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The person charging this material is re- 
 sponsible for its return to the library from 
 which it was withdrawn on or before the 
 Latest Date stamped below. 
 
 Theft, mutilation, and underlining of books 
 
 are reasons for disciplinary action and may 
 result in dismissal from the University. 
 
 UNIVERSITY OF ILLINOIS LIBRARY AT URBANA-CHAMPAIGN 
 
 NOV +5 1977? 
 
 L161— O-1096 
 
 
 
Digitized by the Internet Archive 
 in 2021 with funding from 
 University of Illinois Urbana-Champaign 
 
 https://archive.org/details/elementarytreatiOOmaxw_1 
 
Clarendon Press Series 
 
 AN 
 
 ELEMENTARY TREATISE 
 
 “ON 
 
 ELECTRICITY 
 
 BY 
 
 JAMES CLERK MAXWELL, M.A. 
 
 LL.D. EDIN., D.C.L., F.R.SS. LONDON AND EDINBURGH 
 HONORARY FELLOW OF TRINITY COLLEGE 
 AND PROFESSOR OF EXPERIMENTAL PHYSICS IN THE UNIVERSITY OF CAMBRIDGE 
 
 EDITED BY 
 
 WILLIAM GARNETT, M.A 
 
 / / 9?» 
 FORMERLY FELLOW OF ST, JOHN’s COLLEGE, CAMBRIDGE b A 
 
 
 
 
 . \ 
 
 SECOND EDITION™; , 
 
 @xford 
 AT THE CLARENDON PRESS 
 
 1888 
 
 [ All rights reserved | 
 

 

 
 Mot of the following pages were written by the late Pro- 
 
 fessor Clerk Maxwell, about seven years ago, and some of 
 them were used by him as the text of a portion of his lectures 
 on Electricity at the Cavendish Laboratory. Very little ap- 
 pears to have been added to the MS. during the last three 
 or four years of Professor Maxwell’s life, with the exception 
 of a few fragmentary portions in the latter part of the work. 
 This was partly due to the very great amount of time and 
 thought which he expended upon editing the Cavendish papers, 
 nearly all of which were copied by his own hand, while the 
 experimental investigations which he undertook in order to 
 corroborate Cavendish’s results, and the enquiries he made 
 for the purpose of clearing up every obscure allusion in 
 Cavendish’s MS., involved an amount of labour which left 
 him very little leisure for other work. 
 
 When the MS. came into the hands of the present Editor, 
 the first eight chapters appeared to have been finished and 
 were carefully indexed and the Articles numbered. Chapters 
 IX and X were also provided with tables of contents, but the 
 Articles were not numbered, and several references, Tables, ete., 
 were omitted as well as a few sentences in the text. At the 
 end of the table of contents of Chapter X three points to be 
 treated were mentioned, viz. :—the Passage of Electricity at the 
 surfaces of insulators ; Conditions of spark, etc.; Electrification 
 by pressure, friction, rupture, etc.: no Articles corresponding 
 to these headings could be found in the text. Some portions 
 of Chapters IX and X formed separate bundles of MS., and 
 
v1 EDITORS PREFACE. 
 
 there was no indication of the place which they were intended 
 to fill. This was the case with Arts. 174-181 and 187-192. 
 Arts. 194-196 and 200 also formed a separate MS. with no 
 table of contents and no indication of their intended position. 
 
 It was for some time under consideration by the friends of 
 Professor Maxwell, whether the MS. should be published in 
 its fragmentary form or whether it should be completed by 
 another hand, so as to carry out as far as possible the author's 
 original design; but before any decision had been arrived at 
 it was suggested that the book might be made to serve the 
 purposes of students by a selection of Articles from Professor 
 Maxwell’s LHlectricity and Magnetism, so as to make it in a 
 sense complete for the portion of the subject covered by the 
 first volume of the last-mentioned work. In accordance with 
 this suggestion, a number of Articles have been selected from 
 the larger book and reprinted. These are indicated by a * 
 after the number of the Article. Arts. 93-98 and 141 are 
 identical with Arts. 118-123 and 58 of the larger treatise, but 
 these have been reprinted in accordance with directions con- 
 tained in Professor Maxwell’s MS. 
 
 In the arrangement of the Articles selected from the Hec- 
 tricity and Magnetism care has been taken to interfere as 
 little as possible with the continuity of the MS. of the present 
 work, and in some cases logical order has been sacrificed to 
 this object, so that some subjects which are treated briefly in 
 the earlier portions are reintroduced in the latter part of the 
 book. In Chapter XII some articles are introduced from the 
 larger treatise which may appear somewhat inconsistent with 
 the plan of this book; this has been for the sake of the prac- 
 tical value of the results arrived at. The latter part of the 
 note on pages 149 and 150 may be taken as Professor Maxwell's 
 own comment on the method proposed in Art. 186, written a few 
 years subsequently to that Article. 
 
 All references, for the accuracy of which Professor Maxwell 
 is not responsible, and all Tables, notes, or interpolations in- 
 
PREFACE TO SECOND EDITION. Vil 
 
 serted by the Editor, are enclosed in square brackets. This 
 system has not been carried out in the table of contents, but 
 _the portion of this contained in Professor Maxwell’s MS. is 
 stated above. 
 
 Of the Author’s Preface the portion here given is all that 
 has been found. 
 
 bate 
 CAMBRIDGE : 
 August, 1881. 
 
 PREFACE TO THE SECOND EDITION, 
 
 WEN it became necessary to reprint this Work, it appeared 
 to some desirable that certain changes should be made, 
 and especially that the articles taken from the larger work of 
 Professor Clerk Maxwell should be omitted. When the first 
 edition was published, very few books on electrical measurements 
 were available to the student ; but since that time, the literature 
 of the subject has developed enormously, and there is no longer 
 the same reason for extending this book beyond the limits of 
 the Author’s MS. In addition to this some of the Articles taken 
 from the larger book assume a knowledge on the part of the 
 student which is not to be obtained from the chapters of this 
 work. On careful consideration it was, however, thought best 
 that no change should be made, and, except for a few slight cor- 
 rections, the present edition is simply a reprint of the former. 
 W.G 
 NEWCASTLE-UPON-TYNE : 
 August, 1888. 
 
FRAGMENT OF AUTHORS PREFACE. 
 
 4 aes aim of the following treatise is different from that of my 
 larger treatise on electricity and magnetism. In the larger 
 treatise the reader is supposed to be familiar with the higher 
 mathematical methods which are not used in this book, and his 
 studies are so directed as to give him the power of dealing 
 mathematically with the various phenomena of the science. In 
 this smaller book I have endeavoured to present, in as compact 
 a form as I can, those phenomena which appear to throw light 
 on the theory of electricity, and to use them, each in its place, 
 for the development of electrical ideas in the mind of the reader. 
 In the larger treatise I sometimes made use of methods which 
 I do not think the best in themselves, but without which the 
 student cannot follow the investigations of the founders of the 
 Mathematical Theory of Electricity. I have since become more 
 convinced of the superiority of methods akin to those of F araday, 
 and have therefore adopted them from the first. 
 
 In. the first two chapters experiments are described which 
 demonstrate the principal facts relating to electric charge con- 
 sidered as a quantity capable of being measured. 
 
 The third chapter, ‘on electric work and energy, consists of 
 deductions from these facts. To those who have some acquain- 
 tance with the elementary parts of mathematics, this chapter 
 may be useful as tending to make their knowledge more precise. 
 Those who are not so prepared may omit this chapter in their 
 first reading of the book. 
 
 The fourth chapter describes the electric field, or the region in 
 which electric phenomena are exhibited. 
 
eet 
 aonrrwonre © © 
 
 Lis 
 18. 
 19. 
 
 20. 
 BL. 
 
 22. 
 
 SONATA WHY eG 
 
 CON TEN Is. 
 
 CHAPTER I. 
 
 . Exp. I. Electrification by friction 
 
 » LI. Electrification of a conductor 
 » II. Positive and negative electrification 
 » LV. Electrophorus 
 
 . Electromotive force 
 
 . Potential 
 
 . Potential of a oaths 
 
 . Of metals in contact .. 
 
 . Equipotential surfaces : 
 . Potential, pressure, and “aban = 
 . Exp. V. Gold-leaf electroscope .. 
 
 » Vv. Gold-leaf Bree renes nee Uaaieiad 
 
 . Quadrant electrometer 
 . Idio- and Hetero-Static 
 . Insulators 
 
 . Apparatus .. 
 
 CHAPTER II. 
 
 ON THE CHARGES OF ELECTRIFIED BODIES. 
 Exp. VI. Electrified body within a closed vessel 
 5, WII. Comparison of the charges of two bodies .. 
 » VIII. Electrification of inside of closed vessel eee 
 and opposite to that of enclosed body 
 , LX. To discharge a body completely ; 
 , XX. To charge a body with a given number of Hunan a 
 particular charge 
 Five laws of Electrical phenomena 
 I. In insulated bodies. 
 IT. In a system of bodies during conduction. 
 III. In a system of bodies during electrification. 
 
 1X 
 
 = 
 » 
 NIWNOHHAKRwWneY 
 
 a ee | 
 mo 1 BRB FS © 
 
 16 
 Wi 
 
 18 
 18 
 
 19 
 20 
 
 TY. Electrification of the two electrodes of a dielectric equal 
 
 and opposite. 
 
 V. No electrification on the internal surface of a conducting 
 
 vessel. 
 
23. 
 24. 
 
 25. 
 26. 
 27. 
 28. 
 29. 
 30. 
 31. 
 32. 
 
 33. 
 
 34. 
 35. 
 
 36. 
 
 37. 
 38. 
 39. 
 40. 
 41. 
 
 42. 
 43. 
 
 44. 
 45. 
 46, 
 47. 
 48. 
 49. 
 50. 
 51. 
 52. 
 
 CONTENTS. 
 
 CHAPTER III. 
 
 ON ELECTRICAL WORK AND ENERGY. 
 
 Definitions of work, of energy, of a conservative system .. 
 
 Principle of conservation of energy. Examples of the measure- 
 ment of work .. 
 
 Definition of electric potential 
 
 Relation of the electromotive force to the cquiporeeael suri 
 
 Indicator diagram of electric work .. .. 2. «2 « ow 
 
 Indicator diagram of electric work—continued .. 
 
 Superposition of electric effects .. 
 
 Charges and potentials of a system of sondern 
 
 Energy of a system of electrified bodies ate: 
 
 Work spent in passing from one electrical state to anothae a 
 
 
 
 dQ. 
 la dH 
 = (£P’) = = (£ P) ;—Green’s theorem 7 
 Increment of energy under increments of pone 
 
 AQ» | 
 L=a> ee eo * «ee as oe ae >> «eo ee ee 
 Reciprocity of potentials 
 
 Reciprocity of charges 
 
 Green’s theorem on potentials a heer 
 
 Mechanical work during the displacement of an ‘arated aan 
 
 Mechanical work during the displacement of a system the poten- 
 tials of which are maintained 
 
 CHAPTER IV. 
 THE ELECTRIC FIELD. 
 
 Two conductors separated by an insulating medium .. : 
 This medium called a dielectric medium, or, the electric field .. 
 
 _ EXPLORATION OF THE ELECTRIC FIELD. 
 Exp. XI. By a small electrified body 
 Exp. XII. By two disks .. 5 
 Electric tension .. s 
 Exp. XIII. Cininitea eet ahs . ye 
 Exp. XIV. Electromotive force ata point .. eo 
 Exp. XV. Potential at any point in the field. Two phe 
 
 sExp. XVI"= One'sphere® |...) ej ee) 
 
 Equipotential surfaces ysl dn oe ene 
 Reciprocal method. Exp. XVII. MEA AP. 
 
 Page 
 22 
 
 23 
 23 
 24 
 25 
 25 
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 29 
 
 29 
 
 30 
 30 - 
 
 31 
 
 31 
 32 
 32 
 oo 
 
 34 
 
 36 
 36 
 
 37 
 38 
 39 
 39 
 41 
 
 42 
 42 
 42 
 
Art, 
 53. 
 54. 
 
 55. 
 56. 
 57. 
 58. 
 59. 
 60. 
 61. 
 62. 
 63. 
 64. 
 65, 
 66. 
 67. 
 68. 
 69. 
 70. 
 (ae 
 72. 
 73. 
 74, 
 75. 
 
 Pte 
 78. 
 (ALP 
 80. 
 
 81. 
 82. 
 83. 
 84. 
 
 CONTENTS. 
 
 Exp. XVIII. Method founded on Theorem V. 
 Lines of electric force .. 
 
 CHAPTER V. 
 
 FARADAY’S LAW OF LINES OF INDUCTION. 
 
 Faraday’s Law 
 
 Hollow vessel 
 
 Lines of force 
 
 Properties of a tube of aatcnen 
 
 Properties of a tube of fad ote 
 
 Cells 
 
 Energy 
 
 Displacement 
 
 Tension 
 
 Analogies Fe 
 
 Analogies—continued .. 
 
 Limitation .. 
 
 Faraday’s cube 
 
 Faraday’s on rials 
 
 Current 
 
 Displacement 
 
 Theorems 
 
 Induction and force 
 
 + and — ends 
 
 Not cyclic : 
 
 In the inside of a Hollanr rintine 4505 a Sone any 
 electrified body the potential is uniform and there is no 
 electrification . 
 
 . In the inside of a hollow canbe ae te ii Sentai any 
 
 electrified body the potential is uniform and there is no 
 electrification—continued 
 Superposition 
 Thomson’s theorem 
 Example Ee 
 Induced electricity of ‘ aad Ara species .. 
 
 CHAPTER VI. 
 
 PARTICULAR CASES OF ELECTRIFICATION. 
 Concentric spheres 
 Unit of electricity. Law of eres 
 Electromotive force at a point .. 
 Definition of electromotive force 
 
 xl 
 Page 
 43 
 44 
 
 45 
 45 
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 55 
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 57 
 
 57 
 
 58 
 58 
 58 
 59 
 59 
 
 62 
 63 
 63 
 64 
 
Xi 
 
 Art. 
 
 85. 
 86. 
 Bi. 
 88. 
 89. 
 90. 
 SHE. 
 92. 
 93. 
 
 94. 
 
 95. 
 
 96. 
 
 SWE 
 US: 
 
 99. 
 100. 
 AV ORe 
 102. 
 108. 
 104. 
 105. 
 106. 
 
 107. 
 108. 
 109. 
 a0. 
 a1), 
 
 12. 
 113. 
 
 CONTENTS. 
 
 Coulomb’s law .. 
 
 Value of the potential dvel 0 a oan electrified ‘sphere 
 
 Capacity of a sphere.. 
 
 Two concentric spherical ried Teen iis ar 
 
 Two parallel planes .. 
 
 Force between planes as r 
 
 Thomson’s attracted disk electr eats ¥ 
 
 Inverse problem of electrostatics 
 
 Equipotential surfaces and lines of force for chara of 20 and 
 5 units (Plate I) 7 
 
 Equipotential surfaces and tines of ies for Oppente chanees 
 in the ratio of 4 to —1 (Plate IT) 
 
 Equipotential surfaces and lines of force for an electrical noun 
 in a uniform field of force (Plate IIT) . 
 
 Equipotential surfaces and lines of force for chareed of chee 
 electrified points (Plate IV) .. 
 
 Faraday’s use of the conception of lines of forte 
 
 Method employed in drawing the diagrams .. 
 
 CHAPTER VII. 
 
 ELECTRICAL IMAGES. 
 Introductory 
 Idea of an image ieecered an Pee 
 Electrical image at centre of sphere 
 External point and sphere 
 Two spheres 5 sph Set ene sa cee: 
 Calculation of potentials when enren are given a 
 Surface density induced on a sphere by an electrified point .. 
 Surface density on two spheres and condition for a neutral line 
 
 CHAPTER VIII. 
 
 CAPACITY. 
 Capacity of a condenser 
 Coefficients of condenser .. 
 Comparison of two condensers .. . 
 Thomson’s method with four condensers .. 
 Condition of null effect 
 
 CHAPTER IX. 
 
 ELECTRIC CURRENT. 
 
 Convection current with pith ball 
 Conduction current in a wire .. 
 
 Page 
 64 
 65 
 67 
 68 
 70 
 71 
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 72 
 
 73 
 74 
 75 
 
 75 
 76 
 (ie 
 
 80 
 80 
 81 
 82 
 84 
 85 
 86 
 87 
 
 89 
 90 
 91 
 91 
 93 
 
 96 
 96 
 
Art. 
 
 113. 
 114. 
 115. 
 
 116. 
 Ee by 
 
 118. 
 Pi, 
 120. 
 
 122. 
 123. 
 
 124. 
 125. 
 126. 
 127. 
 128. 
 129. 
 
 130. 
 Dob. 
 132. 
 133. 
 
 134. 
 135. 
 136. 
 137. 
 138. 
 139. 
 140. 
 141. 
 
 CONTENTS. Xl 
 
 Page 
 No evidence as to the velocity of electricity in the current .. 96 
 Displacement and discharge... aoe | 6 OR 
 Classification of bodies through eich slob niene passes... .. 98 
 Definition of the conductor, its electrodes, anode, and cathode 98 
 ieeornar electromotive force ..' 1. © 4. jase tee ee | 98 
 Metals, electrolytes, and dielectrics .. .. .. . .« « 99 
 1. Metals. 
 Ohm’s Law +) <b Se oy et Sede a mee a ee ne man!) 
 ET OLSON UR ee se tase cee ese te eee ae? RE 99 
 2. Hlectrolytes. 
 Anion and cation .. . ik a LP aA Camara eee TEL, 
 Electrochemical piealente ce aM (oct eet Ue neo LOL) 
 Faraday’s Laws 4 Pe Carat, SP Sit tec a EN 
 Force required for Porinlets Sunset Sa! Be pce Soe ree 
 Polarization es ee ares ie wry Meh sn) EN melee EOL 
 . Helmholtz’s penerimrenia eae ty Sa nce UY 
 Supposed inaccuracy of Faraday’s ihe ror moni ee OU 
 Measurement of resistance PL eee eee ce as ee LOS 
 Olim’s Law true for electrolytes .. .. .. «« ««  «. 1038 
 MermCTeC AUSIUA: 4) sts: h sc te) te ek ee ieee LOA 
 Theory of Clausius—continued yo CE) MEET OV es ASL 
 Velocities of ions 3 sei, tepod ones ho 
 Molecular conductivity he an SeGEBEE SP eri ay eee Ure ERIS 
 Kohlrausch’s experiments Re ee test os be (aL OG 
 EC TLOUS i 5 hea ee 2. ee. ae gh, ae an Ahm 
 3. Dielectrics. 
 Displacement .... Mos i eee LOG 
 Dielectric capacity of aris acmaine qeeeals he eee LOS 
 Dielectric capacity of solids, liquids, and gases te 109 
 Disruptive discharge. Mechanical and electrical araleeeat 
 inmate streneth. Brittleness-.... - ..+ .. «+ ©*.. ‘109 
 Residual charge ee rs rk ey mie, iia MELO 
 Eee et iatration we) oi ir. me aee re a a ASE 
 PETAR MELON Eth Ol (used. es (2). 55 Seep ia ees ont hog CLLS 
 Se Sea CEDAR GDSEI pag PS oy ei ga ee Ps RT 08 
 reer iy GR at cutee ee SE lel ioc a) te oes Sd DTS 
 Beer E Wea C: SOCIUITS VAPOULS) se ckhes cn fee oe has) a ITS 
 Meme TORLNGOLY (Ol (HSCS) Gsm ccm Cr cc ep! ee age eee G 
 Electric phenomena of Tourmaline .. .. .. « « « 117 
 
X1V CONTENTS. 
 
 Art. Page 
 142, Electric glow .. 0... 9% -\. +. [s. sy "3 
 148. Electric windmill .. 9...) 4.0 6s 0) 
 Electrified air .. .. 23) ager 
 Motion of france algae not ane to slacteee oo ne ee ees 
 144. To detect the presence of electrified air.. .. RA hod kis 
 
 145. Difference between positive and ae alectticneal eiiaria 318 2) 
 146. Discharge by a point on a conductor electrified by induction 120 
 
 147. Theelectric brush .: 5.) .. 405. ue, ae 
 148. The electric spark... :. 4. «. .. 1. “Se 
 149. Spectroscopic investigation .. .. .. « «« «. «8 sae 
 150*. Description of the voltaic battery ..  -.  .. . « «+ 122 
 151*. Electromotive force . - err 
 152*. Production of a ceric eee Me 
 153*. Magnetic action of the current ..  .. .. “= ]) eee 
 154*. The galvanometer 2... |... -- 6. wm, ee 
 155*. Linear conductors :. 9... 4.2 5. as a ee 
 156%, Ohm’s law =... Pele er 
 157*. Lisiear conductors in series ..  .. os. ++ se 
 158*. Linear conductors in pee ATC 6.0 we |. 
 
 Kirchhoff’s Laws .... 10 
 159*. Resistance of conductor of cue Paden ae . aah 
 
 CHAPTER X. 
 
 PHENOMENA OF AN ELECTRIC CURRENT WHICH FLOWS THROUGH 
 HETEROGENEOUS MEDIA. 
 
 160. Seebeck’s discovery Terre ll” 
 161. Law of Magnus... . 3 ee 
 162. Thermoelectric diagram re dermniion of thermoel siete power 130 
 163. Electromotive force measured by an area on the diagram .. 131 
 
 164. Cumming’s discovery .. -- «s+. ta 
 165. Thermal effects of the current ee 
 166. Peltier’s effect’... 2... > ve §o0 > 
 167. Thomson’s effect .. .. lee ee 
 168. Thomson’s analogy with a Ania 4 in a etate £ ery) 
 169. Le Roux’s experiments .. .. . . | ve) es 
 170. Expression of Peltier’s and Thomson’ 8 affect We 
 171. Heat produced at a junction depends on its temperature .. 135 
 172. Application of the second law of thermodynamics.. .. .. 136 
 173. Complete interpretation of the diagram .) 23 
 174. Entropy in thermodynamics .. =<... 1: ) \s) ee ee 
 
 175. Electric entropy) .) .. ha. (cas) nat ieee) hes 
 
Art. 
 
