 29. Find the mean diameter of a single turn tangent galvanometer coil, such that one C.G.S. unit of current (10 am- peres) will produce a deflection of 45, the needle being controlled by the earth's horizontal field at a place where #=0-1852. Answer. 67-8 centimetres. 26-66 inches. It is not necessary that the coil of a tangent galvanometer should be circular, but in order to obtain the straightness of the lines of force in the neighbourhood of the axis, as seen in Figs. 38 and 49, and not merely for points actually on the axis, of which we could only avail ourselves by using an infinitely short magnet, the diameter of all parts of the coil must be large. Hence, if an elliptic or other non-circular coil were used, its smallest diameter would have to be large, and consequently its largest diameter unnecessarily large. 37. Pivot and Fibre Suspensions. There are two principal methods of supporting the needles of galvanometers. These are illustrated in Fig. 280. In D the little magnet has a jewel in its centre, and rests on a sharp pivot, as in an ordinary pocket compass ; whereas in G the needle is supported by a fine fibre of unspun silk, the upper end of which is fastened in one of the ways illustrated in Fig. 27, so that it can be lowered on to the 100 PRACTICAL ELECTRICITY card on which the scale is marked, when the instrument is being carried about, and raised again so as to be in the centre of the coil when the instrument is in use. The fibre suspension in- troduces far less resistance to the motion of the needle than the best jewel and pivot ; but with a fibre suspension it. is generally necessary that the instrument should have levelling screws, such as are seen attached to G, Fig. 280, and that it should be levelled before being used. Fig. 64.- Section of Galvanometer with Silk Fibre Suspension, Pivoted for Turning round its Centre. A galvanometer needle should therefore be supported by a pivot when the instrument has to be moved about, and used quickly in different positions. But when the galvanometer is employed in a fixed position, and great accuracy is desired, the needle ought always to be suspended by a fibre of unspun silk. 38. Sine Galvanometer. As the tangent galvanometer re- quires a coil of large size compared with the length of the needle, the form is not well suited for instruments of very great sensibility. There is, however, another kind of galvanometer which is free from this defect, viz., the sine galvanometer. In this type of instrument there is no restriction as to size or shape of coil, the only conditions being that the controlling field be constant and uniform, and that the coil and needle always occupy the same relative position when the readings are taken. In Section 6, it was shown that when the needle and coil are in the same relative positions, the couple exerted between them is propor- tional to the rate of chemical decomposition, and therefore to SINE GALVANOMETER: the current strength. When a needle is deflected by a current in a coil, and the coil turned to follow up the needle until the relative position of the two is a definite one, the torque exerted on the needle by the current, when equilibrium exists, being equal to that exerted on the needle by the controlling field, is propor- tional to the sine ot the angle of deflection. Obviously, the current strength is therefore proportional to the sine of the angle through which the needle is deflected from the magnetic meridian. A Fig. 65. Apparatus for Mechanically Testing the Sine Law. Adjustment made by Altering the Direction of the Deflecting Force. Fig. 64 shows a section of a galvanometer arranged so that it can readily be turned about its centre for making relative measure- ments of current strength by the sine method, and in Fig. 65 is illustrated an apparatus for mechanically testing the sine law. Here a rod, N N', representing a needle, is pivotted at o and counterbalanced by a nut c on the screwed end of N N'. From the lower end, N, hangs a weight, w, and to the same point is attached a thread, k k, supporting a scale pan and weight w'. An arm, o D, pivotted at o has another arm, E, clamped to it by a nut n, and E carries a pulley, P Q, over which k k passes. The arm o D is fixed to a tangent wheel and can be turned about o by the screw T. At the lower end of o D is a piece of mirror glass, G, with a scratch on it ; a pointer on the lower end of N N' can be sighted and the arm o D adjusted until the pointer is directly opposite the scratch, by turning the screw T. A PRACTICAL ELECTRICITY horizontal scale, s s, with a mirror behind it, enables distances from a vertical plane through the axis o to be measured ; these distances being proportional to the sines of the angles of deflection of N N' from the vertical position. To make an experiment the weights w and w' and scale pan are removed, N N' balanced by the counterpoise c ; the weight w is then put on and the scale s s adjusted until its zero is directly behind the thread supporting w. The thread k k is then put over the pulley P Q, and a weight w' placed in the scale pan. The arm o D is now adjusted so that the mark on G is directly opposite the pointer on N, by means of the tangent screw T. On taking different values of w' and the corresponding readings 5, it is found that the two quantities w' and s are proportional, i.e., the weight w' is pro- portional to the sine of the angle through which the rod N N' is deflected by w'.* Proportionality between w' and 5 will be found to exist, whatever the direc- tion N k of the deflecting force relative to the rod N N', provided this be unaltered during a set of experi- ments. The ratio of w' to s will, however, alter when this direction is changed. 39. E!ectrodynamometers. Another form of current measur- ing instrument for which the law connecting the deflection and strength of current is known, is the electrodynamometer. It consists essentially of two coils of wire carrying the same current, and the force, or torque, exerted between the coils depends on the strength of the current passing. As we have already seen (Sect. 24), a coil carrying a current creates a magnetic field in its neighbourhood, just as a magnet does ; we may, therefore, regard such a coil as a magnet, and two adjacent coils having currents passing through them will usually exert a force on each other. If the coils are kept in the same relative position, the magnitude of this force will be doubled if the strength of current in either coil be doubled, and if the current in both coils be doubled, the force will be quadrupled. When the two coils are in series with each other, doubling the current in the circuit will double it in both coils, and hence make the mutual force four times as great * w' here includes weight of scale pan. Fig. 66. Simple Electro- dynamometer. ELECTRODYNAMOMETERS 103 We may therefore conclude that the force exerted between the two coils of an electrodynamometer, whose coils are in a fixed relative position to each other, is proportional to the square of the strength of the current flowing through them. An electrodynamcmeter of a simple form is shown in Fig. 66, whilst Fig. 67 illustrates an instrument^ used in practice. ID both instruments one of the two coils is suspended by a silk thread, and the fixed relative position of the stationary and moving coils is brought about by means of a spiral spring shown at N Fig. 66. This spring is at- tached to the torsion head T at its upper end, and to the suspended coil E F G at its lower end, and by turning T the pointer P fixed to the moving coil can be brought to the zero mark on the scale shown in plan in Fig. 68. When so adjusted the relative position of the stationary and suspended coils is perfectly definite.! Stops s s prevent P moving far away from the required position. Usually the planes of the two coils are perpendicular when P is at zero. Mercury cups, m m f , Figs. 66 and 67, are used for leading the current to and from the moving coil, the path of the current, starting from the left hand terminal, being as follows : Through the fixed coil, A B c D to the mercury cup m, then through the moving coil, E F G, to the mercury cup m' and the right hand terminal. When a current passes through the instrument a couple exists between the coils, tending to place the moving coil parallel to the fixed one. This turns the moving coil away from zero in a counter- clockwise direction, and by turning the head T clockwise, the spring exerts a torque in the opposite direction, which can be Fig. 67. Siemens Electrodynamometer. io 4 PRACTICAL ELECTRICITY adjusted so as to exactly balance the couple due to the current in the coils. The torque of the spring is proportional to the angle through which its upper end is twisted, so that the angle of torsion measures the square of the strength of the current. We may, therefore write 7 2 ::a where a is the angle T is turned through. or I 2 = k* a, / = kV*. (14) k being called the constant of the electrodynamometer, and which may be determined by comparison with a voltameter or an absolute tangent galvano- meter. When an electrodynamometer is intended to measure very small currents, say less than J an ampere, mercury cups are not necessary, as flexible wires can be substituted. Such an instrument is shown in Fig. SSb. Example 30. An electrodynamo- meter is used to measure the relative values of two currents which neces- sitate rotations of the torsion head of 25 and 144 respectively ; find the ratio of their strengths. Let /! and 7 2 denote the strength of the two currents, then I 2 = = 5 k, = 12 k t 12 Example 31. A current of 25 amperes gives a deflection of 140 on an electrodynamometer ; what current will produce a deflection of 320 ? 100 Answer. --- ==- = 37-8 V? amperes Example 32. A current which deposits copper at the rate of 1-97 grammes per minute produces a deflection of 225 on an electrodynamometer; find the constant of the instrument. Answer. 6-6. PROPORTIONAL GALVANOMETERS 105 The Siemens electrodynamometer shown in Fig. 67 has one moving coil and two fixed ones of different numbers of turns, and by this device the range of current which the instrument is capable of measuring is considerably extended. The outer or thick fixed coil has 4 turns, and the inner or thin fixed one about 60 turns. They are connected together and to the moving coil at D, and their free ends joined to the left and right hand terminals respectively. When it is desired to employ the thick coil the middle and left-hand terminals are used, and for the thin coil, middle and right. In using an electrodynamcmeter care should be taken to place it so that the plane of the moving coil is perpendicular to the magnetic meridian, otherwise the reading of the instrument will be influenced by the earth's magnetic field,, and a current through the dynamometer in one direction will give a different reading from that produced by an equal current in the other direction. Example 320. The constants of the two windings of the dynamometer in Fig. 67 are 3-40 and 0-920 for the " thick " and " thin " coils respectively. What deflections will be caused by a current of 16 amperes, passed (a), from middle to left-hand terminal, and (b), middle to right-hand terminal. Answer. (a) 22-1 divisions. (b) 302 divisions. 40. Construction of Galvanometers in which the Angular Deflection is directly Proportional to the Current. We have already seen (Section 15) that the current is proportional to the tangent of the deflection of the galvanometer needle, when neither the magnitude nor direction of the controlling force is altered as the needle moves into a new position on being deflected, and when, in addition, the direction of the controlling force is at right angles to the direction of the force with which the current passing round the coil acts on the needle. In order, therefore, that the angular deflection may be directly proportional to the current, we must either cause the needle on being deflected to move into a position in which the current passing round the coil acts more powerfully on it, or into a position in which the controlling force becomes weaker or we may arrange that both these results may be produced. The first condition may be obtained in a rough way by employ- ing the very defect of construction previously referred to in the adjustment of the tangent galvanometer, which made the de- flection on one side of the zero larger than that produced by the same current on the other viz., not putting the coil so that its io6 PRACTICAL ELECTRICITY plane was parallel to the suspended magnet when no current was passing through the coil. The needle, when deflected to that side on which the greater deflection is obtained, will, instead of moving from a stronger to a weaker part of the magnetic field Fig. 69. Walmsley and Mather's Proportional Galvanometer. produced by the current, move at first into a stronger part, and then afterwards into a slightly weaker part. The effect of this arrangement is to make the proportional law connecting current and deflection approximately true for a much larger deflection from the undeflected position of the needle than if we com- menced with the needle parallel to the plane of the coil for no current. But this arrangement has the disadvantage that it can only be used for currents deflecting the needle to one side of the scale, for, if the current be flowing in the opposite direction, the defect of want of proportionality between current-strength and deflec- tion will be increased. This plan, by means of which the proportionality on one side of the scale is sacrificed to increase that on the other, has been employed by one of the authors (W. E. A.), and later on by MM. Carpentier and Deprez, and others, for making proportional galvanometers. Another device for causing the strength of the deflecting field to increase as the needle deflects, is employed in the galvano- meter originally devised by Professor Walmsley and one of the authors (T.M.), and in use for many of the experiments of the first-year students at the City and Guilds College. This instrument, as illustrated in Figs. 69 and 6ga, consists of two coils shaped as Fig. 69*. Walmsley-Mather Galvanometer. PROPORTIONAL GALVANOMETERS 107 shown, and fixed so that they are separated by a distance a little less than the length of the needle. The galvanometer is placed so that when no current is passing through the coils the needle hangs symmetrically between them, and when the controlling field is a uniform one, the current is directly proportional to the angular deflection up to 45 or 50. Even although the controlling magnet of a galvanometer be rather near the needle, the controlling field may be regarded as an approximately uniform one if the deflections of the needle be all very small. Similarly for very small deflections the deflecting field may be re- garded as approximately uniform what- ever be the shape and size of the coil or of the needle. If then, in addition, the controlling magnet be so placed that when no current is passing the needle makes about the same small angle with the plane of the coil on one side of it that it makes with that plane on the other side, for the greatest deflection employed, the distribu- tion of the forces will be as in Fig. 70, where N p represents the magnitude and direction of the controlling force, P RJ the magnitude and direction of the deflecting force for current I, P R 2 for current 2, P R 3 for current 3, etc., P R 2 being twice p R lf p R 3 three times P Rj, etc. Therefore the angular deflections of the needle for currents i, 2, 3, etc., are P N R lf P N R 2 , P N R 3 , etc., and, as these angles are all very small, and the base lines are proportional to the currents, it follows that the angular deflections are also propor- tional to the currents. Indeed for very small deflections this result will be nearly true, whether the angle NPX is a little less than, or a little more than, or exactly equal to a right angle , that is, whatever be the angle the needle makes with the plane of the coil provided that this angle is small. 41. Galvanometers of Invariable Sensibility. Now that measuring currents in amperes has acquired the same sort of practical importance as weighing coals in tons or finding the number of cubic feet of gas passing through a pipe, it is necessary to have galvanometers which are portable, and whose indications are not affected by moving the galvanometer from one place to another, or by placing it near an iron pipe, a fire-place, or even near the powerful electromagnets of the dynamo machines io8 PRACTICAL ELECTRICITY which are employed for the mechanical production of electric currents. An instrument of this type should be " direct-reading " ; that is, the deflection of the pointer must indicate at once the current in amperes, for in commercial work there is no time to refer to a table of values, not to mention the risk that would be intro- duced by a table of values belonging to some other instrument being used by mistake. Such instruments, by means of which the current can be read off at once in amperes without any calculation or reference to any calibration curve, are called " ammeters,"* and since about the year 1880 so much attention has been given to the design and construction of this class of electrical meter that it is now pos- sible to measure a current with as much accuracy as a leg of mutton can be weighed in a pair of scales, or with a spring- balance, and with even greater facility. The controlling force must necessarily be exerted in such a way that it is the same wherever the ammeter is placed ; indeed, many ammeters are so constructed that, the controlling force is not changed by laying the instrument on its side, or in any other position, so that a current can be read off equally well whether the ammeter is lying on a table, hung up on a wall, held in the hand, or used on board a ship rolling in a heavy sea. There are three distinct ways in which the controlling force is exerted in ammeters. (1) By means of a powerful permanent magnet placed inside the instrument and rigidly fixed to it. (2) By means of a spring. (3) By means of a weight. The first two methods have the advantage that with their use the moving part of the ammeter can be balanced like a wheel in a watch, so that the instrument can be made to read correctly in any position ; the former of these two has also the further advantage that as the control exerted by a powerful magnet close to the needle is very large, outside magnetic disturbances have little effect. But while a magnet or a spring can be made con- stant enough in its action for many practical purposes, its variation with time is of course greater than that of a weight, hence the third method of control is the one adopted when accuracy is of more importance than portability. In the earlier editions of this book several ammeters were described, and their advantages and disadvantages compared. But the methods of constructing the coils and needles, and the * Abbreviation for ampere-meters. AMMETERS 109 various devices that are now adopted ii> applying the controlling force in one or other of the three ways just referred to have become so numerous, that anything like a complete description of all the types of ammeters now in use, and an examination of their relative advantages would alone fill a good-sized book. A mmetvrs, besides differing in the methods used for exerting the controlling force, also differ in design, depending on whether the instrument is intended to measure currents of very different values, or only currents all of about the same value. In the former case the design should be such that the scale is equally or nearly equally divided, so that there is about the same distance between any adjacent pair of division marks, while in the latter the scale should be very " open " ; that 'is, the division marks should be widely separated at the one part of the scale which is in constant use, and crowded together at those parts which correspond with currents which rarely have to be measured. Instruments with this latter type of scale are especially employed when an ammeter is used to measure the currents supplied to single lamps, or groups of lamps. In Section 5 we saw that when a conductor conveying a current is placed near a magnet there is a force exerted between the conductor and the magnet, tending to make them move relatively to one another. The force acts in such a direction that a wire carrying a current tends to move perpendicular to itself and perpendicular to the lines of force due to the magnet. It is only when the wire lies along the lines of force that the action between it and the magnet is naught, however strong be the current and however powerful the magnetic field. With any other position of the wire relatively to the direction of the magnetic field there is some force, and this force has its greatest value for a given length of conductor carrying a given current, and placed in a field of given strength, when the conductor is perpendicular to the lines of force. By employing a very powerful magnet the force exerted on a wire, even when conveying a feeble current, can be made con- siderable, and this action has been employed by Maxwell, Lord Kelvin, Deprez, d'Arsonval, Weston, and others, to obtain galvanometers which are not only very sensitive, but the indi- cations of which are very little affected by extraneous magnetic disturbance. 42. Permanent Magnet Ammeters. The earliest ammeter, having an equally divided scale so that the deflection in degrees was directly proportional to the current, was the " permanent magnet ammeter " devised by Professor Perry and one of the no PRACTICAL ELECTRICITY Fig. 71. Ayrton and Perry's Permanent Magnet Ammeter. Latest Form. authors in 1880. The coil was wound on the two halves of a flat brass tube A A (Fig. 71), shown unwound in the figure, and inside this tube, at its centre, there was pivoted a small soft-iron needle, shaped like a long ellipsoid, n n (Fig. jia) and controlled by a powerful permanent magnet, M M (Fig. 71). The weight of the pointer p and any dissymmetry s cf the needle was ac- curately counterbalanced by a small weight, w, hence no controlling force was introduced by gravity, and the instrument could be used equally well in any position. On a current flowing round the coil there was exerted a greater or less force tending to place the axis of the needle along the axis of the brass tube A A (Fig. 71), while the controlling magnet M M exerted a force tending to place the axis of the needle along the line joining the tips of the soft-iron pieces PP; the needle therefore set itself in the direction of the resultant magnetic f: eld. In addition to giving the pole pieces the shape seen in the figure, the wire was heaped up somewhat near the ends of the coil,- therefore not merely did the controlling force diminish as the needle deflected, but the deflecting force, for a given current, also increased. Thus, as explained in Section 40, there were two causes tending to make the angular deflec- tion vary in direct proportion to the current flowing, and, when sufficient care was exercised in winding the coil, a straight line calibration could be obtained. When, however, these instruments began to be manufactured in large numbers, the labour of**?. 7i. Needle JT j.1- J- X AT. -11 A - -i , M Staff and Pointer modifying the winding of the coil by trial, until of Ay.to,. a r.i direct proportionality was obtained, became too -Sagiet Ammeter? great, and, instead of depending only on the shape of the pole pieces, of the needle, and of the coil for obtaining a straight line law, two soft -iron cores screwing into the ends of PERMANENT MAGNET AMMETERS in the brass tube were added, and by screwing these cores in more or less the rate at which the deflecting force (for a given current) varied with the position of the needle could be altered. Later on a third plan was employed. The soft-iron pole pieces were themselves made adjustable, and to prevent the controlling force produced by the pole pieces, when withdrawn, falling off too rapidly as the needle deflected, the ends of the poles were made concave instead of convex as before. These movable poles intro- duced the power of making another adjustment in addition to that effected by screwing the soft -iron cores previously mentioned in or out of the coil, and by means of these two adjustments not merely could the angular deflection be made nearly proportional to the current, but the deflection for the same current could be increased- or diminished. Thus the sensibility of the instrument could be adapted to suit an engraved direct-reading scale, instead of each scale having to be engraved somewhat differently to suit the sensibility of the instrument. Another method of adjusting the straight line calibration, carried out by Mr. Esson, is shown in Fig. 71. Here the small screws s s which pass right through the pole pieces, could be advanced or withdrawn so as to alter the shape of the magnetic field controlling the needle, and therefore make the controlling force fall off more or less rapidly as the needle was deflected. By means of the brass nuts N N the needle and pointer could be moved relatively to the pole pieces into a stronger or weaker part .of the magnetic field due to the magnet, and thus the sensibility of the instrument could be adjusted. The Ayrton and Perry permanent magnet ammeter had an important advantage over the various types of soft -iron needle ammeters that are at present constructed, in that the deflection of the pointer indicated not merely the strength of the current but also its direction ; for in certain cases, such as the charging of "secondary" cells, the supply of current to " arc lamps" &c., a knowledge of the direction of the current is as important as the measurement of its strength. To cause the deflection of the pointer to be exactly the same on the two sides of the zero when a current was reversed, an adjustment was necessary, but this was easily effected in the Esson method of construction by turning the coil about the brass screw which held it to the bar at the back of the magnet, this screw being placed so that a line drawn through it passed through the centre of the needle. By employing a very short needle, and a very light pointer made of thin aluminium, corrugated to give it mechanical H2 PRACTICAL ELECTRICITY strength, the combination seen in Fig. jia had only a small moment of inertia. This, combined with the very strong per- manent magnet producing the control, rendered the instrument very quick in action. Therefore, instead of the needle being set swinging, and only coming to rest after some time, when a change suddenly occurred in the current, the needle moved sharply into its new position, and all such changes, even if quickly produced, were accurately indicated. The promptness of action of the permanent magnet ammeter, its extreme freedom from extraneous magnetic disturbance, its power to indicate the direction of the current as well as its strength, and the fact that this form of ammeter could be used in a horizontal or vertical position, or even on board a rolling ship or on a rapidly -moving train, led to many thousands of them being employed, in spite of the fact that their sensibility gradually became greater as the permanent magnet grew weaker. 43. Moving Coil Ammeters. The permanent magnet ammeter described in the previous section had a very strong controlling field so as to render its readings practically independent of its position in the earth's field, or of stray magnetic fields in the vicinity of dynamos ; and to make it quick and precise in action. This, however, had its disadvantages, for the use of a strong controlling field reduced the sensibility of the instrument and necessitated the use of a strong deflecting field, so that a con- siderable number of turns of wire wound very near the needle were required to produce the deflection. A little consideration will show that a modification by which the magnetic field of the permanent magnet could be utilised as the deflecting field instead of a controlling field, would make a strong magnet advantageous, for then an increase in the strength of the magnet would produce greater sensibility, and at the same time render the instrument less liable to error from disturbing fields. As mentioned in Sect. 41, instruments of this class (named " moving coil galvanometers ") have been employed by Maxwell, Lord Kelvin, Deprez, d'Arsonval, and others. Of late years great developments have been made in measuring instruments of this type, and the moving coil galvanometer, or ammeter, is one of the commonest forms in commercial use. A very convenient, portable, and accurate moving coil ammeter was brought out by Mr. Weston, of Newark, America, in 1888, and a view, about two-thirds full size, of the working parts of a recent type, arranged for reading milliamperes or thousandths of an ampere, is shown in Fig. 72. The coil c c is wound on a rectangu- lar metal frame, and is pivotted between jewelled centres, one oi MOVING COIL AMMETERS 113 which is seen at j ; it can turn in a narrow air gap between the pole pieces and cylindrical iron core of a magnet, like the one shown in Figs. 47 and 48, the movement being controlled by spiral hair springs, s s, of non -magnetic material, which also serve to lead the current into and out of the coil. Fig. 72. Working Parts of Weston Mil. Ammeter. Figs. 73 and 730 illustrate a form of moving coil instrument made by Messrs. Nalder Brothers and Thompson, of London. The latter figure shows separate views of (a) , the magnet with pole pieces, (b), the cylindrical core and its support, (c), the coil with control springs, pivots and pointer attached, and (d), the brass bar which carries the top jewel, in which the pivot on the pointer end of the coil works, whilst in the former figure the parts are assembled, but the cover removed to show their relative positions, and also the scale, reading from o to 150 milliamperes. To obtain a strong magnetic field in which the coil turns, the air gap is made as short as possible, consistent with freedom of motion, and the coil is made very light in order to prevent damage to pivots by wear and transit, and also to keep its in- ertia small. By these means the movements of the coil when the current through it changes, are made quick and decisive, and oscillations of the coil about its new position of equilibrium are checked by currents induced, in the metal frame or " former " on which the coil is wound. As the strength of the magnetic field in the air gap is very 114 PRACTICAL ELECTRICITY nearly uniform, the turning moment exerted by the coil is practically proportional to the strength of current passing through it, and, as the control springs exert a torque approximately proportional to the angle of twist, a given change in current produces the same change of deflection whatever the initial position of the coil ; consequently the divisions of the scale are uniform. Further, as the coil and pointer are carefully balanced, the ammeter can be used in any position. Fig. 73. Nalder Bros. & Thompson Moving Coil Milliampere Meter. The spiral springs which lead the current to and from the coil are of necessity kept of small cross-section, otherwise they would be too rigid, the control exerted would be excessive, and the instrument would be insensitive. This limits the strength of current, which can be led into the coil to a fraction of an ampere, for with larger currents the springs woul or by gravity acting on a weight fixed to the axis which carries the pointer. The latter form of control is commonly used in instruments which are fixed in position, whilst for portable Figs. 76 and y6a. Nalder Gravity Control Ammeter, two-thirds of full ammeters, or ammeters for use on shipboard, spring control is generally adopted. A repulsion type instrument with gravity control, as made by Messrs. Nalder Brothers & Thompson, Limited, is shown in Fig. 76, and the working parts to a larger scale in Fig. 760. The pointer P is fixed to an axle, E F, which is pivotted at its ends, and a weight w tends to hang vertically and keep the pointer at zero. To the pivotted system an iron rod, or bundle of iron wires, A B, are attached, and lie parallel to a bundle of wires, c D, fixed to the framework in which the axle is pivotted. lao PRACTICAL ELECTRICITY When an electric current passes round the coil, say in a clockwise direction, the ends A and c of the iron wires will be south seeking poles, and the ends B and D, north seeking ; the two electro- magnets will therefore repel each other with a force depending on' the strength of the current, and A B being movable, it will be pushed away from c D, thus causing the pointer to move in a counterclockwise direction. The displacement will go on until the controlling moment exerted by the weight w balances the moment due to the forces of repulsion, when the pivotted system will be in equilibrium, and the position of the pointer, if the in- strument has been properly graduated, will indicate the strength of the cur- rent passing through the coil. To pre- vent undue vibra- tion of the pointer an air dash pot, Fig. 766. Damping Device in Nalder Ammeter. P ' ^^ box-shaped vane v, are arranged as shown in Fig. j6b, in the recent form of instrument, and the resistance to motion, due to displacement of air by the movement of v, soon brings the system to rest. Reversing the current through a soft iron ammeter of the kind described above does not reverse the deflection, for although the polarity of the pieces of iron will be reversed by this change, the force between them is still one of repulsion, so that the direction of deflection of such instruments is the same whether the current passes in one direction or the other. This is true of all soft iron ammeters as of electrodynamometers. Another property common to these instruments is the nature of the scales, for when the pointer is in any given position the deflecting moment is approximately proportional to the square of the strength of current, and owing to this fact the divisions are usually crowded together near the zero and open out higher up the scale. The current which produces full deflection of the pointer of an ammeter can be varied to suit actual requirements by SOFT IRON AMMETERS 121 altering the winding of the coil. When large currents are to be measured only a small number of turns are necessary, whereas if a small current is to be determined, the number of turns on the coil must be large. In fact, with a given size of coil and given working parts, the product amperes multiplied by turns is approxi- Figs, 77 and 770;. Evcrshed Gravity Control Ammeter, two-thirds of lull size. mately constant.* Small variations can be made by altering the controlling force. Another form of soft iron ammeter is shown in Figs. 77, 770, and is made by Messrs. Evershed & Vignoles. Limited. The moving part, or needle, A B, Fig. 770, is a half cylinder of sheet iron mounted concentric with the staff s s, which is pivotted at its ends, and controlled by the weight w. The staff passes along the axis of a brass tube, x T, Fig. 77, the back end of which carries the back jewel, and around the outside of this * To this product the name ampere-turns is given. 122 PRACTICAL ELECTRICITY pig. 76. Perspective View of New Evershed and Vignoles Instrument. Fig. 7&a. Section of Evershed and Vignoles Instrument. SPRING CONTROL AMMETERS 123 tube is wrapped a triangular shaped piece of soft sheet iron shown at c D. When a current passes round the coil the needle A B moves towards the narrower part of c D against the action of the control weight w. By fixing c D to T T at different posi- tions circumferentially, the shape of the calibration curve, and therefore the nature of the scale of the instrument, can be varied to suit different requirements. For example, the scale may be one of nearly equal divisions, say from 5 of the highest Fig- 79- Hartmann and Braun Hot Wire Ammeter. reading, or may have divisions near together at each end and wide apart at some particular place. The latest form of Evershed & Vignoles' instrument is illustrated in Figs. 78 and 780. It has an oval shaped coil c c with a narrow internal cavity, c, into which a volute shaped piece of sheet iron, F F, is attracted when a current circulates in the coil. The iron is fixed on a pivotted staff s, to which the pointer P, damping vane v, and several balance weights are attached, and the movement is controlled by a spiral spring Q. Behind the scale plate is a sector shaped box, B, in which the vane v moves, and the air friction caused by air displacement effectually checks oscillations of the system without introducing errors due to solid friction. The winding of the coil is, of course, arranged to suit the current to be measured. 124 PRACTICAL ELECTRICITY 45- Hot-Wire Ammeter.- -Instead of making use of the magnetic property of electric currents, this form of instrument utilises the heating effect of a current, for its indications depend on the expansion of a wire which is heated by the passage of the current. As the expansion of metals, for mcderate changes of d temperature , is extremely ^rr---- 1 "* small, some method of mag- nifying the extension is necessary. In the instru- Fig. 80. Sagging Wire. i ' T?' this is done by aid of " sagging wires" and depends on the fact that a nearly straight wire fixed at both ends and kept taut by a force at right angles to the wire applied near the middle point, alters its sag, for a given change of length of the wire, by an amount greater than the change of length. Suppose a small force,/ (Fig. 80), to act on the nearly straight wire w w, fixed at Q and Q', the point R will be displaced a little from the straight line Q Q' ; and this displacement is called the " sag " of the wire. If now the wire w w increases in length by a small amount, /, due to heating, say, the wire will take up the position Q R' Q', the distance R R' is the change of sag due to this change of length, and is greater than /. This arrangement gives one magnification, and in the actual instrument double magnification is obtained, as shown in Fig. 8oa. Here the force / (Fig. 80) is pro- duced by the tension of a second wire, w', w', fixed at K and at- tached to w w at R, and which is kept taut by a silk thread s pass- ing round a pulley, p, and attached to a stretched spring, s, anchored at L. When a Current IS passed Fig gofl _ Diagram of Sagging Wire " Magnifying System. from Q to Q' through w w, it is heated, and expands and sags, thus causing w' w' to sag and allow the spring s to contract and turn the pivotted pulley p and pointer p in a clockwise direction. The amount of movement of the pointer depends on the extension of w w, and therefore on the heating produced by the current, so the scale over which the pointer moves can be graduated to read the current directly. HOT WIRE AMMETER 125 Fig. 79 shows an elevation of a modern form of hot wire in- strument, the lettering of which corresponds with that of Figs. 80 and Sou, but the spiral spring shown in Fig. Soa is replaced by a flat spring s in Fig. 79 The wire w w is made of platinum- iridium, and is carried by an iron plate 1 1 and a piece of nickel steel, N, fixed to 1 1. This arrangement is used to prevent changes of sag taking place when the instrument as a whole changes its temperature, for the proportions of iron and nickel steel* are chosen so that the coefficient of expansion of the combination is equal to that of the wire ww. To prevent the pointer oscillating much about its position of equilibrium, an aluminium sector, A, is attached to the axis carrying the pointer, and this passes between the poles of a permanent magnet, M. When the sector is moving in the magnetic field electric currents are produced in the metal which tend to stop the motion, and by this means the pointer is brought quickly to rest. A hot wire instrument, such as shown above, is only suited for measuring small currents, say up to about 0-2 ampere (200 milliamperes) , because the wire w w must be thin in order to be kept taut by a small side-pull. For larger currents " shunts " are required, such as described in Section 19, in order that only a fraction of the whole current passes through w w. It is also customary, in large current instruments, to arrange that two or more parts of the wire w w are electrically in parallel. Until quite recently platinum -silver was used for the working wire w w, because of its comparatively large coefficient of expan- sion, but its relatively low melting point and small tensile strength proved serious disadvantages. Platinum-iridium is far superior in both respects, and permits of the wire being safely heated to a far higher temperature, thus obtaining increased elongation, and at the same time reducing the errors caused by external changes of temperature. * Nickel steel has a very small coefficient of expansion. CHAPTER TV DIFFERENCE OF POTENTIAL, AND RESISTANCE 46. Difference of Potentials 47. Potential of the Earth Arbitrarily called Nought ; Positive and Negative Potentials 48. Measurement of Potential Difference 49. Electrometer 50. Ohm's Law 51. Resistance 52. Ohm : Unit of Resistance 53. Resistance Coils and Resistance Boxes 54. Volt 55. Ohm's Law Applicable to Complete Circuits ; E.M.F. 550. Electromagnetic Definition of E.M.F. 56. Current Method of Comparing P.Ds. 57. Reason for Using High Resistance Galvanometers for P.D. Measurements, and Low Resistance Galvanometers for Current Measurements 58. Voltmeter 59. Resistances of Ammeters and Current Voltmeters 60. Ammeters used as Voltmeters 61. Moving Coil Voltmeter 62. Calibrating a Deflectional Voltmeter 63. Voltmeters used as Ammeters 64. Gold Leaf Electroscope 65. Sensibility of Gold-Leaf Electroscopes. 46. Difference of Potentials. When a current of electricity is flowing through a wire, it has the same strength at all cross- sections of the wire. If, for example, the wire be cut anywhere and a galvanometer be put in circuit, the galvanometer will always show the same deflection while the same current is flowing ; or if several galvanometers, or ammeters, be placed at different parts of the same circuit, each instrument will be found to indicate the same current. In the same way, in the case of a water-pipe, the quantity of water passing every cross-section of the pipe per second is exactly the same as soon as the flow of water becomes steady. Just at the commencement, when, for example, some water has entered at one end of the pipe, and none has flowed out at the other when the pipe is filling in fact the flow at different cross-sections may be different ; so also, in many cases, just at the moment after completing an electric circuit, the current will differ at different cross-sections. But as soon as the flow in each case becomes a steady one this difference disappears, and the strength of the water current that is, the number of gallons of water passing per minute (not, of course, the velocity of the particles of water) is the same at all parts of the pipe, even if the pipe be broad at some points and narrow at others. In the same way the strength of the electric current 126 DIFFERENCE OF POTENTIAL OR P.O. 127 flowing through a single circuit is " imiform "* at all parts of the circuit, independently of the thickness of the conductor, and of the material of which it is made. But although the stream of water is the same at all parts of the pipe, the pressure per square inch of the water is by no means the same, even if the pipe be quite horizontal and of uniform cross- section. This pressure per square inch of the water on the pipe, which is the same as the pressure per square inch of one portion of the water on another portion adjacent to it, becomes less and less as we proceed in the direction of the flow. It is, in fact, this difference of pressures, or " loss of head " as it is some- times called, that causes the flow to take place against the friction of the pipe, the difference of pressures at any two points, in the case of a steady flow through a horizontal pipe of uniform sectional area, being balanced by the frictional resistance of that length of pipe for that particular rate of flow. Quite analogous with this, there is, in the case of an electric current flowing through a conductor, a " difference of potentials " between any two points in the conductor, and this difference of potentials, or " potential difference " (or " P.ZX" as it may be shortly called), is needed to overcome the resistance of the con- ductor, or opposition that it offers to the passage of an electric current through it. In fact the analogy between difference of potentials and difference of fluid pressures is so marked that the name " pressure " is now frequently used to stand for difference of potentials. The pressure per square inch of the water at any point of a tube conveying a stream can be ascertained by attaching a verti- cal stand-pipe to the tube, and observing to what height the water is forced up in this stand-pipe, and if at a number of points, PJ, P 2 , P 3 , P 4 , P 5 , P 6 (Fig. 81), in a glass tube, 1 1, conveying a stream of water, a series of vertical glass stand-pipes, S 1 s 2 . . . s e , be fixed, the height to which the water is forced up in them will show the distribution of pressure along the tube. If the tube it be straight and of uniform cross-section, and if the flow be a steady one, the tops of the water columns in the stand-pipes will be found to lie all in one straight line, Qi Q 2 . . Q 6 ," therefore, if the length P X P 2 of the uniform tube be equal to the length P 4 P 5 , the difference between P X Q L , the height of water in the stand-pipe Sj, and P 2 Q 2 , the height of water in the stand-pipe * Uniform refers to space, constant to time. The height of the houses in a street is generally not uniform, but it is constant so long as there is no change made in the height of the houses. If water be run out of a cistern the level at all parts of the surface of the water is uniform, but it is not constant, since it steadily falls as the water runs out. 128 PRACTICAL ELECTRICITY s 2 , is exactly equal to the difference between P 4 Q 4 and P 5 Q 5 . Also,< if the length P x P 4 be three times the length P 4 P 5 , the difference between P I Q L and P 4 Q 4 is equal to three times the difference between P 4 Q 4 and P 5 Q 5 . Or, in other words, when there is a steady flow of liquid through a uniform tube, the difference o/ pressure between any two points is proportional to the distance Fig. 81. Apparatus for Testing the Distribution of Water Pressure. between these points. And this is true whatever the inclination of the tube 1 1 to the horizontal, provided that the tube is straight and of uniform cross-section everywhere. If the tap T! and the screw pinch -cock s 1 be fully open, and the screw pinch-cock s 2 be fairly open, the stream of water through the tube 1 1 will be rapid, and the slope of pressure that is, the line Q x Q 2 .. . . Q 6 joining the tops of the columns of water in the stand-pipes will be steep. If now the pinch-cock s 2 be screwed up a little so as to impede the passage of the water, the flow will be decreased, and the slope of pressure R R 2 . . . R 6 will be less inclined to the horizontal than Qj Q 2 . . . Q 6 . As the pinch-cock s 2 is more and more screwed up, the pressure line will become more and more horizontal until, when the flow is entirely checked, the line H x H 2 . . . H 6 joining the tops of the columns of water in the stand-pipes becomes quite hori- zontal and at the same level as the water in the cistern c x . From this we see that the pressure is the same at all points along the horizontal pipe P X P 6 , through which no flow is taking place, so there is no difference of pressure between any two points WATER ANALOGY OF ELECTRIC FLOW 129 along the pipe ; and as " difference of potential " is analogous to fluid pressure, we conclude that there is no P.D. between points in an electrical conductor through which no current is passing, or, in other words, " all points of an electrical conductor on which electricity is at rest are at the same potential." It will be noted that if there be any flow, the level of the water in the first stand-pipe Sj is less than that in the cistern itself, which is seen through a little glass window at the right of the cistern Cj. This is on account of the resistance offered to the flow by the tap TJ and by the indiarubber tube T X t. Similarly, if the pinch-cock s l be screwed up so as to check the flow between P 3 and P 4 , there will be a sudden drop in pressure between P 3 and P 4 , so that the tops of the water columns in the standpipes will now be in two different straight lines, u x U 8 U 3 and U 4 U 6 U 6 , parallel to one another, but the latter U 4 u, U 6 , much lower than the former. As the pinch-cock s is screwed up more and more the lines u x U 2 U 3 and U 4 u s U 6 will become more and more horizontal, but at a greater distance from one another, until, when s x is entirely closed, the former line will coincide with H x H 2 H 3 , while the latter will sink down to the level of the tube P 4 P 6 P 6 itself. In a very similar way the " electric pressure " (or " potential" as it is usually called) at different points of a wire conveying a current, can be measured by apparatus which we shall presently describe, and if a number of measurements be made of the potential at different points of a circuit conveying a current, it will be found that the results are smaller and smaller as we proceed in one direction ; and, further, if the conductor be all of uniform gauge and material, and the electric current be a steady one, it will be found that the P.D. between any two points is proportional to the length of the conductor between these points (see Section 49). f Electricity is put in motion, and a current of electricity is produced, as a consequence of the potential varying from place to place, just as a current is produced in water when subjected to pressures which are not uniform. In order to produce and maintain a current of either water, or electricity, work of some kind has to be done. Thus in Fig. 81 the current of water in the tube 1 1 will gradually diminish as the water passes from the upper to the lower reservoir, and will cease entirely as soon as the reservoir q is empty. In order to maintain the current it is necessary to provide some means of keeping up the level of water in the upper reservoir, and the simplest method of doing so is by means of a pump working at such a rate that water is raised from I 130 PRACTICAL ELECTRICITY the lower vessel C 2 to the upper one c t just as fast as it flows from G! to C 2 through the tube 1 1. Exactly analogous with this pump in the water circuit is the " voltaic cell," or the " dynamo machine," or other " current generator," in the electric circuit. A current generator does not create electricity any more than a fire-engine Supply Overflow Fig. 82. to -Alternative Arrangement of the Cistern for the Apparatus in Fig. 81. creates water, it merely sets it in motion, and in either case work has to be done in keeping up the flow (see Chap- ter VII. on Electric Energy and Power). In the case of the water flow we may commence by filling the reservoir c and maintain the level by allowing water to flow from the cistern of the building into the reservoir c x as fast as it flows out. Or, to save trouble, we may let the water run into the reservoir rather faster than it flows out through the tube tt, and allow the surplus to flow out through an 'overflow pipe o (Fig. 82). With the latter arrange- ment the level of the water in the reservoir c x will remain auto- matically constant whatever be the flow through the tube 1 1, provided, of course, that the tap T 2 be opened enough to cause the flow from the house cistern into the reservoir to be never less than the flow out through the tube 1 1. If the substance flowing were a gas, the distribution of pressure could not be measured by stand- pipes, since if the pipes were open at the top, the gas would flow out ; or the outside air would flow in, and, if the pipes were closed they would all be filled with the gas itself, or with a mixture of gas and air. The distribution of pressure along a pipe, p p (Fig. 83), convey- ing a stream of gas might be measured relatively to the outside atmospheric pressure by means of " manometers" M lf M 2 , M 3 attached at the points P 1 ,-P 2 , P 3 of the pipe, the difference of level of the liquid on the two parts of the tube of each manometer Fig. 83. Apparatus for Testing the Distribution of Gas Pressure relatively to the Atmospheric Pressure. DISTRIBUTION OF GAS PRESSURE 131 measuring the excess of the pressure of- the gas at that part of the pipe over the atmospheric pressure. Or, if we desired that our measurements should be independent of the atmospheric pressure and merely indicate the pressure at various parts of the pipe relatively to the pressure at one point P 4 , then the manometers might be arranged as in Fig. 84, in which case the " difference of level of the liquid in the curved tube of any one manometer, M 2 (say), would show how much the nrP^lirf 1 of thp P~a; Fig- 84. Apparatus for Testing the Distribution of Gas Pressure relatively to the Pressure at One Point of the Pipe. at the point P 2 of the pipe p p exceeded the pressure at the point P 4 . Perhaps the most convenient way would be to construct the apparatus as seen in Fig. 85, since then the pressure of the gas at any point P 2 relatively to the pressure at P 4 would be at once seen from the distance the top of the column of liquid in the tube M 2 (Fig. 85) was below the horizontal line H H ; and the difference of pressure of the gas at any two points P 2 and P 3 would be therefore measured by the difference in the depths below the horizontal line H H of the tops of the liquid columns in the manometers M 2 and M 3 . If the pressure of the atmosphere surrounding the apparatus in Fig. 83 were changed, then, although the flow of gas along the Fig. 85. Simpler Apparatus for Testing the Distribution of Gas Pressure relatively to the Pressure at One Point of the Pipe. pipe p p might remain exactly the same, as well as the pressures at its two ends, the difference of level of the liquid in each of the manometers in this figure would change. But the level of the 132 PRACTICAL ELECTRICITY liquid in the manometers in Fig. 85 is wholly independent of the outside atmospheric pressure, and depends solely on the length, cross-section, shape, and internal character of the pipe p p, on the rate of flow, and on the nature of the fluid flowing through the pipe. These manometers tell us nothing about the absolute pressure of the gas at the different points of the pipe through which it is flowing, but only the pressures relatively to the pres- sure at the point P 4 . 47. Potential of the Earth arbitrarily called Nought ; Positive and Negative Potentials. So in the same way the electric po- tential of a point in a wire through which a current is flowing is usually measured relatively to that of some other point of the wire. And even when one point of the wire is connected with the earth and the potentials of different points of the wire are measured above or below the potential of the earth, which is arbitrarily called nought, it is still but a relative measurement, for in thus taking the potential of the earth as the potential level to measure from, no assumption is made as to the earth having no electricity on it. Measuring electrical potentials relatively to that of the earth is, therefore, like measuring heights above the Trinity water- mark, or measuring longitude east or west of Greenwich, the place which is arbitrarily said to have zero longitude. A similar convention is followed in the measurement of tem- perature, for in the centigrade scale the temperature of melting ice is called o, while in the Fahrenheit scale the zero is a tem- perature much below this, and one which is roughly that of a mixture of ice and salt. Now, although Fahrenheit is said to have called this temperature zero because he had an idea that it was the lowest temperature that could be produced artificially, no such assumption is at present made in calling this particular temperature o F. In addition to a P.D. being said to exist between two points in a conductor through which a current is flowing, any two con- ductors are said to differ in potential when there is a tendency for electricity to pass from one of them to the other, just as the contents of two gas-holders would be said to differ in pressure if a tendency for the fluid to pass from one to the other existed, or the tendency for the fluid to pass into the atmosphere was different in the case of one, from what it was in the other. This tendency may manifest itself in four ways : (i) By the production of a current (lasting, it may be, for only the fraction of a second) when the two conductors are touched together, or when they are electrically connected by means of a wire, or other conductor ; EFFECTS OF POTENTIAL DIFFERENCE 133 (2) By a " brush discharge " or an " electric spark " passing between the conductors when they are near together, and when the P.D. between them is high ; (3) By small light bodies, such as grains of dust, pieces of paper, pith, etc., being attracted backwards and forwards between the conductors ; (4) By the conductors trying to approach one another, as if there were an attraction between them. When different pieces of electrical apparatus are enclosed in a metallic box (a not infrequent arrangement) the potential of the box itself is usually called nought, and the potentials of the different bodies inside it are measured relatively to that of the box by the methods subsequently described. This box in such a case is sometimes called, in electrical language, the " earth," but it must not, therefore, be inferred that there is any metallic connection between the box and the ground ; the box and all the apparatus inside it might, indeed, be up in a balloon, and still the joining of some part of the internal apparatus to the metallic box by wire might be called " earth- ing " that piece of apparatus. A conductor is said to have a " positive potential " when on earthing the conductor a current flows from the conductor to the earth, and a "negative potential" when the current flows in the opposite direction. Also when two conductors, A and B, are in such a condition that, if joined with a con- ductor, a current would flow from A to B, then, irrespectively of the actual signs of the potentials of A and B, as denned in the last sentence, the potential of A is said to be " higher " than that of B ; (see Section n for the definition of the direction of a current). Further, if two bodies, whether conductors or not, differ in potential, a positively electrified body, placed in their neighbourhood, tends to move away from the body having the higher potential towards the other body having the lower potential. 48. Measurement of Potential Difference. The question now arises, How are we to measure potential differences ? i.e., How are we to determine whether one P.D. is two or three times another P.D. ? This may be answered in the same way as the similar question discussed in Section 7, which relates to the measurement of current strength, for the magnitude of one of the effects exhibited by bodies at different potentials may be chosen as a measure of the P.D. between them. But as electrical phenomena are manifestations of energy (which, according to the law of conservation of energy cannot be created or destroyed, but only changed in form), it is desirable that the choice be such 134 PRACTICAL ELECTRICITY as will make the relations between the electrical quantities and the mechanical quantity Energy (or work) as simple as possible ; for this purpose the property of attraction (Chapter II., page 81) is the most convenient one to choose. If a quantity of water of volume Q flows through a uniform horizontal pipe, A c B, Fig. 850, the energy lost by the fluid as it passes from A to B (or the work done in overcoming friction in the intervening portion of the pipe), is equal to Q multiplied by the difference of A! ^ C_| B 1 pressure between A Fig. 85*. and B. A current of electricity can also do work, as is shown by the ventilating fan, Fig. 9, shown being driven electrically, and by the heating effect of a current, and the analogy between hydraulic and electrical work will be preserved if the measure of P.D. is such that the work done by a quantity of electricity q passing from a point A' to a point B' in a wire A' c' B' is equal to the product of q and the potential difference between A' and B'. This requirement is satisfied if we take as our definition the following : The force of attraction between two bodies in definite relative positions, but at different potentials, is proportional to the square of the P.D., or in other words, the potential difference between two bodies in definite relative positions is proportional to the square root of the force of attrac- tion between them. A similar definition might have been chosen for current strength, for as already shown in Section 39, the force between two coils carrying the same current is proportional to the square of the strength of the current, and therefore the current strength is proportional to the square root of the force. In the electrodynamometer, altering the current strength alters the magnetic condition of both of the coils in the same ratio, and in consequence alters the force between them in the duplicate ratio ; so also in an instrument for measuring the force of attraction between two conductors at different potentials, altering the P.D. between them alters the electrical condition of both, and thus changes the force between them in a duplicate ratio. 49. Electrometer. The forces between conductors at dif- ferent potentials are called electrostatic forces, because they are believed to be due to quantities of electricity at rest on the surfaces of the conductors, and instruments for measuring these forces are called " electrometers" ELECTROMETERS 135 Electrometers, like galvanometers, are of two kinds, those in which the measurement is made by noting how much a needle is deflected against the action of a controlling force, and those in which we observe by how much the controlling force must be increased to resist the motion of the needle and keep it in a fixed position. The latter or zero type of electrometer has an advan- tage over the former, in that it enables the simple definition of the Fig. 86. Ayrton and Mather's Zero Electrometer, or Zero Electrostatic Voltmeter, one-third of the full size. measurement of difference of potential given above to be made use of in practice. The electrostatic forces between bodies at different potentials are very small in magnitude, unless the potential differences are very large, and it is only within comparatively recent years that instruments for measuring the forces produced by P.Ds., such as are used for electric lighting in houses and for telegraphic purposes, have been constructed. A zero electrometer devised by the authors is shown in Figs. 86 and S6a ; the moving part N, or needle* as it is called, takes the form of two thin narrow A magnetised sewing-needle having been originally used for the suspended magnet in a galvanometer, the name needle came gradually to designate the little magnet in a galvanometer, whether it was long 136 PRACTICAL ELECTRICITY pieces of aluminium a, a (Figs. 86 and S6a), joined together at the top and bottom by cross pieces, b, b, and supported by means of a thin strip of phosphor bronze from a head H, carrying an index c, which can be turned round over a graduated dial. The conductors, 1 1, or the " inductors " as they are called, into which the two parts a, a of the needle are attracted, are shaped as shown, and, by means of a pointer p, carried from the bottom of the needle, the position of the needle can be observed. As usual, parallax is avoided by observing the reflection of this pointer p in a piece of looking-glass g fixed to the base of the instrument. Any P.D. set up between the needle and the inductors is then measured by turning the head H until the pointer p (carried by the needle) is brought into the same position that it occupied when the needle and the inductors had the same potential ; the angle through which the index c has been turned is noted, and its square root taken. For this is the angle through which the strip carrying the needle has been twisted, and, therefore, this angle measures the moment of the force, or the torque, that has been exerted on the needle. The terminals T X T 2 are connected respectively with the needle N and the inductors 1 1, and equality of potential of these two bodies can be secured by connecting these terminals together with a piece of wire, thick or thin. For if there be any difference of potential, a momentary current will flow through this wire which will annihilate the P.D. Further, il the terminals be joined respectively by wires with any two conductors A and B, momentary currents will flow, and the potentials of the needle and inductors will become respectively the same as those of A and B. In fact, we may say generally, that if any number of conductors be touched together, or be joined by wires, and if no current be flowing between any of the bodies, the conductors and wires are all at the same potential. To be strictly correct, this general proposition requires that all the conductors should be made of the same material, and be at the same temperature. and pointed like a sewing-needle, or short and blunt. And now the expression needle is employed for the suspended movable part of an electrical measuring instrument, even when the shape of the moving system in no way resembles that of a sewing-needle, as in the electro- meter shown in Fig. 86, Fig. 86. Details of Needle and Inductors, rather larger than full size. ZERO ELECTROSTATIC VOLTMETER This last proposition can be stated briefly and completely thus : the potential of all parts of a conducting system composed of the same material at the same temperature and on which electricity is at rest, is uniform. In order to ensure that the electric force exerted on the needle shall be wholly due to the P.D. between it and the inductors, and that no part of this force shall be caused by the attraction of external bodies, the in- terior of the glass shade is coated with a conduct- ing transparent varnish devised by the authors, the composition and action of which are ex- plained later in Section 6 4 . The spindle of the needle in the electrometer (Fig. 86) moves in guides top and bottom, the upper guide being clearly seen in Fig. 860, which shows the top of the needle and of the inductors rather larger than full size ; hence the instrument may be turned upside down, or carried about without its being necessary to clamp the needle, and without there being much risk of breaking the thin phosphor - bronze strip supporting it. If, in addition to sending a steady stream of water through the tube it, shown in Fig. 81, the water in the tube be now used as a conductor and a steady electric current b^ sent through it, the various P.Ds. between the pairs of points P! and P 2 , P 2 and P 3 , etc., can easily be measured with the elec- trometer just described by simply dipping wires, attached respectively to the terminals of the electrometer, into the water in the various pairs of standpipes s x and S 2 , S 2 and S 3 , etc. For, since there is no electric current in the water in a stand-pipe itself, there can be no P.Ds. between the different Fig. 866. Ayrton and Mather's Zero Electrostatic Voltmeter (Later Form). 138 PRACTICAL ELECTRICITY parts of the water in the same stand-pipe ; hence the water in the stand-pipes can be used simply as extensions of the wires attached to the terminals of the electrometer. When the screw pinch-cock s 1 is fully open, so that the tube 1 1 is throughout of uniform bore, it will be found that the P.Ds. between the different pairs of points are related to one another in exactly the same way as are the differences between the water pressures for the same pairs of points. Thus the distribution of potential along a uniform conductor conveying a steady electric current is exactly analogous with the distribution of fluid pressure along a uniform tube, through which flows a steady stream of liquid. 50. Ohm's Law. But if instead of measuring the P.D.'s between different points along a conductor through which flows a steady current we measure the P.D.'s between two fixed points in a given conductor through which different currents are flowing, then the P.D. does not vary with the current in the same way that the difference of pressure between two points in a given tube varies with the stream of fluid flowing through it. Let us consider the second case first : Keep the level of water in the reservoir c x (Fig. 81) constant in the way already described, open the screw pinch-cock s 2 a certain amount, the screw pinch-cock s 1 being fully open, and, when the stream has become steady, measure with a graduated glass the number of cubic centi- metres of water that flow through the tube tt per second, also the difference of pressure between two fixed points in the tube P X and P 6 for example. Next open the pinch-cock s 2 a little more, and again measure the number of cubic centimetres of water per second that flow out of the tube, as well as the difference between the height of the water in the stand-pipes s l and s 6 . If such measurements be made for several different steady rates of flow, numbers like the following will be obtained, and when plotted they give the curve seen in Fig. 87, concave to the axis along which difference of level is plotted. Difference of Level Flow in Cubic Ratio of Difference in Centimetres. Centimetres per second. of Level to Flow. 6-9 I-2O 5-75 12-4 , -" 2'OO 6-20 18-7 2-78 6-73 24-0 3'39 7-08 29'5 4-09 7-21 . 36-2 4-76 7*60 4*i 5'26 8-00 FLUID AND ELECTRIC FLOW DISSIMILAR 139 If the numbers in the third column were all the same it would tell us that the ratio of the difference of level to the number of cubic centimetres flowing per second that is, the ratio of pressure to current was a constant for a given pipe. In that case the points on the curve in Fig. 87 would all lie in one straight line, and to double, treble, quadruple the current would require Curve connecting Rate oj Flow of > Water with Loss of Head. V. 5 | 4 | 3 1 3 2, 5 10 15 5 50 55 40 Difference, of level in centimetres Fig. 87. 49 exactly double, treble, quadruple the pressure. But the numbers in the third column steadily increase as the current increases, and if we examine the numbers in the first two columns we find that to increase, for example, the flow from i'20 to 4-76 cubic centimetres per second that is, to make the current not quite four times as great r-we have to increase the difference of level from 6-9 to 36-2 centimetres that is, to increase the pressure more than five times. The quantity of water, therefore, that flows per second through a given pipe does not increase as rapidly as the difference of pressure between two fixed points in it, or in other words, we must more than double, treble the difference of pressure to produce twice, three times the flow, even although the tube through which the water flows remains absolutely unchanged. It might, therefore, have been expected that the same sort of inequality would be found in the ratio connecting the P.D. between two fixed points in a conductor and the current flowing through it. 140 PRACTICAL ELECTRICITY But that is not the case, for it the conductor K (Fi~s. 88 and 880) remains at the same temperature, and be not changed in any way, experiment shows that the P.D. between two fixed points, KJ, K 2 in it, measured by the electrometer E (in the way already described in Section 49), is directly proportional to the current flowing through this conductor, the currents being measured relatively to one another by any suitable galvano- meter G,* for which the law connecting current and deflection Fig. 88. Apparatus for Testing Ohm's Law. has been obtained by a relative calibration, as described in Section 19. For carrying out these tests the current can be conveniently produced with a battery, B B, of what are known as " dry cells " or of " accumulators " (for both of which see later Sections) ; and its strength can be varied by altering the number of cells employed. This alteration in the number of cells that are used in the different tests, .can be easily effected by means of the mercury switch -board, s s, seen in front of the battery of cells in Fig. 88. * For the details of the construction of the galvanometer illustrated in Fig. 88, see Section 43. A zero electrodynamometer (Fig. 886) may with advantage be sub- stituted for the galvanometer G, for then the current will be propor- lional to the square root of the reading of the dynamometer just as the P.D. is proportional to the square root of the reading of the zero electro- meter. With these instruments, as their laws are known from first principles, no preliminary calibration is necessary. VERIFICATION OF OHM'S LAW J4T o ( 00 oo -M .bp *5 -2 .2 *! 13 i ^ o o o o o o 6666 CO 6 3 *- i + 1 + + 1 * > -d 5 I 1 1 s g tO H CO O O" O ON ON O^ O O\ ON IX ; HH en 6 H 6 6 6 ' ^ C3 o S 1 ' PH ^~* CO H OO (X d |l or V = IR lt and E = I (R l + # 2 )> = IRi + IR 2 , = V +IR 2 . (16) Hence the E.M.F. of the battery is equal to the potential difference between its terminals, plus the product of the current passing and the internal resistance of the battery. If the current be stopped by interrupting the circuit outside the battery, / will be zero and the product I R 2 = o, so under these conditions E = V, or, in words, the E.M.F. of a battery is equal to the potential difference between its terminals when no current is passing through it. 55. Electromagnetic Definition of E.M.F. E.M.Fs. (or P.Ds.) are generally regarded as the causes of electric currents flowing in complete circuits, or the causes of the tendency for flow to take place in incomplete circuits. There are several ways, other than by batteries, of producing E.M.Fs., chief amongst these being the cutting of lines of force by conductors, which forms the basis of a definition of E.M.F. much used in practice, viz., electromagnetic E.M.F. is measured by the rate of cutting of lines of magnetic force, or, in other words, unit E.M.F. is generated in a conductor when it cuts lines of magnetic force at the rate of one line per second. That the above definition is consistent with that of P.D. previously given in Section 48, may be seen from the following considerations. The absolute unit of current was defined as that current which flowing through a conductor of unit length bent into an arc of unit radius exerts unit force on unit pole at the centre (see Section 8). We have also shown (Section 24) that at unit distance from unit pole the strength of mag- netic field is unity, and that the density of magnetic lines over a surface perpendicular to their direction will be one line per square centimetre. Now, by Newton's third law of motion, "action and reaction are equal and opposite," so that if the conductor carrying the current exerts a force of one 152 PRACTICAL ELECTRICITY dyne on the unit pole, the unit pole will exert an equal force on the conductor. The conductor will therefore be subjected to a force of one dyne in a direction perpendicular to the plane containing the conductor and the pole, and the work done in moving the conductor through a mean distance of one centimetre against this force will be one erg. At the same time the con- ductor will have swept over an area of one square centimetre of the surface of a sphere of unit radius surrounding the pole, and will therefore have cut an amount of magnetic flux represented by one line of force.* As the force acting, and therefore the work done, will be proportional to the current flowing, and to the distance the conductor moves, we conclude that the work is proportional to the product of the current and the number of lines of force cut by the conductor, and if suitable units be taken we may write W = !'$, where W is the work done in ergs, /'the current in C.G.S. units, and $ the number of lines cut. What becomes of the work done in moving the conductor ? Experiment shows that the current in the circuit is slightly increased whilst the movement is taking place and additional heat is produced in the circuit equivalent to the work done. This change of current must be due to some cause, and as the only change made in the circuit is the movement of the con- ductor, we attribute the change of current to this movement, and say that the movement generates an E.M.F. The equation W= I'3> may be written $ W= I't - . ' -* : , - . in which Q = I't, is the quantity passing in time t ; and in order that the analogy between electrical and hydraulic work mentioned in Section 48 may be maintained, viz., work = quantity x pressure, $ the pressure must be represented by . Hence the pressure < (or E.M.F., as it is called in this case) is equal to , i.e., equal $ to the rate of cutting lines of force. Writing = E we have W = I'tE or Y =rE ' (I7) * Since the number of lines of force emanating from unit pole is 417-, Section 24, and the area of the surface of a sohere of unit radius is also 471-. COMPARING P.Ds. AND RESISTANCES 153 from which we see that the rate at Mich work is done in an electrical circuit is equal to the product of the current and the pressure. If the rate of working in a circuit in which unit (C.G.S.) current is flowing be one erg per second, the pressure (or E.M.F.) must be unity in C.G.S. measure. This condition therefore fixes the magnitude of the C.G.S. electromagnetic unit of E.M.F. The magnitude so derived is far too small for practical purposes, so the unit adopted in practice (the volt) is one hundred million times as large as the C.G.S. unit, or 1 volt = 10 8 C.G.S. electromagnetic units of E.M.F., so an E.M.F. of one volt is produced when io 8 lines of force are cut per second. In the foregoing reasoning we have considered unit length of conductor and unit magnetic field, but it will be understood that in a given magnetic field the force exerted on a conductor lying perpendicular to the field will be proportional to the length of the conductor, and also to the strength of the field as well as propor- tional to the strength of the current flowing in the conductor. The number of lines of force cut during a given movement in a uniform field will also be proportional to the length of the con- ductor and to the strength of the magnetic field so that the equation W = I' $ (18) is true for any length of conductor and any magnetic field, all the units being C.G.S. units. It is of fundamental importance in electrical engineering. 56. Current Method of Comparing P.Ds. and Resistances. From Ohm's law it follows that the current flowing through anj ? conductor at constant temperature is directly proportional to the P.D. between its terminals. Such a conductor may be a coil of a galvanometer, or it may consist of a galvanometer G together with a wire w (Fig. 90) in series with it. And no matter how the shape of the circuit composed of G and w may be altered, pro- vided that the joint re- sistance of G and w together is not Changed, Hg, 90. Galvanometer with Added Resistance for ., Measuring Potential Differences. the current passing through the galvanometer will be directly proportional to the P.D. which is maintained between T X and T 2 , the terminals of the arrangement. If then the galvanometer has been calibrated relatively for current, it is calibrated for the relative measurements 154 PRACTICAL ELECTRICITY of any P.D. which may be set up between TJ and T 2 by con* necting the terminals with any conductors, or points in the same conductor, between which a P.D. exists. In place then of employing the zero electrometer (Fig. 86), we may use the combination of galvanometer and auxiliary resistance w to compare, for example, the P.D. between the points A and B (Fig. 91) with the P.D. between the points c and D in the conductor A B c D conveying a steady current. For the P.Ds. in question will be simply proportional to the two currents that flow Fig- 91. through the galvanometer when the terminals TJ, T 2 (Fig. 90) are connected respectively first with the points A and B and then with the points c and D, provided these P.Ds. are not altered by the points being connected with T 1 and T 2 . Further, since the resistance of a conductor is the name given to the ratio of the P.D. between its ends to the current that flows through it, and, since the current that flows through A B is necessarily the same as that flowing through c D, when arranged as shown, it follows that resistance of A B _ potential difference between A and B resistance of c D potential difference between c and D therefore resistance of A B_current when T t and T 2 are joined to A and B resistance of c D~~current when TJ and T 2 are joined to c and D' the current in each case being the current through the galvano- metric arrangement (Fig. 90), the above proviso being understood. Consequently, if the value of one of the resistances A B or c D be known in international ohms, the value of the other in inter- national ohms can be at once found by the method of testing just described. 57. Reason for Using High Resistance Galvanometers for P.D. Measurements, and Low Resistance Galvanometers for Current Measurements. When using a galvanometer for the comparison of two P.Ds., or for the comparison of two resistances by the method described in Section 56, it is not necessary that the galvanometer should be calibrated absolutely in amperes, for, as we have just seen, all that is required to be known is the ratio of the currents that produce different deflections, not the actual value of these currents in amperes. But there is one condition in connection with the galvanometric arrangement G w P.O. AND CURRENT MEASUREMENTS 155 (Fig. 90), that it is most important to fulfil, viz., that the applica- tion of the terminals T p T 2 to the points A and B or to the points c and D shall not alter the distribution of potential that previously existed in the conductor A B c D. In fact, the test must not alter the thing tested, an all-important rule to remember in experi- menting. Whenever a galvanometer, properly constructed and calibrated, is introduced into any circuit the galvanometer measures the current flowing after the galvanometer has been inserted, but this is not necessarily the same as the current that flowed before the galvanometer was inserted. These two currents will only be the same in value when the resistance of the galvanometer is small compared with that of the rest of the circuit, and when the other conditions remain unchanged. It will, therefore, be only under these special circumstances that the deflection of a galvano- meter will measure the current that passed through the circuit before the circuit was disturbed by the insertion of the galvano- meter into it. Similarly, whatever be the resistance, small or large, of a galvanometer, or of a galvanometric arrangement G w (Fig. 90) , provided that this resistance remains quite constant, the relative P.Ds. between two pairs of points A and B, c and D, can be accurate!} 7 compared by means of this galvanometric arrangement ; only it must be carefully remembered that the P.Ds. that are thus compared are the values existing after the joining of the terminals T I , T 2 , to the points A and B, or to the points c and D, and not the values of these P.Ds. before the application of the measuring instrument. And it will be only when the resistance of G and w combined is very large compared with the resistance of the conductor A B, and also with the resistance of the conductor c D, that the application of the galvanometer will produce no appreciable disturbance in the distribution of poten- tial along the conductor A B c D. Therefore for P.D. measurement it is desirable that the gal- vanometer G and the auxiliary conductor w should together have a high resistance, and that the required sensibility of the galvanometer should be attained by winding the galvanometer with a large number of convolutions of fine wire* 58. Voltmeter. A " voltmeter " is an instrument which enables the P.D. between its terminals to be read off directly in volts. Whether the voltmeter be of the electrostatic type and its action depend on the attraction of electrified bodies, or whether it be * Fine wire should be used in order that a large number of turns may be put near the needle. 156 PRACTICAL ELECTRICITY of the galvanometer form and the P.D. be indirectly measured by the current it produces through a fixed resistance, it is obviously necessary that the sensibility of the instrument should not be affected by moving it from place to place. In fact, a voltmeter must possess the constancy of an ammeter, with the addition that its resistance must be constant, and any ammeter of practically constant resistance when graduated to indicate the P.D. between its terminals in international volts instead of the current passing through it in amperes, becomes a voltmeter. The electrometer described and illustrated in Section 49 gives the same reading for the same P.D. between its terminals if the instrument be levelled each time after being moved. Its relative calibration is, of course, known, since our fundamental definition of the relative value of P.Ds. is based on the use of this electrometer. If, then, we ascertain the angle a lf through which the index c has to be turned to bring the pointer p to the zero position when a known P.D.* say V^ international volts, is*set up between the terminals T l and T 2 of the instrument, the P.D. in international volts F 2 corresponding with any other angle a 2 , through which the index c must be turned to bring p to zero is known from the equation or F 2 = yi i being a constant for the particular instrument. A P.D. whose value is known in international volts can be applied to the terminals x lt T 2 of the electrometer (and so the constant ^ can be experimentally found) by connecting T A and Va, T 2 to the ends of a conductor, c (Fig. 94), whose resistance R in international ohms has been ascertained, and through which flows a current of / amperes, as measured by the ammeter A. For this P.D. is equal to / x R international volts. * A.P.D. of known value may be obtained by passing a current whose strength is measured by an ammeter through a resistance, whose value in terms of the international ohm can be determined by the method of Section 56. ELECTROSTATIC VOLTMETER The constant ^is about 2*37 for the zero electrometer illus- Vaj trated in Fig. 86, and that is to say that the index c has to be turned through about 360 to bring the pointer p to zero when a P.D. of 45 volts is maintained between the terminals of this instrument. The dial at the top of the electrometer is initially graduated in degrees or other divisions of equal value. But after the constant of the instru- ment has been experi- mentally determined, in the way just described, this degree scale may conveniently be replaced by one graduated in square roots with which the P.D. can be read off directly in international volts. The electrometer then becomes a direct- reading " electrostatic volt- meter " of the zero type. Another form of electro- static voltmeter intended for measuring compara- tively small P.Ds. is shown in Fig. 92. The instrument is of the de- flectional reflecting type. Instead of the needle being brought back to the zero position before taking a reading as described in Section 49, it is allowed to deflect until the torsion of the fine wire suspending the needle balances the attraction between the needle and the inductors ; the magnitude of the deflection is measured by the movement of a beam of light reflected from a small mirror attached to the needle, which forms a " spot " on a fixed scale. This latter device enables very small deflections to be measured, for the reflected beam turns through an angle equal to double the angular move- ment of the needle, and acts like a massless pointer of length equal to twice the distance between the mirror and scale. Fig. 92. Ayrton and Mather's Reflecting Electrostatic Voltmeter (case removed). i5 PRACTICAL ELECTRICITY On this account reflecting instruments are used in many kinds of delicate measurements. 59. Resistances of Ammeters and Current Voltmeters. From what has been said in Section 57 it will be understood that the resistance of an ammeter should be small compared with that of the rest of the circuit in which it is to be used, and that the resistance of a voltmeter should be large, compared with that of the circuit on which it is employed. The magnitudes of the resistances, however, are purely relative ; for use on a circuit of very low resistance a voltmeter of only a few ohms would be quite suitable, whereas for high resistance circuits, instruments having many thousands of ohms resistance would be necessary to measure P.Ds. with reasonable accuracy. Similarly, ammeters for measuring currents flowing in high resistance circuits, such as long telegraph lines, may be many ohms in resistance, and yet not cause much change of current when inserted, whilst ammeters for use in low voltage circuits conveying large currents (hundreds or thousands of amperes) must have extremely low resistance, only a few millionths of an ohm, if their insertion into the circuit is not to change the current appreciably. 