 176. 
 i Oe 
 178. 
 179. 
 180. 
 181. 
 182, 
 
 183. 
 184. 
 185. 
 186. 
 187. 
 188. 
 189. 
 190. 
 19f. 
 192. 
 
 133*. 
 
 194. 
 195. 
 196. 
 
 ac. 
 
 198*, 
 
 Loos 
 200. 
 
 201*. 
 202*. 
 203%. 
 
 204%. 
 205*. 
 206*. 
 207%. 
 208%. 
 
 CONTENTS. 
 
 Definition of entropy 
 
 Electric entropy equivalent to er aclenerict power 
 
 Thermoelectric diagram ., 
 
 Specific heat of electricity 
 
 Difference between iron and copper 
 
 Complete interpretation of the diagram 
 
 Thomson’s method of finding the E. M. F. at a forte in a 
 circuit Ls 
 
 Determination of the cat of ieee rues ra 
 
 E. M. F. between metal and electrolyte .. a 
 
 Electrolysis. Deposition of metal. Solution of metal.. 
 
 Heat generated or absorbed at anode and cathode 
 
 On the conservation of energy in electrolysis 
 
 Joule’s experiments 
 
 Loss of heat when current does emerald aoe 
 
 Electromotive force of electrochemical apparatus .. 
 
 Reversible and irreversible effects .. 
 
 Example from electrolysis of argentic ie 
 
 On constant voltaic elements. Daniell’s cell 
 
 CHAPTER XI. 
 
 METHODS OF MAINTAINING AN ELECTRIC CURRENT. 
 
 Enumeration of methods 
 
 The frictional electric machine i 
 
 On what the current depends. Use of silk laps - 
 
 Production of electrification by mechanical work. Nicholson! s 
 revolving doubler : 
 
 Principle of Varley’s and mhomeon: s slot ng maee 
 
 Thomson’s water-dropping machine 
 
 Holtz’s electrical machine ty 
 
 Theory of regenerators applied to eee aoohiniee ee 
 
 Coulomb’s torsion balance for measuring charges .. 
 
 Electrometers for measuring potentials. Snow-Harris’s and 
 Thomson’s .. 
 
 Principle of the Sneteaie Homer S erent Eleriromete 
 
 Heterostatic method 
 
 Measurement of the electric psntinl of a mall ambi 
 
 Measurement of the potential at a point in the air 
 
 Measurement of the potential of a conductor without seats 
 it 
 
 . 148 
 
 a LGOk 
 
 XV 
 Page 
 138 
 138 
 139 
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 140 
 141 
 
 142 
 143 
 
 144 
 145 
 146 
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 148 
 149 
 150 
 
 155 
 155 
 156 
 
 158 
 158 
 161 
 
 162 
 165 
 
 167 
 168 
 LOL 
 173 
 174 
 
 175 
 
XV1 
 
 209*. 
 
 210*. 
 
 BS. 
 212s, 
 PA ae 
 214”. 
 eto). 
 216 
 Zila: 
 218%. 
 219. 
 220%. 
 22a 
 222*. 
 B23") 
 224*. 
 225". 
 226%. 
 2276. 
 
 228%, 
 229*. 
 2307. 
 231 
 232 
 233%. 
 234”, 
 PaO. 
 236%. 
 23 Ta 
 238%. 
 
 CONTENTS. 
 
 CHAPTER XII. 
 
 ON THE MEASUREMENT OF ELECTRIC RESISTANCE. 
 
 Advantage of using material standards of resistance in elec- 
 
 trical measurements .. e Bs Ne 
 
 Different standards which have been eed and different syotene 
 
 which have been proposed .. 
 The electromagnetic system of units 
 Weber’s unit, and the British Association unit or Onn! 
 
 Professed value of the Ohm 10,000,000 metres per second .. 
 
 Reproduction of standards... 
 
 Forms of resistance coils 
 
 Coils of great resistance .. 
 
 Arrangement of coils in series 
 
 Arrangement in multiple arc .. ee 
 On the comparison of resistances. (1) Ohm’s method .. 
 (2) By the differential galvanometer 
 
 (3) By Wheatstone’s Bridge ost Rae ia het 
 Estimation of limits of error in the Wie: mination 
 
 Best arrangement of the conductors to be compared 
 
 On the use of Wheatstone’s Bridge . 
 Thomson’s method for the resistance of a ralyanoneg 
 
 Mance’s method of determining the resistance of a battery .. 
 
 Comparison of electromotive forces 
 
 CHAPTER XIII. 
 ON THE ELECTRIC RESISTANCE OF SUBSTANCES. 
 
 Metals, electrolytes and dielectrics 
 Resistance of metals 
 
 Table of resistance of metals .. 
 Resistance of electrolytes 
 
 Experiments of Paalzow .. 
 
 Experiments of Kohlrausch and Nippot 
 
 Resistance of dielectrics .. 
 
 Gutta-percha .. 
 
 Glass 
 
 Gases ; Biel om 
 Experiments of Waedamann aa Ratlmanne . 
 
 Note on the determination of the current in the galvanemeee 
 
 of Wheatstone’s Bridge 
 
 tue 
 
 Page 
 
 176 
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 197 
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 - 200 
 
 200 
 
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 . 205 
 
 205 
 
 206 
 
AN ELEMENTARY TREATISE 
 
 _ ON 
 
 ELECTRICITY. 
 
 \ 
 
 \ 
 \ 
 
 . SLT A 
 CHAPTER I. Sg OLS 
 
 
 
 EXPERIMENT I. 
 
 Electrification by Friction. 
 
 1.] Tax a stick of sealing-wax, rub it on woollen cloth or 
 flannel, and then bring it near to some shreds of paper strewed on 
 the table. The shreds of paper will move, the lighter ones will 
 raise themselves on one end, and some of them will leap up to the 
 sealinge-wax. Those which leap up to the sealing-wax sometimes 
 stick to it for awhile, and then fly away from it suddenly. It 
 appears therefore that in the space between the sealing-wax and 
 the table is a region in which small bodies, such as shreds of paper, 
 are acted on by certain forees which cause them to assume par- 
 ticular positions and to move sometimes from the table to the 
 sealing-wax, and sometimes from the sealing-wax to the table. 
 
 These phenomena, with others related to them, are called electric 
 phenomena, the bodies between which the forces are manifested are 
 said to be electrified, and the region in which the phenomena take 
 place is called the electric field. 
 
 Other substances may be used instead of the sealing-wax. A 
 piece of ebonite, gutta-percha, resin or shellac will do as well, and 
 so will amber, the substance in which these phenomena were first 
 noticed, and from the Greek name of which the word electric is 
 derived. 
 
 The substance on which these bodies are rubbed may also be 
 varied, and it is found that the fur of a cat’s skin excites them 
 better than flannel. 
 
 It is found that in this experiment only those parts of the 
 surface of the sealing-wax which were rubbed exhibit these phe- 
 4b, B ? 
 
2 ELECTRIFICATION BY FRICTION. Ne. 
 
 nomena, and that some parts of the rubbed surface are apparently 
 more active than others. In fact, the distribution of the electri- 
 fication over the surface depends on the previous history of the 
 sealing-wax, and this in a manner so complicated that it would be 
 very difficult to investigate it. There are other bodies, however, 
 which may be electrified, and over which the electrification is 
 always distributed in a definite manner. We prefer, therefore, in 
 our experiments, to make use of such bodies. 
 
 The fact that certain bodies after being rubbed appear to attract 
 other bodies was known to the ancients. In modern times many 
 other phenomena have been observed, which have been found to be 
 related to these phenomena of attraction. They have been classed 
 under the name of electric phenomena, amber, 7Aextpov, having 
 been the substance in which they were first described. 
 
 Other bodies, particularly the loadstone and pieces of iron and 
 steel which have been subjected to certain processes, have also been 
 long known to exhibit phenomena of action at a distance. ‘These 
 phenomena, with others related to them, were found to differ from 
 the electric phenomena, and have been classed under the name of 
 magnetic phenomena, the loadstone, payrys, being found in 
 Magnesia”. 
 
 These two classes of phenomena have since been found to be 
 related to each other, and the relations between the various pheno- 
 mena of both classes, so far as they are known, constitute the 
 science of Electromagnetism. 
 
 EXPERIMENT II. 
 
 Hlectrification of a Conductor. 
 
 2.| Take a metal plate of any kind (a tea-tray, turned upside 
 down, is convenient for this purpose) and support it on three dry 
 wine glasses. Now place on the table a plate of ebonite, a sheet 
 of thin gutta-percha, or a well-dried sheet of brown paper. Rub it 
 lightly with fur or flannel, lift it up from the table by its edges 
 and place it on the inverted tea-tray, taking care not to touch the 
 tray with your fingers, 
 
 * The name of Magnesia has been given to two districts, one in Lydia, the other in 
 Thessaly. Both seem to have been celebrated for their mineral products, and several 
 substances have been known by the name of magnesia besides that which modern 
 chemists know by that name. The loadstone, the touchstone, and meerschaum, 
 seem however to have been the principal substances which were called Magnesian 
 and magnetic, and these are generally understood to be Lydian stones. 
 
a ELECTRIFICATION OF A CONDUCTOR. 8 
 
 It will be found that the tray is now electrified. Shreds of 
 paper or gold-leaf placed below it will fly up to it, and if the 
 knuckle is brought near the edge of the tray a spark will pass 
 between the tray and the knuckle, a peculiar sensation will be felt, 
 and the tray will no longer exhibit electrical phenomena. It is 
 then said to be discharged. If a metal rod, held in the hand, be 
 brought near the tray the phenomena will be nearly the same. 
 The spark will be seen and the tray will be discharged, but the 
 sensation will be slightly different. 
 
 If, however, instead of a metal rod or wire, a glass rod, or stick 
 of sealing-wax, or a piece of gutta-percha, be held in the hand and 
 brought up to the tray there will be no spark, no sensation, and 
 no discharge. The discharge, therefore, takes place through metals 
 and through the human body, but not through glass, sealing-wax, 
 or gutta-percha. Bodies may therefore be divided into two 
 classes: conductors, or those which transmit the discharge, and 
 non-conductors, through which the discharge does not take place. 
 
 In electrical experiments, those conductors, the charge of which 
 we wish to maintain constant, must be supported by non-conducting 
 materials. In the present experiment the tray was supported on 
 wine glasses in order to prevent it from becoming discharged. 
 Pillars of glass, ebonite, or gutta-percha may be used as supports, 
 or the conductor may be suspended by a white silk thread. Solid 
 non-conductors, when employed for this purpose, are called ¢cusu- 
 ators. Copper wires are sometimes lapped with silk, and some- 
 times enclosed in a sheath of gutta-percha, in order to prevent 
 them from being in electric communication with other bodies. 
 They are then said to be insulated. 
 
 The metals are good conductors; air, glass, resins, gutta-percha, 
 caoutchoue, ebonite, paraffin, &c., are good insulators ; but, as we 
 shall find afterwards, all substances resist the passage of electricity, 
 and all substances allow it to pass though in exceedingly different 
 degrees. For the present we shall consider only two classes of 
 bodies, good conductors, and good insulators. 
 
 EXPERIMENT III. 
 
 Positive and Negative Electrification. 
 
 3.] Take another tray and insulate it as before, then after 
 discharging the first tray remove the electrified sheet from it and 
 place it on the second tray. It will be found that both trays are 
 
 B2 
 
+ POSITIVE AND NEGATIVE ELECTRIFICATION, [ 4. 
 
 now electrified. Ifa small ball of elder pith suspended by a white 
 silk thread * be made to touch the first tray, it will be immediately 
 repelled from it but attracted towards the second. If it is now 
 allowed to touch the second tray it will be repelled from it but 
 attracted towards the first. The electrifications of the two trays 
 are therefore of opposite kinds, since each attracts what the other 
 repels. Ifa metal wire, attached to an ebonite rod, be made to 
 touch both trays at once, both trays will be completely discharged. 
 If two pith balls be used, then if both have been made to touch 
 the same tray and then hung up near each other they are found 
 to repel each other, but if they have been made to touch different 
 trays they attract each other. Hence bodies when electrified in 
 the same way are repelled from each other, but when they are 
 electrified in opposite ways they are attracted to each other. 
 
 If we distinguish one kind of electrification by calling it positive, 
 we must call the other kind of electrification negative. We have 
 no physical reason for assigning the name of positive to one kind 
 of electrification rather than to the other. All scientific men, 
 however, are in the ‘habit of calling that kind of electrification 
 positive which the surface of polished glass exhibits after having 
 been rubbed with zine amalgam spread on leather. This is a 
 matter of mere convention, but the convention is a useful one, 
 provided we remember that it is a convention as arbitrary as 
 that adopted in the diagrams of analytical geometry of calling 
 horizontal distances positive or negative according as they are 
 measured towards the right or towards the left of the point of 
 reference. 
 
 In our experiment with a sheet of gutta-percha excited by 
 flannel, the electrification of the sheet and of the tray on which 
 it is placed is negative; that of the flannel and of the tray from 
 which the gutta-percha has been removed is positive. 
 
 In whatever way electrification is produced it is one or other of 
 these two kinds. | . 
 
 EXPERIMENT LV. 
 
 The Electrophorus of Volta. 
 
 4.| This instrument is very convenient for electrical experiments 
 and is much more compact than any other electrifying apparatus. 
 * J find it convenient to fasten the other end of the thread to a rod of ebonite 
 
 about 3mm. diameter. The ebonite is a much better insulator than the silk thread, 
 especially in damp weather. 
 
“a THE ELECTROPHORUS. 5 
 
 It consists of two disks, and an insulating handle which can be 
 screwed to the back of either plate. One of these disks consists 
 of resin or of ebonite in front supported by a metal back. In the 
 centre of the disk is a metal pin*, which is in connection with the 
 metal back, and just reaches to the level of the surface of the 
 ebonite. The surface of the ebonite is electrified by striking it 
 with a piece of flannel or cat’s fur. The other disk, which is 
 entirely of metal, is then brought near the ebonite disk by means 
 of the insulating handle. When it comes within a certain distance 
 of the metal pin, a spark passes, and if the disks are now separated 
 the metal disk is found to be charged positively, and the disk of 
 ebonite and metal to be charged negatively. 
 
 In using the instrument one of the disks is kept in connection 
 with one conductor while the other is applied alternately to the 
 first disk and to the other conductor. By this process the two 
 conductors will become charged with equal quantities of electricity, 
 that to which the ebonite disk was applied becoming negative, 
 while that to which the plain metal disk was applied becomes 
 positive. 
 
 ELECTROMOTIVE FORCE. 
 
 5.] Definition —Whatever produces or tends to produce a transfer 
 of Electrification is called Electromotive Force. 
 
 Thus when two electrified conductors are connected by a wire, 
 and when electrification is transferred along the wire from one 
 body to the other, the tendency to this transfer, which existed 
 before the introduction of the wire, and which, when the wire is 
 introduced, produces this transfer, is called the Electromotive 
 Force from the one body ¢o the other along the path marked out 
 by the wire. 
 
 To define completely the electromotive force from one point to 
 another, it is necessary, in general, to specify a particular path from 
 the one point to the other along which the electromotive force is 
 to be reckoned. In many cases, some of which will be described 
 when we come to electrolytic, thermoelectric, and electromagnetic 
 phenomena, the electromotive force from one point to another may 
 be different along different paths. If we restrict our attention, 
 
 * [This was introduced by Professor Phillips to obviate the necessity of touching 
 the carrier plate while in contact with the ebonite. ] 
 
6 ELECTRIC POTENTIAL. [6. 
 
 however, as we must do in this part of our subject, to the theory of 
 the equilibrium of electricity at rest, we shall find that the electro- 
 motive force from one point to another is the same for all paths 
 drawn in air from the one point to the other. 
 
 Evectric POTENTIAL. 
 
 6.] The electromotive force from any point, along a path drawn 
 in air, to a certain point chosen as a point of reference, is called 
 the Electric Potential at that point. 
 
 Since electrical phenomena depend only on differences of poten- 
 tial, it is of no consequence what point of reference we assume for 
 the zero of potential, provided that we do not change it during 
 the same series of measurements. 
 
 In mathematical treatises, the point of reference is taken at an 
 infinite distance from the electrified system under consideration. 
 The advantage of this is that the mathematical expression for the 
 potential due to a small electrified body is thus reduced to its 
 simplest form. 
 
 In experimental work it is more convenient to assume as a point 
 of reference some object in metallic connection with the earth, such 
 as any part of the system of metal pipes conveying the gas or 
 water of a town. 
 
 It is often convenient to assume that the walls, floor and ceiling 
 of the room in which the experiments are carried on has conducting 
 power sufficient to reduce the whole inner surface of the room to 
 the same potential. This potential may then be taken for zero. 
 When an instrument is enclosed in a metallic case the potential 
 of the case may be assumed to be zero. 
 
 Potential of a Conductor. 
 
 7.| If the potentials at different points of a uniform conductor 
 are different there will be an electric current from the places of 
 high to the places of low potential. The theory of such currents 
 will be explained afterwards (Chap. ix), At present we are dealing 
 with cases of electric equilibrium in which there are no currents. 
 Hence in the cases with which we have now to do the potential 
 at every point of the conductor must be the same. This potential 
 is called the potential of the conductor. 
 _ The potential of a conductor is usually defined as the potential 
 
10. | EQUIPOTENTIAL SURFACES. 7 
 
 of any point in the air infinitely close to the surface of the con- 
 ductor. Within a nearly closed cavity in the conductor the 
 potential at any point in the air is the same, and by making the 
 experimental determination of the potential within such a cavity 
 we get rid of the difficulty of dealing with points infinitely close 
 to the surface. 
 
 8.| It is found that when two different metals are in contact and 
 in electric equilibrium their potentials as thus defined are in general 
 different. Thus, if a copper cylinder and a zine cylinder are held 
 in contact with one another, and if first the copper and then the 
 zine cylinder is made to surround the flame of a spirit lamp, the 
 lamp being in connection with an electrometer, the lamp rapidly 
 acquires the potential of the air within the cylinder, and the 
 electrometer shews that the potential of the air at any point within 
 the zine cylinder is higher than the potential of the air within the 
 copper cylinder. We shall return to this subject again, but at 
 present, to avoid ambiguity, we shall suppose that the conductors 
 with which we have to do consist all of the same metal at the same 
 temperature, and that the dielectric medium is air. 
 
 9.| The region of space in which the potential is higher than 
 a certain value is divided from the region in which it is lower than 
 this value by a surface called an equipotential surface, at every 
 point of which the potential has the given value. 
 
 We may conceive a series of equipotential surfaces to be de- 
 scribed, corresponding to a series of potentials in arithmetical order. 
 The potential of any point will then be indicated by its position in 
 the series of equipotential surfaces. 
 
 No two different equipotential surfaces can cut one another, for 
 no point can have two different potentials. 
 
 10.] The idea. of electric potential may be illustrated by com- 
 paring it with pressure in the theory of fluids and with temperature 
 in the theory of heat. 
 
 If two vessels containing the same or different fluids are put into 
 communication by means of a pipe, fluid will flow from the vessel 
 in which the pressure is greater into that in which it is less till the 
 pressure is equalized. This however will not be the case if one 
 vessel is higher than the other, for gravity has a tendency to make 
 the fluid pass from the higher to the lower vessel. 
 
 Similarly when two electrified bodies are put into electric com- 
 munication by means of a wire, electrification will be transferred 
 from the body of higher potential to the body of lower potential, 
 
8 POTENTIAL, PRESSURE, AND TEMPERATURE.  [1I0O. 
 
 unless there is an electromotive force tending to urge electricity 
 from one of these bodies to the other, as in the case of zine and 
 copper above mentioned. 
 
 Again, if two bodies at different temperatures are placed in 
 thermal communication either by actual contact or by radiation, 
 heat will be transferred from the body at the higher temperature 
 to the body at the lower temperature till the temperature of the 
 two bodies becomes equalized. | 
 
 The analogy between temperature and potential must not be 
 assumed to extend to all parts of the phenomena of heat and 
 electricity. Indeed this analogy breaks down altogether when it is 
 applied to those cases in which heat is generated or destroyed. 
 
 We must also remember that temperature corresponds to a real 
 physical state, whereas potential is a mere mathematical quantity, 
 the value of which depends on the point of reference which we may 
 choose. To raise a body to a high temperature may melt or 
 volatilize it. To raise a body, together with the vessel which sur- 
 rounds it, to a high potential produces no physical effect whatever 
 on the body. Hence the only part of the phenomena of electricity 
 and heat which we may regard as analogous is the condition of the 
 transfer of heat or of electricity, according as the temperature or 
 the potential is higher in one body or in the other. | 
 
 With respect to the other analogy—that between potential and 
 fluid pressure—we must remember that the only respect in which 
 electricity resembles a fluid is that it is capable of flowing along 
 conductors as a fluid flows in a pipe. And here we may introduce 
 once for all the common phrase The Electric Kluid for the purpose 
 of warning our readers against it. It is one of those phrases, 
 which, having been at one time used to denote an observed fact, 
 was immediately taken up by the public to connote a whole system 
 of imaginary knowledge. As long as we do not know whether 
 positive electricity, or negative, or both, should be called a sub- 
 stance or the absence of a substance, and as long as we do not 
 know whether the velocity of an electric current is to be measured 
 by hundreds of thousands of miles in a second or by an hundredth of 
 an inch in an hour, or even whether the current flows from positive 
 to negative or in the reverse direction, we must avoid speaking of 
 the electric fluid. 
 
11. ] GOLD-LEAF ELECTROSCOPE. 9 
 
 ‘On ELEcCTROSCOPES. 
 
 11.] An electroscope is an instrument by means of which the 
 existence of electrification may be detected. All electroscopes are 
 eapable of indicating with more or less accuracy not only the 
 existence of electrification, but its amount. Such indications, how- 
 ever, though sometimes very useful in guiding the experimenter, 
 are not to be regarded as furnishing a numerical measurement of 
 the electrification. Instruments so constructed that their indi- 
 cations afford data for the numerical measurement of electrical 
 quantities are called Electrometers. 
 