60. Ammeters used as Voltmeters. If an ammeter with its scale graduated in volts instead of (or in addition to) its being graduated in amperes has a low resistance, it will be suitable for measuring any small P.D. that may exist between two points separated by a very small resistance. For example, it may be used to measure the P.D. between two points near together in a thick copper electric -light main through which a current is flowing, or to measure the P.D. between the terminals of a gal- vanic cell of very low internal resistance. On the contrary, if the resistance of the instrument alone, or the resistance of the instrument and its auxiliary wire w, combined, (Fig. 90), be high, it may be used to test a larger P.D. between two points separated by a larger resistance ; for example, the P.D. between the positive and negative electric -light mains in a house. Beginners sometimes feel mystified that the same instrument is sometimes employed to measure a current and at other times a P.D. ; that in the former case, when it is called an ammeter, it may be "short-circuited" with impunity, but must not be disconnected, whereas when it is called a voltmeter it may be disconnected but on no account may it be short-circuited. c The difference arises not from any intrinsic dissimilarity between an ammeter and a current voltmeter, but from the differ- ent ways in which the instruments are employed, An ammeter AMMETERS USED AS VOLTMETERS 159 is put into the main circuit in series with, the rest of the apparatus, as is the galvanometer shown at G in Fig. 88, and the ammeter A in Fig. 94, whereas a voltmeter is placed as a branch circuit in parallel with the part of the circuit, the P.D. between the terminals of which is to be measured ; for example, the zero electrostatic voltmeter E in Figs. 88 and S8a, and the voltmeter v in Fig. 94. If the voltmeter be of the current type, then both it and the ammeter simply measure a current directly, but the current that the instrument G in Fig. 88 and A in Fig. 94 measures is the current flowing through the main conductor, K in Fig. 88 and c in Fig. 94 respectively, whereas the current that the voltmeter v, Fig. 94, measures is the current that the P.D. between the terminals of the main conductor c will send through a resistance which is quite external to the main circuit, viz. the resistance of the voltmeter itself. If the resistance of an ammeter be but a small fraction of the resistance of the rest of the circuit in which it is placed, the cnly result of short-circuiting the ammeter by bridging its terminals with a short piece of thick wire, is to electrically remove the instrument from the circuit, for the current remains practically unchanged in strength, and practically the whole of it now passes through the short circuit : whereas in short-circuiting a voltmeter we short-circuit all that part of the circuit with the terminals of which the voltmeter is connected, and thus cause a great, and possibly a dangerous, increase in the current in the remainder of the circuit. For example, the short-circuiting of an ammeter which is used to measure the electric -light current passing through a house will simply cut this particular ammeter out of circuit, whereas short-circuiting the voltmeter, which is placed across the house mains for measuring the P.D. supplied to the house, would momentarily extinguish all the lamps in the neighbourhood and compel the electric current-generating-station to produce an enormous current. Almost instantaneously either the piece of wire used to make the short circuit would itself be burnt up, or one of the "fuses," the name given to the pieces of easily-fusible metal placed in the circuit to diminish the damage caused by such accidents, would itself be volatilised by the excessive current. On the other hand, disconnecting one or both of the voltmeter wires from the main circuit stops, of course, the current through the voltmeter itself, but produces practically no effect on the main current, whereas disconnecting the ammeter stops the main current altogether, unless the ammeter has been short- circuited before being disconnected. 160 PRACTICAL ELECTRICITY 61. Moving Coil Voltmeter. The moving coil ammeter, de- scribed in Section 43, lends itself extremely well for use as a portable voltmeter in consequence of its freedom from out- side magnetic disturbance, its quickness of action, its capability of being used in any position, and its great sensibility, so that the resistance of the coil and of the auxiliary wire w (Fig. 90) combined, can be very high. Indeed, in some moving coil voltmeters, intended to measure a maximum P.D. of about 140 volts, the resistance of the moving coil is about 100 ohms, and that of the auxiliary stationary wire about 16,000 ohms, which is a resistance far higher than that of any other type of volt- meter of the same range and quickness of action. The instru- ment, however, can only be employed to measure small currents, which is a disadvantage when it is desired to use it directly as an ammeter, but this becomes an advantage when the instrument is used as a voltmeter, since the smaller the current taken by a voltmeter, other things being equal, the better the voltmeter, for the smaller is the disturbance of the circuit caused by applying the voltmeter. 62. Calibrating a Deflectional Voltmeter. If the law of the instrument be unknown as well as the P.D. in volts that pro- duces any particular deflection, we can calibrate the instrument throughout the scale in volts in one or other of five distinct ways.* 1. Place the voltmeter v to be calibrated in parallel with a zero electrostatic voltmeter E and apply different P.Ds. between the common terminals of the two instruments. Measure each P.D. in international volts by means of the electrostatic volt- meter and observe the corresponding deflection on the deflectional voltmeter. 2. If the voltmeter to be calibrated has a very much longer, or a very much shorter, range than the voltmeter with which it is to be compared for example, if the one reads from o to 500 interna- tional volts, while the other reads from o to 60 inter- national volts Fig. 93. Comparing Two Voltmeters of Very Different then W6 may prO- ceed as follows : Place two conductors A B, c D (Fig. 93) in series, and, by using the method previously described in Section 56, or a * For methods in which a standard cell is employed see Chapter IX, CALIBRATING VOLTMETERS 161 modification of it, determine the resistance of the two conductors in series A D relatively to that of one of them, A B. For example, let it be found that the resistance of A D is ten times that of A B. The actual resistance of the conductors need not be known, but we must make sure that the resistance of the low reading volt- meter shunting A B is large compared with that of A B. Attach the terminals of the voltmeter of the shorter range to the points A and B respectively, and the ter- minals of the other volt- Fig. 94. Calibrating a Voltmeter by using an meter to the points A and D Ammeter and One Known Resistance. , . , -> , .. . -. respectively. Send different currents of suitable, but not necessarily of known, values through the conductor A D. Observe the corresponding readings of the two voltmeters, and remember that the P.D. between the points A and D is always ten times the corresponding P.D. between the points A and B. 3. Join the voltmeter v (Fig. 94) to be calibrated to the termi- nals of a conductor c whose resistance R is known in international ohms. Send different currents in succession through this con- ductor, and measure the currents with the ammeter A. Observe the deflections of the voltmeter which correspond with each of the currents, 7 lf / 2 , 7 3 , etc., amperes, and note that they are produced by P.D.s of ^R, I 2 R, I 3 R, etc., international volts. If the voltmeter v be an electrostatic one, so that no current whatever passes through it, the deflection of the ammeter A will measure the true current passing through the conductor c. If, however, v be a voltmeter that takes a current, then it must not be forgotten that the current passing through the ammeter is the sum of the currents passing through the conductor c and through the voltmeter. The error introduced by assuming that the ammeter measures simply the current passing through c will be the smaller the leSS is the resistance Of C Fig. 94*- Calibrating a Voltmetei by using an , . . J . . J . Ammeter and One Known Resistance, compared with that of the voltmeter. It will be better, therefore, that c should have a comparatively small resistance, and that the necessary P.D. should be produced between its terminals by sending a strong current through it. If, however, there be a risk that such a current will warm the conductor c and so change its resistance, then it is better to join L 162 PRACTICAL ELECTRICITY up the apparatus as in Fig. 94^. In that case the resistance that must be used in calculating the P.Ds. set up between the terminals of the voltmeter v is R + R a international ohms, where R, as before, is the resistance of the conductor c, and R a is the resistance of the ammeter A. So that when the currents are /-,, 7 2 , 7 3 , etc., amperes respectively, the P.Ds. are I^R + R a ), I 2 (R + R a ), I 3 (R + R a ), etc., international volts. The connection shown in Fig. 94 may be used without introducing error, if for R the resistance of c in parallel with that of the voltmeter be taken. 4. Let BJ, B 2 , B 3 , etc. (Fig. 95), be binding screws attached to different points of a conductor which may be composed all of one wire of uniform cross-section, or of different pieces of wire of any cross-sections joined up to one another in series. Compare Fig. 95. Calibrating a Voltmeter by using Several Known Resistances in Series, with One Known Current passing through them. the resistances of the sections with one another by the method described in Section 56, and compare the resistance of some one of the sections with a standard international ohm, or with some conductor whose resistance is known in international ohms, then the resistance of each of the sections B X B 2 , B 2 B 3 , B 3 B 4 , etc., will be known. Let these resistances be respectively R lt R 2 , R 3 , etc., international ohms. Send a current through the conductor B X B 2 B 3 , etc., and keep the current quite constant at some convenient numbej of amperes, as measured by the ammeter A. Then the P.D. between any pair of the binding screws attached to different points of the conductor is known in international volts ; for example, the P.D., between binding screws B! and B 4 is / (R l + R 2 + R s ) international volts. By connecting, therefore, the terminals of the voltmeter to be calibrated with each of the pairs of binding screws in succession a series of deflections is obtained, the P.D. corresponding with each of which is known in international volts. 5. If the voltmeter be a galvanometric one it may be calibrated by measuring its resistance R g *, ascertaining the currents I lt I 2 , * For the sake of brevity the word international will, throughout the remainder of this book, be omitted before the words volt and ohm, but it is to be understood that in all cases where no prefix is mentioned the word international is implied. VOLTMETERS USED AS AMMETERS 163 7 3 , etc., in fractions of an ampere that produce the deflections d lt d 2 , d 3 , etc. These deflections will then correspond with P.D's. of IiRg, / 2 ^ I*Rg> etc -> volts maintained between the terminals of the voltmeter, or with P.D's. of /j (R g + R w ), I 2 (R g + R w ), 7 3 (R g + R w ), etc., volts maintained between the terminals T I and T 2 (Fig. 90) where R w is the resistance of the auxiliary wire w placed in series with the galvanometer. ,4 Example 41. An ammeter of 17 ohms resistance has been graduated to read milliamperes (thousandths of an ampere) directly. What external resistance must be added to the instru- ment so that the same scale will measure P.Ds. directly in volts ? If a resistance of 1000 17 i.e. 983 ohms be added to the gal- vanometer, a P.D. of V volts, maintained between the terminals of the ammeter and resistance combined, will send V milliamperes through the arrangement, and will, therefore, produce a deflection of V on the scale. Answer. 983 ohms.* Example 42. A voltmeter having 2,475 ohms resistance has been calibrated to read off volts. It is desired that a deflection of d divisions shall correspond with a P.D. of 5 d volts instead of d volts. What external resistance must be added to the voltmeter to obtain the result ? Answer. 4 x 2,475 or 9,900 ohms. 63. Voltmeters used as Ammeters. Any voltmeter, whether electrostatic or of the current type in combination with a constant resistance, can be used and graduated as an ammeter. For, consider the arrangement No. 3, Section 62, used for calibrating a voltmeter, and illustrated in Fig. 94. With every current which is measured in amperes with the ammeter A there is a certain deflection of the voltmeter. If, then, these deflections be marked not in volts but with the numbers of amperes as measured with the ammeter, the reading on the scale of v will at any time give the current in amperes passing through it and the conductor c together, when the two are used in combination as shown. The * This question may also be worked out as follows : Let R be the resistance required, then a P.D. of V volts will cause a current of X 1000 milliamperes to flow, i.e., IOO V milliamperes; and as A + 17 the arrangement is to read directly in volts, this current must give a deflection of V. But a deflection of V means a current of V milliamperts . 1000 ' y _ y 1000. and R = 1,000 17 = 983. 164 PRACTICAL ELECTRICITY graduation of the voltmeter scale in amperes will not, however be correct if the voltmeter be used as a shunt to some other con- ductor having a different resistance from that of c. The device just mentioned enables a moving coil instrument, such as was described in Section 61, through which only a small current can be passed, to indirectly measure any current no matter how large. In such a case, and generally when the voltmeter used as an ammeter is to be portable, the conductor c may be placed inside the case of the voltmeter. It is to be noticed that the combination of voltmeter and conductor c, of fixed resistance, can be graduated and employed, as an ammeter, whatever the relative resistances of the voltmeter and the conductor may be. One important advantage, however, is gained by making the resistance of c very low compared with l-ig. 96. Shunted Voltmeter used as Ammeter. the voltmeter, and that is the facility for altering the sensibility of the arrangement. For, suppose that the conductor c of Fig. 94 takes the form of a short, wide strip (Fig. 96), having therefore a very low resistance, and that the voltmeter joined up as a shunt to it has a resistance of R v ohms, large compared with that of the strip ; further, suppose that a current of 7 amperes sent through the arrangement as measured by the ammeter A deflects the pointer of the voltmeter to the end of its scale. Next, let a resistance of R v ohms be put in series with the voltmeter (Fig. 96), then it will require twice the P.D. to be maintained between the points x and Y to produce the same deflection as before on the voltmeter. Therefore it will require twice the current to flow through the strip, and, since by hy- pothesis the resistance of the voltmeter is very high compared with that of the strip, the current passing through the voltmeter is inappreciable compared with that flowing through the strip. Therefore twice the current flowing through the strip means practically twice the current in the main circuit H j. In other words, by adding to the voltmeter branch a resistance of R v ohms we have halved the sensibility of the arrangement which is, used, EXAMPLES 165 as an ammeter, for measuring the current in the main circuit H j. And, generally, if a resistance of n R ohms be added to the volt- meter branch, the current in H j that produces any particular deflection of the voltmeter will be n -}- I times the current required to produce the same deflection when the voltmeter terminals are joined direct to the points x and Y.* Example 43. A strip of platinoid of resistance 0-017 onm is shunted with a galvanometer of 305 ohms' resistance in series with a variable resistance. The galvanometer is of such sensi- bility that a P.D. of 0-5 volt causes a deflection of 270 scale divisions when the resistance in series with the galvanometer is 1000 ohms. If the scale is a proportional one, what must be the resistance in series with the galvanometer in order that when 10 amperes pass through the strip the deflection shall be 100 scale divisions ? When 10 amperes pass through the strip the P.D. between its terminals is 10 X 0-017, or ' I 7 vo ^- Therefore the current * that this P.D. produces through the galvanometer is - - where R is the resistance in ohms to be put in series with the galvanometer. But by hypothesis a current of - - , or 305 + 1000' 0*0003833 ampere produces a deflection of 270 scale divisions, and therefore, since the scale is a proportional one, a current of - X 0-0003833, or 0-000142, ampere will produce a deflection of 100 scale divisions. Hence 0-17 = 0-000142 305 +R or R = 892 ohms. Answer. 892 ohms. Example 44. Calculate for the strip and galvanometer re- ferred to in the previous question the resistances that must be placed in series with the galvanometer in order that 20, 30, and 50 amperes through the strip may produce 100 divisions' deflection. Answers. 2,089; 3,286; 5, 680 ohms respectively. * If R s be the resistance of the shunt between the points x and Y, then adding a resistance R v + R s to the galvanometer circuit will halve the sens- ibility of the arrangement, whatever the relative values of R v and R s , and in general adding resistance n (R v + R s ) to the galvanometer circuit will reduce the sensibility to of the original value. This should be proved as an exercise by the student after reading Chapter VI. 166 PRACTICAL ELECTRICITY 64. Gold-Leaf Electroscope. If we desire to measure the P.D. between two insulated bodies which have been electrified by touching them, for example, one with a rubbed piece of ebonite, and the other with a rubbed piece of glass, it would be impossible to employ any form of current voltmeter. For no matter how fine or how long were the wire used in winding the galvanometer, or how large was the resistance of the added wire w (Fig. 90), the flow of electricity which enabled the P.D. to be indicated would at once destroy the very P.D. we desired to measure. An electrostatic voltmeter must, therefore, be employed in such a case, but as there is no difficulty in producing a P.D. of many hundreds of volts by means of rubbed ebonite or rubbed glass, the voltmeter may, for many purposes, be of a much rougher kind than the one already described. When it is only required to know whether one potential is higher, or lower, than another, or whether the potential of a body is plus or minus, that is to say, whether a positive current would flow from the body to the ground, or from the ground to the body, if they were connected together by a wire, such a qualitative test can be conveniently made with a " gold-leaf electroscope." This instrument, as formerly constructed, had a variety of faults, but the illustrated description that was given, in the earlier editions of this book, of the proper way to construct a gold-leaf electroscope, has induced some manufacturers, at any rate, to cease reproducing instruments possessing the glaring defects of the older types. In the present edition of the book it will be, therefore, sufficient to describe the way in which a gold-leaf electroscope may be satisfactorily constructed. A glass shade G G (Fig. 97) rests on a wooden base, and is covered inside with the conducting varnish devised by the authors*, or with strips of tin- foil T, placed only just * When the metallic foil is stuck on the glass shade, as indicated in Fig. 97, so that the moving system can be fairly well seen at a distance through the openings between the s1rips, the screening action, although considerable, is by no means complete, and when the area of the metallic coating becomes sufficiently large compared with the area of the glass as to render the screening practically perfect, there is considerable difficulty in seeing the moving system sufficiently well to enable small changes in the deflection to be observed at a distance. The authors, therefore, experimented on methods of coating the whole of the interior of the glass shade with a transparent varnish that should be sufficiently conducting to act perfectly as an electrostatic screen, and yet hard enough that the inside of the glass could be cleaned when desired without risk of the varnish being rubbed off. And this, they find, can be satisfactorily accomplished in either of the following ways : Method No. i. Dissolve ounce of transparent gelatine in i ounce of glacial acetic acid by heating them together in a water bath at 100 C. To this solution add half the volume of dilute sulphuric acid, which has GOLD-LEAF ELECTROSCOPE 167 so far apart as is necessary to enable the gold leaves to be easily seen. These strips of tin-foil are bent round the bottom of the glass shade, and connected electrically with a brass ring, which encircles the outside of the bottom of the glass shade. 'To this ring three horizontal brass lugs are attached for enabling the shade to be screwed to the wooden base, and to one of them is fixed a binding screw, s, for holding any wire which we wish to electrically connect with the tin- foil coating. Inside the glass shade G, G, thin rods of good insu- lating glass gg are cemented into two short l>rass tubes, or sockets, fixed to the base, and the glass rods are joined together at the top by being cemented into a little metallic tube / 1, carrying the thick wire w w, and the gold leaves L. This wire w passes through the top of the instru- ment without touching it, and may carry at its top a little knob or a little binding- screw, v is a glass vessel containing lumps of pumice- stone soaked in strong sulphuric acid, which absorbs any water vapour in the interior of the electroscope, Fi s- 97> ~ Ayr E 1 e C ? r oico^ y s Gold " Leaf and thus keeps the glass rods g g dry. '. When the instrument is not in use the little metal plug or stopper p (which is made to slide a little stiffly on the wire w been prepared by mixing i part of strong acid with 8 of distilled water by volume, and apply the mixture while still warm to the glass shade, which should be previously polished and warm. When this film has become very nearly hard apply over it a coating of Griffith's anti-sulphuric enamel, the chief ingredient of which is resin dissolved in fusel oil. Method No. 2. Thin the gelatine solution, prepared in the manner previously described, by the addition of acetic acid (say, 2 volumes of -acid to i of solution), and after polishing the glass, float the thinned solution over the glass cold. Drive off the excess of acetic acid by warming, allow the glass to cool, and repeat the floating process, say, twice. Thin the anti-sulphuric enamel by the addition of ether, and float it over the gelatine layer applied as just described. Expel the ether by heating, and apply a second layer of this thinned anti-sulphuric enamel. It is advisable to varnish the inside of the glass shades or glass fronts, not merely of electrostatic voltmeters, in one of the ways just described, but of current voltmeters, ammeters, or indeed of any instrument where the electrification of the glass produced by cleaning it on a dry day might cause a deflection of the pointer of the instrument a cause of error that has been noticed with electrical measuring instruments placed in hot dry engine-rooms of electric-light stations. 168 PRACTICAL ELECTRICITY by the hole in the stopper being lined with cork) should always be pushed down, and the hole at the top of the instrument thus closed to keep out dust and damp. If this precaution be carefully attended to on every occasion that the electroscope is left unused, even for a short time, and the surface of the glass rods g, g, be initi- ally carefully cleaned, the insulation of the instrument will remain so good, even for a year after the acid has been put on to the pumice stone, that an electric charge given at any time to the gold leaves will remain practically undiminished by leakage during an hour even on a very damp day. With a given gold-leaf electroscope the divergence of the gold leaves depends simply on the P.D. between the gold leaves L and the tin-foil coating T. For the gold leaves constitute a flexible needle corresponding with N in Fig. 86, and the tin-foil coating is the stationary inductor (called I in the same figure) to which the gold leaves are attracted with a force depending on the P.D. between them and the tin-foil coating. This attraction causes the leaves to diverge, and to be, therefore, lifted ; the angle of divergence for any particular P.D. being such that the attractive forces exactly balance the controlling forces introduced by the weight of the leaves which are slightly displaced from the vertical position. A gold-leaf electroscope is, therefore, a " deflectional gravity-voltmeter. ' ' 65. Sensibility of Gold-Leaf Electroscopes. As already ex- plained, gold-leaf electroscopes are frequently used merely as qualitative instruments, but, employing method No. 2, Section 62, a gold-leaf electroscope may be calibrated, if desired, by comparison with the zero electrostatic voltmeter (Fig. 86). The law connecting the divergence of the leaves with the P.D. set up between them and the case depends on three things (i) the length of the leaves (2) , the weight per square inch of the leaf, and (3) the size of the case. If the length of the leaves and the size of the case be fixed, it follows, from our original definition of what is meant by one P.D. being twice another, that the P.D. required to produce any particular divergence is simply proportional to the square root of the weight of the leaf per square inch. Specimens of gold leaf from different gold-beaters appear to vary as much as 20 per cent, in the weight per square inch, but the lighter the leaf the lower will be the price, provided that it is not much below 40 shillings per book of 1000 leaves, in which case cheapness may result from the impurity and not from the thinness of the gold. At 40 shillings per thousand sheets of 22 carat gold, the sheets being 3j inches square, the weight per square inch is about 0-013 grain. With leaves, each 2j inches SENSIBILITY OF ELECTROSCOPES 169 long, cut from this quality of material and suspended in a con- ducting case 4| inches internal diameter, a divergence of about 56 is obtained for a P.D. of 1000 volts, set up between the leaves and the case. Reducing the length of the leaves to ij inches increases the divergence for the same P.D. to 60 and in addition it renders the various divergences between the leaves in degrees more nearly directly proportional to the P.D. in volts. The calibration curve can also be rendered much more nearly a straight line by increasing the diameter of the case, but this has the counterbalancing effect of diminishing the sensibility for the same leaves, as may be seen from the following table : LEAVES EACH ij INCHES LONG. P.D. OF 1000 VOLTS MAINTAINED BETWEEN LEAVES AND CASE. Internal Diameter of Case in Inches. Divergence between Leaves in Degrees. 41 6 8 10 60 54 48 44-5 Plotting a curve to represent the above four pairs of values and continuing the curve forwards, it is seen that the divergence rapidly approaches 40, which means that however large may be the diameter of the conducting case the divergence will be about 40 when a P.D. of 1000 volts is maintained between this case and a pair of leaves each i J inches long cut from a 40 -shilling book of 22-carat gold leaf. With the leaves each ij inches long the case can be made as narrow as 4! inches in diameter and still nearly direct pro- portionality of P.D. and divergence be obtained up to 70 whatever be the weight of the leaves. This is the size of leaf and case, therefore, that may be conveniently adopted, and the con- stant of instruments so constructed will vary from about 6 per 100 volts to 6 per 225 volts, as the material used in making the leaves costs 40 shillings per 1000 sheets, or a few pence when the material is " Dutch metal." For measuring P.Ds. of 2,000 volts, or higher, such as are now maintained between the underground mains with certain electric light systems, the leaves may be conveniently made out of lead foil instead of gold-leaf. CHAPTER V GALVANIC CELLS 66. Chemical Action in a Simple Voltaic Element: Polarisation 67. Daniell's Use of a Depolariser : Two-Fluid Cell 68. Local or Pre- judicial Action 69. Gravity Daniell's Cells 70. Minotto's Cell 71. Resistance of Daniell's Cells 72. Grove's and Bunsen's Cells 73. Potassium Bichromate Cell 74. Storage or Secondary Cell 75. Leclanchg Cells 76. Dry Cells 77. Hellesen and Dania Dry Cells 78. G. E. C. and Obach Cells 79. Blue Bell and Columbia Cells 80. Extra-Sec and Inert Cells 8 r. Edison-Lalande Cell 82. Standard Cells, Clark's and Weston's Cells 83. Calculation of the E.M.F. of a Cell from the Energy Liberated by the Chemical Action. 66. Chemical Action in a Simple Voltaic Element. A simple voltaic element is illustrated in Fig. i, and a battery of five elements in Fig. ib. When the terminals of such a cell are joined by a conductor a current flows through the conductor, hydrogen is given off at the copper plate and the zinc plate is gradually dissolved, forming zinc sulphate. The zinc in dis- solving in the sulphuric acid liberates some of the energy used in reducing the metal from its ores and part of this energy ap- pears in the electric circuit usually in the form of heat produced in the wires through which the current flows. In fact a cell may be regarded as a sort of furnace in which the zinc takes the place of coal as fuel, the chief difference being that the fuel is burnt at a low temperature instead of a high one. In primary batteries the plate which 'is dissolved when the current flows is called the " positive plate," for the current passes through the liquid from this plate to the one unattacked, this being called the " negative plate." As, however, the current in the outer circuit passes from the unattacked plate to the one dissolved, the terminal on the former is called the " positive terminal," and that on the latter the " negative terminal." The gradual replacing of the sulphuric acid in the liquid by zinc sulphate lowers the E.M.F. of the cell ; but a more serious falling-off of the E.M.F. of a simple voltaic element, when sending a current, arises from the polarisation which is caused by some of the hydrogen gas which is liberated at the copper plate sticking 170 POLARISATION IN SIMPLE CELLS Fig. 98, to it and setting up an opposing or back E.M.F., in consequence of the tendency of the hydrogen to recombine with the SO 4 , or with the oxygen from which it has been separated. That the E.M.F. of a copper- zinc-dilute-sulphuric-acid cell falls when the cell is allowed to send a current can be tested by using a high-resistance voltmeter, v, and a suitable ammeter, A, placed in an external circuit whose resist- ance, R, can be varied (Fig. 98). We commence by making R infinite, so that the reading of the voltmeter gives the E.M.F. of the cell (Section 55). Next, R is made to have some convenient constant value, and the reading of the ammeter watched ; gradually this will be found to fall, the fall being fairly rapid if the value of R be not large. The reading of the voltmeter also falls, and, since the value of R is constant, the ammeter and voltmeter readings fall at the same rate, each instrument telling us the same thing by the falling-off of its deflection viz., that either the resistance of the cell has increased or its E.M.F. has diminished. If, however, we now again make R infinite, we can ascertain which of these two causes it was that made the current fall off ; for, if the diminution of the current was due to an increase in the resistance of the cell only, then on making R infinite the reading of the high-resistance voltmeter v will be the same as it was originally ; whereas, if the diminution ob- served in the current was wholly, or partly, caused by a falling-off in the E.M.F. of the cell and not entirely by an increase in its internal resistance, the voltmeter will read lower when R has been made infinite at the end of the experiment than it did when R was made infinite at the beginning. And this result is found to occur. To ascertain at which of the plates of the cell the opposing E.M.F. is set up, we may use the cell seen in Fig. 99, consisting of two copper plates, c x and C 2 , and two zinc plates, z l and Z 2 , dipping.into dilute sulphuric acid. If the plates are all quite clean and no current has passed between any pair of them, the two Fig. 99. Cell arranged for Experiments on Polarisation. 172 PRACTICAL ELECTRICITY copper plates will be practically the same, so that if they be joined together metallically through even a delicate galvano- meter, G, no current will be observed, or if there be any current, arising from some minute difference in the two copper plates, it will be but a very slight one. And so with the two zinc plates, on joining them together through a delicate galvanometer, no current, or only a very small current, will be observed. If now, however, one of the copper plates, Cj, and one of the zinc plates, Z A , be used to send a current for a short time through some conductor, and then, after breaking the circuit, the two copper plates Cj and C 2 be joined through the galvanometer, G, Fig. 100. No Current CL,ean Copper. Fig. 101. Clean I (Clean Zinc 1 i Zinc, Strong Current Strong Current Weak Current Fig. 102. Fig. 104. it will be found that a polarisation current flows for a short time from C 2 to c 1 through the external circuit, as if c t , the copper plate, that has been used, were positive, or like a zinc plate relatively to C 2 , the unused copper plate. Similarly, if the two zinc plates, instead of the two copper plates, be joined together through the galvanometer, a current will flow through the ex- ternal circuit from Z 1} the zinc plate that has been used, to Z 2 , the clean zinc plate ; but this polarisation current will be very small compared with the one obtained on joining the two copper plates. Indeed, it is so small that we may say without appreci- able error that the diminution of the current in a simple voltaic element is due to polarisation at the copper plate. These tests and the results obtained are shown symbolically in Figs. 100 104. It is to be noticed that while the " primary current " flows from Zj to c x through the liquid, and the " secondary current " DANIELL'S TWO-FLUID CELL 173 flows from c 1 to C 2 or from Z 2 to z x through the liquid, the hy- drogen gas in all three cases moves in the direction of the current, the result obtained with the sulphuric acid voltmeter (see Section n). 67. Daniell's Use of a Depolariser : Two-Fluid Cell. Numer- ous devices were tried to prevent the hydrogen gas sticking to the negative plate ; Smee, for example, used a roughened platinum plate instead of copper, the roughening being for the purpose of enabling the hydrogen bubbles to become detached. But no great improvement was introduced until Prof. Daniell, in 1836, hit on the idea of surrounding the negative plate with a " de- polariser " to prevent the hydrogen gas liberated from the dilute sulphuric acid reaching this plate. Instead of putting both the copper and the zinc plates in the dilute sulphuric acid, he sur- rounded the copper plate with a solution of copper sulphate, the two liquids being prevented from mixing together by a porous diaphragm placed between them as shown in Fig. 2. As before, the dilute sulphuric acid, acting on the zinc plate, forms zinc sulphate and liberates hydrogen gas, but this hydrogen gas arriving at the copper sulphate solution forms sulphuric acid and deposits metallic copper on the copper plate. Omitting, for simplicity, the water used to form the solutions as well as the water of crystallisation of the copper and zinc sulphate crystals, this chemical action may be represented as follows : Before sending the current p A(Cu)+/(CuSO 4 ) .2 w(H 2 SO 4 )+w(Zn), After sending the current (k + 1) (Cu) + (li] (CuSO 4 ) g (SO 4 H 2 ) + (m i) (H 2 SO 4 ) + (ZnS0 4 ) + (w i)Zn, k and n being any arbitrary quantities of copper and zinc used to form the copper and zinc plates, / and m any arbitrary quantities of the copper sulphate and the sulphuric acid employed in the two portions of the cell, and the arrow showing the direction of the current in the cell itself. Substituting the atomic weights for the various substances employed, and remembering that the complete formulae for crystals of copper and zinc sulphate are respectively CuSO 4 + 5H 2 O and ZnSO 4 + 7H 2 O, we find that for every gramme of zinc that is dissolved off the zinc plate about 4-4 grammes of zinc sulphate crystals are formed, about 3-8 174 PRACTICAL ELECTRICITY grammes of copper sulphate crystals are decomposed, and about 0-96 gramme of copper added to the copper plate of a Daniell's cell. Hence, since we know that about 0-0003286 gramme of copper is deposited per second per ampere in a copper voltameter (Section 9), it follows that in each hour for each ampere flowing through a Daniell's cell about 1-18 grammes of copper is de- posited, about 1-22 grammes of zinc is used up, about 4-62 grammes of copper sulphate is consumed, and about 3-0 grammes of zinc sulphate is formed, which latter will become 5-4 grammes when crystallised out, since the complete formula for zinc sul- phate is ZnSO 4 + 7H 2 O. Therefore, in twenty-four hours, for each ampere flowing through a Daniell's cell about I ounce of copper is deposited, about 1-03 ounces of zinc is used up, about 3-94 ounces of copper sulphate are consumed, about 2-55 ounces of zinc sulphate are formed, which be- come 4-54 ounces when crystallised out. In the preceding no allowance is made for materials wasted on ac- count of local action. When a current is pro- duced by a Darnell's cell, 1 copper is deposited on the copper plate, copper sul- phate is used up, the Fig. zos.-Porous Pot Daniell's Cell. su lphuric add remains unchanged in quantity, zinc sulphate is formed, and zinc is used up. If, however, the copper sulphate solution is too weak, the water is decomposed instead of the copper sulphate, and hy- drogen is deposited on the copper 'plate. This deposition of hydrogen lowers the E.M.F., and care should, therefore, be taken to keep up a sufficient supply of crystals of copper sulphate. Daniell originally used a membranous tube made of ox gullet as his porous separator, but this was shortly replaced by a "porous pot" made of unglazed earthenware, indicated by p in Fig. 105, which illustrates a common form of Daniell's cell. The zinc may be in the form of a rod, z, placed in the dilute sulphuric acid which is put inside the porous pot, or in the form of a hollow cylinder surrounding the porous pot, in which case DANIELL'S CELL 175 the dilute sulphuric acid is, of course, placed outside the porous pot and the solution of copper sulphate inside. The former arrangement is the more usual. Electric connection is made with the zinc by means of a copper wire, w, cast into it. The copper plate c, which is usually cut out of sheet copper, is placed in the solution of copper sulphate, and the whole is contained in a glass, or glazed and highly vitrified stoneware jar, j. Electric connection is made with the copper plate by means of a copper wire insulated along its length with gutta-percha or indiarubber, and having its end riveted, or soldered, to the top of the copper plate. If solder be used the joint should be covered over with wax, pitch, or other adhesive matter to prevent the copper sul- phate coming into contact with the joint. For if this were to happen the copper and solder being in metallic contact with one another, and also both coming into contact with the solution of copper sulphate, they would together form a little short-circuited cell, galvanic action would take place and the solder would be rapidly eaten away. The E.M.F. of a Daniell's cell varies from about 1-07 volts to 1-14 volts, depending on the density of the copper sulphate solu- tion and on the amount of zinc sulphate present in the dilute sulphuric acid. As the copper sulphate is used up, and as the density of the copper sulphate solution is thereby diminished, when no steps are taken to maintain it constant, the E.M.F. of the cell falls. It also falls because the sulphate of zinc, which is formed by the eating away of the zinc rod, or plate, dissolves in the dilute sulphuric acid. The cell has, therefore, its highest E.M.F., 1-14 volts, when we start with the sulphate of copper solution saturated and no sulphate of zinc yet formed 'and dis- solved in the dilute sulphuric acid. The falling off of the E.M.F. due to the weakening of the copper sulphate solution can be pre- vented by having crystals of the sulphate placed in the liquid to maintain the saturation, but we cannot so readily withdraw the sulphate of zinc from the dilute sulphuric acid. Hence, if we desire that the E.M.F. shall remain constant while the Daniell's cell is sending a current, it is better to start with both solutions saturated., The resistance of the cell will be higher and its E.M.F. lower than when dilute sulphuric acid is used, but this lower value of about i-io volts will be maintained nearly con- stant while the cell is sending a current. 68. Local or Prejudicial Action. If a piece of chemically pure zinc be placed in strong, or in dilute, sulphuric acid, no chemical action takes place, and no chemical action occurs if a piece of copper or carbon also be introduced into the liquid, 176 PRACTICAL ELECTRICITY provided that the zinc be not touched inside or outside the liquid by the other solid. If, however, the conducting solid be now touched against the zinc, either inside or outside the liquid, there is a rapid evolution of hydrogen bubbles from the solid, and the zinc is turned into zinc sulphate. We have, in fact, a short-circuited cell consisting of an oxidisable metal zinc in contact with a less oxidisable substance copper or carbon and both the oxidisable and the non-oxidisable substances in contact with the liquid. Now, ordinary commercial zinc has impurities in it, such as lead, iron, and graphitic matter, so that when commercial zinc is placed in dilute acid a number of short-circuited galvanic cells are formed by the zinc, impurity, and liquid in contact, hydrogen gas is rapidly evolved, the zinc is speedily converted into zinc sulphate, and the energy that would otherwise be available for generating a useful electric current is frittered away in the heat produced by all these " local currents." It is, in fact, this " local action " which enables the chemist to make hydrogen gas by placing scraps of commercial zinc in dilute sulphuric acid. With a cell, on the contrary, we desire that the zinc shall only be used up when a useful electric current is produced that is, a current that passes through the wire joining the zinc and copper plates outside the liquid. Or, in other words, we desire that no chemical action shall take place when the terminals of the cell are insulated from one another. We must either, therefore, employ chemically pure zinc, or in some way prevent local action taking place with commercial zinc. The normal price of such zinc is about 2d. a pound, while that of redistilled chemically pure zinc is from 35. 6d. to los. a pound, the labour of effectively removing all the impurities from the zinc costing many times as much as the zinc itself. To employ such zinc for ordinary cells is, therefore, out of the question, and is indeed unnecessary, since Sturgeon showed in 1830 that local action can be nearly as well prevented by coating the surface of the zinc with an " amal- gam " of zinc and mercury, or " amalgamating " the zinc, as it is shortly called, as by employing the purest redistilled zinc. To " amalgamate " a piece of zinc dip it into dilute sulphuric acid to clean its surface, then rub a little mercury over it by means of a piece of rag tied on to the end of a stick, and lastly, leave the zinc standing for a short time in a dish to catch the surplus mercury as it drains off. The action of the amalgamated zinc is not well understood ; some consider that amalgamating the zinc prevents local currents by the amalgam mechanically covering up the impurities on the surface of the zinc and preventing their coming into contact with LOCAL ACTION 177 the liquid. By others it is thought that amalgamating the zinc protects it from local action by causing a film of hydrogen gas to adhere to it. This theory is based on the fact that while no action takes place when amalgamated zinc is placed in dilute sulphuric acid at ordinary atmospheric pressure, the creation of a vacuum above the liquid causes a rapid evolution of hydrogen, which, however, stops on the readmission of the air. The addition of a very small amount of zinc to mercury causes the mercury to act as if it were zinc alone, arising perhaps from the amalgam having the effect of bringing the zinc to the surface. A second prejudicial effect is produced by the copper sulphate diffusing through the porous partition, coming into contact with the zinc, and being changed into zinc sulphate, the copper which is thus displaced from the sulphate being deposited on the zinc in a metallic form, or as black cupric oxide, CuO, with the evolu- tion of hydrogen. This impairs the action of the cell, as the zinc partially coated with cupric oxide acts more like copper, and less like zinc, than if it were not so coated ; the E.M.F. of the cell is, therefore, lowered. Diffusion can be retarded by constructing the porous partition so that it is only slightly porous, but this has the disadvantage of causing the cell to have a high resistance. A formation of metallic copper is also produced in the pores of the porous partition at any spot where the zinc rod comes into contact with it, and, the copper so deposited being in metallic contact with the zinc rod, while both are in contact with the liquid, the arrangement forms a short-circuited cell, leading to rapid waste of the battery material, growth of the metallic copper in the pores of the partition, and probable disintegration of the wall of the partition itself. To avoid this the partition must be rendered non- porous, by being dipped into paraffin wax melted in warm oil, at any point where it is likely to be touched by the zinc. For example, the bottom of the porous pot p (Fig. 105), on which the zinc rod rests, should be so treated before the cell is put together. The tendency of the copper sulphate solution to diffuse to the zinc plate, and the possibility of retarding this by diminishing the porosity of the partition at the expense of increasing the resistance of the cell, necessitates our considering, when we make Fig. 106. Meidinger Cell. PRACTICAL ELECTRICITY a Daniell's cell, whether low resis- tance or constancy and portability are desired. And as examples of the two opposite types of Daniell's cells we may instance the " gravity Daniell's cell " and the " Minotto's cell." 69. Gravity Daniell's Cells. Figs. 106, 107, and 108 show three forms of Daniell's cells in which no porous partition is employed, the copper sulphate and the zinc sulphate solu- Fig. 107. caiiaud Ceii. tions being kept separated solely by the action of gravity ; and as the zinc sulphate solution is the lighter of the two, it is therefore put at the top. Fig. 106 shows a type of the " Meidinger " cell, in which the copper plate, e (Fig. 106), is put inside a small inner glass tumbler, d d, so that the particles of zinc which may become detached from the zinc plate shall fall clear of the copper plate and be prevented from coming into contact with it. In this type of Meidinger cell the crystals of copper sulphate are in a glass tube, h, with only a small hole at the bottom ; while in another type crystals are contained in an inverted flask open at the neck. Contact is made with the copper plate by an insulated copper wire, fg (Fig. 106), and the zinc plate, z z, which is in the form of a cylinder, is supported on a shoulder, b b, formed by a contraction at b b of the lower part of the outer glass vessel, A A. The " Callaud " cell (Fig. 107) is a simplification of the Meidinger, being without the reservoir for the copper sulphate crystals and the small glass tumbler to hold the copper plate. The zinc cylinder hangs from the upper edge of the jar. Figs. 108, loSa, and 1086 show a form of gravity Daniell used by the Exchange Telegraph Company, and made under the direction of the late chief engineer to the Fig. 108 Exchange Telegraph Co. s Gravity Danieii. Flan of copper electrode Fig. io8a. GRAVITY DANIELL'S CELLS Company, Mr. F. Hig- gins. It resembles the Callaud cell in some respects, but possesses the advantage of cheapness in construc- tion, and exposes large areas of surface of the electrodes to the liquids, thereby reduc- ing the internal resist- ance of the cell. A gravity Daniell's cell must, of course, not be moved about, or if moved great care must be taken to avoid the two liquids being mixed together. To prevent the copper sulphate wandering to the zinc plate, it is well to allow the cell to send a weak current through an external circuit of considerable resistance even when the cell is not in ordinary use. 70. Minotto's Cell. In the " Minolta's "* cell the porous pot is replaced by a layer of sand or sawdust of comparatively high resistance, and it is constructed as snown in Fig. 109. At the bottom of a glass, or glazed and highly vitrified stoneware, jar, j, there is placed a disc of sheet copper, c, to which is riveted one end of an insulated copper wire, which passes up through the cell. Above this plate are placed some moist crystals of copper sulphate, c s, and on the top a piece cf thin canvas, c, separating the copper sulphate from the layer of sand or sawdust s. On the top of the sawdust rests the zinc plate z, separa- ted from the sand or sawdust by an- other piece of can- vas, c. The cell is completed by pour- ing in some solution of zinc sulphate, so as to cover the zinc Often wrongly spelt i8o PRACTICAL ELECTRICITY Fig. 109. Minotto's Cell. disc, but not so much as to reach up to the brass binding-screw B, cast into the top of a little column of zinc, forming part of the zinc disc. Before putting in the sand or saw- dust it should be soaked in a solution of zinc sulphate and squeezed partially dry, because if put into the cell quite dry a long time must elapse before the liquid will soak through the sand or saw- dust, and until this happens the cell cannot come into action. It is better to employ sand in stationary Minotto's cells, as it sinks down as the copper sulphate is consumed, but if the cells have to be moved about then it is more convenient to use sawdust. 71. Resistance of Daniell's Cells. The resistance of a cell varies with (1) The area of the plates immersed in the liquids ; (2) The distance apart of the plates ; (3) The composition of the liquids ; (4) The thickness and constitution of the walls of the porous partition. A convenient apparatus for experimentally proving these statements is shown in Figs, no and iiofl, and a diagram of the connection in Fig. iio&. The liquid is contained in a long wooden trough rendered water- tight by a lining of gutta-percha or Griffith's anti-sul- phuric enamel, and the copper and zinc plates c and z are supported by stout Wires W W Sliding II0 ._c el r arranged for proving that the E.M.F. is in SCreW Clamps S S, Independent of the distance apart of the Plates and of - , . , J , the Areas Immersed in the Liquids, and that the Resist- OI WniCn Can DC ^ nce depends on dimensions, RESISTANCE AND E.M.F. OF CELLS 181 moved away from the other so as to increase the distance between the plates. The plates can also be raised and lowered so that the area of plate immersed can be varied, and by shaping the plates with a tag at their lower ends, as seen in Fig. noa, this change in immersed area can be made very great. Plate z dips into a porous pot containing zinc sul- phate or dilute sulphuric acid at Fig . Irofl . the right hand end of the trough, and by using several different pots their influence on the resistance of the cell may be studied. Calling the resistance of the cell R&, its E.M.F. E, and the readings of the ammeter and voltmeter 7 and V^ res- pectively, then the equation E V l + IR 2 of Section 55 may be written V V -j - 1 (very approximately), (18) where V is the reading of the voltmeter when the current 7 is reduced to zero by making the external resistance R infinite, and F x is the reading when a current 7 amperes is flowing. The same apparatus may also be used for showing that the E.M.F. of a cell is independent of its size or shape, and depends only on the nature of the materials used in constructing it, for by making the external resistance R, Fig. no&, infinite, and using a voltmeter of very high resistance (say 10,000 ohms), we have E = F , very approximately, and experiment shows that under these con- ditions the reading of the voltmeter is the same whether the plates be near together or far apart, or whether they be fully immersed or only just in contact with the liquid. With a porous pot Daniell's cell, about 7 inches high, of the relative dimensions shown in Fig. 105, the resistance may be as low as J of an ohm when the solution in which the zinc i8a PRACTICAL ELECTRICITY plate is immersed is dilute sulphuric acid of a specific gravity of about 1-15* at 15 C. and the porous pot has a very open grain. Such a cell must, however, be taken to pieces when not in use. If it has to be put on one side for only an hour or two, it will be sufficient to lift the porous pot with the contained zinc rod bodily out of the cell, and to place it in another empty jar, or stand it in a dish while out of use. The porous pot Daniell's cells in the Muirhead type of battery seen in Fig. in may have a resistance of as much as 10 ohms apiece. Such cells were, however, frequently used in telegraph offices on account of the ease with which they can be coupled in series by means of the composite copper and zinc plates, and of the facility with which such a battery can be carried about. For, in ad- dition to the cells being kept in place by the wooden box, the composite copper and zinc plates serve as clips to keep the porous pots in position, and so prevent them shaking about in transport. One of the composite plates is shown, in Fig. ma, flat as received from the manufac- turer, z being the zinc plate, c the copper plate, and c a copper strip, one end of which is cast in the zinc plate and the other riveted to the copper plate. The dotted lines in Fig. ma show the plates, with the strip bent, ready for insertion into the cells. Cells of this type can be left joined up for several weeks, water and crystals of copper sulphate being added from time to time as required. Gravity Daniell's cells have been constructed by Lord Kelvin so as to have a resistance of less than o-i ohm apiece. This * For the percentage of sulphuric acid in solution corresponding with various specific gravities, see Appendix IV. RESISTANCE OF CELLS 183 result is attained by making the zinc and copper each in the form of a large plate, the plates being placed horizontally one above the other at a short distance apart. On the other hand, Minolta's cells have frequently resistances of 20 or 30 ohms each, this high resistance being of little importance when the cells are employed to send a current through a large external resistance, compared with the constancy that is attained by employing a partition of sawdust some inches thick. Indeed, it Fig. ma. Composite Copper and Zinc Plates for Muirhead's Telegraph Battery. (Flat, as received from manufacturers, in dotted lines with connecting strip bent, for insertion in battery.) is only necessary to pour a little water into such cells every few days to make up for that lost by evaporation, in order that they may be used for many months without any other attention being given to them. The resistance of a Daniell's cell, like that of liquids generally, diminishes with increase of temperature ; hence, as its E.M.F. is practically independent of changes of temperature, the current sent by a Daniell's cell through a constant external resistance increases as the temperature rises. Example 45. Calculate the weight of zinc sulphate formed during 2| hours in a Daniell cell when a steady current of 0-5 ampere passes through it, assuming that no zinc is consumed by local action. Answer. 376 grammes. Example 46. In the last question it is found that 6-47 grammes of copper sulphate have been used up. Calculate how much per cent, of the copper sulphate has been wasted through local action. , Answer. n-6 per cent. 72. Grove's and Bunsen's Cells. In the " Grove's " cell a zinc plate is placed in dilute sulphuric acid, as in the 184 PRACTICAL ELECTRICITY Darnell's, but the copper plate is re- placed by one of -platinum and the copper sulphate solution by strong nitric acid, HNO 3 , which is generally said to act as the depolariser. The Bunsen's cell differs from the Grove's only in the use of carbon in place of platinum. These cells are shown in Figs. 112 and 113 respectively. Both cells give a high E.M.F. 1-9 to 1-95 volts, and have low internal resistances, so they may be used for producing fairly large cur- rents. During working, the cells give off dark brown fumes of nitric peroxide, NO 2 , and should be placed in the open air or under a chimney. The chemical action in a Grove's cell may be represented as follows, omitting the water used to dilute the sulphuric acid : Before sending a current q Fig. 112. Grove's Cell. (Pt) +/(HNO 3 ) After sending a current m(H 2 SO 4 ) + k(Pt) + (l 2)(HN0 3 ) +2(N0 2 )+2(H 2 0) (m i)(H 2 S0 4 ) + (ZnS0 4 ) A Grove's or Bunsen's battery must be taken to pieces at the end of each day's use, since the mixing of the liquids through the walls of the very porous pots used to separate them, would render the battery practically useless the next day. The porous pots should be placed in water after use, so that all the zinc sul- phate solution may be dissolved out of the pores of the earthen- ware, for, otherwise, when the pots are dried the zinc sulphate Fig. 113 Bunsen's ceii. CHROMIC ACID CELLS i5 solution will crystallise in the pores and cause the pots to fall to pieces. Example 47. If 4 Ibs. of zinc have been consumed in a Grove's battery, how much sulphuric acid has been used up, assuming that no local action has taken place ? Answer. 6 Ibs. Example 48. 25 Grove's cells in series are sending a current of 8 J amperes ; in what time will 2 Ibs. of nitric acid* be consumed ? Answer. I hour n minutes approximately. 73. Potassium Bichromate Cell. This is a form of cell devised by Prof. Poggendorff, in which the depolariser is chromium trioxide (CrO 3 ), popularly called chromic acid, since chromium trioxide dissolved in water has a strong acid reaction. But, as the chromium trioxide used formerly to be prepared, by the user of the cell, by acting on potassium bichromate, K 2 Cr 2 O 7 , with strong sulphuric acid, the cell is frequently called the " potassium bichromate " cell. Now, however, crystals of chromium trioxide containing 5 per cent, of water of crystallisa- tion can be purchased ready prepared, and when these are used the cell may be shortly called a " chromic acid " cell. The cell is constructed in two forms, one without and one with a porous pot, seen in Figs. 114 and 1140 respectively. The plates employed are of carbon K, and amalgamated zinc, z (Fig. 114), two carbon plates being generally used with the former type of cell to diminish its resistance. The zinc plate z is supported by the rod a, and should be pushed into the liquid only when the cell is required to give a current, and with- drawn directly the current is interrupted, other- wise an insoluble chromium salt forms on the surface of the zinc and interferes with the action of the cell. The chemical change which takes place when a current passes through a single fluid chromic acid cell, containing chromium trioxide dis- solved in dilute sulphuric acid, is as follows : Before sending a current, A(C) + J(CrO 3 ) + w(H 2 SO 4 ) + w(Zn). After sending a current, <: ft (C) + (l-a) (CrO 3 ) + (Cr l3 SO 4 ) + (m-6) (H 2 SO 4 ) +6(H 2 O)+3(ZnSO 4 ) + (w 3)(Zn). out Porous Pot. * The strong nitric acid may be assumed to contain 65% HNO S . i86 PRACTICAL ELECTRICITY Fig. 114*. Fuller's Mercury Bichromate Cell In the type of potassium bichromate cell, having a porous pot, the zinc, (Fig. 114^), is frequently cast in the form of a thick cylinder attached to a stout copper wire, carrying the binding screw, and both the zinc and the wire are well amalga- mated, or the rod is coated with gutta-percha to insulate it. In the porous pot con- taining the zinc, there is put a quantity of mercury to maintain the amalgamation, and either dilute sulphuric acid or a solution of common salt, NaCl. Sodium bichromate, may be used with advantage instead of potassium bi- chromate. This cell has an E.M.F. of about two volts, and is suitable for producing a fairly strong current for a short time. When much used the cell becomes saturated with the potassium and chromium sulphates, and a double salt, chrome alum, KgCr^SO^ crystallises out and sticks so firmly to the bottom of the cell that it is somewhat difficult to remove. Example 49. A single fluid potassium bichromate cell is used to produce a current of i ampere for 10 hours. How much sulphuric acid is consumed in the preparation of the necessary amount of chromium trioxide and in the working of the cell, and how much zinc sulphate and water are formed ? Allow 33 per cent, additional for waste. Answer. Sulphuric acid, about 2 oz. Zinc sulphate 1-4 oz. Water ,, 0-27 oz. Example 50. What is the mean value of the current that a chromic acid cell has been producing for 4 hours if the zinc, which originally weighed 8 oz., has been reduced to 7! oz. ? Also, how much sulphuric acid and chromium trioxide have been used up ? Answer.- Current, about 2-9 amperes. Sulphuric acid, about 1-5 oz. Chromium trioxide crystals, about 0-53 oz. Example 51. How much zinc, sulphuric acid, and chromium trioxide would be consumed in a chromic acid cell having an STORAGE CELLS 187 E.M.F. of 1-8 volt and an internal resistance of 075 ohm, it used for 3 hours to send a current through an external resistance of i J ohms ? Answer. Zinc, about 0-103 oz. Sulphuric acid, about 0-308 oz. Chromium trioxide crystals, about O'li oz. 74. Storage or Secondary Cell. When strong and steady currents are required, it is now customary to use storage cells, which consist usually of lead plates in sulphuric acid ; such a cell is shown in Fig. 115. The plates are generally made in the form of grids or grooved sheets, those intended for positive plates* being covered with red lead (Pb 3 O 4 ) made into a paste with sulphuric acid, and the negative ones with litharge (PbO) paste. When an electric current is passed through diluted sulphuric acidt from the plate pasted with red lead, to that pasted with litharge, the red lead is oxidised to lead peroxide PbO 2 , and becomes of choco- late colour, whilst the lith- arge is reduced to spongy metallic lead and becomes slaty grey ; the cell is then said to be " charged" and will act as a current gen- erator until the peroxide and spongy lead are transformed into lead sulphate (PbSO 4 ). The current on discharge flows from the peroxide plate to the spongy lead plate through the outer cir- cuit. By causing a current to : pass through the cell in the opposite direction it be- comes " recharged." Second- ary cells or " accumulators " as they are sometimes called, are now used in thousands for many purposes, such as electric lighting and traction, propulsion of submarines when submerged, electric cabs, delivery vans and trucks, gas, oil, and petrol engine * The naming of the plates of secondary cells differs from that of primary cells, for in secondary cells the plate to which the current passes through the liquid when the cell is discharging is called the "positive " plate. f Acid of density about i'z is usually employed. Fig. i. T5. Storage or Secondary Cell. 188 PRACTICAL ELECTRICITY POLE FILLING APERTURE CELL COVC* ignitions, electro-plating, and for working telegraph and telephone systems. They possess the advantages of high E.M.F. (about 2 volts), and very low internal resistance, and after being well charged will produce strong currents of constant strength for long periods. To prevent rapid deterioration of secondary cells of the form above described it is desirable that they be recharged before the P.D. has fallen below 1-8 volts. Within recent years a new form of storage cell has been de- veloped by Mr. Edison, specially intended for " accu- mulator traction," in which the grids are of nickel-plated steel and the elec- trolyte a solution of potassium hydrate (density 1-21), with a small amount of lithium hydrate. The active material on the positive plate is nickel hydrate, in which flakes of pure nickel are embedded to increase the con- ductivity, and the perforations on the negative plate are filled with a mixture Fig. lisa Edison Nickel-iron Alkali of iron Oxide and Accumulator. mercury. Fig. 1150 shows the arrangement adopted ; the positive plates are made of numerous perforated tubes of nickel-plated steel containing the nickel hydrate and flaked nickel, whilst the nega- tive plates are formed of numerous flat pockets, with finely per- forated sides containing the iron oxide. The can, or container, is made of corrugated nickel-plated steel, and is therefore light and strong. The changes in P.D. which occur during charge and discharge of secondary cells are shown in Fig. 1156, the upper two curves of which refer to a lead-acid accumulator, made by the Electrical Power Storage Company, and the lower curve to a nickel-alkali or Edison cell of similar ampere-hour capacity. It will be noticed LECLANGHfe CELLS 189 that the lead cell has an average discharge P.D. of about 1-94 volts, whilst that of the nickel cell is about 1-2 volts. 75. Leclanche Cells. In the primary cells we have hitherto dealt with, the liquid acting on the positive plate is an acid and the depolariser a fluid, but an important type of cell was devised by Leclanche in 1866, in which the liquid acting on the zinc, or positive plate, was a neutral liquid, viz. a solution of ammonium chloride, popularly called sal ammoniac, NH 4 C1,* 2-4 6O Fig. 8O IOO 12O Ampere hours 340 160 ISO 200 1156. Charge and Discharge Curves of Acid and Alkaline Storage Cells of about equal capacities, t and the depolariser was a solid, manganese peroxide, MnO 2 , packed with bits of gas carbon round the carbon or negative plate. The " Leclanche " cell is, therefore, a single-fluid cell, the porous pot seen in Fig. 116, which illustrates one of the earlier forms of this type of cell, being used merely for the purpose of keeping the mixture of manganese peroxide and broken gas carbon in contact with the carbon plate ; and, to prevent the mixture being shaken out of the pot, it is closed at the top with pitch. A small vent hole is left in the pitch to allow air and gas to escape from the porous pot. The zinc is made in the form of a rod * This salt can now be obtained in the form of large pellets, or buttons ; they are sometimes called " Voltoids." t The quantity of electricity (expressed in ampere hours) which a fully charged cell will produce before re-charge is necessary, is called the "capacitv" of the cell. PRACTICAL ELECTRICITY with a copper wire cast into the top of it, and the rod rests in a recess in the corner of the glass jar made to receive it. Electric contact with the carbon plate is sometimes made by means of a lead cap cast on to it, firm connection being made between them by the lead run- ning into two small holes drilled side- ways through the top of the plate, and thus riveting the cap on the plate. To prevent the liquid creeping up by capil- lary action between the top of the carbon plate and the lead cap, where it would form a salt of lead and intro- duce a high resistance between the two, the top of the carbon plate, after the holes have been drilled in it, is heated for one hour in paraffin wax at a temperature of 110 C., and thus rendered non-porous. Improved methods of making contact with the carbon electrodes are shown in Figs. 119 and 120 ; carbon heads are formed on the plates during manufacture and holes provided in the heads to receive the metal terminals. The chemical action of the Leclanche cell is as follows : Before sending a current, A(C) -f ;(MnO 2 )-fw(NH 4 Cl)-j-(Zn). After sending a current, Fig. 1 1 6. LeclanchS Cell with Porous Pot. (w 2) (NH 4 Cl) + (ZnCl 2 ) + ( i) (Zn). Manganese peroxide is therefore re- duced to manganese sesqui-oxide, Mn 2 O 3 , sal ammoniac and zinc are used up, water and zinc chloride, ZnCl 2 , are formed, and ammonia gas, NH 3 , is given off. Substituting the atomic weights we see that for every 50 grains of zinc used up about 82 grains of sal ammoniac are consumed, and about 134 grains of the manganese peroxide are reduced to managnese sesqui-oxide. If, however, too little sal ammoniac be present, zinc oxide, or zinc oxychloride, is formed instead _. ;. ., , ,. Fig. 117. Leclanche Agglomerate of zmc chlonde, and the solution Cel i- LEGL4NCHE CELLS 191 becomes milky ; hence, when this happens, more sal ammoniac should be added. The E.M.F. of a Leclanche cell is about 1-5 volts, but in the case of the porous pot form (Fig. 116) the E.M.F. falls rapidly when the cell is used to send a strong current. It will, how- ever, regain its value if the cell be left for some time unused, and it does not sensibly diminish when the cell is put on one side, even for some months. Hence, while the Leclanche cell is much inferior to the Daniell's for the purpose of sending a steady current for an hour or two, it is much superior to the Daniell's cell for pro- ducing intermittent currents at any time during the course of a year or more for example, such currents as are employed for the ringing of electric bells, for house telephones, and for railway signalling. The objections to this simple form of Leclanche cell, in addition to its rapid polarisation, are (i) the use of the porous pot, which increases the resistance of the cell ; (2) the evaporation of the liquid indicated by the liquid filling only half the cell in Fig. 116 ; (3) the eating away of the zinc rod which occurs at the surface of the liquid, thus rendering the rod useless before the lower part is used up ; and (4) the creeping of the salts, this latter defect being, however, * partly counteracted by dipping the top of the porous pot and of the glass j ar as well as the upper part of the carbon plate into melted ozokerite, or, best of all, into paraffin wax melted in warm oil. Various modifications of the Leclanche cell have been introduced to overcome the first two defects. M. Leclanche in 1871 dispensed with the porous pot by replacing the mixture of manganese peroxide and gas carbon with a solid agglomerate composed of 40 parts of granulated manganese peroxide, 52 of granulated carbon, 5 of gum shellac, 3 of potassium sulphate, and a small quantity of sulphur. This mixture is heated to 1 00 C, and pressed into moulds under great pressure : the sulphur volatilises and leaves the blocks in a porous condition, so that the liquid can soak into them. The negative plate is formed by binding a block of the agglomerate, a, on each side of the carbon plate with indiarubber bands (Fig. 117). Fig. 119. "Carsak Cell." (General Electric GJ ) PRACTICAL ELECTRICITY Other modifications of the Le- clanche cells are shown in Figs. 118, 119, and 120, which represent " Six block agglomerate," " Carsak," " and " Lacombe central zinc " cells respec- tively. Cells of the latter type are by some firms termed " Car porous " elements. In the " Carsak " cell (Fig. 119) and other forms of " Sack " cell the chief objections to the ordi- nary porous pot form are eliminated by using a large hollow cylinder of zinc instead of a rod, substituting powdered MnO 2 for the granular variety, and replacing the porous pot by a wrapping of canvas or sacking. These features reduce the internal resistance of the cells con- siderably and greatly increase the current they can produce. The curves in Fig. 121 give the results of tests made on three types of Leclanche cell, when the outside resistance in each case was maintained constant at 10 ohms, the plan adopted by the Post Office for testing cells, and we see that the current under these circumstances fell to half its value in seven, fifteen, and twenty days respectively with the ordinary porous pot Leclanche cell, the " Six block agglomerate " cell, and the " Carsak " cell. It will be ob- served that in the first two cases the polarisation is somewhat rapid at the beginning and especially rapid in the case of the ordinary porous pot Leclanche cell. With the " Carsak " cell the fall of current is much more gradual. It should, however, be mentioned that the " life " (i.e., the time in which the current the cell sends through an external resistance of 10 ohms falls to half its initial value) of many forms of agglomerate block cell is con- Fig. 1 20. Lacombe Central Zinc Cell. Perforated Carbon Cylinder with head ; B, Cylinder of Porous Porcelain ; c, Glass Base uniting A and B ; M, Mix- ture of Ca-rbon and Manganese Di- oxide (Depolariser) ; N, Insulator ; o, Binding Screw ; R, Washer ensuring contact ; s, Solution of Sal Ammoniac ; z, Zinc Rod. LECLANCHE CELLS D siderably less than that of porous pot cells made by the same firm. Using powdered, instead of granulated manganese peroxide, increases the " life " of a cell. Example 52. If 2 Ibs. of zinc have been consumed in a Leclanche battery, how much sal ammoniac has been utilised in the same time ? Answer. About 3-3 Ibs. Example 53. Compare the rates of using up manganese per- oxide and sal ammoniac in a Leclanche cell. Answer. Approximately, as 163 to 100. Example 54. What is the cost of the material consumed in 6 Leclanche cells in series when developing a current of o-i ampere for three hours a day for 200 days, if 10 per cent, of the material used is wasted through local action ? Take the price of zinc as 2^d. per lb., of sal ammoniac 455. per cwt., and of manganese peroxide as 145. per cwt. Answer. Cost of zinc, 3d. ; sal ammoniac, 8Jd. ; manganese peroxide, 4|d. A new form of Leclanche cell, devised by M. Fery, which dis- penses with porous pot and solid depolariser, is shown in Fig. 1200. The carbon c takes the form of a hollow cylinder with slotted sides, and the zinc z is a thick plate resting on the bottom of a glass jar v ; a wooden cross s separates the carbon from the zinc. The oxygen of the air, some of which is dissolved in the electrolyte, acts as depolariser. On closing the circuit, hydrogen forms on the lower end of the carbon tube, as this end, being nearest the zinc and therefore carrying most of the current, and the hydrogen with the carbon and the dissolved oxygen (which is most concentrated near the surface of the liquid) forms a local gas battery which causes the hydrogen to recombine. Carbon of a definite porosity is found to give the best results. In this cell sal-ammoniac is regenerated during working, and thus enables cells of large ampere-hour capacity to be produced. 76. Dry Cells. The commonest form of primary cell used at present is a variety of Leclanche called the " Dry cell." The N Fig. izoa. Section through Fery's modified Leclanche" Cell. 194 PRACTICAL ELECTRICITY name, although a convenient one, is not correct, for a really " dry" cell would produce no appreciable current. A certain amount of moisture must be present, otherwise the resistance of the cell would be extremely high. Moisture is also necessary to the S9J9cllU1Pl||IUI UJ chemical action involved. Many attempts have been made to construct a cell which could be turned upside down or used in any position without interfering with its action. Volta constructed a battery of zinc and copper plates with pieces of moist cloth in- DRY CELLS 195 serted between them. Zamboni used discs of paper covered on one side with tin and on the other with manganese peroxide ; but batteries of this type, although they could produce a large E.M.F. when a sufficiently large number of elements was employed, were only able to furnish an extremely small current in consequence of their large internal resistance. Wolf, Keisen, and Schmidt tried to make a " dry cell " of moderate resistance by mixing sawdust with cellulose. Desruelles filled a Leclanche cell with asbestos fibre and spun glass ; Pollak employed a gelatine glycerine ; but the first to construct a dry cell which could be successfully used to produce an appreciable current was Gassner in 1888. The " Gassner' s " dry cell was a form of Leclanche cell, the plates being formed of carbon and zinc, the latter being made in the shape of a pot to contain a. jelly which surrounded the carbon rod. This jelly was composed of sal ammoniac, zinc chloride and oxide, calcium sulphate, and water, the zinc oxide being possibly added to give porosity to the jelly. The E.M.F. was about 1-3 volts, the internal resistance of different cells of the same size was very different, and the resistance of any one cell varied in an irregular way during working. The cells polarised rapidly when used, and were also liable to short-circuit internally. Nevertheless, their compactness, portability, freedom from all creeping of the salts, and the fact that they did not dry up, led people to consider whether cells constructed somewhat on the principle of the Gassner dry cell might not be manufactured so as to be commercially useful. In most modern dry cells a carbon rod or plate surrounded by a depolariser, consisting mainly of manganese peroxide, carbon and graphite, forms the inner electrode, and a zinc pot or case containing a semi-solid electrolyte containing sal ammoniac and zinc chloride forms theouterone. Their E.M.F.s are about 1-5 volts and their resistances average from about o-i to 0-5 ohms, according to size. They may be classified in four main types as follows : (1) Those in which the electrolyte is in the form of a jelly between the depolariser and the zinc. (2) Those in which the electrolyte is a nearly solid paste usually containing plaster of Paris. (3) Cells having the electrolyte held in an absorbent paper or pulp lining to the zinc container, and into which the depolariset is packed. (4) " Desiccated " or " Inert " cells which are made active when required by the addition of water. Examples of type (i) are the Hellesen cell described in Sect. 77 196 PRACTICAL ELECTRICITY and the well-known Dania cell made by the Atlas Battery Co. Of type (2) the " G. E. C." cell, the " E. C. C." cell, and the " Obach " cell are makes in common use. Type (3) includes the Blue Bell and Columbia cells of American make. Cells of type (4) have been devised to minimise the chief defect of ordinary dry cells, viz. : serious deterio- ration when kept in stock for any considerable time, ._. particularly in hot places. In these cells the electro- lytic materials take the form of dry powders, which __ 2t remain unaltered and in- active until water is added when the cells are required for use. Some of the best known makes are the ' B " Extra Sec " cell made by the General Electric Co., the Dura, made by Messrs. Siemens Bros., the Inert, made by the India Rubber and Gutta Percha Co., the H 2 O, the Reliable, and W. O.* cells made by the Edison Swan Co., the Asso- ciated Battery Co., and the Atlas Battery Co. respec- tively. Dry cells are now manu- Fig. xa2. Heiiesen Dry Ceil. f actured in huge numbers for use in electric torches. Usually two or three are placed side by side connected in series and put in a cardboard case, the combination forming a " refill." 77. Heiiesen and Dania Dry Cells. In the Heiiesen dry cell, which was introduced into England by Messrs. Siemens Bros., about 1890, a carbon rod c, Fig. 122, is surrounded by a black paste D, composed of manganese peroxide, graphite, and ammonium chloride wrapped in calico and tied with string. Outside this is a white paste, E, of ammonium chloride, water, flour, and plaster of Paris. These materials are contained in a round zinc pot z placed in a square millboard case B, the corners of which are packed with sawdust s d. The tops of the depolariser * Signifying "water only." V- "O.K. C." Cell. z - G.E.G. AND OBAGH CELLS and excitant are covered by a layer* of plaster of Paris, P, and the whole sealed with a bituminous compound s. An air tube A is provided to carry off any gas that may be generated in the cell. In the Dania cell the carbon is sur- rounded by the depolarising paste enclosed in a form of sack, and the gelatinous electrolyte is placed between the sack and the zinc. Numerous wooden pegs driven into the sack and projecting therefrom prevent contact between sack and zinc. A layer of insulating material separates the sack from the bottom of the zinc container, which latter is sur- rounded by a cardboard case. A layer of sawdust rests on an annulus of paraffined paper, and the cell is sealed, except for a vent, by a bituminous compound. 78. G. E. C. and Obach Cells. In the G. E. C. cell, shown in Fig. 123, the carbon c is in the form of a flat plate and is surrounded by de- d polarising paste F of carbon and manganese peroxide. Between the cylinder F and the zinc cylinder z is a white electrolytic paste G of plaster of Paris and sal ammoniac. The pastes are covered with sawdust s over which is a bituminous seal H. The Obach cell, made by Messrs. Siemens Bros., is shown in Fig. 1230. Here a carbon rod c is surrounded by a depolarising paste D containing about 55 per cent, manganese peroxide, 44 per cent* plumbago, and i per cent, of gum ; it rests on an insulating layer inside the zinc container z and the electrolytic paste E, com- 3. Obach Dry Cell. 198 PRACTICAL ELECTRICITY Carboa Fig. 123*." Blue Bell " Cell. posed of 85 per cent, plaster of Paris, 15 per cent, flour moistened with sal ammoniac, is poured between the depolariser and the zinc. Sawdust or ground cork resting on a canvas disc c d is placed between the pastes and the seal s. A cardboard box B surrounds the zinc and seal. 79. Blue Bell and Columbia Cells. The Blue Bell cell, made by the Western Electric Co., is largely used in the telephone work of the firm. The carbon is of fluted cross section (Fig. 1236), and rests on several layers of pulpboard E. The zinc container z is lined with absorbent paper G Section of which is folded over the top of the depolariser paste F after the latter has been rammed in between the carbon and the paper lining. The latter is, of course, moistened with sal ammoniac solution. A layer of sand s is placedbetween the paste and the seal H. A section of the Columbia cell is given in Fig. 1230, the lettering of the parts corresponding with those in Fig. 1236, excepting that the layer of sand s in the Blue Bell cell is replaced by layers of sand and sawdust shown at s x and S 2 respectively, and a corru- gated cardboard collar S 2 between them. The air space in S 2 gives room for expan- sion of the electrolyte caused by the passage of large currents. The letter G in Fig. I23C indicates absorbent pulpboard. 80. Extra-Sec and Inert Cells. These cells are good examples of type 4, viz., Desic- cated Cells. The Extra-Sec cell, made by the General Electric Company, is very similar in construction to the Dania, de- scribed in Section 77. Instead, however, of the electrolyte being in a gelatinous form between the sack and the zinc, this space is Fig. y. Columbia Ceil, partly filled with the electrolytic salts in a dry state, together with gum in powder form, the whole material being specially treated by a method which makes it non-hygroscopic. A small ebonite tube passing through the seal of the cell communicates with the space above EDISON-LALANDE CELLS 199 mentioned, and through this tube the space is filled with water when it is desired to make the cell active. The water dissolves the salts and forms with the gum an electrolyte of the gelatinous type. The Inert cell, made by the India Rubber and Gutta Percha Company, is shown in section in Fig. 124. A carbon rod A is surrounded by the depolarising mixture contained in a sack B bound up with string ; a rubber band c separates the sack from the zinc container D. Between B and D is rammed the electrolyte E in the form of dry powder. A cardboard case F surrounds the zinc. G indicates two cork covers through which the stopper s H and vent tube j pass. K and L are the positive and negative terminals respectively, and the space s is filled with water when the cell is required for use. 81. Edison-Lalande Cell. This cell, shown in Fig. 125, consists of plates of black oxide of copper and zinc immersed in a solution of caustic potash, a layer of heavy oil being poured over the solution to prevent evaporation and " creep- ing." No local action or polari- sation takes place in this cell ; under normal conditions it is an easy matter to set it up to give any required number of ampere hours, and to so proportion the constituents that they are all exhausted at the same time. This is a matter of considerable importance where closed circuit working is employed, as in some systems of telegraphy and in " alarm " circuits. Although the E.M.F. of the Edison-Lalande cell is low (0-75 volt), its resistance is also low, and the cell is capable of producing large currents. A strong solution of caustic potash, one to three by weight, is usually employed. Example 55. Assuming the chemical action in the Edison- Lalande cell to be represented by the equation k (CuO) + / (KHO) + m (Zn) = Cu + (k - i) (CuO) + OH 2 + (1-2) (KHO) + (K 2 ZnO 2 ) + (m i)Zn, calculate the amounts of Fig. 124. Section through " Inert" Cell. 200 PRACTICAL ELECTRICITY copper oxide, caustic potash and zinc required per cell to pro- duce i ampere continuously for 30 days. Answer. The quantity of electricity =i x 24 x 30 ampere hours, =720 and the molecular weights of the substances are in the approxi- mate ratio, 63-5 + 16 ; 39 + i + 16 ; 65-5, i.e., 79-5 : 56 : 65-5. Now the electrochemical equivalent of zinc is 0-000339 grammes per coulomb (approximately) or 1*22 grammes per ampere hour. Weight of zinc required = 720 x 1-22 grammes, = 878 = 1-94 Ibs. approx. Weight of copper oxide = i'94 x 79-5. 6 5 -5 = 2-35 Ibs. Weight of caustic potash _ 1-94 x 56 x 2*, 65-5 Fig. 125. Edison-Lalande Cell. = 3-31 Ibs. A modified type of this cell, called the " Neoiherm " cell, is now made, in which the copper oxide, the depolariser, forms a lining to the containing vessel, which is of iron. When the cell has become discharged, the deposited copper can be re-oxidised rapid- ly by heating the iron vessel in an oven. By this means the cost of re-charging is greatly reduced. A cell weighing 12 Ibs, will give i ampere for 150 hours ; the E.M.F. is approximately i volt, and its initial resistance about o-i ohm. 82. Standard Cells, Clark's and Weston's Cells. The cells previously described are intended for use as current generators, and for this purpose high E.M.F. and low internal resistance are desirable features. Those described in this section are designed as standards of E.M.F., so that uniformity and constancy are the principal requirements for this class of cell. Cells which give E.M.Fs. whose values are accurately known, enable many elec- trical measurements to be conveniently made (see Chapter IX.), and much time and trouble have been devoted by many experi- menters to the production of such cells. The best known and * From the formula it will be seen that 2 molecules of caustic potash are used up per i molecule of copper oxide. CLARK'S STANDARD CELL 201 most useful standards are those devised by Mr. Latimer Clark, of London, in 1872,* and by Mr. 'Edward Weston, of Newark, New Jersey, U.S.A., in 1893! respectively. Both types of cell have been made up in many shapes and forms, but the H-form, introduced by Lord Rayleigh in 1882, Fig. 126, or some modifica- tion of it, is now generally used. In the Clark's cell the materials used are pure mercury, mercurous sulphate, solution of zinc sulphate, and zinc amal- gam, the latter being made by dis- solving pure zinc in pure mercury. Such an amalgam behaves electrically like pure zinc. The pure mercury, M, Fig. 126, previously distilled in vacuo, is placed in one leg of the H tube and covered with a layer of paste, M s, made by mixing mercu- rous sulphate with a saturated solu- tion of zinc sulphate. The zinc amalgam A is placed in the other leg of the tube and both legs and the cross tube are nearly filled with zinc sulphate solution z z, crys- tals of zinc sulphate being added to H F rm - About one-half of full size. ensure that the solution may be saturated at all ordinary temperatures. Electrical connection with the mercury and with the amalgam is made by platinum wires, w w, sealed into the lower ends of the legs. The upper ends of the vertical tubes are closed, either by corks and marine glue, or, preferably, by drawing out the tubes in a blowpipe flame and hermetically sealing them. Cells set up in this way, using pure materials, have E.M.F.s remarkably equal in value under specified conditions. Equality within one-tenth of one per cent, is easily obtained, and cells set up with great care will not differ in E.M.F. by more than a few parts in one hundred thousand. This E.M.F. is, for normal cells, given by the expres- sion E*=l-432 8 0-0011 (t 15), international volts, (19) where t is the temperature in degrees centigrade. From this formula it will be seen that the E.M.F. of a Clark's cell at normal temperature, 15 C., is 1-433 (very approximately) volts, and that a rise of temperature produces a fall of E.M.F. of i-i millivolts per degree C. * Proc. Roy. Soc. , vol. xx., p. 444. f The Electrician, vol. xxx.. D. 741. 202 PRACTICAL ELECTRICITY In using the Clark's cell as a standard of E.M.F. it is desirable that its temperature be maintained fairly constant, for if the temperature be altering rapidly the E.M.F. does not change as quickly as the temperature, so there is a lag of E.M.F., and the actual E.M.F. may differ appreciably from that given by the for- mula above. Should the cell be allowed to generate much current Marine glue* -Zinc rod. -Cork Zirtc sulphate solution, crystals. Me rcu rous sulphate Mercury. Fig. 127. Clark's Cell, Board of Trade (1894) foira (full size). polarisation occurs, and the E.M.F. is temporarily reduced, but after a period of rest the cell becomes normal. Usually a few minutes' rest will permit the cell to recover its E.M.F. to within one part in a thousand, unless the cell has been left on closed circuit for a long time. Several other forms of Clark's cell are shown in Figs. 127, 128, and 129. For transport the form devised by the late Dr. Alex. Muirhead, and shown in Figs. 129 and 130, where the mercury is replaced by a well amalgamated platinum wire, has many advantages. The chemical action which occurs in a Clark's cell may be represented as follows : Before sending a current A(Hg) + J(Hg 2 S0 4 ) + w(ZnS0 4 ) + (Zn) After sending a current (k + 2) (Hg) + (/-i) (Hg 2 S0 4 ) + (m + 1) (ZnS0 4 ) + (- 1) (Zn), the mercurous sulphate acting as the depolariser. CLARK AND WESTON CELLS 203 The chief objection to the Clark's cell as a standard of E.M.F. is its comparatively large temperature co-efficient, which, as before stated, amounts to 0-08 per cent, (i-i millivolts) per degree, and to overcome this defect Mr. Weston replaced the zinc of the Clark's cell by cadmium. The Weston cell, Fig. 131, Fig. 128. Kahle's Modification of the Raylei^h H form of Clark Cell (full size) ZS.s, zinc sulphate solution ; ZS.c, zinc sulphate crystals ; MZ.S, mercurous sulphate and zinc sulphate paste ; M, mercury ; A , amalgam of zinc and mercury. therefore contains pure mercury, mercurous sulphate paste, a saturated solution of cadmium sulphate, and cadmium amalgam ; crystals of cadmium sulphate being also added to keep the solu- tion saturated. By this substitution of cadmium for zinc the temperature co-efficient is reduced to about one-twentieth its value for the Clark cell, mainly owing to the solubility of cadmium sulphate changing much less with temperature than that of zinc sulphate. The substitution also reduces the E.M.F. consider- ably, but this is no disadvantage ; in fact, for many purposes a 204 PRACTICAL ELECTRICITY standard cell with an E.M.F. as low as ^ of a volt would be very convenient. Of late years numerous measurements have been made of the E.M.F. of the West on cadmium cell, both in America, Ger- many, and Great Britain, the most accurate determination having been carried out by the authors and Mr. F. E. Smith, O.B.E., F.R.S., at the National Physical Laboratory, Tedding- ton.