 An electrometer may of course be used as an electroscope if it is 
 sufficiently sensitive to indicate the electrification in question, and 
 an instrument intended for an electroscope may, if its indications 
 
 are sufficiently uniform and regular, be used as an electrometer. 
 _ he class of electroscopes of simplest construction is that in 
 which the indicating part of the instrument consists of two light 
 bodies suspended side by side, which, when electrified, repel each 
 other, and indicate their electrification by separating from each 
 other. 
 
 The suspended bodies may be balls of elder pith, gilt, and hung 
 up by fine linen threads (which are better conductors than silk or 
 cotton), or pieces of straw or strips of metal, and in the latter case 
 the metal may be tinfoil or gold-leaf, thicker or thinner according 
 to the amount of electrification to be measured. 
 
 We shall suppose that our electroscope is of the most delicate 
 kind, in which gold leaves are employed (see Fig. 1). The indi- 
 cating apparatus /, /, is generally fastened to one 
 end of a rod of metal Z, which passes through an . 
 opening in the top of a glass vessel G. It then 
 hangs within the vessel, and is protected from 
 currents of air which might produce a motion of 
 the suspended bodies liable to be mistaken for 
 that due to electrification. 
 
 To test the electrification of a body the electrified 
 body is brought near the disk LZ at the top of the 
 metal rod, when, if the electrification is strong 
 enough, the suspended bodies diverge from one 
 another. 
 
 The glass case, however, is liable, as Faraday pointed out, to 
 become itself electrified, and when glass is electrified it is very 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
10 GOLD-LEAF ELECTROSCOPE. [er 
 
 difficult to ascertain by experiment the amount and the distribution — 
 of its electrification. There is thus introduced into the experiment 
 a new force, the nature and amount of which are unknown, and this 
 interferes with the other forces acting on the gold leaves, so that 
 their divergence can no longer be taken as a true indication of 
 their electrification. 
 
 The best method of getting rid of this uncertainty is to place 
 within the glass case a metal vessel which almost surrounds the 
 vold leaves, this vessel being connected with the earth. When the 
 vold leaves are electrified the inside of this vessel, it is true, becomes 
 oppositely electrified, and so increases the divergence of the gold 
 leaves, but the distribution of this electrification is always strictly 
 dependent on that of the gold leaves, so that the divergence of the 
 gold leaves is still a true indication of their actual electrical state. 
 A continuous metal vessel, however, is opaque, so that the gold 
 leaves cannot be seen from the outside. A wire cage, however, 
 may be used, and this is found quite sufficient to shield the gold 
 leaves from the action of the glass, while it does not prevent them 
 from being seen. , 
 
 The disk, J, and the upper part of the rod which supports the 
 gold leaves, and another piece of metal M/, which is connected with 
 the cage m, m, and extends beyond the case of the instrument, are 
 called the electrodes, a name invented by Faraday to denote the ways 
 by which the electricity gets to the vital parts of the instrument. 
 
 The divergence of the gold leaves indicates that the potential of 
 the gold leaves and its electrode is different from that of the sur- 
 rounding metal cage and its electrode. If the two electrodes are 
 connected by a wire the whole instrument may be electrified to any 
 extent, but the.leaves will not diverge. 
 
 EXPERIMENT V. 
 
 The divergence of the gold leaves does not of itself indicate 
 whether their potential is higher or lower than that of the cage; 
 it only shews that these potentials are different. To ascertain 
 which has the higher potential take a rubbed stick of sealing-wax, 
 or any other substance which we know to be negatively electrified, 
 and bring it near the electrode which carries the gold leaves. If 
 the gold leaves are negatively electrified they will diverge more as 
 the sealing-wax approaches the rod which carries them ; butif they 
 are positively electrified they will tend to collapse. If the electri- 
 fication of the sealing-wax is considerable with respect to that of 
 
aol GOLD-LEAF ELECTROSCOPE. 11 
 
 the gold leaves they will first collapse entirely, but will again 
 open out as the sealing-wax is brought nearer, shewing that they 
 are now negatively electrified. If the electrode M/ belonging to 
 the cage is insulated from the earth, and if the sealing-wax is 
 brought near it, the indications will be exactly reversed ; the leaves, 
 if electrified positively, will diverge more, and if electrified nega- 
 tively, will tend to collapse. 
 
 If the testing body used in this experiment is positively elec- 
 trified, as when a glass tube rubbed with amalgam is employed, the 
 indications are all reversed. 
 
 By these methods it is easy to determine whether the gold leaves 
 are positively or negatively electrified, or, in other words, whether 
 their potential is higher or lower than that of the cage. 
 
 12.| If the electrification of the gold leaves is considerable the 
 electric force which acts on them becomes much greater than their 
 weight, and they stretch themselves out towards the cage as far as 
 they can. In this state an increase of electrification produces no 
 visible effect on the electroscope, for the gold leaves cannot diverge 
 more. If the electrification is still further increased it often happens 
 that the gold leaves are torn off from their support, and the instru- 
 ment is rendered useless*. It is better, therefore, when we have to 
 deal with high degrees of electrification to use a less delicate in- 
 strument. A pair of pith balls suspended by linen threads answers 
 very well; the threads answer sufficiently well as conductors of elec- 
 tricity, and the balls are repelled from each other when electrified. 
 
 For very small differences of potential, electroseopes much more 
 sensitive than the ordinary gold-leaf electroscope may be used. 
 
 THOMSON’S QUADRANT ELECTROMETER. 
 
 13.] In Sir William Thomson’s Quadrant Electrometer the 
 indicating part consists of a thin flat strip of aluminium (see 
 Fig. 2) called the needle, attached to a vertical axle of stout 
 platinum wire. It is hung up by two silk fibres w, y, so as to 
 be capable of turning about a vertical axis under the action’ of 
 the electric force, while it always tends to return to a definite 
 position of equilibrium. The axis carries a concave mirror ¢ by 
 which the image of a flame, and of a vertical wire bisecting the 
 flame, is thrown upon a graduated scale, so as to indicate the 
 motion of the needle round a vertical axis. The lower end of 
 
 * [For the sake of safety the cage is often so arranged that the gold leaves touch 
 it and become discharged before diverging to their extreme limit. | 
 
12 QUADRANT ELECTROMETER. [13. 
 
 the axle dips into sulphuric acid contained in the lower part of 
 the glass case of the instrument, and thus puts the needle into 
 electrical connection with the acid. The lower end of the axle 
 also carries a piece of platinum, immersed in the acid which serves 
 to check the vibrations of the needle. The needle hangs within 
 a shallow cylindrical brass box, with circular apertures in the 
 centre of its top and bottom. This box 
 is divided into four quadrants, a, J, ¢, d, 
 which are separately mounted on glass 
 H stems, and thus insulated from the case 
 : and from one another. The quadrant 4 
 is removed in the figure to shew the 
 needle. The position of the needle, when 
 in equilibrium, is such, that one end is 
 half in the quadrant @ and half in e¢, 
 while the other end is half in 4 and 
 half in d. 
 
 The electrode / is connected with the 
 quadrant a and also with d through the 
 wire w. The other electrode, m, is con- 
 nected with the quadrants 4 and c. 
 
 The needle, w, is kept always at a 
 
 Fig. 2. high potential, generally positive. To 
 
 test the difference of potential between 
 
 any body and the earth, one of the electrodes, say m, is connected 
 to the earth, and the other, /, to the body to be tested. 
 
 The quadrants 4 and ¢ are therefore at potential zero, the 
 quadrants a and d are at the potential to be tested, and the needle 
 wis at a high positive potential. 
 
 The whole surface of the needle is electrified positively, and 
 the whole inner surface of the quadrants is electrified negatively, 
 but the greatest electrification and the greatest attraction is, other 
 things being equal, where the difference of potentials is greatest. 
 If, therefore, the potential of the quadrants a and d is positive, 
 the needle will move from @ and ¢ towards 4 and ¢ or in the 
 direction of the hands of a watch. If the potential of a and d is 
 negative, the needle will move towards these quadrants, or in the 
 opposite direction to that of the hands of a watch. 
 
 The higher the potential of the needle, the greater will be the 
 force tending to turn the needle, and the more distinct will be the 
 indications of the instrument. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
15.] IDIOSTATIC AND HETEROSTATIC INSTRUMENTS. 13 
 
 Idiostatic and Heterostatic Instruments. 
 
 14.] In the gold-leaf electroscope, the only electrification in 
 the field is the electrification to be tested. In the Quadrant 
 Electrometer the needle is kept always charged. Instruments in 
 which the only electrification is that which we wish to test, are 
 called Idiostatic. Those in which there is electrification inde- 
 pendent of that to be tested are called Heterostatic. In an 
 idiostatic instrument, like the gold-leaf electroscope, the indications 
 are the same, whether the potential to be tested is positive or 
 negative, and the amount of the indication is, when very small, 
 nearly as the square of the difference of potential. In a hetero- 
 static instrument, like the quadrant electrometer, the indication 
 is to the one side or to the other, as the potential is positive or 
 negative, and the amount of the indication is proportional to the 
 difference of potentials, and not to the square of that difference. 
 Hence an instrument on the heterostatic principle, not only in- 
 dicates of itself whether the potential is positive or negative, but 
 when the potential is very small its motion for a small variation 
 of potential is as great as when the potential is large, whereas in 
 the gold-leaf electroscope a very small electrification does not cause 
 the gold leaves to separate sensibly. 
 
 In Thomson’s Quadrant Electrometer there is a contrivance by 
 which the potential of the needle is adjusted to a constant value, 
 and there are other organs for special purposes, which are not 
 represented in the figure which is a mere diagram of the most 
 essential parts of the instrument. 
 
 On INSULATORS. 
 
 15.] In electrical experiments it is often necessary to support 
 an electrified body in such a way that the electricity may not 
 escape. Jor this purpose, nothing is better than to set it on a 
 stand supported by a glass rod, provided the surface of the glass 
 is quite dry. But, except in the very driest weather, the surface 
 of the glass has always a little moisture condensed on it. For 
 this reason electrical apparatus is often placed before a fire, before 
 it is to be used, so that the moisture of the air may not condense 
 on the warmed surface of the glass. But if the glass is made 
 too warm, it loses its insulating power and becomes a good 
 conductor. 
 
14 INSULATORS. [ 16. 
 
 Hence it is best to adopt a method by which the surface of the 
 glass may be kept dry without heating it. 
 
 The insulating stand in the figure consists of a glass vessel C, 
 with a boss rising up in the middle to which 
 is cemented the glass pillar aa. To the upper 
 part of this pillar is cemented the neck of the 
 bell glass B, which is thus supported so that 
 its rim is within the vessel C, but does not 
 touch it. The pillar @ carries the stand 4 on 
 which the body to be insulated is placed. 
 
 In the vessel C is placed some strong sul- 
 phurie acid ¢c, which fills a wide shallow moat 
 round the boss in the middle. The air within 
 the bell glass 4, in contact with the pillar a, is 
 thus dried, and before any damp air can enter 
 this part of the instrument, it must pass down between Cand 5 
 and glide over the surface of the sulphuric acid, so that it is 
 thoroughly dried before it reaches the glass pillar. Such an in- 
 sulating stand 1s very valuable when delicate experiments have to 
 be performed. For rougher purposes insulating stands may be 
 made with pillars of glass varnished with shellac or of sealing-wax 
 or ebonite. 
 
 16.] For carrying about an electrified conductor, 1t is very 
 convenient to fasten it to the end of an ebonite rod. Ebonite, 
 however, is very easily electrified. The slightest touch with the 
 hand, or friction of any kind, is sufficient to render its surface so 
 electrical, that no conclusion can be drawn as to the electrification 
 of the conductor at the end of the rod. 
 
 The rod therefore must never be touched but must be carried 
 by means of a handle of metal, or of wood covered with tinfoil, 
 and before making: any experiment the whole surface of the ebonite 
 must be freed from electrification by passing it rapidly through a 
 flame. ; 
 
 The sockets by which the conductors are fastened to the ebonite 
 rods, should not project outwards from the conductors, for the 
 electricity not only accumulates on the projecting parts, but creeps 
 over the surface of the ebonite, and remains there when the 
 electricity of the conductor is discharged. The sockets should 
 therefore be entirely within the outer surface of the conductors 
 as in Hie. 4, 
 
 It is convenient to have a brass ball (Fig. 4), one inch in 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
16. | APPARATUS. 15 
 
 diameter, a cylindrical metal vessel (Fig. 5) about three inches 
 in diameter and five or six inches deep, a pair of disks of tin 
 plate (Figs. 6, 7), two inches in diameter, and a thin wire about 
 a foot long (Fig. 8) to make connection between electrified bodies. 
 These should all be mounted on ebonite rods (penholders), one 
 eighth of an inch in diameter, with handles of metal or of wood 
 covered with tinfoil. 
 
 
 
CHAPTER II. 
 
 ON THE CHARGES OF ELECTRIFIED BODIES, 
 
 Experiment VI. 
 
 17.| Take any deep vessel of metal,—-a pewter ice-pail was used 
 by Faraday,—a piece of wire gauze rolled into a cylinder and set 
 on a metal plate is very convenient, as it allows any object within 
 it to be seen. Set this vessel on an insulating stand, and place 
 an electroscope near it. Connect one electrode of the electroscope 
 permanently with the earth or the walls of the room, and the 
 other with the insulated vessel, either permanently by a wire 
 reaching from the one to the other, or oceasionally by means of 
 a wire carried on an ebonite rod and made to touch the vessel 
 and the electrode at the same time. We shall generally suppose 
 the vessel in permanent connection with the elec- 
 troscope. The simplest way when a gold-leaf 
 electroscope is used is to set the vessel on the 
 top of it. 
 
 Take a metal ball at the end of an ebonite rod, 
 electrify it by means of the electrophorus, and 
 carrying it by the rod as a handle let it down into 
 the metal vessel without touching the sides. 
 
 As the electrified ball approaches the vessel the 
 indications of the electroscope continually increase, 
 I : but when the ball is fairly within the vessel, that 
 
 Fig. 9. is when its depth below the opening of the vessel 
 
 becomes considerable compared with the diameter 
 
 of the opening, the indications of the electroscope no longer in- 
 
 crease, but remain unchanged in whatever way the ball is moved 
 about within the vessel. 
 
 This statement, which is approximately true for any deep vessel, 
 is rigorously true for a closed vessel. This may be shewn by 
 
 
 
18.] COMPARISON OF CHARGES. 17 
 
 closing the mouth of the vessel with a metal lid worked by means 
 
 of a silk thread. If the electrified ball be drawn up and let 
 
 down in the vessel by means of a silk thread passing through a 
 
 hole in the lid, the external electrification of the vessel as in- 
 
 dicated by the electrometer will remain unchanged, while the ball 
 
 changes its position within the vessel. The electrifi- 
 
 cation of the gold leaves when tested is found to be | 
 
 of the same kind as that of the ball. | 
 Now touch the outside of the vessel with the finger, . 
 
 so as to put it in electric communication with the 
 
 floor and walls of the room. ‘The external electrifica- 
 
 tion of the vessel will be discharged, and the gold 
 
 leaves of the electroscope will collapse. If the ball be 
 
 now moved about within the vessel, the electroscope 
 
 will shew no signs of electrification; but if the ball 
 
 be taken out of the vessel without touching the sides, 
 
 the gold leaves will again diverge as much as they did Fig. 10. 
 
 during the first part of the experiment. Their electri- 
 
 fication however will be found to be of the opposite kind from 
 
 that of the ball. 
 
 Experiment VII. 
 
 To compare the charges or total Electrification of two 
 electrified balls. 
 
 18.] Since whatever be the position of the electrified bodies 
 within the vessel its external electrification is the same, it must 
 depend on the total electrification of the bodies within it, and 
 not on the distribution of that electrification. Hence, if two balls, 
 when alternately let down into the vessel, produce the same diver- 
 gence of the gold leaves, their charges must be equal. This may 
 be further tested by discharging the outside of the vessel when 
 the first ball is in it, and then removing it and letting the second 
 ball down into the vessel. If the charges are equal, the electro- 
 scope will still indicate no electrification. 
 
 If we wish to ascertain whether the charges of two bodies, 
 oppositely electrified, are numerically equal, we may do so by 
 discharging the vessel and then letting down both bodies into 
 it. If the charges are equal and opposite, the electroscope will not 
 be affected. 
 
 C 
 
18 TO DISCHARGE A BODY COMPLETELY. [19. 
 
 EXPERIMENT VIII. 
 
 When an electrified body is hung up within a closed metalhe vessel, 
 the total electrification of the inner surface of the vessel 1s equal 
 and opposite to that of the body. 
 
 19.| After hanging the body within the vessel, discharge the 
 external electrification of the vessel, and hang up the whole within 
 a larger vessel connected with the electroscope. 'The electroscope 
 will indicate no electrification, and will remain unaffected even 
 if the electrified body be taken out of the smaller vessel and moved 
 about within the larger vessel. If, however, either the electrified 
 body or the smaller vessel be removed from the large vessel, the 
 electroscope will indicate positive or negative electrification. 
 
 When an electrified body is placed within a vessel free of charge, 
 the external electrification is equal to that of the body. This 
 follows from the fact already proved that the internal electrifica- 
 tion is equal and opposite to that of the body, and from the cir- 
 cumstance that the total charge of the vessel is zero. 
 
 But it may also be proved experimentally by placing, first the 
 electrified body itself, and then the electrified body surrounded 
 by an uncharged vessel, within the larger vessel and observing 
 that the indications of the electroscope are the same in both 
 cases. 
 
 EXPERIMENT IX. 
 
 When an electrified body is placed within a closed vessel and then 
 put into electrical connection with the vessel, the body 1s com- 
 pletely discharged. 
 
 20.| In performing any of the former experiments bring the 
 electrified body into contact with the inside of the vessel, and 
 then take it out and test its charge by placing it within another 
 vessel connected with the electroscope. It will be found quite 
 free of charge. This is the case however highly the body may 
 have been originally electrified, and however highly the vessel 
 itself, the inside of which it is made to touch, may be electrified. 
 
 If the vessel, during the experiment, is kept connected with the 
 electroscope, no alteration of the external electrification can be 
 detected at the moment at which the electrified body is made to 
 touch the inside of the vessel. This affords another proof that 
 the electrification of the interior surface is equal and opposite to 
 
24, | MULTIPLE OF A GIVEN CHARGE. 19 
 
 that of the electrified body within it. It also shews that when 
 there is no electrified body within the surface every part of that 
 surface is free from charge. 
 
 EXPERIMENT X, 
 
 To charge a vessel with any number of times the charge of a given 
 electrified body. 
 
 21.| Place a smaller vessel within the given vessel so as to be 
 insulated from it. Place the electrified body within the inner 
 vessel, taking care not to discharge it. The ex- 
 terior charges of the inner and outer vessels will 
 now be equal to that of the body, and their in- 
 terior charges will be numerically equal but of 
 the opposite kind. Now make electric connection 
 between the two vessels. The exterior charge of 
 the inner vessel and the interior charge of the 
 outer vessel will neutralise each other, and the 
 outer vessel will now have a charge equal to that 
 of the body, and the inner vessel an equal and op- 
 posite charge. 
 
 Now remove the electrified body; take out the 
 inner vessel and discharge it; then replace it; 
 place the electrified body within it; and make contact between the 
 vessels. ‘The outer vessel has now received a double charge, and 
 by repeating this process any number of charges, each equal to 
 that of the electrified body, may be communicated to the outer 
 vessel. 
 
 To charge the outer vessel with electrification opposite to that 
 of the electrified body is still easier. We have only to place the 
 electrified body within the smaller vessel, to put this vessel for a 
 moment in connection with the walls of the room so as to dis- 
 charge the exterior electrification, then to remove the electrified 
 body and carry the vessel into the inside of the larger vessel and 
 bring it into contact with it so as to give the larger vessel its 
 negative charge, and then to remove the smaller vessel, and to 
 repeat this process the required number of times. 
 
 We have thus a method of comparing the electric charges of 
 different bodies without discharging them, of producing charges 
 equal to that of a given electrified body, and either of the same 
 
 C2 
 
 
 
 Fig. 11. 
 
20 LAWS OF ELECTRICAL PHENOMENA, By 
 
 or of opposite signs, and of adding any number of such charges 
 together. 
 
 22.] In this way we may illustrate and test the truth of the 
 following laws of electrical phenomena. 
 
 I. The total electrification or charge of a body or system of 
 bodies remains always the same, except in so far as it receives 
 electrification from, or gives electrification to other bodies. 
 
 In all electrical experiments the electrification of bodies is found 
 to change, but it is always found that this change arises from 
 defective insulation, and that as the means of insulation are im- 
 proved, the loss of electrification becomes less. We may therefore 
 assert that the electrification of a body cut off from electrical com- 
 munication with all other bodies by a perfectly insulating medium 
 would remain absolutely constant. 
 
 II, When one body electrifies another by conduction the total 
 electrification of the two bodies remains the same, that is, the one 
 loses as much positive or gains as much negative electrification as 
 the other gains of positive or loses of negative electrification. 
 
 For if the electric connection is made when both bodies are 
 enclosed in a metal vessel, no change of the total electrification 1s 
 observed at the instant of contact. 
 
 III. When electrification is produced by friction or by any other 
 known method, equal quantities of EE and of negative elec- 
 tricity are produced. 
 
 For if the process of electrification is conducted within the 
 closed vessel, however intense the electrification of the parts of 
 the system may be, the electrification of the whole, as indicated by 
 the electroscope connected with the vessel, remains zero. 
 
 IV. If an electrified body or system of bodies be placed within 
 a closed conducting surface (which may consist of the floor, walls, 
 and ceiling of the room in which the experiment is made), the in- 
 terior electrification of this surface is equal and opposite to the 
 electrification of the body or system of bodies. 
 
 V. If no electrified body is placed within the hollow conducting 
 surface, the electrification of this surface is zero. This is true, not 
 only of the electrification of the surface as a whole, but of every 
 part of this surface. 
 