* Marine Oiue HB a S0 4 Paste ~ Fig. 129. Section of Portable Clark Cell (Muirhead's Form). The form of cell used in experiments is shown in Fig. 132, and the value obtained, after reducing to 20 C., and allowing for the difference between the international ampere and the true ampere, is E = 1-0183 volts at 20 C. (20) This value, at 20 C., was adopted by the International Con- ference on Electrical Units and Standards which met in London in October, 1908. The following formula was also agreed to for calculating the E.M.F. at temperatures between o C. and 40 C. E; = E 20 - 0-0000406 (t - 20) - 0-00000095 (t - 20) 2 + 0-00000001 (t 20) 3 . (21) For many purposes no temperature correction is necessary, for a change of 10 alters the E.M.F. by less than I part in 2,000, and for a change of 20 C., the alteration only slightly exceeds I in 1000. * On a New Current Weigher, and the Determination of the E.M.F. of the Normal Weston cadmium cell. Phil. Trans. , 1907. CLARK AND WESTON CELLS 205 The cadmium cell therefore pos- sesses a marked advantage over the Clark as regards variation of E.M.F. with temperature, and on this ac- count is being adopted internation- ally as a secondary standard of electric pressure. A specification for setting up cells of this type is given in Appendix I. In Mr. Smith's form of cell the constrictions in the sides of the vertical tubes prevent the crystals being displaced even if the cell be turned upside down, and thus renders the cell much more portable than it would otherwise be. Example 56. What is the E.M.F. of a normal Clark's cell at I2.6 C., and i8.5 C. respectively ? Answers. 1'435 4 volts and 1-4290 volts. Fig. 130. Mail-head's Portable Clark's Cells (Mounted). Fig. 131. Weston's Cadmium CelL Example 57. At what temperature will the E.M.F. of a Clark's cell be 1-434 volts ? Answer. 14 C. Example 58. Find the E.M.F. of a Weston normal cell at the 206 PRACTICAL ELECTRICITY following temperatures, 10 C., 15 C., 25 C., and 30 C. to five significant figures. Temperatures. volts. Answer. 15 C. 25 C. 30 C. E.M.F. 1-0186 1-0185 1-0181 1-0178 83. Calculation of the E.M.F. of a Cell from the Energy Liberated by the Chemical Action. We have seen that a cell can cause an electric current to flow round a circuit, and that chemical changes occur in the cell during the time the current is passing. If the external circuit consists of a simple wire, there is heat generated in the wire, and this heat is produced at the expense of Fig. 132. Weston Cadmium Cell (F. E. Smith's Form). A, amalgam; Af t mercury; P, mercurous sulphate paste; C, crystals of cadmium sulphate; S, solution of cadmium sulphate. the chemical energy of the constituents of the cell. For example, in a Daniell's cell zinc is dissolved in the sulphuric acid and copper deposited on the copper plate. Now when metals are acted on by acid outside a cell, heat is generated. Experiments made on the amount of heat generated during solution have shown that 106,000 calories,* approximately, are produced by dissolving 65 grammes * A calorie is the amount of heat required to raise i gramme of water CALCULATION OF E.M.F. OF CELL 207 of zinc in dilute sulphuric acid,* and about 56,000 calories by dissolving 63-5 grammes of copper. In the actual cell the copper is removed from solution, heat being absorbed in the process, so the nett amount of heat generated during the time 65 grammes of zinc are dissolved will be 106,000 56,000 calories i.e., 50,000 calories. This, if the law of conservation of energy be true, must be equivalent to the electric energy or work produced by the cell, if none is wasted by local action or otherwise. In Section 48 we have denned P.D. (or E.M.F.) so that the product of P.D. and quantity of electricity which flows under T32. Weston Cadmium Cell i". E. Smith's Form, Mounted). that P.D. shall represent work or energy. Calling the E.M.F. E and the quantity Q we may write EQ = Energy and the energy in h calories is hj, where / is the mechanical equivalent of heat (42 million ergs per calorie approximately). If, therefore, we equate the electric energy to the heat energy we get or E =- an expression which gives the maximum possible value for E, as this assumes no waste. Now the value of Q can be determined by finding the quantity * Thomson's Thermo-Chemistry (translated by Burke, 1908), p. 325. 208 PRACTICAL ELECTRICITY of electricity required to deposit 63-5 grammes of copper or 65 grammes of zinc. This = 63-5/0-0003286 coulombs (see Section 10), = 63-5/0-003286 C.G.S. units of quantity. Fig. 1323. Tinsley Cell. Inserting the numbers in the above equation we have E = ,000 x 42,000,000 x 0-003286 C.G.S. units, 63-5 = 1-086 x io 8 C.G.S. units, = 1-086 volts, (since i volt = io 8 C.G.S. units). In this way we deduce from purely mechanical and thermal experiments and our definitions of E.M.F. and quantity, the approximate value of the E.M.F. of a galvanic cell, a matter of great scientific and practical importance, as it shows the intimate relation that exists between mechanical, thermal, chemical and electrical quantities. That the E.M.F of a Daniell's cell is about 1-08 volts is now a well known fact. Calculations similar to the CALCULATION OF E.M.F. OF CELL 209 above were first made by the late Lord Kelvin in 1851 to ascer- tain the E.M.F. of the Daniell cell in terms of the absolute electro- magnetic unit of P.D., io 8 of which were, several years afterwards, viz., in 1862 called I volt. At the present time the E.M.F. of any cell can be measured directly by a high resistance voltmeter, but in 1851 no voltmeters or ammeters or resistance coils adjusted in ohms, or standard cells existed. NOTE. Persons who desire further information about primary batteries, and the cost of electric energy produced by such means, should consult the 1896 edition of this work. CHAPTER VI RESISTANCE ; ITS LAWS AND MEASUREMENT 84. Comparing Resistances: Voltmeter and Ammeter Method 85. Ohmmeter: Megger 86. Simple Substitution Method of Comparing Resistances 87. Differential Galvanometer ; A Null Method 88. Wheatstone's Bridge: its Principle 89. Wheatstone's Bridge: its Use and Simple Method of Constructing 90. Bridge Key 91. Use of a Shunt with the Bridge 92. Meaning of the Deflection of a Bridge Galvanometer 93. Conditions Affecting the Resistance of a Conductor 94. Variation of Resistance with Length 95. Variation of Resistance with Cross Section 96. Variation of Resistance with Material 97. Resistance of Metals and Alloys per Centimetre Cube and per Inch Cube. Specific Resistance or Resistivity 98. Resistance of Metals and Alloys for a given Length and Weight 99. Variation of Resistance with Temperature 100. Conductors of Large Specific Resistance have Small Temperature Coefficients 101. Conductivity and Conductance 102. Comparison of Electric and Heat Conductivities 103. Resistance and Conductance of Several Conductors in Series or in Parallel 104. Currents in Parallel Conductors 105. Kirchnoff's Rules 106. Shunts 107. Multiplying Power of a Shunt 108. Usual Method of Constructing a Shunt Box 109. Increase of the Main Current Produced by Applying a Shunt no. Principle of Universal Shunts in. Method of Constructing a Universal Shunt Box ; Advantages of Universal Shunts 112. Stand- ard Resistance Coils 113. Ordinary Forms of Wheatstone Bridge 114. Portable Forms of Wheatstone Bridge 115. Dial and Bar Patterns 'of Bridge. 84. Comparing Resistances : Voltmeter and Ammeter Method. By the method described in Section 56, and illustrated in Fig. 91, two resistances can be compared if the relative calibration of a voltmeter only be known. Further, any of the methods described in Section 62 for calibrating a voltmeter in volts, which depend on using a conductor whose resistance is known in ohms, can be used for measuring a resistance in ohms, if the voltmeter has been previously calibrated in volts. The one of these methods which is illustrated in Figs. 94 and 940 is par- ticularly useful when we desire to know the resistance of a con- ductor which is much heated by the passage of a current through it for example, the resistance of the luminous carbon filament of a glow lamp, or the apparent resistance of the " electric arc." The name " resistance " here means, as before, the ratio of 210 . OHMMETERS 211 the P.D. in volts to the current in Amperes, but in these two instances it is no longer a constant quantity and independent of the current passing, so that it is only by a sort of extension of the name " resistance " that it can be used at all in such cases. Indeed, had the early experience of currents passing through conductors been always with currents large enough to produce considerable warmth in the conductor, it is probable that we should never have acquired the conception we now possess of a conductor having a definite resistance as it has a definite length or a definite cross-section. If in Figs. 94 and 940, the readings of the ammeter and volt- meter are / amperes and V volts respectively, then the resistance of the conductor c, will be given by the formula y R = (approximately), the approximation arising from the current through the voltmeter in Fig. 94, and the resistance of the ammeter in Fig. 940, being neglected. Calling the resistance of the voltmeter R v , and that of the ammeter R a the correct expressions for R are : y R = - - , for Fig. 94. for Fig. or 85. Ohmmeter : Megger. Frequently, when we are measuring the resistance of a conductor traversed by a strong current, as, for example, the apparent resistance of an electric arc, we desire to know in addition the current which is flowing. In such a case the necessity of having to take simultaneous readings of an ammeter and a voltmeter in order to ascertain the resistance, is no disadvantage, since two things have to be ascertained, and, therefore, two measurements must necessarily be made at the same time. But in other cases, when the resistance alone has to be ascertained, it may be a disadvantage to have to take readings of two distinct instruments simultaneously. Hence an instrument called an " ohmmeter " was devised by Professor Perry and one of the authors (W. E. A.) to enable the resistance of any part of a circuit, through which a current is passing, to be measured by making a single observation. 212 PRACTICAL ELECTRICITY A simple ohmmeter contains a " current coil " cc (Fig. 133) and a P.D. or " pressure coil " cc placed usually at right angles to one another, and both acting on the same magnetic needle. The former coil has its terminals T T connected with the circuit, the resistance of some portion of which it is desired to measure, so that c c is in series with the circuit, while t, t, the terminals of the pressure coil, are joined with the points H and j, the ends of that bit of the circuit whose resistance, R ohms is wanted, in the same way as a voltmeter, would be placed in parallel with H j. The resistance of the current coil is made as low as possible, while the portion of the ohm- Fig. lasDiagram of ohmmeter. meter between the terminals t and / is made relatively very high, either by the pressure coil c c itself being wound with a very long fine wire, or by an auxiliary resistance being added to this coil and included in the instrument between the terminals t, t. If the needle be short, the force due to the current passing round either of the two coils will be perpendicular to the plane of that coil (Figs. 38, 49). Further, if the needle be made of hard steel so that its magnetism is not altered by the currents in the coils, these two forces will be directly proportional to the currents respectively. Hence the needle will be acted on by two forces at right angles to one another ; one directly propor- tional to V, the P.D. in volts between the points H and j, the other directly proportional to 7, the current in amperes passing through the conductor H j. Consequently, if matters be so arranged that no other magnetic forces than the two just mentioned act on the needle, it will place itself so that the tangent of the angle it makes with the plane of the pressure coil will be directly proportional to the ratio of V to 7, that is to R, the resistance in ohms of the conductor HJ (see Section 31). Further, if all extraneous magnetic action be avoided, then, whether the needle be short or long, made of soft iron or of hard steel, it will place itself at right angles to the plane of the current coil, that is, parallel to the plane of the pressure coil, when t, t are both connected with the same point H, that is, when the resistance of the part of the main circuit included between the two terminals t and t is nought. As the leads to terminals t, t are separated, so as to make contact with points of the main circuit farther apart, say, EVERSHED'S OHMMETER 213 with H and K, the P.D. between the terminals of the pressure coil will increase, and the needle will deflect away from the plane of the pressure coil. And, although the tangent of this deflection may not be directly proportional to the ratio that the P.D. between the points H and K bears to the current passing through the conductor H j K, the deflection will be quite constant as long as the terminals t t are connected with the points H and K respectively, or with any two other points in the main circuit separated by the same resistance, whatever may be the current passing through the main circuit. For if the main current be doubled, the P.D. between the points Line C K Generator Earth Fig. 134. Diagram of Connections of Evershed Ohmmeter. H and K will be also doubled, therefore both the forces acting on the needle will be increased in the same ratio, the resultant will be in the same direction, and the deflection will remain as before. Hence, whatever the shapes and sizes of the two coils and of the needle, the scale of the ohmmeter can be graduated to read off resistances directly in ohms, provided that the only forces acting on the needle be those due to the currents flowing round the pressure and current coils respectively. The principle of the ohmmeter has been employed by Mr. Evershed in constructing a commercial instrument that has been much used for measuring the resistance to leakage of electric -light wires and fittings. The connections of the Evershed ohmmeter are shown in Fig. 134, the lettering being arranged to correspond with that in Fig. 133. The current necessary to work the instru- ment is obtained by means of a portable generator G, or by a battery of a large number of cells. By comparing Fig. 134 with Fig 940 it will be seen that the connections are the same, and 73 therefore we have R + R a = j (see Section 84), where R a is the resistance of the current coil c c, Fig. 134. For simplicity, the magnet and pointer have been omitted in the latter figure. As R a is constant for any given instrument, the position the pointer 2i 4 PRACTICAL ELECTRICITY takes up when R= o can be marked o, so the resistance of the current coil can be allowed lor in this way, and the scale graduated to read off directly the resistance of R. To obviate errors of the readings which may be caused by magnetic forces other than those produced by the currents in the coils c c, and c c, Mr. Evershed has introduced a new form of instrument called the "Megger" (abbreviation for megohm - meter)* in which a fixed magnet and moving coils are used, instead of fixed coils and moving magnets. In fact he has applied the principle of the moving coil galvanometer to the ohmmeter, and thereby obtained the comparative immunity from disturbance by external magnetic fields which is a prominent feature of moving coil instruments (see Sections 43 and 61). He also uses the same fixed magnet to form part of the portable generator employed to produce the necessary currents, and thus combines in a single instrument the functions of generator and ohmmeter. In Fig. 134 an arrangement for increasing the range of resist- ance which the ohmmeter will measure satisfactorily, is shown. By moving the switch arm from A to B, thus joining j with B, the current coil c c is shunted by a resistance K B, so that only a fraction of the main current flows through it. The field of the current coil is therefore weakened and a higher reading is obtained on the instrument. Usually K B is made so that moving the switch from A to B gives a tenfold reading. 86. Simple Substitution Method of Comparing Resistances. If we merely wish to cut off a length of wire which shall have exactly the same resistance as that of some other conductor , for example, if we desire to make a resistance exactly equal to that of a standard ohm, or a standard ten-ohm coil, the following method may be adopted : In circuit with the conductor whose resistance we wish to reproduce, place any convenient current- generator and a galvanoscope. Neither the resistance nor the relative calibration, nor the absolute calibration, of this galvanoscope need be known. Observe the deflection. Next remove this conductor, and put in its place a piece of the wire out of which we desire to construct the resistance, of sufficient length that a smaller deflection of the galvanoscope is obtained with the same current -generator. Gradually diminish the length of this wire until the original deflection is obtained, then the resistance of this wire must be equal to that of the conductor, if no other change has occurred in the circuit. * A megohm is one million ohms. The instrument is called a megohm- meter because it is intended to measure very high resistances. COMPARING RESISTANCES 215 To detect any possible change in the sensibility of the galvano- scope, or in the strength of the current -generator during the test a change in either of which would, of course, destroy the accuracy of the reproduction it is well, after the wire has been shortened nearly sufficiently, to substitute the original conductor and see whether the deflection now obtained with it is exactly the same as it was at first. If it be found to be slightly different, then the final adjustment of the length of the wire must, of course, be made with the new deflection of the galvanoscope. Care must be taken not to shift accidentally the controlling magnet of the galvanoscope between the interchange of the conductor and the wire; further, the current -generator should not be allowed to Unknown Fig. 135. Comparing Resistances by Substitution Method. send a current for so long a time through either the conductor or the wire that there is any evidence of a falling -off of its power. In order to connect the galvanoscope and current -generator quickly, and conveniently, with either the known or the unknown resistance, a " plug key, or switch " (Fig. 135), may be conveni- ently employed. It consists of three sectors of brass, each carrying a terminal, fastened to a slab of ebonite, or hard wood, and a brass taper plug, P, which fits tightly into either of the holes, H or h, this plug being provided with an ebonite or a wooden handle. If, therefore, the plug P is put into the hole h, the current will pass through the known resistance, while if the plug be put into the hole H, the current will pass instead through the unknown. The current ^generator B, galvanoscope G, and the resistances R and R' are shown symbolically in the figure, whilst the plug key is in perspective. The preceding method of comparing the equality of two resistances is exactly analogous with Borda's method of double weighing, by means of which the weight of a body can be accurately compared with that of known standard weights, no matter how unequal be the lengths of the two portions of the beam of the balance, or how unequal be the weights of the scale pans. 2l6 PRACTICAL ELECTRICITY If the known resistance R r consists of a resistance box such as that shown in Fig. 89, then any unknown resistance within the range of the box may be measured by first observing the deflection of the galvanoscope produced when the unknown resistance is in circuit and then substituting the resistance box and finding by trial which plugs have to be taken out of the box to reproduce the deflection.* 87. Differential Galvanometer, A Null Method. The measure- ment of resistance by the method just described is not susceptible of great accuracy, for this depends on the exactness with which B G Fig. 136. Diagram of Differential Galvanometer Circuit. the deflections of the galvanoscope can be read and reproduced, as well as on the constancy of the battery supplying the current. To get over these disadvantages methods have been devised in which equality of two resistances is indicated by absence of deflection of a galvanoscope or galvanometer. Such methods are called " Null Methods," one of the simplest of these is that of the differential galvanometer. If the galvanoscope G in Fig. 136 be wound with two coils c and c' which exert equal forces on the needle when a given current passes through either of them, then, if equal currents be sent through them in opposite directions, there will be no deflection of the needle. If, further, coil c be put in series with the unknown re- sistance R, and c' in series with R 1 ', and in opposition to c, as in- dicated in Fig. 136, then, if the resistance of c is equal to that of c', there will be no deflection of G when R r = R, for under these conditions the currents through R and R' , and therefore through c and c' will be equal. The absence of deflection will thus in* dicate the equality of R' and R, and if R' be known then R also is known. By using sufficient battery power the currents through * In the practical use of resistance boxes and plug keys, it is important that the plugs and holes be kept quite clean, as well as the ebonite supporting the blocks. It should also be remembered that each plug acts like a wedge, and forces the blocks apart to some extent when it is inserted. When a plug is taken out it allows the blocks to approach each other, and thereby loosens the plugs in adjacent holes. //, therefore, any plug be withdrawn from a resistance box those on opposite sides of it should be re-tightened. DIFFERENTIAL GALVANOMETER ^ 217 the two circuits R c and R' c' would be large enough to cause a very small percentage difference in the two currents to produce quite an appreciable deflection of the galvanometer, so the method can be made very sensitive. Any change in the E.M.F. or resist- ance of the battery would affect both circuits equally, so this method of testing does not depend on the constancy of the battery. A galvanometer with two coils fulfilling the conditions stated above, viz. equality of magnetic effect and equality of resistance, is called a differential galvanometer. The two conditions are realised as follows : Two reels of silk -covered copper wire are chosen so that the diameter of the wire on each is as nearly as possible the same,* and the two wires are wound side by side on the galvanometer bobbin until it is nearly full ; the wires are then tested and cut, so that the resistance, but not, of course, necessarily the length, of each wire is the same. A current is now sent in opposite directions through the two coils in series, when it will be found that, although the wires have been wound on side by side, one of them will have a slightly greater magnetic effect than the other, partly perhaps because, being a trifle thicker, it has to be longer than the other, so as to have the same resistance, or partly because it is, on the whole, nearer the sus- pended needle than the other. To remedy this, a small portion of the wire having the greater magnetic effect is unwound, and without being cut, which would, of course, destroy the equality of the resistances of the two coils, the portion so unwound is doubled back on itself and coiled up out of the way in the base of the instrument. Thus, by unwinding more or less from the coil that was magnetically the more powerful, a very good balance can be obtained. In the use of differential galvanometers in which the needle is suspended by a silk fibre, a final and most delicate adjustment can be obtained by raising or lowering one of the levelling screws slightly, so as to tilt the needle nearer to or farther from one of the coils. And the spirit level attached to the instrument should then be permanently adjusted so that the bubble is in the centre of the glass cover of the level, after the instrument has been tilted in the manner just described. When a differential galvanometer is in adjustment no deflection will be produced if a current be passed through the two coils in parallel opposing, or in series opposing. A differential galvanometer can be used not only to indicate the equality of two resistances, but also to show when one resist- ance is any multiple or submultiple of another. For example, if * The wire on the two reels may, with advantage, have been cut from the same long length of wire. 2i8 PRACTICAL ELECTRICITY the terminals of the coil c (Fig. 136) be connected by a wire whose resistance is equal to c but which is arranged to exert no magnetic force on the needle of the galvanometer, then to produce balance R' must be equal to 2R ; for the coil c will only carry half the current passing through R (the other half passing through the wire in parallel with it) so that to give balance the current in R must be twice that in R'. This condition will be satisfied when R' 2R, for then the resistance of the path R' and c' will be double that of R and c with c " shunted " by a resistance equal to itself, and as the two paths are subjected to the same P.D., viz., the P.D. between p and Q, the currents in them will, by Ohm's law, be inversely as the resistances of the two paths ; the current in R will therefore be twice that in R'. Similarly if we put a second " shunt " on the coil c of resistance equal to the coil itself, balance would result when R' = $R. A single " shunt " of resistance equal to half that of either would produce exactly the same result as the two together. From the foregoing we can formulate a rule relating to shunted differential galvanometers, viz., if one oj its coils be shunted by a resistance -ih of its own resistance then n J balance will be produced when the resistance in series with the unshunted coil is n + i times that in series with the shunted coil, 88. Wheatstone's Bridge : its Principle. The differential gal- vanometer is a very convenient apparatus for ascertaining whether one resistance is a certain definite multiple of another ; but for accurately and rapidly comparing any two resistances, whether equal to one another or whatever may be their ratio, the " Wheatstone's bridge," or " Wheatstone's balance," as it is sometimes called, is more convenient. As the late Sir Charles Wheatstone explained, when he first gave a public description of the balance method of comparing resistances, the credit of its conception was due to Mr. Christie. The name of the better i K ^-^jz^^_^> /, #j known man, however, Fig. 137 . has been universally attached to the ar- rangement, which is shown symbolically in Fig. 137. Two conducting branches, P s Q, P T Q, are joined in parallel, and a current sent through the arrangement, as indicated by the arrows, then in passing from p to Q, either along the conductor p s Q, or along the conductor P x Q, there are points having all potentials between the potential of P and that of Q ; therefore it WHEATSTONE'S BRIDGE 219 follows that for every point in the conductor P s Q, there must be a point in the conductor P T Q having the same potential. Let s and T be two such points ; then, if they were joined with the terminals of an electrostatic, or of a current voltmeter, or indeed with the terminals of any galvanometer, there would be no deflection. Given one point s, the corresponding point T can, therefore, be experimentally found by joining one ter- minal of an electrostatic voltmeter, or of any gal- vanometer, to s and touching the other conductor P T Q at different points with a wire attached to the other ter- minal of the voltmeter or galvanometer, until a point T is found for which there is no deflection. In practice a galvanometer is generally employed, since a galvano- meter can be constructed so as to be a much more sensitive detector of a P.D. than an electrostatic voltmeter. Let I a be the current flowing along P s, then I a must be the current flowing along s Q also, since no current passes through a galvanometer connecting the points s and T (Fig. 138). Let /& be the current flowing along P T Q, and let R a , R^, R c , Rj, be the resistances respectively of P s, s Q, P T, T Q* ; then, since the potential difference between P and s is the same as the potential difference P and T, I a Ra h RC- Similarly, since the potential difference between s and Q is the same as the potential difference between x and Q, Fig. 138. Simple Diagram of Wheatstone's Bridge. Therefore, combining these two equations, we have R a - R c / x ~FT ~ 7T \ 22 } Kf) t\d which is the law connecting together the resistances of the four " arms " of the Wheatstone's bridge when balance exists, f * p s, s Q, P T and T Q, are called the " arms " of the bridge, t If no current passes through the galvanometer, when current flows through the arms, the bridge is said to be " ^ '* " balanced. 220 PRACTICAL ELECTRICITY This law may also be written in the form, R a R d -R b R c (23) or in words, the products of the resistances of opposite arms in a balanced bridge are equal. This law may also be proved graphically, thus : Let o, A, B, c (Fig. 139) be points in a conductor through which a steady current is flowing and let o A, A B, B c be drawn so that the lengths of the lines represent, on some convenient scale, the resist- ances of the parts of the con- ductor between the points o and A, A and B, and B and c respectively, then if lines o P, A Q, B R, c s be drawn per- pendicular to the straight line o A B c and of such lengths that they represent the potentials at the points o, A, B and c respec- tively, it follows from our fundamental definition of resistance that the points P, Q, R and s all lie in one straight line, and that the tangent of the angle this straight line makes with o A B c measures the current. The trigonometrical tangent will, however, only measure the current in amperes if the length of the hori- zontal line that represents an ohm is the same as the length of the vertical line that represents a volt. Suppose now that P P' (Fig. 140) represents the P.D. between the points p and Q in Fig. 138, and suppose that P s repre- sents the resistance R a , SQ represents R b , PX represents R c , and TQ represents R d , then, if the points P' and Q in both the figures be joined by Fig * 140< straight lines, and per- pendiculars s s', x x' be erected, it follows these perpendiculars represent the P.Ds. between the points s and Q and x and Q re- spectively of Fig. 138, on the same scale that P p' represents the P.D. between the points P and Q. But the points s and x are by hypothesis selected such that no current flows through a galvano- meter used to join them, therefore s s' equals x x'. Further, from the properties of similar triangles, we know that- WHEATSTONE'S BRIDGE 221 therefore, since s s' equals T x', we have R b Rd R a +R b Rc+Rd' ; ; . or #f = ^ j the same relationship as was previously arrived at as the law of the Wheatstone's bridge. The last equation may also be written in the form _Ra Rp_ RC " Rd' and this is the equation that we should have obtained for no current through the galvanometer, had its terminals joined P and Q, and the current generator been placed between s and T. Hence, when balance is obtained with a Wheatstone's bridge, the balance will nut be disturbed by interchanging the galvanometer and battery. 89. Wheatstone's Bridge : its Use and Simple Method of Constructing. Any one of the four resistances, R a , R^, R c , Rd can be expressed in terms of one of the other resistances multiplied by the ratio of the two remaining resistances to one another. For example, or d = oX-, K a etc. If then the bridge be " balanced," that is, if two points s and x have been found of the same potential, and we know the resistance of one of the arms, say R^, in ohms, and the ratio of the resistance of two of the other arms, say R c to Rd, but not neces- sarily the values of either R c or Rd in ohms, we can, from the first equation given above, find at once the value of the resistance of the fourth arm, R a , in ohms. Similarly, if we know R c in ohms and the ratio of R^ to R a , but without necessarily knowing either Rb or R a , we can at once find the value of R^ in ohms, from the 222 PRACTICAL ELECTRICITY third equation, etc. Hence, one mode of using the bridge to measure the resistance of R a is to keep the ratio of R c to R# con- stant, and simply vary the resistance of R b until no current passes through the galvanometer. Another method consists in keeping R^ constant and varying the ratio of R c to Rj. For ex- ample, the resistances R c and R& may be the resistances of different lengths of the same kind of wire, in which case we know that R c will be to Rd simply as the ratio of these lengths whatever be the abso- lute resistance in ohms of the two parts (see Section 94). In both the above cases R c and R& are called the " ratio arms," or the " pro- portional arms." A form of Wheat- stone's bridge in which P T Q, of Fig. 137, was one piece of stretched wire, and the ratio of the " proportional arms " R c to Rg, varied by moving the connec- tion of the wire leading to one terminal of the galvanometer, was originally employed by the Electrical Committee of the British Association, and is, for this reason, sometimes called the " British Association bridge " ; at other times, the " metre bridge" from the stretched wire being often a metre long. The wire may be made of platinum, or better still, of platinum-iridium which re- sists wear.* In Fig 141, P Q represents the stretched wire and K a sliding key which can make contact with it at any point. Fig. 141. Diagram of Metre Bridge. Fig. 1410;. Commercial Form of Metre Bridge. To protect the platinum-iridium wire from being accidentally knocked or damaged, it may conveniently be placed in a groove cut in the edge of an ebonite or slate disc, D (Fig. 142), and contact made with any point of it by means of the spring key K carried at the end of the movable radial arm A, and shown in detail in Fig. I42. The small pin under the knob K is to prevent the knob being pressed down so much as to damage the platinum- * On account of their comparative cheapness german silver, platinoid or manganin are frequently used for bridge wires. METRE BRIDGE 223 indium wire. The circuit of the battery B (Fig. 142) is closed by a separate key K'. The scale round the edge of the disc in Fig. 142 is divided into centimetres and millimetres, but for rapid work it is more con- venient to have this scale divided into ratios, as indicated for a few points in the following table, where the top line of numbers gives the length of the bridge wire measured from the left hand, the second line of figures the ratio of the length on the left Figs. 142 and 1433. Circular Metre Bridge. to the length on the right, and the third line the ratio of the length on the right to the length on the left : o 10 20 30 40 50 60 70 80 90 zoo o o-in 0-250 0-429 0-667 l I- 5 2-333 4 9 oo oo 9 4 2-333 1-500 i 0-667 0-429 0-250 o-iii o A form of metre bridge of greater range is shown in Fig. 143. It has three stretched wires w w, each a metre in length, and so arranged that either one of them alone, or two of them in series, or all three in series, can be made use of to form the two sides R c and Rj of the Wheatstone's bridge (Fig. 141). When the plug E is, as in the figure, placed in the hole H, the current simply passes through the stretched wire which is nearest to the observer. If, on the other hand, the plug E be put in the hole h, then, since the brass plate P is permanently connected with the plate p' by a thick copper strip under the base of the instrument, the 224 PRACTICAL ELECTRICITY stretched wire nearest to the observer is short-circuited and the middle wire is in series with the one farthest from him. Lastly, if the plug be removed altogether, the three wires are in series. The object of thus lengthening the wire is to increase the accuracy of the test when desired (provided the galvanometer is sufficiently sensitive ), and a still further increase in the accuracy can be effected by removing the short-circuiting pieces s lt S 2 , and inserting coils of known resistance in place of them. For example, suppose that the ratio of the unknown to the known resistance be f , then the slider K must be placed so as to divide the stretched wire into two parts having this ratio. Hence, if one of the three wires only be used, the lengths of the two parts which will give exact balance will be 60 and 40 centimetres, and BRIDGE KEYS 225 an error of I centimetre in the position of the slider will correspond with an error in the determination in the ratio of 6i_6o 30 40 X 100 per cent., or 4-3 per cent. If, on the other hand, the three wires in series be employed, then the lengths into which the three metres of wire must be divided to obtain exact balance will be 180 and 120 centimetres, and an error of one centimetre in the position of the slider will correspond with an error in the determination of the ratio of 181 180 no 120 r - X 100 per cent., or 1-4 per cent. If now two coils, each having a resistance equal to, say, 500 centimetres of the stretched wire, be inserted in place of the short circuit pieces s x and S 2 , an error of a centimetre in the position of the slider will only correspond with an error of 781 780 510 520 - x 100 per cent., or 0-32 per cent. Contact between the platinum-tipped knife-edge k and one or other of the stretched wires, is produced by depressing the knob K, which causes the lever, L L, to which this knife-edge is attached, to turn on an axis A A. On removing the pressure, the lever is pressed up by a spring underneath it. The slider should never be moved with the knife-edge k depressed, as this would scrape the stretched wire and alter its cross-section. To prevent the wire being cut by the knife edge k, if K be pressed down with great force, it is desirable that k be carried on a spring so that the force between the knife-edge and the wire is limited. In order to enable k to make contact with either the first, second, or third wire, the knob K is not fastened rigidly to the lever, but can slide along it in a slot, and be so placed that the near end of the spring s rests in either one of three grooves on the top of the lever, L L, corresponding with the three positions of k when it is in contact with the three stretched wires respec- tively. 90. Bridge Key. In using a Wheatstone's bridge it is desirable to send the current through the four arms of the bridge R a , Rfr, R c , Rj, before it is allowed to pass through the galvanometer, and this is especially important when testing the resistance of P 226 PRACTICAL ELECTRICITY the copper conductor of a long submarine cable, since the current in such a case takes an appreciable time to reach its final value and become steady, due to the cable acting as a "condenser." Hence, if the galvanometer circuit were completed when the battery was attached to the bridge, an instantaneous swing of the galvanometer would be produced, even if the ratio of R a to R^ be the same as the ratio of R c to Rj. And although, since the ratio of resistances having been effected, the current through the galva- nometer would become nought as soon as the currents in the four branches of the bridge became steady, great delay in the testing would be caused by this first swing of the needle. A similar difficulty would occur in measuring the re- sistance of an electro -magnet or even of any coil without an iron core, if it were not specially wound , because whenever Fig. 144-Bridge Key. a Coil ls SO WOUnd that a current passing through it produces magnetic action, a short interval of time has to elapse, after putting on the battery, before the current reaches its maximum, or steady, value, arising from what is called the " self-induction "* of the coil. A key for sending the current through the four arms of the bridge before it is allowed to pass through the galvanometer is shown at K (Fig. 144) , and is a modification of the one originally em- ployed by the Electrical Committee of the British Association. On pressing down the button, contact is first made between the flexible piece of brass A and the flexible piece of brass B. This completes the battery circuit, and causes the current to flow through the four arms of the bridge shown symbolically in Fig. 144 by the spiral lines. On the button being still further pressed down, B is brought into contact with a little knob of ebonite E on the top of the flexible piece of brass c. This does not complete * Defined in Section 195. BRIDGE GALVANOMETER SHUNT 227 any other electric current ; but on the button being still further depressed c is brought into contact with D, and the galvanometer circuit is completed. This form of key is to be preferred to the ordinary bridge key, because all the connections are above the base of the key and in sight, whereas when the connections are made under the base, it occasionally happens that, without its being noticed, the pieces of gutta-percha covered wire used to make the connections are either badly insulated, or are loosely connected at their ends with the terminals of the key, and so introduce unnecessary and unsteady resistance. 91. Use of a Shunt with the Bridge. It is desirable to employ also another key k (Fig. 144), which may be quite simply made of a twisted bit of hard brass wire, bent so as to press up against a sort of bridge of wire. When this key is not depressed, a portion of the current is shunted past the galvanometer through any convenient shunt 5. the resistance of which need not be known, as it does not enter into the calculations. The object of this shunt is merely to diminish the sensibility of the galvanometer when the first approximation to balance is being made. As soon as this has been done the key k should be depressed, and all the current in the galvanometer circuit arising from want of perfect balance allowed to pass through the galvanometer itself, and the resistances adjusted until perfect balance is obtained. Another device to expedite the testing, and also to prevent powerful currents being sent through the galvanometer, consists in not holding the key K down when the first rough approximation is being made, but merely giving it a tap, which has the effect, when the balance is far from perfect, of giving the needle of the galvanometer a slight impulse to one side or the other, according as the ratio of R a to R^ is larger or smaller than that of R c to R#, instead of causing the needle to violently swing against the stops on one side or the other as it would do if the key K were held down before the balance was approximately arrived at. ' 92. Meaning of the Deflection on a Bridge Galvanometer. A considerable amount of time will be saved in testing if the meaning of a deflection of the galvanometer needle, say to the right, be once for all definitely ascertained, and a note be made whether it means that the ratio of R a to R^ is too large or too small. The simplest way of recording this, if we assume, for example, R a to be the unknown resistance, is to put the words " increase R^ " and " diminish R^ " one on each side of the gal- vanometer, these being the directions to be followed according 228 PRACTICAL ELECTRICITY as the needle deflects towards one or other of them.* The position of these two directions must, of course, be reversed if the terminals of the galvanometer, or of the testing battery, be reversed.! Example 59. In measuring a resistance on the Wheatstone's bridge the resistances of the arms p T and T Q (Fig. 138) are 1000 and 100 ohms respectively. The unknown resistance is placed in the arm P s, and the resistance in s Q is adjusted until balance is as nearly as possible obtained. It is found that when the variable resistance s Q is 546 ohms the galvanometer deflection is 15 divisions to the left, while if s Q is made 547 ohms the deflec- tion is 27 divisions to the right. Find the value of the unknown resistance, assuming proportionality of deflection for small changes in the resistance s Q. A change of i ohm in s Q produces a change of 42, i.e. (15 + 27) divisions in the deflection, hence a change of , or 0-36 ohm in s Q would cause a change of 15 divisions in the deflection. Con- sequently, if s Q were 546-36 ohms the galvanometer deflection IOOO would be zero, therefore the resistance tested is x 546-36, or 5463-6 ohms.J 93. Conditions Affecting the Resistance of a Conductor. The resistance of a conductor depends on four distinct conditions : (1) Its length. (2) Its cross-section. (3) The material of which it is composed, Hie purity of the material, and the hardness and density. (4) The temperature. It is therefore important that the student should ascertain by experiment how much change is produced in the resistance by varying each of these four conditions separately. And gener- ally, in experimenting, it is to be remembered that when it is possible to change several of the conditions under which the experi- ment can be made t it is of the utmost importance that only one of the conditions should be varied at one time. The effect produced by the variation of one condition should be fully inquired into before any one of the other conditions is in any way altered, * The words " unplug " or " plug " are also commonly used when plug resistance boxes are employed in the adjustable arm. t When the coils of a bridge do not enable exact balance to be obtained the method of " proportional parts " explained in Example 59 may be used. J This method of " proportional parts " can also be used with the differential galvanometer, in cases where the resistance coils available do not permit of exact balance being attained. RESISTANCE :: LENGTH 229 otherwise it will often be quite impossible to gather from the results, what portion of the variation of the effect was produced by a particular change in the conditions. 94. Variation of Resistance with Length. In Section 49 we saw that when a steady current passed througn a uniform conductor the P.D. between any two points was proportional to the length of the conductor between the points Combining Fig. 145. Apparatus for Proving that Resistance is Proportional to Length. this fact with the fundamental definition of resistance (Section 51), page 142), it follows at once that the resistance of a uniform conductor is proportional to its length. This law may be proved independently by using a high -resist- ance galvanometer as a voltmeter. For it is to be remembered that, although it would not be justifiable to prove that Ohm's law were true by using a cwm^-voltmeter, seeing that the possibility of employing a galvanometer as a voltmeter depends on the fact that Ohm's law is true, galvanometers could be used as accurate voltmeters when once Ohm's law has been proved to be true, even if the distribution of potential along a uniform wire conveying a steady current followed some law other than it actually does. Fig. 145 shows a simple arrangement for testing the distribu- tion of potential in such a case. A current is sent through a uniform wire, say of German silver, stretched along a graduated 230 PRACTICAL ELECTRICITY bar between the points w w'. A tangent galvanometer, whose coil has a high resistance compared with that of the straight wire w w', has one of its terminals, B, connected with one end of this wire, w, while its other terminal, B', can be connected with any point of the stretched wire by means of the loose flexible wire and the sliding key s'. Then experiment shows, if the sensibility of the gal- vanometer is kept unchanged by the adjusting magnet not being moved during the experiment, and if the cur- rent flowing through the wire w w' be kept quite con- stant, that the tangent of the deflection of the galvano- meter, and therefore the P.D. between its terminals, is directly proportional to the length of the wire w s'. In using this apparatus the contact must be loosened be- fore the slider is moved along, otherwise the wire will be scraped and its cross-section no longer remain perfectly uniform. If it be desired to try this experiment with a longer wire than can be conveniently used in a straight form, we may employ the frame (Fig. 146), consisting of six or more wooden cylinders having a screw groove cut on each. Lengths of say 5, 10, 20, 30, 40 and 50 feet of wire of the same material and having exactly the same thickness throughout, say o-oi of an inch, may be wound in the grooves on the respective cylinders, and by connecting the binding screws together in pairs the whole of the wire may be joined up in series. If a current be sent through the whole of the wire joined up in series from left to right through the wire on the first cylinder, right to left through that on the second, etc., and if the current be maintained constant, it will be found that the P.D. between the terminals at the ends of any one of the cylinders is proportional to the length of wire on that cylinder, thus proving that resistance is proportional to length. 95. Variation of Resistance with Cross-Section. For ascer- taining the law of variation of the resistance of a conductor with its cross-section, the spiral grooves in another set of six cylinders (similar to those in Fig. 146 except that all screw grooves are of the same pitch) have wound in them wires all composed of the same RESISTANCE OF METALS 23* material and of exactly the same length (say, twenty feet), but having diameters respectively of, say, 0-0076, 0-0092, 0-0108, 0-0136, 0-0164, 0-02 of an inch.* The resistances of these wires may be tested by any of the methods described in the Sections 84 to 89, or 94, in terms of some one resistance taken as a standard ; and when this is done, it is found that the resistances of the different conductors of the same material are inversely as the squares of their diameters that is, inversely as their sectional areas. 96. Variation of Resistance with Material. The cylinders in this case have wound on them wires of exactly the same length (say, twenty feet) and having exactly the same diameter (say, o-oi of an inch), but made of the following materials respec- tively copper, brass, platinum, iron, lead, and German silver ; and when the resistances are tested by any of the methods described in Sections 84 to 89, it is found that the metals, as given in this list, are arranged in increasing order of resistance, this being, roughly, as the numbers I, 4, 5}, 6, 12, 13. 97. Resistance of Metals and Alloys per Centimetre Cube and per Inch Cube. The " specific resistance " or " resistivity " of a material is usually expressed as the resistance in " microhms " or millionths of an ohm, at o C. of a centimetre cube,| or of an inch cube that is, the resistance from one face to the opposite face across the cube. It has been customary hitherto in books to give a table of the specific resistances of a number of pure materials and alloys expressed to four significant figures as deter- mined by Dr. Matthiessen nearly fifty-five years ago ; and such a table will be found in the early editions of " Practical Elec- tricity." But during recent years a number of investigations have been carried out on the resistance of copper the material generally employed for electric conductors and it has been found that a diminution of from 3 to 4 'per cent, can be produced in the resistance of copper by compressing it, without any change being made in its chemical composition. Electrolytic refining of copper has led to the production of the metal on a large scale of a high degree of purity, so it is now quite common to find " commercial copper " of greater purity arid smaller specific resistance than the " pure copper " tested by Matthiessen. Difference in density of a material alters the specific resistance as also does the mechanical treatment or annealing to which it * These sizes correspond with No. 36, 34, 32, 29, 27 and 25 Standard Wire Gauge. f We may point out that the expression " centimetre cube," as here used, is merely an abbreviation for " a conductor one centimetre long and one square centimetre cross-section." A similar meaning applies to " inch cube." 232 PRACTICAL ELECTRICITY has been subjected, so the numbers given in the following table (No. VI.) must be regarded as being only approximately correct, and they are, in most cases, stated only to three significant figures. The substances are arranged in order of increasing specific resistance, and the unit employed is the international microhm. From the table on page 233 we see that of the various pure metals, annealed silver is the one having the least, and bismuth the one having the greatest, resistance for a given length and sectional area. The numbers given in the table can be used to ascertain the resistance of a wire or rod of any length and of any cross - section composed of any one of the materials at oC. For example, if p be the specific resistance per centimetre cube, in microhms, / the length, and d the diameter of the wire in centi- metres, the resistance is - ^ microhms. (24) or more generally we may write, length Resistance = : X specific resistance, (25) cross-section Example 590. Find the resistance at o C. of an annealed copper wire ^th of an inch in diameter and 100 yards long. Taking the specific resistance as 0-61 microhm per inch cube and using the formula R .= -^ we have 4 x 100 x 36 x 0-61 R 3-1416 x x io 6 100 = 0*279 ohm, approximately. Example 60. What length of hard drawn copper wire, No. 16 gauge (diameter 0-064 inch) will have a resistance of I ohm at oC. (assume p =0-640) ? Answer. 5,020 inches. 140 yards, approx. Example 61. What must be the diameter of a platinum silver wire so that it may have a resistance of one ohm per metre at oC.? Answer. 0-0555 cm. 0-555 mm. SPECIFIC RESISTANCES 233 TABLE VI: PURIFIED SUBSTANCES ARRANGED IN ORDER OF INCREASING RESISTANCE FOR THE SAME LENGTH AND SECTIONAL AREA. Name of MetaL Resistance in International Microhms at Centigrade. Relative Resistance. Centimetre Cube. Inch Cube. Silver, annealed ... 48* 0-583 ( from Copper, annealed j tQ '55 6l o - 6io 0-633 4 09 ,, ,, (International, 1913)! 588 0-6250 07 ,, (Matthiessen) 594 0-6277 077 Silver, hard drawn 58 0-622 07 Copper, hard drawn ... < tQ '59 1-64 0*626 0-646 07 II ,, (Matthiessen) ... 1-630 0-6418 io Gold, annealed ... 2-05 0-807 1-38 Gold, hard drawn 2-089 0-822 1-41 Aluminium, annealed 2'43 0-96 1-64 Silicium Bronze ... (about) 2'5 0-98 1-69 Zinc, pressed 5-01 2-21 379 Tungsten ... ... 6-4! 2'S 4' 3 Nickel, annealed ... T- + 6 '94 J 2 73 T" j 4-69 Phosphor bronze ... (about) 7-8 3-07 5-27 Platinum, annealed 9-04 3'55 6-09 Iron, annealed 9'7 3-82 6-56 Gold-silver alloy (2 oz. gold, i oz. silver), hard or annealed 10-8 4-27 7-33 Tin, pressed ... 13-2 5'9 8-9 Lead, pressed 19-6 7-71 13-2 German silver { f m 19-0 42-O 7-48 n-8 12-8 2O'2 Platinum-iridium alloy, Density 21-32 22'2 8-73 I5-0 Platinum-silver alloy (i oz. platinum, 2 oz. silver), hard or annealed 2 4 '3 9-58 16-4 Platinoid (about) 34 i3H 2 3 Antimony, pressed 35'4 I3-9 23-8 Manganin (about) 42 167 28-7 Nickelin . 43 17 2Q Eureka T" J 47 18-5 ~\7 32 Constantan 48 j 19 J 6 33 Ta Ta e;i 20 34*e KruDcin J x 8; 33 OH- J CT Nichrome D 89 3*5 ~/ / 60 Mercury 94-08 j j 37'4 63-6 Bismuth, pressed ... 1 08 42-5 73 Carbon ... ... (about) 4,000 to i, 600 to 2,700 to 10,000 4,000 6,700 * Profs. Dewar and Fleming give 1-468, and Mr. Fitzpatrick 1-481. t The International Standard for Annealed Copper at 2oC is : i metre length of i square millimetre cross section has a resistance of ^ ohm. (0-017241) . | Varies between 5-0 and 6-6, 234 PRACTICAL ELECTRICITY Example 62. Determine the cross- section of a column of mercury at o C. whose resistance will be o-i ohm per metre ? Answer. 0-094 sq. cm. 9-4 sq. mm. Example 63. Which has the greater resistance, a copper wire 20 feet long, 0-015 inch in diameter, or a platinum -silver wire 10 feet long, 0-037 mcn m diameter at o C. ? The resistance of the copper wire will be to that of the platinum - 20 X 1-55 . 10 X 24-3 silver as ~ is to - , or as 0-79 to I. 0-0152 0-0372 Hence, the copper wire has rather more than three-quarters of the resistance of the platinum -silver wire. 98. Resistance of Metals and Alloys for a Given Length and Weight. As metals are usually sold by weight it is frequently convenient to know, not the resistance of a given volume of a material of specified length, but the resistance of a given length having a given weight. In the following table (No. VII.) will be found the resistances in international ohms at o C. of wires one foot long weighing one grain, and one metre long weighing one gramme, the substances being arranged in increasing order of resistance for a given length and weight, this order being different from that employed in Table VI., where the sub- stances were arranged in increasing order for the same length and cross -section. From Table VII. we see that of the metals aluminium has the least resistance for a given length and weight ; whereas we saw from Table VI., that for a given length and cross-section it was annealed silver that had the least resistance. Since weight = volume X density, = length X cross-section X density, we have cross-section =- : ^3 r , and by substituting in formula length X density (25) we get as the relation between resistance, length, and weight, w where A is the density in grammes per cubic centimetre and w the weight in grammes. Example 64. What will be the weight of an iron wire 100 yards long, having a resistance of i ohm at o C. ? An iron wire I foot long, weighing I grain, has I -08 ohms RESISTANCE, LENGTH AND WEIGHT 235 TABLE VII. PURIFIED SUBSTANCES ARRANGED IN ORDER OF INCREASING RESISTANCE FOR THE SAME LENGTH AND WEIGHT. Name of Metal. Resistance in International Ohms at o Centigrade of a wire Relative Resistance. i foot long weighing igrn. i metre long weighing i grm. Aluminium, annealed 0*090 0-063 I Copper, annealed* { f m 0-199 O-2O9 0-139 0-143 2'2I 2-27 ,, (International 1913) O-2026 0-1413 225 (Matthiessen) ... 0-2037 0-1421 2-26 Copper, hard drawn* { r 0-208 0-218 0-142 0-146 2-26 (Matthiessen)... 0-2078 0-1449 2-3I Silver, annealed 0-218 0-I52 2-42 Silver, hard drawn 0-238 0-166 2-64 Zinc, pressed 0-575 0-401 6-4 Gold, annealed 0-402 6-4 Gold, hard drawn ... 0-587 0-409 6-5 Nickel, annealed . 0-84 '59 9 '4 Phosphor bronze ... (about) i-o 0-70 in Iron, annealed I -08 075 12-0 Tin, pressed ... 1-38 O'Q6 15-3 Tungsten * J 1*64 \S VJW 18-2 Gold-silver alloy (2 oz. gold, W T i oz. silver), hard or annealed 2-36 1-65 262 German silver { rom 2-37 1-66 26-4 I to 2 "87 2"OI 32-0 Platinum, annealed 2-74 i '93 307 Lead, pressed 3' 19 2-22 35-3 Antimony, pressed 2*38 37-8 Platinum-silver (i oz. platinum 2 oz. silver), hard or annealed... 4-19 2-92 46-5 Platinoid (about) 4-40 3-03 48 Manganin (about) 57 Nickelin _ ,g VQO 59 Eureka 6-1 j y w 4'2 C J 64 Constantan ... 6-2 *t ** j 4-33 66 " la fa" 4cc 69 Platinum-iridium (Density 21-32) 6-76 JJ 4-73 V/^J 75 Kruppin Q'9 6-9 1 08 Nichrome . ... ... -7 y 10*6 7*4 118 Bismuth, pressed 15-2 T- io-6 168 Mercury ... 18-36 12 '80 204 * The standard adopted by the International Electro-technical Commission in 1913 for a length of one metre weighing i gramme, is as follows : Annealed copper, 0-15328 ohm at 2oC. For commercial purposes a temperature coefficient (constant mass) of 0-00428 per degree C. from o C. is adopted and 0-00393 when 20 C. is taken as standard temperature. This gives Annealed copper, 0-1413 ohm at o C, Hard drawn copper, 0-1443 ,, 236 PRACTICAL ELECTRICITY resistance at o C. Hence, an iron wire x feet long, weighing x grains, has % x 1-08 ohms at o C. If the weight is y grains, the resistance is - x x 1-08. Now x is here 300, and the resistance is r ohm 300' therefore, x 1-08=1, or y y : 97,200 grains. Answer. 13-9 Ibs. Example 65. What is the resistance of a mile of hard drawn copper wire weighing 20 Ibs. ? (Assume I footgrain 0-210 ohm). Answer. 41-8 ohms. Example 66. The weight of wire to be used for a 10 ohm platinum silver coil is not to exceed 5 grammes ; find (a) , the length and (b), the diameter of the wire required ? Answers. Length =4- 13 metres. Diameter =0-358 mm. 99. Variation of Resistance with Temperature. To ascertain the way in which the resistance of metals and alloys varies with the temperature, small coils of silk-covered wire composed of the different materials may be conveniently wound on paper cylinders and inserted in narrow glass test-tubes, G V G 2 , G 3 , and G 4 (Fig. 147), the test-tubes being supported from a wooden disc. One end of each of the coils may be soldered to a common terminal, T, while the other ends of the coils are soldered to the terminals T V T 2 , T 3 and T 4 . The test-tubes are inserted in the water- bath w (Fig. 148) , which can be warmed with the Bunsen burner B, standing on a sheet of asbestos, A A, to a tem- perature which is indicated by the ther- mometer 1 1, enclosed in a brass tube to prevent mechanical injury ; and the resistances of the different coils of wire can be measured with a Wheatstone's bridge, differential galvanometer, or other suitable arrangement, the measur- ing apparatus being protected from the heat of the burner by means of the double screen s. In carrying out experiments of this kind, it must be borne in mind that, as the glass bulb of a thermometer is very thin, and as mercury is a substance Fig. 147. Coils of Wire used in the Apparatus for Measuring the Variation of Resistance with Temperature. RESISTANCE AND TEMPERATURE 237 having a very small " specific heat," * a thermometer rapidly acquires the temperature of the liquid in contact with it ; whereas a mass of metal inserted in the same liquid may have a very different temperature from the liquid which immediately surrounds it, especially if the temperature be Fig. 148. Calorimeter for Measuring the Coils of Wire shown in Fig. 147. rapidly rising or falling. Further, a liquid, being a bad conductor of heat, the temperature in different parts of it will be different, unless it be kept constantly in motion ; therefore a stirrer, s s, is provided with the heating vessel seen in Figs. 147 and 148. Lastly, the water-bath w is made in two separate parts in order that the current of hot water which rises by " convection " from the heated bottom of the water-bath may not come directly into contact with the glass tubes. By using petroleum in the inner vessel and perforating the tubes G, Fig. 147, the tem- perature of the wires can be altered more rapidly and determined with greater accuracy. This arrangement, however, is more i * The specific heat of a substance is the ratio of the amounts of heat required to raise equal masses of the substance and of water through i. 238 PRACTICAL ELECTRICITY dangerous than the water bath, and for elementary work is not necessary. $ -;> : Before taking a measurement of the resistances of the coils of wire at any particular temperature, it is well to adjust the flame of the Bunsen burner so as to maintain the temperature of the thermometer constant for some minutes, the liquid being con- stantly agitated with the stirrer s s during the time. For the longer the time during which the temperature of the water in the bath is all kept at a uniform and constant temperature, the greater is the probability that the coils of wire have acquired the temperature indicated by the thermometer /. The exact law connecting the variation of the resistance of a metal with the temperature depends not only on its chemical constitution, but on its molecular condition, such as its hardness, density, etc. To a first approximation the law is a linear one, i.e., the resistance increases uniformly with the temperature, the rate of increase for common pure metals being about 0-38 per cent, per i C. Nickel and iron have larger " coefficients " as will be seen from Table VIII., whilst mercury and alloys have smaller ones. Copper is the material of most importance to electricians, and for this metal the approximate simple rule, the resistance of copper increases about 0-4 per cent, per i C., should be remembered. This simple rule is practically exact if 15 C. in- stead of o C. be taken as the temperature of reference. More accurately the relation between resistance and tempera- ture may be represented by the formula, Rt=R (i + at + bt z ), (27) where Rf is the resistance at temperature t and R its resistance at o C., and a and b are coefficients depending on the material. For copper, platinum and mercury the approximate values are : TABLE VIII. Material. a. b. Copper, hard drawn . . annealed Platinum > Mercury + 0-00408* +0-00427 +0-00345 +0-0036 + 0-000888 + 0-000,001, 12f 0-000,000,53 J 0-000,000,11) to U 0-000,000,64 ) +0-000,00103 It should be understood that the values of the coefficients differ somewhat for every specimen, and if it be necessary to * Swan and Rhodin. f Clark, Forde and Taylor. J Callendar. Ayrton and Kilgour. RESISTANCE THERMOMETERS 239 make use of the relation between resistance and temperature for accurate work, the law of variation for* the actual piece employed should be determined experimentally. One of the principal uses to which the relation between resistance and temperature has been put is the measurement of temperatures electrically. When once the law connecting resistance and temperature has been determined for a given specimen, a measurement of its resistance is in effect a measurement of its temperature, and by this means temperatures of ovens, flues, furnaces, etc., can be readily found. Platinum, is the material generally used for these purposes, and "platinum thermometers " form at the present day, one of the most accurate means of measuring temperatures. Copper is sometimes employed for temperature measurements, and in electrical machinery the temperature of the copper coils is frequently deduced from the resistances of the coils themselves ; the resist- ance at some known temperature having been previously deter- mined. 100. Conductors of Large Specific Resistance have Small Temperature Coefficients. On comparing Table VI. with Table IX. it will be observed that, if the metals and alloys be arranged in increasing order of specific resistance, they are arranged roughly in decreasing order of temperature variation, or, in other words, the poorer the conductor the smaller its variation of resistance with temperature. And not only does the tempera- ture variation become less and less as the specific resistance of the material increases, but it passes to the other side of zero and is negative in the case of a bad conductor like carbon, which, in the form used in electric arc -lamps, has a specific resistance of about o-oi ohm per centimetre cube a value, roughly, 6,000 times as great as that of copper. That is to say, the resistance of carbon diminishes with increase of temperature ; for example, the resistance of the carbon filament of a glow-lamp, when glowing at its normal brilliancy, is only about three-quarters of the resist- ance it possesses when cold. This property of carbon has been utilised by making a resistance of a metallic wire in series with a carbon filament, so arranged that the increase of the resistance of the wire caused by rise of temperature was practically balanced by the simultaneous diminution in the resistance of the carbon filament. When we came to still poorer conductors, such as gutta-percha or indiarubber, which are, therefore, usually termed insulators, the temperature coefficient is not only negative, but is numerically much larger than it is for any metal. For example, the gutta- percha which is usually employed in the manufacture of 240 PRACTICAL ELECTRICITY submarine cables sometimes has a specific resistance of about 350 x io 12 ohms per centimetre cube at 24 C., or about 200 million million million times the specific resistance of the copper conductor ; but this high resistance is diminished to one-ninth by an increase of temperature of only 15 C. TABLE IX. SPECIFIC RESISTANCE AND PERCENTAGE TEMPERATURE VARIATION OF MATERIALS USED FOR RESISTANCES. Material. Approximate s Resistance in Inter- national Microhms per Centimetre Cube at oC. Percentage Variation of Resistance per iC. Tungsten 6*4 6*9 7*8 9-0 97 io'8 19 to 30 22*2 24 '3 34 to 36 39 to 42 43 47 4 8 Si 85 89 94 4,500 to 10,000 0-3 to 0-5 o'6 0*08 '3S o'S 0-065 0*04 to 0*028 0-13 0*027 to 0*03 0'02 tO O'OlS 0*0017 to o'ooo o "0024 0-0048 0*0014 to 0*0029 0-0029 to 0*0076 0*074 0*024 0*089 0-03 Nickel Phosphor bronze ... Platinum ... Iron Gold-silver alloy (2 oz. gold, i oz. silver) German silver Platinum-indium (Density 21*32) ... Platinum-silver alloy (i oz. plati- num, 2 oz. silver) Platinoid ... Manganin (85 oz. copper, 12 oz. manganese, 3 oz. nickel) Nickelin Eureka Constantan "Ja-/a "wire Kruppin . . Nichrome Mercury ... Carbon The connection between high specific resistance of a metallic alloy and low temperature coefficient has led people to seek for metallic alloys of higher and higher specific resistances. About 1888 Martino found that adding a trace of tungsten to German silver raised its specific resistance from about 20 microhms to 34 microhms per centimetre cube, and lowered its temperature coefficient from about 0-044 to ' 2 P er cent * P er l0 C. The substance thus produced is called " platinoid," and has been much used in the construction of resistances. Going still farther, Mr. Weston, by adding manganese to copper instead of, or in addition to, nickel, succeeded in pre- paring alloys whose resistance does not vary appreciably for ordinary changes of temperature, or, like the resistance of carbon, actually diminishes with rise of temperature. TEMPERATURE COEFFICIENTS 241 These manganese alloys have been very fully investigated at the Physikalische Technische Reichsanstalt, the German Government physical laboratory at Berlin, and meltings, with as much as 30 per cent, of manganese and a specific resistance over 100 times that of copper, have been prepared. The par- ticular alloy, however, which the work of this Institute has shown to be the best for ordinary purposes is one containing 85 per cent, of copper, 12 per cent, of manganese, and 3 per cent, of nickel by weight, and is called " manganin." Manganin, which has a specific resistance of about 42 microhms per centi- metre cube, or about 28 times that of copper, is now manu- factured commercially, and, excepting when the most minute accuracy is desired, the variation of the resistance of com- mercial manganin may be regarded as zero for ordinary changes of temperature. To protect the wire of standard resistance coils from oxidation, it is customary to coat them with wax or shellac. Manganin coils protected by shellac have been found to vary very slightly in resistance, according as the atmosphere is moist or dry, and to prevent these changes such coils must be hermetically sealed. With an improved form of standard coil devised by Dr. C. V. Drysdale, this sealing is unnecessary. The wire is of constantan* having a negative temperature coefficient, and is electroplated with nickel of a thickness sufficient to make the temperature coefficient of the plated wire practically nil. Example 67. To find the resistance of a wire 52 metres long, i square millimetre in section at 22 C., made of pure copper, hard drawn, specific resistance 1-63. Resistance required in ohms. 1-63 52 x 100 , = - -~> ~> -jT> are respectively the resistances of the several parts ; hence, we can write R=R a + R b + R c -\-R d (28) The same may be proved for any number of conductors. If the conductors be connected in parallel as shown in Fig. 150, then the P.D. between the ends of all the conductors is Fig. 150. Branch Circuits in Parallel. the same. Calling this potential difference V, the current V through the conductor of resistance R a will be given by I a = - R* Similarly, the currents /&, I c and I d , through the conductors of V V V resistance, R^, R c , and R d , will be , and - respectively; whilst Kb K c Kd the main or total current / will be equal to the sum of the several 246 PRACTICAL ELECTRICITY currents (see Section 7). Hence we have JL L Jl Jl ^# fl + fy+l^^ R d ' J i i i i r = i^ + ^ + ^ + ^- Now represents a current divided by a potential difference, and is therefore the reciprocal of a resistance, or a conductance, whilst - and are the conductances of the several RO, Kb KG branches, hence we see that the conductance of a number of conductors in parallel is equal to the sum of the conductances of the several conductors. A single conductor which would allow a current / to flow through it when a P.D. equal to V existed between its ends, would have the same conductance as the several conductors in parallel, and consequently the same resistance as these conductors so arranged. Such a single conductor is said to be " equivalent " to the several conductors and its resistance is equal to the " combined resistance " of these conductors. Calling the com- bined resistance R, the above equation becomes and en taking the reciprocals of both sides ' . (30) From this we deduce the rule : the combined resistance of several conductors (or resistances) in parallel is equal to the re- ciprocal of the sum of the reciprocals of the several resistances. The particular case of two conductors in parallel occurs so 'fre- quently that it is convenient to remember the relation in another form. From the above we have T R = i i PARALLEL RESISTANCE 247 or. in words, the combined resistance of two conductors in parallel is equal to the product of the resistances divided by their sum. The truth of the rule for the resistance of conductors in parallel can be proved experimentally. A set of four coils on one bobbin, designed for this purpose, is shown in Fig. 151, one end of each coil is connected with the binding screw s x and the other ends respectively with the small brass mercury cups c lt C 2 , C 3 , C 4 , as indicated symbolically by the zig-zag lines on the face of the bobbin. A brass bar R R carries another terminal S 2 , and has four holes in it opposite c 1 , C 2 , C 3 , c d respectively, which R Fig. 151. Set of four Coils used for Testing the Resistance of Conductors in Parallel. contain mercury. By using one of the wire bridge pieces BJ, B 2 , etc., any of the four coils can be connected between s 2 and s lt and its resistance measured. As shown in the figure the coils Cj and C 2 are in parallel, and the resistance between s x and S 2 will give the combined resistance of Cj and C 2 , when in parallel. By suitably placing the bridge pieces any combination of the four coils can be arranged, and measured, and the formulae given above experimentally verified. Example 75. Resistances of 25, 32, 17, and 40 ohms are put in parallel with one another. What is the combined resistance ? i i i I i - V = A .'. x = 6-4 ohms. x 25 32 17 40 Answer. 6-4 ohms. 248 PRACTICAL ELECTRICITY Example 76. A coil of wire has 1,125 ohms resistance. What resistance placed in parallel with it will make the combined resistance 1000 ohms? Answer.-g.ooo ohms. Example 77. The wire of a resistance coil has 10,000 ohms resistance, but the surface of the ebonite between the terminals, having been imperfectly cleaned, has a resistance of only 870,000 ohms. What is the combined parallel resistance between the terminals ? Answer. 9,886 ohms. 104. Currents in Parallel Conductors. If I a , I b , I c , 1$, etc., be the currents in the various branch circuits (Fig. 150), and / be the current in the main, we have _ ft *' d &c.; la i a + i b + i c + id + &c. -i- + -L + .L + -1.+ & c . * *& *c K d i ^0 *o~ or Similarly, ~^ I J- + JL + -j- __ -j- &c. ^a ^6 ^c ^rf Example 78. Resistances of 12, 7, 2, and 30 ohms are placed in parallel with one another, and a current of 10 amperes, as measured by an ammeter in the main circuit, passes through the combination. What are the currents in the respective branches ? Answer. 1-097, 1-881, 6-583, and 0-439 amperes respectively. 105. Kirchhoff 's Rules. In the foregoing section we have made use of the fact demonstrated in Section 7 that when a current divides into two or more parts, the whole is equal to the sum KIRCHHOFFS RULES 249 of the parts. The statement of this fact is usually known as Kirchhoff's First. Law, or Kirchhoff's iFirst Rule, and is given in two forms, viz. : (i) The sum of the currents flowing to any point is equal to the sum of the currents flowing away from that point, or (2) the algebraical sum of all the currents meeting at a point is zero. Symbolically, the rule may be written, 2/=o. (32) It is of fundamental importance and of great use in electrical calculations. Another useful rule formu- lated by Kirchhoff is known as Kirchhoff's Second Rule, which says that in every closed circuit the algebraical sum of the products of the currents into the resistances equals the algebraical sum of the EM.Fs. in the circuit w (33) B I'K E W This rule is really a de- Fig . 152 . duction from Ohm's Law. For a simple circuit its truth is self-evident, for the expression may in this case be written : or ?-* which is merely the algebraic representation of Ohm's Law. Consider next the closed circuit w G w' w, in Fig. 152. In this circuit there is no electromotive force, therefore = o. Let the letters I b , I g , and I r in this figure represent currents, and Rfj, R g and R resistances, let the P.D. between w and w' be V, and the E.M.F. of the battery, E. By Ohm's Law we have V or V=IgR g , = JL, or V = I r R, Hence 7^ = I r R. 250 PRACTICAL ELECTRICITY If we trace out the closed circuit, w G w' w, in a clockwise direction, the direction of the current I g must be considered positive, whilst that of I r is negative. ? (IR) thus becomes I g R g + ( - I r R), or IgRg and as V = I g R g = I r R, we have I g R g I r R = o, In the closed circuit B w R w' B, the E.M.F. is E, and is IR + IR Fig. 153- where 7? 6 is the resistance of the battery branch between w and w'. Applying the rule we therefore get I b R b + I r R = E, but I r R = V, by Ohm's Law, /. E =V+I b R b . or, in words, the E.M.F. of the battery is equal to the potential difference between the terminals, plus the product of the current passing and the resistance of the battery, a result previously arrived at in Section 55. Again, for the closed circuit B w G P w' B (Fig. 153), ?(IR) is I b R b + I g R g> where R b is the resistance of the path P w' B w, and R g that of WGP, 2E = -'. Hence, according to KirchhofFs second rule we have 'I b R b + I g Rg = E-E'. SHUNTS 251 This may be written' E-I b R b = E' + I g R g . Now E I b R b gives the P.D. between the points w and p, and E' + IgRg, also represents the same P.D., for if the current Ig passes in the direction of w G p, the potential difference V must exceed the electromotive force E', which acts in the opposite direction ; and the excess of V over E' must be such as will cause a current I g to pass through the resistance Rg, viz., IgRg- Hence V - E' = I g R g , or V = E' + I g R g) But V=E-I b R b , or I b R b + I g R g = E-E'. or 2 (IR) = 2E. From these examples we see that KirchhofFs Second Rule is consistent with Ohm's Law. This rule enables us to write down simultaneous equations which represent the relations be- tween currents, resistances, and E.M.F.s in circuits constituting more or less complicated networks, and from these equations the unknown quantities can be expressed in terms of the known ones. 106. Shunts. One of the commonest instances of parallel circuits occurs when a galvanometer is " shunted." We have already seen (Section 19), when calibrating a galvanometer by comparing it with a standard galvanometer, and again when using a Wheatstone's bridge (Fig. 144), that it is sometimes convenient to employ a bypath, or shunt, to convey a portion of the current, so that the current passing through the galvanometer is less than the current in the main wires connected with it. We will now consider what must be the relative resistances of the shunt and galvanometer to allow any particular fraction of the whole current to pass through the galvano- Fig. 154. meter. Let R g be the resistance of a galvanometer, R s that of the wire shunting it, and let V be the P.D. between the terminals of the shunted galvanometer which is joined to the mains M x and M 2 (Fig. 154). Then if I g and I s be the currents that pass respectively through the galvanometer and shunt, 252 PRACTICAL ELECTRICITY or the currents in the galvanometer and shunt bear to one another the inverse ratio of the resistances. This relation may be deduced by Kirchhoffs second rule, as follows : ^ (IR) = ^E = Q (in this case). or I g R g I S R S = o, la R s whence -f- = . l s K g Also, by a well-known rule of proportion, it follows that R > _ I g + I S R g + R s I s but Ig + I s , the sum of the currents flowing through galvano- meter and the shunt respectively, is equal to the current / in the mains MJ or M 2 , hence I* K. R g + R (34) and T = pV (35) / Kg + K s or the current in either branch bears to the whole current, the ratio of the resistance of the other branch, to the sum of the resistances of the two branches. 107. Multiplying Power of a Shunt. Since _ R g + R 5 j R s * 7? -4-7? the fraction g - is frequently called the " multiplying power KS of the shunt "^that is, the number by which the current flowing through the galvanometer must be multiplied to obtain the total current, or current in the main. MULTIPLYING POWER OF SHUNTS 253 As an example of the last equation, let us suppose that we desire that I g shall be one-tenth of /; then or generally, if we desire that -th of the whole current shall pass through the galvanometer, R g + R s or R s = -I R g . (36) n i Example 79. A galvanometer of 2,572 ohms resistance is shunted with a resistance of 285-8 ohms. What fraction of the main current passes through the galvanometer ? Answer h _ Rs . 285'8 _ _!_ f -R g + R s ~ 2857-8 ~ 10* Example So. A galvanometer has 5,461 ohms resistance, what must be the resistance of the shunt in order that yjo^h of the main current may pass through the galvanometer ? 7? T Answer. ,. 5 = , therefore R & = 55-16 ohms. 5461 + R s 100 Example Si. A galvanometer and its shunt are both wound with copper wire. The multiplying power of the shunt is 100 when the temperatures of the galvanometer and of the shunt coils are the same. What is the multiplying power when the tem- perature of the galvanometer coils is 5 C. above that of the shunt ? Answer. 102. 108. Usual Method of Constructing a Shunt Box. Three coils, having respectively the Jth, ^th, and Jgth of the resistance of the galvanometer, are usually inserted in a small box b (Figs. 155-6), which generally accompanies a galvanometer. The ter- minals of the galvanometer, as well as the two wires which connect the galvanometer with the rest of the circuit, are joined to the bind- ing screws, s, s on the shunt box, and each of the first three shunt coils has one of its ends connected with the brass piece c, while the 0ther ends are connected respectively with the brass pieces D, 254 PRACTICAL ELECTRICITY Fig. 155. High Insulation Shunt Box. E, and F, as indicated symbolically in Fig. 156. If, then, the brass plug p' be inserted in the hole between the brass bar A B and the brass piece c, all the current will pass from A B to c, through the plug, and practically none will pass through the galvano- meter, since the resistance of the path from A B to c through the plug is ex- tremely small compared with that through the gal- vanometer. If, on the other hand, the plug be inserted in the hole be- tween A B and D, as in Fig. 155, current will pass from AB to D through the plug, and from D to c through the coil in the shunt box, which connects with c. And as this coil has jjth of the resistance of the galvanometer, r \jth of the total current will pass through the galvanometer. Simi- larly, if the plug be inserted in the hole between A B and E or in the hole between A B and F, ^ooth or ^ 1 o * ^ e whole current will pass through the galvanometer. Instead of employing three coils whose resistances respectively are Jth, ^gth, and 5 J^th of that of the galvanometer, and joining one end of each of these coils to the brass piece c, the coils may be joined up in series between the brass pieces c and D, D and E, E and F respectively, like the coils of an ordinary resistance box (Fig. 157). In this case the coils must have resistances ^R gt (fo--fffa)Rg, and ( J V ) Rg and the b l ck marked F will correspond with the Jth shunt, while that marked D will corre- spond with the g^-gth shunt, as indicated symbolically in Fig. 157. In order to obtain very good " surface insulation " the brass pieces, A B, c, D, E, and F are, in the particular shunt box shown in Fig. 155, mounted on ebonite pillars P, P, P, Fig. 156. Top of Shunt Box, showing Parallel Arrange- ment of Shunts. SHUNT BOXES 255 p , and, to avoid the insertion of the, plug into one or other of the holes pushing these pillars outwards and so preventing the plug making firm contact with the pieces of brass on each side of it, there is a spring cap c c, sliding on the plug, which passes over the two vertical pins on each side of the hole, and so holds the brass pieces together against the wedging action which tends to force them asunder when the plug is pressed in. The plug has a long ebonite handle i, which should be held by the flat part at the end to minimise the leakage taking place along the surface of the handle and through the body of the experimenter to the ground. 109. Increase of the Main Current Produced by Applying a Shunt. Although the current passing through an unshunted galvanometer is the same as the current in the main, and although the current passing through a shunted galvanometer is always times the current in the main, it must not be assumed Fig. 157. Top of Shunt Box, showing Series Arrange- ment of Shunts. K that the application of a shunt to the current passing through it in the ratio of unity to a galvanometer diminishes RS R g + R< For the application of the shunt diminishes the resistance in the 7? 7? circuit by the difference between R and -^ ^-^ , and this diminution of the resistance of the cir- cuit increases the current in the main, more or less, depending on the arrange- ment of the circuit ; so that the current in the main after the application of the shunt is greater than the current in the main before the shunt was applied by an amount that may be very small or may be very large. Let the circuit consist of a resistance R m in series with a galvanometer of re- sistance R g , and let a fixed P.D. of V volts be maintained between the terminals of this circuit (Fig. 158), then I gl , the current passing through the main or through the galvanometer, equals V l 256 PRACTICAL ELECTRICITY Next let the galvanometer be shunted with a shunt or resistance R s (Fig. 159), and let the P.D. of V volts be still maintained be- tween the outer terminals of the circuit shown in Fig. 159, then the current now passing along the main equals V and Ig 2 , the current now flowing through the galvanometer equals _ _ R S V _ , -~ ' U7) + T Z? / Z? i U \ i ZP ~D 2 gl K m ( K g + Kg) + KgK s Now the value of this ratio depends on the value of R m as well as on R g and for example, if R m be very large compared with R g , ^2 R s whereas if R m be very small compared with R g and R s , -f^- = unity (approx.). 2 gi That is to say, if the resistance external to the galvanometer lie very large, the galvanometer current after the application of the shunt, bears to the galvanometer current before its application the ratio ofR s to Rg + R s ; while, on the other hand, if the resistance external to the galvanometer be very small, shunting the galvanometer produces very little effect on the current passing through it. And this arises from the fact that on applying the shunt in the first case the main current is not appreciably changed, while in the second it is increased by an amount almost exactly equal to the current that is shunted past the galvanometer. For example, let R g be 1000 ohms and R s Jth of R g ; (i) let R m be 100,000 ohms, then the true ratio of I g2 to I gl is in-i x 101,000 r -- . or 0-1009 about, 100,000 x 1,111-1 + iii,m" T-) whereas the value of ? is o-i, which differs by about i per KS + K CIRCUITS IN PARALLEL 257 cent, from the true ratio, so that the current through the galvano- meter is reduced practically to one-tenth of its previous value ; (2) let R m be 10 ohms, then the true ratio of I S2 to I SI is in-i X 1,010 , or oQ2 about, IO X I,III'I + 111,111 whereas the approximate value of the ratio is unity, which differs by about 8 per cent, from the true ratio, so that the current Fig. 1 60. Part of the Plan of an Electrically Lighted House. s, Street Mains ; H, Mains to House ; M, Supply Meter ; D, Distribution Board ; /, Leads to the Rooms ; /', Branch Leads ; L, Glow Lamps. through the galvanometer remains nearly unchanged by the application of the shunt. An important example of this independence of currents in parallel circuits that can be produced by making the value of R^ in Fig. 159 very small, occurs in the wiring of a house for electric lighting. -The glow lamps are all connected in parallel with the house mains as indicated in Fig. 160, which represents a portion of the plan of the ground floor of a house, and shows the way in which the electric lighting mains and branch mains are run. At the place where the house mains, H, are connected with the street mains, s, a constant, or nearly constant, P.D. is maintained by the Electric Supply Company, the value of this 258 PRACTICAL ELECTRICITY nearly constant P.D. being frequently 100 volts. Each lamp, L, or each group of lamps, is provided with a switch so that the current can be turned on to, or off from, each lamp, or group of lamps, independently ; and it is obviously important that the turning on, or off, of a switch in one part of a house shall not sensibly affect the light given by the glow lamps in some other part of the house. Now a glow lamp is a very sensitive in- dicator of any variation of the current passing through it, for the light given out by a glow lamp, when glowing at about its normal brilliancy, varies about four per cent, for each one per cent, variation of the current passing through it. Hence it is extremely important to arrange matters so that the current passing through each lamp shall be practically independent of the current passing through any other lamp, and this result is attained by making the resistance of the house wires H, /, /' small compared with the resistance of the filaments of the lamps, in accordance with the principle discussed in this section for a galvanometer and shunt. Example 82. A galvanometer of 8,100 ohms' resistance is in a circuit having 500,000 ohms' resistance external to the gal- vanometer. What is the percentage change in the main current made by shunting the galvanometer with a |th shunt ? Answer. 1-46 per cent. Example 83. If a galvanometer have 1,980 ohms' resistance, and a shunt be attached so that the current passing through the galvanometer is only ^th f * ne total current, what will be the resistance of the shunt, and by how many ohms will the resist- ance of the circuit be diminished by employing the shunt ? Answer. Resistance of shunt =20 ohms. Diminution of resistance=ig6o-2 ohms. Example 84. A pair of " leads " or branch conductors runs from the street mains, where a P.D. of 100 volts is maintained, to a hall where 150 glow lamps are in use. Each of the lamps would take 0-5 ampere at 100 volts. What must be the resist- ance of the leads in order that, when all the lamps are burning in parallel, the P.D. between their terminals is 98 volts ? Answer. The resistance of each lamp is = 200 ohms. o-5 08 Hence the current taken by each lamp at 98 volts is -^ , or 0-49 ampere, and the total current through 150 lamps in parallel is 150 X 0*49, or 73-5 amperes. The resistance of the leads must UNIVERSAL SHUNTS 259 be such that there is a " drop " of pressure of 2 volts when the 2 current is 73-5 amperes : hence the resistance is -- , or 0-0272 / o o ohm. In some cases it is desirable that shunting a galvanometer should not alter the current passing in the main circuit. This necessitates the insertion of an additional resistance to compen- sate for the diminution produced by shunting the instrument. Shunt boxes arranged to effect this compensation are called " constant total current shunts " ; one form is shown diagrammatically in Fig. 161, and in plan in Fig. 162.* no. Principle of Uni- versal Shunts. When using a shunt to compare the relative strengths of two currents, it is un- necessary to know what is the exact fraction of the main current that / passes through the gal- / vanometer, for all that iJ s H 3jj^ fatal size Figs l6lA . l6a ._ Constant Total Current Shunt Box . has to be known is the way in which this fraction is varied when the shunt is altered. Carrying out this idea, the authors have devised a method of applying shunts to a galvanometer in which the resistances of the coils of the shunt box need have no special relation to the resistance of the galvanometer itself. Hence the same shunt box can be used with any galvanometer. For example, let a galvanometer of any resistance Rg ohms be permanently shunted with any resistance R s ohms (Fig. 163), and when a current of / amperes conies along the main M 2 and leaves the main M X , let the deflection of the shunted galvano- meter be a. Next, let the main M 2 be moved from the point d to the dotted position at the point c, the fraction of R s between the points b and c being -. Now when a stronger current of /' * A useful exercise for the student is to work out the values of R sl , R sz , R s3 , R lt R 2 , and R 3 , say, for a galvanometer of 1000 ohms resistance. 260 PRACTICAL ELECTRICITY amperes comes along the dotted main M 2 at the point c and leaves by the main M lf let the deflection be a' ; then, if the deflections of the galvanometer are directly proportional to the currents passing through it, /' a 9 = n - / a whatever be the values of R g and of R s . Fig. 163. Principle of Ayrton and Mather's Universal Shunt. For let the galvanometer current in the first case be I g , and in the second I' g , then D 7 g = P , 5 p I [formula (34)], and P g = R = L L "la n ' I ' and when = -/ a Ig I' a' n n For example, if n be 10 or 100, tne ratio of the currents /' to / will be exactly 10 times or 100 times the ratio which the de- flection a' bears to a, independently of the values of R g and of R s . in. Method of Constructing a Universal Shunt c Box, and its Advantages. A " universal shunt box " constructed on this principle is seen in Fig. 164. The terminals A and B of the shunt UNIVERSAL SHUNT BOX 261 box are permanently connected respectively with the terminals of the galvanometer, while the terminals B and c of the box are connected with the two main wires which lead the current up to and away from the galvanometer and shunt. The ends of a coil of any resistance R s ohms are permanently connected as shown, 7? 7? 7? and at points in this coil corresponding with ^ 100 1000 ohms, Fig. 164. Plan of Ayrton and Mather's Universal Shunt Box. permanent connections are made with the several blocks of the shunt box as illustrated. Then, whatever be the resistance of this coil R s compared with the resistance of the galvanometer R g (either, or both, of which may, therefore, be unknown), it is easy to show that if I g amperes be the current flowing round the galvanometer when a plug is h amperes placed in the hole marked d, it will be -, - 10 100 1000 respectively when the plug is put instead into the holes marked c, b and a respectively, if there be the same current in the main circuit. 262 PRACTICAL ELECTRICITY This method of altering the shunting of a galvanometer by using a fixed resistance R s and varying the position of the mains, instead of keeping the mams fixed and varying the resistance of the shunt, as in Fig. 155, has several important advantages, viz. : (1) The same shunt box can be used with any galvanometer, etc. (2) Variations of the temperature of the room produce no error, for if all the coils be of the same material, change of temperature will not alter the ratios of the resistances. (3) The coils of the universal box can, by a proper choice of R s , be made integral numbers of ohms, and therefore more easily and cheaply adjusted than fractional values such as are necessitated by J, ^, and ^^ of R g . (4) Lastly, whatever be the value of R g , n, and R s (Fig. 164), provided that R s is less than R K (n + ^n 2 + n), the use of the universal shunt produces less change in the total resistance of the circuit than would be caused by an ordinary shunt of equal multiplying power.* The one disadvantage of the universal shunt, which, how- ever, is usually of little im- portance, is that the application of the shunt to the galvano- meter reduces the sensitiveness of the instrument. For the very few cases in which full sensibility is necessary, pro- _ vision is made for unshunting the galvanometer. Thus in Fig. 165. Universal Shunt Box. ,-,. -big. 104, removing the plug from the box and connecting the right hand terminal of the galvanometer to c instead of to A, will cause all the current to pass through the instrument. A more recent form of Universal Shunt Box is shown in Fig. 165, having multiplying powers of i, 3, 10, 30, 100, 300 and 1000. Instead of plugs, a switch arm touching metal studs is used to alter the point of contact of the main wire with the shunt. Fig. 166 shows a universal shunt intended for carrying large currents (up to 100 amperes), with multiplying powers of i, 2, 5, 10, 20, and 5o.f A great advantage of this shunt is that the sensibility can be changed by merely turning the switch handle. * A proof of this is given in the 1896 edition of this book, on page 308 t A diagram of connections of this shunt is given in Fig. 2560 LARGE CURRENT SHUNT 263 The main circuit is not in- terrupted by this movement, and the ammeter remains connected to the same two points of the shunt, which- ever multiplying power is used. Example 85. A Universal Shunt, 7,000 ohms in resist- ance, is employed with a galvanometer having a resist- ance of 1,270 ohms. What fractions of the main current pass through the galvano- meter if the part of the shunt included between the ,| mains is 10 ohms, 70 ohms, J 700 ohms, and 7,000 ohms| successively ? Answer. The ratio of the galvanometer Current tO the Fig. 166. Ayr ton and Mather Shunt for Strong main current is Currents ' designed by Mr ' DuddeU ' IO 70 700 , 7000 and 8270' 8270' 8270 ' 8270 respectively, or the fractions are in the ratio ^, ^~, ~ to I. Example, 86. Taking the galvanometer and shunt referred to in the preceding question, find the percentage difference in maximum sensibility between the galvanometer used with the universal shunt and used in the ordinary way. Answer. If the universal shunt is used, the maximum sensibility is obtained when the mains are across the galvanometer terminals, and the galvanometer takes 7000 8270 = 0-846 of the main current. If the ordinary method is employed, the galvanometer takes the whole of the main current for maximum sensibility. Hence, the universal shunt gives 15-4 per cent, less maximum sensibility. Example 87. If a galvanometer of 1,270 ohms resistance be employed, and if the resistance of the circuit external to the galvanometer be 200,000 ohms, calculate the percentage variation that will be made in the main current when the sensibility of the galvanometer is diminished from its maximum to one-hundredth of the maximum, first by using a shunt specially constructed 264 PRACTICAL ELECTRICITY for the particular galvanometer, secondly a universal shunt of 7,000 ohms in resistance. Answer. The percentage change in the main current will be 0-62 when using the ordinary shunt, and 0-50 when the universal shunt is employed. 112. Standard Resistance Coils. A resistance coil, when used as an accurate standard, is wound inside a brass box, B (Fig. 167), which is inserted in a vessel of water or oil, v v, and the temperature of the liquid is accurately measured by means of the thermometer t. The hollow cylindrical brass box B, which holds the coil, is made of large diameter outside and in- side, so as to expose as much surface as possible to the liquid in order that the coil inside may acquire the temperature of the bath as quickly as possible. It is desirable to provide a stirrer for agitating the liquid and bringing it all to one tem- perature ; and the vessel v v may with advantage have double sides, with an air - space between them, as seen in the figure, to check transference of heat between the water and the outside space. The tubes T, T are to prevent the coil being short-circuited by water getting into the holes through which the rods w, w attached to the ends of the coil, are brought out. These tubes are made of brass, but they are lined with tubes of ebonite to prevent electric contact between the brass tubes and the rods w, w. Electric connection with these rods is made by dipping their ends E E into little cups containing clean mercury. Within recent years the form of standard coil shown in Fig. 168 has come into extended use. It is known as the " Reich- zanstalt " form, and is arranged so that the ends of the copper E Fig. 167. Standard Resistance Coil. STANDARD RESISTANCES 265 rods which dip into the mercury cups may be about level with the top of the bath in which the coils are immersed when in use. Two terminals, known as " potential terminals" are provided on the rising parts of the rods, and the coil is adjusted so that the resistance between these points is of the value marked on the coil. Its advantages are (i) the length of copper terminal rod (which has a high temperature coefficient), especially the part included between the measuring points, is much smaller than in the form shown in Fig. 167 ; (2) this part of the copper may be immersed in the oil bath, and conse- quently its temperature can be controlled and measured more readily than if in air ; and (3) errors which might be produced by defective contacts in the mercury cups are eliminated. Coils of this form are particu- Fig. 768. standard ohm o>a . . . i f tt ^ (Reichsanstalt Form). larly well suited for potentio- meter measurements " described in Chapter IX. As it is frequently necessary to know the resistance of standard coils to a very high degree of accuracy, say, one part in one hundred thousand, it is evidently an advantage to make such coils of metal whose variation of resistance with temperature is very small. For this reason it is customary to construct them of German silver, platinum -silver, platinoid, eureka, constantan, or manganin wire. The coils in resistance boxes are also made of low temperature coefficient alloys. 113. Ordinary Forms of Wheats tone Bridge. In Section 89 two forms of Wire Bridge are described. By such instruments it is theoretically possible to measure any resistance, how- ever large or small, by comparison with a unit coil, but practically there is a limit to the range of measurements, for if the resistance be either very great or very small the slider will have to be moved very near one end of the wire to obtain balance, and it will be impossible to read off the short length of wire between the end and the slider with accuracy. For example, in comparing a resistance of about 100 ohms with a unit coil, the shortest segment of the metre wire would be about I centi- metre, and this length could not be read off from an ordinary scale nearer than a tenth of a millimetre, which would mean a possible inaccuracy of one per cent. Further, the ends of a bridge 266 PRACTICAL ELECTRICITY wire usually differ somewhat in resistance for a given length, from the middle portion, because of their having been heated in soldering to the copper bars, so that the true ratio of the resist- ances of the two arms may differ appreciably from the ratio of their lengths. Also, a bridge arranged unsymmetrically is not so sensitive as one with nearly equal arms, i.e., it requires a greater change made in the ratio of R c to R d (Fig. 141) to pro- duce an appreciable deflection of the galvanometer. Fig. 169. Top of a Commercial Wheatstone's Bridge. For these reasons it is customary to have several coils of values, say, i, 10, 100, 1000, as the known arm of a metre bridge, and use one or other to suit the resistance to be measured. A resistance box is sometimes employed and arranged so that the point of balance comes somewhere near the middle of the stretched wire. We have already seen that a simple bridge cannot be used to measure a resistance whose ratio to the known arm is 100 or ^jo> to within i per cent., so with a unit coil the range of the bridge (measuring to about i per cent.) is from o-oi ohm to 100 ohms. By the aid of coils 10, 100, and 1000, this range can be extended to 100,000 ohms. If, however, it is desired to measure within o of one per cent., the total range will be approximately o-i ohm to 10,000 ohms. In practice it is necessary to measure resistances much smaller than o-i ohm and far greater than 10,000 to a higher accuracy than YQ%, and for these purposes other forms of bridge are employed. One of the commonest forms is the " coil bridge/' such as is shown in Fig. 169. It consists essentially of a resistance box containing coils from i to 10,000 ohms COIL BRIDGES 267 (total), and two sets of coils 10, 100 and 1000 as shown. The latter are called the " proportional arms " or " ratio coils " of the bridge, and the former the " adjustable arm," whilst the resistance R x to be measured, is spoken of as the " ' unknown ' arm." A copper link seen at the top left-hand corner of the box, connects the left-hand proportional arm with the adjustable arm. The range of a bridge of this kind may be taken as T J n to 1,000,000 ohms ; the accuracy attainable depending chiefly on the sensi- tiveness of the galvanometer and on the battery used. In the wire bridge balance is obtained by changing the ratio of the arms of the bridge by varying both of them, whereas in using a coil bridge it is customary to obtain balance by varying one arm only, the adjustable arm. This makes it possible to read off the value of a previously unknown resistance directly ; for if we make the proportional arms equal to each other, either 10 and 10, or 100 and 100, or 1000 and 1000, balance will be obtained when the adjustable arm is made equal to the unknown arm ; the resistance of the latter can therefore be read off Fig. 170. Post Office Wheatstone's Bridge. directly. For measuring resistances not exceeding 100 ohms, we may make the proportional arm adjacent to the adjustable arm greater than the arm opposite, say, 1000 and 10 respectively, in this case balance will exist when the adjustable arm is 100 times the unknown. By this means the value can be read off directly to two decimal places, i.e., to o-oi ohm. For resistances 268 PRACTICAL ELECTRICITY GALV. & LINE INFlN NFIN 300 GALV. LOGO IOO IO 10 100 1,000 20O ' IOO 410 30 20 Fig. i7oa. Plan of Post Office Bridge. Fig. 1706. Improved " Post Office Bridge." COIL BRIDGES 269 between 100 and 1000 ratio of 1000 to 100 would be convenient, and for coils between 1000 and 10,000 a ratio of 1000 to 1000. When the unknown resistance exceeds 10,000, it is necessary, with such a bridge, to make the adjacent arm less than the opposite one, and to multiply the value of the adjustable arm which gives balance, by 10 or 100, according to the ratio used, to obtain the resistance of the " unknown " arm.* I Fig. 171. Portable Wheatstone's Bridge with Battery and Galvanometer combined. The proportional coils of a bridge can be adjusted to equality or to ratios of 10 to i, or 100 to i, with a very high degree of accuracy (about one part in 100,000), so the chief objection to using high ratios in a wire bridge does not exist in a coil bridge, and for this reason coil bridges are more generally useful. In all forms of bridge intended for very accurate measurement provision is made for measuring the temperature of the coils, for in such cases variation of resistance with temperature must always be taken into account. 114. Portable Forms of Wheatstone Bridge. Another form of coil bridge is the one adopted by the Post Office Telegraphs Depart- ment, and known as the " Post Office Bridge," or " Post Office Box." It is shown in Fig. 170, and Fig. ijoa gives a diagram of connections of the instrument. From these it will be seen that keys for the battery and galvanometer circuits respectively are placed at the front of the box, the proportional arms at the back, and the terminals to which the ends of the resistance to be measured are to be joined, are marked LINE. Sometimes they * When measuring a resistance on a given bridge, with a given gal- vanometer and battery, the most sensitive arrangement is generally obtained by making the four arms as nearly equal as possible, and joining the galvanometer or battery, whichever has the greater resistance, between the junction of the two highest arms and the junction of the two lowest arms, 270 PRACTICAL ELECTRICITY are marked LINE and EARTH respectively, the term " earth " being used because the currents sent through most telegraph lines return through the earth, and one terminal of the bridge is Fig. iyia. Diagram of Connections of Portable Bridge. connected to earth in most of the resistance measurements made in the service. An infinity plug (INFIN.), i.e., a plug which breaks the circuit of the adjustable arm when removed from its Fig. 172. Dial Pattern of Bridge (Silvertown Co.). hole, is provided. It is useful in testing whether a line is broken or disconnected, and for other purposes. Fig. 1706 shows in plan an improved form of Post Office Bridge in which the units, tens, hundreds and thousands are arranged PORTABLE BRIDGES 271 in separate columns. This facilitates reading off, and also makes the box-easier to clean. The values of the coils are I, 2, 3, 4, and decimal multiples of these numbers, a system which is gradually replacing the i, 2, 2, 5, arrangement formerly in common use, and in each proportional arm there are four coils, i, 10, 100, and 1000 ohms. Fig. 173. Bar Pattern of Bridge (Gambrell Bros.). A portable bridge, complete with double key, for galvanometer and battery, is illustrated in Fig. 171. The battery is contained in the space below the galvanometer, access to which is provided for by the door D at the near end of the box. There is space for four small dry cells. The plugs on the box belong to the adjust- able arm, whilst the ratio coils are fixed inside the box and arranged like a universal shunt ; the small switch seen on the top of the box connects the galvanometer to either one of three points on the shunt, so that the bridge reads direct, or multiplies or divides by 10, as indicated in the diagram, Fig. 1710. With 14 - coils in the adjustable arm, o-oi to 20 ohms, the bridge can be used from o-ooi to 400 ohms, and forms a very handy instrument in an electrical engineering laboratory. 115. Dial and Bar Patterns of Bridge. These forms are shown in Figs. 172 and 173 respectively. In both of them the ad- justable arm consists of sets of nine equal coils, units, tens, hundreds and thousands arranged in a ring or alongside a bar. Only one plug is required for each dial, but the number of coils necessary to obtain a given resistance is more than doubled.* * The system i, 2, 4, 8, 16, &c., advancing by powers of 2, is the one which requires the smallest number of separate coils, but it is incon- venient for use with the common scale of notation. 272 PRACTICAL ELECTRICITY The bar pattern has an advantage over the dial form because of the greater ease in cleaning the insulation. Fig. 174. Portable Bridge,~with Switch Contacts. For industrial purposes dial bridges with switch contacts, Fig. 174, instead of plugs, are frequently employed. They are easy to use, and there are no plugs to get lost, but the resistance of a switch contact is not so small or constant as that of a well- fitting clean plug. The bridge shown in Fig. 174 is of the latest design, having been made during the autumn of 1910 by Messrs. Gambrell Bros., for the City Guilds Engineering College. It is fitted with a single pivot moving coil galvanometer, battery, and keys, and has a Mather's ratio-coil switch, as indicated diagrammatically in Fig. 1710. CHAPTER VII ELECTRIC ENERGY AND POWER 116. Work done by a Current 117. Electric Unit of Energy: Joule 118. Heat Produced by a Current 119. Measuring the Heat Equi- valent of Electric Energy 120. Power 121. Electric Unit of Power: Watt 122. Joule's Law 123. Instruments for Measuring Power : Wattmeter 124. Commercial Forms of Wattmeters 125. Joule, or Energy Meter : Clock Form 126. Board of Trade Unit of Energy 127. Energy Meter : Motor Form 128. Quantity or Ampere-hour Meters 129. Electric Transmission of Energy 130. Power Developed by a Current Generator 131. Connection between the E.M.F. of a Battery, the P.D., between its Terminals, the Resistance and the Current 132. Electromotive Force of any Current Generator 133. Power Absorbed in the Circuit Exterior to the Generator: Back E.M.F. 134. Distribution of Power in an Electric Circuit 135. External Circuit that Receives Maximum Power from a Given Current Generator 136. Arrangement of n Cells to give Maximum Power to an External Circuit of Fixed Resistance 137. Minimum Number of Cells required to give a Fixed Amount of Power to a given External Circuit 138. Importance of Low Resistance and High E.M.F. for Large Powers 139. Modifications introduced into the Previous Results by Limitation of the Maximum Current a Cell may Produce 140. Efficiency 141. Efficiency of Electric Trans- mission of Energy 142. Connection between Electrical Efficiency of Transmission and the Ratio of Power Received to the Maximum Power Receivable 143. Economy in Electrical Transmission of Energy: Kelvin's Law. 116. Work Done by a Current. Whenever an electric current flows through a circuit work is done, just as whenever a water current flows through a pipe or along a river bed the flowing water does work on the obstacles that obstruct its passage. When a water stream of Q cubic feet per second falls down a height of / feet, the work done in t seconds equals 62-43 Q / t foot pounds very approximately, 62*43 being approximately the weight of a cubic foot of water in pounds. So when an electric current of / amperes flows from a point L to a point M through any circuit, the potential at M being V volts lower than the potential at L, the work done on the part L M of the circuit by the electric current in t seconds equals 0-7372 I V t foot pounds, very approximately, or 44-23 IV t' foot pounds, very approximately, in t' minutes. s 2 73 274 PRACTICAL ELECTRICITY The constant 0-7372 is derived from the fundamental definition of the ampere and volt (see Sections 8, 48, and 54), and the known relation between the erg and the foot pound given in Appendix II. In Section 48 we explained that the unit of potential difference was chosen so that the product of the P.D. between two points and the quantity of electricity passed from one to the other should be equal to the work done by the electric current between those points. Now the expression IV t can be written V (It). Here (It) represents the quantity of electricity in coulombs, and to fulfil the above condition the product of V and (//) must represent work or energy in joules (see Section 52). Now i joule = io 7 ergs, by definition, and i foot Ib. = 30-48 x 453-6 x 981 ergs (Appendix II.) = 1-356 x io 7 ergs, = 1-356 joules ; /. 77* joules - ^ 7/Hootlbs., 1-356 = 0-7372 77*foot Ibs. Neither the current of water nor the current of electricity mentioned at the beginning of this section is changed, but the current of water in falling from one level to a lower level, and the current of electricity in falling from one potential to a lower potential, gives up energy, provided that there is no apparatus in the part of the circuit in question which gives energy to the current instead of receiving* energy from it. When the stream of water is a steady one, and when it flows through a uniform tube such as / 1 (Fig. 81), all the energy lost by the water between any two points PJ and P 3 is converted directly into heat, and is employed in slightly warming the water and the tube ; so, in the same way, when a steady electric current flows through a wire, the wire and the surrounding bodies being at rest relatively to one another, the energy lost by the current is turned directly into heat and the wire is warmed. If, however, the obstruction to the passage of the water be produced not merely by objects at rest but by the buckets of a water-wheel which can be moved by the falling water, then a portion of the energy lost by the water appears as mechanical energy given to the water-wheel ; so, in the same way, when there is a magnet or a piece of iron near the wire conveying the steady electric WORK DONE BY A CURRENT 275 current, and when the relative positions of the wire and the magnet or iron can be changed by electromagnetic attraction, then a portion of the energy given up by the current is employed in doing work on the movable system. For example, when a current is sent through a galvanometer with a pivoted needle, or through a coil of wire suspended in a magnetic field, or through the coil of an electromagnet with a movable armature, or, generally, through any " electromotor," the current not only does work in heating the wire through which it flows, but it also does work in producing mechanical motion against the controlling or resisting force. As soon as the galvanometer needle or the suspended coil has been deflected to such a position that the force due to the current is balanced by the controlling force, or when the armature of the electro -magnet has been pulled down against some stop, or the electromotor has been brought to rest by some opposing force becoming greater than the electromotor can overcome, no more mechanical work is done by the current, and all the energy it subsequently loses is directly turned into heat and goes to warm the wire through which the current flows. The expression 44-23 IV V foot pounds may be divided into two parts, one part representing the energy which is lost by the current and turned directly into heat, and the other the energy lost by the current which is converted into some form of energy other than heat. If an electromotor be driven by the current and be employed to grind corn or to turn a grindstone, this second portion of the energy will also be turned into heat ; but this heat will not be produced by a direct conversion of electric energy into heat, but by a conversion first of electric energy into mechanical energy, and secondly of mechanical energy into heat. If, on the other hand, the electromotor be used to raise blocks of stone to the top of a scaffolding for building purposes, then this second part of the energy will not be turned into heat at all. If the circuit through which the current flows contains an electrolytic cell, then, although no mechanical work will be done by the current in this cell, chemical change will be effected, and when, as a consequence, chemical energy is added to the elec- trolytic cell, the work done by the current in producing this chemical energy is analogous with the work done in producing mechanical energy, and must be added to the work done by the current in directly heating the conductor to obtain the equivalent of the expression 44-23 IV t' foot pounds. If, on the contrary, chemical energy disappears from the cell on the passage of the current, this energy is transformed into electric energy, and the electrolytic cell, therefore, acts as a 276 PRACTICAL ELECTRICITY current generator and introduces electric energy into the circuit. In this case the amount of electric energy thus introduced into the circuit must be subtracted from the amount spent in heating the portion of the circuit considered, to obtain the energy transferred to that portion from the remainder of the circuit. The net amount of energy may, therefore, be either positive or negative according as the energy introduced by the cell is less or greater than the energy spent in heating the conductor.* When the current flows from L to M and the potential of the point L is higher than that of M, the current flows in the direction of the P.D. and the energy transferred to LM is positive, but when L is at a lower potential than M and the current flows from L to M, i.e., in opposition to the P.D., in that part of the circuit, the energy transferred to L M is negative. This means that the portion L M of the circuit generates more energy than it dissi- pates, and therefore it causes energy to be transferred from L M to the remainder of the circuit. In all cases, however, we may say that if a P.D. of V volts be maintained between any two points L and M in a circuit, the amount of electric energy transferred in t' minutes between the portion of the circuit L M and the res,t of the circuit by a current of I amperes, equals in all cases 44-23 IVt' footpounds. In certain exceptional cases the electrolytic cell may act simply as a resistance and be merely warmed by the passage of the current, but for that to be the case the work done in producing chemical action at one plate of the cell must be exactly balanced by the work given out in the same time by the chemical action at the other plate. Example 88. An arc lamp takes 12 amperes at 50 volts pressure. How many foot pounds of energy does it receive per minute ? Answer. 26,538. Example 89. A resistance coil of 1,500 ohms has a P.D. of 12 volts maintained between its terminals. How many foot pounds of energy does it receive per minute ? Answer. 4-25. Example 90. What current at 100 volts' pressure will supply 1000 foot pounds per second to a given circuit ? * From this it will be seen that a current generator may, if its internal resistance be high enough, abstract energy from a circuit even when its E.M.F, helps the current. THE JOULE 277 Foot pounds per second = - -- x 77, ,. , 1000 x 60 therefore I = - - amperes. 44-23 x 100 Answer. 13*56 amperes. 117. Electric Unit of Energy : Joule. In the previous section we have shown that the work done in t seconds by a current of / amperes flowing in a path L M between the ends of which there is a P.D. of V volts is given by the expression, 0-7372 IV t foot pounds (approximately), but as one foot pound equals 1-356 x io 7 ergs, we may write, work done in t seconds = IV t x io 7 ergs, or =IVt joules. By choosing io 7 ergs as the practical unit of electrical work or energy, no numerical coefficient other than unity is required in the expression for electrical work done. This is a distinct advan- tage, for we can now write, W=IVt, (38) where W is the number of joules produced in t seconds by a current of I amperes at a P.D. of V volts. If I and V and / be all unity, then W = i, from which we see that the work done in one second by a current of one ampere flowing through a circuit between the terminals of which a P.D. of one volt is maintained, is one joule. The joule is therefore the practical unit of electrical energy corresponding with the ampere, and volt, and second, and is consequently of great importance. Its relation to the foot pound is expressed by the equations 1 joule = 0-7372 ft. Ibs., very approximately, and 1 foot pound = 1-356 joules, very approximately. Example 91. A pressure of no volts is maintained between the electric -light mains of a house, and twenty glow lamps in parallel, each taking a current of 0-3 ampere, are turned on for five hours nightly for thirty nights. How much energy in joules does the house receive ? Answer. 20 X 0-3 x no X 5 X 3600 x 30, or 356-4 million joules. 118. Heat Produced by a Current. When a circuit acts simply like a resistance, so that the whole of the energy given up by a current flowing through it is converted directly into heat, Ohm's law holds in its simple form. Hence, if R be the resistance in 278 PRACTICAL ELECTRICITY ohms of the circuit, / the current flowing through it in amperes, and V be the P.D. between its terminals in volts, V = IR or the work in joules done by a current of / amperes in t seconds in heating a circuit of R ohms equals I 2 Rt. But we know from the investigations carried out by Joule which have been repeated subsequently, with even greater accuracy, by Prof. Rowland, Prof. Reynolds, and others that the heat required to raise the temperature of one pound of water by i C. when the water is at 15 C. is the equivalent of 1,400 foot pounds of work.* Therefore, if we take this as our unit of heat, it follows, since one joule equals 07372 foot pound very approximately, that h, the number of these heat units generated in t seconds in the circuit, is given by h = 0-000,526,6 I 2 Rt, very approximately ; or if t' be the time in minutes, h = 0*031,60 I 2 Rt' ', very approximately. Lastly, if a " calorie " be denned as the heat required to raise the temperature of i gramme of water by i C. when the water is at 15 C., then c., the number of calories generated in t seconds by a current of / amperes in a resistance of R ohms, is given by c 0-2390 I 2 Rt, very approximately, (39) or the number of calories generated in t* minutes is given by c = 14*34 I 2 Rt' , very approximately, and i calorie = 4-184 joules. 119. Measuring the Heat Equivalent of Electric Energy. The formulae given in the last section may be verified by sending a known current for a certain time through a coil of wire of known resistance immersed in a measured mass of water, and by ob- serving the rise of temperature with a delicate thermometer. As, however, a portion of the current passes through the water, the resistance in the circuit is a little smaller than that of the coil of wire ; also the resistance may vary by warming during the course of the experiment. Hence greater accuracy will be obtained if, instead of attempting to measure the resistance of the circuit directly, we observe from time to time the current that flows, say / amperes, and the P.D. between the terminals of the coil, say V volts ; then, if I' and V be the mean values of the current * The "British Thermal Unit" is the amount of heat required to raise i Ib. of water from 60 F. to 6i p F., and is the equivalent of 778 ft. Ibs. of work, very approximately. HEAT EQUIVALENT OF ENERGY 279 and the pressure during a period of ^seconds, the electric energy that has been given to the coil and water during that time is I'V't joules, which must therefore be proportional to the amount of heat produced in that time. If the product I'V be small, electric energy will be given to the circuit slowly; therefore the heat will be produced in it slowly, and it will not be possible to accurately ascertain the amount of heat generated in a given time, without allowing for the heat that is lost by radiation, convection, and conduction during the experi- ment. If, however, the pro- duct TV be made fairly large, and the quantity of water employed in the ex- periment be not too great, the time taken for a rise of temperature to be produced that can be accurately read on a sensitive thermometer need not be long enough for any serious loss of heat to occur. Further, if the vessel containing the water be made of very thin glass, the heat ^ absorbed in raising the tern- - perature of the vessel may be " : neglected unless very great accuracy is desired ; also, if Fi the wire be composed of a substance of high specific resistance and small temperature co- efficient, not only will the change of resistance of the coil through warming become negligible, but its mass may be small and still a considerable amount of power may be given to it. Hence the heat absorbed by the coil to raise its own temperature may be so small compared with the heat absorbed by the water that the former may be neglected, unless great accuracy be desired. The problem of properly proportioning the parts, and of gener- ally arranging the apparatus so that a beginner may obtain con- siderable accuracy by using it, without its being necessary to make any corrections for the loss of heat by radiation, convection, and conduction, was worked out by Mr. Haycraft, formerly one of the staff at the Central Technical College, and one of the authors ; and they found that with the apparatus illustrated in Figs. 175 280 PRACTICAL ELECTRICITY and 176, which fulfils the conditions they have theoretically arrived at, students can easily obtain results not differing by as much as one per cent, from the truth. A strip of manganin about J inch wide, 0-03 inch thick, and about 10 feet long, is wound so as to form the top and bottom of a sort of cylindrical box, M M, about 5 inches across and 3 inches high (Figs. 175, 176), the convolutions of the strip being kept Fig. 176. Apparatus for Measuring the Heat Equivalent of Electric Energy. from touching one another by being screwed to a light frame- work composed of two horizontal strips of vulcanised fibre, F, F, joined by three thin vertical rods of ebonite, E, E, E. The two ends of the strip are soldered to two stiff vertical copper wires, c, c, about J inch thick and 6 inches long, the soldered joints being covered over with varnish to prevent galvanic action taking place at the joint (see Section 68), and the strip M M, and the upper wires c, c are also varnished to prevent electrolysis being produced by the current leaking through the water. The whole is immersed in about 122 cubic inches or 2 litres of water contained in a thin glass beaker, GG (Figs, 175, 176), which is just wide enough to take the framework of manganin strip, and, to diminish the risk of this beaker being broken, a piece of felt N is placed between it and the base board o o. Electric connection is made with the stiff wires c c by means of two insulated very flexible leads, L L, each composed of a strand of about 210 thin copper wires, the copper wires being each about o-oii inch thick. The current is measured with an accur- HEAT EQUIVALENT OF ENERGY 2 Si ate ly -calibrated ammeter, A, and the P.D. set up between the upper ends of the stiff copper wires by means of an accurately- calibrated voltmeter, v (Fig. 176). The object of using a flat conducting strip and forming it into the box shape seen in the figures is to enable the conductor itself to act as an efficient stirrer when it is moved up and down in the water, the flexible leads L, L, which are fastened to a wooden rod p p fixed to the base board o o, as shown in Fig. 176, serving as a handle to the box M M. The heat generated in the strip is, therefore, given off fairly uniformly to the water, and the mean temperature can be read with considerable accuracy on a single stationary thermometer, t. With an apparatus constructed as above described, and used with a current of 30 amperes, the temperature of the water rises at approximately the same rate as that of the leads L L and wires c c, so that the conduction of heat to or from the water by the copper is practically nil, and error from this cause is eliminated. Below we give a set of results actually obtained by students of the Central Technical College, using 2 litres of water. In calcu- lating the last column allowance has been made for the water equivalent of the glass vessel and stirrer, which amounted to 47 grammes. Time in Temperature C. Current in Mean P.D. Calories Seconds. Initial. Final. Rise. Amperes. in Volts. per Joule. 120 18-40 22-O2 3-62 30 8-634 0-2383 1 80 I3-25 18-70 5'45 30 8-634 0-2390 1 80 13-60 19-00 5'4 30 8-648 0-2367 I2O 12-97 16-58 3-6i 3 8-656 0-2375 I2O 12-64 16-26 3-62 30 8-698 0-2365 I2O 12-89 16-49 3-60 30 8-662 0-2364 120 12-11 15-72 3-6i 3<> 8-666 0-2368 120 I2-IO 15-74 3-6 4 30 8-642 0-2395 I2O I3-I3 16-75 3-62 30 8-692 0-2367 Mean 0-2375. Average deviation from the mean = o-ooi = 0-42 per cent. Now we saw in Section 118 that the true number of calories per joule was about 0-2390, hence only two of the preceding results obtained by the students differs by more than I per cent, from the truth, while the mean of the nine observations gives a result which has an error of only about one half per cent. Conse- quently the result aimed at in designing this apparatus has been achieved. In carry out the investigation we may vary either (i) The time during which the current is allowed to flow ; 282 PRACTICAL ELECTRICITY (2) The current made to flow through the strip ; (3) The resistance of the conductor, by using similar stirrers made of somewhat thicker or thinner manganin strip ; and when a series of experiments is made varying each of these three conditions, one at a time, it is found that the rise of tempera- ture of the water, and therefore the amount of heat produced, is proportional to the time, proportional to the square of the cur- rent, and proportional to the ratio of V to / that is, to the resistance of the arrangement. Further, if we take as the calorie the heat required to raise the temperature of i gramme of water by i C. when the water is at a temperature of about 15, we find that the relationship between the number of calories, the current in amperes, the resistance in ohms, and the time, is practically that given in Section 118. Example 92. A current of 30 amperes is passed through a coil of wire immersed in water for five minutes, a voltmeter reading 10-3 volts at its terminals. The volume of water is 2,000 cubic centimetres, and the temperature rises from 15-7 to 26-66 C. What result does the experiment give for the heat equivalent of one joule in calories ? Answer. 0-2364, a result about one per cent, too low, no corrections having been made for cooling during the experiment. Example 93. A temporary resistance is made by putting a coil of wire of 4 ohms resistance into a wooden bucket containing 37 pounds of water. If a current of 40 amperes be sent through the coil, what about will be the rise of temperature of the water in the first three minutes ? Answer. 16 C. 120. Power. " Power " is the name given to the rate of doing work that is, the rate of transformation of one form of energy into another and it must be carefully distinguished from the amount of work done, there being the same sort of difference be- tween power and work that there is between a velocity and a distance. The word power was, however, used in the older books on dynamics to stand for the applied force, and that is the meaning of the word power in such expressions as " the mechani- cal advantage of a machine is the ratio of the weight to the power." Again, the word power is sometimes wrongly used for energy, as in the expression the " storage of power/' Beginners must, therefore, be on their guard against being misled by such loose expressions, and they should never employ the name power, or " activity," as suggested by Lord Kelvin, in any other meaning than the rate of doing work. In that sense, of course, power cannot be stored, for while a certain quantity of water in a reser- POWER 283 voir at the top of a hill represents a certain store of energy, the power that this water can exert at any time when flowing out of the reservoir will depend on the rate at which it is allowed to flow. When work is being done at a constant rate, the power is constant, and it is measured by dividing the number which expresses the work done in any time by the number expressing the time. If, however, the rate of doing work at one moment is greater than at another for example, when a person runs upstairs quickly at first and then more slowly we do not mean by the power expended at any moment, the actual work done in a minute, or even in a second, for the rate of doing work may be changing very rapidly. In such a case the power at any time is the limiting value of a ratio obtained thus : Measure the work done in a very short time, a portion of which precedes, and the remainder of which follows, the instant at which we wish to measure the power ; divide the work done in the very short time by that time, then this ratio more and more nearly represents the power being expended at the moment in question, as we make the very short time shorter and shorter. When, however, electric energy is being transformed into some other form of energy, the power may be very easily ascertained, whether the rate of doing work is constant or not, without its being necessary to measure a small time. For the work done in t seconds by a constant current of I amperes flowing through a circuit under a constant P.D. of V volts equals IVt joules, provided that there is no apparatus in the circuit that gives energy to the current instead of receiving energy from it ; therefore the rate of doing work in joules per second equals simply IV. Hence, if at any moment we measure the current and the P.D. simultaneously, the product of the two measurements gives us the instantaneous value of the power being expended at that moment, and no measurement of time need be made. Conse- quently the rate of transformation of electric into some other form of energy may be varying, but as long as it is not varying so rapidly as to prevent accurate readings of an ammeter and voltmeter being taken, the instantaneous value of the power can be ascertained at any moment. 121. Electric Unit of Power : Watt. When work is being done at the rate of one joule per second the power exerted is 284 PRACTICAL ELECTRICITY called a " watt " ; therefore the power of one watt is developed when work is done at the rate of 10 7 ergs per second, or 1 joule per second, or 0-7372 foot pound per second, very approximately, or 44-23 foot pounds per minute, very approximately ; and since when work is being done at the rate of 550 foot pounds per second, or 33,000 foot pounds per minute, one " horse-power " is said to be exerted, 33 000 1 horse-power = TJ- watts, very approximately, ' 1 horse-power = 746 watts, very approximately, 1 watt = 1 /746th of a horse-power, very approximately, 1 kilowatt = 1000 /746th, or 1-340 horse-power, very ap- proximately. .'. 1 kilowatt = 1J horse-power, roughly. Further, if P be the power in watts expended in a circuit between the ends of which a P.D. of V volts is maintained and through which a steady current of I amperes is flowing, P = IV (40) provided that the circuit contains no apparatus that gives energy to the current, instead of receiving energy from it. Example 94. What power in watts is expended in the arc lamp and in the resistance coil referred to in Examples 88 and 89. Answer. 600 watts, and 0-096 watt respectively. Example 95. What power in kilowatts is expended in the coil in Example 92 ? Answer. 0-309 kilowatts. Example 96. The adjoining figure shows the " load diagram " LOAD DIAGRAM. DECEMBER. 1000 of a central station for December i.e., the curve giving the out- put of the station in amperes throughout the twenty-four hours. JOULE'S LAW 285 If the station pressure is 440 volts, what is the output in H.P. (horse-power) at 7 a.m., 12 noon, 6.30 p.m., and 10 p.m. ? Answer jj H.P., 23 H.P., 612 H.P., and 286 H.P. respectively. Example 97. Two glow lamps, each giving 16 -candle power, take 1-75 and 1-25 watts per candle respectively. How many lamps can be supplied per horse-power expended in the two cases, and how many candles per horse-power will they give ? Answer. Twenty-seven and thirty-seven lamps respectively ; 426 and 597 candles. Example 98. How many candles per horse -power are given by an arc lamp taking n amperes and 50 volts, and giving a mean candle-power of 1,750 in all directions ? Answer. 2,374 candles per horse-power. 122. Joule's Law. From the above it follows that if P be the power in watts expended in heating a circuit of resistance R ohms through which a current of / amperes is flowing, then P - I*R t for P=IV, and V = IR, by Ohm's Law, /. P = I*R; (41) or, the rate at which heat is generated in a resistance through which a current is flowing, is proportional to the product of the square of the current and the resistance ; this is known as Joule's Law. y Since / = - the above expression may be written K a form which is useful in many cases. Further, the energy transformed into heat in t seconds is by Section 117, given by the equation W = IVt joules, and this may be written W = I 2 Rt joules, or F 2 W=~.t joules R according as / or V is given. Expressed in calories we have c = 0-2390 I 2 Rt, (43) V' 2 and = 0-2390*, (44) 2 56 PRACTICAL ELECTRICITY as the forms most convenient to use when the current is given, or the P.D. is given, respectively. Example 99. Calculate the power in watts expended in a con- ductor whose resistance is 2-5 ohms when a current of 20 amperes is flowing through it. Determine also the energy used in one hour. Answers (i) 1000 watts, or i kilowatt (2) 3,600,000 joules, or i kilowatt hour; Example 100. One of the lamps of an electric radiator con- sumes 250 watts at 210 volts ; what current passes through it, and what is its resistance ? Answer (i) 1-19 amperes approx. ,, (2) 176-4 ohms approx. Example 101. An ammeter graduated to 150 amperes has a resistance of 200 microhms ; find the power expended in the instrument at maximum reading. Answer. 4-5 watts. Example 102. What power is used in a voltmeter of 8,000 ohms resistance when a P D. of 220 volts exists between its terminals ? Answer. 6-05 watts. 123. Instruments for Measuring Power : Wattmeters. The electric power used in a circuit can be determined, as shown in the preceding sections, by finding the current and P.D., and multi- plying them together ; when the current is steady this method is comparatively simple, but in many cases it is more convenient to use an instrument, the reading of which gives the power direct - ly. Instruments for this purpose (called wattmeters) were first made in England by Professor Perry and one of the authors in 1881. One of the simplest forms of wattmeter resembles the electrodynamometer described in Section 39, in having a fixed coil and a moving coil, and a torsion head whereby the moving coil can be brought into a definite relative position to the fixed coil. In the electrodynamometer both coils carry the same current, but in the wattmeter one of the coils carries the main current of the circuit in which the power is to be measured, whilst through the other coil a current proportional to the P.D. between the terminals of the circuit passes. They are called the current coil and the pressure coil respectively. The current coil c c, Fig. 177, which is made of a few turns of thick wire, is inserted in the main circuit ; while the other coil, c c, consisting either of many turns of fine wire, or, better, of a few turns of fine wire in series with a stationary high resistance, w, is connected as a shunt to that portion of the circuit L M the power given to which we desire to measure. The current passing through c c is therefore proportional to the P.D. between the WATTMETERS 287 ends of L M, while the current passing through c c is the sum of the currents flowing through L M and' through c c. If, however, the resistance of the fine-wire circuit of the wattmeter is very large, the current passing through it will be very small compared with the current flowing through L M, so that the current passing through c c will be practically that flowing through L M. Hence the part of the wattmeter between the terminals T lt T 2 acts as an ammeter, while that between the terminals t lt t 2 serves as a voltmeter. Consequently the product of the currents in c c and c c is proportional to the power given to L M. But this Fig. 177. Diagram of Wattmeter. product is directly proportional to the couple exerted between these two coils if the coils be always brought into the same position relatively to one another. Hence the power to be measured is proportional to the torque that must be exerted on the movable coil of the wattmeter to keep it in a fixed position relatively to the stationary coil. The torque required to be exerted on the suspended coil c c in order to maintain it in a fixed position relatively to the station- ary coil c c, may be conveniently produced by turning the head H and the pointer p attached to it. This twists the thin vertical wire supporting the movable coil, as the upper end of this wire is rigidly fastened to the head H. And, since the angle through which one end of a wire is twisted relatively to the other end is directly proportional to the torque exerted, the power given electrically to the portion of the circuit L M will be directly pro- portional to the angle through which the pointer p has been turned to keep the coil ccm the position it occupied when no current was passing through the coils. 288 PRACTICAL ELECTRICITY Another way of joining up a wattmeter is to connect t v t 2 , the terminals of the fine-wire circuit, to L and T 2 respectively, so that the fine-wire circuit is a shunt to both L M and the thick wire coil c c of the wattmeter. In that case the current passing through c c will be accurately the current that flows through L M, but the current passing through c c will now be proportional to the P.D. between the points L and T 2 , and not between the points L and M. The difference between these two P.Ds. will, however, be very small if the power spent in sending the cur- rent through c c is very small compared with the power spent in sending it through L M, and this result can be practically attained by making the resist- ance of the coil c c as small as possible. 124. Commercial Forms of Wattmeters. Commercial watt- meters based on the principle described in the last section have been constructed by several people. A compact form, de- signed by Mr. Swinburne, is ,seen in Fig. 178, which shows 'the instrument with the outer cylindrical cover removed so that the interior may be visible. The stationary coil c is made in two sections, the front one having been removed in the figure so that the suspended coil c can be better seen. The position of this suspended coil is sighted by means of a small pointer which is rigidly attached to the bottom of the vertical rod hanging down from the small moving coil c, and when a measurement is made, the milled head H is turned until the small pointer is exactly over a black line marked on a silvered plate which is fixed to the base of the instrument just under the little pointer. Parallax is avoided by the pointer and this line being looked at through a small window w w in the dial plate at, the top of the wattmeter. Instead of measuring the torque that has to be exerted to keep the suspended coil in its initial position, as in using the zero watt- meters shown in Figs. 177 and 178, we may observe the angle through which the moving coil is turned against the action of a Fig. 178. Swinburne Wattmeter, with cover removed. WATTMETERS 289 spring or gravity. A deflectional wattmeter with spring control is shown in Fig. 179. The case has been removed to show the working parts more clearly. From the figure it will be seen that the moving coil to which the pointer is attached, is outside the fixed or thick wire coil c c being the ends of the thick wire coil, and c c those of the thin coil. Near the bottom of the figure are four magnets between the poles of which an aluminium disc fixed to the moving coil passes. They serve to " damp " the move- Fig. 179. Elliott's Deflectional Wattmeter. ment, and bring the pointer quickly to rest, but otherwise do not affect the indications of the instrument. The power to be measured will not be directly proportional to the angle through which the pointer attached to the movable coil turns, but the scale seen in Fig. 179 has been constructed by sending various known currents through the two coils respectively, and making a direct reading scale by a process similar to that described in Section 22. Example 103. In a certain wattmeter it is found that the head must be turned through 125 to bring the pointer to zero, when the current in the main coil is 20 amperes and the P.D. between the ends of the shunt coil is 120 volts. How much must the head be turned to bring the pointer to zero if the wattmeter is measuring the power taken by a resistance of 7-3 ohms through which a current of 30 amperes is passing ? The wattmeter reading is proportional to the product of the currents in the two coils, and the current in the shunt coil is proportional to the P.D. between its terminals, which is 7-3 x 30, PRACTICAL ELECTRICITY or 219 volts in the second case. Hence, if is the angle through which the head must be turned, _0_ _ 30 x 219 125 ~~ 20 X 120* Answer. The head must be turned through 342. Example 104. If the resistance of the shunt coil in the above wattmeter is 6,542 ohms, what additional resistance in the shunt circuit will make the constant of the instrument 20 watts per degree ? Answer. 272-6 ohms. Example 105. If the current passing through the circuit, the power given to which we desire to measure, is 20 amperes, while the P.D. maintained between its terminals is 30 volts, and if the resistances of the thick -wire coil and of the fine-wire circuit of a wattmeter are o-oi and 1000 ohms respectively, calculate the error that will be made by using the wattmeter when joined up in the two ways described in Section 123. Answer. When the wattmeter is joined up as shown in Fig. 177 the current passing through the thick -wire coil cc will be 30 20 H amperes instead of 20 amperes that is, will be 0-15 per cent, too large ; therefore the power measured by the watt- meter will be 0-15 per cent, greater than the power given to the circuit L M. If, on the other hand, the wattmeter be joined up as described at the end of Section 123, the current passing through the fine-wire circuit of the wattmeter will be produced by a P.D. of 30 + 20 X o-oi, or 30-2 volts instead of 30 volts, the 0-2 volt being the P.D. expended in sending the current through c c. Hence the current through the fine-wire circuit, and therefore the power measured by the wattmeter, will be 0-67 per cent, too large. Consequently the former method of joining up the wattmeter would give the more accurate result in this particular case. It should here be noted that in both cases the wattmeter reading is greater than the true power given to L M. 125. Joule-or Energy Meter: Clock Form. As shown by Professor Perry and one of the authors in 1882, any pendulum clock can be easily converted into a " joulemeter ; " that is, into an instrument which records the energy given to an electric circuit in any definite time, and Fig. 180 illustrates the first electric energy meter, called originally an " ergmeter," that was constructed in this way. The ordinary pendulum bob is replaced by a bobbin B, on which is wound a coil of fine wire, the coil being wound on in two ENERGY METER 291 parts, c, c, for convenience of attachment of the bobbin to the pendulum rod. These two halves of the fine-wire coil are joined in series with one another, and the terminals of this coil, t lf t 2 , are connected as a shunt with that portion of the circuit LM, the energy given to which we desire to record. Fixed to the clock case in the position shown is a coil consisting of a few turns of thick wire ; this coil being also constructed in two parts, c c, so that the pendulum coil may swing symmetrically between them. These two halves of the thick -wire coil are joined in series with one another, and . the terminals of this coD, TJ, T 2 , are connected up as shown, so that the main current through L M passes through the station- ary coil c c. Cur- rents, then, flow both round the moving coil cc and the stationary coil c c, and produce, therefore, an at- traction or a repul- sion between these coils, depending on whether the coils are joined up so that the currents circulate round them in the same direction or in opposite directions. The force exerted between the coils will vary with their relative posi- tions, but its mean value will be proportional to the product of the currents flowing in the two coils ; that is, it will be propor- tional to the power given to the part of the circuit L M. The action of this force on the swinging pendulum will be approximately the same as if the action of gravity had been increased or diminished ; hence, if the coil be joined up so that there is an attraction, the clock will be caused to gain, whereas if the ends of the fine-wire coil be interchanged, the clock will lose. And if this force is small compared with the weight of the pendu- lum bob it may be shown in the following way, that the total gain or loss of the clock in any period is directly proportional to the energy given electrically to the circuit in that period. Fig. 180. Ayrton and Perry's Original Gaining Clock Joulemeter. 292 PRACTICAL ELECTRICITY From the binomial theorem we know that when b is small compared with a, then (a b) n = a n n a n ~ l b, approximately, so that the difference between (a + b) n and a n is proportional to b, when a and n are constants. Now it follows from Section 27, that the number of vibrations a pendulum makes in t seconds is proportional to the square root of the controlling force, and therefore n equals J in this case, and if the con- trolling force be increased or decreased by a small amount, the difference in the number of vibrations in a given time caused by this change will be proportional to the change. We thus see that the gain or loss of the clock produced by the magnetic force between the fixed and moving coils will, if this force be small compared with the weight of the pendulum, , be proportional to the power given to the circuit L M, and to the time-. In other words the gain, or loss, is proportional to ' K ' the energy expended in the circuit. Further, if instead of observing the gain or the loss of the . l ..-er,n p aeann 8 o e ro D meter ^ ^ j^^j by comparing its indication at the beginning and end of the interval with a good clock or watch, we place two clocks inside the " supply meter," one clock being an ordinary one, and the other a clock having for its pendulum a pressure coil swinging near a stationary coil, through which passes the main current flowing through the house, the energy given to the house in any time will then be directly proportional to the difference between the number of vibrations that the two clocks have made in the interval. This difference can be read off on a counting mechanism like that used on a gas meter if the staff F F (Fig. 181), driving this counting mechanism be connected by means of " differential gearing" with the two clocks. The staff F F (Fig. 181), is rigidly connected with the balanced arm, A A, which carries at one end a planet wheel P P. This gears into two crown wheels K K, K' K', turning loosely on the staff F F. These crown wheels have also teeth cut on their circum- ferences like ordinary toothed-wheels, and are geared with the two clocks respectively, one wheel being rotated by one of the clocks right-handedly and the other left-handedly. When no electric energy is being supplied to the circuit LM (Fig. 180), ARON ENERGY METER 293 the crown wheels are driven by the clocks at equal rates in oppo- site directions, the wheel p p therefore is simply turned round on the arm A A as an axis, but the arm itself is not moved. But when energy is supplied to the circuit the clock with the magnetic pen- dulum goes faster, the crown wheel driven by it, therefore, also ro- tates faster than the other crown wheel, and the pinion p p not only is rotated on the arm A A, but the arm itself, and the staff F F at- tached to it, are driven round, and move on the dial hands at a rate depending on that at which electric energy is supplied to the circuit. In the modern form of clock energy meter designed by Dr. Aron, the two clocks are driven by the same mainspring, each pen- dulum carries a coil c, c, whose axis is vertical when in the mid-posi- tion, and these swing over and near to fixed coils c, c, as seen in Figs. 182 and 1820. The coils are so con- nected that one of the pendulums gains and the other loses, when current passes through the meter, and the difference in the number of vibrations in a given time is doubled by this device, and this difference is registered on the meter dials by differential gearing as described above. Two difficulties experienced with the original form of energy meter, viz., winding the clocks, and the necessity Fig. 182. Aron Energy Meter (cover removed). PRACTICAL ELECTRICITY of the clock keeping correct time when no current was passing through the current coils, have been overcome by (a) fitting an automatic winding device operated electrically, and (b) reversing the current through the pendulums at equal intervals of time, and at the same instant reversing the connection of the differential gearing with the registering dials. Both the latter operations are ^ WW^(k,m r tatojy^ nEsSFTri [( O 1 L5 i i -^ ^^ I 1 r LJVX U Winding- g-ear c D endulums c c^z: [-"-| 1 < EV ; i ^din 1 H^C p \ _J ci r .oils ^1 rf i?H j 1) (S r i 8 e ) 1 i + c V. J l^J f J Gen era tor [ LarnDs);x| Fig. 1820. Connections of Aron Energy Meter. performed automatically by electrically driven mechanism, and thus the effect of any want of synchronism in the two pendulums when no current is passing is eliminated. By properly choosing the toothed gearing between the differential gear and the regis- tering dials, and by adjusting the position of the stationary coils relative to the pendulums, the dials can be arranged to read the energy in joules, or any convenient multiple thereof. 126. Board of Trade Unit of Energy. The units of energy we have hitherto considered are the " erg," the " foot pound," and the BOARD OF TRADE UNIT 295 " joule/' the former being the unit in the "C.G.S." system, and the latter the unit in the " practical " system of electrical units. Although the joule (or the " watt second ") is ten million ergs, the joule is found inconveniently small for commercial purposes, and when electric energy began to be supplied to the public by electric lighting companies in the early eighties, the Board of Trade adopted a much larger unit, viz., the " kilowatt hour " or 1000 watt hours as the commercial unit. The word " kelvin " has been suggested as a name for this unit, but is not yet generally adopted. " The Board of Trade Unit," the words " of energy " being generally omitted, is the name given to the work done in a circuit when the power exerted in watts multiplied by the time during which it is exerted in hours equals 1000, or 1 Board of Trade unit = 1000 watt hours, = 3,600,000 joules, = 36 x 10 12 ergs, = 2,653,800 foot pounds, very approxi- mately, = 1-340 horse-power hour, very approxi- mately. = 1 horse-power hour, roughly. A Board of Trade unit is, therefore, a thing that can be bought and sold at a specified price, like a ton of iron, and this price can be regulated by agreement or by law, just as the price of gas per 1000 cubic feet, or as cab fares are regulated. The average price per Board of Trade unit supplied to large consumers in London for lighting purpose is about sixpence, whilst for heating, cook- ing and electric-driving of machinery, i|d. per unit is a common price. Example 106. If electrical energy is supplied at 4d. per Board of Trade unit, determine whether it is more economical to use 16 -candle power carbon lamps taking 2-5 watts per candle and lasting 500 hours, or 16 -candle power lamps taking 3-5 watts per candle and lasting 900 hours, the cost of a new lamp being in each case lod. Answer. Using 2-5 watt lamps : 2-5 Cost for energy per candle hour = X 4 IOOO = o-oio penny. Cost for lamp renewals per candle hour . = ID X 5OO = 0-0012 penny. Total cost per candle hour = 0-0112 penny. 296 PRACTICAL ELECTRICITY Using 3-5 watt lamps : Cost for energy per candle hour = 5 - x 4 Cost for lamp renewals per candle hour 1000 = 0-014 penny. 10 16 x 900 = 0-0007 penny. Total cost per candle hour = 0-0147 penny. Therefore, in this particular case, it is more economical to use the lamp having a shorter life but taking less power per candle. * Example 107. Is the same conclusion true if the lamps are 8 -candle power, all other things remaining the same ? Answer. The total costs per candle hour become 0-0125 and 0-0154 penny, so that the shorter-life lamp is still the cheaper. Example 108. Compare the cost for equally lighting the same area with gas at 2s. 6d. per 1000 cubic feet (the mantles used giving 60 candles for 5 cubic feet per hour) with metal filament incandescent lamps using electric energy at 4d. per unit (the lamps taking 1-25 watts per candle), and with flame arc lamps supplied with electric energy also at 4d. per unit (the lamps taking J watt per candle). The cost of renewal for broken mantles and glow lamps, and the carbons for the arc lamps, not to be included. Answer. Relative costs : Gas, 2-5 ; electric glow lamps, 5-0 ; arc lamps, i. Example 109. What is the reduction in a consumer's bill of 80 per annum for electric energy supplied, (a), if the price of a unit is reduced from 6d. to 5d. ; (b), if lamps taking 1-2 watts per candle are used instead of lamps taking 3-5 ? Answer. (a) A reduction of 13 6s. 8d. ; (b) a reduction of 52 us. 5d. Example no. How many Board of Trade units are consumed by a loo-volt 20-candle power lamp taking 28 watts burning continuously for one year ? What is the cost at id. per unit ? Answer. 245 units ; i os. 5d. 127. Energy Meter: Motor Form. In the last section was described the method of recording the sum of the products of the power into the time ; that is, the total amount of electric energy given to a circuit, by using the attraction between the current and pressure coils of a wattmeter to alter the rate of going of a clock. * The costs in examples 106-110 are based on pre-war prices. MOTOR ENERGY METERS 297 But, as pointed out by Professor Perry and one of the authors in 1882, in the same patent specification, this attraction may, in- stead, be employed to drive the counting mechanism, and give a direct record of the energy supplied to any circuit if the current and pressure coils be made to form the stationary and moving parts respectively of an electromotor without iron, and if the rotation of the motor be resisted by a torque proportional to the velocity of rotation. This principle has been used by Professor Elihu Thomson in the construction of a very large number of joulemeters. For some reason this instrument as constructed by Professor Elihu Thomson has been called a " recording wattmeter " ; this name is, however, a misnomer, since it is the total amount of energy in Board of Trade units, and not the variations of the power in watts, which the instrument records. It is impossible to obtain continuous motion by the mutual action of the currents in two coils unless the current in one of the coils, at any rate, be periodically reversed. For, suppose currents flow round two coils in such directions that the coils attract one another, the coils, if one or both of them be free to move, wilJ approach one another, the force of attraction will rapidly increase, causing them to finally rush together, when they will press against one another, and any further motion will be clearly impossible. On the other hand, if the directions of the currents be such that the coils tend to repel one another, either it will happen that one of the coils will turn round, when they will approach as before, or, if neither of the coils be free to turn, they will recede from one another until the distance separating them becomes so great that the force of repulsion is too small to overcome any fri^tional resistance that may oppose the motion. To keep up a continuous motion, then, of one coil relatively to another, there must be employed some form of " commutator " or arrangement for reversing the current through one of the coils ; further, if we wish that the force producing the motion shall remain fairly constant, either the moving or stationary part of the motor must consist of a number of coils so arranged that, as the rotation of the motor changes the position of one coil in the magnetic field, its place in the field is taken by the next coil. This part of the motor is called the " armature" while the other part is called the " field," and if the armature has a sufficient number of coils on it the torque exerted between the field and the armature remains practically constant, in spite of the motion of the one relatively to the other. The armature of the Elihu Thomson energy meter is the rotating portion, and it consists of eight coils, c v c 2 . . . c g , wound on 298 PRACTICAL ELECTRICITY Fig. 183. Rotating Armature of the Elihu Thomson type of Motor Energy Meter. this shunt current is led into and out of the commu- tator by two station- ary " brushes," B, B, the current dividing into two parts at each brush and fol- lowing the paths shown by the arrows (Fig. i83).* * In the modern form of House Service Meter illustrated in Fig. 184, the armature is made spherical in shape, and the brushes B, B, Fig. 183, are made of small silver tubes supported on fine brass wires. a light framework, as seen in Fig. 183, which shows the armature detached from the complete meter in order that the construction of the former may be clearly seen. The end of each coil is electrically connected with the beginning of the next, and is also connected by means of one of the wires ze>i, ze> 2 . . . . W Q (Fig. 183) with one piece of the eight- part commutator k lt k z . . . . k 8 . The armature, which is in series with a stationary resistance, is joined as a shunt to the portion of the circuit the energy given to which it is desired to record, like the pressure coil of the wattmeter, Fig. 177, and Fig. i83. Diagram showing the Directions of the Currents in the Armature of the Elihu Thomson type of Energy Meter. THOMSON ENERGY METER 299 This figure is a diagrammatic sketch of the armature, commuta- tor, and brushes at the moment when the two pieces k 7 and k a of the commutator are touching the brushes, and to avoid con- fusion in this sketch only these two pieces of the commutator are shown connected with the coils. But in reality each of the eight commutator pieces ^ .... k 8 is joined respectively with the end of one coil and the beginning of the next, and, since the brushes are stationary while the armature and commutator re- volve, the direction of the currents in the coils would appear exactly the same whether the pair of commutator pieces touching the brushes were k L k 5 , The result is, that although the armature rotates, the current flowing round it pro- duces a magnetic field, in a nearly fixed posi- tion, indicated by the dotted line N s. The stationary field coils c c, seen in perspective in Figs. 1836 and 184, are placed in series with the portion of the circuit the power given to which we desire to measure, so that the main current passes through these field coils and produces another stationary magnetic field, which is almost at right angles to that produced by the armature, and the action of the one field on the other causes a continuous rotation of the armature. As these two fields have always the same relative position, the torque exerted will be directly proportional to the product of the strengths of the fields, and, as no iron is used in either the armature or the field coils, the magnetic fields will be directly proportional to the currents producing them ; hence the torque producing the rotation will be directly proportional to I V, the power in watts given to the portion of the circuit under consideration. The motion of the armature c (Fig. 184) is resisted by the horizontal aluminium disc D. which is rigidly attached to the Fig. 183$. Interior of Thomson Energy Meter (Old Form). 300 PRACTICAL ELECTRICITY armature spindle E, being rotated in the magnetic field produced by four stationary permanent magnets M, M, M, M. The south pole of each of these magnets (Fig. 184) is above, and the north pole below, the disc, so that the lines of force produced by these permanent magnets are vertical and at right angles to the plane of rotation of the disc. This rotation between the poles of Fig. 184. Thomson House Service Meter (Modern Form). the magnets causes currents, called " Foucault currents " or " eddy currents," to be induced in the disc, and the attraction between these currents and the stationary magnets impedes the turning of the armature. Now, the strength of these induced currents is proportional to the angular velocity a, so that the torque which resists the motion is proportional to a. We have, therefore, as shown by Professor Perry and one of the authors, a driving torque proportional to IV and a retarding LAW OF ENERGY METER 301 torque proportional to a ; hence, if the frictional resistance to motion introduced by the bearings of the armature, the rubbing of the commutator against the brushes B, B (Fig. 183), and the train of wheels in the counting mechanism driven by the screw or worm s (Fig. 183), be very small, the armature must rotate at such a speed that the electromagnetic driving torque, which is proportional to IV, is exactly equal to the electromagnetic retarding torque, which is proportional to a, or IV oc a. If, now, during any time t seconds, the power supplied to the circuit be constant, IV will be constant for that time, and so also will a ; therefore IV t a at, but IV t is the energy in joules and a t is the angle turned through by the armature in that time. Consequently for each period of time during which the energy is supplied at a constant rate, the angle turned through by the armature, and therefore the advance of the counting mechanism, is directly proportional to the energy supplied in that time. Therefore, adding together all the amounts of advance of the counting mechanism and ail the amounts of energy supplied for each of the periods during which energy is supplied at various constant rates, we may conclude that the total advance of the counting mechanism in any interval will be directly proportional to the total amount of energy supplied in that interval, whether the energy has been supplied at a uniform or at a variable rate. The friction at the bearings of the armature may be rendered small by using a very light armature, and by forming the ends of the armature spindle of hard metal, carefully pointed, and by supporting them in jewels, as is done in good watches. The friction and inertia of the counting mechanism can be overcome by making the parts small and light ; and the friction of the commutator & t . . . . k 8 against the brushes B, B, Professor Elihu Thomson has found, can be reduced to a workable limit by constructing the commutator of silver, as well as the parts of the brushes that rub against it, and by making the diameter of the commutator very small. The clock type of meter has the great advantage over the motor form that, no matter how small be the rate at which electric energy is supplied to a circuit, the clock meter actually records the total amount of energy supplied, whereas, in conse- quence of friction, a motor meter will not start until the currents passing round its coils reach a certain value. Hence, if the 302 PRACTICAL ELECTRICITY electric power that a circuit receives be always very srr.all, the armature of a motor meter may never move, and so the meter will record no energy received, even though the period during which this very small amount of power has been supplied has been so long that the total amount of energy that ought to be recorded is considerable. To overcome this defect it is now customary to put a "starting coil" j, Fig. 184, near the main coils, consisting of a moderate number of turns of fine wire in series with the arma- ture. These turns produce a magnetic field in the space oc- cupied by the arma- ture and give a torque nearly equal to the friction torque. The meter is therefore on the point of starting when no current is being used in the lamps, and switching on a single small lamp causes the armature to turn. 128. Quantity or Ampere-hour Meters. In practically all cases where electric energy is supplied to consumers the supply authorities are re- Fig. 185'. Bastian Meter (front cover removed). quired to maintain a constant P.D. between the mains which convey the currents, and as the energy is Vlt the quantity (It) is proportional to the energy supplied when V is constant. If, therefore, we can measure It, the whole energy can be determined by multiplying by an appropriate constant. For example, if the supply pressure be 200 volts a current of one am- n r x 200 x i x i pere flowing for one hour will mean a consumption of 1000 i.e., 0*20 Board of Trade units. AMPERE-HOUR METERS 303 The simplest form of quantity meter is the voltameter (Sections 7, 8, n, 12, and 13), and both copper and acid (or alkali) volta- meters are used for this purpose. In one form of the latter, the Bastian Meter (Fig. 185'), a solution of caustic soda is electrolysed, Fig. 185. Chamberlain and Hookham Metet. the resulting gases being allowed to escape, and the diminution of volume of the liquid measures the quantity of electricity passed through it. A layer of oil about half an inch deep is placed on the surface of the electrolyte to prevent evaporation, and the current is led into and out of the liquid by rods R', R termina- ting in cylindrical nickel electrodes near the bottom of the glass vessel. The scale of the instrument is graduated to read 304 PRACTICAL ELECTRICITY Board of Trade units at a particular pressure. Most ampere- hour meters are, however, of the motor form. Two of these, the " Chamberlain and Hookham," and the " Ferranti " meters are shown in Figs. 185 and 186. A section of the former meter is given in Fig. 1850, and a part plan in Fig. 1856. A metal disc D, supported by the spindle F, is situated in a circular cavity formed between two blocks of ebonite, E E, and containing mer- cury, and on the opposite faces of this cavity are conical poles, B B, made magnetic by the large permanent magnet A. Wires -((< Fig. 1856. Chamberlain and Hook Meter (Plan of Disc). Fig. 1 8 5 a. Chamberlain and Hookham Meter (Section). conveying the current project slightly into the cavity at j and K, so that the direction of current flow is mainly radial, and be- tween the poles B B. Here we have conductors, the mercury and disc, conveying a current in a magnetic field ; a force will therefore exist causing the mercury and the disc to revolve, and this force will be proportional to the strength of the current flow- ing. The speed of the disc will become constant when the retard- ing torque due to Foucault currents induced in the disc by the poles B is equal to the driving torque, and, as seen in the last section, the retarding torque due to eddy currents is proportional to the velocity. The velocity of rotation will, therefore, be pro- portional to the current strength, and the number of revolutions in a given time proportional to the product of the current and AMPERE-HOUR METERS 305 the time, or to the quantity passed. By a suitable choice of gear, depending, amongst other things, on the pressure of supply, the revolution of the spindle F is transmitted to the registering Fig. 186. Ferranti Meter (Front View). dials shown in Fig. 185, which read off direct in Board of Trade units. A coil i on an' iron core, seen to the right of Fig. 1850, is used to correct the slight error arising from the fluid friction of the mercury. Fluid friction increases more rapidly than the velocity, so that unless this was compensated in some way the 3o6 PRACTICAL ELECTRICITY velocity of rotation would not be quite doubled by doubling the current. The coil i, however, is arranged to cause more lines of force to pass through its core as the current increases, and thereby weaken the magnetic field between the poles B, thus reducing the retarding torque due to the eddy currents in C, Fig. i86a. Ferranti Meter. (Section through Disc.) the disc. At first sight one might imagine that weakening the field between the poles would decrease the driving torque just as much as it decreases the retarding torque, but this is not the case, for the retarding torque varies as the square* of the strength * That the retarding torque is proportional to the square of the strength of field may be seen by remembering that the force exerted on a conductor carrying a current in a magnetic field is proportional to the product of the strength of current and the strength of the field, and that the strength of the induced eddy current is proportional to the strength of the field ; the product, therefore, varies as the square of the strength of the field, all other things remaining constant. AMPERE-HOUR METERS 307 of field, and the driving torque as the first power ; reducing the field strength by one per cent, would therefore lessen the driving torque by one per cent, and the retarding torque by two per cent, approximately. The Ferranti continuous current meter is illustrated in Figs. 186 and i860. In principle it resembles the Chamberlain and Hookham instrument, and differs mainly in having an additional magnet with poles s B, Fig. i860, which acts as a brake only, whilst the magnet with poles s D, s D, produce a magnetic field which acts both as driver and brake. Compensation for fluid friction is effected by a coil, c, c, which strengthens the poles s D, and weakens s B. The driving force is, therefore, in- creased more rapidly than the current as the current increases, whilst the retarding torque remains practically unchanged owing to one pair of poles being weakened as much as the other is strengthened. The path of current through the, meter is from the positive terminal Cj, through the copper disc c D, and mercury in a radial direction to C 2 , and by way of the coil c c to the negative terminal. A worm and wheel, seen at the top of the vertical spindle supporting the disc, are used for recording the number of revolutions, and therefore the quantity passed through the meter. It will, of course, be understood that meters of the motor type cannot be exactly correct under all circumstances, and to secure reasonable accuracy without necessitating exceptional perfection in construction with its attendant high cost, errors not exceeding -for 2 per cent, are tolerated.* This is the permissible error between full load and ^ of full load. For currents between ^ and 1*0 full load, the percentage error must not exceed 2-5 per cent, and at ^o fc^ l a d 4*5 per cent. On the other hand, all meters, to comply with the specification, must start running when a cur- rent equal to i per cent, of full load current passes through the main circuit, provided this current be not less than - of an am- pere; if this current be less than ^o of an ampere then the meter must start and run steadily with ^o ampere. Example in. A current of 68 amperes at no volts is passed through a meter for twenty minutes, and the change of reading produced is 2-54 Board of Trade units. Calculate the percentage error of the meter. Answer. 2 per cent. fast. Example 112. The gearing between the revolving disc of a meter, and the recording dials is such that the former makes 48,000 revolutions whilst the " Units " dial makes one complete * " Btsh. Eng. Stand. Specification for Consumer's Electric Supply Meters. " 3o8 PRACTICAL ELECTRICITY revolution (10 units). On being tested by passing 5 amperes at 200 volts through the meter, the disc makes 60 revolutions in 46 seconds. Find the error of the meter. Answer. Meter reads 1-3 per cent. slow. Example 113. A quantity meter correctly adjusted for a pressure of 220 volts is used temporarily on a 100 volt circuit ; what multiplier must be employed to convert its readings into true Board of Trade units ? 100 Answer. , or 0-4545. 