 For if a conductor be placed within the surface and in contact 
 with it, the surface of this conductor becomes electrically continu- 
 ous with the interior surface of the enclosing vessel, and it 1s found 
 that if the conduetor is removed and tested, its electrification is 
 
22.| LAWS OF ELECTRICAL PHENOMENA. 21 
 
 always zero, shewing that the electrification of every part of an 
 interior surface within which there is no electrified body is zero. 
 
 By means of Thomson’s Quadrant Electrometer it is easy to 
 measure the electrification of a body when it is a million times less 
 than when charged to an amount convenient for experiment. 
 Hence the experimental evidence for the above statements shews 
 that they cannot be erroneous to the extent of one-millionth of the 
 principal electrifications concerned. 
 
CHAPTER III. 
 ON ELECTRICAL WORK AND ENERGY. 
 
 23.| Worx in general is the act of producing a change of con- 
 figuration in a material system in opposition to a force which 
 resists this change. 
 
 Energy is the capacity of doing work. 
 
 When the nature of a material system is such that if after the 
 system has undergone any series of changes it is brought back 
 in any manner to its original state, the whole work done by 
 external agents on the system is equal to the whole work done 
 by the system in overcoming: external forces, the system is called 
 a Conservative system. 
 
 The progress of physical science has led to the investigation of 
 different forms of energy, and to the establishment of the doctrine, 
 that all material systems may be regarded as conservative systems 
 provided that all the different forms of energy are taken into account. 
 This doctrine, of course, considered as a deduction from experiment, 
 can assert no more than that no instance of a non-conservative 
 system has hitherto been discovered, but as a scientific or science- 
 producing doctrine it is always acquiring additional credibility 
 from the constantly increasing number of deductions which have 
 been drawn from it, which are found in all cases to be verified. 
 In fact, this doctrine is the one generalised statement which ‘is 
 found to be consistent with fact, not in one physical science only, 
 but in all. When once apprehended‘it furnishes to the physical 
 enquirer a principle on which he may hang every known law 
 relating to physical actions, and by which he may be put in the 
 way to discover the relations of such actions in new branches of 
 science. Tor such reasons the doctrine is commonly called the 
 Principle of the Conservation of Energy. 
 
Bs. ELECTRIC POTENTIAL. 23 
 
 GENERAL STATEMENT OF THE CONSERVATION OF ENERGY. 
 
 24.| The total energy of any system of bodies is a quantity 
 which can neither be increased nor diminished by any mutual 
 action of those bodies, though it may be transformed into any of 
 the forms of which energy is susceptible. 
 
 If, by the action of some external agent, the configuration of the 
 system is changed, then, if the forces of the system are such as to 
 resist this change of configuration, the external agent is said to do 
 work on the system. In this case the energy of the system is 
 increased. If, on the contrary, the forces of the system tend to 
 produce the change of configuration, so that the external agent has 
 only to allow it to take place, the system is said to do work on 
 the external agent, and in this case the energy of the system is 
 diminished. Thus when a fish has swallowed the angler’s hook 
 and swims off, the angler following him for fear his line should 
 break, the fish is doing work against the angler, but when the fish 
 becomes tired and the angler draws him to shore, the angler is 
 doing work against the fish. 
 
 Work is always measured by the product of the change of 
 configuration into the force which resists that change. ‘Thus, when 
 a man lifts a heavy body, the change of configuration is measured 
 by the increase of distance between the body and the earth, and 
 the force which resists it is the weight of the body. The product 
 of these measures the work done by the man. If the man, instead 
 of lifting the heavy body vertically upwards, rolls it up an inclined 
 plane to the same height above the ground, the work done against 
 gravity is precisely the same; for though the heavy body is moved 
 a greater distance, it is only the vertical component of that 
 distance which coincides in direction with the force of gravity 
 acting on the body. 
 
 25.| If a body having a positive charge of electricity is carried 
 by a man from a place of low to a place of high potential, the 
 motion is opposed by the electric force, and the man does work on 
 the electric system, thereby increasing its energy. The amount 
 of work is measured by the product of the number of units of 
 electricity into the increase of potential in moving from the one 
 place to the other. 
 
 We thus obtain the dynamical definition of electric potential. 
 
24 ELECTROMOTIVE FORCE. [26. 
 
 The electric potential at a given point of the field is measured by 
 the amount of work which must be done by an external agent im 
 carrying one unit of positive electricity from a place where the potential 
 is zero to the given pornt. 
 
 This definition is consistent with the imperfect definition given 
 at Art. 6, for the work done in carrying a unit of electricity from 
 one place to another will be positive or negative according as the 
 displacement is from lower to higher or from higher to lower 
 potential. In the latter case the motion, if not prevented, will 
 take place, without any interference from without, in obedience to 
 the electric forces of the system. Hence the flow of electricity 
 along conductors is always from places of high to places of low 
 potential. 
 
 26.| We have already defined the electromotive force from one 
 place to another along a given path as the work done by the 
 electric forces of the system on a unit of electricity carried along 
 that path. It is therefore measured by the excess of the potential 
 at the beginning over that at the end of the path. 
 
 The electromotive force at a point is the force with which the 
 electrified system would act on a small body electrified with a unit 
 of positive electricity, and placed at that point. 
 
 If the electrified body is moved in such a way as to remain on 
 the same equipotential surface, no work is done by the electric 
 forces or against them. Hence the direction of the electric force 
 acting on the small body is such that any displacement of the body 
 along any line drawn on the equipotential surface is at right angles - 
 to the force. The direction of the electromotive force, therefore, is 
 at right angles to the equipotential surface. 
 
 The magnitude of this force, multiplied by the distance between 
 two neighbouring equipotential surfaces, gives the work done in 
 passing from the one equipotential surface to the other, that is to 
 say, the difference of their potentials. 
 
 Hence the magnitude of the electric foree may be found by 
 dividing the difference of the potentials of two neighbouring equi- 
 potential surfaces by the distance between them, the distance 
 being, of course, very small, and measured perpendicularly to 
 either surface. The direction of the force is that of the normal 
 to the equipotential surface through the given point, and is 
 reckoned in the direction in which the potential demdnishes. 
 
28.] DIAGRAM OF WORK, 25 
 
 Inpicator Dracram or Evectric Work. 
 
 27.| The indicator diagram, employed by Watt for measuring 
 the work done by a steam engine*, may be made use of in investi- 
 gating the work done during the charging of a conductor with 
 electricity. 
 
 Cc 
 
 
 
 Let the charge of the conductor at any instant be represented by 
 a horizontal line OC, drawn from the point O, which is called the 
 origin of the diagram, and let the potential of the conductor at 
 the same instant be represented by a vertical line C4, drawn from 
 the extremity of the first line, then the position of the extremity 
 of the second line will indicate the electric state of the conductor, 
 both with respect to its charge, and also with respect to its 
 potential. 
 
 If during any electrical operation this point moves along the 
 line AYGHB, we know not only that the charge has been increased 
 from the value OC to the value OD, and that the potential has 
 been increased from Cd to DB, but that the charge and the 
 potential at any instant, corresponding, say, to the point / of the 
 curve, are represented respectively by Ow and vf. 
 
 28.| Theorem. The work expended by an external agent in 
 bringing the increment of charge from the walls of the room to 
 the conductor is represented by the area enclosed by the base line 
 CD, the two vertical lines CA and DB, and the curve 4YGHB. 
 
 For let CD, the increment of the charge, be divided into any 
 number of equal parts at the points, 7, y, 2. 
 
 * Maxwell’s ‘ Theory of Heat,’ 4th ed., p. 102, 
 
26 WORK DONE IN CHARGING A CONDUCTOR. [ 29. 
 
 The value of the potential just before the application of the 
 charge Cx is represented by AC. Hence if the potential were to 
 remain constant during the application of the charge Cx, the work 
 expended in charging the conductor would be represented by the 
 product of this potential and the charge, or by the area ACrQ. 
 
 As soon as the charge Cz has been applied the potential is #f. 
 If this had been the value of the potential during the whole 
 process, the work expended would have been represented by 
 KCaF. But we know that the potential rises gradually during 
 the application of the charge, and that during the whole process 
 it is never less than CA or greater than wf. Hence the work 
 expended in charging the conductor is not less than 4CxQ, nor 
 greater than KCaxl. 
 
 In the same way we may determine the lower and higher 
 limits of the work done during the application of any other portion 
 of the entire charge. 
 
 We conclude, therefore, that the work expended in increasing 
 the charge from OC to OD is not less than the area of the figure 
 CAQKRGSHTD, nor greater than CKPYIGMHNLD. The differ- 
 ence between these two values is the sum of the parallelograms 
 KQ, LR, MS, NT, the breadths of which are equal, and their 
 united height is BV. Their united area is therefore equal to that 
 of the parallelogram NvV B. 
 
 By increasing without limit the number of equal parts into 
 which the charge is divided, the breadth of the parallelograms will 
 be diminished without limit. In the limit, therefore, the difference 
 of the two values of the work vanishes, and either value becomes 
 ultimately equal to the area CAYGHBD, bounded by the curve, 
 the extreme ordinates, and the base line. 
 
 This method of proof is applicable to any case in which the 
 potential is always increasing or always diminishing as the charge 
 increases. When this is not the case, the process of charging 
 may be divided into a number of parts, in each of which the 
 potential is either always increasing or always diminishing, and 
 the proof applied separately to each of these parts. 
 
 SUPERPOSITION OF ELEectric EFFECTS. 
 
 29.| It appears from Experiment VII that several electrified bodies 
 placed in a hollow vessel produce each its own effect on the 
 electrification of the vessel, in whatever positions they are placed. 
 
30.] SUPERPOSITION OF ELECTRIC EFFECTS. 27 
 
 From this it follows that one electric phenomenon at least, that 
 called electrification by induction, is such that the effect of the 
 whole electrification is the sum of the effects due to the different 
 parts of the electrification. The different electrical phenomena, 
 however, are so intimately connected with each other that we are 
 led to infer that all other electrical phenomena may be regarded 
 as composed of parts, each part being due to a corresponding: part 
 of the electrification. 
 
 Thus if a body A, electrified in a definite manner, would produce 
 a given potential, P, at a given point of the field, and if a body, B, 
 also electrified in a definite manner, would produce a potential, Q, 
 at the same point of the field, then when both bodies, still elec- 
 trified as before, are introduced simultaneously into their former 
 places in the field, the potential at the given point will be P+ Q. 
 This statement may be verified by direct experiment, but its most 
 satisfactory verification is founded on a comparison of its conse- 
 quences with actual phenomena. 
 
 As a particular case, let the electrification of every part of the 
 system be multiplied ~ fold. ‘The potential at every point of the 
 system will also be multiplied by x. 
 
 30.] Let us now suppose that the electrical system consists of a 
 number of conductors (which we shall call 4,, 4,, &c.) insulated from 
 each other, and capable of being charged with electricity. Let the 
 charges of these conductors be denoted by /,, #,, &c., and their 
 potentials by P,, P,, &e. 
 
 If at first the conductors are all without charge, and at potential 
 zero, and if at a certain instant each conductor begins to be charged 
 with electricity, so that the charge 
 increases uniformly with the time, 
 and if this process is continued till 
 the charges simultaneously become 
 F, for the first conductor, /, for the 
 second, and so on, then since the in- 
 erement of the charge of any con- 
 ductor is the same for every equal 
 interval of time during the process, 
 the increment of the potential of 
 each conductor will also be the same 
 for every equal increment of time, so that the line which represents, 
 on the indicator diagram, the succession of states of a given con- 
 ductor with respect to charge and potential will be described with 
 
 O C 
 Fig. 13 
 
28 ENERGY OF AN ELECTRIFIED SYSTEM. [ 31. 
 
 a velocity, the horizontal and vertical components of which remain 
 constant during the process. This line on the diagram is therefore 
 a straight line, drawn from the origin, which represents the initial 
 state of the system when devoid of charge and at potential zero, to 
 the point 4, which indicates the final state of the conductor when 
 its charge is /,, and its potential P,, and will represent the process 
 of charging the conductor 4,. The work expended in charging 
 this conductor is represented by the area OCA, or half the product 
 of the final charge /, and the final potential P,. 
 
 ENERGY OF A System oF ELEctrirtep Boptss. 
 
 31.] When the relative positions of the conductors are fixed, the 
 work done in charging them is entirely transformed into electrical 
 energy. If they are charged in the manner just described, the 
 work expended in charging any one of them is +#P, where £ re- 
 presents its final charge and P its final potential. Hence the work 
 expended in charging the whole system may be written 
 
 t2,P,+3L,P, + &e., 
 there being as many products as there are conductors in the system. 
 
 It is convenient to write the sum of such a series of terms in the 
 form | 3 3 (EP), 
 where the symbol = (sigma) denotes that all the products of the 
 form /P are to be summed together, there being one such product 
 for each of the conductors of which the system consists. 
 
 Since an electrified system is subject to the law of Conservation 
 of Energy, the work expended in charging it is entirely stored up 
 in the system in the form of electrical energy. The value of this 
 energy is therefore equal to that of the work which produced it, or 
 32 (HP). But the electrical energy of the system depends al- 
 together on its actual state, and not on its previous history. Hence 
 
 THroremM I, 
 
 The electrical energy of a system of conductors, in whatever way 
 they may have been charged, is half the sum of the products of the 
 charge into the potential of each conductor, 
 
 We shall denote the electric energy of the system by the symbol 
 
 Q, where = 2D (EP). i ives (1) 
 
33.] WORK DONE IN ALTERING CHARGES, 29 
 
 Work done in altering the charges of the system. 
 
 32.] We shall next suppose that the conductors of the system, 
 instead of being originally without charge and at potential zero, are 
 originally charged with quantities Z,, 2,, &c. of electricity, and are 
 at potentials P,, P,, &c. respectively. 
 
 When in this state let the charges of the conductors be changed, 
 each at a uniform rate, the rate being, in general, different for 
 each conductor, and let this process go on uniformly, till the 
 charges have become L,’, H,’, &c., and the potentials P,’, P,, &e. 
 respectively. 
 
 By the principle of the superposition of electrical effects the in- 
 crement of the potential will vary as the increment of the charge, 
 and the potential of each con- 
 ductor will increase or diminish 
 at a uniform rate from P to P’, 
 while its charge varies at a uni- 
 form rate from / to LH’. Hence 
 the line 4d’, which represents 
 the varying state of the con- 
 ductor during the process, is the 
 straight line which joins 4, the 
 point which indicates its original 
 state, with 4’, which represents § i ~ 
 its final state. The work spent Fig. 14, 
 in producing this increment of 
 charge in the conductor is represented by the area ACC’A’, or 
 CC’ (CA+C’A’), or (H’—F) 4 (P+ PP’), or, in words, it is the pro- 
 duct of the increase of charge and the half sum of the potentials 
 at the beginning and end of the operation, and this will be true for 
 every conductor of the system. 
 
 As, during this process, the electric energy of the system changes 
 from Q, its original, to Q’, its final value, we may write 
 
 GeO ES (HSIN (BE CEPY}, votes ss. cvcsges (2) 
 
 
 
 hence, 
 Tueorem IT, 
 
 The increment of the energy of the system is half the sum of the 
 products of the imerement of charge of each conductor into the 
 sum of its potentials at the beginning and the end of the process. 
 
 33.| If all the charges but one are maintained constant (by the 
 
 insulation of the conductors) the equation (2) is reduced to 
 
30 GREEN S THEOREM. [ 34. . 
 
 Q-Q=F-H) (P+), 
 or C8 4 (P'4 Py a so I) 
 
 If the increment of the charge is taken always smaller and smaller 
 till it ultimately vanishes, P’ becomes equal to P and the equation 
 may be interpreted thus :— 
 
 The rate of increase of the electrical energy due to the increase 
 of the charge of one of the conductors at a rate unity 1s numerically 
 equal to the potential of that conductor. 
 
 In the notation of the differential calculus this result is expressed 
 by the equation dQ, 
 
 dH 
 in which it is to be remembered that all the charges but one are 
 maintained constant. 
 
 34.| Returning to equation (2), we have already shewn that 
 
 Q=i2(£P) and QO =F (07); fee 
 
 we may therefore write equation (2) 
 
 
 
 
 
 —are ee ee ere i (4) 
 
 23 (PP) =143(EP)4+43 (LP —#LP+YP—EP). ... (6) 
 Cutting out from the equation the terms which destroy each 
 other, we obtain 3 (EP) = SH P) oe ome belte 
 
 or in words, 
 
 TueroremM III. 
 
 In a fixed system of conductors the sum of the products of the original 
 charge and the final potential of each conductor is equal to the sum 
 of the products of the final charge and the original potential, 
 
 This theorem corresponds, in the elementary treatment of electro- 
 statics, to Green’s Theorem in the analytical theory. By properly 
 choosing the original and the final state of the system we may 
 deduce a number of results which we shall find useful in our after- 
 work. 
 
 35.] In the first place we may write, as before, 
 
 43 {((P-BD(P'+P)} =43("P—EP+HP—EP); ... (8) 
 adding and subtracting the equal quantities of equation (7), 
 
 O = 3 (HP — EP), os. .s.s0es tee (9) 
 and the right-hand side becomes 
 SDLP —EP—EP+EP’), ecsceccsecaee (10) 
 
 or 32 {HW -A)\P+P)} = O-Q =e 2 {K+ 4) (LP —P), (11) 
 
 or in words, 
 
37.] RECIPROCITY OF POTENTIALS. aH! 
 
 THrorEM IV. 
 
 The increment of the energy of a fixed system of conductors is equal 
 to half the sum of the products of the increment of the potential 
 of each conductor into the sum of the original and final charges 
 of that conductor. 
 
 36.] If all the conductors but one are maintained at constant 
 potentials (which may be done by connecting them with voltaic 
 batteries of constant electromotive force), equation (11) is reduced to 
 
 YQ = FF AL) (P—P)y vrccccsnrseesee (12) 
 or ee Peel ten. sary eeek othe sateees (13) 
 
 If the increment of the potential is taken successively smaller and 
 smaller, till it ultimately vanishes, 4” becomes at last equal to H 
 and the equation may be interpreted thus :— 
 
 The vate of increase of the electrical energy due to the increase 
 of potential of one of the conductors at a rate unity is numerically 
 equal to the charge of that conductor. 
 
 In the notation of the differential calculus this result is expressed 
 by the equation 
 
 y 1 ive = E, Tree Sots eta tiaccetaetiataes (14) 
 in which it is to be remembered that all the potentials but one are 
 maintained constant. 
 
 37.] We have next to point out some of the results which may 
 be deduced from Theorem III. 
 
 If any conductor, as 4,, is insulated and without charge both in 
 the initial and the final state, then #, = 0 and H,’= 0, and therefore 
 
 HAR dA, 2a Bree Pe ye (15) 
 
 so that the terms depending on 4, disappear from both members 
 of equation (7). 
 
 Again, if another conductor, say 4,, be connected with the earth 
 both in the initial and in the final state, P,, = 0 and P,/ = 0, so that 
 HP. = 0 and #,/P,, = 0; 
 so that, in this case also, the terms depending on 4,, disappear from 
 
 both sides of equation (7). 
 
 If, therefore, all the conductors with the exception of two, say 
 A, and A,, are either insulated and without charge, or else connected 
 with the earth, equation (7) is reduced to the form 
 
 aD uerOP Und Pee BP ee, oie (O86) 
 
32 RECIPROCITY OF POTENTIALS AND CHARGES. [38. 
 
 Let us first suppose that in the initial state all the conductors 
 except 4, are without charge, and that in the final state all the con- 
 ductors except 4, are without charge. The equation then becomes 
 
 EP(=2'P,, (17) 
 jhe 
 or on = LE’ ’ 
 [If, therefore, B= EY, Pe eee 
 
 or in words, 
 
 THEOREM V. 
 
 In a system of fixed insulated conductors, the potential (P,) produced 
 in A, by a charge E communicated to A, is equal to the potential 
 (P,’) produced in A, by an equal charge EL communicated to A,. 
 
 This is the first instance we have met with of the reciprocal 
 relation of two bodies. There are many such reciprocal relations. 
 They occur in every branch of science, and they often enable us 
 to deduce the solution of new electrical problems from those of 
 simpler problems with which we are already familiar. Thus, if 
 we know the potential which an electrified sphere produces at a 
 point in its neighbourhood, we can deduce the effect which a small 
 electrified body, placed at that point, would haya in raising the 
 potential of the sphere. 
 
 38.| Let us next suppose that the original potential of 4, is P, 
 and that all the other conductors are kept at potential zero by 
 being connected with the walls of the room, and let the final 
 potential of 4, be P,’, that of all the others being zero, then in 
 equation (7) all the terms involving zero potentials will vanish, 
 and we shall have in this case also 
 
 BPH EP i, (18) 
 If, therefore, Pi=P,, E.= 2). (19) 
 
 or in words, 
 Turorem VI. 
 
 In a system of fixed conductors, connected, all but one, with the walls 
 of the room, the charge (E,) induced on A, when A, is raised to 
 the potential P, rs equal to the charge (E,/) induced on A, when 
 A, 8 raised to an equal potential (P,’). 
 
 39.] Asa third case, let us suppose all the conductors insulated 
 and without charge, and that a charge is communicated to A, 
 
40.] GREEN’S THEOREM ON POTENTIALS AND CHARGES. 33 
 
 which raises its potential to P. and that of 4, to fF, Next, let A, 
 be connected with the earth, and let a charge H,’ on A, induce the 
 charge ZL,’ on A,. 
 
 In equation (16) we have #.=0 and P’= 0, so that the left- 
 hand member vanishes and the equation becomes 
 
 DiS sald OS OT eee eee AN Pay (20) 
 or £ =— Ey 
 | tke ide. 
 Hence, if 1d ee ts oe He verte (29 
 
 or in words, 
 THEOREM VII, 
 
 If in a system of fixed conductors msulated and originally without 
 charge a charge be communicated to A, which raises its potential 
 to P.., unity, and that of A, to n, then if in the same system of 
 conductors a charge unity be communicated to A, and A, be 
 connected with the earth the charge induced on A, will be —n. 
 