129. Electric Transmission or Energy. In order to maintain a steady electric current we must have a closed electric circuit such as K L M N K (Fig. 187) , and any complete circuit usually consists of two essentially distinct parts. In the one part K L M N the current flows in the direction in which the potential diminishes that is to say, the potential at K is greater than that at L, the potential at L greater than that at M, and so on and at every point throughout this portion of the circuit electric energy is being turned into heat, or into heat and also into some other form of energy, such as chemical or mechanical energy. This part of the circuit corresponds with the overhead telegraph wires and the telegraph instruments which are placed at the ends of the wires used to receive the telegraph messages, or it corresponds with the electric-light insulated copper mains under the streets, the wires, the glow and arc lamps in the houses, and the electro- motors used to do work in houses and factories which are supplied with current from the street mains. And in nearly all the calcu- lations which have been made in this book hitherto regarding current, P.D., energy, and power, it is this part of the circuit K L M N only that we have been dealing with. So in the same way we might have been studying the flow of water in the water mains under the streets or in the water pipes in our houses, or the flow of water along a river where the water moves under the action of gravity. The water which produces the stream may be obtained from a reservoir or an elevated cistern, or from some pond at the top of a hill ; but, unless there be some contrivance for keeping the reservoir filled by raising the water from a low level to a high level against the action of gravity, the reservoir will run dry TRANSMISSION OF ENERGY 309 and the water stream will cease. Hence to maintain a continuous stream, the water must continuously, or at any rate from time to time, be carried up in buckets, or be raised by some form of pump, or by the evaporating power of some hot body like the sun ; in fact, as much work must be done on the water in raising it as it does in its descent through the pipes or along the river bed. So in the same way in some part N K (Fig. 187) of any complete electric circuit there must be some apparatus for raising the electricity from a low to a high potential, and the energy which this apparatus thus puts into the electric circuit must be withdrawn by the apparatus from some outside store of energy. In the sending of currents to produce telegraphic signals, or to ring an electric bell, the battery forms the pump which raises the electricity from a low to a higher potential as the current passes through it, and the chemicals placed in the battery constitute the store of energy on which the battery draws ; while in the sending of a current to produce the electric light or to work electromotors in a town, the dynamo at the "Electric Light Station" is the pump, and the coal in the bunkers at the " Generating Station," which is used to generate steam, is the store of energy on which the dynamo indirectly draws, through the medium of the steam engine and boiler. A complete electric circuit which includes an electromotor is something like one of the pipes of the London Hydraulic Company under the streets with a pump at one end and a water- motor at the other. The pump takes energy from some outside source and gives it to the water, this energy is partly wasted in heating the running water and the pipes, in consequence of friction, but the greater part of the energy is given out by the water to the water-motor at the other end of the pipe. Here the pipe corresponds with the electrical conductor, the pump with the bat- tery or the dynamo, and the water-motor with the electromotor. There is, no doubt, an important difference in the two cases, the water which flows out through the water-motor at one end of the pipe need not be immediately returned to the pump at the other end, indeed it may not be the same water at all which is pumped up again by the pump to maintain the water stream, whereas in the electric circuit the same electricity is regarded as flowing round and round the circuit. But there is this important similarity, that, just as the water is not the energy, so electricity is not energy, in spite of erroneous statements that have been sometimes made to the contrary. Pressure is given to the water by the pump at one end of the pipe, and pressure is given out by the water to the motor at the other end, so potential is given 3io PRACTICAL ELECTRICITY to the electricity by the battery at one end of the wire, and potential is lost by the electricity at the other end of the wire, where the electricity flows through the electromotor. Probably the closest analogy with the electric transmission of energy is the driving of one pulley by another by means of an endless belt (Fig. 188). Energy is put into the belt as it passes round the driving pulley p lt energy is given out by the belt as it passes round the driven pulley P 2 . The running belt corre- sponds with the electric current, the driving pulley PJ with the battery or dynamo, and the driven pulley P 2 , with the electro- motor. The model (Fig. 189) shows in a rough symbolical way what takes place in the transmission of energy with pressure- water, compressed air, an endless belt, or elec- tricity. The working stuff, water, air, belt, Fig. 188. Transmission of Power with an Endless Belt. r electricity, IS first raised in pressure, and has energy given to it symbolised by the ball, B, being raised in the carrier c through the height N K against the action of gravity; the ball then gradually loses pressure (or height) as it proceeds along the tube or wire K L, which conveys it to the other end of the system, shown by the ball falling as it proceeds from K to L, and the energy thus lost is spent in heating the tube or wire. At the other end there is a great drop of pressure as the ball falls., in the carrier c', through L M, corresponding with a transference of energy to the motor m which drives a little air-fan, and finally the ball comes back along the return pipe or wire M N, losing, as it returns, all that remains of the energy given to it initially in the pump, or elevator at N K. The ball has, in fact, come back to its original level. If the circuit external to the battery is simply a resistance containing no electromotor nor electrolytic cell, then the circuit is analogous with the model seen in Fig. 190, the balls B, B falling by gravity along the rails K L M N, and being raised against the action of gravity through the height N K. The balls are lifted by their being picked up by the hooks attached to the endless belt b b, the right-hand side of which is made to rise continuously by the handle H being turned. There is another way of transmitting energy through a pipe which is wholly different from the methods previously consideried, and that is by means of coal gas, but in this case the quality of MECHANICAL ANALOGIES Ihe material sent through the pipe and not its pressure is the important consideration. The energy contained in coal gas is not pressure-energy, but chemical energy ; therefore, as long as the pressure of the gas is sufficient to make it come out of the pipe at a suitable rate, it does not matter, as far as the amount of energy contained in a given weight of gas is concerned, whether the pressure be small or large. But the chemical constitution of the coal gas is of great importance. On the other hand, when energy is trans- mitted by water, or by air or electricity, the pressure is as important a factor in estimating the amount of energy delivered as the quantity of the working substance. Apart from the action of the tide, water at sea -level is quite useless for working machinery, no matter how much water be available, so also, electricity at zero potential is useless for working an electromotor or producing an electric light. It is, therefore, all important to the user of the water supplied by the London Hydraulic Company whether its pressure is 750 pounds per square inch or 500 pounds per square inch, but it is of no importance to him whether the water be ordinary river water or be chemically pure. Hence, while practically no restriction is imposed by law on the pressure that the Gas Companies must maintain in the gas as supplied to a house, the public Electric Light Companies are prohibited by law from allowing the P.D. between the electric light mains, where they join the house mains, from varying more than 4 per cent, from the standard pressure. This fundamental difference between the transmission of energy by coal gas and by electricity must be fully grasped, for it is probably a want of appreciation of this important difference that has led people to make such erroneous statements as that electricity is a form of energy. Fig. 189. Mechanical Model illustrating the Transmission of Energy from a Generator, N K, to a Motor, L M. 312 PRACTICAL ELECTRICITY 130. Power Developed by a Current Generator. If I be the current in amperes flowing round the circuit K L M N K (Fig. 187), and if V be the P.D. in volts between the points K and N, the work done per second on the part of the circuit K L M N equals / V joules. In addition, if the resistance of the portion of the circuit between N and K be R b ohms, the current will do work in heating this resistance at the rate of I z Rb joules per second. Hence, the total power developed by the current equals (/7+/ 2 # & )watts. (45) Now, from the conservation of energy it follows that the work done per second by the current on the circuit must equal the work IL Fig. 190. Mechanical Model illustrating an Electric Circuit composed of a Current Generator and an External Resistance. done per second on the current by the apparatus between N and K, which converts some form of energy into electric energy. Therefore, whatever be the construction of this apparatus, the rate at which the transformation of energy takes place in it, the rate, in fact, at which it introduces electric energy into the circuit, must equal There are three distinct classes of apparatus that may be employed for introducing electric energy into a circuit, viz. : (1) A battery, which transforms chemical energy into electric energy ; (2) A " the* "mo-pile," which transforms heat into electric energy ; (3) A dynamo, which transforms mechanical energy into electric energy ; and in all cases, whether the current generator be of the battery, thermo-pile, or dynamo type, the rate at which the current generator withdraws energy from some outside source and introduces it into the electric circuit equals (IV+ I 2 R b ) watts, or / (V + IR b ) ; E.M.F. AND P.O. OF BATTERY 313 where the quantity (V + IR b ) is called the E.M.F. of the generator. 131. Connection between the E.M.F. of a Battery, the P.D. between its Terminals, the Resistance, and the Current. If E be the E.M.F. of a battery in volts, R b its resistance in ohms, V the P.D. between its terminals in volts, R the external resistance in ohms, and / the current in amperes produced by the battery, we have from the last section E=V+IR b ; (46) therefore, since V equals 7 R, we have or = + b . These equations can be most conveniently written in the following forms : V = E - IR b . V- R E iT~I E>~ K+ K b V and as /= , K From the last equation it follows, since E is a constant for a given battery, that when R is very large I is very small, and from the first equation we see that when / is very small, V is equal to E. Hence to find the E.M.F. of a battery we must measure the P.D. between the terminals when the battery is sending no current at all, or but an extremely small one. A voltmeter whose resistance is very high compared with that of the battery must, therefore, be used in measuring E, and the only current that the battery is allowed to send must be that passing through the voltmeter. If a current / amperes be sent through a battery of resistance R b ohms, in the direction opposed to that in which the battery would itself send a current, then the P.D. of V volts maintained between the battery terminals has to send this current against the battery resistance R b , as well as to overcome the E.M.F. of the battery, say E volts. Hence in this case V = E+IR b . (460) Example 114. A battery having an E.M.F. of 15 volts, and an internal resistance of 25 ohms, is sending a current through 314 PRACTICAL ELECTRICITY an external resistance of 5 ohms ; what is the P.D. between the battery terminals ? Answer. 2\ volts. Example 115. What current must the battery in the last question send so that its terminal P.D. may be 7-5 volts ? Answer. 0-3 ampere. Example 116. The P.D. between the terminals of a battery is 15 volts when the battery is sending a current of 2 amperes, and 12 volts when the current is 3 amperes ; what is its internal resistance ? If E be the unknown E.M.F. of the battery, and R^ its resist- ance, we have 15 = E zR^, also 12 = E-^R b , or Rb 3 ohms. Answer. 3 ohms. Example 117. A battery having an E.M.F. of 55 volts, and an internal resistance of 0-25 ohm, is sending a current of -20 amperes through an external resistance. How many watts are spent in the external resistance, and in the battery itself ? Answer. The total watts developed are 20 x 55, or 1,100. The watts taken by the battery itself, due to its resistance, are 2O 2 x 0*25, or 100. Hence, the watts spent in the external circuit are 1,000. Example 118. A battery having an E.M.F. of 2-2 volts, and a resistance of 0-18 ohm, is opposing a current sent through it by a more powerful battery. If the current passing through it is 15 amperes, what is the P.D. between its terminals ? Since, generally, V = E + IR^, we have V = 2-2 + 15 x 0-18 ; .'. V = 4-g. Answer. 4-9 volts. 132. Electromotive Force of any Current Generator. If R b be the internal resistance of any current generator, and V be the P.D. in volts between its terminals, when the current that the generator is producing, or is helping to produce, is / amperes, it is customary to call the expression (V+ I Revolts, the E.M.F. of the generator, even when the expression is not independent of the value of I. In such a case the E.M.F. of the generator is not a constant, as it is very approximately in the case of a battery, but varies with the current passing, and it must then be regarded merely as a name for the value that (V + I R b ) E.M.F. OF ANY GENERATOR 315 may happen to have. A dynamo is an example of a very impor- tant type of current generator, the E.M.F. of which often varies greatly with the current passing, and the name E.M.F. which, like the name resistance, originally came into existence to desig- nate a constant property which was not altered by varying the current, is now used in an extended sense in connection with a dynamo, as is the name " resistance " when speaking of the apparent resistance of the electric arc (see Section 84). When the E.M.F. of a current generator varies with the current, we cannot find its value, as we did in the case of a battery, by stopping the current and measuring the P.D. between the terminals of the generator, since the stoppage of the current would alter the value of the thing to be measured. The values of V and / can, however, be measured at any moment by means of a voltmeter and an ammeter, and if the generator be, for example, a dynamo, whose resistance is practically independent of its E.M.F. and of the current passing (except in so far as the current warms the coils of the machine), we can stop the rotation of the armature, which reduces the E.M.F. to nought, and then measure R^, the resistance of the dynamo, by means of a Wheat- stone's bridge or ammeter and voltmeter, as we would measure the resistance of any other coil of wire. In Section 130 it was shown that the power developed by a current generator equalled (IV+I 2 R b ) watts, or I (V + IR b ) watts, therefore, if we decide in all cases to call the expression (V + IR^) the E.M.F. of the generator, whether it be constant and inde- pendent of the current or not, it follows that the electric power developed by any current generator equals the product of the current, that is flowing, into the E.M.F. of the generator at the time. Hence, we may define the E.M.F. of any current generator in volts as the ratio which the electric power developed by the generator, in watts, bears to the current flowing through it, in amperes, this ratio being a constant in the case of a good battery, but varying greatly with the current in the case of other types of current generators, such as dynamos. We then have E-P T or P = IE. 133. Power Absorbed in the Circuit Exterior to the Generator ; Back E.M.F. When power is given by a current of / amperes 316 PRACTICAL ELECTRICITY to a circuit between the ends of which a P.D. of V volts is main- tained, the power so given equals IV watts. Of this a portion, I 2 R watts, will be spent in heating the circuit where R is its resistance in ohms, and if the circuit acts as a simple resistance, the whole of the electric energy given to it being converted directly into heat. If, however, no thermo-pile be in circuit, and if there must be some apparatus in the circuit which transforms electric energy into some form of energy other than heat, and the rate at which this transformation takes place equals IV -I 2 R. Two classes of apparatus may be employed for removing electric energy from a circuit without directly converting it into heat, viz. : (1) An electromotor ; * (2) A cell, or battery, placed in the circuit so that its E.M.F. opposes the current, and we know from formula 460 that, if the E.M.F. of an opposing cell has a constant value of E volts, then E=V-IR; (47) so that (IV I 2 R), which represents the rate at which electric energy is withdrawn from the circuit by the cell and not converted into heat, equals IE, where E is the " back E.M.F/' in the circuit, or the E.M.F. of the cell opposing the current. When there is an electromotor, or thermo-pile in the circuit, the expression (V IR) will not usually be a constant and independent of the current, as it is in the case of a good cell, but we are led by analogy to call the expression (V IR) in all cases the back E.M.F. in the circuit, whether it be constant and independent of the current or not. So that in all cases, apart from the heating due to resistance, the rate of conversion of electric energy in a circuit into some other form of energy, equals the product of the current into the back E.M.F. in the circuit at the time ; or we may define the back E.M.F. of any apparatus, in volts, as the ratio which the rate of conversion, in watts, of electric * The type of electromotor dealt with throughout in this chapter is the "series," or single circuit electromotor, having its armature and the field magnet in series with the main circuit. BACK E.M.F. 3 i 7 energy into some other form of energy bears to the current, in amperes, flowing through the apparatus. If the back E.M.F. is independent of the current, when, for example, it is produced by a battery which is inserted in the circuit so as to oppose the current, we can find its value by stop- ping the current and measuring the P.D. between the ends of the circuit containing the back E.M.F. When, however, the back E.M.F. varies with the current while the resistance of the appara- tus producing it does not, as, for example, in the case of a motor, the value of the back E.M.F. can be ascertained at any moment by taking simultaneous observations of a voltmeter and ammeter, to determine the values of V and I, then, having stopped the rotation of the armature of the motor to reduce the back E.M.F. to nought, the resistance of the motor, R, can be measured with a Wheatstone's bridge, or in any other convenient way. When there is a back E.M.F. of E volts in a circuit of resistance R ohms, and between the ends of which a P.D. of V volts is maintained, the current T V ~ E 1= 5 amperes. K If, now, R and V be kept constant, and E be increased, / will diminish ; when E becomes equal to V the current will be nought ; when E is made larger than V the current becomes negative, the change of sign meaning that the current begins to flow in the opposite direction ; and the apparatus that previously had a back E.M.F., and was withdrawing electric energy from the circuit and transforming it into some other form of energy, begins to act as a generator, exerting a forward E.M.F. and introduces electric energy into the circuit. Example 119. A current generator having a resistance of 0-3 ohm, maintains a P.D. of 100 volts between its terminals when producing a current of 45 amperes. What is its E.M.F. ? Answer. 113^ volt. Example 120. A current generator has an E.M.F. of 67 volts, and maintains a P.D. of 63 volts between its terminals when it is producing a current of 12 amperes. What will be the current when the E.M.F. is 105 volts and the terminal P.D. 98 volts ? Answer. If the resistance of the generator is constant the difference between the E.M.F. and the terminal P.D. is propor- tional to the current, therefore the required current is J X 12 or 21 amperes. Example 121. A battery of 3 cells in series, each having 1-08 volts E.M.F., is joined up in circuit with two lead plates immersed 3i8 PRACTICAL ELECTRICITY in dilute sulphuric acid. The resistance of the whole circuit, including the battery and the lead cell, is 2-7 ohms, and the cur- rent is found to be 0-385 ampere. What is the back E.M.F. of the lead cell ? Answer. 2-2 volts. Example 122. A battery sends current through a cell consist- ing of two lead plates in dilute sulphuric acid, the cell having a back E.M.F. of 2 volts. What is the resistance of the cell if the P.D. between the terminals is 5 volts and the current 1-5 amperes ? Answer. 2 ohms. Example 123. The resistance of a motor is 0-24 ohm, and when a P.D. of 83 volts is maintained between its terminals a current of 25 amperes passes. What is the back E.M.F. of the motor ? Answer. 77 volts. Example 124. If the resistance of a motor is 1-2 ohms, and when a P.D. of 100 volts is maintained between its terminals it runs at such a speed that its back E.M.F. is 91 volts, what is the current flowing through the motor ? Answer. 7^ amperes. Example 125. A current of 30 amperes is flowing through a motor of J ohm resistance, and it is running at such a speed that its back E.M.F. is 76 volts. What is the P.D. that is maintained between the motor terminals ? Answer. 91 volts. 134. Distribution of Power in an Electric Circuit. When a current generator sends a current of / amperes through a simple circuit consisting of several resistances in series, the power spent in any part of the circuits is given by I 2 R, where R is the resist- ance of the part considered. We therefore see that in such a cir- cuit the power is distributed amongst the several parts, in propor- tion to the resistances of those parts. Calling the resistance of the generator R^ and that of the remaining parts R v R 2 , R s , etc., the electric power spent in heating the generator is I 2 Rb, and that in the parts R lf R 2 , R 3 , etc., is I*R lt I 2 R 2 , I 2 R 3 , etc., respectively. If the total resistance of the circuit be R', the total power is I 2 R', and the fraction of the total power used in a part of the circuit, say, R 2 , is given by This may be also written - -2 TR' OT JT' CIRCUIT TAKING MAXIMUM POWER 319 where F 2 is the P.D. between the terminals of the part of the circuit of resistance R 2 , and E is the .M.F. of the generator. When the circuit contains an electromotor or battery of back E.M.F., E', the effective E.M.F., in the circuit is E-E' y p _ -pi and the current I (48) K The rate at which electric energy is transformed into some other kind of energy, mechanical or chemical, will be IE', or^'.'. (49) 135. External Circuit that Receives Maximum Po'wer from a Given Current Generator. Let E be the E.M.F. of the current generator in volts, and R^ its resistance in ohms, then, if / is the current in amperes produced when the terminals of the generator are connected to the ends of some external circuit, and if P is the power in watts given to this external circuit, P l the electric power in watts produced in the generator, and P 2 the power in watts wasted in heating the generator, P, = IE, (50) in all cases. The change, however, produced in the value of P by varying the external circuit so as to alter the value of /, will depend on whether the values of E and R^ are constant and independent of the value of /, or whether one or both vary with /. If the generator be one having a fixed E.M.F. and resistance, an examin- ation of the change of the value of P with a variation in the cur- rent, /, is quite simple. For when the external circuit is so selected that / is very small, then P is obviously very small ; if, now, the circuit be gradually altered so as to make / increase, then P increases ; on the other hand, when / has nearly its maximum 77 value, viz. - , which is, of course, attained on the generator being Rb short-circuited, P is very small again. As I is continuously increased there must, therefore, be some value of / at which P ceases to increase and begins to diminish, or, in other words, there must be some value of / which makes the expression just given for P a maximum. 320 PRACTICAL ELECTRICITY 0-36 that / s To ascertain this value of I we may employ various methods ; for example, we may give arbitrary values to E and R^, plot a curve connecting P and 7, and find out by inspection the approx- imate value of 7 for which P is a maximum. Such a curve is seen in Fig. 191, the values of 2 volts and 3 ohms having been arbi- trarily given to E and R^ respectively in calculating the values of the expression for P. From this curve we see that P is a maxi- mum for some value of 7 between 0-32 ampere and ampere, and .this value of somewhat nearer 0-32 than 0-36 am- pere. If the curve be drawn on a much larger scale, so that the value of 7 that makes P a maxi- mum can be read off with still greater accuracy, it is found that this value of 7 is 0-33 or J ampere. Now J ampere is half | ampere,which is the current that the generator would produce if short-circuited, and the same result would be arrived at whatever values were given to E and R b ; there- fore, generally, we may conclude that the external circuit which receives maximum power from a current generator, of fixed E.M.F. and resistance, is the circuit which makes the current half as great as it would be if the generator were short- circuited. The following is another way of obtaining the same result : Since P = 7 (E - IR b ), formula (50), or Fig. 191. Curve showing the Value of the Current that gives ; the Maximum Power to an External Circuit. IE \. E* therefore, subtracting and adding we have CONDITIONS FOR MAXIMUM POWER 321 = (l* - ~ y Rb --('- 2R b J CE \ 2 / - =r J is a square it can never be negative, 2 J\ b j p therefore - will be a maximum when The above relations may also be written 2 IR b = E, or IR b =(E-IR b ). Now 7.R& is the voltage required to send a current / through the generator, and (E R b ) the P.D. on the external circuit, so we see that when maximum power is given to the external circuit, half the E.M.F. of the generator is used in the generator and half in the external circuit. Since I = " _, we see that Kfr + jR. when I = R = R b , (52) or the resistance of the external circuit which receives maximum power from a given generator of fixed E.M.F. and resistance, is equal to the internal resistance. Fig. 192 shows a curve between P and R plotted for the case E = j volts and R b = 2-5 ohms, from which it will be seen that P is maximum when R = 2-5. Further, when E Now the maximum power that the generator can produce is . E, for - is the maximum current, or short-circuit current. Hence, the maximum power the generator can develop is - , v 322 PRACTICAL ELECTRICITY from which we conclude that the greatest power a generator of fixed E.M.F. and resistance can give to an external circuit is one quarter the power which the generator would develop if short-cir- cuited. Should the external circuit include another generator of E.M.F., E f , the current will be given by , E+E' where R' is the resistance of the external circuit, and the power Fig. 192. Curve connecting the Power received by an External Circuit and the Resistance of that Circuit, given to the external circuit by the primary generator will be I (E IR b ) as before. This, as shown previously, has a maxi- mum value when 7 = ^, so the condition for maximum power 2K b in the external circuit is R b + R' 2R b f or E' 2Kb or (54) (55) CONDITIONS FOR MAXIMUM POWER 323 Similarly, if an electromotor or battery of back E.M.F., E' be in the external circuit, the value of E' which permits of maxi- mum power being given to the external circuit of resistance R ', is E'=^.E. ( 5 6) and when both E and E' are fixed and R' alone is variable we have Rf = E ~E E> ' Rb ' (57) It is to be observed that the preceding results are all generally true whatever be the nature of that portion of the external circuit which we desire shall receive maximum power. For example, the reasoning would be exactly the same whether the portion of the external circuit under consideration were composed of a variable resistance, or whether it contained in addition a forward E.M.F. produced by some current generator that could be altered, or a back E.M.F. produced by some electrolytic cell, or by a running electromotor, the E.M.F. of which could be adjusted to bring the current to the required value. From what precedes, then, we may conclude : (1) If an external circuit be a simple resistance of R' ohms, then in order that it may receive maximum power from a generator having a fixed E.M.F. of E volts and a fixed resistance of R b ohms R' must equal R b . (2) If the external circuit contain in addition a forward E.M.F. of E' volts, E+E' E must equal 2R b ' _. , R Rh ,-> or E must equal - x E. 2 R b (3) If it contain instead a back E.M.F. of E' volts, E' must equal -^. x E. 2R b If we desire that a current generator of fixed E.M.F. and re- sistance shall give maximum power to a portion of an external circuit for example, if the generator be connected by long leads of fixed resistance RI ohms to a motor or to lamps at a distance, and we desire to arrange the motor or the lamps so that they shall receive the maximum power then the fixed resistance of the leads must be added to the fixed resistance of the generator ; hence for R b in what precedes we must substitute R b + RI. 3 2 4 PRACTICAL ELECTRICITY Example 126. What is the maximum horse-power that can be given to any external circuit by a battery of 50 cells in series, each having a resistance of 0-05 ohm and an E.M.F. of 2-2 volts ? Answer. The maximum power will be one-quarter of the power which the battery would develop if short-circuited, on the assumption that short-circuiting the battery did not affect its E.M.F. or resistance. Therefore the maximum power that can be given to any external circuit equals I 50 X 2-2 - X - - X 50 X 2-2, or 1,210 watts, 4 50 x 0-05 which equals 1-622 horse-power. Example 127. How many glow lamps, each requiring a current of J ampere and a P.D. of 100 volts between its terminals to make it glow properly, can be used with the above battery of cells, and how should the lamps be arranged ? Answer. In order that the battery may give maximum power to the lamps, the lamps, as they contain no forward or back E.M.F., must be grouped so that the resistance of the group equals the resistance of the battery. The latter is 2-5 ohms, while that TOO of one lamp is , or 300 ohms ; hence the lamps must be placed in parallel, and, if p be the number of lamps arranged in parallel, the lamps will receive maximum power when . that is, when p = 120. It does not follow, however, that 120 lamps can be used in parallel and each receive J ampere ; indeed, all the preceding shows us is that arranging lamps in parallel up to the number of 120 is the method for causing the group of lamps to receive the maximum power from the battery of 50 cells. To find the actual number of lamps, p, that can be employed in parallel, each lamp receiving a current of J ampere, we have p 50 X 2-2 .*. p = 12, or twelve is the greatest number of lamps that can be used, and they must be arranged in parallel. Example 128. A current generator having a fixed E.M.F. of 80 volts and a resistance of 0-7 ohm is used to drive an electro- EXAMPLES 325 motor having a resistance of J ohm, the electromotor being con- nected with the generator by mains having a resistance of 2 ohms. What should be the back E.M.F. of the motor so that it will receive the maximum power ? Answer. ^~ - x 80, or 32-6 volts. 2 (07 + 2) Example 129. What should be the back E.M.F. of the motor in the last question so that it may develop the greatest mechanical power ? Answer. 40 volts. Example 130. If a battery of 50 cells in series, each having an E.M.F. of 2-0 volts and a resistance of 0-05 ohm, be giving the maximum power to an external circuit, what is the current that flows, and by how much per cent, will the power given to the outside circuit be reduced if the circuit be altered so that the current flowing is diminished by 20 per cent. ? Answer. 20 amperes ; By 4 per cent. Example 131. If the external circuit in the last question consist of a simple resistance, what is the value of this resistance when it receives maximum power, and by how much per cent, will the power given to the external circuit be reduced if its resistance is (a) 50 per cent, smaller, (b) 40 per cent, larger, than that which corresponds with maximum power ? Answer. 2-5 ohms; By 1 1 -i per cent., By 2-8 per cent. Example 132. A generator having a fixed E.M.F. of 220 volts drives a motor. What should be the back E.M.F. of the motor so that it may develop the greatest mechanical power, and by how much will the power it develops be reduced if the back E.M.F. be increased by f rds above this value ? Answer. no volts ; By|ths. 136. Arrangement of n Cells to give Maximum Power to an External Circuit of Fixed Resistance. Since the power expended in a resistance R is equal to I 2 R, the problem resolves itself into finding the arrangements of the cells which will produce maxi- mum current through the circuit. A number of cells may be grouped in various ways ; they may be put all in Series Or all Fig. i 93 .-Four Cells joined in SerieT 3*6 PRACTICAL ELECTRICITY Fig. 194. Four Ceils joined in Parallel. in parallel with each other, as shown in Figs. 193 and 194 respectively, or they may be arranged partly in series and part in parallel as shown in Fig. 195. Fig. 196 indicates another of the many possible groupings of cells.* We will, however, confine our attention for the present to " regular " groupings in which all the cells have equal currents through them. Suppose the cells n in number are all alike, and each have an E.M.F. E volts and resistance R b ohms, and let them be arranged s in series and p in parallel. Only certain values of s and p are possible, for to satisfy the assumed condition s p must equal n. The internal resistance of s cells in series is sR b , and of p cells in 7~> parallel , consequently, for a P battery with s cells in series and p in parallel, the internal resist- . sR h ance is - -, or ', and the Fig. 195. Six Cells joined Three in Series and Two in Parallel. E.M.F. is sE. The current through an external resistance R will, by Ohm's Law, be sE I = - or = (58) , n s As E is constant, 7 will be a maximum when the denominator is a minimum. This occurs when the two terms are equal to ead other, for n * s/ * n * As a general rule, cells having different E.M.Fs. should not be con- nected in parallel. GROUPINGS OF CELLS 327 R Fig. 196. Mixed Grouping of Cells. and the right hand side is a minimum when i.e., when or -p- But - is the internal resistance of the battery when arranged s P 46 8 10 12 Values of 5 Fig. 197. Curve connecting the Current and the Number of Cells in Series when the total number of Cells and the External Resistance are fixed. cells in series and p in parallel, hence the current through the circuit is a maximum when the cells are grouped in such a way 328 PRACTICAL ELECTRICITY (if possible) that the internal resistance of the battery is equal to the resistance of the external circuit. It may happen that no regular grouping of the cells may make the internal resistance equal to the external resistance, in which case the two possible arrangements which give internal resist- ances nearest to R should be tried to determine which of the two gives the largest current. Usually if the two arrangements give values of - equally near to R the one with the largest number of cells in series gives the largest current.* Fig. 197 is a curve showing the relation between s, the number of cells in series, and the current in amperes, produced when n = 12, E = i, Rb = i, and R = 0-5, from which it will be seen that s = 2 gives the best practical arrangement. Example 133. What arrangement of 24 cells, each having a resistance of 0-47 ohm, will send the maximum current through an external resistance of 1-2 ohms ? We have s p = 24 and Rb = 0-47, also, when the cells are arranged to send the greatest current through the external circuit, $Rb -1-2 - L Z, P hence s = 7-83, and p = 3*06. Answer. 8 cells should be placed in series and 3 in parallel. Example 134. What is the maximum current that can be sent by 100 cells, each of 1-4 volt E.M.F. and 3 ohms resist- ance, through an external resistance of 20 ohms ? Answer. 0-904 ampere. Example 135. What is the maximum current that can be sent by 80 such cells through the same resistance ? Answer. 0-800 ampere. Example 136. Would it be possible to arrange 48 Grove's cells, each having an E.M.F. of 1-87 volts, and a resistance of 0-14 ohm, so as to develop J a horse-power in an external resistance of 0*1 ohm ? We have s p = 48, and Rfr = 0-14 ; * Problems of this nature are treated in detail in a small book on " The Grouping of Electric Cells," by W. F. Dunton. GROUPINGS OF CELLS 329 also, when the cells are arranged to, give the greatest power to the external circuit, P s = 6 and p = 8. With this arrangement of cells the current will be 54-7 amperes ; consequently, the power developed in the external circuit will be about 299 watts, which is about 0-4 of a horse- power. Answer. It is not possible to develop \ a horse-power in the external circuit in question with any arrangement of the particular cells ; but if they be placed 6 in series and 8 in parallel, the power given to the external circuit will be about 0-4 of a horse-power. Another way of treating the problem is to find the maximum E 2 power that one cell can give to an external circuit, viz., , and 4^b multiply this by the number of cells. We then have Total external power = - - watts. 4 x 0-14 = 300 watts, = 0-40 horse-power. This shows that half a horse-power cannot be obtained from the battery. 137. Minimum Number of Cells required to give a Fixed Amount of Power to a given External Circuit. This problem may be approached in two ways. In the first place we may find the maximum power P one cell can give to an external circuit, and divide the required power P' by P. This will give n, the mini- mum number of cells. From formula (53), Section 135, P=P', -. -^ (59) Secondly, since the external circuit is given, its resistance R is known, and the power spent in the circuit is or 330 PRACTICAL ELECTRICITY The problem, therefore, resolves itself into finding the minimum number of cells required to send a current of / amperes through a resistance of R ohms. When the number of cells is a minimum, then each one cell must be giving out as much power as possible, and its P.D. must therefore be half its E.M.F. If we have s cells in series, the total E.M.F. is sE, and the P.D. = . 2 Hence IR = , 2V E' The conditions also necessitate the internal resistance being equal to the external resistance : P or /. n,orsp= , 2lR \ E J R 4P'R b = - , as before. The second method of treatment enables us to find the arrange ment of cells required, 2lR for s = E /P 7 R = 2 V P -P' 7^ = 2 E (60) EXAMPLES ON CELLS 33 i Example 137. It is required, by .means of cells each having an E.M.F. of 1-8 volts and a resistance of 0-3 ohm, to maintain a terminal P.D. of 12 volts when a current of 8 amperes is flowing. What is the least number of cells that must be used, and how should they be arranged ? 2 X 12 and p = = 2 . 6 2 8 Take, therefore, p equal to 3, and recalculate s from the equation which gives s = 12. Answer. 36 cells, 12 in series and 3 in parallel. Example 138. What is the least number of Daniell's cells, each having an E.M.F. of i-i volts and a resistance of 0-5 ohm, that will send a current of 4 amperes through a resistance of i ohm ? Answer. 27 cells, arranged 9 in series and 3 in parallel, will send a current of 3-96 amperes through the external circuit ; while 28 cells, arranged 7 in series and 4 in parallel, will send a current of 4-1 amperes through the external circuit. Example 139. A circuit is to receive 250 watts at a pressure of 20 volts from cells having an E.M.F. of 1-5 volts each, and a resistance of o-i ohm. What is the least number of cells required, and how should they be arranged ? The power developed by one cell, if short-circuited, would be or 22-^ watts. Hence, when the least number of cells is o-i ' 22"^ used, each cell will give -, or 5-625, watts to the external 4 2^0 circuit ; and, therefore, at least | , or 44-44 cells, are necessary. 2 X 2O The number that must be placed in series equals - . or 26-7 ; practically, then, 23 cells in series and 2 in parallel is what is necessary. Answer. 46 cells, 23 in series and 2 in parallel. Example 140. 18 glow lamps, each requiring 5 volts and I ampere to glow properly, are to be used with cells each having 332 PRACTICAL ELECTRICITY an E.M.F. of 2 volts and a resistance of 0-2 ohm. Calculate the minimum number of cells required, and give the arrangements of lamps and cells that may be employed with about that number of cells. Answer. We have R^ = 0-2, P' = 5 x i x 18 = 90, E = 9, ; hence _ E* = n, and 4xo-2x9O_ ~ ~^~ ' or 1 8 is the smallest number of cells necessary. To give maximum external power, each cell must supply a current of amperes, i.e., 5 lamps, and have a P.D. of 2 iv 2, i.e., i volt. As each lamp requires i ampere at a P.D. of 5 volts, the lamps should be arranged not less than 5 in parallel, and the cells not less than 5 in series. From this it is clear that the practicable arrangements of lamps and cells requiring a number of cells not differing much from the minimum are Number of Lamps. Number of Cells. Total In Parallel. In Series. In Parallel. In Series. Cells. 18 I 4 5 20 9 2 2 9 18 6 3 I J 9 T 9 Example 141. It is desired to expend 100 watts for heating pur- poses in a coil of wire, the current being supplied by cells having each an E.M.F. of 17 volts and a resistance of 0-3 ohm. What is the least number of cells that must be employed, and what are the various resistances that can be given to the coil so that the required amount of power can be developed in it with the least number of cells ? Answer. The minimum number of cells equals 2* ,that 4 X 0-3 X IPO "" or 41-5 ; so that practically 42 cells must be is, - / used. These 42 cells may be arranged either 42, 21, 14, 7, 6, 3, 2, or i in parallel, and the corresponding resistances of the coil A. ^* O*Q^ "V TOO must equal - , where p has the values just given. Hence we have CELLS FOR LARGE POWERS 333 Number of Cells Resistance of Coil in Parallel. in Ohms. 42 . . . . . . . . 0-007065 21 . . . . . . . . 0-02826 14 .. .. .. 0-06357 7 0-2543 6 . . . . . . . . 0-3460 3 i'385 2 .. .. .. 3-II5 I . . . . . . . . 12-46 138. Importance of Low Resistance and High E.M.F. for Large Powers. On examining the equation n = 4 P'R^/'E 2 we observe that n is proportional to Rj, and inversely proportional to E 2 . Hence the smaller the internal resistance of the type of cell used, and the larger its E.M.F., the smaller the number of cells required. Halving the internal resistance would halve the num- ber of cells required, whilst doubling the E.M.F. of such would reduce the number to one quarter. High E.M.F. and low internal resistance are, therefore, factors of great importance where cells are required for producing large powers* 139. Modifications Introduced into the Previous Results by Limitation of the Maximum Current a Cell may Produce. With some types of cell, particularly secondary cells or accumulators, the internal resistance of which is usually very low, the maximum safe value of the current is not limited by the E.M.F. and resist- ance of the cell, but by the fact that the plates are liable to disin- tegrate, if currents exceeding certain values per square foot of plate surface are permitted to flow. Primary cells polarise rapidly, and therefore vary in effective E.M.F., if currents approxi- mating to are taken from them, unless R^ is large. A limita- ?6 tion of the permissible current is consequently necessary in many cases, and this leads to a modification in the solution of the problem considered in Sections 135, 136, and 137, when the j maximum current allowable is less than -. zR b For example the arrangement of a given number of cells n, to produce the maximum current through a given external resist- ance, must in such a case be limited by the condition that / must not exceed p I' , where p is the number of cells in parallel, and /' the maximum current which may pass through one cell. * A modified form of two-fluid Bichromate Cell (the Bleeck-Love Cell), having the very high E.M.F. of 27 volts, was put on the market by the Silvertown Co. about 1910. 334 PRACTICAL ELECTRICITY Now in Section 136, we have n T *E P JL = ~ sR b R nR b | R = p I' as a maximum ; gives the smallest permissible number of cells in parallel. To determine the minimum number of cells required to produce a current of / amperes and a given terminal P.D. of V volts ^ when /' is less than we have, 2 K S-r/'Jfc' F7 /. n=sp=- -. (63) 1 (b - I K b ) The above expression gives also the solution to the problem of the number and arrangement of cells required to supply a power of IV ( = P) watts to an external circuit. It may also be written and as (pP) is the total current, we have /' (64) Example 142. What arrangement of 20 cells, each having an E.M.F. of i-i volts, and a resistance of 0-5 ohm, will send the largest current through an external resistance of 4 ohms, if no cell is to produce a larger current than I ampere ? What is the value of this maximum current ? / 20 /i-i \ / Answer. p = \J ( 0-5 1 = y 3 ; therefore, the cells must be arranged 2 in parallel and 10 in series. The current will be 1-69 amperes. EFFICIENCY 335 Example 143. With the cells referred to in the last question, and with the same condition as to the maximum current a cell may produce, what is the least number of cells that will maintain a P.D. of 10 volts between the terminals of an external circuit when sending a current of 3 amperes through it ? Answer. p = 3 10 ,-.. s = - = 16-6 ; i-i 0-5 therefore 48 or 51 cells must be employed, the former maintaining a P.D. of rather less than 10 volts, and the latter a P.D. of more than 10 volts, when producing a current of 3 amperes. Example 144. It is desired to give a power of 125 watts to an external circuit by means of storage cells, each having an E.M.F. of 1*9 volts and a resistance of o-oi ohm, on the condition, how- ever, that a cell may not produce a larger current than 10 amperes. What is the least number of cells required, how should they be arranged, and what should be the resistance of the outside circuit ? I2 - Answer. sp = - - = 6^94 ; 10 x i '9 100 x o'oi therefore, 7 cells must be used, and as 7 is a prime number the cells must, for " regular " grouping, be placed all in series or all in parallel. There will consequently be two values of R, viz., the highest values given by the equations - I25> and +K* = I2S Hence, the external circuit must have a resistance of 1-271 ohms when the cells are in series, and 0-0259 ohm when the cells are in parallel. 140. Efficiency. When, by means of any machine, or contriv- ance, one form of energy is converted into another form, some heat is produced ; hence, if heat energy is not the form in which the energy is required after the conversion, some portion of the energy which has been used up has been converted into a useless form as far as the object in question is concerned, and may, therefore, be regarded as wasted. Consequently, in all cases the amount of useful energy produced is less than the amount of energy used up. For example, when a machine is used to do work there is a waste of energy in the heating of the bearings of the machine ; if falling water is employed to turn a water wheel there is in addition waste of energy in the eddies set up in the water, in the splash of the water against the blades of the wheel as well as in the friction of the water stream against the sides of 33 6 PRACTICAL ELECTRICITY the channel which guides it to the wheel. When oil, wax, gas, etc., are consumed as illuminants only a very small fraction oi the available energy is converted into the special form of energy, called light, which affects the retina of the eye, and the greater part is wasted in heat, whose action on the eye does not differ from its action on other parts of the body. Again, in a battery a certain amount of chemical energy is wasted in the heat pro- duced by " local action " (see Section 68), which goes on even when the battery is producing no useful current ; further, on the battery being used to send a current through some exter- nal circuit a portion of the chemical energy that is converted into the electric energy is always wasted in heating the battery in consequence of its resistance. The value of any machine or contrivance for effecting a conver- sion of one form of energy into another depends first on the rate at which energy in a useful form is evolved by it that is, the useful power the machine develops, and which is sometimes called its " useful activity " secondly, the value of the contrivance depends on the ratio of the amount of useful energy produced to the amount used up in the process, and this ratio is called the " effioiency." Efficiency, then, is expressed by a number, less than unity, such as J, 0-63, 75 per cent., 84 per cent. Sometimes, how- ever, it is found convenient to employ different units of energy, or of power, in the numerator and denominator of the fraction which represents the efficiency. For example, while the true efficiency of a carbon glow lamp does not generally exceed o-oi that is, not more than one -hundredth of the electric energy supplied to it is converted into light the efficiency of such a glow lamp is sometimes spoken of as J candle per watt, meaning that an electric power of 4 watts must be supplied to the lamp to produce as much light as is given out by i standard candle.* When any current generator developing an E.M.F. of E volts and having a resistance of R^ ohms is sending a current of I am- peres round any circuit, the ratio which the power in watts given to the external circuit bears to the electric power in watts de- veloped in the generator is I (E IR^) IE or (65) where V is the P.D. Therefore, this fraction is the efficiency of the generator. * It is customary to speak of the "efficiency" of a lamp as so many " watts per candle " ; the word " inefficiency " would be more correct. EFFICIENCY 337 The name " electrical efficiency " is sometimes given to the preceding expression to distinguish ' it from the " commercial efficiency " of the generator, which means the ratio of the useful electric power it produces to the total power it consumes. Now, the total power consumed is always greater than the total electric power the generator produces. For example, if the generator be a battery some of the chemicals will often be wasted in local action, or if it be a dynamo there will be power wasted in friction at the bearings of the machine, etc. Hence the commercial efficiency of a generator is always less than I(E-IR b ) V IE E' Similarly, if E' be the back E.M.F. of a motor in volts, R m its resistance in ohms, and / be the current in amperes flowing, the electrical efficiency of the motor is IE' E' or v> while its commercial efficiency, or the ratio of the useful mechani- cal power it produces to the electric power it receives from the mains, will be less than this, since some of the mechanical power which the motor produces will be wasted in the friction at its bearings, as well as in the friction between the rotating commu- tator and the brushes, etc. The commercial efficiency, however, of very large well-made dynamos and motors is as high as 96 per cent. Another useful definition of commercial efficiency is Output Input ' Where the word Output means the useful power which the appara- tus gives out and Input is the power put in. The difference between the input and the output is converted into some form of energy (usually heat), other than the form desired, and is there- fore spoken of as " loss." From the law cf Conservation of Energy we know that Output -f loss = Input. .. the above expression may be written, . Output Efficiency == F , Output + loss Input loss or = ^ . Input w 368 PRACTICAL ELECTRICITY Example 145. What must be the resistance of a current gener- ator so that 95 per cent, of the power produced by it shall be given to the outside circuit, consisting of a simple conductor having a resistance of 35 ohms ? We have 35 n = -25-, 35 + R b 100 if Rb be the resistance of the generator ; .*. R b = 1-842. Answer. 1*842 ohms. Example 146. loj horse-power is spent in driving a dynamo which maintains a P.D. of 100 volts between its terminals when it is generating a current of 65 amperes. What is the commercial efficiency of the machine ? Answer. loj horse-power equals loj X 746, or 7646-5 watts, while a current of 65 amperes at a P.D. of 100 volts equals 6,500 watts ; therefore, the commercial efficiency is 85 per cent. Example 147. A motor having i ohms resistance develops a mechanical power of J a horse when a P.D. of 60 volts is main- tained between its terminals and a current of 9 amperes is sent through it. What are the electrical and the commercial efficiencies of the motor ? Answer. The power wasted on account of the resistance of the motor is 121-5 watts, while the power received is 540 watts ; therefore, the electrical efficiency is =, or 77-5 per cent. 540 The mechanical power developed is 373 watts ; therefore, the o/^o commercial efficiency is 2L2. or 6q-i per cent. 540 Example 148. An electromotor is required to work a pump raising water through a height of 120 feet. If 15,000 gallons are to be raised per day of ten hours, what current will the motor take at 200 volts' pressure, supposing the " combined efficiency " of motor and pump to be 60 per cent. ? A gallon of water weighs 10 Ibs. ; hence the work to be done in ten hours equals 15,000 X 10 X 120 ft. Ibs., so that the power exerted in foot pounds per minute equals 30,000. But as 40 per cent, of the power given to the motor is wasted in the machinery, the motor must receive X 30,000 or 50,000 ft. Ibs. per minute, o Hence 50,000 = 44-23 X I X 200, .-. / = 5-65. Answer. 5-65 amperes. EFFICIENCY OF TRANSMISSION 339 Example 149. If electric energy is supplied by public mains to a factory at id. per Board of Trade unit, and an electromotor works with an efficiency of 80 per cent., how much does the energy used to drive the machinery in the factory cost per horse -power hour ? Answer. One Board of Trade unit equals 1-340 horse-power hour, and of this 80 per cent, is delivered by the motor to the machinery ; therefore, 1-072 horse-power hour costs id., or one horse-power hour costs o-93d. Example 150. If a glow lamp gives light equal to 16 candles when a current of 0-21 ampere passes and a P.D. of 100 volts is maintained between its terminals, how many watts are required per candle ? Answer. 1-31. 141. Efficiency of Electric Transmission of Energy. If a stream of water be used to work a turbine that drives a dynamo producing a current which- flows through long leads and works an electromotor at the other end of the leads, the commercial effi- ciency of the whole arrangement is the ratio of the useful mechani- cal power developed by the motor at the one end of the system to the power of the falling water supplied to the turbine at the other. The whole power given by the falling water to the turbine would not, however, be available for driving the machinery in a factory even if the factory were built close to the falling water, for some of the power will be wasted in the turbine itself ; hence the " commercial efficiency of transmission " from one end of the sys- tem to the other may be taken as the ratio that the useful me- chanical power given out by the distant electromotor bears to the mechanical power given by the turbine to the dynamo at the near end. The " electrical efficiency of transmission " in such a case is the ratio that the electric power which is converted into mechanical power in the motor bears to the electric power which is produced in the dynamo, or the electrical efficiency of transmission equals JF' F' if OI E ' (66) where E is the E.M.F. of the dynamo and E' that of the motor If Rd, RI, and R m be the resistances in ohms of the dynamo, the leads, and the motor respectively, . E -E' E-E' when R equals the total resistance of the circuit, therefore, eliminating E' , the electrical efficiency of transmission equals 340 PRACTICAL ELECTRICITY (67) Now, whether E, R$, and R m are constant and independent of the current, or whether they change their values with the current, the preceding expression varies from zero when the external p circuit is such that / equals ~ (which will happen when the R motor is held at rest so that it has no back E.M.F. and acts simply like a resistance) to unity when the external circuit is such that / is zero, provided, of course, that neither R^ nor R m becomes extremely large when / becomes very small. The electrical efficiency of transmission is, therefore, the greater the smaller is the current. Diminishing the current, how- ever, diminishes the power developed by the generator unless its E.M.F. be increased. Similarly, diminishing the current diminishes the power that can be received by the distant motor unless its back E.M.F. is increased. Hence, to electrically transmit a large amount of mechanical 'power over a long distance with high efficiency we must employ a dynamo producing a large E.M.F. at the one end and a motor producing a large back E.M.F. at the other, and the current that flows must be kept small. For precisely similar reasons, if we desire to employ water to transmit a large amount of power through a long pipe with high efficiency, the water must be at a high pressure and the stream must be small. Hence the London Hydraulic Company use water at 750 pounds per square inch pressure in their pipes, and boiler makers employ a pressure of as much as 1,400 pounds' pressure per square inch with portable tools for riveting, etc., by hydraulic pressure. It is interesting to consider how the E.M.F. of the generator must increase with the amount of power to be transmitted and with the resistance of the circuit, in order that the loss of power due to the resistance of the circuit may not exceed a certain percentage of the power to be transmitted. The electric power P l developed in the _ , generator - x E watts. the electric power P/ converted into E E' mechanical power by the distant motor = x E' watts, R therefore the power PI lost on account of _ ,. 2 the resistance of the circuit = -J- watts ; K EFFICIENCY OF TRANSMISSION 341 hence p l = (Y x R, (68) so that - , the percentage of the power developed in the generator which is lost on account of the resistance of the circuit, equals "fl- (69) Consequently, if this percentage loss is to be a constant, E 2 must increase proportionately to the product of P 1 into R. For example, if we desire to transmit 10,000 watts along a circuit having a resistance of 2 ohms, and to keep the loss of power down to 4 per cent., /IOO X IO,OOO X 2 E= \/ - 4 = 707 volts, or the generator must have an E.M.F. of 707 volts. If in addition to, or instead of, the motor at the other end of the leads there be some apparatus of resistance R' ohms in which we wish to develop heat or light, then this resistance R' must not be included in the preceding expressions, for the heat developed in this resistance is what we desire shah 1 be produced, and there- fore must not be regarded as energy wasted in heat. For example, if the arrangement receiving energy at the other end of the leads be simply a group of glow lamps, having any resistance of R' ohms, it follows, from what precedes, that the percentage of the power developed by the generator, which is lost on account of the resistance of the circuit, equals ioo-| (R b + RI). Although the transmission of signals by electricity through wires many hundreds of miles in length has been successfully carried on for more than half a century, the history of the electric transmission of considerable amounts of power is all comprised within the past forty years. In the following table are given the results of attempts to accomplish this object, and it is seen how the employment of higher and higher P.Ds. has enabled larger and larger amounts of power to be transmitted over longer and longer distances with increasing efficiencies. During 1919 a scheme for transmitting 500,000 kilowatts 570 miles in California was designed, the transmitting pressure being 220,000 volts. CO , i 6-: >' ^ sb'g CO x S N .2 S 8 ^^o o>O > o ^ ,0 ^ .i: p 'c ^ 0_ R o o" .Sffc 273 ^ b * 00 jO fljfc N 00 IIS H ^ 6 tH CO M R o M I||F i OT o tin g. II? ! o" Tt" f .fill gvT) s QJ 1/5 il 1 8 M } d H ^ 33 M ON vg g en 00 CJj.SS o 8 8 CO ^^ ^ ^ 00 ^* H ^^ CO |||| a 1 I >0 ill! N 8 CO 8 00* 2 +" f 00 H '^ HH 1 . 13 0) Ui OJ A W 0) IH fl N 8 ||| V ^ CO o 8 "S & o ^ TJ -H . d o oo ' 5 ' CO 6 a ^ ^0 8^ MCJ ^ 3 * vO rt t-^ ig Q oo 'N -2 G CO o ^ "N o M j> tH o ^^ vO CO cJ ^3 .0 Hn 00 00 ^o o oo ^ ** CO 6 vO , M 5 a in i a ^T? 0) S * PS O C D O T^j T .s 8 Wi O **i J, : I'll ""S t>^ ^ rt ^ oo '821 * I 111 "cS ^ O T3 "o3 1 .2 a 5 ffi CJ PU ELECTRICAL EFFICIENCY 343 142. Connection between Electrical Efficiency of Transmission and Ratio of the Power Received to the Maximum Power Receiv- able. When the current generator has a fixed E.M.F. and resistance like a battery, we have seen in Section 135, that whether we desire the whole of the external circuit, or a portion of the external circuit, to receive maximum power, or whether we desire that the transformation of electric energy into non- heat energy shall proceed most rapidly in an electromotor or electrolytic cell possessing a back E.M.F., the electric power usefully employed must be equal to the electric power wasted in heating the circuit, so that the electrical efficiency of trans- mission must be one-half. If, however, the part of the circuit under consideration be arranged so that it receives less than the maximum power receivable from the given current generator, the electrical efficiency of transmission may be greater than one-half, and the following calculation gives the connection between the electrical efficiency of transmission and the ratio of the power received to the maximum power that could be received. If E is the fixed E.M.F. of the current generator in volts and R^ its resistance in ohms, R x the resistance in ohms of the part of the circuit under consideration, and RI the resistance in ohms of the leads connecting it with the generator, we have D electrical efficiency = ~, where R is the total resistance K of the circuit, and _ power rece maximum power receivable ~ R power received _ _ \^ R 4 (-*,) = 4 efficiency (i efficiency). (70) Now this is a quadratic equation connecting the efficiency with the ratio of the power received to the maximum power receivable, which ratio we will call r for brevity, therefore each value of r will be given by two different values of the electrical efficiency of transmission, the sum of the two values being equal to unity ; for example, whether the efficiency is J, or f , the 344 PRACTICAL ELECTRICITY 8 M o O CO co o CO o VO co vO CO "* o CO ro ? CO HO 6 >o . * o H CO o o f * 6 oo 6 CO 6 6 PO . N & o o o 1 M * M 1 r Electrical efficiency ) : transmission . . j r Ratio of power] ceived to maximum I >wer receivable . . j IH P* wlw K preceding equation gives r equal to -J . In the following table, however, which rv T~> gives corresponding values of , or -=' h K the electrical efficiency, and of r, only the larger value of the efficiency is given corresponding with any particular value of r. From this table we see that when R x , the resistance of the part of the circuit under consideration, is increased until the electri- cal efficiency of transmission is 75 per cent., the power this part of the circuit receives is | of the possible maximum, also that R x may be increased until the electrical efficiency of transmission is over 85 per cent, without the power received being less than J of the possible maximum. The figures given in the first two lines of the preceding table are equally true whether the E.M.F.- and resistance of the generator are constant, or whether they vary with the current, but the figures in the third line, for the ratio of the power received to the maximum power receivable, are only true when both the E.M.F. and the resistance of the generator are con- stant ; indeed, it may be shown that an external circuit receives maximum power from a dynamo when the external resist- ance is smaller than that of the dynamo, and when the electrical efficiency is there- fore less than J. If the external circuit, instead of being a simple resistance, contains an apparatus of resistance R m ohms and back E.M.F. of E' volts, and if it receives power through leads of fixed resistance RI ohms from a generator having a fixed E.M.F. of E volts and resistance of R^ ohms, we know that the ratio which the rate of trans- formation of electric energy due to the back E.M.F. in this apparatus bears to the maxi- mum rate of such transformation, equals ELECTRICAL EFFICIENCY 345 E-E' xE , where R = R b + RI + R m , is* E-E' F_/ _&\ E' and as the electrical efficiency of transmission = , therefore the ratio which the rate of transformation of electrical energy due to the back E.M.F. bears to the maximum rate of such transformation, equals 4 x efficiency (i efficiency). This is exactly the same equation as was obtained in the previous case, see (70) above, and therefore must lead to the same numerical connection between the values of the electrical efficiency of transmission and the ratio which the rate of trans- formation of electrical eneigy due to back E.M.F. bears to the maximum rate of such transformation. Example 151. A battery having an E.M.F. of 30 volts and a resistance of 4 ohms is sending a current through an outside circuit consisting of leading wires having a resistance of i ohm and 4 glow lamps arranged in parallel. A P.D. of 12 volts is maintained between the lamp terminals, and each lamp produces 3j candles. Calculate the number of candles that is produced per watt, and the percentages of the power generated in the battery that are given to the lamps and wasted in the battery and leading wires. Answer. The current = , or 3-6 amperes. 4 + 1 The power given to the 4 lamps = 3-6 X 12, or 43-2, watts. Therefore, as the total illumination is 4 X 3|, or 14, candles, 0-324 candle is produced per watt. Also the power generated by the battery 3*6 X 30, or 108, watts. The power wasted in the battery = (3 : 6) 2 X 4, or 51-84, watts. The power wasted in the leading wires = (3*6) 2 X i, or 12-96, watts. 346 PRACTICAL ELECTRICITY Therefore, of the 108 watts produced by the battery 43-2 watts, or 40 per cent., is given to the lamps, and 64-8 watts, or bo per cent., is wasted in heating the battery and the leading wires. Example 152. A dynamo of 0-2 ohm resistance is supplying current to a group of glow lamps in parallel placed at the ends of leads having 1-8 ohms resistance. The lamps take 75 amperes, and a P.D. of 100 volts has to be maintained between their terminals. If 32 horse- power is spent in driving the dynamo, what are the electrical and commercial efficiencies of the trans- mission, and what are the electrical and commercial efficiencies of the dynamo alone ? Answer. 40, 31-4, 94 and 73-8 per cent. Example 153. A dynamo haying a resistance of 2j ohms, and an E.M.F. of 1000 volts, develops 40 horse-power. What may be the resistance of the leads so that 60 per cent, of the power developed by the dynamo is delivered to some apparatus at the other end of the leads ? Answer. 10-9 ohms. 143. Economy in Electrical Transmission of Energy. Kelvin's Law. The question of efficiency has been dealt with in the pre- ceding paragraphs, from which it will be seen that the higher the P.D. and the lower the resistance of the lines the greater the efficiency of transmission of a given amount of electrical energy will be. It might therefore be surmised that by using the highest possible P.D., and lines of very small resistance, maxi- mum economy would be secured. This, however, is not the case, for although such an arrangement would give a " high efficiency," the previous reasoning takes no account of the cost of the apparatus and transmission line, or of the serious leakage of current that takes place from wires when very high P.Ds. are used. The cost of machines per kilowatt output increases considerably when the pressure is very large, and the cost of the line would be greatly increased if wires of large cross-section were used, with a view to making their resistance small. Very high pressures enable smaller, and therefore qheaper, conductors to be used without excessive I 2 R loss in the conductors, whilst the cost of apparatus and loss by leakage are increased. For very long lines the cost of the line becomes of prime importance, whilst for short distances the cost of the machinery is the principal item. It will, therefore, be evident that there will be some particular pressure and some particular size of wire for which, in a given case, the total yearly cost, made up of interest on capital, upkeep, and cost of the energy wasted in the circuit, will be a minimum. KELVIN'S LAW 347 Transmission at medium pressure is most economical for short distances, whilst for long distances/ such as those mentioned in columns 8 n of the table on page 342, very high pressures are essential to economy. Considering the transmission line alone we may notice that small wires would be cheaper than large ones, both to buy and to erect, but the energy wasted in heating would be increased by their use. As energy has a money value, the total yearly cost of a line (interest on capital, and cost of energy wasted in heating) will not be a minimum if either very small or very large conductors be used. In fact, as was shown by the late Lord Kelvin, in 1881, the total cost will be. least when the interest on capital spent in erecting the line is equal to the cost of the energy wasted in heating the line. This is " Kelvin's Law," and is of great commercial importance. CHAPTER VIII QUANTITY AND CAPACITY. 144. Electric Quantity and its Measurement 145. Ballistic Galvanometer 146. Measurement of Quantity by Ballistic Galvanometer 147. Correction of Ballistic Galvanometer for Damping 148. Determina- tion of Decrement and Logarithmic Decrement 149. Constant of a Ballistic Galvanometer 150. Comparison of Quantities 151. Capacity 152. Condensers : Mechanical Analogies 153. Units ot Capacity : Farad : Microfarad 154. Variation of the Capacity of a Condenser with the Area of its Coatings and the Distance between them 155. Relation between Electrostatic Unit of Capacity and the Farad 156. Capacity of Spherical and Plate Air Condensers in Farads 157. Capacity of Cylindrical Condensers 158. Specific Inductive Capacity 159. Dielectric Strength of Insulators 160. Resistivity of Insulators 161. Construction of Condenser of Large Capacity 162. Condensers for Large P.Ds. ; Leyden Jars 163. Comparison of Condensers 164. Potential Divider 165. Combined Capacity of Several Condensers 166. Charged Condensers are Stores of Electric Energy, not of Electricity 167. Energy Wasted in Charging a Condenser from a Source of Constant P.D. 168. Absolute Measurement of Capacity 169. Measurement of Specific Inductive Capacity and Resistivity of Insulators 170. Standard Air Condensers 171. Ratio of Electromagnetic and Electrostatic Units of Quantity 172. Use of Condensers for Comparing E.M.Fs. of Cells, or other Current Generators 173. Condenser Method of Measuring the Resistance of a Cell. 144. Electric Quantity and its Measurement. We have already defined (Section 10) the coulomb or unit of quantity, as the quantity that flows in one second through a conductor conveying a current of one ampere, or the " ampere second." When the cur- rent is a steady one it is quite easy to find the quantity which passes in a long interval, by observing the current in amperes and the time in seconds, and multiplying them together, just as one can calculate the quantity of water that passes a given point in a channel of known cross-section in a given time by observing the velocity of the stream and the time, supposing, of course, the stream to be a steady one. If the electric current be varying then the quantity may be determined by summation (or integra- tion). The whole time is supposed to be divided into a very large number of very small intervals, during any one of which the current will be practically constant (say / amperes), and the pro- 348 Fig. ig8a. Needle and Coils of Simple Ballistic Galvanometer. MEASUREMENT OF QUANTITY 349 duct of the current during that interval and the length of the interval (81 seconds, say) is the quantity that passes during the interval. The whole quantity is then expressed by Q = 2 / 8t coulombs. To measure such a quantity practically, one way is to pass it through a voltameter and observe the amount of chemical decomposition produced, from which the number of coulombs could be calculated (see Sections 9 and 10). Another way is to pass the current through a quantity meter, such as described in Section 128. These methods can, however, only be used when the quantities are fairly large (say many coulombs), for the decomposi- tion produced by a single coulomb in the most sen- sitive voltameter known is very small. For ex- ample, in a silver voltameter one coulomb deposits 0-001118 gramme of silver, and in a copper voltameter 0-0003286 gramme Of COpper, Whilst Fig . I9 8. Simple Ballistic Galvanometer. in an acid volta- meter one coulomb liberates 0-1734 cubic centimetre of mixed gases at normal temperature and pressure. 145. Ballistic Galvanometer. To measure small fractions of a coulomb, especially if the quantity passes in a very short time, it is evident from the above that some other method must be used ; and for this purpose galvanometers may be employed. Instruments intended for this purpose are generally called " ballistic galvanometers," and should fulfil the following con- ditions : (a) the periodic time of vibration should be great compared with the duration of the current, so that the whole quantity passes before the needle has moved appreciably from its zero position ;* (b) the frictional (or damping) forces tending to bring the needle to rest after displacement, should be small, i.e., the needle should make many oscillations before coming to rest. * A current of short duration is called a " transient current," and the quantity conveyed by such a current is spoken of as a " charge " or a " dis- charge." 350 PRACTICAL ELECTRICITY In other words, the amplitude of one swing should not differ much from the next.* An instrument satisfying these conditions fairly well is shown in Figs. 198, 1980. The needle is formed of a " bell-shaped magnet " (see Fig. 1980) and a fine glass pointer p serves for reading the deflection. The magnet is suspended by a single fibre of silk, so that it hangs centrally in the axis of the two Fig. 199. Reflecting Ballistic Galvanometer. circular coils of wire which form the winding. The coils are drawn open in Fig. 1980 to show the magnet. Another ballistic galvanometer intended for measuring very small quantities is illustrated in Fig. 199. Here the coils are separated to show the magnets forming the needle. This needle consists of three bell magnets fixed to a vertical wire, the magnets being placed so that the upper and lower ones have their north poles pointing in a direction opposite to that of the middle magnet. When the magnetic moment of the one magnet is nearly equal to that of the other two, the system * The ratio of the amplitude of one swing to the next is called the " decrement," and the Napierian logarithm of this ratio, the " logarithmic decrement" LAMPSTAND AND SCALE 351 Fig. 200. Lampstand and Scale for Reflecting Galvanometer (Front View). Fig. 2ooa- Lampstand and Scale (Back View). An electric lamp is contained within the conical shade. 352 PRACTICAL ELECTRICITY has little directive force in a uniform magnetic field, and this lengthens the period of vibration, and increases the sensitiveness. Such a system of magnets is called an " astatic needle," and has the great advantage of being little affected by stray magnetic fields. By placing one of the magnets inside the coil and the others outside, the forces exerted on the magnets by a current in the coil all tend to deflect the needle in the same direction. To magnify the movement of the needle caused by the quantity passing through the coils, a mirror is attached to a wire stem support- ing the suspended magnets, and a spot of light from a lamp is reflected from this mirror to a finely graduated scale placed several feet away. By this means a very small deflec- tion of the needle can be measured. A con- venient form of lampstand and scale Fig. zooft. Reading Telescope and Scale. ] s shown in FigS. 200 and 2000. Another method of observing such deflection is by viewing in the mirror the reflection of a scale by the aid of a telescope, such as indicated in Fig. 2006. Moving coil galvanometers can often be used with advantage for ballistic work, and one of the narrow-coil form, devised by the authors, is illustrated in Fig. 201. The coil M is shown separately with its mirror L attached, and the mirror is also seen between F and F in the complete instrument. The coil hangs inside a brass tube in a narrow gap between the poles of the magnet A. Tubes containing suspended coils of different resist- ances are supplied for use with the same magnet ; one of these is shown at E D H, near the base of the galvanometer. Where fine adjustment of the zero of the instrument is required, a tan- gent screw is fitted to the torsion head, as seen in the upper right hand corner of the figure. BALLISTIC GALVANOMETER 353 146. Measurement of Quantity by Ballistic Galvanometer. Suppose a quantity Q is passed through a tangent galvanometer in correct adjustment, and that at time t the current has a value of / amperes, we shall then have 81 being a very short interval of time. Fig. 201. Ayrton and Mather's Reflecting Galvanometer. If the coil has n convolutions of mean radius r centimetres the strength of field at the centre will be given by 2 TC / n F = 10 r (see Sections 30, 35) and this will produce a turning moment on a small magnet at the centre of FM dyne-centimetres, where M is the magnetic moment of the needle (Section 26). The forces exerted on the needle will, at any instant, be proportional x 354 PRACTICAL ELECTRICITY to the value of the current at that instant, and if the whole time during which the current flows is very short, the result is that the needle receives a sudden twist, or impulse, which produces an angular momentum Kco (where K is the moment of inertia of the needle and &> is the angular velocity produced by the impulse) equal in value to the sum 2 FM 8t* Hence K co = 2 F MM, lor 10 r 2 nn M Q, coulombs, 10 r = gMQ, (71) TT n where = 2 10 r' the " constant " of the galvanometer coil, or the strength of field produced at the centre of the coil by a current of i ampere. From this we see that the angular velocity o> is proportional to the quantity Q discharged through the instrument, and if we can measure this velocity we have then a measure of Q. When there is no frictional resistance offered to the movement of the needle, ay can be measured by observing the angle through which the needle swings before it begins to return towards the zero position. This is called the "first swing," the " throw " or the " kick " of the needle. The kinetic energy stored up in the needle when its angular velocity is w is given by the expression , and as the displacement increases, this energy is trans- formed into potential energy of position. At the instant of maximum elongation &> = o, and all the kinetic energy has been converted into the potential form. Frictionless motion being postulated, no waste of energy occurs, so the potential energy must equal - , and the work done against magnetic forces as the needle moves from the zero position through an angle s t where s is the "first swing " of the needle, is - . Now the work done in turning a magnet of moment M through * The case is analogous with the momentum generated by a force /', acting as a mass m for a short time 5t, for m v = 2j'5t. BALLISTIC GALVANOMETER 355 an angle s from its position of equilibrium in a uniform magnetic field of strength H is ME (i - cos s), or 2 MR sin 2 -, 2 Hence - = 2 MHsm*-, 2 , IMH , s and co = 2sm- I j # o> = 2 V MHK sin -, 2 but Kcu= gMQ, from (71) gMQ - 2 y/MHK sin -, =w HK . s sin , gM M 2 (72) As H, g, K, and Af , are constants in a given case, we learn that the quantity discharged through a galvanometer is proportional to the sine of half the first swing which it produces, the needle being controlled by a uniform magnetic field. Now in Section 27 we have shown that when a magnet of magnetic movement M, and moment of inertia K oscillates in a magnetic field of strength H, its time of small vibrations is so the above equation may be written H T s These corrections can only be used with accuracy when the amount of damping is small, and for cases where the decrement is large a more complicated correcting factor is necessary.* When the angles of swing and steady deflection are small, as is usually the case with reflecting instruments, the formulae reduce to Q = - ~1 SI+ ' a PP roxima tely, (78) and Q = - s(i-\ j, approximately, respectively. (79) 148. Determination of Decrement and Logarithmic Decrement. The motion of a swinging needle may be represented by a curve as shown in Fig. 202, where vertical distance represents " displacement " and horizontal distance " time." This figure corresponds with a moderately damped oscilla- tion, the amplitude being reduced to half its initial value in about * This correcting factor, the calculation of which is too elaborate for an elementary work, is TT * where e is the base of Napierian logarithms. Two approximations are (i + 0-5 y 0-277 y* + 0-130 y), and (i + 0-5 A 0-027 A - a '54 * 3 )> respectively. 358 PRACTICAL ELECTRICITY fourteen swings. An undamped vibration is represented in Fig. 203, and a well damped one in Fig. 204, the motion being nearly destroyed in six swings. 1 Time * Fig. 202. Diagram representing a moderately damped vibration. The motion of a vibrating needle is very nearly isochronous, i.e., the time of a complete vibration is the same whether the swings are fairly small or very small, and the ratio of one swing to the succeeding one is found to be constant. In the above figure the decrement (i -f- y) = = = = etc., etc., and also = etc. Hence and Time ^ Fig. 203. Diagram representing an undamped vibration. so we see that the decrement can be found by observing the magnitude of one swing and the nth swing reckoning from the one first observed, and finding the (n i)th root of the ratio. Time Fig. 204. Diagram representing a well-damped vibration. thus BALLISTIC GALVANOMETER CONSTANT 359 and x = 5 log.A* n i s n or x = 3 3 log-io ~ (approximately). H I 5|| for log. 10 = 2-303 approximately. 149. Constant of a Ballistic Galvanometer. In Sections 146 and 147, formulae are given connecting quantity and the swing produced by discharging that quantity through the coil of a galvanometer. The expressions include T, IT, - > or and y tan d d or \, all of which are constant for a particular instrument used under definite conditions. We may therefore write the equations (74) and (75) of Section 146 as and Q = k r s' , respectively, where k and A/ are constants, and have the iollowing meanings : viz., k is the quantity which will produce a swing of 180, k f the quantity which will produce a swing of i division of the scale of the reflecting instrument, and s' the first swing expressed in scale divisions. When there is damping, the correcting factor fi-f--jor(i-f--j being constant under given conditions, may be included in " the constant " of the instrument, so that whether there is damping or not the formula 0=sin|. (80) expresses the quantity passed through a ballistic galvanometer, when the swing produced is so large that the sine differs appreci- ably from the angle, while Q=k'J. (Si) expresses the quantity approximately when the angles of swing are small. The constants k and k' differ considerably. In fact, for the same instrument, k is usually a much larger number than k' t but the ratio of the two depends on the angle corresponding with i division of the scale in the second type of formula. * The percentage accuracy to which A is determined by this method spends on the value of , a value approxin Sn Napierian logarithms, 2-718 gives the best result. depends on the value of , a value approximating to e, the base of 360 PRACTICAL ELECTRICITY Example 154. A ballistic galvanometer gives a swing of 60 when o-ooi coulomb is discharged through it. Find the constant k of the instrument. Answer. k 0-002 coulomb, i.e., a quantity of 0-002 coulomb would produce a swing of 180. Example 155. If the galvanometer mentioned in the last question were fitted with a mirror and used with a millimetre circular scale placed at i metre away from the mirror find the constant k' of the instrument. Answer. Here we must remember that when the mirror moves through any angle 0, the ray of light reflected from it turns through double the angle, viz., 26. A quantity of -^ oi o-ooi coulomb would produce a swing of where 6 60 sin 30 sin - = sm ~ 100 2 2 100 . 6 i /. sin - = = 0-005 2 200 and = 34' Now an angle of i will be subtended by 27T X IOOO .... ,, millimetres on the scale, 360 .'. i = 17-4 divisions, and iiV = jZ divisions, = 19-8. Hence T J of o-ooi coulomb produces a swing of 19-8 divisions, and the quantity per division is o-ooooi f _ = 0-000000506, or k' = 0-506 microcoulombs. Example 156. The periodic time of an undamped reflecting galvanometer is 10 seconds and a current of ^ milliampere gives a steady deflection of 200 scale divisions. Find the quantity of electricity which produces a swing of 100 divisions. T I Answer. Using the formula Q = - - ~ s , (Section 146), 2 71" w 10 i we have Q = 7 X 100 6-28 10,000 x 200 = 80-5 microcoulombs. COMPARISON OF QUANTITIES 361 Example 157. Calculate the quantity in the last example if the instrument had been damped so that its decrement was 1-06. Answer. 80-5 X 1-03 approximately, = 83 micrccoulombs. Example 158. The following readings of the swings of a galvanometer were taken to find the decrement and logarithmic decrement. Determine their values. (Odd suffixes denote swings to the right of zero and even ones swings to the left.) Values of \ Sl S * Sa $ * S * SG S? S * Sj> SIQ SH ( 312 281 253 227*5 205 184 166 149 134 121 io8 ; 5 Answer. Decrement (i + y) = i-in. Log e dec. X = 0-105. 