 If, instead of supposing the other conductors 4, &e. to be all 
 insulated and without charge, we supposed some or all of them 
 to be connected with the earth, the theorem would still be true, 
 only the value of ~ would be different according to the arrange- 
 ment we adopted. 
 
 This is one of Green’s theorems. As an example of its applica- 
 tion, let us suppose that we have ascertained the distribution of 
 electric charge induced on the various parts of the surface of a 
 conductor by a small electrified body in a given position with unit 
 charge. Then by means of this theorem we can solve the following 
 problem :—The potential at every point of a surface coinciding in 
 position with that of the conductor being given, determine the 
 potential at a point corresponding to the position of the small 
 electrified body. 
 
 Hence, if the potential is known at all points of any closed 
 surface, it may be determined for any point within that surface if 
 there be no electrified body within it, and for any point outside if 
 there be no electrified body outside. 
 
 Mechanical work dene by the electric forces during the displacement 
 of a system of imsulated electrified conductors. 
 
 40.] Let 4,, 4, &e. be the conductors, #,, H, &c. their charges, 
 which, as the conductors are insulated, remain constant. Let P,, 
 P, &c. be their potentials before and P,’,P,' &c. their potentials 
 
 D 
 
34 MECHANICAL WORK DURING DISPLACEMENT. [4I. 
 
 after the displacement. The electrical energy of the system before 
 the displacement is = 3D (EP). .4.).0eeeee (22) 
 
 During the displacement the electric forces which act in the 
 same direction as the displacement perform an amount of work 
 equal to WV, and the energy remaining in the system is 
 
 Q'= 13 (EP). .....04: (23) 
 
 The original energy, Q, is thus transformed into the work 7 and 
 the final energy Q’, so that the equation of energy is 
 
 Q = WA Q, ccsscescosovecsnnastyeermaeine (24) 
 or : Wet 2 \E(P—P)). eee ee (25) 
 This expression gives the work done during any displacement, 
 small or large, of an insulated system. To find the force, we must 
 make the displacement so small that the configuration of the 
 system is not sensibly altered thereby. The ultimate value of the 
 quotient found by dividing the work by the displacement is the 
 value of the force resolved in the direction of the displacement. 
 
 Mechanical work done by the electric force during the displacement 
 of a system of conductors each of which is kept at a constant 
 potential. 
 
 41.] Let us begin by supposing each conductor of the system 
 insulated, and that a smad/ displacement is given to the system, by 
 which the potential is changed from P to P,. The work done 
 during this displacement is, as we have shewn, 
 
 W=13([B(P—P,))...000e (26) 
 
 Next, let the conductors remain fixed while the charges of the con- 
 ductors are altered from # to F,, so as to bring back the value of 
 the potential from P,te~P. Then we know by equation (7) that 
 
 * 
 
 “. > (LP— FE P,) = 0. .3.:.5++00 eee (27) 
 ray substituting for = (HP) in (26), | 
 : laese: W=+2|[(4,—H?P,]. .....eeeeeee 
 
 Performing these two operations alternately for any number of 
 times, and distinguishing each pair of operations by a suffix, we 
 find the whole work : 
 
 Wo Wy WAC. vs eden van sees vagenn soy tent (29) 
 = $2[(2,-—F)P,]+43[(4,—4)P,]+&e. ....., (30) 
 
 By making each of the partial displacements small enongh, the 
 
 @ 
 
A4I.] MECHANICAL WORK DURING DISPLACEMENT. 35 
 
 values of P,, P, &c. may be made to approach without limit to P, 
 the constant value of the potential, and the expression becomes 
 
 W=t3|[(L,—F)P]+42[( 4,—£,)P] + &e.4+23[( — #,_,)P],(31) 
 where / is the value of / after the last operation. The final result 
 is therefore USS 1 eres Repecres (32) 
 
 which is an expression giving the work done during a displacement 
 of any magnitude of a system of conductors, the potential of each of 
 which is maintained constant during the displacement. 
 We may write this result 
 WS SIE) ESC EP )y eo cisscscecssensee (33) 
 
 or LE Bast (ee A is yo St WER ERE TREE ». (34) 
 
 or the work done by the electric forces is equal to the zucrease of 
 the electric energy of the system during the displacement when 
 the potential of each conductor is maintained constant. When the 
 charge of each conductor was maintained constant, the work done 
 was equal to the decrease of the energy of the system. 
 
 Hence, when the potential of each conductor is maintained con- 
 stant during a displacement in which a quantity of work, VV, is 
 done, the voltaic batteries which are employed to keep the poten- 
 tials constant must do an amount of work equal to 2W. Of this 
 energy supplied to the system, half is spent in increasing the 
 energy of the system, and the other half appears as mechanical 
 work. 
 
 
 
 D 2 
 
CHAPTER IV. 
 THE ELECTRIC FIELD. 
 
 42.] WE have seen that, when an electrified body is enclosed in 
 a conducting vessel, the total electrification of the interior surface 
 of the surrounding vessel is invariably equal in numerical value 
 but opposite in kind to that of the body. This is true, however 
 large this vessel may be. It may be a room of any size having 
 its floor, walls and ceiling of conducting matter. Its boundaries 
 may be removed further, and may consist of the surface of the ~ 
 earth, of the branches of trees, of clouds, perhaps of the extreme 
 limits of the atmosphere or of the universe. In every case, where- 
 ever we find an electrified insulated body, we are sure to find at 
 the boundaries of the insulating medium, wherever they may be, 
 an equal amount of electrification of the opposite kind. 
 
 This correspondence of properties between the two limits of 
 the insulating medium leads us to examine the state of this 
 medium itself, in order to discover the reason why the electrifica- 
 tion at its inner and outer boundaries should be thus related. In 
 thus directing our attention to the state of the insulating medium, 
 rather than confining it to the inner conductor and the outer sur- 
 face, we are following the path which led Faraday to many of his 
 electrital discoveries. 
 
 43" | To render our conceptions more definite, we “sal begin by 
 supposing a conducting body electrified positively and insulated 
 within a hollow conducting vessel. The space between the body 
 and the vessel is filled with air or some other insulating medium. 
 We call it an zsulating medium when we regard it simply as 
 retaining the charge on the surface of the electrified body. When 
 we consider it as taking an important part in the manifestation 
 of electric phenomena we shall use Faraday’s expression, and call 
 it a dielectric medium. Finally, when we contemplate the region 
 
44. | EXPLORATION OF THE ELECTRIC FIELD, 3” 
 
 occupied by the medium as being a part of space in which electric 
 phenomena may be observed, we shall call this region the Electric 
 Field. By using this last expression we are not obliged to figure 
 to ourselves the mode in which the dielectric medium takes part in 
 the phenomena. If we afterwards wish to form a theory of the 
 action of the medium, we may find the term dielectric useful. 
 
 EXPLORATION OF THE Exxectric Fre. 
 
 EXPERIMENT XI. 
 (a) Exploration by means of a small electrified body. 
 
 44.| Electrify a small round body, a gilt pith ball, for example, 
 and carry it by means of a white silk thread into any part of the 
 field. If the ball is suspended in such a way that the tension of 
 the string exactly balances the weight of the ball, then when the 
 ball is placed in the electric field it will move under the action of a 
 new force developed by the action of the electrified ball on the 
 electric condition of the field. This new force tends to move the 
 ball in a certain direction, which is called the direction of the 
 electromotive force. 
 
 If the charge of the ball is varied, the force is sensibly pro- 
 portional to the charge, provided this charge is not sufficient to 
 produce a sensible disturbance of the state of electrification of the 
 system. If the charge is positive, the force which acts on the ball 
 is, on the whole, from the positively electrified body, and towards 
 the negatively electrified walls of the room. If the charge is 
 negative, the force acts in the opposite direction. 
 
 Since, therefore, the force which acts on the ball depends partly 
 on the charge of the ball and partly on its position and on the 
 electrification of the system, it is convenient to regard this force as 
 the product of two factors, one being the charge of the ball, and 
 the other the electromotive force at that point of the field which is 
 occupied by the centre of the ball. 
 
 This electromotive force at the point is definite in magnitude 
 and direction. A positively charged body placed there tends to 
 move in the positive direction of the line representing the force. A 
 negatively charged body tends to move in the opposite direction. 
 
a8 EXPLORATION OF THE ELECTRIC FIELD. [45. 
 
 EXPERIMENT XII. 
 
 (6) Exploration by means of two disks. 
 
 45.] But the electromotive force not only tends to move elec- 
 trified bodies, it also tends to transfer electrification from one part 
 of a body to another. 
 
 Take two small equal thin metal disks, fastened to handles of 
 shellac or ebonite ; discharge them and place them face to face in 
 the electric field, with their planes perpendicular to the direction of 
 the electromotive force. Bring them into contact, then separate 
 
 them and remove them, and 
 
 test first one and then the 
 
 { other by introducing them 
 
 : into the hollow vessel of Ex- 
 periment VII. It will be 
 
 + 
 found that each is charged, 
 - and that if the electromotive 
 force acts in the direction 4B, 
 . the disk on the ide of 4 will 
 Fig. 15. be charged negatively, and 
 
 that on the side of B posi- 
 tively, the numerical values of these charges being equal. This shews 
 that there has been an actual transference of electricity from the one 
 disk to the other, the direction of this transference being that of 
 the electromotive force. This experiment with two disks affords a 
 much more convenient method of measuring the electromotive force 
 at a point than the experiment with the charged pith ball. The 
 measurement of small forces is always a difficult operation, and 
 becomes almost impossible when the weight of the body acted on 
 forms a disturbing force and has to be got rid of by the adjust- 
 ment of counterpoises. The measurement of the charges of the 
 disks, on the other hand, is much more simple. | 
 The two disks, when in contact, may be regarded as forming a 
 single disk, and the fact that when separated -they are found to 
 have received equal and opposite charges, shews that while the 
 disks were in contact there was a distribution of electrification 
 between them, the electrification of each disk being opposite to 
 that of the body next to it, whether the insulated body, which is 
 charged positively, or the inner surface of the surrounding vessel, 
 which is charged negatively. 
 
47.| ELECTRIC TENSION. 39 
 
 Electric Tension. 
 
 46.| The two disks, after being brought into contact, tend to 
 separate from each other, and to approach the oppositely electrified 
 surfaces: to which they are opposed. The force with which they 
 tend to separate is proportional to the area of the disks, and it 
 increases as the electromotive force increases, not, however, in the 
 simple ratio of that force, but in the ratio of the square of the 
 electromotive force. 
 
 The electrification of each disk is proportional to the electro- 
 motive force, and the mechanical force on the disk is proportional 
 to its electrification and the electromotive force conjointly, that is, 
 to the square of the electromotive force. 
 
 This force may be accounted for if we suppose that at every 
 point of the dielectric at which electromotive force exists there is 
 a tension, like the tension of a stretched rope, acting in the direc- 
 tion of the electromotive force, this tension being proportional to 
 the square of the electromotive force at the point. This tension 
 acts only on the outer side of each disk, and not on the side which 
 is turned towards the other disk, for in the space between the disks 
 there is no electromotive force, and consequently no tension. 
 
 The expression Electric Tension has been used by some writers 
 in different senses. In this treatise we shall always use it in the 
 sense explained above,—the tension of so many pounds’, or grains’, 
 weight on the square foot exerted by the air or other dielectric 
 medium in the direction of the electromotive force. 
 
 Experiment XIII. 
 Couloml’s Proof Plane. 
 
 47.| If one of these disks be placed with one of its flat sur- 
 faces in contact with the surface of an electrified conductor and 
 then removed, it will be found to be charged. If the disk 1s 
 very thin, and if the electrified surface is so nearly flat that the 
 whole surface of the disk lies very close to it, the charge of the disk 
 will be nearly equal to that of the portion of the electrified surface 
 which it covered. If the disk is thick, or does not lie very close to 
 the electrified surface, its charge, when removed, will be somewhat 
 greater. 
 
 This method of ascertaining the density of electrification of a 
 surface was introduced by Coulomb, and the disk when used for 
 this purpose is called Coulomb’s Proof Plane. 
 
40 | COULOMB'S PROOF PLANE. [47. 
 
 The charge of the disk is by Experiment XII proportional to the 
 electromotive force at the electrified surface. Hence the electro- 
 motive force close to a conducting surface is proportional to the 
 density of the electrification at that part of the surface. 
 
 Since the surface of the conductor is an equipotential surface, the 
 electromotive force is perpendicular to it. The fact that the elec- 
 tromotive force at a point close to the surface of a conductor is 
 perpendicular to the surface and proportional to the density of the 
 electrification at that point was first established experimentally by 
 Coulomb, and it is generally referred to as Coulomb’s Law. 
 
 To prove that when the proof plane coincides with the surface of 
 the conductor the charge of the proof plane when removed from 
 the electrified conductor is equal to the charge on the part of the 
 surface which it covers, we may make the following experiment. 
 
 A sphere whose radius is 5 units is placed on an insulating 
 stand. A segment of a thin spherical shell is fastened to an in- 
 sulating handle. The radius of the spherical surface of the shell 
 is 5, the diameter of the circular edge of the segment is 8, and the 
 height of the segment is 2. When applied to the sphere it covers 
 one-fifth part of its surface. A second sphere, whose radius is 
 also 5, is placed on an insulating handle. 
 
 The first sphere is electrified, the sezment is then placed in 
 contact with it and removed. ‘The second sphere is then made to 
 touch the first sphere, removed and discharged, and then made to 
 touch the first sphere again. The segment is then placed within 
 a conducting vessel, which is discharged to earth, and then in- 
 sulated and the segment removed. One of the spheres is then 
 made to touch the outside of the vessel, and is found to be perfectly 
 discharged. 
 
 Let e be the electrification of the first sphere, and let the charge 
 removed by the segment be ze, then the charge remaining on the 
 sphere is (1—z)e. The charge of the first sphere is then divided 
 with the second sphere, and becomes }(1—z)e. The second sphere 
 is then discharged, and the charge is again divided, so that the 
 charge on either sphere is +(1—z)e. The charge on the insulated 
 vessel is equal and opposite to that on the segment, and it is there- 
 fore —ne, and this is perfectly neutralized by the charge on one of 
 the spheres; hence 4(1—z)e+(—xe) = 0, 
 from which we find n= +, 
 or the electricity removed by the seement covering one-fifth of the 
 surface of the sphere is one-fifth of the whole charge of the sphere. 
 
49. | ELECTROMOTIVE FORCE AND POTENTIAL. 41 
 
 EXPERIMENT XIV. 
 
 Direction of Electromotive Force at a Point. 
 
 48.| A convenient way of determining the direction of the elec- 
 tromotive force is to suspend a small elongated conductor with its 
 middle point at the given point of the field. The two ends of the 
 short conductor will become oppositely electrified, and will then be 
 drawn in opposite directions by the electromotive force, so that the 
 axis of the conductor will place itself in the direction of the force 
 at that point. <A short piece of fine cotton or linen thread, through 
 the middle of which a fine silk fibre is passed, answers very well. 
 The silk fibre, held by both ends, serves to carry the piece of thread 
 into any desired position, and the thread then takes up the direc- 
 tion of the electric force at that place. 
 
 EXPERIMENT XV. 
 
 Potential at any Powt of the Field. 
 
 49.| Suspend two small uncharged metal balls in the field by 
 silk threads, and then connect them by means of a fine metal wire 
 fastened to the end of an ebonite rod. Remove the wire and the 
 spheres separately, and then examine the charges of the spheres. 
 
 It will be found that the two spheres, if they have become 
 electrified, have received equal and opposite charges. If the poten- 
 tials at the points of the field occupied by the centres of the spheres 
 are different, positive electrification will be transferred from the 
 place of high to the place of low potential, and the sphere at the 
 place of high potential will thus become charged negatively, and 
 that at the place of low potential will become charged positively. 
 These charges may be shewn to be proportional to the difference of 
 potentials at the two places. 
 
 We have thus a method of determining points of the field at 
 which the potential is the same. Place one of the spheres at 
 a fixed point, and move the other about till, on connecting the 
 spheres with a wire as before, no charge is found on either sphere. 
 The potentials of the field at the points occupied by the centres of 
 the spheres must now be the same. In this way a number of 
 points may be found, the potential at each of which is equal to that 
 at a given point. 
 
42 POTENTIAL DETERMINED BY ONE SPHERE.  [50. 
 
 All these points lie on a certain surface, which is called an equi- 
 potential surface. On one side of this surface the potential is 
 higher, on the other it is lower, than at the surface itself. 
 
 We have seen that electricity has no tendency to flow from one 
 part of such a surface to another. An electrified body, if con- 
 strained so as to be capable of moving only from one point of the 
 surface to another, would be in equilibrium, and the force acting on 
 such a body is therefore everywhere perpendicular to the equi- 
 potential surface. 
 
 Experiment XVI. 
 
 50.]| We may use one sphere only, and after placing it with its 
 centre at any given point of the field we may touch it for a moment 
 with a wire connected to the earth. We may then remove the 
 sphere and determine its charge. The charge of the sphere is pro- 
 portional to the potential at the given point, a positive charge, 
 however, corresponding to a negative potential. 
 
 Equipotential Surfaces. 
 
 51.] In this way the potential at any number of points in the 
 field may be ascertained, and equipotential surfaces may be sup- 
 posed drawn corresponding to values of the potential represented 
 by the numbers 1, 2, 3, &c. 
 
 These surfaces will form a series, each, in general, lying within 
 the preceding surface and having the succeeding surface within it. 
 No two distinct surfaces can intersect each other, though a par- 
 ticular equipotential surface may consist of two or more sheets, 
 intersecting each other at certain lines or points. 
 
 The surface of any conductor in electric equilibrium is an equi- 
 potential surface. For if the potential at one point of the con- 
 ductor is different from that at another point, electricity will flow 
 from the place of higher potential to the place of lower potential 
 till the potentials are rendered equal. 
 
 EXPERIMENT XVII. 
 
 52.| To make an experimental determination of the equipotential 
 surfaces belonging to an electrified system we may use a small 
 exploring sphere permanently connected by a fine wire with one 
 electrode of the electroscope, the other electrode being connected 
 with the earth. Place the centre of the sphere at a given point, 
 
53.| RECIPROCAL METHOD: 43 
 
 and connect the electrodes together for an instant. The indication of 
 the electroscope will thus be reduced to zero. If the sphere is now 
 moved in such a manner that the indication of the electrometer 
 remains zero during the motion, the path of the centre of the 
 exploring sphere will lie on an equipotential surface. For if it 
 moves to a place of higher potential, electricity will flow from the 
 sphere to the electroscope, and if it moves to a place of lower 
 potential, electricity will flow from the electroscope to the sphere. 
 
 If the bodies belonging to the electrified system are not perfectly 
 insulated, their potentials and the potentials of the points of the 
 field will tend to approach zero. The path in which the centre of 
 the exploring sphere moves is such that its potential at any poit 
 has a given value at the time when the centre of the sphere passes 
 it. The different points of the path are not therefore on a surface 
 which has the same potential at any one instant, for the potential 
 is diminishing everywhere, and the path must therefore pass from 
 surfaces of lower to surfaces of higher potential so as to make up 
 for this loss. 
 
 Experiment XVIII. 
 
 53.] The following method, founded on Theorem V, Art. 37, is 
 therefore in many cases more convenient, as it is much easier to 
 secure good insulation 
 for the exploring sphere : 
 on an insulating handle 
 
 than for a large electri- Ye ee 
 fied conductor of irregu- 
 lar form. Let it be re- 
 
 quired to determine the 
 
 equipotential surfaces 
 
 due to the electrification 
 
 of the conductor C placed 
 
 on an insulating stand. 
 
 Connect the conductor C 
 
 with one electrode of the 
 
 electroscope, the other 
 
 being connected with 
 
 the earth. Electrify the 
 
 exploring sphere, and, Fig. 16. 
 
 carrying it by the insulating handle, bring its centre to a given 
 point. Connect the electrodes for an instant, and then move the 
 
44 LINES OF ELECTRIC FORCE. [54. 
 
 sphere in such a path that the indication of the electroscope remains 
 zero. This path will lie on an equipotential surface. 
 
 For by Theorem V, the part of the potential of the conductor C 
 due to the presence of the charged exploring sphere with its centre 
 at a given point is equal to the potential at the given point due to 
 a charge on the conductor C equal to that of the exploring sphere. 
 
 By this method the potential of the conductor remains zero, or 
 very nearly zero, during the whole time of the experiment, so that 
 there is very little tendency to change of the charge of this body. 
 The exploring sphere, on the other hand, is at a high potential, but 
 as it 1s not connected by a wire with any other body, its insulation 
 may be made very good. 
 
 Lines of Electric Force. 
 
 54.| If the direction of the electric force at various points of the 
 field be determined, and if a line be drawn so that its direction at 
 every point of its course coincides with the direction of the electric 
 force at that point, such a line is called a Line of Force. By 
 drawing a number of such lines, the direction of the force at 
 different parts of the field may be indicated. 
 
 The lines of force and equipotential surfaces may be drawn, not in 
 the electric field itself, where the mechanical operation of drawing 
 them might produce disturbance, but in a model or plan of the 
 electric field. Drawings of this kind are given in Plates I to VI 
 at the end of the volume. 
 
 Since the electric force is everywhere perpendicular to the equi- 
 potential surfaces, the lines of force cut these surfaces everywhere 
 at right angles. The lines of force which meet the surface of a 
 conductor are therefore at right angles to it. When they issue from 
 the surface the electrification is positive, and when they enter the 
 surface of the conductor the electrification is negative. | 
 
 A line of force in every part of its course passes from places of 
 higher to places of lower potential. 
 
 The extremities of the same line of force are called corresponding 
 points. 
 
 The beginning of the line is a point on a positively electrified 
 surface, and the end of the line is a corresponding point on a 
 negatively electrified surface. 
 
 The potential of the first of these surfaces must be higher than 
 that of the second. 
 
CHAPTER V. 
 
 FARADAYS LAW OF LINES OF INDUCTION. 
 
 55.] Farapay in his electrical researches employs the lines of 
 force to indicate, not only the direction of the electric force at each 
 point of the field, but also the quantity of electrification on any 
 given portion of the electrified surface. 
 
 If we consider a portion of an electrified surface as cut off from 
 the rest by the bounding line which surrounds it, and if from every 
 point of this bounding line we draw a line of force, producing it 
 till it meets the surface of some other body in a point which is 
 said to correspond to the point of the body from which the line was 
 drawn, these lines will form a tubular surface, and will cut off a 
 certain portion from the surface of the other body corresponding: to 
 the portion of the surface of the first body, and the total electrifica- 
 tions of the two corresponding portions are equal in numerical 
 magnitude but opposite in kind. 
 
 56.| A particular instance of Faraday’s law is that which we 
 have already proved by experiment, namely, that the electrification 
 of the inner surface of a closed conducting: vessel is equal and 
 opposite to that of an electrified body placed within it. Here 
 we have a relation between the whole electrification of the inner 
 surface and that of the opposed surface of the interior body. 
 Faraday’s law asserts that, by drawing lines of force from the one 
 surface to the other, points corresponding to each other in the two 
 surfaces may be found ; that corresponding lines are such that any 
 point of one has its corresponding point in the other; and that the 
 electrifications of the two portions of the opposed surfaces bounded 
 by such corresponding lines are equal and opposite. 
 
 57.| We have called these lines ‘lines of force’ because we 
 began by defining them as lines whose direction at every point 
 
46 FARADAY'S LAW OF LINES OF INDUCTION.  [58. 
 
 coincides with that of the electric force. Every line of force 
 begins at a positively electrified surface and ends at a negatively 
 electrified surface, and the points of these surfaces at which it 
 begins and ends are called corresponding points. | 
 
 A system of lines of force forming a tubular surface closed at 
 the one end by a portion of the positively electrified surface and 
 at the other by the corresponding portion of the negative surface, 
 is called by Faraday a Tube of Induction, because electric induction, 
 according to Faraday, is that condition of the dielectric by which 
 the electrifications of the opposed surfaces are placed in that 
 physical relation to one another, which we express by saying that 
 their electrifications are equal and opposite. 
 
 Properties of a Tube of Induction. 
 
 58.] (1) The electrification of the portion of the positively 
 electrified surface from which the tube of induction proceeds is 
 equal in numerical value but opposite in sign to the negative 
 electrification of the portion of the surface at which the tube of 
 induction terminates. 
 
 By dividing the positive surface into portions, the electrification 
 of each of which is unity, and drawing tubes corresponding to 
 each portion, we obtain a system of wit tubes, which will be very 
 convenient in describing electric phenomena. For in this case 
 the electrification of any surface is measured by the number of 
 tubes which abut on it. If they proceed from the surface, this 
 number is to be taken as representing the positive electrification ; 
 if the tubes terminate at the surface, the electrification is negative. 
 
 It is in this sense that Faraday so often speaks of the number of 
 lines of force which fall on a given area. 
 
 If we suppose an imaginary surface drawn in the electric field, 
 then the quantity of electrostatic induction through this. surface 
 is measured by the number of tubes of induction which pass 
 through it, and is reckoned positive or negative accordingly as 
 the tubes pass through it in the positive or negative direction. 
 
 Note. By an imaginary surface is meant a surface which has 
 no physical existence, but which may be imagined to exist in 
 space without interfering with the physical properties of the sub- 
 stance which occupies that space. Thus we may imagine a vertical 
 plane dividing a man’s head longitudinally into two equal parts, 
 and by means of this imaginary surface we may render our ideas 
 
60. | PROPERTIES OF A TUBE OF INDUCTION. zat | 
 
 of the form of his head more precise, though any attempt to con- 
 vert this imaginary surface into a physical one would be criminal. 
 Imaginary quantities, such as are mentioned in treatises on 
 analytical geometry, have no place in physical science. 
 
 59.| In every part of the course of a line of electrostatic in- 
 duction it is passing from places of higher to places of lower 
 potential, and in a direction at right angles to the equipotential 
 surfaces which it cuts, 
 
 We have seen that the electric field is divided by the equi- 
 potential surfaces into a series of shells, ike the coats of an onion, 
 the thickness of each shell at any point being inversely as the 
 electric force at that point. 
 
 We have now divided the electric field into a system of unit 
 tubes of induction, the section of each tube at any point varying 
 inversely as the intensity of the electric induction at that point. 
 
 Each of these tubes is cut by the equipotential surfaces into a 
 number of seements which we may call unit cells. 
 
 60.] If we take the simplest case—that of a single positively 
 electrified body placed within a closed conducting vessel, all the 
 tubes of induction begin at the positively electrified body and end 
 at the negatively electrified inner surface of the vessel. The 
 number of these tubes, since they are unit-tubes, is equal to the 
 number of electrical units in the charge of the electrified body. 
 Each of them cuts all the equipotential surfaces which enclose 
 the electrified body and are enclosed by the vessel. Each tube, 
 therefore, is divided into a number of cells representing the differ- 
 ence of the potential of the electrified body from that of the vessel. 
 If e is the charge of the body and p its potential, # and P being the 
 charge and potential of the vessel, the whole number of cells is 
 
 e(p—FP), 
 
 or, since / = —e, we may write this expression 
 ep + EP, 
 
 Now this is double of the expression which we formerly obtained 
 for the electrical energy of the system (see Art. 31). Hence in 
 this simple case the number of cells is double the number of units 
 of energy in the system. 
 
 If there are several electrified bodies, 4, 6, C, &c., the tubes 
 of induction proceeding from one of them, 4, may abut either 
 on the inner surface of the surrounding vessel or on one of the 
 other electrified bodies. 
 
48 ENERGY OF AN ELECTRIFIED SYSTEM. [61. 
 
 Let F,, #,, £3; be the charges of A, B, C and P,, P,, Ps their 
 potentials, the charge and potential of the vessel being H, and Pp. 
 
 Let Lan, Fac, Fao denote the number of tubes of induction 
 which pass from 4 to the conductors B and C and the vessel O, 
 respectively. Then the whole number of cells will be 
 
 Bap (Py—P.) + Lac(P,— Ps) + E40 (Py — Po); 
 
 + Eng (P,—Ps) + Exo (P,—Py); 
 
 + Evo (P;—Po). 
 By arranging the terms according to the potentials involved in 
 them, and remembering that since #4, denotes the number of 
 
 tubes which pass from dA to 6, Hz4 must denote the number 
 which pass from 6 to A and therefore 
 
 Ena = — Lap, 
 the expression may be written 
 P, (Eant+ Eac+ Lao), 
 +P, (Eee + Lyo + Epa), 
 +P; (Loo+ Hea + Lez), 
 + P, (Lost Lon t+ Loc). 
 
 Now fant Lact £49 18 the whole number of tubes issuing from 
 A and this therefore is equal to £,, the charge of 4, and the co- 
 efficients of the other potentials are also the charges of the bodies 
 to which they refer, so that the final expression is 
 
 PH t+ Pi B+ PH, +P; By 
 and this is double the energy of the system. 
 
 Hence, whether there is one electrified body or several, the num- 
 ber of cells 1s twice the number of units of electrical energy in the 
 system. 
 
 61.] This remarkable correspondence between the number of cells 
 into which the tubes of induction are cut by the equipotential sur- 
 faces, and the electrical energy of the system, leads us to enquire 
 whether the electrical enerey may not have its true seat in the 
 dielectric medium which is thus cut up into cells, each cell being a 
 portion of the medium in which half a unit of energy is stored up. 
 We have only to suppose that the electromotive force, when it acts 
 on a dielectric, puts it into a certain state of constraint, from which 
 it is always endeavouring to relieve itself. 
 
 To make our conception of what takes place more precise, let us 
 
62. | ELECTRIC DISPLACEMENT. 49 
 
 consider a single cell belonging to a tube of induction proceeding 
 from a positively electrified body, the cell being bounded by two 
 consecutive equipotential surfaces surrounding the body. 
 
 We know that there is an electromotive force acting outwards 
 from the electrified body. This force, if it acted on a conducting 
 medium, would produce a current of electricity which would last as 
 long as the force continued to 
 act. The medium however is 
 a non-conducting or dielectric 
 medium, and the effect of the 
 electromotive force is to produce 
 what we may call electric dis- 
 placement, that is to say, the 
 electricity is forced outwards in 
 the direction of the electromo- 
 tive force, but its condition when 
 so displaced is such that, as soon geht: 
 as the electromotive force is removed, the electricity resumes the 
 position which it had before displacement. 
 
 The amount of electric displacement is measured by the quantity 
 of electricity which crosses an imaginary fixed surface drawn parallel 
 to the equipotential surfaces. 
 
 We know absolutely nothing with respect to the distance through 
 which any particular portion of electricity is displaced from its 
 original position. The only thing we know is the quantity which 
 crosses a given surface. The greater the amount of electricity 
 which we suppose to exist, say, In a cubic inch, the smaller the 
 distance through which we must suppose it displaced in order that 
 a given quantity of electricity may be displaced across a square 
 inch of area fixed in the medium. It is probable that the actual 
 motion of displacement is exceedingly small, in which case we must 
 suppose the quantity of electricity in a cubic inch of the medium to 
 be exceedingly great. If this is really the case the actual velocity 
 of electricity in a telegraph wire may be very small, less, say, than 
 the hundredth of an inch in an hour, though the signals which it 
 transmits may be propagated with great velocity. 
 
 62.| The displacement across any section of a unit tube of in- 
 duction is one unit of electricity and the direction of the displace- 
 ment is that of the electromotive force, namely, from places of 
 higher to places of lower potential. 
 
 Besides the electric displacement within the cell we have to 
 
 E 
 
 
 
50 ELECTRIC TENSION. [63. 
 
 consider the state of the two ends of the cell which are formed by 
 the equipotential surfaces. We must suppose that in every cell the 
 end- formed by the surface of higher potential is coated with one 
 unit of positive electricity, the opposite end, that formed by the 
 surface of lower potential, being coated with one unit of negative 
 electricity. In the interior of the medium where the positive end 
 of one cell is in contact with the negative end of the next, these 
 two electrifications exactly neutralise each other, but where the 
 dielectric medium is bounded by a conductor, the electrification is 
 no longer neutralised, but constitutes the observed electrification at 
 the surface of. the conductor. 
 
 According to this view of electrification, we must regard electri- 
 fication as a property of the dielectric medium rather than of the 
 conductor which is bounded by it. 
 
 63.| If we further admit that in every part of a dielectric 
 medium through which electric induction is taking place there is 
 a tension, like that of a rope, in the direction of the lines of force, 
 and a pressure in all directions at right angles to the lines of force, 
 we may account for all the mechanical actions which take place 
 between electrified bodies. | | 
 
 The tension, referred to unit of surface, is proportional to the 
 square of the electromotive force at the point. The pressure has 
 the same numerical value, but is, of course, opposite in sign. 
 
 In my larger treatise on electricity a proof is given of the fact 
 that a system of stress such as is here described is consistent with 
 the equilibrium of a fluid dielectric medium, and that this state of 
 stress in the medium is mechanically equivalent to the attraction 
 or repulsion which electrified bodies manifest. 
 
 I have not, however, attempted, by any hypothesis as to the inter- 
 nal constitution of the dielectric medium, to explain in what way the 
 electric displacement causes or is associated with this state of stress. 
 
 We have thus, by means of the tubes of induction and the 
 equipotential surfaces, constructed a geometrical model of the field 
 of electric force. Diagrams of particular cases are given in the 
 figures at the end of this book. 
 
 The direction and magnitude of the electric force at any point 
 may be indicated either by means of the equipotential surfaces or 
 by means of the tubes of induction. Hence, when it is expressed 
 in both ways, we may by the study of the relation between the 
 equipotential surfaces and the tubes of induction deduce important 
 theorems in the theory of electricity. 
 
64.] ANALOGIES BETWEEN ELECTROSTATICS AND HEAT. 51 
 
 On the use of Physical Analogies. 
 
 64.] In many cases the relations of the phenomena in two 
 different physical questions have a certain similarity which enables 
 us, when we have solved one of these questions, to make use of our 
 solution in answering the other. The similarity which constitutes 
 the analogy is not between the phenomena themselves, but be- 
 tween the relations of these phenomena. 
 
 To begin with a case of extreme simplicity ;—a person slow at 
 arithmetic having to find the price of 52 yards of cotton at 7 pence 
 a yard, if he happened to remember that there are 52 weeks and a 
 day in a year of 365 days, might at once give the answer, 364 
 pence, without performing the calculation. Here there is no re- 
 semblance whatever between the quantities themselves—the weeks 
 and the yards of cotton,—the sole resemblance is beween the arith- 
 metical relations of these quantities to others in the same question. 
 
 The analogy between electrostatic phenomena and those of the 
 uniform conduction of heat in solid bodies was first pointed out by 
 Sir W. Thomson in a paper ‘On the Uniform Motion of Heat in 
 Homogeneous Solid Bodies, and its connection with the Mathema- 
 tical Theory of Electricity,’ published in the Cambridge Mathematical 
 Journal, Feb. 1842; reprinted in the Phil, Mag. 1854, and in the 
 
 reprint of Thomson’s papers on Lectrostatics and Magnetism. 
 
 The 
 
 analogy is of the following nature :— 
 
 Electrostatics. 
 
 The electric field. 
 
 A dielectric medium. 
 
 The electric potential at different points 
 of the field. 
 
 The electromotive force which tends to 
 move positively electrified bodies from 
 places of higher to places of lower po- 
 tential. 
 
 A conducting body. 
 
 The positively electrified surface of a con- 
 ductor. 
 
 The negatively electrified surface of a 
 conductor. 
 
 A positively electrified body. 
 
 A negatively electrified body. 
 
 An equipotential surface. 
 A line or tube of induction. 
 
 Heat. 
 
 An unequally heated body. 
 
 A body which conducts heat. 
 
 The temperature at different pvints in the 
 body. 
 
 The flow of heat by conduction from 
 places of higher to places of lower 
 temperature. 
 
 A perfect conductor of heat. 
 
 A surface through which heat flows into 
 the body. 
 
 A surface through which heat escapes 
 from the body. 
 
 A source of heat. 
 
 A sink of heat, that is, a place at which 
 heat disappears from the body. 
 
 An isothermal surface. 
 
 A line or tube of flow of heat. 
 
 By a judicious use of this analogy*and other analogies of the. 
 same nature the progress of physical science has been greatly as- 
 E 2 
 
52 ANALOGIES BETWEEN ELECTROSTATICS AND HEAT, [65. 
 
 sisted. In order to avoid the dangers of crude hypotheses we must 
 study the true nature of analogies of this kind. We must not con- 
 elude from the partial similarity of some of the relations of the 
 phenomena of heat and electricity that there is any real physical 
 , similarity between the causes of these phenomena. The similarity 
 isa similarity between relations, not a similarity between the things 
 related. 
 
 This similarity is so complete as far as it goes that any result we 
 may have obtained either about electricity or about the conduction 
 of heat may be at once translated out of the language of the one 
 science into that of the other without fear of error ; and in pursuing 
 our ‘Investigations in either subject we are at liberty to make use 
 of the ideas belonging to the other, if by so doing we are enabled 
 to see more clearly the connection between one step and another of 
 the reasoning. 
 
 We must bear in mind that at the time when Sir W. Thomson 
 pointed out the analogy between electrostatic and thermal phe- 
 nomena men of science were as firmly convinced that electric at- 
 traction was a direct action between distant bodies as that the 
 conduction of heat was the continuous flow of a material fluid 
 through a solid body. The dissimilarity, therefore, between the 
 things themselves appeared far greater to the men of that time than 
 to the readers of this book, who, unless they have been previously 
 instructed, have not yet learned either that heat is a fluid or that 
 electricity acts at a distance. 
 
 65.| But we must now consider the limits of the analogy—the 
 points beyond which we must not push it. 
 
 In the first place, it is only a particular class of cases of the 
 conduction of heat that have analogous cases in electrostatics. In 
 general, when heat is flowing through a body it causes the tempera- 
 ture of some parts of the body to rise and that of others to fall, 
 and the flow of heat, which depends on the relation of these tempera- 
 tures, is therefore variable. If the supply of heat is maintained 
 uniform, the temperatures of the different parts of the body tend to 
 adjust themselves to a state in which they remain constant. The 
 quantity of heat which enters any given portion of the body is then 
 exactly equal to that which leaves it during the same time. Under 
 these circumstances the flow of heat is said to be steady. 
 
 Now the analogy with electric phenomena applies to the steady 
 flow of heat only. The more general case, that of variable flow of 
 heat, has nothing in electrostatics analogous to it. LHven the re- 
 
 
 
67.] LIMITATION TO THE USE OF ANALOGIES. 53 
 
 stricted case of steady flow of heat differs in a most important 
 element from the electrostatic analogue. The steady flow of heat 
 must be kept up by the continual supply of heat at a constant rate 
 and the continual withdrawal of heat at an equal rate. This in- 
 volves a continual expenditure of energy to maintain the flow of 
 heat in a constant state, so that though the state of the body’ 
 remains constant and independent of time, the element of time 
 enters into the calculation of the amount of heat required. 
 
 The element of time does not enter into the corresponding case 
 in electrostatics. So far as we know; a set of electrified bodies 
 placed in a perfectly insulating meditim might remain electyitied 
 for ever without a supply of anything from external sources. 
 There is nothing in this case to which’we can apply the term 
 ‘flow,’ which we apply to the case of thé transmission of heat 
 with the same propriety that we apply it to ‘thleease of a cu -uppent 
 of electricity, of water, or of time itself. Ss a 
 
 66.] Another limitation to the analogy is that the temperature 
 of a body cannot be altered without altering its physical state. 
 The density, conductivity, electric properties, &e. all vary when the 
 temperature rises. 
 
 The electrical potential, however, which is the analogue of tem- 
 perature is a mere scientific concept. We have no reason to 
 regard it as denoting a physical state. If a number of bodies 
 are placed within a hollow metallic vessel which completely sur- 
 rounds them, we may charge the outer surface of the vessel and 
 discharge it as we please without producing any physical effect 
 whatever on the bodies within. But we know that the electric 
 potential of the enclosed bodies rises and falls with that of the 
 vessel. This may be proved by passing a conductor connected 
 to the earth through a hole in the vessel. ‘The relation of the 
 enclosed bodies to this conductor will be altered by charging and 
 discharging the vessel. But if the conductor be removed, the 
 simultaneous rise and fall of the potentials of the bodies in the 
 vessel is not attended with any physical effect whatever. 
 
 67.| Faraday* proved this by constructing a hollow cube, twelve 
 feet in the side, covered with good conducting materials, insulated 
 from the ground and highly electrified by a powerful machine. 
 ‘I went into this cube, he says, ‘and lived in it, but though I 
 used lighted candles, electrometers, and all other tests of electrical 
 states, I could not find the least influence upon them, or indication 
 
 * Fup. Res. 1173. 
 
54 FARADAY S CUBE. « [ 68. 
 
 of anything particular given by them, though all the time the 
 outside of the cube was powerfully charged and large sparks and 
 brushes were starting off from every part of its outer surface.’ 
 
 It appears, therefore, that the most sudden changes of potential 
 produce no physical effects on matter, live or dead, provided these 
 changes take place simultaneously on all the bodies in the field. 
 
 If Faraday, instead of raising his cube to a high electric potential, 
 had raised it to a high temperature, the result, as we know, would 
 have been very different. 
 
 68.| It appears, therefore, that the analogy between the con- 
 duction of heat and electrostatic phenomena has its limits, beyond 
 which we must not attempt to push it. At the time when it was 
 pointed out by Thomson, men of science were already acquainted 
 with the great work of Fourier on the conduction of heat in solid 
 bodies, and their minds were more familiar with the ideas there 
 developed than with those belonging to current electricity, or to 
 the theory of the displacements of a medium. 
 
 It is true that Ohm had, in 1827, applied the results obtained 
 by Fourier for heat to the theory of the distribution of electric 
 currents in conductors, but it was long before the practical value 
 of Ohm’s work was understood, and till men became familiar with 
 the idea of electric currents in solid conductors, any illustration of 
 electrostatic phenomena drawn from such currents would have served 
 rather to obscure than to enlighten their minds. 
 
 69.] When an electric current flows through a solid conductor, 
 the direction of the current at any point is from places of higher 
 to places of lower potential, and its intensity is proportional to the 
 rate at which the potential decreases from point to point of a line 
 drawn in the direction of the current. 
 
 We may suppose equipotential surfaces drawn in the conducting 
 medium. The lines of flow of the current are everywhere at right 
 angles to the equipotential surfaces, and the rate of flow is pro- 
 portional to the number of equipotential surfaces which would be 
 cut by a line of unit length drawn in the direction of the current. 
 
 It appears, therefore, that this case of a conducting medium 
 through which an electric current is passing has certain points 
 of analogy with that of a dielectric medium bounded by electrified 
 bodies. 
 
 In both the medium is divided into layers by a series of equi- 
 potential surfaces. In both there is a system of lines which are 
 everywhere perpendicular to these surfaces. In the one case these 
 
70. | CURRENT. 55 
 
 lines are called current lines or lines of flow; in the other they 
 are called lines of electric force or electric induction. 
 
 An assemblage of such lines drawn from every point of a given 
 line is called a surface of flow. Since the lines of which this 
 surface is formed are everywhere in the direction of the electric 
 current, no part of the current passes through the surface of flow. 
 Such a surface therefore may be regarded as impervious to the 
 current without in any way altering the state of things. 
 
 If the line from which the assemblage of lines of flow is drawn 
 is one which returns into itself, which we shall call a closed curve, 
 or, more briefly, a 72g, the surface of flow will have the form of a 
 tube and is called a tube of flow. Any two sections of the same 
 tube of flow correspond to each other in the sense defined in Art. 54, 
 and the quantities of electricity which in the same time flow across 
 these two sections are equal. 
 
 Here then we have the analogue of Faraday’s law, that the 
 quantities of electricity upon corresponding areas of opposed con- 
 ducting surfaces are equal and opposite. | 
 
 Faraday made great use of this analogy between electrostatic 
 phenomena and those of the electric current, or, as he expressed 
 it, between induction in dielectrics and conduction in conductors, 
 and he proved that, in many cases, Induction and conduction are 
 associated phenomena. Lap. Res. 1320, 1326. 
 