150. Comparison of Quantities. Quantities of the same order of magnitude may be compared by noting the swings produced when they are discharged successively through the same ballistic galvanometer, the adjustment of which remains unchanged. If the quantities be Q l and Q 2 and the swings produced be s t and s a then, . s or = -i, approximately, S 2 when the angles are small. To make such comparisons it is not necessary to know H, g, T, y or X, nor the deflection for a known current. If one of the quantities be much greater than the other, and the galvanometer be sufficiently sensitive to give a reasonably large swing with the smaller quantity, the instrument must be shunted when the larger quantity is discharged. Calling the multiplying power of the shunt m^ and assuming Q l to be the greater of the two quantities, we have, JO = m 1 , approximately, Qz s z a relation which is strictly true when a Universal Shunt is em- ployed, for in this case the damping of the instrument is not altered by changing the multiplying power of the shunt. With an ordinary shunt, however, changing the shunt alters the re- sistance between the galvanometer terminals, and the currents induced (see Section 187) by the motion of the magnets near the 362 PRACTICAL ELECTRICITY coils, causes the decrement to increase as the multiplying power of the shunt is increased. Allowance for change of damping must consequently be made by finding the decrements under the two conditions, when an accurate comparison is required. It is sometimes necessary to use a shunt of multiplying power m 2 say, when the smaller quantity is discharged ; then, 0+ m^s* , approximately. Q 2 2 S 2 151. Capacity. On page 82, we have already stated that the name " capacity " has been given to the ratio, Quantity Potential Difference The equation C = ^therefore defines " capacity " C just as the expression of Ohm's Law, R = _ defines " resistance " R. Ohm's Law relates to conductors at constant temperature, through which electric currents are passed, and says in effect, that the ratio of potential difference to current is constant for a given conductor. Experi- ment also shows that when two conductors are in a fixed relative position, and far removed from other bodies (or when one is com- pletely surrounded by the other), the ratio of Quantity to Potential Difference is constant. The value of this constant depends on the sizes, shapes, and proximity of the conductors. This in- fluence of proximity can be readily shown by the ar- rangement of gold leaf electroscope and plate M sketched in Fig. 205. The electroscope may be charged with a definite quantity of electricity by placing the plate M very near p, (but not actually touching), and* charging the arrangement by connect- ing one pole of a battery to p and the other to M, and then Fig. 205. Condensing Gold-leaf Electroscope. CONDENSERS 363 disconnecting it. On moving M away from p the leaves of the electroscope will diverge, showing that the potential of the leaves rises. The quantity of electricity on the electro- scope, however, is not altered by the movement of M, for the metal system attached to p is highly insulated. As Q is con- stant and V increases, -,- decreases, i.e., the capacity decreases. When one conductor is completely surrounded by another, the capacity of the inner one is the number of coulombs required to be given to the inner one to Fig. 206. produce a P.D. of I volt between the two. For ex- ample, the capacity of A, Fig. 206, is the number of coulombs on A when there is a P.D. of i volt between A and B. If a metal plate A, Fig. 207, be surrounded with a flat metallic box B, the top and bottom of the box being parallel to and very near A, then the capacity of A will be very large, since it will require a large charge of electricity to be given to A in order to raise the P.D. between A and B to i volt. 152. Condensers : Mechanical Analogies. An arrangement of conductors such as are shown in Fig. 207, is called a " conden- ser," so a condenser may be denned as two conductors separated by an insulator and so placed relatively to one another that the capacity of the arrange- ment is large compared with the size of the con- ductors. The conductors are called the " coatings " Fig. 207. of the condenser. Condensers behave like mechanical springs. When a spring is subjected to pressure (or tension) the spring is strained, and work is done, this work being stored up as potential energy of deformation of the spring. The energy remains in the spring so long as the pressure is maintained, and on relieving the pressure work is done by the spring. Similarly when a condenser is subjected to electric pressure, electric energy is stored in the condenser, and is given out again when the pressure is removed. Other ways in which springs and condensers behave similarly are (a), the deformation of the spring is proportional to the force applied, and the electric displacement in a condenser is propor- tional to the electric pressure used ; (i), the energy stored in a spring 364 PRACTICAL ELECTRICITY is proportional to the square of the force, and that in a condenser proportional to the square of the potential difference ; and (c), if the force to which the spring is subjected is excessive the spring will break, whilst excessive electric pressure on a condenser will cause the dielectric* to break down. Another analogy exists in gas cylinders used for storing and for transporting compressed gases. The quantity of gas in a cylinder is proportional to the pressure, and the energy due to the compression proportional to the square of the pressure. Excessive pressure would cause the cylinder to burst. r,^-r----~_^T-"j\c Y Fig. 208. Hydraulic Analogue of Condenser. A hydraulic analogue to the action of a condenser is represented in Fig. 208. Let P be a close fitting piston attached by a spring s, to a fixed point A in a smooth horizontal pipe A B, and the whole pipe A B c D filled with water. When the paddle wheel w is stationary, the pressure of water on the two sides of the piston, will be equal, and the piston will be in equilibrium, and the spring s is unstretched. If now w be rotated clockwise, the pressure on the A side of P will be greater than that on the B side and the piston will move until the difference of pressure is balanced by the tension of the spring. For a definite speed of w, p will take up a definite position in the pipe, and a definite quantity of water will have passed a given cross-section of the pipe, say x Y. Increas- ing the speed of w will increase the difference of pressure, increase the displacement of p and also of the water past x Y, whilst a decrease of speed will result in a diminution of the displacement and cause a quantity of water to move in the direction B A. So long as the speed of w remains constant and no leakage past the piston occurs, there will be no movement (or current) of water * The insulating medium between the conductors is called the dielectric. UNIT OF CAPACITY; FARAD 365 in A B, but any change of speed will cause a transient current to pass in one direction or the other, according as the speed rises or falls. In the same way an electric condenser allows transient currents to pass (i.e. permits quantities of electricity to be displaced) when the P.D. between its coatings changes, but allows no current to pass when the P.D. is constant. Increasing the P.D. will cause a transient current in the direction of the P.D., whilst a decrease of P.D. will result in a transient current in the opposite direction. If the pipe A B, Fig. 208, be of small cross-section, the quantity of water displaced for a given change of pressure will be small, whilst if the cross-section be large the displace- ment produced by the same change of pressure will be great. A pipe of small cross section therefore corresponds with a con- denser of small capacity, and large cross-section of pipe with a condenser of large capacit}^. 153. Units of Capacity ; Farad ; Microfarad. A condenser having a large capacity does not mean one that would hold a large quantity (or charge) without its insulation breaking down, but one that will hold a large charge relatively to the P.D. between its coatings ; i.e., is large. If A, Fig. 207, be charged with positive electricity, there will be a charge of negative electricity on B, whereas, if A'S charge be negative, that on B will be positive. Experiment also shows that the quantity on A is exactly equal in amount and opposite in sign to that on the inside of B, for if the outer conductor of a charged condenser be connected momentarily to earth and then insulated, the condenser will show no external signs of electrification, although the condenser remains charged. To make a condenser of large capacity we may either use very large plates, or make the distance of the plates apart very small. There are obviously prac- tical difficulties in making the distance separating the plates extremely small, for the plates might tOUCh, Or a Spark might pig. 209. Diagrammatic Representation of a Condenser. pass across the interven- ing space, if a moderate P.D. was set up between the plates, and so discharge them.* On the other hand if we make the plate A, Fig. 207, and the * To reduce the risk of this occurring, it is customary to place thin sheets of solid insulating material, such as mica or paraffined paper, between the plates. 366 PRACTICAL ELECTRICITY box B very large, the apparatus becomes cumbersome. To over- come this difficulty two series of plates, A and B, connected as shown in section in Fig. 209, are employed, and a condenser is often represented symbolically by such a figure. A simpler diagram, representing a condenser, is given in Fig. 2090. A condenser which holds i coulomb when the P.D. between its coatings is I Fig. 209*. Simple diagram of Condenser. Volt is Said to have U capacity of one "farad." The "farad " is therefore the unit of capacity. For practical purposes a capacity of one farad is far larger than is convenient, so a millionth of a farad, i microfarad, is adopted as the com- mercial unit. As i volt is io 8 C.G.S. electromagnetic units of E.M.F. or P.D., and i coulomb equals ~ of a C.G.S. unit of quantity, it follows that i farad is io~* C.G.S. units of capacity, and i microfarad =icT 15 C.G.S. units. 154. Variation of the Capacity of a Condenser with the Area of its Coatings and the Distance between them. That the capacity of a condenser is directly proportional to the effective area* of either coating hardly needs proof, but an experimental proof can readily be obtained by charging two or more similar condensers to the same P.D. and discharging them first separately, and then putting them in parallel and discharging them all together, through a ballistic galvanometer. It will be found that the quantity in the latter case is equal to the sum of the several quantities in the former, thus showing that the combined capacity of several condensers connected in parallel is equal to the sum of their individual capacities. Now a large condenser is usually formed of a number of smaller parts all connected in parallel, so the experiment described proves the statement in italics above. From the experiment with the condensing electroscope (Fig. 205) described in the previous section we see that increasing the distance between the coatings diminishes the capacity of the condenser, but the law between capacity and distance could not be determined satisfactorily by this apparatus. Plates of large area would be required to obtain quantities measurable by a ballistic galvano- meter unless very high potential differences were employed. We * The effective area of the coating of a plate condenser is usually rather greater than the actual area, for near the edges of the coatings the lines of electric force passing from one to the other spread beyond the edges. An approximate correction may be made by assuming the smaller plate is increased all round by a strip of width 0-4 times the distance between the coatings. CAPACITY AND DIMENSIONS 367 can, however, show theoretically in a simple way that the capacity oj a condenser with plane parallel plates is inversely proportional to the distance between the coatings, and this conclusion is verified by experiment. In Chapter II., page 82, we have shown that the capacity of a condenser formed of concentric spheres is in electrostatic units, where r and r 2 are the radii in centimetres of the inner and outer spherical surfaces respectively. If (r 2 rj be called t, the thickness of the insulator, we have, capacity = ^ ( -. t and the capacity per unit area of inner sphere is which may be written -- 47T* If we now suppose the sphere to become infinitely large the opposing surfaces will become plane, and r L = cc . The term - 47T7-J will become zero and we get the capacity per unit area of two plane parallel plates is - electrostatic units ; i.e., the capacity 47T/ is inversely as the distance between the coatings. Combining the two conclusions we may say that the capacity of a plate condenser is directly proportional to the effective area of the plates and inversely proportional to their distance apart. 155. Relation between the Electrostatic Unit of Capacity and the Farad. On page 81, we have stated that the electro- magnetic C.G.S. unit of quantity is approximately 3 X io 10 electrostatic units of quantity, and a method of proving this experimentally will be found in Section 171. In both the c.G.s. electromagnetic system of units and the c.G.s. electrostatic system the dyne and the erg are the units of force and energy respectively, and in both systems potential difference is defined so that the work done when a quantity of electricity passes * The area of a sphere of radius ?'j = 4 IT r\ -. 368 PRACTICAL ELECTRICITY from one point to another is equal to the product of the quantity and the potential difference between these points (see Section 48). As the work done when one c.G.s. electromagnetic unit of quantity passes between two points whose P.D. is one C.G.S. electromagnetic unit of P.D. is one erg, and the work done when one c.G.s. electrostatic unit of quantity passes between two points whose P.D. is one C.G.S. electrostatic unit of P.D. is also one erg, we see that the magnitudes of the units of P.D. in the two systems must be inversely as the magnitudes of the units of quantity. Accordingly the magnitude of the electrostatic unit of P.D. must be 3 x io 10 times as large as the c.G.s. electro- magnetic unit of P.D., and as I volt is io 8 C.G.S. units (see Section 55a), one electrostatic unit of P.D. must be 3 x io 10 -*- io 8 volts, i.e. 300 volts, approximately. If a condenser of capacity i farad had a P.D. of 300 volts (i electrostatic unit of P.D.) between its coatings, the quantity on each coating would be 300 coulombs, and as i coulomb is 3 x io 9 electrostatic units, this equals 300 x 3 X io 9 electrostatic units of quantity, approximately, i.e. 9 X io 11 electrostatic units of quantity, approximately. The quantity on each coating of a condenser whose capacity is i electrostatic unit, would only be i electrostatic unit of quantity under the same P.D., so that I jar ad is 9 X io 11 electrostatic units oj capacity, and i electrostatic unit of capacity = i-in micro- microfarads. 156. Capacity of Spherical and Plate Air Condensers in Farads. The foregoing numerical relation enables us to express the capacity of spherical and plate condensers in electromagnetic units of capacity (farads) as follows : Capacity of isolated sphere of radius r centimetres equals, - - farads; (82) 9 X io 11 Capacity of concentric spheres of radii r and r 2 centimetres equals, farads ; (83) 9 x io 11 (r 2 - Capacity of parallel plate condenser of effective area A square centimetres and distance apart of t centimetres equals, == farads, 9 x io 11 X 4 TT t ^ or : farads; (84) 1-131 X io 13 / CALCULATION OF CAPACITY 369 Expressed in microfarads (C m ) the formulae become, r (85) 1^2 r 6 ('a-'i) (86) and 1-131 x io 7 * (87) respectively, and if the measurements be taken in inches instead of centimetres we get, For an isolated sphere, _ 2-822 r" For concentric spheres, 2822 r\ r and for plate condensers, (90) the two dashes (") above the letters signifying inches, and square inches in the case of A. 157. Capacity of Cylindrical Condensers. Another form of condenser of great practical importance consists of two long concentric cylinders, for the insulated wires and cables used in the distribution of electrical energy, and in submarine telegraphy, approximate to this shape. The capacity of such a condenser whose axial length / centimetres is very great compared with the diameter, and whose dielectric is air, can be shown to be electrostatic units, (91) 2lg 6 J where D and d are the inner diameter of the outer cylinder and the outer diameter of the inner cylinder respectively.* If common logarithms be used, the formula becomes n 4 ^ 43 , JV electrostatic units, (92) 2 (log D log d} and converting to farads and microfarads we get, ~~~io (log D - log d) ' * D and d should both be measured in term? gf the sa.me unit, but the. unit employed is immaterial, PRACTICAL ELECTRICITY and C m = - (log p log gf (94) / being in centimetres. When / is measured in inches the latter expression becomes, 6-128 I" Cm = io 7 (log D - log d)' Example 159. Express in electrostatic units of capacity and in microfarads (a), the capacity of an isolated spherical conductor of i metre diameter, (b) , that of the earth considered as an isolated sphere whose diameter is 12,756 kilometres. Answers. (a) 50 electrostatic units, -^ microfarads. (b) 6-378 X io 8 electrostatic units, 709 microfarads. Example 160. Find the capacity of a spherical conductor i foot in diameter placed concentric with a hollow sphere of I2j inches inside diameter, air being the dielectric. Answer. 747 electrostatic units, or 0-830 milli-micro farad. Example 161. Determine the capacity of an air condenser, having parallel plates of effective area 2,000 square inches spaced ~ of an inch apart. Answer. 4,043 electrostatic units, 4-492 milli-microfarads. 158. Specific Inductive Capacity. In Section 157 we have con- sidered the coatings of condensers to be separated by layers of air. If instead of air, solid or liquid insulators, such as glass, gutta- percha, indiarubber, oil, etc., be used, we find that the capacity is increased in definite proportions, depending on the nature of the insulator employed. For example, if an air condenser be submerged in paraffin oil, so that the air between the coatings is replaced by the liquid, the capacity is found to be about 2-1 times greater than before, whilst if gutta percha be used as the dielectric instead of air, the capacity will be about 4 times as great. The ratio in which the capacity of a condenser is altered by substi- tuting some other material for air between its coatings is called the " specific inductive capacity " of the substance. In the cases just mentioned we may say that the specific inductive capacity of paraffin oil is 2-1, and the specific inductive capacity of gutta percha, 4. The following table gives a list of the " specific inductive capacities " of many important dielectrics as determined by various experimenters using different specimens of material : TABLE XI APPROXIMATE " SPECIFIC INDUCTIVE CAPACITIES," OR " DIELECTRIC CONSTANTS " OF SUBSTANCES Substance. Specific Inductive Capacity. Air at 760 mm. pressure 5 ' Carbon dioxide at 760 mm. Hydrogen ,, ,, Sulphur dioxide Alcohol Oil, Castor ,, Linseed Olive Paraffin (Light) (Heavy White) . . ,, Resin ,, Sperm ,, of Turpentine Water Amber Balata .. Chatterton's Compound Ebonite Glass, Plate .. ;; .. Flint (Very Light) (Dense) (Double extra dense) Gutta Percha India Rubber, Pure ,, ,, Vulcanized Jute .. ..x> .. Marble Mica Paper Impregnated with oil ,, Dry cellulose Pitch Porcelain Quartz, Fused ,, Crystalline Resin Shellac Sulphur Wax, Paraffin Sealing i-o (Taken as Standard.) 0-9985 to 0-9994 1-00069,, I'QOoS 0-9997 ,. 0-9998 1-0037 26 (about) 4-62 to 4-67 3'35 3'i6 2-04 2'55 37 3'i 2-2 tO 2-43 80 (about) 2-8 2-4 to 3-6 4-0 2-56 to 3-15 6-1 6-57 ' 7'4 io-i 3-6 to 4-43 2-1 27 3 6-1 5-0 1-8 2-8 6-7 1-8 4-4 378 4-27 4'6 2-55- 3'i 2'5 37 2-58,, 4-03 1-92 , 2-47 4'5 5'2 2'3 5'5 4 ,, 6-6 2-2 3-8 6-8 372 PRACTICAL ELECTRICITY From what has been said above it will be evident that the capacities of condensers with dielectrics other than air can be obtained from the formula for air condensers of the same dimen- sions, by multiplying by the specific inductive capacity of the dielectric used. Example 162. Find the capacity of i mile of gutta percha covered wire, 2 millimetres diameter, covered to 6 millimetres, assuming the specific inductive capacity of the material to be 4-2. 6-128 X 1760 X 3 X 12 , , Answer. -- '-? - ^ - X 4-2= 0-342 microfarads. io 7 (log. 6 log. 2) Example 163. What must be the area of each coating of a condenser whose capacity is to be one microfarad, and the dielec- tric mica u\y of a millimetre thick (Spec. Ind. Cap. 5) ? Answer. 1-13 X io 4 sq. cms., or 1-13 sq. metres. 159. Dielectric Strength of Insulators. Not only is the capacity of a condenser increased by using (say) glass, mica, or wax, instead of air, as the " dielectric " or insulating material between its coatings, but the resistance to the loss of charge by sparking from one coating to the other is greatly increased by the change. With a glass condenser far greater P.Ds. can be used than is possible with an air condenser of the same size. The resistance to sparking does not depend on the insulating quality of the substance, but on its rigidity and the resistance it in consequence opposes to rupture.- The property of resisting rupture by electric pressure is spoken of as " dielectric strength " or electric strength, and is usually ex- pressed as the potential gradient* in volts per centimetre, or volts per millimetre, at which breakdown occurs. For example, if a P.D. of V volts exists between two faces of a plate of thickness / the V V potential gradient is , and the value of which causes rupture t t of a substance is called the dielectric strength, or more shortly the electric strength of that substance. Electric strength is influenced by many conditions, such as temperature, time of application of of the P.D., etc., so it is difficult to make precise measurements. In the case of gases the electric strength increases as the pressure increases, and nearly in direct proportion. For air at normal pressure and temperature, the electric strength is approximately 3,800 volts per millimetre. Approximate values for various substances are given in Table XII. * Potential gradient means ' DIELECTRIC STRENGTH & RESISTIVITY 373 TABLE XII APPROXIMATE DIELECTRIC STRENGTHS OF SUBSTANCES IN KlLOVOLTS PER MILLIMETRE Air 3-8 Ebonite . . . . . . . . . . 53 Glass (ordinary) . . . . . . . . . . 16 Mica . . . . . . 50 to 60 Micanite . . . . . . . . . . . . 18 to 40 Paraffined paper . . . . . . . . . . 34 Porcelain (hard) . . . . . . . . . . 18 Press-spahn . . . . . . . . % . . . . 9 to 22 Rubber (pure) . . . . . . . . . . 47 Rubber covered Cable . . . . . . . . 10 to 25 160. Resistivity of Insulators. In making condensers another property of insulators which has to be considered is their resistivity. A perfect condenser is one that allows no current whatever to pass through, when a steady P.D. is maintained between its coatings, and the greater the resistivity of the insulator used in the condenser the nearer this perfection is attained. It is, there- fore, important to use very high resistance materials for the purpose. In the case of metals and other good conductors the resistivity is constant (if the temperature remains constant), however long the current is passed. With insulators this is not so, for in almost all cases where the P.D. used is much below that required to produce rupture of the material, the resistance increases with the time of application of the P.D., but increases more slowly as time goes on. This phenomenon is called " electri- fication," and to obtain more consistent results in measuring the resistances of insulators it is usual to make the necessary read- ings after a constant P.D. has been applied for definite intervals of time. For electric light wires and telegraph cables one minute is now adopted as the standard time of electrification. Temperature has a very large influence on the resistivity of bad conductors (or insulators), their resistance decreasing as the temperature rises. In the case of gutta percha the resistivity is halved by raising the temperature about 5 C., and for indiarubber a rise of about 15 C. halves the resistance. Approximate data relating to the resistivity of insulating materials in common use at about normal temperature will be found in Table XIII. 374 PRACTICAL ELECTRICITY TABLE XIII APPROXIMATE RESISTIVITY OF INSULATORS ( Substance. Ohms per Centimetre Cube. Amber 155 x lo 12 Canada Balsam 280 x io 12 Cellulose (Dry) i, 600 x io 12 Ebonite 450 x io 12 to 30,006 x io 12 Glass . . . . . . , * 50 Xio 12 300 x io 12 Flint (Density 4-1) 250 x io 12 . ( 3'3) 9,900 Xio 12 ,, 2O,000 X IO 12 Gutta Percha 25 xio 12 ,, 5,000 x io 12 India Rubber 1,500 xio 12 ,, l8,OOO X IO 12 Jute (Impregnated) 3,000 x io 12 Marble 500 x io 12 Mica 4 xio 12 8,800 X IO 12 Micanite 2,500 x io 12 Paper 0-5 x io 12 Porcelain 2,100 x io 12 Press-spahn . ^V* . o-oi x io 12 Quartz (Fused) . . i, 600 x io 12 (Parallel to Axis) 153 x io 12 (Perpendicular to Axis) 20,000 X IO 12 Resin 7,000 x io 12 Resin Oil 0-2 XIO 12 Shellac 1,500 XIO 12 ,, 9,000 x io 12 Slate 0-08 xio 12 10 X IO 12 Sulphur 4,000 xio 12 ,, 8,200 XIO 12 Wax (Paraffin) 49,000 XIO 12 ,, 294,000 xio 12 For some of the substances in the previous tables, two values are given representing the variation of resistivity of different specimens of the material. These indicate to some extent the great variations that exist between different samples, and show the necessity of testing in all cases, where it is important to know, even roughly, the actual resistivity of a particular specimen. 161. Construction of Condenser of Large Capacity. When a very large capacity is required the dielectric employed consists usually of sheets of paper or of mica, which have been soaked in melted paraffin wax or in a solution of shellac in alcohol. CONSTRUCTION OF CONDENSERS 375 The sheets of tinfoil are shaped as, shown in a (Fig. 210), one corner being cut off, and the sheets of insulating material b are made about two inches wider and two inches longer, and have two corners cut off. On a sheet of insulating material there is first laid a sheet of tinfoil, as in c, then a sheet of insulating material is laid on the top, then a second sheet of tinfoil with its uncut corner turning the other way, and so on, so that finally there are a number of alternate sheets of tinfoil with their corners projecting over the sheets of insulating material to the right, and the other set of alternate sheets of tinfoil, with their uncut corners projecting over to the left. Each of the exposed sets of corners is soldered together, and forms an electrode or terminal of the condenser. When paraffined paper is employed as the insulating material, the paper is first very carefully examined by holding it up to the Fig. 210. light, sheet by sheet, so that the existence of any small holes may be detected, and any sheet possessing such holes discarded. The good sheets are then placed in a bath of melted paraffin wax warmed by steam to about 110 C., or a little above the boiling point, so that all water may be driven off. On a horizontal slab of cast iron, also warmed by steam to about the same tempera- ture, the sheets of paraffined paper and tinfoil are' laid in the way just described, the sheets being carefully smoothed with a flat strip of wood as they are laid on. Two sheets of paper are placed between each pair of sheets of tinfoil to avoid the possi- bility of a hole in the paper causing leakage, it being, of course, most improbable, even if there were a minute hole in each sheet, that the holes would come exactly opposite one another. After the condenser has been built up in this way it is placed between two warm metal plates, and pressed with a heavy weight while it is cooling, in order that the surplus paraffin wax may be squeezed out and the whole consolidated. It is not desirable to use the paraffin wax in the baths more than once, since even when the temperature is not raised to more 376 PRACTICAL ELECTRICITY than about 110 C. or 120 C, slight decomposition of the wax may occur, which diminishes its high specific resistance. Within recent years " rolled " condensers and "foiled paper " condensers* have come largely into use for telephone and other purposes, for which the precise constancy of capacity is not of prime importance. " Rolled " condensers are formed by taking two long strips of tinfoil (like wide ribbons), separated and covered by lengths of paper somewhat wider than the foil, and rolling them up together. The resulting roll is then flattened under a press. The two strips of foil are insulated from each other by the separating papers, and form the two coatings of the condenser. Foiled paper resembles the paper frequently used for wrap- ping up packets of tea, and is made by coating long strips of paper with a layer of finely powdered tin mixed with an adhesive ; after being dried, the coated paper is passed between rollers and burnished. To make a condenser from foiled paper, two strips of it are interleaved with plain paper, and rolled up together, the length of paper rolled depending on the capacity of the condenser required. Condensers of one microfarad can be made from strips about n feet long and about 7 inches wide. After rolling, the papers are dried, waxed, and pressed. Such a condenser may occupy a volume as small as 4 cubic inches and weigh about 4 ounces. 162. Condensers for Large P.Ds., Leyden Jars. The charge, or quantity, in a condenser of capacity F is given by Q = CV coulombs, and this charge can be made great by making V very large, even if C be of moderate magnitude. Condensers for large P.Ds., such as are produced by frictional electrical machines, must be constructed so that they will not break down under the high pressures, and to fulfil this condition it is desirable that the insulating material should have large dielectric strength and be of sufficient thickness to reduce the potential gradient to a safe working value. The material most used for such condensers is glass, either in the form of sheets, tubes, or jars. A very common form of condenser for large P.Ds. is shown in Fig. 211, which represents a " Leyden Jar." The name is derived from the town of Leyden, at which the property of electric capacity was accidentally discovered in 1746, by Musschenbroek, and his pupil, Cunens. Desiring to collect the supposed electric fluid, the}'- used a bottle partly filled with water, into which * Devised by Mr. G. F. Mansbridge, of the Postal Telegraphs Department. LEYDEN JARS 377 Fig. an. Leyden Jar. dipped a nail, passing through the cqrk, to carry the supposed fluid from the electric machine to the water, and on Cunens touching the nail with one hand, the bottle being held in the other, he received a shock. In the ordinary Leyden jar the coatings are sheets of tinfoil, one pasted inside and the other out- side. Electric connection is made with the inside either by a metal rod or foot resting on the bottom, or more commonly by a chain or flexible wire, hanging from a brass rod supported by a wooden cover, resting on the top edge of the jar. The use of a wooden cover supporting the rod or chain is objectionable from the fact that it facilitates surface leakage of electricity between the two coatings by short-circuiting the inner surface of the glass between the top edge of the jar and the upper edge of the tinfoil inside the jar. An improved form of jar is shown in Fig. 212, in which the outer tinfoil is not carried so high up the jar, and the inner foil is replaced by strong sulphuric acid s s. A lead foot and stem L supports a metal rod I from the bottom of the jar, and serves to make contact with the acid. The rod I passes through a large hole in the wooden cover w w, which hole may be closed by the cork c, sliding on i when the jar is not in use. When the cork is raised, as in the figure, no electricity can pass to the cover from the rod, and the surface leakage path from inside to outside is up the inside of the jar, over the edge, and down the outside. The inner surface of the jar being kept very dry by the ' presence of the strong acid, is, if Fig. 212. An improved form of properly cleaned, an excellent insulator, and enables such a jar to retain its charge for many hours without much loss.* * We may here remark that, with high P.Ds., leakage over surfaces is often far more serious than the passage of electricity from one coating to the other through the dielectric. 378 PRACTICAL ELECTRICITY In Fig. 212, a cone with spherical end is shown supported on the rod i. When the jar is charged, the cone will be charged also, and by means of a " proof plane" the density (quantity of electricity per unit area) at different points of the conductor may be investigated. Fig. 213. Three Ley den Jars in Parallel. Ley den jars are sometimes used for wireless telegraphy, which are charged and discharged many thousands of times a second. Under these conditions tinfoil coatings are not very satisfactory, for want of intimate contact with the glass causes local heating to occur. This can be avoided to a great extent by silvering the glass by chemical deposition, and the coatings on the best jars are formed of deposited silver. Fig. 214. Three Leyden Jars in Series. Where it is necessary to have condensers of capacity greater than that of a single jar, of the largest size obtainable, several jars are connected in parallel, forming a battery of Leyden jars, as illustrated in Fig. 213, and if the P.D. is too great for one jar to withstand, several jars may be connected in series, care, of course, being taken to insulate them in a suitable manner. (See Fig. 214, where the letters I.S. indicate insulating stands.) COMPARISON OF CONDENSERS 379 163. Comparison of Condensers. The simplest way of comparing two condensers of about the same capacity is to charge them to the same P.D. by a suitable battery, and observe the swings produced on discharging them in succession through a ballistic galvanometer. The ratio of the swings pro- duced (or the sines of half the swings when the angles are large) gives the ratio of the capacities, Fig. 215. Charge and Discharge Key. ' c. sin _ = _i f or = sin- any damping that may exist in the instrument, being the same in the two cases, cancels out. A form of key, called a charge and discharge key, suitable for the comparison of condensers, is shown in Fig. 215, and a scheme of connections is indicated diagrammatically in the same figure. Fig. 216. Condenser Circuit in which the Charge only is Measured. Fig. 217. Condenser Circuit in which both Charge and Discharge are Measured, L is a brass spring, supported on a corrugated ebonite pillar, and normally is in contact with the platinum tipped screw s 2 . When the left-hand end of L is depressed by touching the ebonite push p the coatings of the condenser c are charged to the P.D. of the batten^ B, and on allowing P to rise, the quantity on the coatingr 38o PRACTICAL ELECTRICITY of c is discharged through the galvanometer G, thus giving a throw, which is a measure of the capacity. The object in sup- porting the several terminals of the key on ebonite pillars is to obtain very good insulation, and the pillars are grooved to increase the length of surface from terminals to base and thereby lessen surface leakage. Arrangements of condenser circuits, including charge and dis- charge keys, are shown in Figs. 216 and 217 ; in the former the galvanometer is in series with the battery and measures the charge that passes through the condenser when the key is pressed, but the discharge does not pass through the instrument. Fig. 215 illustrates connections whereby the discharge only is measured, whilst in Fig. 217 both charge and discharge pass through the galvanometer. It is usually advisable to measure discharge only by the arrangement in Fig. 215, for less error is introduced if the condenser be imperfect, and there is no risk of damaging the galvanometer if the condenser be short-circuited or very leaky. When the condensers to be compared differ greatly in capacity they may be charged to the same P.D., and shunts, preferably of the " Universal " type, used, as already described in Section 150, to compare the resulting quantities ; or the two condensers may be charged to different P. D.'s so as to make the quantities discharged of the same order of magnitude. Calling the P.D. used on the larger condenser F t and that on the smaller one V^ we have c 2 In cases of extreme inequality it may be necessary to adopt a combination of shunts and different P.Ds., the formula in such cases being sin or r = -r^ , approximately, (98) C 2 v\ m 2 s 2 m t and m 2 being the multiplying powers of the shunts used. 164. Potential Divider. From equations (96) (98) it will be noticed that the ratio of P.Ds. VJV^ must be known, but the absolute value of either V^ or V^ in volts is not required. A simple POTENTIAL DIVIDER means of getting two P.Ds. in known ratio for condenser work is to close the circuit of a battery through a set of resistance coils whose relative values are known. (See Fig. 218.) The P.D. between any two points in the circuit will then be proportional to the resistance between them. For example, if Illlllll jl the resistance between A and c in Fig. 218 be R 1 ohms, and that between A and D, R 2 ohms, then A^^AA/V^JA/VV\AAAAAAAA^- J E R -v, D Fig - btaining tw The arrangement in Fig. 218 may be called a " Potential Divider." Any ordinary resistance box can be used for a potential divider, provided the wire with which the coils are wound is sufficiently thick to carry safely the current which the battery will send through them. A device more convenient than an ordinary box is shown in Fig. 219, where a number of equal Fig. 219. Simple Dial Potential Divider. coils arc connected in series, and their junctions joined to studs arranged in a circle so that a switch arm s, pivoted at P, can make contact with any of them. If there be m coils between A and c and n coils between A and E, the ratio of the P.D. between A and T to that between A and E will m be. With two sets ot 3 8 2 PRACTICAL ELECTRICITY coils, one (say) of 10 coils of unit resistance and the other of 9 coils, each 10 units (Fig. 220), a P.D. can be divided into hundredths ; for when s is at 4 and S 1 at 7, the P.D. between T and ^ will be ga of that between A and E'. Increasing the number of dials or the number of coils per dial enables finer subdivisions to be obtained ; for example, with two dials, one having 100 one unit coils and the other 99 coils, each 100 units, a subdivision to i part in 10,000 is possible. Fig. 220. Two-Dial Potential Divider. 165. Combined Capacity of Several Condensers. Condensers may be connected together either in parallel or series, as shown for Leyden jars in Figs. 213 and 214. When several are joined in parallel (Fig. 221), and a P.D. applied to the terminals D E, each condenser has the same P.D between its coating, viz. V, so the quantities of electricity on Cj, C 2 , C 3 , etc., will be etc., etc., and if the battery be removed and D and E connected together by a wire, each of the condensers will be discharged and the total quantity, Q, that passes between D and E will be the sum of Qi, Q*> Qz> etc., - CjV + C 2 V + C 3 V + etc. = (d + C 2 + C 3 + etc.) V GOiMBINATIONS OF CONDENSERS 383 Hence -^ = C l + C 2 + C 3 + etc. But p. is the capacity of the combination, .*. combined capacity C = C 1 + C 2 + C 3 + etc. D (99) A _L f F, * F, Fig. 221. Three Condensers in Parallel. from which we learn that the combined capacity of a number of condensers joined in parallel is equal to the sum of their several capacities. Condensers connected in series, as represented in Fig. 222, have a combined capacity less than the capacity of either, which may be calculated from the formula ^-+ ^, + etc. (100) i i i i ^r -~ r T; r ;r tj ^ 2 Oj, vSuppose we take three condensers as shown in Fig. 222, and let them all be completely discharged to begin with. When the P.D. D 1 due to the battery is applied, the quantity Q 1 on condenser C]. is the same as that on C 2 , for the lower plate of C l and the upper plate of C 2 , with the connecting wire, form an in- sulated system whose total charge is zero ; there must, therefore, be as much posi- tive electricity on the upper plate of C 2 as there is nega- tive on the lower plate of C lt and as the quantity of nega- tive on C l is the same as the amount of positive on the other coating of C v we see that the quantity on the coating of C 1 is the same as that on C 2 , and also the same as on C 3 . Fig. 222. Three Condensers in Series. the quantity that will pass on discharge. < - CF < = CF, But 384 PRACTICAL ELECTRICITY and the total P.D., F, equals the sum of the P.Ds. on the several condensers. i.e. V = F x + F 2 + F 3 , ^ + + * t by definition of C capacity, Section 151. C 2 Q Q Q _ __ L. JC __ L ^ ~ c c c ' 1 2 3 . I-i+J 1 C 1 + C 2 + C 3 > The same may be proved for any number of condensers. It is interesting to notice that capacities in parallel have the same law of combination as resistances in series, and capacities in series the same law as resistances in parallel. Example 164. Two condensers of capacities 2-3 and 4-2 microfarads are connected first in series and second in parallel. Find the capacities of the two combinations; Answers. 1-486 microfarads. 6-5 microfarads. Example 165. What capacity must be put in series with the two condensers of Example 164, when coupled in parallel so that the capacity of the whole combination may be 1-5 microfarads ? Answer. 1-95 microfarads. Example 166. A submarine telegraph cable 2,300 nauts* long has a capacity of 0-345 microfarads per naut. Find the quantity of electricity required to charge the copper conductor to a potential of 40 volts. Answer. 0-0317 coulombs. Example 167. Three condensers of 20, 10 and 5 microfarads respectively are available, how could they be combined so as to make up a capacity of 12 microfarads approximately ? Answer. The 20 and 10 in series, and these in parallel with the 5. Actual capacity, 11-67 m -^ s - 166. Charged Condensers are Stores of Electric Energy, not of Electricity. If a suitable galvanometer be inserted in each of the wires connecting the two coatings of the condenser c with the two ends of the battery B (Fig. 223), it will be found on completing the circuit by closing a key at K, that the first swings on the two galvanometers are such as indicate equal * A naut is a nautical or geographical mile, =6,087 feet approximately. ENERGY STORED IN CONDENSERS 385 quantities of electricity passing through them. And if when the condenser is charged the battery be removed, and the condenser be discharged by connecting together the wires P and Q coming from the galvanometers, then the first swings of the galvanometer needles will again be such as to indicate that equal quantities of electricity pass through them, but in this case in the opposite direction to that in which the electricity passed during the charge. Hence, both on charging and on discharging A con- denser, as much electricity passes into one coating as passes out of the other, and there is or accumulating, of elcc- " Fig. 223. tricity. In fact, so far as the galvanometer deflections during the charge show, we could not say whether there was a condenser at c or a resistance, the value of which was, from some cause, rapidly increased, to practically infinity, on completing the circuit. The sudden deflections, however, produced on the galvanometer when the wires P and Q are joined together after removing the battery, could not be produced if c were a resistance, since no alteration of the value of a resistance can, by itself, and without any current generator, produce a current. When the condenser has a large capacity and when the P.D. employed in charging it is large, the current obtained on discharging it may produce very powerful effects. Hence, we are led to conclude that, although a charged condenser contains no store of electricity, it contains a store of electric energy, and it can be shown that, if the capacity of the condenser be C farads, and if it be charged with a P.D. of V volts, the store of electric energy, or the work this store can do when the condenser is discharged, equals C x V* footlbs. 2712 For the unit of P.D. is chosen (Section 48) so that the work done when a quantity of electricity passes between two points whose P.D. is V, is equal to the product of Q and V. When Q is expressed in coulombs and V in volts, QV will be in joules. Now a condenser of capacity C farads charged to a P.D. of V volts contains a quantity equal to CV . coulombs on each coating, and if this quantity were discharged at a constant P.D. of V volts the work done would be CV x V, i.e. CV 2 . But the P.D. falls as the discharge proceeds and eventually z 386 PRACTICAL ELECTRICITY becomes zero, the average value being half the initial value, y viz. . The energy of discharge is therefore CV 2 or - - joules, (102) and as I joule equals 07372 foot-pounds (see Section 117), the energy is 0-3686 CV 2 ft. Ibs. approximately. CV 2 167. Energy wasted in charging a Condenser from a Source of Constant P.D. In this case the whole quantity CV passes under a pressure V, so the work done by the source is CV 2 ; but, as proved above, the energy stored in the condenser is only CV 2 , so that half the total energy is wasted in the process, and appears as heat in the circuit. It is possible, however, to charge a condenser without appreciable loss if this be done gradually from a source whose P.D. rises steadily from zero to the maximum value, as can be done by means of a dynamo, or a potential divider. Example 168. How many times per second would a con- denser of 10 microfarads have to be charged with 86 volts and discharged, so that it would give out about Yoob ^ a horse- power ? Answer. About 20. Example 169. If a battery having an E.M.F. equal to 200 volts be used to charge a condenser of 2oXio~ 8 farads, how many foot Ibs. of work are wasted in the charging ? Answer. 0-295. Example 170. Find the energy stored in (a) t a Leyden jar of capacity 3-5^ of a microfarad charged to a P.D. of 10,000 volts, and (b), the 1894 Atlantic cable, whose capacity is 775 microfarads when charged to a P.D. of 50 volts. Answers. (a) ~ joule, or 0-0123 ft.-lb. (b) 0-97 joule, or 0-71 ft.-lb. Example 171. If an air condenser be formed of two parallel metallic plates, each two square feet in area, placed ^th of an inch apart, and charged with a P.D. of 250 volts, what amount of work must be done in separating the plates, so that the distance between them is increased to th of an inch, if the wires used ABSOLUTE MEASUREMENT OF CAPACITY 387 in charging the condenser be removed before the plates are separated, so that the charge in the condenser remains unaltered during the separation ? Answer. As the distance is made three times as great the capacity will be reduced to J, and the P.D. raised to 250 X 3 = 750 volts. If we calculate the energy stored in the condenser before and after the plates are separated, the difference will give the amount of work done. This equals 8-94 Xio~ 5 ft.-lbs. Another way of considering the problem is to notice that as the energy depends on the square of the P.D. and the first power of the capacity, the energy is tripled by the separation, and therefore the work done in the separation is, in this particular case, twice that originally stored. 168. Absolute Measurement of Capacity. If a constant source of P.D. whose value is known in volts, such as may be secured by means of standard cells (Section 82) be used, the capacity of a condenser in absolute measure can be found by measuring the quantity of electricity which passes into or out of the condenser on charge or discharge by the method described in Section 146 ; for if the E.M.F. of the cells battery used be E, then Q = CE, and as C = , E .'. C = 5 ( i -f- )/ E, approximately, (104) 2 7T d \ 2' (formula 79), when the swing and damping are small. The current / may be measured in terms of E if a good resist- ance box is available, for if R 1 be the resistance through which the E.M.F. E will produce a current / amperes, giving a steady deflection d on the galvanometer, we have / = -, and the above formula becomes R i C = (i + -j approximately. (105) 27U K l d\ 2- This shows that it is not necessary to know the value of E when the value of R 1 is known in C.G.S. measure, and the above re- lation between C and R l is one of great interest, for it enables us to measure a capacity in terms of resistance and time, or a resistance in terms of capacity and time.* * If an air condenser be constructed so that its capacity can be cal- culated (see Sections 156, 157), the above relation enables a resistance K v to be determined in absolute measure if v (the ratio of the electro- magnetic to the electrostatic unit of quantity) be known. See Section 171. 388 PRACTICAL ELECTRICITY Fig. 224. In making the above measurement it is seldom possible to get a resistance large enough to prevent the deflection d being too great to measure, when the whole E.M.F. necessary to produce a swing s that can be accur- ately read, is used to send the steady current, so it is usual to employ a potential divider, to obtain a known fraction of the whole E.M.F. (or P.D.). An arrangement for this purpose is shown in Fig. 224, where the left- hand side shows the con- nections for obtaining s, and the right-hand side for determining d. The value of R l can be found from the TD equation R l = R R r + R g * approximately, where R and R' are the resistances indicated in Fig. 224, Rg the resistance of the galvanometer circuit from N to o, and the resistance of the battery is small compared with R and R g . 169. Measurement of Specific Inductive Capacity, and Re- sistivity of Insulators. As denned in Section 158 the specific in- ductive capacity of an insulating material (or dielectric) is the ratio in which the capacity of an air condenser is altered when the air between its coating is replaced by the material. If, therefore, we can construct a condenser in which the change from air to another dielectric can be made without varying anything else, and measure the capacities in the two conditions, the ratio of the capacities will give the value required. The specific inductive capacity of gases and liquids can be measured in this way, but for solid materials difficulty arises in excluding air from between the surface of the metal coating and the dielectric to be tested, and also in obtaining a sheet of the material of exactly the same thickness as the air between the coatings of the air condenser. To avoid these difficulties we may make a condenser of the material by pasting sheets of tinloil of known area opposite each other on the two sides of a sheet of the material, the thickness of * Students should deduce this as an exercise. Fl 's- 225< Plate SPECIFIC INDUCTIVE CAPACITY 389 which has been carefully measured. , Fig. 225 shows a condenser made in this way in which x is one coating and D the dielectric sheet supported on an insulating block B to keep the coating away from the table. From the dimensions of the condenser the capacity of an air condenser of the same size can be calculated by formula (84) in section 156, and the capacity of the actual condenser can be measured either absolutely (Section 168) or by Fig. 226. Circular Plate Condenser with Guard Ring. comparison with one whose capacity is known (Section 163), from which we can find e, the specific inductive capacity or " in- ductivity " of the material, for __ capacity of actual condenser capacity of air condenser of the same size Standard air condensers which may conveniently be used foi the comparison are described in Section 170. To determine the resistivity of an insulating material, which can be obtained or made into sheet form, a condenser like the one shown in Fig. 225 may be constructed, and, by employing a very sensitive galvanometer, the current which a high P.D. will cause to pass from one coating to the other may be measured, after the P.D. has been applied for i minute, and the resistance calculated from - 390 PRACTICAL ELECTRICITY The resistivity or specific resistance of the material is given by area of coatings P =R X tfrckness of sheet a PP rox ' matel y (* Sections 93-95). In practice the values of V and / are not determined, but the galvanometer is " standardised " by observing the deflection Fig. 227. High Resistance Galvanometer with Highly Insulated Coils. produced by a known small fraction of the P.D. through a very high resistance of known value, usually one megohm. Let the deflection produced after one minute electrification of the dielectric be d t when the whole P.D. is used, and suppose a deflection d 2 given by -th of the P.D. through a megohm in series n with the galvanometer, then R = n - (1,000,000 + R g ) ohms, approximately, a i when the deflection of the galvanometer is proportional to the current strength, and R g is the resistance of the galvanometer. RESISTIVITY OF INSULATORS 391 The arrangements of circuits shown on the left and right of Fig. 224 are suitable for observing d l and d 2 respectively. Results of experiments on the resistance of insulating materials are liable to be seriously vitiated by surface leakage, unless great precautions are taken. For example, leakage may occur from the tinfoil T, Fig. 225, over the uncoated surface of D to the foil on the other side, unless this surface be carefully cleaned and dried, Fig. 228. Standard Air Condenser, and any such leakage would cause the resistance measured to be smaller than the true resistance of the material which it is the object of the experiment to measure. One method of avoiding such errors is by use of a guard wire or guard ring, suggested in 1895, by Mr. W. A. Price, and since then considerably developed by the authors. This is illustrated diagrammaticaUy in Fig. 226, as applied to testing a circular sheet of insulating material, D. Here T represents a tinfoil sheet and T' an annulus of tinfoil or other conductor, in contact with the surface of the dielectric and joined to one galvanometer terminal as shown. With this arrangement there will be no tendency for current to leak from T to T', for they are practically at the same potential, on account of the current through the galvanometer being so very small, and any leakage from T' to the lower electrode is not 392 PRACTICAL ELECTRICITY measured by the galvanometer, so that error from this cause is eliminated. To determine the resistivity of good insulators in this way necessitates the use of very sensitive galvanometers having a very large number 0} convolutions of wire wound as near as possible to the needles, so that the force exerted on the needle by the Fig. 229. Improved Form of Plate Air Condenser. extremely small current which passes through the insulator may be as large as possible. Fig. 227 shows a reflecting galvanometer constructed for the authors for testing insulators, which has a total length of wire on the four coils of about thirteen miles, and a resistance 360,000 ohms. To insulate the coils from earth they are supported from corrugated ebonite rods, which hang from a brass ring, R, carried on three corrugated ebonite pillars from the slate base, and these rods are artificially dried by strong sulphuric acid contained in the vessel v. 170. Standard Air Condensers. Fig. 228 shows a form of air condenser which can be made quite easily and whose dimensions may be measured with moderate precision. Sheets of plate glass about 12 inches square, are used to support tinfoil coatings, but do not act at the dielectric of the condenser. The top sheet, STANDARD AIR CONDENSERS 393 Fig. 229. T T, in the figure, which is removed from the condenser to show the second one, is covered all over with tinfoil, as is also every alternate sheet in the pile of plates. The intermediate ones, i.e. the even numbers counting from the top, are only partially covered on both sides, as is seen at p p, the sheets of foil being ten inches square. Small pieces, F F F, of " patent plate " glass about ~ of an inch thick serve to keep the plates apart, and thus determine the distance between adjacent plates, allowance being made, of course, for the thickness of the foil. The two sheets of foil on opposite faces of the even numbered plates are connected together and all joined to the terminal B, whilst the sheets on the odd numbered plates are all joined to terminal A. The smaller sheets, P p, etc., therefore, form the inside coating of the condenser, and x T, etc., the outside coating. As there are thirteen glass plates altogether, seven form outer coatings and six inner coatings, and this gives an approximate area of the inner coating, 6 X 2 X 100, i.e. 1,200 square inches, and the distance apart being ^ of an inch approxi- mately, the capacity will be [formula (90) ], 2-246 X 1,200 . , , ^ microfarads, approx. io 7 X^ i.e. = 2-7 milli-microfarads, approx. An improved way of making a plate air- condenser is to silver the glass plates all over * by the ordinary process, and cut a narrow circular groove in the deposits on both * Or platinise them by covering them with " plat- inizing liquid " and then Jbig. 2296. Fig. 229c. applying heat. 394 PRACTICAL ELECTRICITY faces of even numbered plates, leaving a tongue projecting over the same edge, as indicated in Fig. 2290. The deposit will thus be divided into two parts insulated from each other by the groove, the inner parts on the two faces forming circular discs united by a narrow strip of conductor, as shown developed in Figs. 2296 and 229$, and acting as the inner coatings of the con- denser when the plates are assembled as in Fig. 229. The outer portion of the deposit on the even numbered plates is used as a " guard ring " like T', Fig. 226, whilst the deposit on the odd numbered plates acts as the outer coating of the condenser as described above. Con- nections between ter- minals and electrodes are made by means of metal springs, s s s, pressing against the edges of the plates at suitable places, as Fig. 230. Diagram of Connections for Testing Guard Ring Condenser. shown in Fig. 229. The use of a guard ring practically eliminates leakage error in testing, and at the same time enables the capacity to be calculated with greater accuracy. For determining the capacity of such a condenser in electro- magnetic measure, or for standardising a ballistic galvanometer, connections as shown in the diagram, Fig. 230, may be used, the inner coating, guard ring, and outer coatings being designated by T, T' T', and i" respectively. Another form of standard air condenser is illustrated in Fig. 231, which is employed at the National Physical Laboratory, Teddington. As will be seen from the vertical section the con- denser is made up of. many concentric cylinders, alternate ones being connected together to form one coating, and the remainder forming the other coating. Its capacity is about 20 milli- microfarads. Fig. 232 shows a condenser formed by two con- centric spheres used by Dr. Rosa, at the Bureau of Standards, Washington, in determining " v." (See Section 171.) 171. Ratio of Electromagnetic and Electrostatic Units of Quantity. This ratio is of fundamental importance in many branches of electrical work, such as the calculation of capacities RATIO OF UNITS 395 of transmission lines and cables, and in telegraphy and telephony, both ordinary and " wireless." Its value has already been stated as 3 x io 10 approximately, and used in Sections 155 and 156 ; the ratio is generally designated as " v." Fig. 231. Cylindrical Standard Air Condenser. Although the accurate determination of "" requires very delicate instruments and great experimental skill, it is possible to give a simple explanation of one of the best methods of making the experiment. In Section 154 and Chapter II., page 82, we have shown how the capacities of air condensers of certain simple forms 396 PRACTICAL ELECTRICITY can be calculated in elec- trostatic units, and in Section 168, a method of measuring the capacity of a condenser in farads is given. Since one farad equals io" 9 C.G.S. (electro-magnetic units of capacity), the result of the measurement divided by io 9 gives the capacity in terms of this unit. Let c be the calculated capacity of an air con- denser in C.G.S. electro- static units, and C its measured capacity in C.G.S. electro - magnetic units, whilst v' and V represent in electrostatic and electro-magnetic units respectively the P.D. to which the condenser is charged in the experiment. If q and Q denote the quantity discharged expressed in these two systems of units, then Fig. 232. Standard Spherical Condenser. ^ - = v (definition of v) and also Q = cv. Hence = . Now in both systems of units the unit of P.D. is chosen so that one erg of work is done when unit quantity passes between two points in a circuit, between which unit P.D. exists. qv' = or Consequently, the above equation may be written COMPARING E.M.F.s BY CONDENSER 397 or c C' C (106) When c has been calculated and C measured as described in Section 168, the value of v is determined. Experiments carried out by Prof. Perry and one of the authors in 1878 gave v = 2-98 x io 10 , whilst the value obtained by Rosa and Dorsey in 1907 at the American Bureau of Standards is 2-9963 x io 10 . This number differs very little from 2-9986 x io 10 , the velocity of light in centimetres per second. For ordinary purposes, 3 x io 10 , the number employed in Section 155, is sufficiently exact. 172. Use of Condensers for comparing E.M.Fs. of Cells or other Current Generators. A diagram of connections suitable for the above purpose is given in Fig. 233. A and B indicate the generators whose @ E.M.Fs. are to be compared, c the con- denser, G a galvano- meter, K a charge and discharge key (Fig 215), and P a three- way plug key (Fig. 135), whereby either A or B may be con- nected to D. When A is connected to D, and the key K pressed and released, the con- denser discharges through the galvano- meter a quantity pro- portional to the E.M.F. of A, producing a swing s lf and when B is joined to D the swing s 2 on discharge is proportional to the E.M.F. of B ; if G is a reflecting instrument we have E s = approximately (107) ^B S 2 This method can be used satisfactorily with cells that polarise rapidly when on closed circuit, and also with cells of high internal C Fig. 233. Comparison of E.M.Fs. by Condenser Method. 398 PRACTICAL ELECTRICITY resistance, such as standard Clark and Weston cells. By using a universal shunt on the galvanometer, generators whose E.M.Fs. are of different orders of magnitude can be satisfactorily compared. 173. Condenser Method of Measuring the Resistance of a Cell. In Section 131, we have seen that when a genera- tor of constant E.M.F. E and resistance R^ is on closed circuit through an external resistance R, the P.D. between the terminals is I'ig. a34. Arrangement of Key and Condenser for Testing Resistance of Battery. V= E ' c/ and when R is infinite, V = E. If, therefore, we charge a condenser Cj C 2 (Fig. 234) from the generator B on open circuit, and obtain a swing s lf on discharging it through the galvanometer, and a swing s 2 , when the circuit of the generator is closed through a resistance R s l will be pro- portional to E and s 2 to V. We can therefore write the above relation R (108) K from which we deduce With cells that polarise quickly, the circuit should not be closed longer than necessary ; Fig. 235 shows an arrange- ment of circuits in which the circuit is opened by the act of releasing the key K'. The expression for Rb contains (s l s 2 ), and it is interest- ing to notice that this difference may Fig< 23S ._ Finding Resistance of Battery by Condenser Method. EXAMPLES 399 be measured directly by observing the^swing produced on break- ing or making the circuit of R whilst the key K is pressed. This may be conveniently done by lifting or depressing the key K' in Fig. 235. Example 172. Find the resistance of a cell which produces a swing of 250 divisions when an open circuit and one of 200 divisions when its circuit is closed through 5 ohms. Answer. 1-25 ohms. Example 173. What swing would be produced on the ballistic galvanometer in the previous question, by changing the external resistance from five to infinity ? Answer. 50 divisions. Example 174. A storage battery on open circuit causes a swing of 340 divisions on a ballistic galvanometer shunted with a j^ shunt, and one of 282 divisions on the unshunted instru- ment, when the circuit of the battery is closed through 0*05 ohm, the key K, Fig. 235, being previously pressed. Find the resistance of the battery. Answer. 0*0045 ohm. CHAPTER IX POTENTIOMETER MEASUREMENTS 174. Poggendorff's Method of Comparing the E.M.Fs. of Cells or Batteries 175. Principle of the Potentiometer 176. Calibration of Potentio- meter Wire 177. Industrial Forms of Potentiometer 178. Modern Form of Crompton Potentiometer 179. Dial Potentiometer 1 80. Calibration of Voltmeters. Volt (or Ratio) Boxes 181. Stand- ard Resistances for Current Measurements 182. Calibration of Ammeters 183. Comparison of Resistances by Potentiometer 184. Measurement of Power 185. Advantages and Disadvantages of Potentiometer Measurements. 174. Poggendorff's Method of Comparing the E.M.Fs. of Cells or Batteries. A way of measuring the E.M.F. of cells by means of a voltmeter has been described in Section 131. This method, although convenient and moderately accurate for cells having small internal resistance, and which do not polarise on sending a current, cannot be used for comparing E.M.Fs. of " standard cells," the internal resistance of which is usually very high, unless a sensitive electrostatic voltmeter or electrometer be available. The condenser method, Section 172, removes to a great extent objection arising from resistance and polarisation of the cells tested, but if either condenser or electrometer be used, the accur- acy of the measurements would be limited by the exactness with which the deflection of the instrument could be read, just in the same way as the accuracy of measuring a resistance by the substitution method of Section 86 depends on the unavoidable errors in observing the deflection of the galvanoscope employed. Now in measuring resistances we saw (Section 87) that by using a " null method " much greater precision could be obtained, and in the comparisons of E.M.Fs., the introduction of null methods by Poggendorff in 1841 contributed greatly to the accuracy of such measurements. The principle of the method is to balance an E.M.F. against the P.D. produced between two points in a circuit through which a current flows from an independent source. Suppose B is a constant battery sending a current through the long stretched 400 POGGENDORFFS METHOD 401 wire D o, Fig. 236 ; there will be a certain P.D. between D and o, D being at a higher potential than o, because the current flows from D to o, and there is no source of E.M.F. between them. I I B ill o 1 c Fig. 236. Simple Circuit through Stretched Wire. The P.D. between the point o and a point c between o and D, will be less than that between o and D, and if the point c be supposed to move gradually from o to D, the P.D. between c and m- B 1 E Fig. 237. Balancing an E.M.F. against the P.D. between two points in a wire, o will gradually increase from zero to V, where V is the P.D. between D and o. If, therefore, we have a cell of E.M.F., E, not greater than V, and connect its negative terminal to o, as x 1 E A C, E Fig. 238. Poggendorffs Method of Comparing E.M. Fs. shown in Fig. 237, it will be possible to find a point c between o and D, such that the P.D. between c and o is equal to E. The potential of this point c will then be the same as the potential of c', a conductor connected to the positive pole of the cell, and if 2 A 402 PRACTICAL ELECTRICITY c' be brought into contact with c, there will be no current through the galvanoscope G, because E is balanced by the P.D. between c and o. Another cell or battery, of E.M.F. E lt Fig. 238, may similarly have its E.M.F. balanced by the P.D. between c^ and o, and if both E and E l are balanced simultaneously, we have E = IR, and E l = IR lt where 7 is the current passing through the wire D o, and R and RI, the resistances of the wire between c and o, and Cj and o respectively. Hence E _ R 1 i' I = /? if the wire o D be uniform, and / and ^ are the lengths o c and OC G E, Fig. 239. PoggendorfPs Method, using only one Galvanometer. When the current I is quite constant, there is no need to use two galvanometers, or even two sliding contacts c' and c/, for by using a two-way key or switch, K, as shown in Fig. 239, the balance points corresponding with E and E lt may be found in succession. After obtaining the second balance, the first one should be tested again, and if any change has occurred, the balancings should be repeated. The accuracy attainable in the above tests depends, of course, on the precision with which / and ^ can be measured, and on the sensibility of the galvanometer. Usually the galvanometer can be made to give a large deflection for a small alteration of / (especially if a reflecting one be used), so the measurement of the lengths of wire is the controlling factor as regards accuracy of the test. It is, therefore, when great accuracy is required, desirable to make the lengths to be measured as large as con- venient, in order that a given error of reading the lengths, say a PRINCIPLE OF POTENTIOMETER 403 fraction of a millimetre, may introduce a very small error in the ratio of E to E r A long straight wire would occupy much bench room, and be awkward to use, so to avoid this inconvenience, the wire may be arranged in zigzag fashion, or several lengths connected in series, as shown in Fig. 240, where five metres of wire are placed on a board about no centimetres long. The slider s can be moved along the graduated scale, and the contact B Fig. 240. Five Wire Potentiometer. piece c brought over any one of the five wires by moving it along the slot L in the spring -key part of the slider. To utilise the wire to the best advantage, the battery B should be chosen so that the P.D. between the ends of the wire is only slightly greater than the largest of the two E.M.Fs. to be com- pared. A variable resistance may be inserted at R x Figs. 239 and 240, for reducing the P.D. on the wire when necessary. We may here point out that in all balance measurements of the kind above referred to, the question of " polarity " is of great im- portance, for unless the two voltages oppose each other, as regards the galvanometer, no balance can possibly be obtained. 175. Principle of the Potentiometer. From the previous section it will be understood that any E.M.F. or P.D., not greater in value than the P.D, between D and o, can be balanced by the P.D. between two points on the wire, through which a current is flowing, and if the drop of potential per unit length of the wire, or the resistance and current, be known, the balanced P.D. or E.M.F. may be measured in this way. This is the principle of potentiometer .measurements. The drop of potential per unit length of wire may be found 404 PRACTICAL ELECTRICITY in several ways. The usual one is to employ a standard cell, either Clark or Cadmium, whose E.M.F. is known, and to find the length of wire / (o c, Fig. 239, say), the P.D. between the ends of which balances E, the known E.M.F. ; then the drop of potential per unit length is . This may give an inconvenient number, so it is customary to adjust the strength of current flowing through Fig. 241. Knife Edges for Calibrating Wire. the wire by means of a variable resistance R x in series with the battery B, until the P.D. per unit length is a round number, say, foo' food" or Toooo f a V0 ^- For example, if we use a cadmium cell for which E = 1-0184 volts at 17 C., the P.D. drop per cm. of the wire in Fig. 240 may be made equal to one-hundredth of a volt by placing the slider s and contact piece c so that it touches the wire at a point 101-84 centimetres from o, and adjusting R x until no deflection of the galvanometer occurs on pressing the key c against the wire. By balancing the cadmium cell at a point 2x101-84 (203-68) cm. from o, a drop of ~o vo ^ per centimetre can be obtained. If a Clark cell be used, for which E = 1-433 volts at 15 C., the balance points must be 143-3 cms. and 286-6 cms. from o, to get P.D. drops of ^ and ao of a volt per centimetre respectively. 176. Calibration of Potentiometer Wire. The potentiometer measurements above described depend for their accuracy on the uniformity of resistance of the stretched wire, as also do the measurements of resistance made by a metre bridge. It is, therefore, of importance to have some convenient means of testing the uniformity. This may be done roughly by measuring the P.Ds. on equal lengths of the wire, say, by a high resistance reflecting galvanometer, when a constant current is passing through the wire. The deflection will then be approximately proportional to the resistances of the parts of the wire tested. A convenient appliance for making this test is a rectangular bar of wood w, say 10 centimetres long, with a metal plate p P CALIBRATING POTENTIOMETER WIRE 405 having a knife-edged notch in it, fixed to each end, as shown in Fig. 241. Wires from the terminals T, T lead to the galvanometer G, Greater accuracy can be obtained by employing a high resist- ance differential galvanometer arranged as in Fig. 242, where R is a fixed resistance, approximately equal to that of the wire between the knife edges of the bar w. The difference between i\\ D.G. Fig. 242. Calibrating Wire by Differential Galvanometer. R and the resistance of the length of wire under test may be found by observing the deflection of the instrument, the sensitiveness of the galvanometer being determined by observing the change of deflection caused by shunting R with a known resistance. Supposing R to be ~ of an ohm, shunting it with 100 ohms would produce a change of resistance of 10 Q 00 of an ohm Fig. 243. Calibrating Wire by Differential Galvanometer. approximately, so that the deflection produced by the act of shunting would correspond approximately with a ten -thousandth of an ohm. Another method of calibrating the wire is to measure the P.D. on successive equal lengths by a second potentiometer, the initial test being repeated at intervals, to ascertain whether any change is taking place in either of the circuits. Instead of the wood bar w, with contacts at a fixed distance apart, we may use, either with differential galvanometer or potentiometer, two independent sliders, s, s', Fig. 243, similar to the one shown in Fig. 240, to subdivide the wire into parts 406 PRACTICAL ELECTRICITY of equal resistance. Putting s at the zero point of the wire, we may move s' to a point such as will give true balance on the differential galvanometer, the resistance between s and s' will then be equal to R. Now move s to the position of s' and again adjust s' to give balance, then the wire between the new positions of s and s' will have a resistance R t * Proceeding in this way the whole length of wire may be tested, and a relative calibration curve drawn between length of wire and resistance, taking R as the unit, which unit need not be known in ohms, so far as the relative calibration of the wire is concerned. To lessen the necessity of using a calibration curve for poten- tiometer and bridge wires, great care is taken in drawing wires intended for these purposes, in order that they may be very nearly uniform. It is also important that the wire be not easily oxidised, hard enough to resist wear and indentation by contact with the slider, that it be of material having a small temperature coefficient for resistance, and have small thermo- electric force,f with respect to copper and brass. German silver, manganin, platinum -silver, and platinum -iridium are frequently employed for these purposes. 177. Industrial Form of Potentiometer. Instead of using a very long wire, some forms of potentiometer are provided with short wires 100 to 105 units J long, and a number of coils in series with the wire, each of which has a resistance equal to that of 100 units of length of the wire. Such an arrangement is represented diagrammatically in Fig. 244. Each coil is then equivalent to 100 divisions of the wire, and 14 such coils (as shown in the figure) have a resistance equal to 1,400 units of the wire ; the coils and the wire have therefore a combined resistance equal to 1,500 1,505 units. The choice of fourteen coils was made * In order that this test may be correct, the index marks on the two sliders s and s" should be placed so that when they are put successively at a given point on the scale, their contact makers touch the wire at exactly the same point along its length. The index error, if any, may be found by interchanging the positions of s and s' and reversing the wires to the galvanometer. Index error may be avoided by leaving s' in the first position, which gave balance, and moving s to s" (Fig. 243) and adjusting it so that balance is produced when the connections between R and the differencial galvanometer are reversed, as shown in dotted lines. t When a circuit includes different metals and the junctions are not all at the same temperature, an electric current usually flows round the circuit. Bismuth and antimony give comparatively large currents, and for this reason are used in " thermopiles," or " thermo- junctions," instruments for detecting radiant heat or indicating differences of temperature. J Some makers take the centimetre as the unit of length for the potentio- meter wire, whilst others use of an inch as the unit. SIMPLE POTENTIOMETER 407 to permit of the Clark cells (E.M.F. 1-433 at 15 C) being conveni- ently used for adjusting the P.D. drop per unit length of wire to Y^th of one volt. For by connecting the negative terminal of the cell to 14 and the positive one through the galvano- meter to the slider placed at 33 on the wire, and varying the current through the wire by resistance R x until balance exists, the required adjustment is made. When so adjusted the potentio- meter can be used to measure any E.M.F. between zero and 1-5 50 100 Fig. 244. Diagram of Simple Potentiometer. volts, for there is a P.D. drop of one-tenth of a volt on 100 divisions of the wire, and on each of the fourteen coils. P.Ds. not exceeding o-i volt can be balanced on the wire itself, those between o-i and 0-2 by the aid of the first coil and the wire. P.Ds. between 0-2 and 0-3 require the use of the first two coils and the wire, and so on. Now that the cadmium cell is much used as a standard of E.M.F. (1-0184 volts at 17 C.) ten coils would be sufficient, but the limit of P.Ds. measurable would then be reduced to about i-i. As the addition of a few coils is not very costly, it is undesirable to reduce the range of the instrument by omitting the four coils n to 14 ; in fact, the addition of four coils, making 18 in all, is to be recommended, for the range of the instrument is then increased to 1-9 volts. A number of coils greater than 18 would necessitate the use of more than one storage cell for producing the constant current through the circuit,* because the E.M.F. of such a cell, in dis- charging, falls below 2 volts, but if allowed to fall below 1-9 becomes unsteady. Some potentiometers have been constructed with 25 to 35 coils for special purposes, and two cells employed for supplying the constant current. The range of an ordinary potentiometer may, however, be doubled by using two storage cells and balancing the standard cell at half value, e.g. 0-7165 (seven coils and 16-5 divisions* of the wire) for the Clark cell at 15 C., and 0-5092 for the cadmium cell * Storage cells have been found to be by far the most convenient source of current for this purpose. f Here the word " division " means a main division of the scale ; these are often subdivided into ten parts. 408 PRACTICAL ELECTRICITY at 17 C. Under these conditions the drop of P.D. per division of the wire is T Q 2 oo f a vo ^ ( 2 millivolts), and the potentio- meter readings have to be doubled. In practice the equal coils of the potentiometer are arranged (between contact studs) round a circular dial, and a switch arm makes contact with any one of them desired, as shown at Q in Fig. 245. The variable resistance R x , Fig. 244, is generally made in two parts, R xl and R xz , Fig. 245, one for rough and the other oFo Fig. 245. Crompton Potentiometer Diagram. for fine adjustment, each section of R xl being nearly equal to the whole of R X2 ; the latter is usually a circular slide wire con- tinuously adjustable. Another device fitted to potentiometers for measuring a number of P.Ds. in quick succession is a multiple double pole switch M, Fig. 245, sometimes called a selector switch, and several pairs of terminals (usually six pairs) connected with this switch, whereby any pair of these terminals may be joined to the measuring points, Q and s. The pairs of terminals are marked A, B, c, D, E, F, in the figure, but to avoid confusion the con- nections of only one pair (c) are shown. 178. Modern Form of Crompton Potentiometer. The scheme of connections described in the last paragraph is substantially that adopted in the Crompton Potentiometer, of which large numbers are in actual use. The latest form, however, differs from this in several details, chief amongst which is that Q, Fig. 246, carries another contact arm insulated from Q and joined to the lower of the two terminals marked POTENTIOMETER COILS, in Fig. 246, which gives an outside view of the instrument. Q is joined to the upper of these two terminals, and by using them the resistance of any of the coils can be readily tested. To keep the contacts free from dirt they are all placed under glass, and the slider s is moved by a handle H, outside the case. A triple successive contact spring key is placed INDUSTRIAL POTENTIOMETER 409 in the galvanometer circuit. Pressing the key lightly com- pletes the circuit through a high resistance ; greater pressure brings the second contact into operation and short circuits most of the resistance, whilst still greater pressure cuts all of it out of circuit, thereby permitting the full sensibility of the galvanometer as a voltmeter to be utilised. Example 175. A Clark cell at 18 C. is balanced by the P.D. on 103-5 centimetres of the potentiometer wire in Fig. 240 ; Fig. 246. Crompton Potentiometer. (General View.) find the P.D. drop between the extreme ends of the stretched wire. Answer. 6*907 volts. Example 176. What current must be passed through a poten- tiometer wire having a resistance of 0-436 ohm per metre, so that the P.D. drop per division (quarter of an inch) may be one milli- volt ? Answer. 0-3613 ampere. Example 177. The wire of a potentiometer has a resistance of thirteen milliohms per centimetre, and is 106 centimetres long ; find (a) , the resistance of each of the fourteen coils in series with the wire, (b), the resistance required external to the wire and coils when the drop per centimetre of wire is o-ooi volt, and the P.D. of the storage cell used is 2-05 volts. Answers. (a) 1-3 ohms, (b) 7-07 ohms. Example 178. By how much must the external resistance in the case above be reduced when the P.D. of the cell falls to 1-95 volts? Answer. 1-3 ohms. 179. Dial Potentiometer. In this form of instrument the slide wire is replaced by a series of coils arranged around dials as shown in Fig. 247. Here the dial on the right takes the place of one-tenth the slide wire in Figs. 244 and 245, and enables steps of P.D. of yi^ of the P.D. between adjacent studs on the left-hand dial to be obtained. The connection between the dials is exactly the same as that of the two-dial potential divider, shown in Fig. 220, the 4 io PRACTICAL ELECTRICITY only difference being that the left-hand dial has 150 coils instead of 9, and the right-hand dial 100 coils instead of 10 ; the whole 100 coils of the right-hand dial have a resistance equal to that of Fig. 2-17. N.C.S. Dial Potentiometer. one coil in the left-hand dial. Variable resistances R xl , R X2 , Fig. 2470, corresponding with R X1 and R X2 of Fig. 245, are provided for adjusting the current through the coils to give a drop of Fig. 247*. N.C.S. Potentiometer. (Diagram.) P.D. of o-oi volt per coil in the left-hand dial, and there- fore of o-oooi volt per coil in the right-hand dial. This adjustment is carried out by putting the arms of the main dials at readings corresponding with the E.M.F. of the stan- VOLTMETER CALIBRATION 411 dard cell used, and varying R xl and R% 2 until balance is obtained. For a Clark cell (E = 1-433) at 15 C. the arm on the left-hand dial would be set at 143, and that on the right-hand one at 30, whilst for a Weston cadmium cell (E = 1-0183 at 20 C.) the corresponding position would be 101 and 83 respectively. The N.C.S.* potentiometer has three pairs of measuring termi- nals A, B and v, Fig. 2470;, joined to a selector switch M. To the pair A -f , A , the standard cell is usually attached ; B + and Fig. 248. Calibration of Low Reading Voltmeter by Potentiometer. B can be used for any P.D. within the range of the dials (1-51 volts), whilst the pair v + and v are connected with a sub- divided resistance N, Fig. 2470, by means of which any voltage up to 300 times the range of the dials may be measured. This subdivided resistance is a single dial potential divider, with un- equal coils giving ratios of I, 3, 10, 30, 100, and 300. 180. Calibration of Voltmeter by Potentiometer: Volt (or Ratio) Boxes. In nearly all cases a separate source of current is used to produce the necessary P.D. between the voltmeter ter- minals, and this P.D. is measured in one of two ways, depending on whether the maximum reading of the instrument is below or above the range of the potentiometer. For low reading volt- meters (say below 1-5) the terminals of the instrument may be con- nected with one of the pairs of measuring terminals on the potentio- meter, and the P.D. which produces any given deflection of the voltmeter measured directly. Different scale readings on the voltmeter may be obtained by altering the rheostat R x , in series with the generator B 2 , Fig. 248, which gives a scheme of con- * The letters N.C.S. are the initials of the partners of the firm of Nalder Bros., Ltd. (Nalder, Crawley & Soames), the original makers of the instrument. 412 PRACTICAL ELECTRICITY nections suitable for the purpose, when the resistance of the volt- meter is not very high. For high resistance instruments a rough potential divider, indicated in Fig. 249, a form used at the Gity Guilds College, would be more convenient. By aid of it any desired reading can be produced on the voltmeter. Switch To potentiometer Fig. 249. Potential Divider for Voltmeter Calibration. The coils in the rough potential divider are made of open spirals of bare platinoid wire, No. 20 S.W.G., and will carry currents up to three amperes without excessive heating. It can therefore be used on any voltage not exceeding 750, but the power used is considerable at high voltages. To voltmeter /To voltmeter Fig. 250. Diagram of " Volt-Box " Connections. To calibrate a voltmeter reading higher than 1-5 on a Crompton potentiometer it is necessary to have some means of obtaining a known fraction of the P.D. between the voltmeter terminals.* This is generally done by means of a high resistance with tappings * The N.C.S. instrument is, as already explained, provided with a subdivided resistance by which P.Ds. up to 450 volts (300 times 1-5) can be measured. VOLT- OR RATIO-BOXES along its length at points giving convenient ratios. Thus, in Fig. 250, if R be the total resistance between p and Q, and con- nections be made at points R, s, and T such that the resistance 1-5 4-5 15 45 ISO 450VOlt5 To poCenbiomefcer Fig. 251. Ratio-Box Connections. PR, PS, and PT are |, J, and ^ of R respectively, the P.Ds. on these portions, when a current is passing through the whole resistance, will be, half, a fifth, or a tenth that between p and Q. If, therefore, the whole resistance be connected across the terminals of the voltmeter v, Fig. 249, the P.D. between P and T will be ^Q that on the instrument, whilst the P.Ds. between P and s, and p and R will be ^ and ^ respectively. Such an arrangement is called a " Volt Box," or " Ratio Box," and by connecting p and T to the " potentiometer " the range of the instrument could be increased tenfold. Volt boxes with ratios i, 2, 5, 10, 20, 50, 100, 200, and 500 are frequently met with, whilst other boxes are made with ratios I, 3, 10, 30, 100, and 300. A universal shunt, such as shown in Fig. 165, serves admirably as a volt box for P.Ds. which will not produce undue heating of the coils. In another form of volt box the con- nections from the potentiometer re- main fixed, and those from the volt- meter (or P.D.) altered to suit the pressure to be measured. A volt box with ratios I. 3, 10, 30, 100, and 300, would have terminals marked 1-5, 4-5, 15, 45, 150, and 45 volts respec- tively, as indicated diagrammatically in Fig. 251. An outside view of a Paul ratio box is given in Fig. 252 and its connections Fig. 252. Outside View of Ratio Box. 414 PRACTICAL ELECTRICITY in Fig. 25 2a. By means of this and a potentiometer, any voltage up to 1,500 can be determined. Potential differences greater than the E.M.F. of a standard cell and capable of supplying steady currents can be measured 5OO CsJ 5,OOOte) IO.OOOCD 25,000(s) 5O,000(fl> POSITIVE MAIN TERMINALS READ VOLTS TOI5(DIRLCT) I5O(DIRECT) 3OO(RLADING*3 750(* E *, D ' NG ) IOOf DIRECT) 450 (o) X \ 4,500 fo> X~X 5,000 .lS.OOO (a) X^\25,000 (a) S~\ AA/\AM()VV\/W^ + FOR POTENTIOMETER O COMMON 5O OHMS ' MAIN Fig. 2520. Paul's Ratio Box. Diagram of Connections. without the aid of a storage cell to send the current through the potentiometer, by using the P.D. to be measured to produce the potentiometer current, and varying resistance in series with it until balance is obtained with the standard cell, the slider being placed at the reading corresponding with its E.M.F. A diagram of the arrangement is given in Fig. 253. It is necessary, of course, in this case, to know the value of R in terms of the resistance of the potentiometer wire. S.C Q rl (?) X 1 1 ED.'to be measured R 1 w Fig. 253. 181. Standard Resistances for Current Measurements. As the P.D. between the terminals of a resistance R ohms with a current / amperes passing through it is IR volts, we can deter- mine / by measuring V when R is known. V can be readily measured on a potentiometer, as described above, and the combination of a known resistance and potentiometer form a very convenient arrangement for measuring currents with LOW RESISTANCE STANDARDS considerable accuracy. The value and form of resistances for potentiometer measurements of current depend on the magnitude of the currents to be measured ; usually they are made for a P.D. drop of o-i to 1-5 volts when carrying maximum current, so that the P.D. can be easily read on a potentiometer. Resist- ances intended for large currents are frequently arranged to give smaller P.D. drops than those for small currents, for other- wise the power (I 2 R) dissipated in the resistance and the resulting rise of temperature becomes excessive, unless the dimensions are very great, or artificial cooling is employed. For a given P.D. drop F,g. 254. Standard Low Resistance crooi Ohm to carry 120 Amperes. Fig. 255. Standard Low Resistance o'ooi Ohm to carry 1,500 Amperes. (Elliott Bros.) the power spent is proportional to the current, so a resistance for 1000 amperes has 1000 times as much heat generated in it per second as one for one ampere. Resistances for carrying large 416 PRACTICAL ELECTRICITY currents are usually made from sheet metal having a large surface, so that the heat produced may be readily got rid of by radiation and convection. Two forms of standard resistance are shown in Figs. 254 and 255. Both are fitted with " potential terminals " as well as main terminals, and their resistances are measured between the potential terminals. The re- sistances here illustrated are intended for maximum currents of 120 amperes and 1,500 amperes respectively, and P.D. drops of 0*12 and 1-5 volts. Fig. 256 shows a particular arrangement of the Ayrton- Mather universal shunt useful for potentiometer work. It consists of six resistances suitable foi maximum currents of 2, 4, 10, 20, 40, and 100 amperes respectively, each of which gives a drop of one volt with maximum current. The arrangement is thus equiva- j lent to six separate resist- ances, and possesses the further advantage that no changes of connections, either to ma i ns or potentiometer, are required over the whole range of current from 0-2 to 100 amperes. With this resistance and an ordinary potentiometer reading ^ of a volt to i part in 1000, any current between 0-2 and 100 amperes can be measured to an accuracy of one-tenth per cent. When the switch is placed as in diagram, Fig. 2560, the arrangement is suitable for currents up to 20 amperes ; M and M being the main current terminals, and p! and p 2 the potential terminals. 182. Calibration of Ammeters. Fig. 257 shows an arrange- ment for this purpose ; a known resistance, R, whose current- carrying capacity is suited to the range of the ammeter, being joined in series with the ammeter A, a separate source of current, and a variable resistance, R x . The variable resistance R x may conveniently consist of a number of carbon plates in, but insulated from, a metal frame, Fig. 258, pressed together more or less by a screw. Movable metal plates, with terminals attached, enable the Fig. ^.- AMMETER CALIBRATION 417 number of carbon plates in circuit to be varied, and also permit of their being arranged in two or more' parallel circuits when very large currents are required. For small currents, say below two amperes, the form shown in Fig. 259 is very useful. It is built up of numerous discs of sail-cloth, c c c c, carbonised at a very high temperature, whereby the flexibility and elasticity of the cloth ^Potentiomeber terminals Fig. as6a, Diagram of Connections of Universal Shunt for Strong Currents. are retained. Brass plates p lt p 3 , p t at top and bottom of the pile and at an intermediate place serve to make contact with the discs, and more or less pressure is exerted on the discs by the nut n and wooden washer e. An insulating sleeve is slipped over the brass rod h to prevent the discs being short circuited. 183. Comparison of Resistances by Potentiometer. If two resistances be connected in series and a steady current passed through them the P.D. drops on them will be proportional to their resistances. If, therefore, we measure the P.Ds. on a po- tentiometer we get the ratio of their resistances directly, and if one be of known value the other is determined. The method, illustrated diagrammatically in Fig. 260, is particularly useful for low resistances having potential terminals, for the resistances of contact between the leads and the main terminals, which may be quite considerable compared with the whole resistance, are 2 B 4i8 PRACTICAL ELECTRICITY eliminated in this test. To obtain satisfactory results the two resistances should be of the same order of magnitude, and when Fig. 257. Calibration of Ammeter by Potentiometer. an ordinary potentiometer, used in the ordinary way, is employed, it is desirable that the smallest P.D. be not less than about half a volt. There is no necessity, however, to have the current through the potentiometer of strength sufficient to give a drop of o-i volt per 100 divisions of the wire, for whatever the current passing, so long as it is steady, the P.Ds. are proportional to the potentiometer readings which give balance, and in cases where Fig. 258. Carbon Plate Rheostat. P.Ds. of the order J a volt on either of resistances to be com- pared would produce undue heating, the current through the potentiometer wire may be reduced by putting resistance in series with the storage cell supplying the current. This reduction COMPARISON OF RESISTANCES 419 must not be carried too far, otherwise the galvanometer ma) 7 not indicate an appreciable movement of the slider. When the two resistances to be compared are very unequal, the Fig. 259. Carbon Cloth Rheostat. P.Ds. to be measured on the potentiometer differ widely, so one reading would be small and difficult to observe accurately. A way of improving the conditions of the test is to shunt the largest of the two resistances, by a volt -box, universal shunt, or similar subdivided high resistance, Fig. 261, and measure a fraction of the P.D. on the larger resistance, which is of the same order Fig. a6o. Comparison of Resistances by Potentiometer. of magnitude as that on the smaller resistance. To increase the potentiometer readings one of two things is necessary, either the current through the two resistances to be compared 420 PRACTICAL ELECTRICITY must be increased, or, if this be not permissible, the current through the potentiometer must be diminished by inserting resistance at R X) Fig. 261. Shunting R will slightly reduce the K AWWwJ o 66 66 ~ 09 oo oo . Fig. 261. Comparison of Resistance by Potentiometer. P.D., but the error will often be negligible. When this is not the case a proper correction can easily be made. 184. Measurement of Power. We have already explained how P.D. and current can be measured by a potentiometer, and as the power used in a given circuit is the product of the two, it is evident that the instrument can be employed for power measurements. A scheme of connections for this purpose is given in Fig. 262, where T 1 and T 2 are the terminals of the part of a circuit in which the power is to be determined. Here R represents the volt box and R 1 the resistance by means of which the current is measured. 185. Advantages and Disadvantages of Potentiometer Measure* ments. Amongst the advantages of the instrument must be mentioned the universal nature, and the wide range over which measurements can be made, for with a potentiometer and its adjuncts, a ratio box and standard resistances, pressures, currents, resistances and powers can be determined with great facility and high accuracy. For pressures and currents the range is very large, suitable ratio boxes and resistances enabling values of either quantity, from a fraction of a volt (or ampere) to many hundreds of volts (or amperes) to be measured. For testing resistances, the elimination of the resistances of contacts and connections which the method renders possible, is of great importance, and on this account it is much used for low POTENTIOMETER MEASUREMENTS 421 resistance measurements. Moderate or large resistances can be more easily compared by bridge methods. The disadvantage of the potentiometer, as compared with the bridge for measuring resistances, lies chiefly in the fact that two sources of current, both of which must be very steady for appre- ciable times, are required with the former instrument, whilst a bridge needs only one battery, the current from which need nol T, R To poCenDiometer Fig. 262. Measurement of Power by Potentiometer. To potentiometer be exactly constant. Two adjustments and two readings are also necessary in the former case, and only one in the latter. Another disadvantage is that the effects of thermo-electric forces are not so easily eliminated as in bridge measurements. Example 179. In comparing an unknown resistance with a standard of o-i ohm, without using a standard cell, the potentio- meter readings were 1-265 and 0-832 respectively; find the value of the resistance tested. Answer. 0-1521 ohm. Example 180. Two unequal resistances are compared and the standard (o-oi ohm) is shunted by a ratio box of 100 ohms total, a 10 ohm section of which is used for the potentiometer measurement. The readings obtained on the unknown and known are 0-0430 and 0-0695 respectively. What is the value of the unknown resistance uncorrected for shunting error of the standard ? Answer. 619 microhms. Example 181. What is the approximate error in the resistance measured caused by shunting the standard in the last example ? Answer. i part in 10,000, i.e. 0-06 microhms approx. 422 PRACTICAL ELECTRICITY Example 182. Supposing the shunt of 100 ohms in Example 1 80 to have been placed across the unknown resistance and that the potentiometer readings on the known and unknown resistances were 0-0541 and 0-0928 respectively ; calculate the unconnected value of the latter, and the approximate correction for shunting. Answers. 0-1715 ohm,and 0-0003 approximately. CHAPTER X INDUCED CURRENTS 186. Introduction 187. Direction of Induced Currents due to Magneto- Electric Induction 188. Lenz's Law : Fleming's Rule 189. Rela- tion between Quantity Induced and the Resistance of the Circuit 190. Determination of Constant of Ballistic Galvanometer by Earth Inductor Method 191. Distribution of Magnetism in a Bar Magnet 192. Flux Density over Cross-section, and over Surface of Magnet 193. Mutual Induction 194. Unit of Mutual Induction : Henry 195. Self-Induction 196. Induction Coil 197. Induction of Currents in Parallel Wires. 1 86. Introductory. Of the several means of producing electric currents, batteries, thermopiles, frictional machines and dynamos, the last-named is by far the most important, for without this method of transforming mechanical energy into electrical energy, electric lighting, electric traction on tramways and railways, the electric driving of workshops and factories, the electric refining of copper, the production of aluminium, and other electrochemical products, would be commercially impossible. The subject of induced currents on which the action of the conversion depends is, therefore, of prime importance to electrical engineers. Faraday in 1831 discovered that electric currents could be produced in wires and coils by the relative motion of magnets and wires. Had the principle of " conservation of energy " been thoroughly under- stood when Romagnosi in 1802 and Oerstedt in 1819 observed the effect of electric currents on a magnetic needle, Faraday's discovery would probably have been anticipated by many years. For the deflection of a magnetic needle by a wire conveying a current, proved that mechanical energy could be produced by the action of a current on a magnet, and, conversely, the mechanical energy used in moving a magnet, near wires should, on the prin- ciple of conservation of energy, produce equivalent effects in the electric circuit. This we now know to be the case, and the history of the progress and developments which have led from the mere shifting of a compass needle when a wire connected with two plates of metal dipping in a liquid was brought near it, to the building of 40, ooo -horse -power dynamo machines, capable of 423 424 PRACTICAL ELECTRICITY lighting a whole city, forms one of the most interesting examples of the beneficial association of science and engineering. 187. Direction of Induced Currents due to Magneto -Electric Induction. The laws of magneto -electric induction may be investigated, both as regards direction of currents and quantities of electricity produced, by the apparatus shown in Fig. 263, which consists of a bar magnet, a coil of wire, some resistances, and a simple galvanometer. First as regards direction of current, we find out by means of a small cell the direction in which the needle of the galvanometer deflects when a current is sent through it in Fig. 263. Magneto-Electric Induction Apparatus. a known direction. To make matters definite we will suppose the deflection is clockwise when the terminal TJ is positive. On connecting the coil terminals to those of the galvanometer by means of long wires,* and bringing the magnet near the coil, the galvanometer needle is deflected momentarily and returns to zero if the motion of the magnet ceases. Removing the magnet produces a swing in the opposite direction. If the magnet be turned end for end, and the experiment repeated, the effects are as before, except that the directions of deflection are reversed. From these tests we learn ist. Induced currents flow only when relative motion is taking place ; 2nd. The direction of the current is reversed when the relative motion is reversed ; and 3rd. Reversal of the magnet causes the current induced by any given relative motion to be reversed ; 4th. By observing the polarity of the magnet, the direction of its motion and of the deflection of the needle, and the direction of winding of the coil, we find that bringing the north -seeking pole of the magnet towards a face of coil, a counter-clockwise current flows in the coil when looking at the face to which the pole is brought near ; the same is true on removing a south -seeking pole, whilst if the north pole be removed or a south pole approaches the coil, the currents induced are in a clockwise direction in the coil when looking at the face concerned * To enable the coil to be placed far enough from the galvanometer so that the magnet, when near the coil, has no direct action on the needle. LENZ'S LAW 425 Now the lines of force produced by a magnet are regarded as emanating from the north pole and entering the south pole, so that when we look at the face of the coil when a north pole is brought towards this face, we are looking along the lines of force, i.e. in the direction of the magnetic force. Further, when the pole of the magnet is near the face of the coil, a number of the lines of force of the magnet will link through the winding of the coil, and we may say that as the magnet approaches the coil, the number of lines linked with the coil increases. The results of the above observations may therefore be expressed as follows ; When we look along the lines of force and the number of lines of force linked with a coil increases, a counterclockwise current is induced in the coil. This rule can be applied practically to all cases of magneto -electric induction. When the number of linkages decreases, the current will, of course, appear clockwise in direction, when we look at the coil along the lines of force. 188. Lenz's Law ; Fleming's Rule. As we have already men- tioned (Section 5) when an electric current passes round a coil on an iron bar the iron becomes a magnet, and even if the iron is removed the coil exhibits magnetic properties. A simple test with a compass needle shows that the north pole of the coil is that end or face looking at which the current circulates counter- clockwise, so we see that when a north pole approaches a coil and induces a current in it, the direction of this current is such as to produce north polarity at the end of the coil nearest the north pole of the magnet. The magnet will therefore be repelled. The experimental facts may be summed up in the statement that the direction of induced currents produced by relative motion oj coil and magnet is such as to oppose the motion producing it. This is one way of stating Lenz's Law of induced currents. Another rule by which the directions may be remembered was stated by Prof. J. A. Fleming and is known as Fleming's Right- hand rule. Here we consider the magnet stationary and the coil moving, and confine our attention to a small part of the wire crossing the lines of force. Put the thumb, the index finger, and middle finger of the right hand mutually perpendicular, and place the hand so that the index finger points along the lines of force, the thumb in the direction in which the wire moves, the middle finger will then point in the direction of the induced E.M.F. in the portion of wire considered. If the magnet and coil in Fig. 263 be placed co-axial, and the magnet be steadily passed through the coil, and away on the other side, the induced current will be in one direction until the magnet is halfway through ; it will then reverse, increase, and diminish as 426 PRACTICAL ELECTRICITY the magnet moves away from the coil. If the movement of the magnet be rapid the galvanometer needle will give a kick in one direction and then jerk back, and when the motion is extremely quick no perceptible movement of the needle will occur. This proves that the quantity of electricity which passes round the circuit as the magnet approaches the coil is equal and opposite to the quantity that passes as the magnet goes away. This is equally true whether the magnet passes right through the coil, or is brought up to the coil and then taken back to its initial position. 189. Relation between Quantity Induced and Resistance of the Circuit. A ballistic galvanometer connected with the coil in Fig. 263 enables the quantities of electricity produced by definite relative movements of the coil and magnet to be measured. Suppose we place the magnet inside the coil, so that the middle of the magnet is at the middle of the coil and the two are coaxial. Withdrawing the magnet to a distance quickly, so as to approxi- mate to the condition that the whole quantity passes through the galvanometer before the needle has moved appreciably from its zero position, will produce a swing of the needle, from which the quantity of electricity can be calculated by the formula c Q k sin (see Section 149), where k is the " constant " of the instrument. Instead of withdrawing the magnet from the coil by one move- ment, we may do it in two or more steps, and observe the swing produced by each step. When this is done, and the several quan- tities of electricity added together, it is found that their sum is equal to that produced by the withdrawal in a single movement, provided the initial and final positions are the same in the two cases. Expressed symbolicallv, we have Q = 0i + (?2> (?3 e tc., being the quantities produced by the several steps. This is a most important result, for it proves that the quantity of electricity induced depends on the initial and final relative positions of the magnet and coil, and not on the inter- mediate positions they may have occupied. The above statement presupposes that the resistance of the circuit remains constant during the experiments. Changing the resistance alters the quantity produced by any given movement. To enable the law of variation to be experimentally determined, resistances are placed between the mercury cups shown to the right in Fig. 263, and more or less resistance can be included in LAW OF INDUCED QUANTITIES 427 the circuit by altering the position of the copper bridge piece b. To simplify the experiment, each of these resistances is made equal to the sum of the resistances of galvanometer, coil, and connecting leads, which, in the apparatus shown, is about 2 ohms. The experiment is best carried out by placing the magnet centrally within the coil, and, when the galvanometic needle is quite at rest, suddenly withdrawing it, and observing the swing. This is done first with no added resistances in the circuit, and then with i, 2, 3, etc., coils inserted. It is convenient to tabulate the results observed and calculations made from them as indicated below Total Resistance of Circuit. First Swing Produced. Sine of Half Swing. Product of Total Resistance and Sine of Half Swing, 2 4 etc. when it will be found that the numbers in the last column are practically equal. In this way we can prove that the quantity of electricity induced in a circuit by a given relative movement oj magnet and coil, is inversely proportional to the resistance of the circuit* Putting this statement into symbols we have R where N is some constant. We may now enquire what the constant N represents ? In the experiment as performed, the initial and final positions of the magnet relatively to the coil have been maintained constant. Now in the initial position, there were a certain number of lines of force of the magnet linked with the coil, and in the final position another number (generally zero if the magnet has been taken far enough away) were linked with it, so we see that the change of linkages has been maintained constant in the several experiments. Further we know that if the magnet be only partly withdrawn, the swing, and therefore the quantity, will be less than before, * The small differences from equality obtained in a carefully made set of observations arise from the change produced in the " damping " of the galvanometer swings when the resistance of the circuit is altered, the damping decreasing as the resistance increases. The differences dis- appear if we determine the decrements and correct the several quantities by multiplying by 1 1 4 1 (see Section 148). 428 PRACTICAL ELECTRICITY so we are led to the conclusion that the constant N in the above equation represents the change of linkages of lines of force with the coil.* The expression is similar in form to the usual method of writing Ohm's Law, and it is easily remembered by regarding it as the Ohm's Law of in- duced quantities. In fact, it can be deduced directly from Ohm's Law and the definition of E.M.F. given in Section 550, viz. the rate of cutting of lines of force. Suppose at the beginning of a short time, t, the number of lines linked with the coil to be N v and at the end of the interval N 2 . The change of linkage is N 1 N 2 , and the average rate of cutting of lines will be - - - -, which may be written SN t ' where m = N - N 2 , where E is the average E.M.F. during the interval t. Writing Ohm's Law as '=! T 8N we have / = , -! but It = quantity, or the quantity of electricity which passes in a given time equals the change of number of lines which occurs in that time divided by the resistance of the circuit. As this is true for any short interval of time, it is true for the whole time, so the whole quantity equals the whole change of lines, divided by the resistance, i.e., * When the coil consists of many turns, as is usually the case, the number of lines linked with one turn may differ from the number linked with another, and the whole number of linkages is the sum of numbers of lines linked with the several turns. EARTH INDUCTOR METHOD 429 This relation between Q, N, and R is of great importance, and should be thoroughly understood. If the quantity is to be expressed in coulombs and R is in ohms, we have (since I coulomb = -fo C.G.S. unit of quantity, and i ohm = io 9 c.G.s. units of resistance) Q = - 5 coulombs, (no) N being expressed in C.G.S. lines. 190. Determination of Constant of a Ballistic Galvanometer by Earth-Inductor Method. The value of N in the last equation may be found by observing the swing produced on a ballistic galvanometer, if we know, or can determine, the constant of the instrument. This may be done by measuring the periodic time of the needle, the sensitiveness to steady currents, and the decre- ment as described in Sections 146-8, or by discharging through it a known quantity from a condenser of known capacity charged to a known P.D. But the simplest way of determining the constant is the Earth Inductor Method, in which a coil of known area and number of turns is connected by flexible leads with the galvano- meter and suddenly turned through 180 in the earth's magnetic field, the strength of which may be measured by methods de- scribed in Chapter II. (see Section 27). Values of the horizontal component H of the earth's magnetic field for several places in England are given in Table III, Section 36. These strengths are rather small, and in this country, where the " magnetic dip "* approximates to 70, it is convenient to make use of the vertical component of the earth's magnetic force. Calling the vertical component U we have U = H tan d, where d is the angle of dip, and when H and d are known, U can be calculated. Taking H = 0-185 an ^ ^ = ^7'5> the approximate values for undisturbed areas near London, we get U = 0-185 tan 67-5 = 0-447, approximately. A coil suitable for use as an earth inductor is shown in Fig. 264. It is wound with 100 turns No. 18 wire of mean area 1000 square centimetres approximately. When held in a horizontal plane, each turn is linked with 0-447 x IOO unes f f rce due to the vertical component of the earth's field, so the total linkage is 447 x 100 or 44,700. When the coil is turned upside down, * The "magnetic dip" is the inclination to the horizontal at which a truly balanced, freely-suspended needle sets itself when magnetised (see Section 15). 430 PRACTICAL ELECTRICITY there are an equal number of linkages in the opposite direction as regards the coil, the lines now passing from face B to face A instead of from face A to face B. The total change of linkage is therefore, zUnA, where n is the number of turns in the coil and A the average area, so in this case N = 89,400. If the circuit of the coil be Fig. 2 6 4 .-Simple Earth- Inductor. completed through the galvanO- meter when the movement is made, a quantity of electricity _. 89,400 Q = -2g x io~ 8 coulombs K will pass through the circuit, where R is the total resistance. Since Q k sin , where s is the swing produced, we have, 89,400 x io" 8 and R and s being known, k is determined. Should the resistance of the circuit when the earth inductor is used be made the same as when the observations were made in measuring N (formula no), there is no need to calculate k, for in one case DISTRIBUTION OF MAGNETISM 431 Either of the last two formulae may be written N 1 = A 7 ' sin S - , (in) where N' is the change of linkages which would produce a swing of 180 under the then existing conditions. The constant N' may be termed the linkage constant of the ballistic galvanometer and circuit. For a reflecting galvanometer, as in Section 149, the " linkage constant " may be taken as the change of linkage which produces a swing of one division. Fig. 265. Apparatus for Testing the Distribution of Magnetism in a Bar Magnet 191. Distribution of Magnetism in a Bar Magnet. The induc- tion of electric currents due to change of linkage of magnetic lines may be used to find the distribution of magnetism in a magnet. A convenient form of apparatus is illustrated in Fig. 265. The cylindrical bar magnet B, 30 centimetres long and 1-67 centimetres diameter, is supported in a vertical position in a wood block, w. A coil of 100 turns of fine wire on a thin brass tube which fits closely, but slides freely on the bar, is connected by long flexible leads to a ballistic galvanometer, G. The brass sleeve s, on which the coil rests, can be clamped at any point of the bar, and the bar is graduated in centimetres, so that the distance of the centre of the coil from the middle of the bar can be easily read off. In commencing the experiment we fix s so that the centre of the coil c is at the middle of the bar when c rests on s. On suddenly sliding the coil off the bar a swing is produced on G, from which the quantity of electricity may be determined, the constant of the galvanometer being conveniently found by the earth inductor method described above. As the coil fits close 432 PRACTICAL ELECTRICITY to the magnet and is of small axial length and radial depth, we may say that the lines of force linked with each turn will be approximately the same, and if we designate by $ * the number of lines of force passing through the central section of the bar, the change of linkage produced by sliding the coil off and away from the magnet will be 100 $ , and the quantity of electricity which flows round the circuit, in consequence of this change, is given by TOO ^ Q = ^ - x io~ 8 coulombs, K and <1> = R sin - x io 8 , 100 2 s being the swing produced, and R the resistance of the circuit. By moving the sleeve s one centimetre higher, and repeating the experiment, * x the flux through the cross-section of the bar which is one centimetre from the centre can be found. Similarly the fluxes at distances 2, 3, 4, etc., . . 15 centimetres, from the centre may be determined and a curve plotted showing the relation of flux to distance from the centre. Turning the bar upside down, the distribution in the other half of the bar can be investigated, and a curve for the complete magnet obtained, such as is shown in Fig. 266. Instead of turning the bar the other end up, it is advisable in testing the second half of the bar to fix it by the upper end and slide the coil off the bottom, for reversing the bar in the earth's field affects the magnetism slightly and the curves for the two halves do not quite join at M, Fig. 266. A better plan still is to support the magnets in a horizontal direction perpendicular to the magnetic meridian, so that the earth's field will have little or no effect on the longitudinal magnetisation of the bar. 192. Flux Density over Cross-sections and over Surfaces of a Magnet. On dividing the values of the flux obtained in the experi- ment just described, by the area of the cross-section in square centimetres, numbers are obtained to which the name " Flux density " or " Induction density " are given, this quantity being usually denoted by B. If we consider the fluxes through two sections at one centimetre apart, say for definiteness $ 3 and 3 $ 4 , must have emerged from the cylindrical surface of the bar between the two cross-sections, and if we divide this difference * The number of lines of force passing through a given area is often spoken of as the magnetic flux through that area, so in this case $> is the flux through the middle section of the magnet. FLUX DENSITY IN MAGNET 433 c 0) "T3 X CL ^ *^* ! rf k- F *"*x ^ 1 / ^ N ^s | w i < t\ y s 7 / \ / s, 1 r / \ . / \ / 1 I / \ / \ / \ J ! \ I 14 12 10 8 64 2+0-24 o 8 Distances from centre of magnet Fig. 266. Distribution of Magnetism over cross sections of bar. 10 12 Tauu <^1 c8 +<&OO *t- "S J_ 3 s 1 <0 s "^ A j_ HOO ~o X E \ g> s V % V 03 n ^ S X 3i ^>s X3 >* "s by the area of the cylindrical surface of the bar between these cross-sections, viz. TT d, where d is the diameter of the bar in centimetres, the resulting number will be the average density of lines emerging from this portion of the magnet's surface. This value may be plotted on a vertical line at a distance representing 3 J centi- metres from the centre of the bar, and by obtaining a number of such points a curve showing the approximate surface distribution of flux along the length of the bar can be drawn. To avoid great inaccuracy in the values of surface density, owing to their being calculated from the difference of two relatively large quantities, either or both of which may be in error, the difference should be taken from the smooth curve drawn amongst the points, shown in Fig. 266, instead of from the numerical values obtained. When this is done a curve like that shown in Fig. 267 results. As the cross-section of the bar is the same throughout its length, the curve in Fig. 266 represents the flux, as well as the flux-density at different cross-sections of the bar, to a certain scale, and from it we conclude that the flux, and also the flux density, in a cylindrical bar magnet, is greatest about the middle, and falls off rapidly towards the ends, whilst Fig. 267 shows that the surface density is very small in the centre and increases as the ends are approached. It is interesting to notice that the greatest density over the middle cross-section is about 4,600, whilst the greatest surface density is approximately 200 lines per square centimetre, in this particular case. Example 183. The swing produced on a ballistic galvano- meter was 15 when the earth inductor, Fig. 264, was quickly turned over in the earth's field. Find the " constant " of the galvanometer, having given that the resistance of the galvano- meter is 0-87 ohm and of the earth inductor and leads, 1-65 ohms. (U 0-447) Answer. &= 0-00272. Example 184. Withdrawing the magnet from the centre of the coil c in Fig. 263 causes a swing of 80 degrees on the galvano- meter when the total resistance of the circuit is 2 ohms. Cal- culate the flux through the magnet, having given that the coil has 400 turns and that the galvanometer is constant 0-00283. Answer. 910 C.G.S. lines. Example 185. What is the induction density at the centre of the magnet mentioned in Example 184, its dimensions being 15 X 1-6 X 0-25 centimetres. Answer. 2,280 c.G.s. lines per square centimetre, approxi- mately. MUTUAL INDUCTION 435 Example 186. Find the " linkage constant " of the galvano- meter and circuit- in Example 184. Answer. 910 X 400 -f- sin 40 = 566,000. 193. Mutual Induction. Not only can currents be induced in circuits by bringing a magnet near them, but, as a coil carrying a current has magnetic properties, relative motion of two circuits, one of which is conveying an electric current, produces a current B T, T Fig. 268. Mutual Induction Apparatus. in the other circuit. This is called mutual induction, and the two circuits are generally spoken of as the primary and secondary circuits respectively. Relative motion is, however, not essential in this case, for stopping the current in the primary is equivalent to removing it far away from the secondary circuit. A change of current in the primary circuit thus produces an induced current in the secondary. This only occurs when the relative position of the two circuits is such that some of the lines of force produced by the primary are linked with the secondary circuit. The laws of mutual induction may conveniently be studied by the apparatus shown in Fig. 268. Cj and C 2 are two coils, the former of which can be placed inside the latter. The winding on G! (the primary coil) is connected with a storage battery B, through a number of resistances, indicated diagrammatically on the board, by means of which the current through c x may be varied in known proportions. In the actual apparatus, the current can have values proportional to i, 2, 4, 6, 8, 10, by placing the copper bridge piece b in the proper mercury cups. The direction in which the current flows through Cj may be found by an examination of the winding, and testing or observing the polarity of the battery ; and the relation between the direction of deflection of the galvanometer G, and that of the current in C 2 , which is connected by long wires to G, but is entirely insulated from the primary circuit, can be determined as explained in Section 187. 436 PRACTICAL ELECTRICITY If we place c x inside C 2 and then complete the primary circuit, the galvanometer needle will give a swing and then return to zero. It will remain at zero so long as the primary current remains constant and the position of the two coils is unaltered. From the direction of the first swing of the needle it will be seen that the current induced in the secondary coil is in the oppo- site direction to that started in the primary. On stopping the primary current, a swing equal in magnitude to the previous one but in the opposite direction, will be produced. These tests show, first, that starting a current in the primary circuit causes a transient inverse* current in the secondary ; second, stop- ping a current in the primary causes a trans- ient direct current in the secondary. Strengthening the pri- mary current acts in the same way as starting a current, whilst weaken- ing the primary current is 2 68*. Fig . *. qualitatively equivalent to stopping a current. When the above-mentioned experiment is performed with currents of several strengths, and the several quantities of electricity produced determined by a ballistic galvanometer, it is found that the quantity induced in the secondary circuit is proportional to the strength of the primary current which is started or stopped. The quantity induced depends not only on the strength of the inducing current but also on the relative position of the two coils. If the primary coil be placed centrally inside, Fig. 2680, the second- ary effect is greatest, and stopping the current in this case is exactly equivalent to withdrawing the primary suddenly to a distance with the current still flowing. When the coils are placed co -axial but the primary above the secondary, Fig. 2686, the effects are much reduced, whilst when the coils are placed with their axes at right angles and intersecting as shown in Fig. 268c, the effect of starting or stopping the primary current is nil. These facts are expressed by saying that the mutual in- duction of the two coils in position 2680 is zero, that in 2686, small, and in Fig. 2680 the mutual induction is a maximum. * A current in direction opposite to the primary current is called an inverse current, and one in the same direction a direct current. MUTUAL INDUCTION 437 When the two coils are standing side, by side, close together, Fig. 268^, the mutual induction is small and negative, for starting a current in Cj causes a direct current in c. 2 . Another important fact may be demonstrated by interchanging the two coils, i.e. using C 2 as primary and Cj as secondary. When Fig. 268c. Fig. 268^. this is done, and the resistance of the secondary circuit made the same as before, experiment shows that the induced quantity pro- duced by a given change of current is exactly the same in the two cases, whatever the sizes, shapes, or numbers of convolutions in the two coils. This fact suggested the name mutual induction. The mutual induction of two coils is much affected by the presence of iron, and to show this an iron core I, Fig. 268, which can be fixed inside the coil c lf forms part of the apparatus. When this is inserted the quantities induced by a given change of current are greatly increased, and to prevent damage to the galvanometer it is necessary to increase the resistance of the secondary circuit when the iron core is being used. A core of the size and proportion shown (six inches long and f inch dia- meter) when employed in this apparatus, multiplies the effects about twenty times, and in making tests with it, care must be taken to avoid direct action between the core, which becomes an electro -magnet, and the needle of the galvanometer. 194. Unit of Mutual Induction : Henry. A quantitative mean- ing is given to the expression mutual induction by defining it as the linkage of lines of force with one coil due to unit current in the other coil. It can be measured by observing the quantity induced in one of the coils by stopping or starting a measured current (say /' C.G.S. units) in the other. We have then Linkage N = I'M', (112) where M' is the mutual induction, or co-efficient of mutual induction as it is called ; and since N Q = x io~ 8 coulombs (Section 189), 438 PRACTICAL ELECTRICITY I'M' we have Q = x icr 8 coulombs ; K and as /' = - , when 7 is the current in amperes, 10 we get Af' = 5L, x io 9 C.G.S. units, (113) where Q is in coulombs, R in ohms, and 7 in amperes. The practical unit of mutual induction is for convenience taken at io 9 C.G.S. units, and is called the " henry," so the above expression becomes, on writing M x io 9 for M', M = - henrys, , . s k sin ry or M= R henrys, (114) k being the constant of the ballistic galvanometer in coulombs. 195. Self -Induction. Lines of magnetic force may be linked with a coil not only by its being placed in a magnetic field, or near another circuit conveying a current, but also by a current in the coil itself. In this case the linkage is said to be due to self-induction, and the co -efficient of self-induction* of a coil is denned as the number of linkages due to unit current in the coil. The henry is the unit of self-induction as well as of mutual induction, and a coil whose inductance is one henry is such that the linkages of lines of force with the coil when one c.G.s. unit of current is passing through it, is io 9 , and the number due to one ampere, io 8 . 196. Induction Coil. The quantity of electricity which passes round a circuit of fixed resistance due to a given change of linkage is, as we have already seen, independent of the time in which the change occurs. The current being a transient one, must increase and then decrease again, and its average value must be greater the shorter its duration, so that although the quantity is in- dependent of the time in which the change of linkage occurs, this is by no means true of the current. The quicker the change of linkage, the greater the current, and also the greater the E.M.F. If, therefore, we can make a given change of linkage very quickly, a large E.M.F. can be induced in a circuit. Further, the change of linkage and therefore the E.M.F. can be increased " The word " inductance " is now commonly used for the expression, " coefficient of self-induction." INDUCTION COIL 439 by increasing the number of convolutions of wire through which the lines of force link, so that to produce a high E.M.F. a large number of lines of force, linked with a large number of convolu- tions of wire, combined with rapid change of lines, is required. A common piece of apparatus in which these principles are made use of is the " induction coil " of which tens of thousands T, T 2 Fig. 269. Diagram of Induction Coil. are in daily use for motor-car ignitions, X-ray work, wireless telegraphy, etc. A diagrammatic view of a simple form of induction coil is given in Fig. 269. It has two circuits, a primary p p con- sisting of a comparatively few turns of copper wire wound round a bundle of iron wires 1 1, and a secondary coil, s s, of a very great number of turns wound outside the primary and entirely insulated from it. The primary circuit is completed through a contact breaker, A, which acts like the armature of a trembling bell, a switch c and battery B. On closing the switch the current flows round the primary coil and makes the iron core, 1 1, into an electromagnet and therefore produces a large number of lines of force linking through both primary and secondary. The core being an electromagnet, attracts the piece of iron rod or hammer H supported on the spring s, and breaks the contact between a piece of platinum on the spring and the platinum-tipped screw p, 440 PRACTICAL ELECTRICITY thus stopping the current and causing the magnetism in the core to change very rapidly. This produces a high E.M.F. in both coils, but especially in the secondary circuit, which has a large number of turns. Transient E.M.F. 's of tens of thousands of volts can be produced in this way, between the secondary terminals, TJ and T 2 , using a battery whose E.M.F is only a few T 2 Fig. 270. Marconi zo-inch Induction Coil. volts, say 2 to 10, and on this account the secondary winding and secondary terminals must be exceptionally well insulated. The action of an induction coil is improved by shunting the break with a condenser K, as shown dotted in Fig. 269, for by this means the sparking at the platinum contacts is lessened, and the current stopped more quickly. Induction coils are often provided with rocking commutators by which the direction of the primary can be reversed without altering any wires. One is seen at c, Fig. 270, which represents a form of Marconi coil much used for Wireless Telegraphy. The reason for using a bundle of iron wires as the core instead of a solid iron rod, is to obtain quicker magnetisation and demagneti- sation of the core. If the core were solid, the change of magnetic flux which occurs on making or breaking the circuit would induce electric currents in the material of the core, the direction of which would, by Lenz's Law, oppose, and therefore delay the change. These currents would flow in planes at right angles to the axis INDUCED CURRENTS 441 of the core, and by using a bundle of ,wires the resistance of the current paths in these directions is enormously increased, and the eddy currents thereby prevented. Another advantage is also obtained, viz. that heating of the core due to eddy currents is practically eliminated, and a considerable waste of energy avoided. 197. Induction of Currents in Parallel Wires. If we have two wires near each other, and a current is started in one of them, there will be an induced E.M.F. in the adjacent wire, and if the circuit of this wire is closed, a current will, in general, flow in this circuit. The induced current is an inverse one on starting or increasing the primary current, and direct when the current is stopped or decreased in strength. This phenomenon led to considerable inconvenience in telegraph and telephone lines which ran side by side for long distances, until means were taken to reduce the effect by twisting or crossing the wires at intervals, so that the mutual induction between the circuits was positive in some parts and negative in others. Example 187. What would be the average E.M.F. generated in the earth inductor described in Section 190 supposing it to be turned through 180, about a horizontal axis, in T \j of a second. Answer. The change of linkage in T X Q of a second is 2 UnA (see Section 190) .'. the average rate of change = 10 x 2 UnA, and this equals the E.M.F. in C.G.S. units. .'.Average E.M.F. =10x2x0-447x100x1000 c.G.s. units. =8-94 X io 5 c.G.s. units, and dividing by io 8 to bring it to volts, since I volt is eaual to io 8 C.G.S. units of E.M.F., we have Average E.M.F. = 8-94 X io~* volts, or =8-94 millivolts. Example 188. Calculate the mutual induction between the coils of Fig. 268, when q is inside C 2 , having given that stopping a current of 2 amps, in Cj caused a swing of 50 of the galvano- meter, and the swing due to the earth inductor, connected directly with the galvanometer, is 17. Resistance of galvano- meter, coil C 2 , and earth inductor being 1-91, 1-44, and 1-64 ohms respectively. Answer. 0-0012 henry, approximately. Example 189. Find the E.M.F. in the secondary circuit (Example 188), assuming the primary current to fall at constant rate from 2-5 amperes to zero in a ten-thousandth of a second. Answer. 30 volts. CHAPTER XI MAGNETISATION OF IRON 198. Lifting Magnets 199. Relation between Lifting Force and Current- Turns 200. Lifting Force and Flux Density 201. Magnetic Satura- tion 202. Magnetic Field produced by a Current in a Straight Conductor 203. Magneto-Motive Force 204. Testing Magnetic Properties by the Ballistic Method 205. Permeability 206. Hys- teresis of Iron 207. Remanent Magnetism, Coercive Force 208. Loss of Energy due to Hysteresis; Mechanical Analogy 209. The Magnetic Circuit ; Reluctance. 198. Lifting Magnets. One of the properties of the electric current mentioned in the early part of this volume was that an iron rod becomes magnetic when a current passes round it (Fig. 5), and in Fig. 14 a horseshoe electromagnet is shown supporting a weight. This property is now employed in many ironworks and shipyards for handling and transporting material. A properly designed electromagnet hanging from a crane hook can be used to pick up material such as bars, sheets, plates, or rails without using slings or ropes, and can be deposited in any desired position by stopping the current circulating in the electromagnet. A lifting magnet employed in this way is shown in Fig. 271, the advantages of which are that loading and unloading are done simply by starting and stopping the current, thus effecting considerable saving in time and labour. 199. Relation between Lifting Force and Current-Turns. An apparatus such as shown in Fig. 272 may be conveniently used for rinding out how the force of attraction between an electromagnet and its armature depends on the strength of the current and on the number of turns of wire em- ployed. The current may be measured by an ammeter and the force of detachment by a spring balance. To obtain consistent results great care must be taken to have the surfaces of contact as perfectly plane as possible, and that the armature is put on in exactly the same way each time. In carrying out a series of tests on the apparatus shown in Fig. 272, the numbers given 442 LIFTING MAGNETS 443 Fig. 271. Witton Kramer Magnet Lifting Pig Iron, in the following table were obtained by students of the City Guilds College. They are plotted in a curve in Fig. 273. Strength of Current in Amperes. Magnetic Pull in Pounds. 0-3 0-25 0-5 1-5 0-6 3'5 0-8 5'0 i-o 6-9 i-3 975 1-5 12-25 2-0 1575 2'5 20-0 From the shape of the curve we may conclude that when the current is small, the pull increases more rapidly than the current, and when the current is large the pull increases more slowly than the current, for the curve tends to bend over to the right. 444 PRACTICAL ELECTRICITY The coils on the iron in Fig. 272 are wound in twelve sections, numbered i to 12, each of 100 turns, connected with the mercury Fig. 272. Apparatus for measuring Magnetic Pull, cups shown at 1 1', 2 2', etc., Fig. 2720, and the coils may be joined in series or parallel, as indicated in Figs. 2720 and 2726 respect- ively, or partly in series and partly in parallel as in Fig. 272^, which shows three coils in series and four in parallel. With the coils all in series, Fig. 2720, a current passing from T to x' will pass 1,200 times round the iron core, whereas T' *%nrmnn Fig. 2720. Mercury Board. All Coils in Series. when arranged as in Fig. 272^, it will go 100 times round, and in Fig. 2720, 300 times. Experiment shows that to produce a certain pull the current required when the coils LOWER DIVISION MAGNETIC PULL 445 are all in parallel is twelve times as great as that necessary when the coils are in series, and we are thus led to the conclusion that the pull depends on the product of the current and the number of times it passes round the core. In other words, the pull depends on current x turns, and, as current is usually measured T' * */* * 4 * * t * * I2< Fig. 272*. Mercury Board. All Coils in Parallel. in amperes, we may say that the pull depends on the ampere- turns ; the composite word " ampere-turns " meaning amperes multiplied by turns. From this it will be understood that the winding of the magnet may be either a large number of turns of thin wire for carrying a small current, or a small number of turns of thick wire for carrying a large current, depending on the source of current available. Further, from the shape of the curve in Fig. 273, coupled with the fact that the electric power spent in heating a given winding of the magnet varies as the square of the current, it is evident that the pull per watt expended diminishes rapidly when the pull becomes large. The efficiency of the magnet as a lifting agent , efficiency being measured by pull per watt expended, goes down T? Fig. 272C. Mercury Board, showing 3 coils in Series and 4 in Parallel. after a certain excitation (measured in ampere-turns) is reached, so we are led to enquire whether the pull can be augmented in any way other than by increasing ampere-turns. An obvious thing to try is to increase the thickness of the iron core, and thereby increase the area of cross -section of the magnetic material ; and another is to try cores of different lengths. Experiments carried out on these lines prove that for a given length of core and a 446 PRACTICAL ELECTRICITY given number of ampere -turns, the pull is proportional to the area of the core, and that for different lengths of core of given cross- section, and with a given number of ampere-turns, the pull diminishes as the length increases. We are thus led to the con- clusion that the core of a lifting magnet should be short and of large cross-section. zo 15 I 10 E I 5 A 0-5 I-o 1-5 2-0 Current in amperes Fig. 273. Relation between Magnetic Pull and Current. 2-5 200, Lifting Force and Flux Density. We may now enquire into the relation that exists between magnetic pull p, and flux density B. From analogy with electrostatic attraction, page 82, we may surmise that the force will be proportional to the square of the flux density, and this is found by experiment to be the case. The apparatus described in the preceding section (Fig. 272) may be used to prove this statement, for by winding a few turns of fine wire round the poles of the magnet, and connecting this coil with a ballistic galvanometer, the change of flux caused by reversing the current in the main winding can be found. As the flux is reversed by reversing the current, the actual flux is half the change produced by reversal. On plotting the values of half the change of flux measured ballistically, and the square root of the pull per pole produced by the corre- sponding current, the points lie approximately on a straight line passing through the origin, Fig. 274.* Hence v p : : ft P :: B* * In this curve half the change of flux is divided by the area of the iron, so as to get flux density. LAW OF MAGNETIC PULL 447 Combining with this the statement made in the last section, that pull is proportional to the area a of the core, we have or p = B 2 A, x a constant. If, therefore, we can determine the value of this constant, we shall have an important formula relating to lifting magnets. If we measure the area of the cores in the electromagnet in -3 6 O 2 4 6 3 IO 12 Flux densiby in bhousa/nds Fig. 274. Relation between *J Pull and Flux Density. Fig. 272, then by aid of the curve in Fig. 274 an approximate value of the constant can be found. The actual area of each core is 6.74 square centimetre, and from Fig. 274 we get 0-00037 Bt Expressed in dynes, we get , _ 453'6 x 981 y 3-7 ~*~" and as the area A of the two cores is 1-48 square centimetres, p = 0-041 B 2 x 1-48 dynes, = 0-041 B 2 A. This does not differ greatly from the theoretical value I . ( = 0-0398) given below ; in fact, the agreement is within the 448 PRACTICAL ELECTRICITY possible error of experiment, which, in pull tests, is fairly large. That the pull between two surfaces of area A, over which the flux density is B, may be expressed by the formula B*A p (dynes) = - , can be seen from the following considerations : Let L M and L' M', Fig. 275, represent sections of the surface of poles s and N, between which a magnetic flux of density B lines per square centimetre exists. Imagine the surfaces separated by an air gap of in- finitesimal length,shown M 1 much magnified in the figure. A unit pole placed in the middle of M the gap would experi- ence a force of B dynes, for there are B lines per square centimetre, Now the unit S Fig. 275. and the strength of the field is therefore B. pole (unit quantity of magnetism) will be repelled by the pole N just as much as it is attracted by s, and as the sum of these forces is B dynes, we may regard it as being attracted by s r> with a force equal to dynes ; and as the lines of force in the gap will be parallel to each other, this force will be the same whether the unit pole is at the middle or not. Consider now a small surface of area a' on the face of N. The number of lines emanat- ing from this area will be B a', and, as unit pole (unit quantity of magnetism) emits 4 K lines, the quantity of magnetism on this area equals Ba' and the force exerted on the magnetism on this area by the pole s will be an attraction of Ba' B . dynes, J i.e., - dynes. O 7T This is true of every small area* of the pole N. * For parts very near the edges this will not be exactly true, but when the gap is infinitesimal the error introduced by these parts will be inappreciable. CALCULATION OF PULL 449 So the force for the .whole area A is 1 given by B 2 or p= ^ dynes, (115) O 7T = - kilogrammes weight approximately, = 4-05 B*A io- 8 (116) If the area of the pole be expressed in square inches instead of square centimetres, B still being expressed in lines per square 12,000 10,000 8,000 4,000 2,OOO o O-5 i-o 1-5 2-0 z-z Current in amperes Fig. 276. Relation between Flux Density and Current. centimetre, we have ^(inlbs.) = 577# 2 .4"io- 8 . When B = 10,000 the formula becomes (117) so that the magnetic pull per square inch of surface is roughly half a cwt. when the flux density is 10,000 C.G.S. lines per square centimetre. As iron cannot readily be magnetised to a higher flux density than 20,000, we may say that the maximum pull per square inch is about two cwts. 2 D 450 PRACTICAL ELECTRICITY 201. Magnetic Saturation. Having now obtained a relation between pull, flux density and area of polar surface, we can replot the curve of Fig. 273 in terms of flux density instead of pull. This has been done in Fig. 276, from which we see that when the current is large, a given increase in current produces only a small increase in the flux density. The effect is more Fig. 377. Iron Filings diagram, showing lines of force around a straight wire, carrying a current. marked in Fig. 284, where the flux density is carried to a higher value. This phenomenon is described as " magnetic saturation." for the greater the flux density is, the greater the increase in exciting current required to produce a given change of flux ; and the curve between B and / becomes nearly parallel to the current axis. This fact is of great importance in electrical engineering, as it seriously limits the flux densities that can be economically employed in practice. The flux density at which iron becomes practically saturated differs with different specimens, but as a rough rule we may say that the values lie between 10,000 and 21,000 lines per square centimetre, the lower value being for cast iron, and the higher for wrought iron or mild steel. 202. Magnetic Field produced by Current in a Straight Con- ductor. From the well-known fact that a small magnet tends to set itself at right angles to a wire conveying a current, and MAGNETIC FIELD from considerations of symmetry, we may conclude that the lines of force produced are concentric circles with the axis of the wire as centre. This can also be shown experimentally by mapping out the field either with iron filings, Fig. 277, or by the Fig. 278. Lines of Force (Circles) and Equipotential Surfaces (Planes), due to long straight current of 7-95 i.e., 1^ amperes. compass needle method. Equipotential surfaces being every- where perpendicular to the lines of force, they will, in this case, be planes containing the axis of the wire and at equal angles apart, Fig. 278. As the work done in conveying unit pole from one equipotential surface to an adjacent one is of fixed amount, it follows that the magnetic force varies inversely as the distance from the axis of the wire, because the distance between adjacent equipotential surfaces measured along a line of force is pro- portional to the distances from the axis. Further, Work = force X distance, and as the work is constant, the force must vary inversely as distance. 452 PRACTICAL ELECTRICITY From the definition of current strength stated in Section 8, the magnetic force is proportional to the current, so at a point at distance c centimetres from a straight current, we have or Z, N, R where k is a constant. To find the value of the constant k, suppose unit pole to be placed at the point P, Fig. 279, distance c centi- metres from the long wire z l Z 2 carry- ing a current of / amperes, and consider the force exerted on the pole by a very short length M N of the current, which subtends a very small angle M p N at P. The current may be supposed to be resolved into two components, one along and the other at right angles to P Q, the point Q being at the middle of M N. Only the latter component will exert a force at P, and this force will be normal to the plane containing Z 1 Z 2 and P, and its magnitude is given by 10 PQ 2 Draw a quarter of a circle O Q 2 R Fig. 279. calculation of Magnetic with centre at P, and a line P R in Parallel to oz, ; we may write / _ / MiN t J IO I PQ M 2 N 2 = 10 I I * If IO I IO i PO 2 - 2 M 3 N 3 ; PO I PO M 2 N 2 COS 0, V P Q 2 = where M 3 N 3 is the projection of M 2 N 2 on p R, and p o = c. FIELD OF STRAIGHT CURRENT 453 From this we see that the force exerted on unit pole at P by a length M N is equal to -- - multiplied by the length M 3 N 3 derived from M N as shown in Fig. 279. A similar construction can be used for any part of the wire o z lt and as all the forces are perpendicular to the plane of the paper, to get the total force we add them together. Denoting by F l the total force due to the part o z l (supposed to be very long), we have but the sum of all the parts M 3 N 3 , etc., will equal p R when o Zj is very long, and therefore will equal p o. Hence F, = - . p o 10 c' 2 IOC 2 I c, TO C For the part o Z 2 of the wire, the force will be equal to that produced by o z lf F - 7 2 ~i^' and the force due to the whole long wire z l Z 2 is the sum of Fj and F 2 , 2 The constant k is therefore , and as the magnetic force exerted on unit pole is taken as the measure of the strength of a magnetic field (page 36) we learn that the strength of field at a distance c centimetres from a long straight wire carrying a current of I amperes is equal to IOC If the current be expressed in C.G.S. units, then we have F=^l. (119) 454 PRACTICAL ELECTRICITY 203. Magneto-Motive Force. The magnetic force at a distance 27 c from a long straight wire being , as shown above, the work io o done on unit pole in moving once round the wire at a distance c from it will be 2! X 2 7T C, IO C for the force will be the same at every point of the path and the length of the path is 2 TT c. Fig. 280. Lines of Force around a Straight Wire carrying 12? amperes. XT ?! 4 7T 7 NOW X 2 7T C = , IO C TO which is independent of c. Hence we see that the work done on unit pole in travelling once round a long conductor conveying a current 7 amperes, is the same whatever the path followed, and is equal to -- . 7 . (120) MAGNETO-MOTIVE FORGE 455 The pole may be moved along circular paths of radii o P X , o P 2 , o P 3 , Fig. 280, or along the irregular path, p l R s P t or any other path whatever ; the work done will be exactly the same if the initial and final positions of the pole lie in the same radial plane, and the path passes round the wire once only. This is a 'most important result. The same conclusion follows from a consideration of the work done when a conductor carrying current /' (C.G.S. units) cuts $ lines of force (see Section 550). There it was shown that W = I'* ergs. Now if unit pole moves once round a very long wire, or the wire moves once round the pole, the whole of the lines of force eman- ating from the pole will cut the wire, and if the wire be conveying current /' the work SS' tlm^with' Sciiil done will be /' *. But for unit pole * = 4 TT (Section 24), W = 4 TT /', ergs per unit pole, = 4 TC , ergs per unit pole. Now this is true not only of a long straight wire but of any closed circuit conveying a current /, for if unit pole is moved through the circuit to its starting-point, all the lines of force emanating from the pole will have cut the circuit, and the work done will be I' $ as before, and therefore for unit pole IO If the circuit has several convolutions in it, Fig. 281, and the path of the pole links once with one or more of them (say s), the work done will be s times as great, so in this case TT7 4 TT s I . W = - , ergs per unit pole, (121) s / being the total current through the closed curve which forms the path of the pole. Now in the electric circuit, Fig. 187, the work done when unit quantity of electricity passes once round the circuit is called the electromotive force in that circuit, and by analogy the name magneto-motive force* is given to the magnitude - - , the * Magneto-motive force is often written M.M.F. for brevity. 456 PRACTICAL ELECTRICITY work done on unit quantity of magnetism (unit pole) in passing once round a path through which a current / links s times. Magneto-motive force (M.M.F.) is a quantity of fundamental importance in magnetic work, just like E.M.F. in electrical prob- lems, and its meaning should be thoroughly mastered by the student. The name is not a happy one, for M.M.F. is'not really a force, but a line integral of the force acting on unit pole, i.e. Z force X distance, or work per unit pole. Fig. 282. Iron Ring Wound for Ballistic Tests; Magneto-motive force is sometimes called Magnetic potential, and magnetic lines of force (magnetic flux) pass between points which have a difference of magnetic potential between them, just as electric currents tend to flow between points whose electric potentials differ. Difference of magnetic potential there- fore causes magnetic flux (lines of force), whilst difference of electrical potential (P.D.) tends to cause a flow of electric current. Current actually flows when the two points considered are joined by a conductor, but does not flow if no conducting path exists. On the other hand, magnetic lines of force always exist where a difference of magnetic potential exists, for no substance which acts as a magnetic insulator has yet been discovered. Some substances, however, such as iron, nickel and cobalt, allow magnetic flux to pass through them much more readily than others, and are, in consequence, regarded as good magnetic conductors. But there is this great difference between electric conductors and so called magnetic conductors, viz. that heat is generated, and therefore power wasted, in an electric conductor whenever electric current passes through it, whereas a constant magnetic flux may pass through a substance without causing a TESTING IRON 457 waste of energy. Nevertheless, the similarity between the two phenomena of electric current and magnetic flux has led to the conception of the magnetic circuit, the laws of which are closely related to those of the electric circuit. 204. Testing Magnetic Properties of Iron by the Ballistic Method. If we take a uniform ring of iron and wind it uniformly with a single layer of insulated wire of s, turns, Fig. 282, we have an arrangement in which the M.M.F. is easily calculated, for when a current I is passing, 47TSJ/ M.M.F. = 10 and the M.M.F. per unit length of iron 10 / ' where / is the mean length of the lines of force in the iron in centimetres. Fig. 283. Primary and Secondary Circuits on Iron Ring. Now M.M.F. per unit length is, in such an arrangement as just described, equal to the magnetic force,* or strength of field inside the winding, and this is called the " magnetising force " because it is that which causes magnetic flux in the iron. It is customary to use the letter H to represent the magnetising force, so we may write fi = 4^f (i 22 ) 10 / and this is often expressed in words as H = - times the ampere turns per centimetre length. If another winding, say of s 2 turns, has been put on the ring below the one above mentioned, we can connect this with a ballistic galvanometer G, and resistance box R, as in Fig. 283. In this .figure, to avoid confusion between the two windings, they * This follows from the fact that M.M.F. = S magnetic force x distance. PRACTICAL ELECTRICITY are shown on separate halves of the ring, but it should be under- stood that the winding of Sj turns, called the primary winding, covers the whole ring, and it is advisable, although not essential, that the other winding of s 2 turns, called the secondary winding, should do so too. In both figures, c represents a "rocking commutator," the "rocker" of which resembles that shown at B in Fig. 286. When placed to the right, the current passes in the direction shown by the arrows, Fig. 282, and when turned over to the left-hand side the current round the winding of the ring is reversed, whilst its direction in the ammeter circuit remains unchanged. When a current flows through the primary P, Fig. 283, mag- netic flux passes round the iron ring, and links with the secondary winding s, and if the primary current be reversed, by means of the commutator c, the flux is reversed, and a quantity of electricity, TV 2 BAs 2 Q = j; g = -ft g- coulombs, (see Section 189), where B is the flux density in the iron, A the area of cross-section of the iron, and R the total resistance' of the secondary circuit, will pass round the secondary circuit, if the key K be closed when the reversal occurs. The multiplier 2 arises from the reversal of the flux. As s 2 is known and A and R can be measured, the above equation enables B to be calculated when the constant of the ballistic galvanometer is known. Assuming for convenience that a reflecting galvanometer is used, and that its constant is k', we have (Section 149), Q = k's', where s' is the first swing, and therefore p _ k'Rio* , LJ - - - S . Now for a given arrangement, the quantity - is a constant, and may be calculated once for all and called k", the formula then reducing to B = k"s r . A series of observations may be made by using different primary currents, starting with small values and then increasing to large ones, the corresponding values of H and B being calculated from 4?r s l n - 2 , 10 / and B = k"s', respectively. Plotting H and B on squared paper we get a curve such as is B-H AND B-M CURVES 459 O 2 10 M 20 30 Values of H Fig, 284. Magnetisation Curve for soft Iron Ring. given in Fig. 284, which represents observations made on a wound iron ring, having the following constants : A = 1-974 square centimetres, s i = 145, s 2 = 50, / = 31-4 centimetres. I5poo JO 'in I x 10,000 o 5,000 500 1,000 Values of jju Fig. 285. Permeability of soft Iron Ring. 2,000 460 PRACTICAL ELECTRICITY The secondary circuits had a resistance of 3,500 ohms, and for the ballistic galvanometer k' = 1-571 X 10 ~ 8 coulombs. From these data we get, 4-2<_145 7 g/ 10 x 31-4 B I s' = 27-0 s' approximately. 2 X 1-974 X 50 The rapid rise in B for values of H between i and 4 is well shown in the curve, as well as the bending over for larger values of H, due to approaching " saturation." 205. Permeability. The presence of the iron core inside the winding, Figs. 282 and 283, greatly increases the magnetic flux, for if there were only air present, the flux density would be equal to H, the magnetising force. If we draw a straight line o H through the origin o, Fig. 284, satisfying the equation B = H* the intercept M' M between o H and o x will represent the flux density which would exist in the space within the winding if no iron were present. The effect of the iron for H = OM, is therefore to increase the flux density from a tenth of MM' to MP. This increase is usually very large, and the ratio in which the flux density is increased under any given conditions by the presence of the iron, is called the " permeability " of the iron under those con- ditions, and is denoted by the Greek letter /*. Hence we have Flux density Permeability = magnetising force ' or A* = jj, (123) We may also write B = jmH. Calculating the values of fi for several points on the curve in Fig. 284 we get the results plotted in Fig. 285, which show that as B increases from zero upwards, jut. first increases and then decreases, reaching, in the particular specimen here dealt with, a maximum value of 2,360 when B = 8000. 206. Hysteresis of Iron. The ballistic method of investigating the magnetic properties of iron, although one of the most accurate, is more troublesome, and perhaps less easy for beginners to under- stand, than the magnetometer method, described below, which is suitable for specimens in the form of wire. The wire i is bent into the form of a long hairpin, the legs of which pass through two long thin coils or solenoids Cj c lf C 2 C 2 , Fig. 286, into two soft * In the actual figure the vertical ordinates of the line o H are made ten times too large, otherwise o H would not be distinguishable from o x. HYSTERESIS OF IRON 461 iron spheres Sj S 2 . A short magnetic needle, with pointer attached, is suspended at N in the plane of the coils, and equidistant from the spheres S A and S 2 , and the apparatus is placed so that the solenoids lie in the magnetic meridian. When the iron wire is re- moved the pointer stands at zero on the scale, but when the wire is replaced and magnetised by passing current through the magnet- ising coils Cj C 2 , the needle is deflected. The tangent of the deflection is an approximate measure of the flux emanating from the spheres, and, therefore, of the flux through the iron wire. To Fig. 286. Simple Apparatus for Testing Hysteresis. simplify the experiment, a set of resistances with mercury cups is provided, whereby currents of known values pass through the magnetising coils, when two storage cells (E.M.F. 4 volts) are joined to the terminals TJ T 2 , as described in Section 193. Placing the copper bridge pieces p successively between the middle mercury cup o and the holes marked abode, the corre- sponding currents are 0-2, 0-4, 0-6, 0-8, i-o ampere. Having first demagnetised the iron, say by heating it to redness in a flame, the wire is put in position, a small current 0-2 ampere passed through the coils by inserting the copper bridge piece p in the holes o and a, and the steady deflection of the magnetometer- needle observed. The current is next increased to 0-4 ampere by placing a bridge piece between o and b without removing the first one, and the corresponding deflection noted. The object of this procedure is to ensure that the current shall not decrease between the two readings. Further increments of current can be obtained in a similar manner, until the maximum value i-o ampere is reached. Plotting the results we obtain a curve o c, Fig. 287, resembling the lower part of Fig. 284. If now we decrease the current from i-o to 0-8 ampere, the deflection will be found to be larger than that produced by the 462 PRACTICAL ELECTRICITY same current in the preceding set, and from a series of obser- vations with gradually decreasing currents from 0-8 to zero, the curve c D, Fig. 287, is obtained instead of c o. This shows a very important property of iron, viz. that the flux produced by decreasing values of current is greater than that produced by the same current when increasing ; in other words, the flux produced by a given magnetising force depends on whether the force has risen or fallen to the value in question, i.e. it depends on the previous history of the iron. The name Hysteresis * has been given to this phenomenon. After bringing the iron to the state represented by the point D, Figs. 287 and 288, the direction in which current is passed through 5 _ / the magnetising coil may be re- versed by rocking the commutator B, f Fig. 286, from left to right in the cups at c, and inserting a bridge piece at a, thus causing a current of- 0-2 ampere to pass through G! and C 2 . This gives a Currenb in amperes i-2point between D and E, Fig. 288, Fig. 287. Curve showing effect of Previous Historv. and by increasing the current step by step as before, up to its maximum negative value i-o, points on the curve D E F are obtained, so that by changing the current continuously from -{- i-o ampere to i-o ampere we get the curve c D E F. Reducing the current from i-o to zero gives points on F G, Fig. 288, and by rocking the commutator from right to left, and increasing the current step by step from zero to + i-o ampere, the curve G K c results. We have now subjected the iron to a complete cycle of magnetising force by reducing the current from + i-o ampere to zero, from zero to i-o ampere, then i-o ampere to zero, and from zero to + i-o ampere, its original value, the result of which is the curve c D E F G K c, which encloses an appreciable area, and is called the " Hysteresis Loop." If the cycle of operation be repeated, points lying on the curve c D E F G K c will be again obtained, provided the maximum * Meaning lag, f Rocker shown resting on board at B. HYSTERESIS LOOP 463 value of the current remains unchanged, thus showing that the loop represents a Definite property of the particular specimen of iron. Considering the points p and p' on the curve we notice that for the same value of the magnetising force, -f- o L, there are two values of the flux density, viz. + L P and L P', the positive values occurring after the irtm has been previously magnetised more strongly in a -f direc- tion, and the negative value after its being magnetised in a nega- tive direction, the arrowheads on the curve showing the direction in which the cycle of operation has been carried out. The points Q and Q' show that the same flux density -j- M Q may result from two widely different magnetising forces, viz. + o M and - o M' respectively. These facts show that, unlike an electrical circuit (in which a definite electromotive force always causes a definite current to flow), the flux in a magnetic circuit depends not only on the magneto -motive force existing at the time, but also on that which existed previously. On this account the calculations relating to magnetism are some- what more complicated than analogous electrical problems. Loops such as that shown in Fig. 288 can only be obtained when the length of the iron is very great compared with its cross-section, or when the iron is formed into practically closed rings. 207. Remanent Magnetism : Coercive Force. In the hysteresis loop, Fig. 288, the points D and G and E and K are of consider - -B Fig. 288. Hysteresis Loop. 464 PRACTICAL ELECTRICITY able importance. The length o D represents the flux density in the iron when the magnetising force has been reduced to zero from the value existing at c, and the name " residual magnetism'' or " remanent magnetism" is given to this flux density, whilst that represented by N c is spoken of as " temporary magnetism.'" Similarly the points F and G show the temporary and residual magnetism respectively, when magnetised in the negative direction. At the point E the flux density is zero, but the magnetic or magnetising force has a negative value represented by the length OE, so we see that after the iron has been magnetised in one direction it is necessary to apply a magnetising force in the other direction in order to reduce the magnetism to zero. Similarly after the iron has been magnetised in a negative direction, represented by the point F, a positive magnetising force OK is required to remove the negative magnetism previously existing. The magnitude of this force is a criterion of the magnetic " hardness " of the iron, and the name " coercive force " is given to the lengths o E and o K when expressed in terms of H. In soft iron the coercive force is small, whereas in hard iron, and more especially in hardened steel, it has large values. On the other hand, soft iron when forming continuous magnetic circuits, generally has a larger ratio of remanent to temporary magnetism than hard iron or steel, but a comparatively small demagnetising force will reduce the residual magnetism to a very small value. For the construction of permanent magnets, a material having high coercive force and large residual magnetism is desirable, but of the two properties, the former, high coercive force, is the more essential, otherwise comparatively weak magnetising forces, such as the earth's field, will greatly affect the mag- netism of a magnet of bar or horse-shoe form. Hardened steel is the material hitherto found most suitable for use as permanent magnets, and of the many kinds of steel tried, Tungsten steel has proved most satisfactory. 208. Loss of Energy due to Hysteresis, Mechanical Analogy. The fact that when iron is subjected to a cycle of magnetising force, the magnetism lags behind the force, gives rise to a loss of energy, for the material behaves as if it were imperfectly elastic. Before showing how the loss may be calculated, it may be helpful to describe a mechanical analogue of the phenomenon. Suppose we take a piece of poor indiarubber, a commodity of very common occurrence when rubber is dear, and that we stretch it a given amount / by applying a gradually increasing force and then LOSS OF ENERGY 465 1200 allow it to contract again. It will not contract to its original length but will be somewhat longer ; the amount of contraction is therefore less than /, and as the distance through which the stretching force acted is greater than that over which the contractile force operates, the work done in stretching the rubber is greater than that done by the rubber in con- tracting. Conse- quently there is some work or energy lost in the process. If we now compress the rubber with a force equal to the stretching force previously used, and then gradually re- duce the compress- ing force to zero, the rubber will not t2OQ quite recover the length it had before compression, and energy will again be lost. From this we see that when im- perfectly elastic rubber is subjected to a cycle of opera- tions, viz. stretch- ing, release, com- pression, release, more work is done On the rubber than Fig> ^-"Calculation of Energy Loss by Hysteresis. the rubber gives out again, the difference being the loss already referred to above. When any imperfectly elastic substance is subjected to a cycle of mechanical forces, a loss of energy will occur ; this lost energy generally appears as heat in the sub- stance.* Similarly when iron or steel is subjected to a cycle of magnetic forces, a loss of energy is caused by the imperfect magnetic elasticity of the material, as shown by the hysteresis * A familiar instance occurs in the tyres of racing automobiles, which become very hot at high speeds,, 466 PRACTICAL ELECTRICITY loop, and, as a matter of fact, the area of the hysteresis loop is a measure of the energy lost during a complete cycle. To prove this statement let the closed curve c D E F G K c, Fig. 289, represent the relation between $, the total flux in, and I Si t the ampere turns acting on a specimen of iron during a com- plete cycle, and consider what occurs when the iron changes from the state represented by the point P to that represented by P', the two points being near together on the curve. The flux changes from o M to o M', whilst the ampere-turns change from o L to o L'. There is, therefore, a change of linkage with the magnetising coil of s t . M M' lines, whilst a current, whose mean value is , is flowing. Now in Section 550, we have 2Sj shown that when a conductor carrying a current /', C.G.S. units, cuts $ lines of force, the work done is given by W = I' $ ergs, so in this case the work done when the flux changes from o M to o M' is given by W= S!.MM' 10 - . M M', . MM' IO O L + O I/ 2 x 10 Now - - is the mean length of the figure PMM'P', and M M' is its height, so x M M' is equal to the area 2 of the figure, therefore w _ area of figure P M M' P' 10 Hence we see that the work done by the iron* during the change is proportional to, and may be represented by, the figure p M M' p'. By employing similar reasoning to other parts of the * That the work done by the iron and not on the iron may be seen from the consideration that the iron is being released from a strained condition, and therefore doing work against the straining force. It may also be seen from Lenz's Law, Section 188, for the change of linkage will induce a current in the magnetising circuit in a direction tending to prevent the change, i.e. it will tend to increase the magnetising current ; consequently the iron will act as a current generator, and therefore give out energy. HYSTERESIS LOSS 467 line c D, it is seen that the work done during the change of state from c to D is represented by the area of the figure c R D c. During the change of state represented by the part D E F of the loop, the flux is changing in the same direction as before, but the current is in the reverse direction. This means that work is being done on the iron, and the area of the figure D E F s D represents the amount. In the stage F G, which is like c D, the work done by the iron is given by the area F s G F, whilst for the stage G K c, the area G K c R G represents the work done on the iron. Consequently, during the whole cycle we have Work done by the iron = area CRDC + FSGF, and work done on the iron = area DEFSD + GKCRG; and as the latter is greater than the former by the area of the figure c D E F G K c, the net work done on the iron is represented by the area of the hysteresis loop. When the hysteresis curve is plotted in terms of flux and ampere turns, the area expressed in these units, divided by 10 ($ee formula 124) gives the work lost in the whole specimen in ergs. If, however, B and H be plotted, since 10 I BHIA and H = we have : - = 10 4^ IA i.e. Total loss == area of (B-H) loop x . 47U Now IA is the total volume of the magnetic material, so the loss per unit volume per cycle is given by area of (B-H) loop , . , . , . ergs per cubic centimetre. (125) 209. The Magnetic Circuit : Reluctance. In Section 203, and near the end of Section 206, the analogies between electro- motive force and magnetomotive force, and between an electric circuit and the path traversed by magnetic flux, have been mentioned. If we consider the case of a ring uniformly wound with wire, such as that shown in Fig. 282, and imagine the iron replaced by air, we have a simple case of a magnetic circuit in which the path of the flux is in air, the flux density B being equal to the magnetising force H. 468 PRACTICAL ELECTRICITY Now, in such a case TT 47T/S, H= -^, page 454- and if A be the area of cross-section of the path inside the solenoid, the total flux ***BA = HA, in this case. 10 r IO / A M.M.F. This expression is similar to Ohm's Law, written in the ordinary form of " flux " taking the place of " current," magnetomotive force that of " electromotive force," and " " the place of " resistance." A In Chapter VI., Section 97, we have shown that electrical resistance of uniform conductors is proportional to their length and inversely as their sectional area, and the fact that I _ Magnetomotive force A magnetic flux suggested the name " magnetic resistance " for (in air). As, A however, an electric current flowing through a resistance always generates heat, whilst magnetic flux can pass continuously through air without dissipation of energy, the name " magnetic resistance " has been abandoned and the word reluctance adopted. We therefore write Magnetomotive force Magnetic flux = - 5 - (126) Reluctance an expression which is sometimes called " the law of the magnetic circuit." THE MAGNETIC CIRCUIT 469 Lines of magnetic force always form closed curves, so the flux in a magnetic circuit is the same at every cross-section, just as the current in an electric circuit is the same at every cross-section. The cross-section of the magnetic path may, however, differ at different parts, just as the cross-section of an electric circuit is not necessarily uniform. In the electric circuit the total resistance is equal to the sum of the resistances of the several parts, so also in a magnetic circuit the total reluctance is the sum of the reluctances of the several parts, and to find the reluctance of a magnetic circuit the cross-section of which is not uniform, we suppose it divided into parts each of which is of nearly uniform section, calculate the reluctance of each part, and add them together. If we now suppose the iron core to be inside the coil in Fig. 282, the flux will be increased in the proportion of ^ to i, where //, is the permeability of the iron under the given conditions, for the flux density would become B, where B = /* H. The same M.M.F. would therefore produce n times the previous flux, so the pre- sence of the iron reduces the reluctance of the circuit. If, there- fore, the reluctance of an air path of length / and sectional area A be represented by , the reluctance of an iron path of the same A. dimensions will be - , where /x is the permeability. A/* Further, when the path is made up of several parts of lengths /! 1 2 1 3 , etc., and sectional areas A^A 2 A^, etc., the total reluct- ance of the whole path will be equal to *i . ^ _i_ ^ , f " " ~T~ . I M \ CLLx. where /i lf jtx 2 , /w 3 , etc., are the permeabilities of the several parts. For air /* = i, so the formula giving the magnetic flux in a circuit consisting of air and iron can be written M.M.F. A A si ** i A^i where / and A refer to the air parts, and l lt A and ^ to the iron parts of the circuit. An exactly similar formula is true for the electric circuit if the air parts be omitted,* and we write conductivities instead of * Otherwise no current would flow, as air is a very perfect electric insulator. 470 PRACTICAL ELECTRICITY permeabilities of the other parts of the circuit. There is, how- ever, this difference between the two cases, viz. that the electric conductivity of conductors does not depend on the current density, whereas the permeability of magnetic materials is greatly influenced by flux density, as will be seen from Fig. 285, Section 205. Example 190. Calculate the strength of field at points distant i, 2, 5, 10, 50, and 100 inches from a long straight conductor carrying a current of 100 amperes. Answers. 7-87, 3-94, 1-57, 0-787, 0-157 an d 0-0787. Example 191. Find the area of the poles of a lifting magnet to support one ton, assuming the flux density to be 18,000 lines per square centimetre. Answer. 77-3 sq. cm. or 12 sq. inches. Example 192. The iron of the magnet in example 191 has a mean length of 40 centimetres, and at a flux density of 18,000 the permeability is 100. How many ampere turns will be required to produce the required flux, neglecting the reluctance between the surfaces in contact ? Answer. 5,730. Example 193. An iron ring whose mean diameter is 12 cms. is uniformly wound with 200 turns of wire. Find (a) , the magneto- motive force when a current of 10 amperes flows through the winding, (b) , the current required to produce a magnetising force of 200. c.G.s. units. Answer. (a) 2,512. (b) 30 amperes. Example 194. A thin circular ring of iron 2 cms. in diameter, is threaded over a long straight wire carrying 50 amperes, and placed concentric with it. Find (i), the magnetomotive force acting on the ring, (2), the magnetising force, and (3), the flux density in the ring, assuming the permeability of the iron to be 1,300. (i) M.M.F. = 62-8 Answers. (2) H = 10 (3) B = 13,000. Example 195. If a narrow air gap 0-2 m/m wide be made across the ring in example 194, calculate the value of B in the iron, assuming the permeability of the iron to remain unaltered, and that the flux density in the air gap is the same as in the iron. Answer. The total M.M.I 7 , is x 50, i.e. 62-8, and this is spent partly in the iron and partly in the air gap. The M.M.F. EXAMPLES D in the air gap is B x length of air path, and that in the iron is x length of iron path. Taking this length as 2 n we have HT nr T B X 2 7T M.M..b. in iron = -- 1,300 .. 62-8 = B X 0-02 + B X 1,300 5 = 2,530 approx.* Example 196. Find the magnetic reluctance of a square iron bar 20 centimetres long and of 4 centimetres side, the permeability being taken as 1000. T Answer. C.G.S. units. 800 Example 197. Calculate the approximate reluctance of an air gap 2 m/m long in a magnetic circuit the cross-section of which is 10 square cms. Answer. 0-02 C.G.S. units. Example 198. Assuming the ring in Example 194 to be a cylinder i m/m thick and 15 m/m long, find its reluctance. 6-28 Answer. -- =0-0322 (approx.). o-i x 1-5 x 1300 Example 199. Find the relation of the reluctance of the air gap in Example 195 to that of the iron portion of the magnetic circuit, assuming the lines of force to be circles (i.e. spreading of lines at the gap is to be neglected) and that the permeability of the iron is 1880. _/ A in* ,. Answer. -- - = "y* = 6 approx. Example 200. In a uniformly wound magnetic circuit of uniform cross-section express the magnetic force H in terms of " ampere turns per centimetre length, "f 4nsww. -Here #=-, and as 10 / = ampere turns per c.m. length, we have H 1*256 X " ampere turns per cm. length." * Observe that the introduction of an air gap of only one-fifth of a millimetre, in an iron circuit of over 60 millimetres long, reduces the flux density from 13,000 to about 2,500. t " One ampere turns per centimetre length," is frequently used as a unit of magnetising force. " One ampere turn per half-inch length " is also a convenient approximate unit, when the length of the magnetic circuit is given in inches. 