 We must remember, however, that the electric current cannot 
 be maintained constant through a conductor which resists its 
 passage except by a continual expenditure of energy, whereas 
 induction in a perfectly insulating dielectric between oppositely 
 electrified conductors may be maintained in it for an indefinitely 
 long time without any expenditure of energy, except that which 
 is required to produce the original electrification. ‘The element of 
 time enters into the question of conduction in a way in which it 
 does not appear in that of induction. 
 
 -70.| But we may arrive at a more perfect mental representation 
 of induction by comparing it, not with the instantaneous state of 
 a current, but with the small displacements of a medium of in- 
 variable density. 
 
 Returning to the case of an electric current through a solid 
 conductor, let us suppose that the current, after flowing for a 
 very short time, ceases. If we consider a surface drawn within 
 the solid, then if this surface intersects the tubes of flow, a certain 
 quantity of electricity will have passed from one side of the surface 
 
56 TUBES OF INDUCTION AND LINES OF FORCE. [7I. 
 
 to the other during the time when the current was flowing. This 
 passage of electricity through the surface is called electric dis- 
 placement, and the displacement through a given surface is the 
 quantity of electricity which passes through it. In the case of 
 a continuous current the displacement increases continuously as 
 long as the current is kept up, but if the current lasts for a finite 
 time, the displacement reaches its final value and then remains 
 constant. The lines, surfaces, and tubes of flow of the transient 
 current are also lines, surfaces, and tubes of displacement. The 
 displacements across any two sections of the same tube of dis- 
 placement are equal. At the beginning of each unit tube of 
 displacement there is a unit of positive electricity, and at the end 
 of the tube there is a unit of negative electricity. 
 
 At every point of the medium there is a state of stress con- 
 . sisting of a tension in the direction of the line of displacement 
 through the point and a pressure in all directions at right angles 
 to this line. The numerical value of the tension is equal to that 
 of the pressure, namely, the square of the intensity of the electric 
 foree divided by 4 7. 
 
 71.] By the consideration of the properties of the tubes of 
 induction and the equipotential surfaces we may easily prove 
 several important general theorems in the theory of electricity, 
 the demonstration of which by the older methods is long and 
 difficult. The properties of a tube of induction have already 
 been stated, but for the sake of what follows we may state them 
 again :— 
 
 (1) If a tube of induction is cut by an imaginary surface, the 
 quantity of electricity displaced across a section of the tube is the 
 same at whatever part of the tube the section be made. 
 
 (2) In every part of the course of a line of electrostatic force 
 it cuts the equipotential surfaces at right angles, and is proceeding 
 from a place of higher to a place of lower potential. : 
 
 Note. This statement is true only when the distribution of 
 electric foree can be completely represented by means of a set of 
 equipotential surfaces. This is always the case when the electricity 
 is In equilibrium, but when there are electric currents, though in 
 some parts of the field 1t may be possible to draw a set of equi- 
 potential surfaces, there are other parts of the field where the 
 distribution of electric foree cannot be represented by means of 
 such surfaces. or an electric current is always of the nature of 
 a circuit which returns into itself, and such a circuit cannot in 
 
75.| ELECTRIFICATION AT ENDS OF INDUCTION TUBE, 57 
 
 every part of its course be proceeding from places of higher to 
 places of lower potential. 
 
 72.| It may be observed that in (1) we have used the words 
 ‘tube of induction, and in (2) the words ‘line of electrostatic 
 force. In a fluid dielectric, such as air, the line of electrostatic 
 force is always in the same direction as the tube of induction, and 
 it may seem pedantic to maintain a distinction between them. 
 There are other cases, however, in which it is very important to 
 remember that a tube of induction is defined with respect to the 
 phenomenon which we have called electric displacement, while a 
 line of force is defined with respect to the electric force. In fluids 
 the electric displacement is always in the direction of the electric 
 force, but there are solid bodies in which this is not the case*, and 
 in which, therefore, the tubes of induction do not coincide in direc- 
 tion with the lines of force. 
 
 73.] It follows from (1) that every tube of induction begins 
 at a place where there is a certain quantity of positive electricity 
 and ends at a place where there is an equal quantity of negative 
 electricity, and that, conversely, from any place where there is posi- 
 tive electricity a tube may be drawn, and that wherever there is 
 negative electricity a tube must terminate. 
 
 74.| It follows from (2) that the potential at the beginning of a 
 tube is higher than at the end of it. Hence, no tube can return 
 into itself, for in that case the same point would have two different 
 potentials, which is impossible. 
 
 75.] From this we may prove that if the potential at every 
 point of a closed surface is the same, and if there is no electrified 
 body within that surface, the potential at any point within the 
 region enclosed within the closed surface is the same as that at 
 the surface. 
 
 For if there were any difference of potential between one point 
 and another within this region, there would be lines of force from 
 the places of higher towards the places of lower potential. These 
 lines, as we have seen, cannot return into themselves. Hence they 
 must have their extremities either within the region or without it. 
 Neither extremity of a line of force can be within the region, for 
 there must be positive electrification at the beginning and negative 
 electrification at the end of a line of force, but by our hypothesis 
 there is no electrification within the region. On the other hand, a 
 
 * See the experiments of Boltzmann on crystals of sulphur. Vienna Sitzungsb. 
 9 Jan. 1873. 
 
58 NO ELECTRIFICATION WITHIN A HOLLOW CONDUCTOR. [ 76: 
 
 line of foree within the region cannot have its extremities without 
 the region, for in that case it must enter the region at one point 
 of the surface and leave it at another, and therefore by (2) the 
 potential must be higher at the point of entry than at the point 
 of issue, which is contrary to our hypothesis that the potential 1s 
 the same at every point of the surface. 
 
 Hence no line of force can exist within the region, and therefore 
 the potential at any point within the region is the same as that at 
 the surface itself. 
 
 76.] It follows from this theorem, that if the closed surface is 
 the internal surface of a hollow conducting vessel, and if no elec- 
 trified body is within the surface, there is no electrification on the 
 internal surface. For if there were, lines of foree would proceed 
 from the electrified parts of the surface into the region within, 
 and we have already proved that there are no lines of force in 
 that region. 
 
 We have already proved this by experiment (Art. 20), but we 
 now see that it is a necessary consequence of the properties of 
 lines of force. 
 
 Superposition of electric systems. 
 
 77.| We have already (Art. 29) given some examples of the 
 superposition of electric effects, but we must now state the principle 
 of superposition more definitely. 
 
 If the same system is electrified in three different ways, then if the 
 potential at any point in the third case ts the sum of the potentials im 
 the first and second cases, the electrification of any part of the system 
 in the third case will be the sum of the electrifications of the same part 
 in the first and second cases. 
 
 By reversing the sign of the electrifications and potentials in 
 one of these cases, we may enunciate the principle with respect 
 tio the case in which the potential and the electrification are at 
 every point the excess of what they are in the first case over what 
 they are in the second. 
 
 . 78.| We may now establish a theorem which is of the greatest 
 importance in the theory of electricity. 
 
 If the electric field under consideration consist of a finite portion 
 of a dielectric medium, and if at every point of the boundary of 
 this region the potential is given, and if the distribution of electri- 
 fication within the region be also given, then the potential at any 
 
80. | THOMSON 'S THEOREM. 59 
 
 point within the region can have one and only one value consistent 
 with these conditions. 
 
 One value at least of the potential must be possible, because the 
 conditions of the theorem are physically possible. Again, if at 
 any point two values of the potential were possible, then by 
 taking the excess of the first value over the second for every 
 point of the system, a third case might be formed in which the 
 potential is everywhere the excess of the first case above the second. 
 At the boundary of the region the potential in the third case is 
 everywhere zero. Within the region the electrification is every- 
 where zero. Hence, by (Art. 75), at every point within the region 
 the potential in the third case is zero. 
 
 There is, therefore, no difference between the distribution of 
 potential in the first case and in the second, or, in other words, 
 the potential at any point within the region can have only one 
 value. 
 
 If in any case we can find a distribution of potential which 
 satisfies the given conditions, then by this theorem we are assured 
 that this distribution is the only possible solution of the problem. 
 Hence the importance of-this theorem in the theory of electricity. 
 
 79.] For instance, let 4 be an electrified body and let B be 
 one of the equipotential surfaces surrounding the body. Let the 
 potential of the surface 6 be equal to P. Now 
 let a conducting body be constructed and placed 
 so that its external surface coincides with the 
 closed surface B, and let it be so electrified that Ca) 
 its potential is #. Then the conditions of the per 
 region outside 6 are the same as when it was 
 acted on by the body 4 only. For the potential Fig. 18. 
 over the whole bounding surface of the region is P, the same as 
 before, and whatever electrified bodies exist outside of B remain 
 unchanged. Hence the potential at every point outside of B may, 
 consistently with the conditions, be the same as before. By our 
 theorem, therefore, the potential at every point outside B must be 
 the same, when, instead of the body A, we have a conducting 
 surface B, raised to the potential P. 
 
 80.] The charge of every part of the surface of a conductor is 
 of the same sign as its potential, unless there is another body 
 in the field whose potential is of the same sign but numerically 
 greater. 
 
 Let us suppose the potential of the body to be positive; then, 
 
 5B 
 
60 INDUCED ELECTRIFICATION. [80. 
 
 if on any part of its surface there is negative electricity, lines of 
 force must terminate on this part of the surface, and these lines 
 of force must begin at some electrified surface whose potential is 
 higher than that of the body. Hence, if there is no other body 
 whose potential is higher than that of the given body, no part 
 of the surface of the given body can be charged with negative 
 electricity. 
 
 If an uninsulated conductor is placed in the same field with a 
 charged conductor, the charge on every part of the surface of 
 the uninsulated conductor is of the opposite sign to the charge of 
 the charged conductor. 
 
 For since the potential of the uninsulated body is zero, there 
 ean be no line of force between it and the walls of the room, or 
 infinite space where the potential is always zero. The line of force 
 which has one end at any point of the surface of this body must 
 therefore have its other end at some point of the charged body, 
 and since the two extremities of a line of force are oppositely 
 electrified, the electrification of the surface of the uninsulated body 
 must be everywhere opposite to the charge of the charged body. 
 
 The charged body in this experiment is called the Inductor, and 
 the other body the induced body. 
 
 When the induced body is uninsulated, the electricity spread 
 over every part of its surface is, as we have just proved, of the 
 opposite sign to that of the inductor. 
 
 The total charge, 14, of the induced body, which we may call 4, 
 may be found by multiplying Pz», the potential of the inductor B, 
 by Qza, the mutual coefficient of induction between the bodies, 
 which is always a negative quantity. 
 
 This. electrification induced on an uninsulated body is called 
 by some writers on electricity the Induced Electrification of the 
 First Species. Since the potential of A is already zero, it is 
 manifest that if any part of its surface is touched by a fine wire 
 communicating with the ground there will be no discharge. 
 
 Next, let us suppose that the body, 4, instead of being unin- 
 sulated is insulated, but originally without charge. Under the 
 action of the inductor & part of its surface, on the side next to B, 
 will become electrified oppositely to B; but since the algebraic 
 sum of its electrification is zero, some other part of its surface must 
 be electrified similarly to 5. 
 
 This electrification, of the same name as that of B, is called by 
 writers on electricity the Induced Electrification of the Second 
 
80. ] COEFFICIENTS OF CAPACITY. 61 
 
 Species. If a wire connected with the ground be now made to 
 touch any part of the surface of <A, electricity of the same name 
 as that of B will be discharged, its amount being equal and op- 
 posite to the negative charge (of the first species) which remains 
 on the body 4, which is now reduced to potential zero. 
 
 In order to obtain a clearer idea of the distribution of electricity 
 on the surface of 4A under various conditions, let us begin by 
 supposing the potential of 4 to be zero and that of B to be unity. 
 Let the surface-density at a given point P on the surface of 4 
 be —o,, and let the whole charge of 4 be—qaz. The negative 
 sign is prefixed to the symbols of these quantities because the 
 quantities themselves are always negative. 
 
 The charge of 6 in this case is 7p. 
 
 Let us next suppose the potential of 4 to be unity and that of B 
 to be zero, and let the surface-density at the point P be now o,, and 
 the whole charge on A, g4. 
 
 These quantities are both essentially positive, and g,4 is called 
 the capacity of A. The value of both is increased on account of 
 the presence of J in the field. 
 
 Let us now suppose that the potentials of 4 and B are Py and 
 
 Px respectively ; then the surface-density at the point P is 
 ed oa eae 
 and the charge of dis #4 = P4qa—Pp an, 
 and that of B is Ep = Ps qp—P 4 Yap. [See Art. 39.] 
 
 If A is insulate and without charged #4 = 0, which gives 
 
 ee Pete. 
 
 qA 
 
 
 
 and the surface-density at P is 
 Px 
 o = —-(Y4B%—94%). 
 GA 
 
 On a region of the surface of A next to B, o will be of the 
 opposite sien from Pz; and on a region on the other side from B, 
 o will be of the same sign with P,. The boundary between these 
 two regions forms what is called the neutral line, the form and 
 position of which depend on the form and position of 4 and B, 
 
CHAPTER VI. 
 PARTICULAR CASES OF ELECTRIFICATION. 
 
 81.| A sPHERICAL conductor is electrified and insulated within 
 the concentric spherical internal surface of a conducting vessel. 
 
 On account of the perfect symmetry of this system in all direc- 
 tions, it is manifest that the distribution of electricity will be 
 uniform over each of the opposed spherical surfaces, that the lines 
 of force will be in the directions passing through the common 
 centre of the spheres, and that the equipotential surfaces will be 
 spheres having this point for their centre. 
 
 If e is the quantity of electricity on the inner sphere and /# that on 
 the eta surface of the outer ae then by Experiment VIII 
 
 | i PEE (1) 
 
 If + and #& are the radii of the ney s and S their surfaces, 
 and o and & the surface-densities of the electricity on these 
 surfaces, then by geometry, 
 
 Vee wg S=. 47 KR... eee (2) 
 where 7 denotes the ratio of the circumference of a circle to its 
 diameter. 
 
 The whole charge on either sphere is found by multiplying: the 
 surface into the surface-density, or 
 
 Sas ie E= 8>.......3. (3) 
 e EH | 
 Hence, ar wise = ge? ee (4) 
 ie 
 and by COR 2= 7p 0:0 0:40. 0,e7efelate slate nature (5) 
 
 It appears, therefore, that when the charge, e, of the inner 
 sphere is given, the surface-density, 5, on the internal surface of 
 the vessel is inversely as the square of the distance of that surface 
 from the centre of the electrified sphere. 
 
 Hence by Coulomb’s law (Experiment XIII, Art. 47) the elec- 
 tromotive force at the outer spherical surface is inversely as the 
 square of the distance from the centre of the sphere. 
 
8:3. | ELECTROSTATIC UNIT OF ELECTRICITY. 63 
 
 This is the law according to which the electric force varies 
 at different distances from a sphere uniformly electrified. The 
 amount of the force is independent of the radius of the inner 
 electrified sphere, and depends only on the whole charge upon it. 
 If we suppose the radius of the inner sphere to become very’ small 
 till at last the sphere cannot be distinguished from a point, we 
 may imagine the whole charge concentrated at this point, and 
 we may then express our result by saying that the electric 
 action of a uniformly electrified sphere at any point outside the 
 sphere is the same as that of the whole charge of the sphere would 
 be if concentrated at the centre of the sphere. 
 
 We must bear in mind, however, that it 1s physically impossible 
 to charge the small sphere with more than a certain quantity of 
 electricity on each unit of area of its surface. If the surface- 
 density exceed this limit, electricity will fly off in the form of the 
 brush discharge. Hence the idea of an electrified point is a mere 
 mathematical fiction which can never be realised in nature. The 
 imaginary charge concentrated at the centre of the sphere, which 
 produces an effect outside the sphere equivalent to that of the 
 actual distribution of electricity on the surface, is called the 
 Electrical Image of that distribution. See Art. 100. 
 
 Measurement of Electricity. 
 
 82.]| We have already described methods of comparing the 
 quantity of electrification on different bodies, but in each case we 
 have only compared one quantity of electricity with another, 
 without determining the absolute value of either. To determine 
 the absolute value of an electric charge we must compare it with 
 some definite quantity of electricity, which we assume as a unit. 
 
 The unit of electricity adopted in electrostatics is that quantity 
 of positive or vitreous electricity which, if concentrated in a point, 
 and placed at the unit of distance from an equal charge, also 
 concentrated in a point, would repel it with the unit of mechanical 
 force. The dielectric medium between the two charged points is 
 supposed to be air. 
 
 83.| Let us now suppose two bodies, whose dimensions are nieriall 
 compared with the distance between them, to be charged with 
 electricity. Let the charge of the first body be e units of electri- 
 city and that of the second ¢’ units, and let the distance between 
 the bodies be 7. 
 
64 ELECTROMOTIVE FORCE AT A POINT. [84. 
 
 Then, since the force varies inversely as the square of the 
 distance, the force with which each unit of electricity in the 
 first body repels each unit of electricity in the second body will 
 
 1 : : : ; 
 be a ,and since the number of pairs of units, one in each body, 
 
 is ce’, the whole repulsion between the bodies will be 
 ee 
 
 ye 
 
 If the charge of the first or the second body is negative we 
 must consider ¢ or é negative. If the one charge is positive and 
 the other negative, f will be negative, or the force between the 
 bodies will be an attraction instead of a repulsion. If the charges 
 are both positive or both negative, the force between the bodies 
 will be a repulsion. 
 
 84.] Definition.—The electric or electromotive force at a point 
 is the force which would be experienced by a small body charged 
 with the unit of positive electricity and placed at that point, the 
 electrification of the system being supposed to remain undisturbed 
 by the presence of this unit of electricity. 
 
 We shall use the German letter © as the symbol of electric 
 force. | 
 
 85.] Let us now return to the case of a sphere whose radius 
 is r, the external surface of which is uniformly electrified, the 
 surface-density of the electrification being o. As we have already 
 - proved, the whole charge of the sphere is 
 
 e= 47770. 
 At any point outside the sphere such that the distance from 
 the centre of the sphere is 7’ the electromotive force, ©, is directed 
 from the centre, and its value is 
 
 e 
 Ca 
 rh 
 
 If the point is close to the surface of the sphere, 7’ = 7, and 
 e 
 
 G—3=47s, 
 
 or the electric force close to the surface of an electrified sphere is 
 at right angles to the surface and is equal to the surface-density 
 multiplied by 4 7. 
 
 We have already seen that in all cases the electric force close 
 to the surface of a conductor is at right angles to that surface, and 
 is proportional to the surface-density. We now, by means of this 
 
86.] VALUE OF THE POTENTIAL. 65 
 
 particular case, find that the constant ratio of the electric force 
 to the surface-density is 47 for a uniformly electrified sphere, and 
 therefore this is the ratio for a conductor of any form. 
 
 The equation G = 420 
 
 is the complete expression of the law discovered by Coulomb and 
 referred to in Arts. 47 and 81. 
 
 Value of the Potential. 
 
 86.] We must next consider the values of the potential at 
 different distances from a small electrified body. 
 
 Definition. The electric potential at any point is the work which 
 must be expended in order to bring a body charged with unit 
 of electricity from an infinite distance to that point. 
 
 If y is the potential at 4 and y’ that at B, then the work 
 which must’ be spent by the external agency in overcoming 
 electrical force while carrying a unit of electricity from 4 to B 
 is y’—y. 
 
 The quantity y’— would also represent the work which would 
 be done by the electrical forces in assisting the transfer of the unit 
 of electricity from B to A if the motion were reversed. 
 
 If the force from B to A were constant and equal to ©, then 
 In general, the electric force varies as the body moves from B to A, 
 so that we cannot at once apply this method of finding the differ- 
 ence of potentials. But, by breaking up the path £4 into a 
 sufficient number of parts, we may make these parts so small that 
 the electric force may be regarded as uniform during the passage 
 of the body along any one of these parts. We may then ascertain 
 the parts of the work done in each part of the path, and by adding 
 them together, obtain the whole work done during the passage 
 
 from B to A. 
 
 Z C B A 
 | | l | 
 
 Hig. 19; 
 
 Let us suppose a unit of electricity placed at O, and let the 
 distances of the points 4, B, C,.,, Z from O be a, 4, ¢,...% The 
 
 electric force at A is 5 at B a and so on, all in the direction 
 
 from O to A. 
 
66 POTENTIAL AT A POINT. [86. 
 
 To find the work which must be done in order to bring a 
 unit of electricity from 4 to B we must multiply the distance 46 
 by the average of the electromotive force at the various points 
 
 between 4 and #&. The least value of the force is 5 and the 
 
 yor ee: AB 
 greatest value is 72° Hence the work required is greater than a 
 
 and less than are Now AB is a—6, and the true value of the 
 
 work is the excess of the potential at B over that at 4. Hence 
 if we now write 4, 5, C,... Z for the potentials at the correspond- 
 ing points, we may express the work required to bring the unit 
 of electricity from 4 to B by B—A. This quantity therefore is 
 greater than oe; 1 1 cap 
 
 a OG 
 
 * 
 
 a—b se 7 
 but less than ees Ge a) se 
 We may express this ie the double inequality 
 Can e358) 
 Similarly E22). <C—B< ee 
 
 b 
 and so on. The ratios ~ a , &c., are all greater than unity. Let 
 
 us suppose that the yore of these ratios is equal to p. The 
 
 ratios &e., are the reciprocals of these; they are therefore all 
 
 less than unity, but none less than Hence 
 aa 5 <B—A< G= ~)p 
 
 als <0 < Bai 
 
 (5-3); a /e— ¥<(-—<)p, 
 
 Adding these inequalities we find 
 
 (5-7) 5<4-4<(5-2)p. 
 