472 PRACTICAL ELECTRICITY Example 201. Find the ratio of the C.G.S. unit of magnetising force to " one ampere turn per half inch." Answer. H = 1*256 x ampere turns per cm. length, *2. H = 1-256 x X ampere turns per half inch, 2'54 = x ampere turns per half inch. I-OII Hence one C.G.S. unit of magnetising force is about I per cent- greater than " one ampere turn per half inch." APPENDIX I SHORT HISTORY OF THE ABSOLUTE UNIT OF RESISTANCE,* AND OF THE ELECTRICAL STANDARDS OF THE BOARD OF TRADE IN 1821 Sir Humphry Davy published the results of his experi- ments, proving that metals varied in their power of conducting electricity, and that this conducting power diminished as the temperature rose ; but the idea of resistance being a property of a conductor was due to Ohm, who published the mathematical proof of his famous law in 1827. The writers, however, who immediately followed Ohm did not employ a unit of resistance, but contented themselves with reducing, by calculation, the resistance of all parts of a heterogeneous circuit to a given length of some part of that circuit, so that Lenz, for example, in his paper of 1833, calls the resistance of a conductor its " reduced length." The next step, when comparing different circuits, was naturally to refer these " reduced lengths " to the length of some one standard wire, although the wire might not form part of any of the circuits under test, and to consider the resistance of unit length of this standard wire as the unit resistance : thus, we find Lenz, in 1838, stating that one foot of No. n copper wire was his " unit " of resistance a unit, however, which he appeared to have selected at random, and without any idea of suggesting that it should be used by others. In 1840 Wheatstone constructed the first instrument by which definite multiples of a resistance unit could at will be added to or subtracted from, a circuit. And in 1843 he proposed that the resistance of one foot of copper wire weighing 100 grains, which was selected with reference to the British standards of * The earlier part of this History is abstracted from a " Report to the Royal Society on the New Unit of Electrical Resistance, &c.," by the late Prof. Fleeming Jenkin, which, together with the Reports from 1862 to 1869 of the British Association Committee on Standards of Electrical Resistance, and with his Cantor Lectures, were issued by him in 1873 in the form Of a very useful book. 473 474 PRACTICAL ELECTRICITY length and weight, should constitute the standard of resist- ance. Later on other wires were proposed as units of resistance ; and, to avoid the inconvenience arising from the multiplicity of standards, Jacobi, in 1848, sent a certain copper wire to Poggendorff and to others, requesting that copies might be taken of it. For Jacobi pointed out that the mere definition of a standard of resistance in terms of the length and weight of a wire of some material was not sufficiently definite, and that good copies of a standard, even if that standard had been origin- ally chosen at random, would be more exact. Until about 1850, measurements of resistance were confined, with few exceptions, to the laboratory, but about that time underground wires, followed shortly afterwards by submarine cables, began to be employed ; and, since it was impossible to ascertain the position of a defect in such a telegraph line by inspection, electrical methods of " localising the position of a fault " by measuring the resistance of the wire between the testing station and the faulty spot, had to be developed. As early as 1847 C. F. Varley is said to have used a rough method of distance testing, while in 1850 Werner Siemens published two methods, and in 1852 Charles Bright patented a plan for deter- mining the position of a fault by the use of resistance coils. The first effect of this commercial use of resistance was to turn the " foot " of the laboratory into the " mile " : thus, the unit of resistance in England became that of a mile No. 16 copper wire ; in Germany, of a German mile of No. 8 iron wire ; and in France, of a kilometre of iron wire 4 millimetres in diameter. Next, Marie Davy and De la Rue pointed out that, as it was possible by chemical cleaning and subsequent distilling to remove practically all impurities from mercury, this metal was specially suitable for selection as a standard substance ; and in 1860 Werner Siemens constructed standards in which his unit was the resistance of a column of chemically pure mercury i metre long, i square millimetre in cross-section at a temperature of oC. The definition of the " Siemens unit " of resistance was a very simple one ; and, since mercury in a nearly pure state is not very difficult to obtain, it might be thought that the unit pro- posed by Siemens would have been finally adopted. The sim- plest way, however, to obtain a column of mercury of uniform cross-section is to place mercury in a tube of uniform bore, and the cross-section of the bore of such a tube can be most accurately determined by finding the weight of mercury that is contained in a given length, and deducing the volume from a knowledge ABSOLUTE SYSTEM OF UNITS 475 of the specific gravity of mercury. Although, then, the defini- tion of Siemens units is apparently based simply on length, cross- section, and temperature, it really depends on weight, specific gravity, and temperature. In the specimens of this unit originally issued there was an error of 2 per cent., and even in later issues an error of over one- quarter per cent, was introduced up to 1873, through Werner Siemens having adopted 13-557 as the specific gravity of mercury instead of 13-596. The labour, however, bestowed by the late Werner von Siemens on perfecting electrical measurements merits special recognition, as it materially helped in introducing strict accuracy. All the preceding units of resistance are based on the more or less arbitrary size and weight of some more or less suitable material ; but measurements of resistance can be conceived and carried out entirely without reference to the special qualities of any particular material. In 1849 Kirchhoff effected a measure- ment of this kind ; but it is to W. Weber that we owe the first distinct proposal, made in 1851, of a system of electrical and magnetic measurement in which an electrical resistance would be expressed as an absolute velocity, were " magnetic permeability " a simple numeric. Previous to this, Gauss, desiring to make precise measurements of the distribution of terrestrial magnetism, found it necessary at the outset to decide on a unit of force which, unlike the weight of a given mass, should not be affected by the position of the place at which the experiment was made, and on a magnetic pole whose strength should be independent of any molecular change in steel. He therefore devised what has since become well known as Gauss's " absolute unit of force," and the " unit magnetic pole," the former being defined as the force which, acting on unit mass, generates unit acceleration, and the latter as the pole which repels an exactly similar pole at unit distance with unit force. : - Following Gauss's nomenclature, Weber called the two systems of units to which he was led the " absolute electromagnetic " and the " absolute electrostatic " systems ; but the name " derived " would have conveyed the meaning better than " absolute," since the essence of Weber's system consisted in the various electrical and magnetic units being derived from those of length, mass, and time. As soon as the proposal of Weber appeared, W. Thomson (now Lord Kelvin) accepted and extended it by showing that the abso- lute unit of work formed part of the same system. And ten years later, at the meeting of th^ British Association in 1861, 476 PRACTICAL ELECTRICITY W. Thomson proposed that a Committee of that Association should be formed to determine the best standard of electrical resistance. This Committee, which consisted of only six names at the outset, gradually increased its numbers as it enlarged the scope of its work. A few of the members of thirty years ago*are still taking an active part in the labours of the Committee on Elec- trical Standards of to-day, but the Committee has lost by death Clerk Maxwell, Cromwell Varley, Fleeming Jenkin, Joule, Matthiessen, and others whose names are distinguished for the active part they took in the development of electrical science. The principle of the method employed by the British Associa- tion Committee in 1863 for the determination of the unit of resist- ance was, briefly, as follows : If a coil like that of a tangent galvanometer for example, c c (Fig. 62, page 91) be spun in a uniform magnetic field round a vertical axis passing through the centres of the coil and of the needle, an E.M.F. is induced in the coil, this E.M.F. reaching its maximum when the plane of the coil is parallel to the lines of force, and becoming zero when it is perpendicular to the lines of force. If, then, the coil be short- circuited a current will be induced in it, and, although the E.M.F. reverses its direction each time the plane of the coil is perpen- dicular to the lines of force, and although, therefore, as regards the coil the current flows in opposite directions during the two halves of its revolution, it flows in the same direction as regards the needle. Hence, for a uniform speed of rotation of the coil there will be a constant mean value of the deflecting force exerted on the suspended needle, and, therefore, if the time taken by the coil to make one revolution is small compared with the time of vibration of the needle, the needle will remain steadily deflected as if it were acted on by a perfectly constant deflecting force. Further, since for a given angular velocity of the coil the aver- age value of the induced current is directly proportional to the strength of the uniform magnetic field, while the controlling force exerted on the needle is also directly proportional to the strength of this magnetic field, it follows that the magnitude of the deflec- tion is independent of the field. And, as proved in Section 15, the deflection is also independent of the strength of the needle. In fact, when the equations connecting the various electric and magnetic magnitudes are written in their simplest forms, without the introduction of useless coefficients, it can be shown that, to the first degree of approximation, * Written in 1896. -T, M. BA. UNIT OF RESISTANCE 477 7T 2 r n 2 co tan a = - -- ; where d is the angular deflection produced by a coil of mean radius r, wound with n convolutions of wire, and having a resist- ance R, when spun with a uniform angular velocity 432 g at I5 <> C . In accordance with the recommendations made in Schedule D of the Report of the 1908 Conference, 3, page 505, further investigations were made in several National Laboratories, notably at the Bureau of Standards, Washington, the Laboratoire Central d'Electricite, Paris, the Physikalisch-Technischen, Reichs- anstalt, Charlottenburg, and the National Physical Laboratory, Teddington, on Resistance Standards, Silver Voltameter and Weston Cadmium Cell. For the better co-ordination of the re- searches it was agreed that the chief experimenters of the several laboratories should work together and carry out joint and separ- ate experiments on the cadmium cell and silver voltameter. Further work on the silver voltameter had been shown to be necessary on account of differences in values for the E.M.F. of cadmium cells in International volts as determined in the several laboratories. The persons entrusted with the work were as follows : 512 PRACTICAL ELECTRICITY Dr. E. B. Rosa and Dr. F. A. Wolff, of the Bureau of Standards, Washington ; Prof. F. Laporte, of the Laboratoire Central d'Electricite, Paris ; Dr. W. Jaeger, of the Physikalisch-Tech- nischen Reichsanstalt, Charlottenburg ; Mr. F. E. Smith, of the National Physical Laboratory, Teddington. About seven weeks (April 4th to May 25th, 1910) was devoted to the work, which resulted in the definite recommendation (i) that 1-0183 be taken as the E.M.F. of the Weston cell in International volts at 20 C, and (2) that further experiments on the mercury ohm, silver voltameter and standard cells were necessary before completing or changing the specifications of the London Conference (1908). In the Washington determinations (1910) the mean of the mercury ohms as realised at the Reichsanstalt and N. P. L. respectively was taken as the International ohm. These differ by about i part in 100,000, the former being the greater. The work on the silver voltameter indicated that the use of filter paper caused a slight increase in the deposit. Combining the latest absolute determination of the ohm at the National Physical Laboratory (page 499) with the Wash- ington results we get the following relation between the In- ternational units and absolute units. i International ohm = i-ooo5 2 true ohm (approx.) i ampere = 0-99988 ampere i volt = i-ooo4 volt i watt = i-ooo2 8 watt A determination of the E.M.F. of the Weston cell of great pre- cision was published in 1914 (Phil Trans., Vol. 214) by Mr. A. Norman Shaw, who used a Maxwell- Weber bipolar electro- dynamometer to measure the current, and the international ohm as resistance standard. The result obtained was 1-01831 semi-absolute volts at 20 C., the semi-absolute volt being taken as the P.D. between the 'terminals of an international ohm (Britain, America and Germany) when it is traversed by a current of one absolute ampere. REGENT DETERMINATIONS 5i3 NOTE re Silver Voltameter (New Form). In the latest and most satisfactory form of voltameter yet devised the anode, instead of being wrapped in filter paper as described on page 494, rests in a shallow glass cup, having a ground edge supported clear of the platinum bowl. A glass cylinder is ground to fit over the edge of the cup, thus forming a chamber for the anode. Before making an experiment the anode and cathode chambers are filled with pure electrolyte to the same level, and the glass cylinder raised until its lower edge is just below the surface of the liquid. After the deposit has been made the glass cylinder is lowered so as to close the anode chamber before removing the anode system from the platinum bowl. The anode is coated with electro-deposited silver before use in a determination. PRACTICAL ELECTRICITY APPENDIX II COMPARISON OF C.G.S. AND BRITISH SYSTEMS OF UNITS. FUNDAMENTAL UNITS. Unit. C.G.S. System. British System Relations between the Units. Length Mass I centimetre i gramme i foot i pound* {i cm. =0-03281 ft. = 0-3937 inch, i ft. =30-48 cms., i inch = 2-54 cms. /I gramme = 0-0022046 Ib. \i Ib. =453-6 grammes. Time i second i second * British engineers commonly take g pounds as the unit of mass (where g = 32-19 approximately), so that the weight of a pound may be used as the unit of force. DERIVED UNITS (MECHANICAL). C.G.S. System. British System. Ratios of Units. C.G.S. Unit. British Unit. British Unit. C.G.S. Unit. Velocity i cm. per second . . i foot per second 0-03281 30-48 Acceleration i per sec. i per sec. 0-03281 30-48 i' i gramme moving at i pound moving at a^ Momentum I a velocity of i cm. velocity of i foot per I 0-00007233 13,826 I per second. second. j (i dyne, g fc T of the 11 poundal (53^3 the ) weight of a gramme weight of a pound at j 0-00007233 13,826 Force 4 at London, approx. London (approx.) i pound, the weight of } ( Ditto a pound at London > 0-000002247 445,000 (approx.) Moment of 1* force about axis or of a i dyne centimetre (i poundal foot i pound footf 0-000002373 0-00000007372 421,400 13,560,000 couple . Work or energy j- i cm. dyne or i erg 3 f i foot poundal. . ( i foot pound . . 0-000002373 0-00000007372 421,400 13,560,000 Powee 1i erg per second i watt, or 10 7 ergs per second J i foot poundal per sec. \ i foot pound per sec. 1 1 foot poundal per sec. i foot pound per sec. . . 1 i horse-power, or \ 33,000 foot pounds per > \ minute, 1 0-000002373 0-00000007372 0-000002373 0-00000007372 0-001340 421,400 13,560,000 421,400 13,560,000 746 t The " pound-inch " or " inch-pound " is frequently employed as the unit of moment of z force about an axis, or moment of a couple. $ Another unit of work frequently used by engineers is the kilogramme-metre or metre- kilogramme, whose relations with the foot-pound are as follows : i metre-kilogramme = 7*233 foot-pounds, i foot-pound = 0*13826 metre-kilogrammes, also i horse-power hour = 1,980,000 foot pounds. ,, ,. ,, = 273,700 metre-kilogrammes, and i kilowatt hour = 2,654,000 foot pounds. ,, = 366,900 metre-kilogrammes. RELATIONS OF UNITS APPENDIX III RELATIONS BETWEEN THE PRACTICAL, C.G.S. ELECTROMAGNETIC, AND C.G.S. ELECTROSTATIC UNITS. Unit of Practical Unit. C G.S. Electro- magnetic Unit. C G.S. Electrostatic Unit. Current i ampere 10 amperes I , Y ^ x 10 9 am P ere l a PP rox -J Resistance i ohm j ohm 9 x 10 n ohms P.D., or ) E.M.F. } Quantity i volt i coulomb a volt 10 8 10 coulombs 3 volts 3 xio ^ouluuib Energy i joule i -. joule I0 7 J 7o^ j ule Power i watt -L watt I0 7 7 watt Capacity i farad 10 farads i f A ^ 9X 10 u Inductance i henry i 9Xio n henry APPENDIX IV SPECIFIC GRAVITIES, SPECIFIC RESISTANCES, AND SPECIFIC CONDUCTIVITIES OF MIXTURES OF PURE SULPHURIC ACID AND DISTILLED WATER. Percentage H,S0 4 by weight. Percentage H.SO, by volume. Specific gravity at i8C. Specific resistance per centimetre cube (ohms). Specific conductiv- ity per centimetre cube at 18 C. 5 2-7 3 4-8! 0-208 10 5-7 07 2-56 0-391 15 8-7 II 84 0-543 20 I2-O M 53 0-653 25 I5'3 18 40 0-717 30 18-9 22 35 0-739 35 22-6 26 39 0-721 40 26-6 31 47 0-680 50 35-2 4 85 0-540 60 44'9 50 2-68 o-373 70 55'9 I'6l 4-63 0-216 80 68-3 i-73 9-00 O-III 90 83 1-82 9-26 0-108 IOO IOO 1*4 52-6 0-019 516 PRACTICAL ELECTRICITY APPENDIX V SHOWING THE DIMENSIONS OF WIRES ACCORDING TO THE APPROXIMATE RELATIONS BETWEEN LENGTHS, WIRE AT A i q CO DIAMETER. AREA. LENGTH AND RESISTANCE. 0.4.WNN | S.W.G. NO. | Mils.* Milli- metres. loooths ol a sq. inch.f Square millimetres. Feet per ohm. Metres per ohm. Ohms per 1000 feet. Ohms per kilometre. i 2 3 4 5 300 276 252 232 212 7-62 7-01 6-40 5-89 5-38 70-6 59'8 50-0 4?-3 35-3 45-6 38-6 32-2 27'3 22-8 8870 7500 6250 5300 4420 2700 2290 1900 1610 1340 0-113 0-133 0-160 0-189 0-226 0-370 0-437 0-526 0-621 0-746 6 I 9 10 192 176 160 144 128 4-88 4-47 4-06 3-66 3-25 29-0 24-3 2O-I 16-3 12-9 18-7 15-7 13-0 10-5 8-30 3640 3050 2520 2050 1620 IIOO 930 770 621 49i 0-275 0-327 0-397 0-487 0-617 0-909 1-08 1-30 1-61 2-04 6 7 8 9 IO ii 12 13 14 15 116 104 Io 72 2-95 2-64 2'34 2-03 1-83 10-6 8-5 6-65 5-03 4-06 6-82 5-48 4-29 3'24 2-63 1320 1060 832 603 5io 405 325 254 192 156 0-758 0-943 1-20 1-66 1-96 2-47 3-08 3'94 5-21 6-45 ii 12 13 14 15 16 17 18 19 20 % 48 8 1-63 1-42 1-22 roi6 0-914 3-21 2-46 1-81 1-26 1-02 2-08 X'59 1-17 0-8 1 1 0-657 404 309 226 158 128 123 94-0 69-1 48-0 38-9 2-47 3-24 4-42 6-32 7-8i 8-13 10-6 14-5 20-8 25-7 16 11 19 20 21 22 23 24 25 ii 24 22 20 0-813 0-711 0-610 o-559 0-508 0-804 O-6l5 0-452 0-380 0-314 0-519 0-397 0-292 0-245 0-203 IOI 76-2 56-8 47-6 39*4 30-7 23-6 17-3 I4'5 I2-O 9-90 I3'i 17-6 21-0 25-4 32-6 42-4 57-8 69-0 83-3 21 22 23 24 25 26 27 28 29 30 18 l6- 4 14-8 13-6 12-4 0-457 0-417 0-376 0-345 0-315 0-254 O-2II 0-I72 0-145 0-120 0-164 0-136 0-111 0-0937 0-0779 3i'9 26-5 21-6 18-2 I5'2 9'73 7-89 6-59 5'53 4-61 31-3 37-7 46-3 54'9 65-8 103 127 152 181 217 26 27 28 29 30 31 32 33 34 35 n-6 10-8 10-0 r; 0-295 0-274 0-254 0-234 0-213 0-106 0-092 0-0785 0-0665 0-0554 0-0682 0-0591 0-0507 0-0429 0-0358 I3'2 n-7 9-85 8-32 6-95 4-04 3'50 3-00 2-54 2-II 75-8 85-5 JOI I2O 144 248 286 333 394 479 31 32 33 34 35 36 I? 39 40 36 H 39 4 7-6 6-8 6-0 n 0-193 0-173 0-152 0-132 O-I22 0-0452 0-0363 0-0282 O-02I2 O'OlSl 0-0293 0-0234 0-0182 0-0137 0-0117 5'7o 4'55 3'55 2-66 2-26 i'73 1-48 1-08 0-8 1 0-69 175 220 282 376 442 578 676 926 1230 1450 41 4* 43 44 45 ~4*~ S 49 50 4'4 g 3 O-II2 O-IO2 0-0914 0-0813 O-O7II 0-0152 0-0126 0-OI02 0-0080 0-O06I5 0-00982 0-00811 0-00656 0-00519 0-00397 1-90 1-58 1-28 i-oi 0-762 0-58 0-48 0-389 0-307 0-236 526 633 78i 990 1310 1720 2080 2570 3260 4240 4i 42 43 44 45 2'4 2-0 1-6 fa I'D 0-0610 0-0508 0-0406 0-0305 0-0254 O-OO452 O-003I4 O-O02OI O-OOII3 0-00078 0-00292 0-00203 0-00129 0-00073 0-00051 0-568 0*394 0-252 0-142 0-098 0-173 O-I2O oo77 0-0432 0-0300 1760 2640 3970 7040 IOIOO 578o 8330 13000 23100 33300 46 47 48 49 50 * A mil is a thousandth of an inch (o-ooi inch), and eauals ^ of a millimetre (approx.). f This column also shows the carrying capacity of the wires on the basi? of 1000 amperes pec square- inch. COPPER WIRE TABLE 517 APPENDIX V (continued}. BRITISH STANDARD WIRE GAUGE (S.W.G.) AS WELL AS THE RESISTANCES, AND WEIGHTS OF PURE COPPER TEMPERATURE OF 15 C. C/j I 2 3 4 5 RESISTANCE AND WEIGHT. WEIGHT AND LENGTH. 1' o" c/i Ohms per pound.} Pounds per ohm. Pounds per 1000 feet. Grammes per metre. Feet per pound. Metres per gramme. 0-000417 0-000581 0-000833 0-00116 0-00167 2400 1720 1200 860 600 272 230 192 163 136 407 3*4 286 242 202 3-64 4-35 5-21 6-14 7-35 0-00246 0-00291 0-00350 0-00413 0-00495 i 2 3 4 5 6 7 8 9 10 0-00247 0-00351 0-00515 0-00781 0-0127 405 285 194 128 79 112 93-7 77-4 62-8 49'6 1 66 140 116 93-5 73-9 8-93 10-7 12-9 15-9 20-2 0-00602 0-00714 0-00862 0-0107 0-0135 6 I 9 10 ii 12 13 14 15 0-0185 0-0286 0-0472 0-0820 0-125 54 35 21'2 12-2 8-0 40-8 32-6 25-6 19-4 15-7 60-7 48-5 38-2 28-8 23'3 24-5 3 0-7 39-1 5i-5 63-7 0-0165 0-0206 0-0262 0-0347 0-0429 ii 12 13 14 15 16 17 18 19 20 0-202 0-344 0-633 1-31 2-O 4-96 2-91 1-58 0-765 0-50 12-4 9-5 7-o 4-85 3-92 18-5 14-1 10-4 7-20 5-8 5 80-6 105 143 206 255 0-0541 0*0709 0-0962 0-139 0-171 16 17 18 19 20 21 22 23 24 25 3*23 5-49 10-2 14*3 2I'I 0-310 0-182 0-098 0-070 0-0475 3-10 a-37 1-74 1-46 I-2I 4-62 3-54 2-60 2-18 1-81 323 422 III 826 0-217 0-283 0-385 0-459 0-553 21 22 23 24 25 26 27 28 29 30 32-3 46-5 70-4 99-0 144 0*0310 0-0215 0-0142 o-oioi 0-00696 0-98 0-8I5 0-662 0-560 0-466 1-46 12I 0-988 0-835 0-693 IO2O I23O 1510 1790 2I5O 0-685 0-826 I-OI 1-20 i-44 26 27 28 29 30 31 32 33 34 35 I8 4 244 336 474 680 0-0054 0-0041 0-00298 0-0021 1 0-OOI47 0-406 0-353 0-303 O-256 O-2I4 O-607 0-525 0-451 0-382 0-3I7 2460 2830 3300 3910 4670 1-65 1-91 2-22 2-62 3-i6 31 32 33 34 35 36 % 39 40 IOIO 1560 2600 4610 6410 0-00099 O-OOO64 0-000385 O-OOO2I7 O-OOOI56 0-175 0-I40 O-IO9 ' 0-0820 0-0698 O-26O 0-208 O-l62 0'122 0-104 5710 7140 9170 I220O 14300 3-85 4-8x 6-17 8-2O 9-62 36 37 38 39 40 4i 42 43 44 45 8930 13100 20000 32400 54900 O-OOOII2 O-OO00765 O-OOOO5OO O-O0003O9 O'OOOOl82 0-0585 0-0485 0-O392 0'03IO 0-0237 0-087 0-072 0-0585 0-0462 0-0354 I7IOO 2060O 2250O 32300 4220O 1 1-5 13-9 17-1 21-7 28-3 4 1 42 43 44 45 46 47 48 49 50 IOIOOO 210000 518000 I62OOOO 340OOOO O'O000099 0-00000476 O-OOOOOI93 0-00000o6l8 0'O00000294 0-0174 O-OI2I 0-00775 0-00436 0-O0303 0-0260 O'OlSo 0-0116 0-0065 0-0045 57500 82600 129000 229000 330000 38-5 55-6 86-2 154 222 46 49 50 To get "ohms per kilogramme" (approx.) double the numbers in this column and add 10%. To get " kilogrammes per ohm " (approx.) halve the numbers in this column and deduct 10%. NOTES re APPENDIX VI These tables are based on data relating to covered copper wires contained in list of London Electric Wire Co. & Smith's, Ltd. Each wire in a winding is assumed to occupy a square whose side is equal to the diameter of the covered wire, as indicated in the figure , and that no bedding occurs. For the resistance columns a temperature of 15 C. is taken, one foot of I mil copper wire having at this temperature a resist- ance of 10-15 ohms. As the thicknesses of coverings are liable, to appreciable variation the calculated values are only given to three significant figures. APPENDIX VI (a) ORDINARY COTTON COVERED (SINGLE). Approximate Approximate Approximate Resistance i Mils Wires per Number of Wires Ohms Dia. in * Dia. + S.W.G. Mils. OI Covg Covg. Lineal inch. Lineal cm. Per sq. inch. Per sq. cm. Per cu. inch Per cu. cm. 10 128 8 136 7'35 2-89 54-i 8-38 0-00279 0-000170 IO II 116 8 I2 4 8-06 3-17 65-0 IO-I 0-00408 0-000249 II 12 104 8 112 8-93 3-52 79-7 12-4 0-00623 0-000380 12 13 92 8 IOO 10-0 3'94 IOO 15-5 O-OIOO 0-000610 13 M 80 8 88 11-4 4'49 129 2O'O 0-0170 0-00104 14 *5 72 8 80 12-5 4'92 156 2 4 -2 0-0255 0-00156 15 16 64 7 7i 14-1 5'55 198 30-7 0-0409 0-00250 16 *7 56 7 63 15-9 6-26 252 39-1 0-0679 0-00415 17 18 48 6 54 18-5 7-28 343 53'2 O-I26 0-00768 18 19 40 6 46 21-7 8-54 472 73-2 0-250 0-0152 19 20 36 6 42 23-8 9'37 567 87-9 0-370 O-O225 20 21 32 6 38 26-3 10-4 692 107 0-572 0-0349 21 22 28 6 34 29-4 n-6 865 !34 0-933 0-0569 22 23 2 4 6 30 33'3 13-1 IIIO 172 1-63 0-0996 23 2 4 22 6 28 35'7 14-1 1280 198 2-23 0-136 24 25 20 6 26 38-5 15-2 1480 229 3'I3 0-191 25 26 18 6 24 41-7 16-4 1740 269 4-53 0-277 26 ,27 16-4 6 22-4 44-6 17-6 1990 309 6-26 0-382 27 28 I 4 -8 6 20-8 48-1 18-9 2310 358 8-92 0-545 28 29 I 3 -6 6 19-6 51-0 2O-I 2600 43 11-9 0-727 29 30 12-4 6 18-4 54'4 21-4 2950 458 16-2 0-991 30 3 1 n-6 6 17-6 56-8 22-4 3230 500 20-3 1-24 31 3 2 10-8 5 15-8 63-3 2 4 -9 4010 621 29-0 1-77 32 33 10-0 5 15-0 66-7 26-3 4440 689 37-6 2-29 33 34 9-2 5 14-2 70-4 27.7 4960 769 49-5 3-02 34 35 8-4 4 12-4 80-6 31-7 6500 1010 78-0 4-76 35 36 7-6 4 n-6 86-2 33'9 7430 1150 109 6-64 36 37 6-8 4 10-8 92-6 36-5 8570 1330 57 9-57 37 3 8 6-0 4 10-0 IOO 39'4 10000 1550 235 4'3 38 39 5'2 4 9-2 109 42-9 11800 1830 370 2-6 39 40 4-8 4 8-8 114 44'9 12900 2OOO 474 8-9 40 WINDINGS TABLE APPENDIX VI (b) ORDINARY COTTON COVERED (DOUBLE). Approximate Approximate Approximate Resistance in Dia. in Mils _* Dia. J- Wires per Number of Wires Ohms S.W.G. Mils. OI Covg. Covg. Lineal inch Lineal cm. Per sq. inch. Per sq. cm. Per cu. inch. Per cu. cm. 10 128 M I 4 2 7-04 2'77 49-6 7-69 0-00256 0-000156 IO ii 116 14 I 3 7-69 3-03 59-2 9-l8 0-00372 0-000227 II 12 104 M 118 8-48 3-34 71-8 II-I 0-00561 0-000343 12 13 92 M 1 06 9-43 3-7I 89-0 13-8 0-00889 0-000543 13 14 80 T 4 94 10-6 4-17 H3 17-6 0-0149 0-000912 14 15 72 J 4 86 n-6 4'57 135 2I'O 0-O22O 0-00134 15 16 64 12 76 13-2 5-20 173 26-8 0-0357 0-00218 16 I 7 56 12 68 14-7 5'79 216 33-5 0-0583 0-00356 17 18 48 10 58 17-2 6-77 297 46-1 O-IO9 0-00673 18 19 40 IO 50 20-0 7-88 400 62-0 O-2II O-OI29 19 20 36 IO 46 21-7 8-55 472 73'2 0-308 0-0188 20 21 32 IO 42 23-8 9'37 567 87-9 0-468 0-0286 21 22 28 10 38 26-3 10-4 692 107 0-747 0-0455 22 23 24 10 34 29-4 n-6 865 134 1-27 0-0775 23 24 22 10 32 3i-3 12-3 977 151 1-71 0-104 2 4 25 20 IO 30 33-3 13-1 IIIO 172 2-35 0-143 25 26 18 IO 28 35'7 14-1 1280 198 3*33 0-203 26 27 16-4 IO 26-4 37'9 14-9 1440 222 4-5i 0-275 27 28 I 4 -8 10 24-8 40-2 15-8 1630 252 6-28 0-383 28 29 13-6 10 23-6 42-4 16-7 1800 .278 8-21 0-501 29 30 12-4 10 22-4 44'7 17-6 1990 309 II-O 0-669 30 31 n-6 IO 21-6 46-3 18-2 2140 332 13-5 0-822 31 32 10-8 9 19-8 50-5 19-9 2550 395 18-5 I-I 3 32 33 10-0 9 19-0 52-6 20-7 2770 429 23-4 i'43 33 34 9-2 9 18-2 54'9 21-6 3020 468 30-2 1-84 34 35 8'4 8 16-4 61-0 24-0 3720 576 44-6 2-72 35 36 7-6 8 15-6 64-1 25-2 4110 637 60-2 3-67 36 37 6-8 8 14-8 67-6 26-6 4560 708 83-5 5-10 37 38 6-0 8 14-0 71-4 28-1 5100 791 1 20 7-32 38 39 5-2 8 13-2 75-8 29-9 574 890 1 80 n-o 39 4 4-8 8 12-8 78-1 30-8 6100 946 224 137 40 520 PRACTICAL ELECTRICITY APPENDIX VI (c) SPECIALLY FINE COTTON COVERED (SINGLE), Dia. in Mils of Dia. + Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms Mils. Covg. Covg. Lineal inch. Lineal cm. Per sq. inch. Per sq. cm. Per cu. inch. Per cu. cm. 10 128 7 135 7-41 2-92 54'9 8- 5 I 0-00283 0-000173 10 II 116 7 123 8-13 3*20 66-1 10-2 0-00415 0-000253 ii 12 104 7 III 9-01 3'55 81-2 12-6 0-00635 0-000387 12 13 92 7 99 IO-I 3-98 102 I5'8 O-OI02 O-OOO622 13 J 4 80 7 B? "5 4'53 132 20-5 0-0175 0-00106 14 15 7 2 7 79 12-7 5-00 160 24-8 O-O26I 0-00159 15 16 64 6 70 14-3 5-63 204 3f6 0-O42I 0-00257 16 J 7 5<5 6 62 16-1 6-34 260 4'3 O-O7O2 0-00428 17 18 48 5 53 18-9 7'44 356 55-2 O'I3I 0-00797 18 19 40 5 45 22-2 8-74 494 76-6 O-26I 0-0159 19 20 36 4 40 25-0 9-85 625 96-9 0-408 0-0249 20 21 32 4 36 2 7 -8 io - 9 772 I2O 0-637 0-0389 21 22 28 4 32 31-3 12-3 977 152 1-05 0.0643 22 23 24 4 28 35'7 14-1 1280 198 I-8 7 0-114 23 24 22 4 26 38-5 15-2 1480 229 2-59 0*158 2 4 25 20 4 24 41-7 16-4 1740 269 3-67 0-224 25 26 18 4 22 45-5 17-9 2070 3 20 5'39 0-329 26 27 16-4 4 20-4 49-0 19-3 2400 373 7-56 0-461 27 28 14-8 4 18-8 53'2 21-0 2830 439 10-9 0-667 28 29 13-6 4 17-6 56-8 22'4 3230 500 14-8 0-901 29 30 12-4 4 16-4 61-0 24-0 3720 576 20-5 1-25 30 3 1 n-6 4 15-6 64-1 25-2 4110 637 25-8 1-58 31 32 10-8 4 14-8 67-6 26-6 4560 708 33-i 2-O2 32 33 IO-O 4 14-0 71-4 28-1 5100 791 43'3 2-64 33 34 9-2 4 13-2 75-8 29-9 574 890 57'3 3-50 34 35 8-4 3 ix-4 87-7 34'5 7700 1190 92-2 5-63 35 36 7-6 3 10-6 94'4 37-2 8900 1380 130 7-96 36 37 6-8 3 9-8 102 40-2 10400 1610 190 n-6 37 38 6-0 3 9-0 III 43'7 12400 1910 290 17-7 38 39 5'2 3 8-2 122 48-1 14900 2300 465 28-4 39 40 4-8 3 7-8 128 50-4 16400 2550 604 36-8 40 WINDINGS TABLE 521 APPENDIX VI (d) SPECIALLY FINE COTTON COVERED (DOUBLE). S.W.G. Dia. in Mils. Mils of Covg. Dia. + Covg. Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms s.w.c. Lineal inch. Lineal cm. Per sq.inch. Per sq. cm. Per cu. inch. Per cu. cm. IO II 12 13 H 15 128 116 104 92 80 72 10 10 10 IO IO IO 138 126 114 102 90 82 7-25 7'94 8-78 9-80 n-i I2'2 2-85 3'13 3-46 3-86 4'37 4-80 52'5 63-0 77-0 98-0 124 149 8-14 9-76 II-9 14-9 I9-I 2 3 -I 0-00271 0-00396 0-0060I 0-00960 0-0163 0-0242 0-000165 0-000241 0-000367 0-000586 0-000995 0-00148 10 II 12 13 M 15 16 17 18 19 20 64 56 48 4 36 9 9 8 8 6 73 65 56 48 42 13-7 15-4 I7-9 20-8 23-8 5-39 6-06 7'5 8-19 9'37 188 237 319 434 567 29-1 367 49'4 67-3 87-9 0-0387 0-0638 O-II7 0-230 0-370 0-00237 0-00389 0-00714 0-0140 O-0226 16 11 19 20 21 22 23 2 4 25 32 28 24 22 20 6 6 6 6 6 38 34 30 28 26 26-3 29-4 33'3 35'7 38-5 10-4 n-6 13-1 14-1 15-2 692 865 IIIO 1280 1480 107 *34 172 198 229 0-572 0-933 1-63 2-23 3-13 0-0349 0-0569 0-0996 0-136 0-I9I 21 22 23 24 25 26 27 28 29 30 18 16-4 I 4 -8 13-6 12-4 6 6 6 6 6 24 22-4 20-8 19-6 18-4 41-7 44'7 48-1 51-0 54'4 16-4 17-6 18-9 2O-I 21-4 1740 1990 2310 2600 2950 269 300 358 43 458 4'53 6-27 8-92 11-9 16-3 0-277 0-382 0-545 0-727 0-991 26 2 7 28 29 30 31 32 33 34 35 n-6 10-8 10-0 9-2 8-4 6 6 6 6 5 17-6 16-8 16-0 15-2 I3H 56-8 59-6 62-5 65-8 74-6 22'4 23-5 2 4 -6 25-9 29H 3230 354 3910 4330 557 500 549 605 671 863 20-3 25-7 33'i 43'3 66-8 1-24 i-57 2-02 2-6 4 4-08 31 32 33 34 35 36 37 38 39 40 7-6 6-8 6-0 5'2 4-8 5 5 5 5 5 12-6 n-8 II-O IO-2 9-8 1 9 A 04-0 90-9 98-0 IO2 31-3 33-4 35-8 38-6 40-2 6300 7180 8260 9610 10400 976 IIIO 1280 1490 1610 92-2 131 194 301 382 5'63 8-02 11-9 18-4 23-3 36 37 38 39 40 PRACTICAL ELECTRICITY APPENDIX VI (e) SILK COVERED (SINGLE). S.W.G. Dia. in Mils of Dia. + Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms Mils. Covg. Covg. Lineal Lineal Per sq. Per Per Per S.W.G. inch. cm. inch. sq. cm. cu. inch. cu. cm 10 128 3 131 7-64 3-01 58-3 9-03 0-00301 0-000183 IO II 116 3 119 8- 4 I 3-31 70-6 II-O 0-00444 0-000271 II 12 104 3 107 9-35 3-68 873 13-5 0-00683 0-000417 12 13 92 3 95 10-5 4-14 III I 7 -2 o-oin 0-000675 13 14 80 3 83 I2-I 4-76 145 22-5 0-0191 O-OOII7 15 72 3 75 13-3 5-24 I 7 8 2 7 -6 0-0290 0-00177 15 16 64 3 67 I 4 -9 5-87 223 34'5 0-0460 O-OO28I 16 17 56 3 59 I6'9 6-66 28 7 44-5 0-0774 0-00473 17 18 48 2 50 2O-O 7-88 400 62-0 0-147 0-00896 18 19 40 2 42 23-8 9-37 567 87-9 0-300 0-0183 19 20 36 2 38 26-3 10-4 692 107 0-452 0-0275 20 21 32 2 34 29-4 n-6 865 134 0-715 0-0436 21 22 28 2 30 33-3 13-1 IIIO 172 I-2O 0-0731 22 23 24 2 26 38-5 15-2 1480 229 2-17 0-133 23 24 22 75 2375 42-1 16-6 1770 275 3-10 0-189 2 4 25 20 75 21-75 46-0 18-1 2110 328 4'47 0-273 25 26 18 75 19-75 50-6 19-9 2560 397 6-69 0-408 26 27 16-4 75 18-15 55-1 21-7 3040 9-55 0-583 27 28 14-8 '75 16-55 60-4 23-8 3650 566 14-1 0-860 28 29 13-6 75 15-35 65-2 25-7 4240 658 19-4 1-18 29 30 12-4 '5 13-9 72-0 28-4 5180 802 28-5 1-74 30 31 n-6 5 13-1 76-3 30-1 5830 903 36-6 2-24 31 32 10-8 5 12-3 81-3 32-0 66lO 1020 47'9 2-93 32 33 IO'O 5 87-0 34-3 7560 1170 64-0 3-90 33 34 9-2 5 10-7 93'4 36-8 8730 1350 87-3 5'33 34 35 8-4 5 9-9 101 39-8 10200 1580 122 7'45 35 36 7-6 5 9-1 no 43-3 I2IOO 1870 I 77 10-8 36 37 6-8 5 8-3 121 47-6 14500 2250 266 16-2 37 38 6-0 5 7'5 133 52-4 17800 2750 417 25-5 38 39 5' 2 '5 6-7 149 58-7 223OO 3450 697 42-5 39 40 4-8 5 6-3 159 62-6 25200 3910 925 56-5 40 41 4'4 5 5'9 169 66-6 28700 4450 I25O 76-6 4 1 42 4' 25 5-25 191 75'2 36300 5620 I92O 117 42 43 3-6 25 4-85 206 81-1 42500 6590 2780 169 43 44 3-2 25 4'45 225 88-6 50500 7830 4170 255 44 3-o 25 4'25 235 92-6 554 8580 5200 45 2-8 25 4-05 247 97-3 6IIOO 9470 6590 402 45 46 2'4 25 3-65 274 108 75100 11600 IIOOO 673 46 47 2-O 25 3-25 308 121 94700 14700 2OOOO I22O 47 48 1-6 25 2-85 35i 138 I230OO 19100 40700 2480 48 1*4 25 2-65 377 148 I42OOO 22IOO 61400 3750 49 1-2 1-25 2-45 408 161 167000 25800 97800 597 49 50 VO 1-25 2-25 445 175 198000 30600 167000 10200 50 WINDINGS TABLE 523 APPENDIX VI (/) SILK COVERED (DOUBLE). Approximate Approximate Number Approximate Resistance in Dia. in Mils _.r Dia. + Wires per of Wires Ohms S.W.G. Mils. OI Covg. Covg. Lineal inch. Lineal cm. Per sq.inch. Per sq. cm. Per cu. inch. Per cu. cm. S.W.G. 10 128 4'5 I32'5 7'55 2-97 57' 8-83 0-00294 0-000179 IO n 116 4'5 120-5 8-30 3-27 69-0 10-7 0-00433 0-000264 II 12 104 4'5 108-5 9-22 3-63 85-0 13-2 0-00664 0-000405 12 13 92 4'5 96-5 10-4 4-10 107 16-7 0-0107 0-000655 13 M 80 4'5 84-5 n-8 4'65 I 4 21-7 0-0185 0-00113 M 15 72 4'5 76-5 13-1 5'i6 171 26-4 0-0278 0-00170 15 16 64 4'5 68-5 14-6 5'75 213 33-o 0-0440 0-00270 16 17 56 4'5 60-5 16-5 6-50 274 42-4 0-0737 0-00449 17 18 48 3'5 51-5 I9'4 7-64 377 58-4 0-139 0-00845 18 19 40 3'5 43'5 23-0 9-06 528 81-9 0-279 0-0170 19 20 36 3'5 39'5 25-3 9-96 641 99'3 0-418 0-0255 20 21 32 3'5 35'5 28-2 n-i 793 123 0-655 0-0400 21 22 28 3'5 3i-5 3i-8 12-5 IOIO 156 1-09 0-0663 22 23 24 3'5 27-5 36-4 M-3 1320 205 1-94 0-119 23 2 4 22 3 25 40-0 15-8 1600 248 2-80 0-171 24 25 2O 3 23 43-5 17-1 1890 293 4-00 0-244 25 26 18 3 21 47-6 18-7 2270 352 5-92 0-361 26 27 16-4 3 19-4 5i-6 20-3 2660 412 8-36 0-510 27 28 14-8 3 I 7 -8 56-2 22-1 3160 489 12-2 *744 28 29 13-6 3 16-6 60-2 23-7 3630 562 16-6 I-OI 29 30 12-4 2'5 14-9 67-1 26-4 4500 698 24-8 '5i 30 31 n-6 2'5 14-1 70-9 27-9 5030 780 31-6 1-93 31 32 10-8 2'5 I3-3 75'2 29-6 5650 876 41-0 2-50 32 33 10-0 2'5 12-5 80-0 31-5 6400 992 54'i 3-31 33 34 9-2 2'5 11-7 85-5 33-7 73io 1130 73-0 4-46 34 35 8'4 2-5 10-9 91-8 36-2 8420 1310 101 6-16 35 36 7-6 2-25 9-85 102 40-2 10300 1600 151 9-22 36 37 6-8 2-25 9-05 110 43'3 12200 1890 223 13-6 37 38 6-0 2-25 8-25 121 47'7 14700 2280 345 2I-I 38 39 5'2 2-25 7'45 T 34 52-8 I8OOO 2790 564 34'4 39 40 4-8 2-25 7-05 142 55-9 20IOO 3120 739 45-1 4 4 1 4*4 2-25 6-65 150 59'i 22600 35io 988 60- 1 4 1 42 4-0 2 6-0 167 65-8 27800 43io 1470 89-6 42 43 3-6 2 5-6 179 7-5 31900 4940 2080 127 43 44 3-2 2 5'2 192 75-6 37000 5730 3060 187 44 3-0 2 5'0 200 78-8 4OOOO 6200 3760 230 45 2-8 2 4-8 208 81-9 43 4 00 6730 4700 287 45 46 2'4 2 4'4 227 89-4 51600 8000 7580 463 46 47 2-0 2 4-0 250 98-4 62500 9690 13200 807 47 48 1-6 2 3-6 278 109 77200 I2OOO 25500 1560 48 I '4 2 3'4 294 116 86500 13400 37300 2280 49 1-2 2 3'2 313 123 97700 15200 574o 35oo 49 50 1-0 2 3-0 333 131 IIIOOO 17200 94000 574 5 l 1 524 PRACTICAL ELECTRICITY APPENDIX VI fe) ENAMEL INSULATED. S.W.G. Dia. in Mils. Mils of Covg. Dia. + Covg. Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms S.W.G. Lineal inch. Lineal cm. Per sq. inch. Per sq. cm. Per cu. inch. Per cu. cm. 16 17 18 19 20 64 56 48 40 36 2'5 2'5 2-5 2-25 2-25 66-5 58-5 50-5 42-25 38-25 I 5 -0 I 7 -I 19-8 23-7 26-1 5-91 6-73 7-80 9-33 10-3 226 292 392 560 685 35-i 45-3 60-8 87-0 106 0-0467 0-0788 0-144 0-296 0-447 0-00285 0-00481 0-00878 0-oi8l 0-0272 16 17 18 19 20 21 22 23 2 4 25 32 28 24 22 20 2-0 2-0 i'75 i'75 i'75 34' 30-0 25-75 23-75 21-75 29-4 33-3 38-8 42-1 46-0 n-6 13-1 I5-3 16-6 18-1 865 IIIO 1510 1770 2110 134 172 234 275 328 0-715 I-I2 2-21 3-10 4'47 0-0436 0-0731 0-135 0-189 0-273 21 22 23 24 25 26 27 28 29 30 18 16-4 14-8 13-6 12-4 i'75 i'5 i*5 i'5 1-25 19-75 17-9 16-3 I5-I i3 <6 5 50-6 55-9 61-4 66-2 73-3 19-9 22-0 2 4 -2 26-1 28-9 2560 3120 3760 4380 537 397 484 583 680 832 6-69 9-8l H-S 20-0 29-5 0-408 0-599 0-887 1-22 1-80 26 27 28 29 30 31 32 33 34 35 n-6 10-8 10-0 9-2 8'4 1-25 1-25 1-25 i i 12-85 12-05 11-25 IO-2 9'4 77-8 83-0 88-9 98-0 1 06 30-6 32-7 35-o 38-6 41-7 6060 6890 7900 9610 11300 94 1070 1230 1490 1750 38-1 5O-O 66-8 96-0 136 2- 3 2 3-05 4-08 5-86 8-28 31 32 33 34 35 36 37 38 39 40 7-6 6-8 6-0 5'2 4-8 i i i o-75 o-75 8-6 7-8 7-0 5'95 5'55 116 128 *43 168 1 80 45-7 50-4 56-3 66-2 70-9 13500 16400 20400 28300 32500 2IOO 2550 3160 4380 5030 198 300 479 883 1190 I2-I 18-3 29-3 53-9 72-7 36 37 38 39 40 4 1 42 43 44 45 4'4 4-0 3-6 3-2 2-8 o-75 o-75 o-75 o'75 o-75 5-i5 4'75 4'35 3'95 3'55 194 211 230 253 282 76-4 83-1 90-6 99-6 in 37700 44300 52800 64100 79400 5840 6870 8190 993 12300 1650 2340 3450 5290 8560 100 143 210 323 522 41 42 43 44 45 WINDINGS TABLE 525 APPENDIX VI (h) ENAMEL-INSULATED AND COTTON COVERED (SINGLE). S.W.G. Dia. in Mils. Mils of Covg. Dia. + Covg. Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms S.W.G. Lineal inch. Lineal cm. Per sq. inch. Per sq. cm Per cu. inch. Per cu. cm 16 J 7 18 19 20 64 56 48 40 36 9 9 8 8 8 73 65 56 48 44 137 I5H 17-9 20-8 22-7 5'4 6-06 7'5 8-19 8-94 1 88 237 319 434 519 29-1 36-7 49'4 67-3 80-4 0-0387 0-0638 0-II7 0-230 0-339 0-00236 0-00389 0-00714 0-0140 O-O2O6 16 17 18 19 20 21 22 23 24 25 32 28 24 22 20 8 8 8 8 8 40 36 32 30 28 25-0 2 7 -8 3I3 33'3 35'7 9-85 10-9 12-3 13-1 14-1 625 772 977 IIIO 1280 96-9 I2O 152 I 7 2 198 0-516 0-832 i-43 1-94 2-70 0-0314 0-0508 0-0875 0-118 0-165 21 22 23 24 25 26 27 28 29 30 18 l6'4 I 4 -8 13-6 12-4 8 7 7 7 7 26 23-4 21-8 20-6 19-4 38-5 42-7 45-9 48-6 51-6 15-2 16-8 18-1 19-1 20-3 1480 1830 2110 2360 2660 229 283 326 365 4 I2 3-86 5'75 8-12 10-8 14-6 0-236 o-35i 0-496 0-658 0-892 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 n-6 10-8 10-0 9-2 8-4 6 6 6 6 5 17-6 16-8 16-0 15-2 13-4 56-8 59-5 62-5 65-8 74-6 22-4 23-4 24-6 25-9 29-4 3230 354 3910 4330 557 500 549 606 671 863 20-3 25-7 33-1 43'3 66-8 1-24 i'57 2-O2 2-64 4-08 31 32 33 34 35 7-6 6-8 6-0 5'2 4-8 5 5 5 5 5 12-6 n-8 II-O IO-2 9'8 79'4 84-8 91-0 98-0 102 31-3 33'4 35-8 38-6 40-2 6300 7180 8260 9600 10400 976 IIIO 1280 1 1490 1 1610 92-2 131 194 301 382 5-63 8-02 n-9 18-4 233 36 37 38 39 40 526 PRACTICAL ELECTRICITY APPENDIX VI (t> ENAMEL INSULATED AND COTTON COVERED (DOUBLE). Mils Approximate Wires per Approximate Number of Wires Approximate Resistance in Ohms Dia. in r>f Dia. + S. W.G. Mils. OI Covg. Covg. Lineal inch. Lineal cm. Per sq. inch. Per sq. cm. Per cu. inch. Per cu. cm. S.W.G. 16 64 T 4 78 12-8 5-04 !6 4 25H 0-0339 O-OO2O7 16 17 56 *4 7 H'3 5-63 20 4 3 I-6 0-0550 0-00336 17 18 48 12 60 16-7 6-58 2 7 8 43-o O-IO2 0-00623 18 19 4 12 52 19-2 7-56 370 57'4 0-196 O-OII9 19 20 36 12 48 20-8 8-19 434 67-3 0-284 0-0173 20 21 32 12 44 22-7 8-94 519 80-4 0-427 0-026I 21 22 28 12 40 25-O 9-85 625 96-9 0-675 0-0412 22 23 24 12 36 2 7 -8 10-9 772 I2O I-I 3 0-0695 23 2 4 22 12 34 29-4 n-6 865 134 i'5i 0-0920 24 25 2O 12 32 31-3 12-3 977 152 2-06 0-126 25 26 18 12 30 33'3 13-1 IIIO 172 2-91 0-178 26 27 16-4 II 27-4 36-5 14-4 1330 207 4-19 0-256 27 28 I 4 -8 II 25-8 38-8 I5-3 1500 232 5-83 0-356 28 29 I 3 -6 II 24-6 40-7 16-0 1650 257 7-56 0-461 29 30 12-4 II 23H 427 16-8 1830 283 IO-O 0-610 3 31 n-6 IO 21-6 463 18-2 2140 333 13-4 0-818 3 1 32 10-8 IO 20-8 4 8-I 18-9 2310 358 16-4 I-OO S 2 33 IO'O IO 2O-O 50-0 19-7 2500 388 21-2 1-29 33 34 9-2 IO 19-2 52-1 20-5 2710 421 27-2 1-66 34 35 8-4 9 17-4 57'5 22-6 3300 5" 39-6 2-42 35 36 7-6 9 16-6 6O-2 23^ 3630 562 53-o 3-24 36 37 6-8 9 15-8 63-3 24-9 4010 620 73'3 4'47 37 38 6-0 9 15-0 66.7 26-3 4440 689 104 6-35 38 39 5'2 9 14-2 70-4 27-7 4960 768 156 9-48 39 40 4-8 9 13-8 72-5 28-6 5250 812 193 11-8 4 TABLE OF SYMBOLS 527 APPENDIX VII. TABLE OF SYMBOLS. ADOPTED BY THE INTERNATIONAL ELECTROTECHNICAL COMMISSION, 1913. Name of Quantity Symbol 1. Length / L \ For 2. Mass . . . . . . m M Y Dimensional 3. Time . . . . . . t TJ Equations. 4. Angles . j$ . - . . a, (3, 7 5. Acceleration of gravity . . g 6. Work < . . . . A or W 7. Energy . . . . . W or U 8. Power . . . , .- P 9. Efficiency . . ... rj 10. Number of turns in unit time . n 11. Temperature Centigrade . t or 12. Temperature absolute . ; . T or @ 13. Period . . . . T 14. Angular frequency, 2:r/T . w 15. Frequency . . . . / 1 6. Phase displacement . . 9 17. Electromotive force . . . E The alternative 18. Current / s y m ^ ol H is f reco f m - D menAed for the 19. Resistance . . . * . R cage in which the 20. Resistivity . . * P principal symbol 21. Conductance . . , . G i s no t suitable. 22. Quantity of electricity . Q 23. Flux-density, electrostatic . D 24. Capacity . . . . C 25. Dielectric constant . . . e 26. Self inductance . . . . L or 5? 27. Mutual inductance . . . M ,, *d* 28. Reactance . . . . X ,, * 29. Impedance . . . . . Z 30. Reluctance . . . . S 8% 31. Magnetic flux . . $ ,, *fl 32. Flux-density, magnetic . . B ,, J$ p//? 33. Magnetic field . . . H ' && 34. Intensity of magnetisation . /**' 35. Permeability . . ^ 36. Susceptibility . . . . K| 37. Difference of potential . . V 4"- INDEX Absolute system of units (see C.G.S. system) , unit of resistance, short history of, 473 Accumulator (see Storage cells) Acid voltameter, Ayrton's form of, 3 1 , McMichael's form of, 33 Action of zinc, local or prejudicial, i?5 Air condenser, standard, 391, 392 , dielectric strength of, 373 , specific inductive capacity of.- 37i Alcohol, specific inductive capacity of, 371 Alloys, resistance of, 231, 234 Aluminium, annealed, resistance of, 233- 235 , relative conductivities of, 244 Amalgamating zinc, method of, 176 Ambers, resistivity of, 374 , specific inductive capacity of, 37 1 American specification defining electrical standards, 497 Ammeter, Ayrton and Mather's moving coil, 115 , Ayrton and Perry's perman- ent magnet, no , calibration of, by potentio- meter, 418 , damping device in Nalder, 120 , Evershed and Vignole's new, 122 , Evershed gravity control, 121 , hot-wire, 124 , hot-wire, Hartmann and Braun, 123, 124 , moving coil, 112 -, Nalder Bros, and Thompson's moving coil, 114 , Nalder gravity control, 119 , needle, pointer and staff of Ayrton and Perry's permanent magnet, no Ammeter, shunted voltmeter used as, 164 , Weston's moving coil, work- ing parts of, 113 Ammeters and current voltmeters, resistances of, 158 , calibration of, 416 , definition of, 108 , permanent magnet, 109 , soft iron, 118 , spring and gravity control, 118 , used as voltmeters, 158 , voltmeters used as, 163 Ampere, definition of, 18 Ampere-hour, definition of, 24 - meter, Ferranti, 305, 306 meters, 302 Amperes, value in, of deflection of tangent galvanometer, 96 Ampere-turns, definition of, 121 Analogies, mechanical, of con- densers, 363 Analogue, hydraulic, of condensers, 364 Angle of dip, definition of, 37 Anode, definition of, 20 Antimony, 233, 235 Apparatus for measuring heat equivalent of electric energy, 280 for testing strength of mag- netic field along axis and in plane of circular coil, 74 Armature, rotating, of Elihu Thom- son energy meter, 298 Aron energy meter, 293 energy meter, connections of, 294 - supply meter, differential gearing of the Aron, 292 Astatic needle, 352 Atomic weights, international, 22 (footnote) Ayrton and Mather's moving coil ammeter, 115 and Mather's non-inductive resistance coil, 148 2 I 529 530 INDEX Ayrton and Mather's reflecting elec- trostatic voltmeter, 157 and Mather's reflecting gal- vanometer, 353 - and Mather's shunt for strong currents, 263 and Mather's shunt, principle of, 260 and Mather's universal shunt, 260, 416, 417 and Mather's universal shunt box, plan of, 261 arid Mather's zero electro- meter or zero electrostatic volt- meter, 135-137 - and Perry's gold-leaf electro- scope, 167 - and Perry's original gaining clock joulemeter, 291 Ayrton's form of sulphuric acid voltameter, 31 B B.A. unit, error in, 479 - unit of resistance, 477 Back, E.M.F., 315, 316 - Balance for finding strength of mag- nets, 63 - for finding strength of pole, 62 , Hibbert's magnetic, 61 Balata, specific inductive capacity of, 37 1 Ballistic galvanometer, 349 galvanometer, constant of, 359, 427 galvanometer, correction for damping, 356 galvanometer, determination of constant of, by earth induc- tor method, 429 galvanometer, measurement of quantity by, 353 galvanometer, needle and coils of, 349 galvanometer, reading tele- scope and scale for, 352 galvanometer, reflecting, 350 galvanometer, reflecting, lamp stand and scale for, 351 galvanometer, simple, 349 tests, iron ring wound for, 456 Bar magnet, lines of force of a, 57 magnets, lines of force with two, 69 - pattern of bridge, 271 Bast-ian energy meter, 302 w Batteries, cells and, 3 Battery, finding resistance of, by condenser method, 398 Battery, Muirhead's telegraph, 182 resistance of, key and con- denser for testing, 398 Berlin conference on electrical standards, 499 Bismuth, 233, 235, 244, 406 Bleeck-Love cell, 333 Blue Bell cell, 198 Board of Trade committee on electrical standards, composition of, 488 of Trade electrical standards, short history of, 473 of Trade (1894) form of Clark's cell, 202 of Trade unit of energy, 294 Box, shunt, method of constructing, 253 Boxes, ratio, 413 , resistance, 145 , volt, 413 Branch circuits in parallel, 245 Brass, relative conductivities of, 244 Bridge, bar pattern of, 271 Bridge, British Association, 222 galvanometer, meaning of deflection on, 227 key, 225, 226 , metre, 222 , metre, circular, 223 , metre, diagram of, 222 , portable, with switch con- tacts, 272 , three-wire, 224 , use of shunt with, 227 , Wheatstone's bar pattern of, 270 , Wheatstone's diagram of, 219 , Wheatstone's dial pattern of, 270, 271 , Wheatstone's method of con- structing, 221 , Wheatstone's, ordinary forms of, 265 , Wheatstone's portable, dia- gram of connections of, 270 , Wheatstone's portable forms of, 269 , Wheatstone's portable, with battery and galvanometer com- bined, 269 Wheatstone's Post Office, 267, 268 , Wheatstone's, principle of, 218 , Wheatstone's top of a com- mercial, 266 Wheatstone's, use of, 221 Bridges, coil, 267 INDEX British Association bridge, 222 Association's recommenda- tions on electrical standards, 485 Bronze, silicum, resistance of, 233 Bunsen's cell, 183, 184 C Cadmium cell, Weston's, 206 , relative conductivities of, 244 Calculation of capacity of conden- sers, 368 - of E.M.F. of cell from energy liberated by chemical action, 206 of magnetic field strength, 452 of magnetic pull, 448 Calibrating deflectional voltmeter, 1 60 - potentiometer wire, 404 potentiometer wire, knife edges for, 404 - voltmeter by using ammeter and one known resistance, 161 - voltmeter by using several known resistances with known current passing through them, 162 - wire by differential galvano- meter, 405 Calibration, absolute, of tangent galvanometer, 42 curve, method of plotting, 45 curve of galvanometer, 45 - curve of tangent galvano- meter, 84 , definition of, 34 of ammeters, 416 - of ammeter by potentiometer, 418 of detector by comparison with tangent galvanometer, 44 - of galvanometer, absolute, 39 of galvanometer by direct comparison with tangent gal- vanometer, 43 of galvanometer, relative^ 39 - of voltmeter by potentio- meter, 411 potential divider for volt- meter, 412 Callaud cell, 178 Calorimeter for measuring coils of wire, 237 Calorimeter used in measuring heat equivalent of electric energy, 279 Canada balsam, resistivity of, 374 Capacity, absolute measurement of, 387 , calculation of, 368 , measurement of specific in- ductive, 388 Capacity of conductors, 362 of several condensers, com- bined, 382 , specific inductive, of di- electrics, 370 , unit of, and farad, relation between, 367 , units of, 365 , variation of, of condenser, 364 Carbon cloth rheostat, 419 dioxide, specific inductive capacity of, 371 - plate rheostat, 418 , resistance of, 233 , specific resistance and tem- perature variation of, 241 " Carsak " cell, 192 Castor oil, specific inductive capa- city of, 371 Cathode, definition of, 19 Cell and circuit, simple, 2 - arranged for experiments on polarisation, 171 - arranged for proving inde- pendence of E.M.F. , 180 , Bleeck-Love, 333 , Blue Bell, 198 , Board of Trade (1894) form of Clark's, 202 , Bunsen's, 183, 184 , calculation of E.M.F. of, from energy liberated by chemi- cal action, 206 , Callaud, 178 , " Carsak," 192 , Clark's, 200 , Columbia, 198 , Dania dry, 196 , Daniell's porous pot, 174 , Daniell's two-fluid, 173 , dry, 193 , Edison-Lalande, 199, 200" , Exchange Telegraph Com- pany's form, 178 , Extra-Sec, 198 , Fery's modified Leclanche, 193 1 86 Fuller's mercury bichromate, , G.E.C., 197 , " Gassner's " dry, 195 , Grove's, 183, 184 , Hellesen dry, 196 , inert, 199 , Kahle's modification of Ray- leigh's H-form of Clark's, 203 , L,acombe central zinc, 192 , L,eclanche, 189 , Leclanche agglomerate, 190 , I/ord Rayleigh's H-form of Clark's, 201 532 INDEX Cell, Meidinger, 177 , Minotto's, 179, 1 80 , Obach dry, 197 , portable Clark's (Muirhead's form), 204 , potassium bichromate, 185 , " Six Block Agglomerate," 191 , Tinsley, 208 , Weston's, 200 - , Weston's cadmium, 204 , Weston's cadmium (F. E. Smith's form), 205, 206 Cells and batteries, 3 , arrangement of, to give maxi- mum power to external circuit of fixed resistance, 325 , charge and discharge curves of storage, 188- , discharge curves for Icelandic, 194 , B.M.P. of Daniell's, 181 , galvanic, 170 , gravity, Daniell's, 178 - joined in parallel, 326 joined in series, 325 joined, partly in parallel and partly in series, 326 minimum number of, required to give fixed amount of power to given external circuit, 329 , mixed grouping of, 327 , modifications introduced in, 333 , polarisation in simple, 170 , resistance of, 180, 182, 183, 196, 313 , standard, 200 storage or secondary, 187 Cellulose, dry, resistivity of, 374 Central station, load diagram of, 284 C.G.S. and British systems of units, comparison of, 512 Chamberlain and Hookham quan- tity meter, 303, 304 Charge and discharge curves of storage cells, 189 Charged condensers stores for energy, 384 Chatterton's compound, specific inductive capacity of, 371 Chemical action in simple voltaic element, 1 70 - property of a current, 7 Chicago conference on electrical standards, 490 Circuit, cell and, simple, 2 , diagram of, for testing Ohm's law, 142 , differential galvanometer, diagram of, 216 Circuit, distribution of power in elec- tric, 318 , electric, linked with paths of pole, 455 , external, receiving maximum power from current generator, 319 , magnetic, 467 et seq. of condenser in which charge and discharge are measured, 379 of condenser in which charge only is measured, 379 Circuits, complete, application of Ohm's law to, 149 in parallel, 257 in parallel, branch, 245 , primary and secondary, on iron ring, 457 Circular plate condenser, 389 Clark's cell, 200 cell, Board of Trade (1894) form, 202 cell, I^ord Rayleigh's H-form, 201 - cell, portable form), 204 (Muirhead's Coercive force, definition of, 464 Coil, Ayrton and Mather's non- inductive resistance, 148 - bridges, 267 , induction, 438 , lines of force due to circular, carrying current, 59 , lines of force due to current in circular, 58 of a tangent galvanometer, adjusting, 84 , standard ohm (Reichsanstalt form), 265 Coils of wire used in apparatus for measuring variation of resistance with temperature, 236 , resistance, 145 , standard resistance, 264 - used for testing the resistance of conductors in parallel, 246 Columbia cell, 198 Combined resistance, 245 Comparing E.M.F.'s. by condenser, 397 K.M.Fs., Poggendorff's method of, 400 Comparison of condensers, 379 of quantities, 361 of resistances by potentio- meter, 417 of resistances by substitution method., 214 Compass needle, mapping out lines of force with, 71 needle, weighted, for measur- ing strength of magnetic field, 73 INDEX 533 Condenser, charge and discharge key, 379 circuit in which both charge and discharge are measured, 379 circuit in which charge only is measured, 379 circular plate, 389 , comparing E.M.Fs. by, 397 , cylindrical standard air, 395 , diagram of connections for testing guard ring, 394 , diagrammatic representation of a, 365 , energy wasted in charging, from a source of constant P.D., 386 , hydraulic analogue of, 364 improved form of plate air, 392 , rectangular plate, 388 , simple diagram of, 365 , standard air, 393, 394 , standard spherical, 396 , variation of capacity of, 366 Condensers, calculation of capacity of, 368 , capacity of spherical and plate air, 368 , charged, stores of energy, 384 , combined capacity of several, 382 , comparison of, 379 , construction of, 374 , cylindrical, capacity of, 369 , for large potential differences, 376 , in parallel, 383 , in series, three, 383 , mechanical analogies, 363 Conditions for maximum power, 320 Conductance and conductivity, 242 - of conductors in series and in parallel, 244 Conductivities, approximate rela- tive table of, 244 , comparison of electric and heat, 243 , specific, of mixtures of sul- phuric acid and water, 515 Conductivity and conductance, 242 Conductor, conditions affecting resistance of, 228 , definition of, 3 Conductors and insulators, 3 , capacity of, 362 , currents in parallel, 248 - in parallel, resistance of, coils used for testing, 246 in series and in parallel, con- ductance of, 244 Conductors in series, 243 - in series and in parallel, resist- ance of, 244 of large specific resistance and small temperature coefficients, 239 Connection between E.M.F. and P.D. of battery, 313 Connections of ratio box, 413 Constant cell, 4 cell, definition of, 3 - of ballistic galvanometer, 359, 429 Constantan, resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Construction of condensers, 374 of Wheatstone's bridge, 221 of shunt box, 253 Controlling force, definition of, 36 Copper, annealed, resistance of, 233, 235 , hard drawn, resistance of , 233, 235 , relative conductivities of, 244 - wire tables, 518 et seq. Coulomb, definition of, 24 Coulomb meter (see Ampere-hour meters) Crompton potentiometer, 408, 409 Current and flux density, relation between, 449 , apparatus for showing pro- perties of, 9 , chemical property of, 7 , defining strength of, 12 , direction of, 26 , electric, definition of, i , electric, direction of flow of, i, 26 , electric measurement of, i , electric method of production of, 2 , electric properties of, 4 , field of straight, 453 , generator, E.M.F. of, 314 , generator, external circuit receiving maximum power from, 319 , generator, power developed by, 312 , heat produced by, 277 , heating property of, 7 , increase of, produced by applying shunt, 255 , magnetic property of, 7 , measurement of, by galvano- meter, 36 - measurements, reason for using low resistance galvano- meters for, 154 534 INDEX Current measurements, resistances for, 416 , measuring effects of, 8 , measuring, with copper volta- meters, 14 measuring, with electro-mag- net, 15 measuring, with galvano- scopes, 14 measuring, with sulphuric acid voltameters, 13, 17 measuring, with thermometer, 16 - method of comparing poten- tial differences and resistances, 153 strength, 7 - turns and lifting force, rela- tion between, 442 unit, definition of, 18 voltmeters and ammeters, resistances of, 158 , work done by, 273 Currents, electro-dynamometer for measuring, 104 , induced, direction of, 424 , induced, introductory remarks on, 423 , induction of, in parallel wires, 441 in parallel conductors, 248 , shunt for strong, Ayrton and Mather's, 263 Curve, calibration of a tangent galvanometer, 84 , calibration, of galvanometer, 45 connecting current and num- ber of cells in series when cells and external resistance are fixed, 327 connecting power received by an external circuit and its resistance, 322 connecting rate of flow of water with loss of head, 139 , discharge, for Icelandic cells, 194 , magnetisation, for soft iron ring, 459 showing value of current giving maximum power to external circuit, 320 Curves, charge and discharge of storage cells, 188 , hysteresis, 462, 463 - of flux density in magnet, 433 , practical value of drawing, to record results of experiments, 147 Cylindrical condensers, capacity of, I 369 Damped vibration, diagrams of, 358 Damping, correction of ballistic galvanometer for, 356 devices in ammeters, 120 Dania dry cell, 196 Daniell's cells, K.M.F. of, 181 cells, gravity, 178 cells, resistance of, 180 - porous pot cell, 174 two-fluid cell, 173 - use of depolariser, 173 Decrement, determination of, 357 , logarithmic, 357 Definition of ampere, 18 of ampere-hour, 24 of Board of Trade Unit, 294 of coulomb, 24 of direction of current, 26 of electric current, i of electromagnetic unit quan- tity of electricity, 24 of electrostatic unit of quan- tity of electricity, 81 of farad, 266 of henry, 438 of joule, 277 of ohm, 143 of volt, 148-153 of watt, 284 Deflection of magnet by conductor carrying current, 4 of tangent galvanometer, value in amperes of, 96 - on bridge galvanometer, meaning of, 227 Deflectional voltmeter, calibrating, 1 60 - wattmeter, Elliott's, 289 Desiccator used with silver volta- meter, 20 Detector, calibration of, by direct comparison with a tangent gal- vanometer, 44 Determination of decrement, 357 Dial pattern of bridge, 270 potentiometer, 409 - potentiometer, N.C.S., 410 Dielectric constant, 389 - strength of insulators, 372 Difference of potentials, 126 Differential galvanometer, 216 galvanometer-circuit, diagram of, 216 gearing of the Aron supply meter, 292 Dip, angle of, definition of, 37 Direct reading scale, 51 Direction of current, de^m'tion of, 26 INDEX 535 Discharge curves for Leclanche cells, 194 Distribution of gas pressure, apparatus for testing, 130, 131 of magnetism, 431 of water pressure, apparatus for testing, 128, 130 Divider, potential, 380 Dry cell, Dania, 196 - cell, " Gassner's," 195 - cell, Hellesen, 196 - cell, Obach, 197 - cells, 193 et seq. Earth inductor, simple, 430 , potential of, arbitrarily called nought, 132 Ebonite, dielectric strength of, 373 , resistivity of, 374 , specific inductive capacity of, 37i Economy in transmission of energy, 346 Edison-I,alande cell, 199, 200 Efficiency, 335 et seq, - of electric transmission of energy, 339 Electric and heat conductivities, comparison of, 243 - circuit linked with paths of poles, 455 current and its measurement, i - current, definition of, i current, direction of flow of, i - current, methods of produc- tion of, 2 - current, properties of, 4 - energy (see Energy, electric) energy and power, 273 - flow, water analogy of, 129 - lines of force (see lyines of force, electric) - power (see Power, electric) - pressure (see Potential differ- ence) - quantity, measurement of, 348 quantity, units of, 394 - transmission of energy, 308 - unit of energy, the joule, 277 Electrical efficiency of transmission and ratio of power received to power receivable, 343 measurements, decisions of (1882) conference on, 482 measurements, decisions of Paris Congress on, 481 Electrically lighted house, part of plan of, 257 Electricity, definition of unit quan- tity of, 24 Electro-chemical equivalents, 21 Electrodynamometer, 102 for measuring very small currents, 104, 142 - , Siemens, 102 , simple, 1 02 , zero for small currents, 142 Electrolysis, 21 Electromagnetic and electrostatic units, relations between, 515 - definition of E.M.F., 151 Electromagnets, measuring current with, 15 Electrometer, 134 , Ayrton and Mather's zero, 135, 137 . Electromotive force, 150-153 of current generator, 314, 315 of standard cells, 204, 205 Electroscope, Ayrton and Perry's gold-leaf, 167 , gold-leaf, 1 66, 362 , gold-leaf, a deflectional gravity voltmeter, 168 , gold-leaf, sensibility of, 168 Electrostatics, electric lines of force and, 8 1 Elliott's deflectional wattmeter, 289 E.M.F. and P.D. of battery, con- nection between, 313 - back, 315, 316 , cell arranged for proving in- dependence of size, 1 80 , electromagnetic definition of, of any current generator, 314 of cell from energy liberated by chemical action, calculation of, 206 - of Daniell's cells, 181 E.M.Fs., comparing by condenser, 397 , Poggendorff's method of comparing, 400 Energy, Board of Trade, unit of, 294 charged condensers, stores for, 384 , electric, and power, 273 -, electric, measuring heat equivalent of, 278 , electric, table of heat equiva- lent of, 281 , electric transmission of, 308 -, electric transmission of, effi- ciency of, 339 loss, calculation of, by hyster- esis, 465 , loss of. due to hysteresis, 464 536 INDEX Energy meter, Aron, 293 meter, Aron, connections of, 294 meter, clock form, 290 meter, law of, 301 meter, motor form, 296 - meter, Thomson, 298, 299, 300 stored in condensers, 384 , transmission of, economy in, 346 transmission of, mechanical of analogies illustrating, 311 , transmission of, table results achieved, 342 wasted in charging condenser from a source of constant P.D., 386 Equipotential surface, 79 - surfaces and lines of force due to long straight current (dia- gram), 451 surfaces due to circular cur- rent, 79 Equivalents, electro-chemical, 21 Eureka, resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Evershed and Vignole's new am- meter, 122 gravity control ammeter, 121 - Megger, 214 ohmmeter, diagram of con- nections of, 213 Exchange Telegraph Co.'s gravity Daniell's cell, 178 Experiment, graphically recording results of, 44 "Extra-Sec" cell, 198 Farad, 365 and unit of capacity, relation between, 367 , definition of, 364 Farads, capacity of spherical and plate air condensers in, 368 Ferranti ampere-hour meter, 305, 306 Fery's modified Leclanche cell, 193 Field, magnetic (see Magnetic field) of straight current, 453 Fields, magnetic (see Magnetic fields) Five-wire potentiometer, 403 Fleming's rule, 425 Flow, electric, water analogy of, 129 of electric current, direction of, i Flux density and current, relation between, 449 Flux density and lifting force, relation between, 446 density in magnet, 432 density in magnet, curves of, 433 Force, coercive, 464 -, controlling, definition of, 36 , laws of magnetic, 61 , lines of (see Lines of force) , magnetic lines of, 56 , magnetomotive, 454 Fuller's mercury bichromate cell, 1 86 Galvanic cells, 170 Galvanometer, absolute calibra tion of, 39 , Ayrton and Mather's reflect ing, 353 , ballistic, 349 , ballistic, constant of, 359 , ballistic, reflecting, 350 , calibrating, by direct com- parison with tangent galvano- meter, 43 , calibrating wire by differ en tial, 405 , calibration curve of, 45 circuit, diagram of differen- tial, 216 , differential, 216 for measuring potential dif- ferences, 153 , high resistance, with highly insulated coils, 390 , meaning of deflection on bridge, 227 -, measurement of current by, 36 - needles, pivot and fibre sus- pensions of, 99 , Paul's single-pivot, 117 , reflecting, 35 , relative calibration of, 39 - scale, constructing a, 50 - scale, protractor used in subdividing a, 50 , section of, with pivot and fibre suspension, 100 , sine, 100 , single-pivot, core, coil and pole pieces of, 1 1 8 , single-pivot moving coil, 117 , single-pivot section of, 118 , tangent, 36, 86 , tangent, absolute calibration of, 42 , tangent, adjusting coil of, 84 , tangent, calibration curve of, 84 INDEX 537 Galvanometer, tangent, comparison with a voltameter, 40 , tangent, constructing scale for, 87 , tangent, showing modes of supporting fibre, 37 , tangent, testing laws of varia- tion of sensibility, apparatus for, 9i , tangent, value in amperes of deflection of, 96 , tangent, variation of sensi- bility of, 90 , torsion, n , Walmsley and Mather's pro- portional, 1 06 Galvanometers, high resistance, reason for using for potential difference measurements, 154 , low resistance, reason for using for current measurements, J 54 of invariable sensibility, 107 , proportional, construction of, 105 , voltameters and, relative advantages of, 33 Galvanoscope, definition of, 34 (footnote) , measuring current with, 14 Gas pressure apparatus for testing distribution of, 130, 131 " Gassner's " dry cell, 195 G.E.C. cell, 197 Generator, E.M.F. of current, 314 , power absorbed in circuit exterior to, 315 , power developed by current, 312 Geometrical construction for find- ing strength of field at point on axis of circular coil, 77 German silver, relative conductivi- ties of, 244 , resistance of, 233, 235 specific resistance and tem- perature variation of, 241 Glass, dielectric strength of, 373 , resistivity of, 374 , specific inductive capacity of, 37 1 Glow lamp, 8, 10, 87 Gold, annealed, resistance of, 233, 235 , hard drawn resistance of, 233. 235 , relative conductivities of, 244 Gold-leaf electroscope, 166, 362 electroscope, Ayrton and Perry's, 167 Gold-leaf electroscopes, sensibility of, 168 Gold-silver alloy, resistance of, 233. 235 alloy, specific resistance and temperature variation of, 241 Gravitational potential gradient, 80 Gravities, specific, of mixtures of sulphuric acid and water, 515 Gravity control ammeter, 118 control ammeter, Evershed, 121 control ammeter, Nalder, 119 , Daniell's cell, 178 Grove's cell, 183, 184 Guard ring condenser, diagram of connections for testing, 394 Gutta-Percha, resistivity of, 374 , specific inductive capacity of, 37i H Hartmann and Braun hot-wire ammeter, 123 Heat and electric conductivities, comparison of, 243 equivalent of electric energy, measuring, 278 equivalents of energy, table of, 281 - produced by current, 277 Heating property of a current, 7 Hellesen dry cell, 196 Henry, definition, 437 Hibbert's magnetic balance, 61 High E.M.F. for large powers, importance of low resistance and, insulation shunt box, 254 resistance galvanometer, rea- son for using for potential difference measurements, 154 resistance galvanometer with highly insulated coils, 390 Hoffman's sulphuric acid volta- meter, 27 Horse-shoe magnet with curved iron pole pieces, 70 Hot-wire ammeter, 124 - ammeter, Hartmann and Braun, 123, 124 House, electrically lighted, part of plan of, 257 service energy meter, Thom- son, 300 Hydraulic analogue of condenser, 364 Hydrogen, specific inductive -capa- city of, 371 Hysteresis, apparatus for testing, 461 538 INDEX Hysteresis, calculation of energy loss by, 465 - curves, 462, 463 loop, 463 , loss of energy due to, 464 of iron, 460 I India rubber, resistivity of, 374 , specific inductive capacity of, Induced currents, direction of, 424 , Fleming's rule on, 425 currents, introductory re- marks on, 423 - currents, Lenz's law of, 425 Induction apparatus, magneto- electric, 424 coil, 438 - coil, diagram of, 439 - coil, Marconi lo-inch, 440 , mutual, 435 , mutual, unit of, 437 - of currents in parallel wires, 441 Inductivity, 389 Inductor, simple earth, 430 Industrial forms of potentiometer, 406 "Inert " cell, 199 Insulator, definition of, 4 Insulators, conductors and, 3 , dielectric strength of, 372 , resistivity of, 373, 388 International atomic weights, 22 (footnote) conference (1908) on units and standards, 500 - ohm, 144 Ion, definition of, 26 , electronegative, definition of, 27 27' -, electropositive, definition of, Iron, annealed, resistance of, 233, 235 filings, diagram showing lines of force round straight wire carrying current, 450 , hysteresis, of, 460 , magnetisation of, 442 et seq. , relative conductivity of, 244 ring, magnetisation curve for soft, 459 - ring, permeability of soft, 459 - ring, primary and secondary circuits on, 457 ring wound for ballistic tests, 456 rod picking up nails when current flows through wire coiled round it, 5 Iron, specific resistance and tem- perature variation of, 241 Ja Ja, resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Jars, Leyden, 376, 377 Joule, definition of, 277 Joule's law, 285 Joulemeter, or energy meter, clock form, 290 , Ayrton and Perry's original gaining clock, 291 Jute, resistivity of, 374 , specific inductive capacity of, 37i K Kahle's modification of Rayleigh's H-form of Clark's cell, 201 Kelvin's law, 346 Key, bridge, 225, 226 Kilowatt hour, 295 Kirchhoff's rules, 248 Kruppin, resistance of, 233, 235 Lacombe central zinc cell, 192 Lamp, glow, 7 stand and scale for ballistic galvanometer, 351 Law, Joule's, 285 , Kelvin's, 346 , Lenz's, 425 of energy, meter, 301 of magnetic pull, 446 , Ohm's, 138 , Ohm's apparatus for testing, 140 , Ohm's, verification of, 141 , sine, apparatus for testing, 101 , tangent, 89 , tangent, improved apparatus for testing, 90 , tangent, simple apparatus for testing, 89 Laws of magnetic force, 61 of resistance, 210 of variation of sensibility of tangent galvanometer, 91 Lead, pressed, resistance of, 233, 235 , relative conductivities of, 244 Leclanche agglomerate cell, 190 cell, 189 cell with porous pot, 190 cells, discharge curve for, 194 Lenz's Law, 425 INDEX 539 Leyden jars, 376, 377, 378 Lifting force and current turns, relation between, 442 - force and flux density, rela- tion between, 446 magnet, Witton Kramer, 439 magnets, 442 Lines of force and equipotential surfaces due to long straight current (diagram), 451 of force of bar magnet, 57 of force due to circular coil carrying current, 59 of force due to circular cur- rent, 79 of force due to current in circular coil, 58 of force, electric, and elec- trostatics, 8 1 of force, mapping out with compass needle, 71 of force round straight wire, 454 of force with two bar magnets, 69 - of magnetic force, 56 Linkage Constant, 431 lines of force, 425, 429 Linseed oil, specific inductive capacity of, 371 Load diagram of central station, 284 Logarithmic decrement, 357 London conference on electrical units and standards, 500 Loop, hysteresis, 463 Lord Rayleigh's H-form of Clark's cell, 201 Loss of energy due to hysteresis, 464 Low resistance and high B.M.F. for large powers, importance of, 333 - resistance galvanometers, reason for using, for current measurements, 154 - resistance, standard, 415 M McMi chad's form of acid volta- meter, 33 Magnet ammeters, permanent, 109 - apparatus for testing distri- bution of magnetism in a bar, 431 , curves of flux density in, 433 deflected by conductor carry- ing current, 4 , distribution of magnetism in bar, 431 , flux density in, 430 , horse-shoe, with curved iron pole pieces, 70 lines of force of bar, 57 Magnet, tinsel coiling itself round, when current flows through the tinsel, 5 Magnetic balance, Hibbert's, 61 circuit, 467 et seq. - field, 51, 55, 66, 68-78, 451 - field, absolute measurement of, 66 field, apparatus for testing strength of, along axis and in plane of circular coil, 74 field, arrangement for neutra- lising uniform, 72 field, comparing by magneto- meter method, relative strengths of different parts of, 74 field, comparing by vibration method, relative strengths of different parts of, 72 field curve of variation of strength along axis of coil, 75 - field, earth's, 37, 38, 97 field, geometrical construc- tion for finding strength of, at point on axis of circular coil, field strength, calculation of, 452 field, weighted compass needle for measuring, 73 fields, 53 - fields, magnetometer method of measuring, 55 fields, mapping, 68 fields, measuring, 5 - flux, 432 flux density, 432 force, laws of, 61 - induction, 424 - lines of force, 56 linkage, 425 moment, 63 - moment, absolute measure- ment of, 66 moment, measuring, 65 - moment, torsion apparatus for measuring, 64 needle, 10, 39, 73 needle, astatic, 352 needles, time of vibration of, 67 poles, strength of, 59 potential difference, definition of, 79 potential gradient, 80 - properties, testing, by ballistic method, 457 - property of a current, 7 - pull, 442 pull, apparatus, for testing, ' 444 540 INDEX Magnetic pull, law of, 446 - pull, mercury board for test- ing apparatus, 444, 445 reluctance, 468 saturation, 450 Magnetisation, curve for soft iron ring, 459 of iron, 442 Magnetism, distribution of, 431 , remanent, 463 Magneto-electric induction appara- tus, 424 Magnetometer, measuring magnetic moment by, 65 method, comparing relative strength of different parts of magnetic field by, 74 - method of measuring mag- netic fields, 55 , principle of, 56 , reflecting, 57 - with pointer, 56 Magnetomotive force, 454 Magnets, balance for finding strength of, 63 , lifting, 442 , lines of force with two bar, 69 Manganese peroxide, 189-191 Manganin, relative conductivity of, 244 , resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Mansb ridge condensers, 376 Mapping magnetic fields, 68 out lines of force with compass needle, 71 Marble resistivity of, 374 , specific inductive capacity of, 37 1 Marconi lo-in. induction coil, 440 Mather, Ayrton and (see Ayrton and Mather) Mather's form of sulphuric acid voltameter, 32 ratio-switch, 270, 272 (dia- gram) - simple apparatus for testing iron, 461 Maximum power, conditions for, 320 Measurement by potentiometer, advantages and disadvantages of, 420 of capacity, absolute, 387 of electric current, i - of E.M.P., 151 of potential difference, 133 of power, 283, 286 et seq., 420 of resistance, 210 Measurement of specific inductive capacity, 388 - of specific resistance, 231, 373 Measurements, decisions of 1882 conference on electrical, 482 , electrical, decisions of Paris congress on, 481 Measuring effects of a current, 8 electric current, 13-17 Mechanical analogies illustrating transmission of energy, 311 - analogies of condensers, 363 - analogy illustrating hysteresis, 464 equivalent of heat, 144, 278 Megger, 214 Meidinger cell, 177 Mercury bichromate cell, Fuller's, 1 66 ohm, 143 - , resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Metals, resistance of, 231, 234 , tables of resistance of, 233, 235 Meter, Aron energy, 293, 294 , Aron energy, connections of, 294 , Ayrton and Perry's, 290 , Bastian, 302 , Chamberlain and Hookham, 303, 304 , Ferranti, 305, 306 , Thomson, 297 et seq. Meters, energy, motor form, 296 , quantity, or ampere-hour, 302 Metre bridge, 222 - bridge, circular, 223 - bridge, diagram of, 222 Mica, dielectric strength of, 373 , resistivity of, 374 , specific inductive capacity of, 371 Micanite, dielectric strength of, 373 , resistivity of, 374 Microfarad, definition of, 366 Miiiotto's cell, 179, 1 80 Model of electric circuit composed of current generator and external resistance, 312 Moment, magnetic (see Magnetic moment) Motor form of energy meters, 296 Moving coil ammeter, Ayrton and Mather's, 115 - coil ammeter, Nalder Bros, and Thompson's, 114 coil ammeter, West on 's, working parts of, 113 INDEX Moving coil ammeters, 112 coil, galvanometer, single- pivot, 117 coil voltmeter, 160 Muirhead's telegraph battery, 182 - telegraph battery, composite copper and zinc plates for, 183 Mutual induction, 435 - induction apparatus, 435 induction, unit of, 437 N Nalder ammeter, damping device in, 120 - Bros, and Thompson's moving coil ammeter, 114 gravity control ammeter, 119 Naut, definition of, 384 N.C.S. dial potentiometer, 410 Needle, astatic, 352 - time of vibration of, 67 Needles, galvanometer, pivot, and fibre suspensions of, 99 Negative and positive potentials, 132 Neutralising uniform magnetic field, arrangement for, 72 Nichrome, resistance of, 233, 235 Nickel, annealed, resistance of, 233, 235 , relative conductivities of, 244 , specific resistance and tem- perature variation of, 241 Nickelin, resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Non-conductor, or insulator, defini- tion of, 4 Obach dry cell, 197 Ohm coil, standard (Reichsanstalt form), 265 , definition of, 143 , international, 144 , Paris (1884) Congress, 484 , the unit of resistance, 143 Ohmmeter, diagram of, 212 diagram of connections of Evershed, 213 Ohmmeters, 211 Ohm's law, 138 - law, apparatus for testing, 140 - law, application of to com- plete circuits, 149 law, diagram of circuit for testing, 142 law, verification of, 141 Olive oil, specific inductive capa- city of, 371 Order in Council defining electrical standards, 492 Paper, resistivity of, 374 Paper, specific inductive capacity of, 37i , squared, use of, 44 Paraffin oil, specific inductive capa- city of, 371 Paraffined paper, dielectric strength of, 373 Parallel conductors, currents in, 248 resistance, 246 , three condensers in, 383 - wires, induction of currents in, 441 Paris (1884) Congress ohm, 484 Paul's ratio box, diagram of con- nections, 414 single-pivot galvanometer, 117 P.D. and E.M.F. of battery, con- nection between, 313 Pendulum, law of, 67 Periodic time of vibration, 64-67, 355 Permanent magnet ammeter, 109 - magnet ammeter, Ayrton and Perry's, no Permeability, 460 - of soft iron ring, 459 Phosphor bronze, resistance, 233, 235 bronze, specific resistance and temperature variation of, 235 Pitch, specific inductive capacity of, 371 Pivot and fibre suspension of galvanometer needle, 99 - and fibre suspension, section of galvanometer with, 100 Plate-air condenser, of improved form, 392 Plates,, composite copper and zinc for Muirhead's telegraph battery, 183 Platinoid, relative conductivity of, 244 , resistance of, 233, 235 , specific resistance and tem- perature variation of, 241 Platinum, annealed, resistance, 233, , relative conductivities of, 244 , specific resistance and tem- perature variation of, 241 Platinum-iridium, alloy, resistance of, 233, 235 , alloy, specific resistance and temperature variation of, 241 542 INDEX Platinum -silver alloy, resistance of, 233, 235 alloy, specific resistance and temperature variation of, 241 Platinum thermometer, 239 Plotting calibration curve, method of, 45 Poggendorff 's method of comparing E.M.F.s, 400 - method, using only one gal- vanometer, 402 Polarisation, cell arranged for experiments in, 171 in simple cell, 170 Pole, balance for finding strength of, 62 - pieces, horse-shoe magnet with curved iron, 70 Poles, magnetic strength of, 59 Porcelain, dielectric strength of, 373 , resistivity of, 374 , specific inductive capacity of, 3?i Portable bridge with switch con- tacts, 272 Positive and negative potentials, 132 Post Office Wheatstone's bridge, 267 Potassium bichromate cells, 185 Potential difference, effects of, 133 difference, gravitational, 80 difference, magnetic, defini- measurement of, tion of, 79 difference, 133 - difference, measurements, reason for using high resistance galvanometers for, 154 difference, ratios of practical units of, 148 - differences and resistances current method of comparing, 153 differences, condensers for large, 375 - differences, diagram of ar- rangement for obtaining two, of known ratio, 381 differences, galvanometer for measuring, 153 divider, 380 divider for voltmeter calibra- tion, 412 divider, simple dial, 381 - divider, two-dial, 382 gradient, magnetic, 80 magnetic, difference of : equi- potential surface, 79 of Earth arbitrarily called nought, 132 Potentials, positive and negative, 132 Potentiometer, calibration of am- meter, by, 418 , calibration of voltmeter by, 411 , comparison of resistances by, 417 et seq. , Crompton, 408, 409 , dial, 409, 410 , five-wire, 403 -, industrial form of, described, 406 measurement, advantages and disadvantages of, 420 , N. C. S., 409, 410 , principle of, 403 , simple form of, 407 , wire, calibrating, 404 -, wire, materials for, 406 Power absorbed in circuit exterior to generator, 315 conditions for maximum, 320 , definition of, 282 developed by current genera- tor, 312 , distribution of, in electric circuit, 318 , electric, 282 , electric energy and, 273 , instruments for measuring, 286-289 , measurement of, by potentio- meter, 420 , transmission of, with an end- less belt, 310 unit of electric, 283 Practical units of potential differ- ence, ratios of, 148 Presspahn, dielectric strength of, ' 373 - , resistivity of, 374 Pressure, electric (see Potential difference) Primary and secondary circuits 011 iron ring, 457 cells (see Cells) Production of an electric current, methods of, 2 Properties of a current, apparatus for showing, 9 magnetic, 457-467 Proportional galvanometers, con- struction of, 105 - galvanometer, Walmsley and Mather's, 106 Protractor used in subdividing a galvanometer scale, 50 Pull, calculation of magnetic, 448 , magnetic, 442 INDEX 543 Quantities, comparison of, 361 Quantity, electric, measurement of, 348 , induced, and resistance of circuit, relation between, 426 , measurement of, by ballistic galvanometer, 353 meters, 302 , ratio of units of, 394 Quartz, resistivity of, 374 , specific inductive capacity of, Ratio boxes, 413, 414 of units of quantity, 394 Ratios of practical units of resist- ance, 145 Rayleigh's, Lord, form of Clark's cell, 201 Reading scale, direct, 51 - telescope and scale for re- flecting galvanometer, 35, 350, 352, 353 Recording results of an experiment graphically, 44 Rectangular plate condenser, 388 Reflecting ballistic galvanometer, 350 electrostatic voltmeter, Ayr- ton and Mather's, 157 -- - galvanometer, Ayrton and Mather's, 353 - galvanometer, making, 35 -- , high resistance, 390 magnetometer, 57 Reichsanstalt form of standard ohm coil, 265 Relative advantages of voltmeters and galvanometers, 33 Reluctance, 467 Remanent magnetism, 461 Residual magnetism, 464 Resin, resistivity of, 374 - oil, resistivity of, 374 - oil, specific inductive capacity of, 37i - , specific inductive capacity of, 37. 1 Resistance, 142 - - , absolute unit of, short history of, 473 - - , arrangement of cells to give maximum power to external circuit of fixed, 325 - , B.A. unit of, 477 > - boxes, 145, 146 - coil, Ayrton and Mather's non-inductive, 148 Resistance coils, 145 coils, standard, 264 combined, 245 , conductors of large specific, and small temperature coeffi- cient, 239 , its laws and measurement, 210 - of battery, finding, by con- denser method, 398 of cell condenser method of measuring, 398 of cells, 1 80 of circuit, relation between quantity induced and, 426 - of conductors, conditions affecting, 228 of conductors in series and in parallel, 244, 246 - of insulator, 373, 393 - of metals and alloys, 231, 234 - of metals, tables of, 233, 235 ohm, the unit of, 143 - parallel, 246 - ratios of practical units of, 145 - specific, 233 - standard low, 415 , variation of, with cross- section, 230 , variation of, with length, 229, 230 ., variation of, with material, 231 , variation of, with tempera ture, 236, 241 Resistances, comparing, by substi- tution method, 214 , comparing, voltmeter and ammeter methods, 210 , comparison of, by potentio- meter, 417 et seq. , potential differences and, cur- rent method of comparing, 153 , specific, of mixtures of sul- phuric acid and water, 515 , standard, for current measure- ments, 416 Resistivity of conductor, 233 - of insulators, 373, 388 Rheostat, carbon cloth, 419 , carbon plate, 418 Rolled condenser, 376 Rubber covered cable, dielectric strength of, 373 , dielectric strength of, 373 Rule, Fleming's, 425 Rules, Kirchhoff's, 248 Sagging wire magnifying system ot hot-wire ammeter, 124 544 INDEX St. Ivouis conference on electrical standards, 495 Saturation, magnetic, 450 Scale, constructing galvanometer, 50 - - , direct reading, 5 1 - - for reflecting galvanometer, 351, 352 - - for tangent galvanometer, constructing, 87 Secondary cells, 187 Self-induction, 438 Sensibilities, comparing two volt- meters of very different, 160 Sensibility, invariable, galvano- meters of, 107 -- of tangent galvanometer, vari- ation of, 90 Series, cells in, 3, 325 -- , condensers in, 383 , resistances in, 244 Shellac, resistivity of, 374 - , specific inductive capacity of, 37 1 Shunt, Ayrton and Mather's, 416 for strong currents, Ayrton and Mather's, 263 , increase of current produced by applying, 255 , principle of Ayrton and Mather's, 260 - with bridge, use of, 227 - box, high insulation, 254 - box, method of constructing, 253 box, plan of Ayrton and Mather's universal, 261 box, top of, showing parallel arrangement of shunts, 254 - box, top of, showing series arrangement of shunts, 255 box, universal, advantages of, 260 - box, universal, construction of, 260 - box, universal, recent form of, 262 Shunted voltmeter used as amme- ter, 164 Shunts, 251 , multiplying power of, 252 , universal, principle of, 259 Siemens dynamometer, 103 Silicum bronze, resistance of, 233 Silver, annealed, resistance of, 233, 235 , hard drawn, resistance of, 233, 235 Sine law, apparatus for testing, 101 - galvanometer, 100 Single pendulum, time of vibration of, 67 Single-pivot galvanometer, 117, in " vSix Block Agglomerate " cell, 198 Slate, resistivity of, 374 Soft iron ammeters, 118 Specific gravities, resistances and conductivities of mixtures of sulphuric acid and water, 515 - inductive capacity, measure- ment of, 388 resistance of insulation, 374 resistance of metals, 233, 235, 241 Sperm oil, specific inductive capa- city of, 371 Spherical condenser, standard, 396 Spring control ammeters, 118 Squared paper, use of, 44 Standard air condenser, 391, 392, 395 cells, 200 low resistance, 415 ohm coil (Reichsanstalt) form, 265 resistance coils, 264, 265 - resistances for current meas- urements, 416 spherical condenser, 396 Standards, Board of Trade elec- trical, short history of, 473 et seq. , Board of Trade committee on electrical, composition of, 488 , British Association's recom- mendations on electrical, 485 , Chicago conference on elec- trical, 490 , electrical, American specifi- cation defining, 497 , electrical, Berlin conference on, 499 , electrical, Orders in Council defining, 492 , electrical, St. Louis confer- ence on, 499 , international conference (1908) on units and, 500 Steel, relative conductivities of, 244 Storage cell, Edison nickel-iron, 188 Storage cells, 187 - cells, charge and discharge curves of, 189 Straight current, field of, 453 Strength, calculation of magnetic field, 452 -, current, 7 of current, defining, 12 of magnetic poles, 59 - of magnetic pull, 51, 66 Substitution method, comparing resistances by, 214 Sulphur, resistivity of, 374 INDEX 545 Sulphur dioxide, specific inductive capacity of, 371 , specific inductive capacity of, 37i Sulphuric acid voltameter, Ayrton's form of, 31 acid voltameter, description of practical forms of, 31 acid voltameter, Mather's form of, 32 acid voltameter, McMichael's form of, 33 Supply meters, 292 Surface, equipotential, 79 Swinburne wattmeter, 288 Symbols, Table oi, 527 Tables, calibration, 43 , chemical decomposition, 41 , dielectric strengths, 373 , dimensions of wires, etc., 517 , fundamental units, 514 , intensity of earth's field, 97 , materials for resistances, 241 , ratio of practical units of resistance, 145 , ratio of practical units of P.D., 148 , relations between electro- magnetic . and electrostatic unit, 515 , relative conductivities of metals, 244 : resistance, weight, length, 235 , resistivity of insulators, 314 , specific gravities, etc., of mixtures of pure sulphuric acid and distilled water, 515 , specific inductive capacities, 371 , specific resistances of metals, 233 . , symbols, 527 . , temperature co-efficients of copper, platinum, mercury, 238 . , values of ohm, 483-485 , windings, 518, 526 Tangent galvanometer, 36, 83, 84 galvanometer, absolute cali- bration of, 42 galvanometer, calibrating any galvanometer by direct com- parison with, 43 galvanometer, comparison with a voltameter, 40 galvanometer, constructing scale for, 87 galvanometer, scale for, 86 galvanometer sensibility, examples, 93 et seq. 2 J Tangent galvanometer, showing modgs of supporting fibre, 37 - galvanometer, value in am- peres of deflection of, 96 galvanometer, variation of sensibility of, 90, 91 law, 89 law, improved apparatus for testing, 90 law, simple apparatus for testing, 89 Telegraph battery, Muirhead's, 182 Telescope, reading and scale for reflecting galvanometer, 352 Temperature, variation of resist- ance with, 236, 241 Temporary magnetism, 464 Testing hysteresis, 457, 461 . - magnetic properties by bal- listic method, 457 - of copper, 238 - Ohm's law, apparatus for, 140 , polarisation, 171 resistance of cells, 181, 313 resistivity, 388 sine law, apparatus for, 101 specific inductive capacity, 388 - tangent law, apparatus for, 89, 90 temperature coefficient of wires, 241 variation laws of sensibility of tangent galvanometer, appara- tus for, 91 Thermo-electric currents, 406 (foot- note) Thermometer, platinum, 239 Thermometers, measuring current with, 1 6 Thomson energy meter, 299 Three-wire bridge, 224 Tin, pressed, resistance of, 233, 235 , relative conductivities of, 244 Tinsley cell, 208 Torque, definition of, 35 (footnote) Torsion apparatus for measuring magnetic moment, 64 electrostatic voltmeter, 135, 137 galvanometer, n Transmission, electrical efficiency of, and ratio of power received to power receivable, 343 of energy, economy in, 346 of energy, efficiency of elec- tric, 339 of energy, electric, 308 of energy, mechanical ana- logies illustrating, 311 546 INDEX Transmission of energy, table of results achieved, 342 - of power with an endless belt, 310 Tungsten, resistance of, 233, 235 Turpentine, oil of, specific inductive capacity of, 371 U Unit current, definition of, 18 of capacity and farad, rela- tion between, 367 of energy, Board of Trade, 294 of energy, electric, the joule, 7 of mutual induction, 437 of potential difference, 148 of power, electric, 283 of resistance, 143 of resistance, absolute, short history of, 473 quantities of electricity, rela- 277 tion between, 24 quantity, 24 Units and standards, International Conference (1908) on, 500 , C.G.S. and British systems of comparison of, 514 , electromagnetic and electro- static relations, between, 515 of capacity, 365 - of quantity, ratio of, 394 of resistance, ratios of prac- tical, 145 Universal shunt, Ayrton and Mather's, 259, 260 et seq., 416, 417 shunt box, advantages of, 260 shunt box, construction of, 260 shunt box, plan of Ayrton and Mather's, 261, 262 shunts, principle of, 259 Variation of resistance with cross- section, 230 of resistance with length, 229 of resistance with material, 231 of resistance with tempera- ture, 236 of sensibility of tangent gal- vanometer, 90 of strength of magnetic field along axis of coil, curve of, 75 Vibration, diagram of moderately damped, 358 , diagram of undamped, 358 , diagram of well-damped, 358 Vibration method, comparing rela- tive strength of different parts of magnetic field by, 72 Voltaic element, chemical action in simple, 170 Voltameter, Ayrton's form of sul- phuric acid, 31 , comparison of, with a tangent galvanometer, 40 , Hoffman's sulphuric acid, 27 , McMichael's form of acid, 33 , Mather's form of sulphuric acid, 32 , silver, desiccator used with, , silver, for measuring cur- rents, 19 (footnote) Volt-boxes, 411 Voltmeter and ammeter method of comparing resistances, 210 , Ayrton and Mather's reflect- ing electrostatic, 157 , Ayrton and Mather's zero electrostatic, 135-137 , calibrating, 160, 161, 162 , calibration of, by potentio- meter, 411 , calibration, potential divider for, 412 , explanation of, 155 , moving coil, 160 , shunted, used as ammeter, 164 Voltmeters, ammeters used as, 158 , comparing two, of very dif- ferent sensibilities, 160 current, and ammeters, re- sistances of, 158 used as ammeters, 163 W Walmsley and Mather's pro- portional galvanometer, 106 Water analogy of electric flow, 129 , curve connecting rate of flow of, with loss of head, 139 pressure, apparatus for testing distribution of, 128, 130 , specific inductive capacity of, 37 1 Watt, definition of, 284 Wattmeter, commercial forms of, 288 , diagram of, 287 , Elliott's deflectional, 289 , Swinburne, with cover re- moved, 288 Wattmeters, 286 Wax, paraffin, resistivity of, 374 INDEX 547 Wax, paraffin, specific inductive capacity of, 371 , sealing resistivity of, 374 , sealing specific inductive capacity of, 371 Weights, international atomic, 22 (footnote) Weston's cadmium cell, 205 cadmium cell (F. E. Smith's form), 206, 207 cell, 200 moving coil ammeter, work- ing parts of, 113 Wheatstone's bridge, bar pattern of, 270 bridge, construction of, 211 bridge, diagram of, 219 bridge dial, pattern of, 270, 271 bridge, ordinary forms of, 265, 266 bridge, portable, diagram of connections of, 270 bridge, portable forms of, 269 bridge, portable, with battery and galvanometer combined, 269 bridge, Post Office, 267, 268 bridge, principle of, 218 bridge, use of, 221 Windings table for cotton- covered (double) wire, 519 table for cotton - covered (single) wire, 518 table for enamel insulated wire, 524 table for enamel insulated and cotton-covered (double) wire, 526 table for enamel insulated and cotton-covered (single) wire, 525 table for silk covered (double) wire, 523 Windings table for silk covered (single) wire, 522 table for specially fine cotton (double) wire, 521 table for specially fine cotton (single) wire, 520 tables, 518 et seq. Wire, calibrating, by differential galvanometer, 405 , lines of force round straight, 454 , relation between lengths, resistances and weights of pure copper, 516 Wires, dimensions of, according to British Standard wire gauge (S.W.G.), 516 , windings tables for, 518 et seq. Witton-Kramer magnet, 443 Work done by current, 273 , electric unit of, 277 Zero electrodynamometer for small currents, 142 electrometer, Ayrton and Mather's, 135-137 Zinc, 233, 235, 244 amalgam, 201 , amalgamating, 176 , consumption of, in batteries, 185, 186, 200 , local or prejudicial action of, 175 , pressed, resistance of, 233, 235 , relative conductivities of, 244 , resistance of, 233, 235 , sulphate of, heat of forma- tion of, 207 PRINTED BY CASSELL & COMPANY, LIMITED, LA BELLE SAUVAGB, LONDON, B.C. 4 30.721 !0935