87.] POTENTIAL AT A POINT. 67 
 
 By increasing the number of points between 4 and Z and making 
 the intervals between them smaller we may make the greatest 
 ratlo, p, as near to unity as we please, and we may therefore 
 assert that, as the line 4Z is more and more minutely divided, 
 
 : : : 1 : . 
 the quantity p and its reciprocal a approach unity as their com- 
 
 mon limit. In the limit, therefore, 
 
 ian faa he 
 @ 
 
 a 
 
 We have thus found the difference between the potentials at 
 Aand Z. To determine the actual value of the potential, say at Z, 
 we must refer to the definition of the potential, that it is the 
 work expended in bringing unit of electricity from an infinite 
 distance to the given point. We have therefore in the above 
 expression to suppose the point 4 removed to an infinite distance 
 from O, in which case the potential A is zero, and the reciprocal of 
 
 , ie eT 
 the distance, or 338 also zero. The equation is therefore reduced to 
 
 the form gz! . 
 
 Z 
 or in words, the numerical value of the potential at a given point 
 due to unit of electricity at a given distance is the reciprocal of the 
 
 number expressing that distance. 
 
 If the charge 1s e, then the potential at a distance z is =. 
 
 The potential due to a number of charges placed at different 
 distances from the given point is found by adding the potentials 
 due to each separate charge, regard being had to the sign of each 
 potential. 
 
 87.| Since, as we have seen, therelectric force at any point 
 outside a uniformly electrified spherical surface is the same as if the 
 electric charge of the surface had been concentrated at its centre, 
 the potential due to the electrified surface must be, for points 
 outside it, 
 
 Wao 
 where ¢ is the whole charge of the surface, and 7 is the distance of 
 the given point from the centre. 
 Let a@ be the radius of the spherical surface, then this expression 
 
 for the potential is true as long as 7 is greater than a, At the 
 F 2 
 
68 CAPACITY OF TWO CONCENTRIC SPHERES. [88. 
 
 surface, 7 is equal to a. The potential at the surface due to its 
 own electrification is therefore 
 
 Y= 
 
 [since there can be no discontinuity in the value of the potential 
 between the surface and a point just outside it]. 
 
 Within the surface there is no electromotive force, and the 
 potential is therefore the same as at the surface for all points 
 within the sphere 
 
 If the potential of the spherical surface is unity, then 
 
 R18 
 
 €=4, 
 or the charge is numerically equal to the radius. 
 
 Now the electric capacity of a body in a given field is measured 
 by the charge which raises its potential to unity. Hence the 
 electric capacity of a conducting sphere placed in air at a con- 
 siderable distance from any other conductor is numerically equal 
 to the radius of the sphere. 
 
 If by means of an electrometer we can measure the potential of 
 the sphere, we can ascertain its charge by multiplying this potential 
 by the radius of the sphere. This method of measuring a quantity 
 of electricity was employed by Weber and Kohlrausch in their 
 determination of the ratio of the unit employed in electromagnetic 
 to that employed in electrostatic researches. Since there is no 
 electric foree within a uniformly electrified sphere the potential 
 
 within the sphere is constant and equal to = . 
 
 88.] We are now able to complete the theory of the electrifica- 
 tion of two concentric spherical surfaces. 
 
 Let a spherical conductor of radius @ be insulated within a 
 hollow conducting vessel, the internal surface of which is a sphere 
 of radius 4 concentric with the inner sphere. Let the charge 
 on the inner sphere be e, then, as we have already seen, the 
 charge on the interior surface of the vessel will be —e At any 
 point outside both spherical surfaces and distant 7 from the 
 centre the electric potential due to the inner sphere will be 
 
 ~, and that due to the outer sphere will be — . Since these 
 
 two quantities are numerically equal, but of opposite sign, they 
 destroy each other, and the potential at every point for which + 
 is greater than 0 is zero. 
 
 a a 
 
88.] LEYDEN JAR. 69 
 
 Between the two spherical surfaces, at a point distant 7 from the 
 centre, the potential due to the inner sphere is =, and that due to 
 
 the outer sphere is =. Hence the whole potential in this inter- 
 
 é 4 ] 1 
 mediate space is e(— — —)- 
 P 7 i) 
 
 At the surface of the inner sphere 7 = a, so that the potential of 
 : ‘ 1 1 
 the inner sphere is ¢ ( — i) : 
 The potential at all points within the inner sphere is uniform and 
 equal to e (- - ) : 
 
 The capacity of the inner sphere is numerically equal to the value 
 of e when the potential is made equal to unity. In this case 
 Lee a0 
 Dae tga 
 
 a 6b 
 
 or, the capacity of a sphere insulated within a concentric spherical 
 surface is a fourth proportional to the distances (,—a@) between the 
 surfaces and radii (a, ¥) of the surfaces. 
 
 By diminishing the interval, 4—a, between the surfaces, the 
 capacity of the system may be made very great without making 
 ~ use of very large spheres. 
 
 This example may serve to illustrate the principle of the Leyden 
 jar, which consists of two metallic surfaces separated by insulating 
 material. The smaller the distance between the surfaces and the 
 greater the area of the surfaces, the greater the capacity of the jar. 
 
 Hence, if an electrical machine which can charge a body up to a 
 given potential is employed to charge a Leyden jar, one surface of 
 which is connected with the earth, it will, if worked long enough, 
 communicate a much greater charge to the jar than it would toa 
 very large insulated body placed at a great distance from any other 
 conductor. 
 
 The capacity of the jar, however, depends on the nature of the 
 dielectric which is between the two metallic surfaces as well as on 
 its thickness and area. See Art. 131 et sqq. 
 
 
 
 
 
70 FORCE BETWEEN TWO PARALLEL PLANES. [89. 
 
 Two PARALLEL PLANES. 
 
 89.| Another simple case of electrification is that in which the 
 electrodes are two parallel plane surfaces at a distance c. We shall 
 suppose the dimensions of these surfaces to be very great compared 
 with the distance between them, and we shall consider the elec- 
 trical action only in that part of the space between the planes 
 whose distance from the edges of the plates is many times greater 
 than c. 
 Let A be the potential of the upper plane in the figure, and B 
 that of the lower plane. Then 
 ON Cal. §Gsser ay oe A the electric force at any point 
 —____________ 7 P between the planese=amsaes 
 Fig. 20. near the edge of either plane, 
 
 
 
 is mi acting from 4 to B. The electric density on the upper 
 plane is found by Coulomb’s Law by dividing this quantity by 47. 
 If o be the surface density 
 
 oA Be (1) 
 47¢ 
 
 The surface density on the plane B is equal to this in magnitude 
 but opposite in sign. 
 
 Let us now consider the quantity of electricity on an area S, 
 which we may suppose cut out from the upper plane by an 
 imaginary closed curve. Multiplying 8 into o, we find 
 
 A—B 
 Ein 47¢ ee (2) 
 The quantity of electricity on an equal area of the plane B taken 
 exactly opposite to S will be —e. The energy of the electrification 
 of these two portions of electricity is, by Art. 31, 
 
 
 
 
 
 Q = 2{4e+ B(—e)} = 3(4—B)e. (3) 
 Expressing this in terms of e it becomes 
 sR: 
 Q= nae (4) 
 
 If c, the distance between the surfaces, be made to increase to c’ 
 the charges of the surfaces remaining the same, the energy will 
 become 
 
 ,__ 27 oy 
 Y= — ee. (5) 
 
gt.] _ ATTRACTED DISC ELECTROMETERS. 71 
 The augmentation of the potential energy is 
 Zap, | 
 Y-Q= ee —e), (6) 
 
 and this is the work done by external agency in pulling the planes 
 asunder against the electric attraction. 
 If Fis the electric attraction between the two areas S, 
 
 F(¢—0) = 28 (eo) (7) 
 2 
 or a mae (8) 
 
 90.| This result gives us the best experimental method of measur- 
 ing the quantity of electricity on the area S, for by this equation 
 
 aA) ae ) 
 
 In this expression / is the force of attraction on the area S deter- 
 mined in dynamical measure from observation of its effects. S is 
 the area of the surface and 7 is the ratio of the circumference of a 
 circle to its diameter. 
 
 The difference between the potentials, 4 and B, of the two planes 
 is easily found in terms of e by means of equation (2), thus, 
 
 ms 
 A-B=tnogmey [22 (10) 
 
 91.] In Sir Wilham Thomson’s attracted disk electrometers a 
 disk is so arranged that when in its proper position the surface of 
 the disk forms part of a much larger plane surface extending for a 
 considerable distance on all sides of the disk. The part of the sur- 
 face outside the moveable disk is called the Guard Ring and the 
 surface of the disk and guard ring together may be considered as 
 the surface of a large disk, part of which, near its centre, is 
 moveable. Opposite this disk is placed another disk having its 
 surface parallel to the first disk and much larger than the move- 
 able disk. The electrification of the moveable disk is then the 
 same as that of a small portion of one of the large opposed planes 
 taken at a considerable distance from the edge of the plane, and 
 only very small corrections are needed to make the formule already 
 given apply to the case of the moveable disk. 
 
 The distribution of electrification and of electric force near the 
 edges of the large disks is by no means so simple. It is calculated 
 
72 INVERSE PROBLEM OF ELECTROSTATICS. [92. 
 
 in Art. 202 of my larger Treatise, and the lines of force and 
 equipotential surfaces are shown in Plate V at the end of this 
 book. 
 
 92.] The direct problem of electrostatics—the problem which 
 the circumstances of every electrostatic experiment present to us— 
 may be stated as follows. 
 
 A system of insulated conductors is given in form and position, 
 and the electric charge of each conductor is given, required the 
 distribution of electricity on each conductor and the electric po- 
 tential at any point of the field. 
 
 The mathematical difficulties of the solution of this problem have 
 been overcome hitherto only in a small number of cases, and it is 
 only by a study of what we may call the inverse problem that the 
 results we possess have been obtained. 
 
 In the inverse problem, a possible distribution of potential 
 being given, it is required to find the forms, positions, and charges 
 of a system of conductors which shall be consistent with this dis- 
 tribution of potential. 
 
 Any number of solutions of this latter problem may be obtained 
 by taking, instead of the electrified bodies of the original distribution, 
 any set of equipotential surfaces surrounding them, and supposing 
 these surfaces to be the surfaces of conductors, the charge of each 
 conductor being equal to the sum of the charges of all the bodies 
 of the original distribution which it encloses. 
 
 Every electrical problem of which we know the solution has been 
 constructed by an inverse process of this kind. It is therefore of 
 great importance to the electrician that he should know what results 
 have been obtained in this way, since the only method by which he 
 can expect to solve a new problem is by reducing it to one of the 
 cases in which a similar problem has been constructed by the 
 inverse process. 
 
 This historical knowledge of results can be turned to account in 
 two ways. If we are required to devise an instrument for making 
 electrical measurements with the greatest accuracy, we may select 
 those forms for the electrified surfaces which correspond to cases of 
 which we know the accurate solution. If, on the other hand, we 
 are required to estimate what will be the electrification of bodies 
 whose forms are given, we may begin with some case in which 
 one of the equipotential surfaces takes a form somewhat resem- 
 bling the given form, and then by a tentative method we may 
 modify the problem till it more nearly corresponds to the given 
 
93.| DIAGRAMS OF EQUIPOTENTIAL SURFACES. 73 
 
 ease. This method is evidently very imperfect, considered from a 
 mathematical point of view, but it is the only one we have, and if 
 we are not allowed to choose our conditions, we can make only an 
 approximate calculation of the electrification. It appears, therefore, 
 that what we want is a knowledge of the forms of equipotential 
 surfaces and lines of induction in as many different cases as we can 
 collect together and remember. In certain classes of cases, such 
 as those relating to spheres, we may proceed by mathematical 
 methods. In other cases we cannot afford to despise the humbler 
 method of actually drawing tentative figures on paper, and select- 
 ing that which appears least unlike the figure we require. 
 
 This latter method, I think, may be of some use, even in cases 
 in which the exact solution has been obtained, for I find that an 
 eye knowledge of the forms of the equipotential surfaces often leads 
 to a right selection of a mathematical method of solution. 
 
 I have therefore drawn several diagrams of systems of equipo- 
 tential surfaces and lines of force, so that the student may make 
 himself familiar with the forms of the lines. 
 
 93.| In the first plate at the end of this volume we have the 
 equipotential surfaces surrounding two points electrified with quan- 
 tities of electricity of the same kind and in the ratio of 20 to 5. 
 
 Here each point is*surrounded by a system of equipotential 
 surfaces which become more nearly spheres as they become smaller; 
 but none of them are accurately spheres. If two of these surfaces, 
 one surrounding each sphere, be taken to represent the surfaces 
 of two conducting bodies, nearly but not quite spherical, and if 
 these bodies be charged with the same kind of electricity, the 
 charges being as 4 to 1, then the diagram will represent the 
 equipotential surfaces, provided we expunge all those which are 
 drawn inside the two bodies. It appears from the diagram that 
 the action between the bodies will be the same as that between 
 two points having the same charges, these points being not exactly 
 in the middle of the axis of each body, but somewhat more remote 
 than the middle point from the other body. 
 
 The same diagram enables us to see what will be the distribu- 
 tion of electricity on one of the oval figures, larger at one end 
 than the other, which surround both centres. Such a body, if elec- 
 trified with a charge 25 and free from external influence, will 
 have the surface-density greatest at the small end, less at the large 
 end, and least in a circle somewhat nearer the smaller than the 
 larger end. 
 
74 DIAGRAMS OF EQUIPOTENTIAL SURFACES [94. 
 
 There is one equipotential surface, indicated by a dotted line, 
 which consists of two lobes meeting at the conical point P. That 
 point is a point of equilibrium, and the surface-density on a body 
 of the form of this surface would be zero at this point. 
 
 The lines of force in this case form two distinct systems, divided 
 from one another by a surface of the sixth degree, indicated by a 
 dotted line, passing through the point of equilibrium, and some- 
 what resembling one sheet of the hyperboloid of two sheets. 
 
 This diagram may also be taken to represent the lines of force 
 and equipotential surfaces belonging to two spheres of gravitating 
 matter whose masses are as 4 to 1. 
 
 94.] In the second Plate we have again two points whose charges 
 are as 4 to 1, but the one positive and the other negative. In this 
 case one of the equipotential surfaces, that, namely, corresponding 
 to potential zero, is a sphere. It is marked in the diagram by the 
 dotted circle Q. The importance of this spherical surface will be 
 seen when we come to the theory of Electrical Images. 
 
 We may see from this diagram that if two round bodies are 
 charged with opposite kinds of electricity they will attract each 
 other as much as two points having the same charges but placed 
 somewhat nearer together than the middle points of the round 
 bodies. . 
 
 Here, again, one of the equipotential surfaces, indicated by a 
 dotted line, has two lobes, an inner one surrounding the point 
 whose charge is 5 and an outer one surrounding both bodies, the 
 two lobes meeting in a conical point P which is a point of equili- 
 brium. 
 
 If the surface of a conductor is of the form of the outer lobe, a 
 roundish body having, like an apple, a conical dimple at one end of 
 its axis, then, if this conductor be electrified, we shall be able to 
 determine the superficial density at any point. That at the bottom 
 of the dimple will be zero. | 
 
 Surrounding this surface we have others having a rounded 
 dimple which flattens and finally disappears in the equipotential 
 surface passing through the point marked J. 
 
 The lines of force in this diagram form two systems divided by a 
 surface which passes through the point of equilibrium. 
 
 If we consider points on the axis on the further side of the point 
 B, we find that the resultant force diminishes to the double point P, 
 where it vanishes. I¢ then changes sign, and reaches a maximum 
 at M/, after which it continually diminishes. 
 
96. | AND LINES OF INDUCTION. 75 
 
 This maximum, however, is only a maximum relatively to other 
 points on the axis, for if we draw a surface perpendicular to the 
 axis, M is a point of minimum force relatively to neighbouring 
 points on that surface. 
 
 95.| Plate III represents the equipotential surfaces and lines 
 of force due to an electrified point whose charge is 10 placed at 
 A, and surrounded by a field of force, which, before the intro- 
 duction of the electrified point, was uniform in direction and 
 magnitude at every part. In this case, those lines of force which 
 belong to A are contained within a surface of revolution which 
 has an asymptotic cylinder, having its axis parallel to the un- 
 disturbed lines of force. 
 
 The equipotential surfaces have each of them an asymptotic 
 plane. One of them, indicated by a dotted line, has a conical 
 point and a lobe surrounding the point 4. Those below this surface 
 have one sheet with a depression near the axis. Those above have 
 a closed portion surrounding 4 and a separate sheet with a slight 
 depression near the axis. ; 
 
 If we take one of the surfaces below A as the surface of a con- 
 ductor, and another a long way below 4 as the surface of another 
 conductor at a different potential, the system of lines and surfaces 
 between the two conductors will indicate the distribution of electric 
 force. If the lower conductor is very far from A its surface will 
 be very nearly plane, so that we have here the solution of the 
 distribution of electricity on two surfaces, both of them nearly 
 plane and parallel to each other, except that the upper one has 
 a protuberance near its middle point, which is more or less pro- 
 minent according to the particular equipotential line we choose for 
 the surface. . 
 
 96.| Plate IV represents the equipotential surfaces and lines 
 . of force due to three electrified points 4, B and C, the charge of 4 
 being 15 units of positive electricity, that of B12 units of negative 
 electricity, and that of C 20 units of positive electricity. These 
 points are placed in one straight line, so that 
 
 Ab Nee 16 ACG = 695% 
 
 In this case,-the surface for which the potential is unity consists 
 of two spheres whose centres are 4 and Cand their radii 15 and 20. 
 These spheres intersect in the circle which cuts the plane of the 
 paper in D and D’, so that B is the centre of this circle and its 
 radius is 12. This circle is an example of a line of equilibrium, for 
 the resultant force vanishes at every point of this line. 
 
76 LINES OF INDUCTION. [97. 
 
 If we suppose the sphere whose centre is 4 to be a conductor 
 with a charge of 3 units of positive electricity, and placed under 
 the influence of 20 units of positive electricity at C, the state of 
 the case will be represented by the diagram if we leave out all the 
 lines within the sphere 4. The part of this spherical surface within 
 the small circle DD’ will be negatively electrified by the influence 
 of C. All the rest of the sphere will be positively electrified, and 
 the small circle DL” itself will be a line of no electrification. 
 
 We may also consider the diagram to represent the electrification 
 of the sphere whose centre is C, charged with 8 units of positive 
 electricity, and influenced by 15 units of positive electricity placed 
 at A. 
 
 The diagram may also be taken to represent the case of a 
 conductor whose surface consists of the larger seements of the 
 two spheres meeting in DD’, charged with 23 units of positive 
 electricity. 
 
 97.| I am anxious that these diagrams should be studied as 
 illustrations of the language of Faraday in speaking of ‘lines of 
 force, the ‘forces of an electrified body,’ &c. 
 
 In strict mathematical language the word Force is used to signify 
 the supposed cause of the tendency which a material body is found 
 to have towards alteration in its state of rest or motion. It is 
 indifferent whether we speak of this observed tendency or of its 
 immediate cause, since the cause is simply inferred from the effect, 
 and has no other evidence to support it. 
 
 Since, however, we are ourselves in the practice of directing the 
 motion of our own bodies, and of moving other things in this way, 
 we have acquired a copious store of remembered sensations relating 
 to these actions, and therefore our ideas of force are connected in 
 our minds with ideas of conscious power, of exertion, and of fatigue, 
 and of overcoming or yielding to pressure. These ideas, which give 
 a colouring and vividness to the purely abstract idea of force, do 
 pot in mathematically trained minds lead to any practical error. 
 
 But in the vulgar language of the time when dynamical science 
 was unknown, all the words relating to exertion, such as force, 
 energy, power, &c., were confounded with each other, though some 
 of the schoolmen endeavoured to introduce a greater precision into 
 their language. 
 
 The cultivation and popularization of correct dynamical ideas 
 since the time of Galileo and Newton have effected an immense 
 change in the language and ideas of common life, but it is only 
 
98. ] CONSTRUCTION OF DIAGRAMS. cif? 
 
 within recent times, and in consequence of the increasing im- 
 portance of machinery, that the ideas of force, energy and power 
 have become accurately distinguished from each other. Very few, 
 however, even of scientific men, are careful to observe these dis- 
 tinctions ; hence we often hear of the force of a cannon-ball when 
 either its energy or its momentum is meant, and of the force of an 
 electrified body when the quantity of its electrification is meant. 
 
 Now the quantity of electricity in a body is measured, according 
 to Faraday’s ideas, by the number of lines of force, or rather of 
 induction, which proceed from it. These lines of force must all 
 terminate somewhere, either on bodies in the neighbourhood, or on 
 the walls and roof of the room, or on the earth, or on the heavenly 
 bodies, and wherever they terminate there is a quantity of elec- 
 tricity exactly equal and opposite to that on the part of the body 
 from which they proceeded. By examining the diagrams this will 
 be seen to be the case. There is therefore no contradiction between 
 Faraday’s views and the mathematical result of the old theory, 
 but, on the contrary, the idea of lines of force throws great light 
 on these results, and seems to afford the means of rising by a con- 
 tinuous process from the somewhat rigid conceptions of the old 
 theory to notions which may be capable of greater expansion, so 
 as to provide room for the increase of our knowledge by further 
 researches. ’ 
 
 98.] These diagrams are constructed in the following manner :— 
 
 First, take the case of a single centre of force, a small electrified 
 
 body with a charge #. The potential at a distance 7 is V = = 
 
 hence, if we make 7 = - » we shall find 7, the radius of the sphere 
 
 for which the potential is 7. If we now give to V the values 
 1, 2, 3, &c., and draw the corresponding spheres, we shall obtain 
 a series of equipotential surfaces, the potentials corresponding to 
 which are measured by the natural numbers. The sections of these 
 spheres by a plane passing through their common centre will be 
 circles, which we may mark with the number denoting the potential 
 of each. These are indicated by the dotted circles on the right 
 hand of Fig. 21. 
 
 If there be another centre of force, we may in the same way draw 
 the equipotential surfaces belonging to it, and if we now wish to 
 find the form of the equipotential surfaces due to both centres 
 together, we must remember that if /, be the potential due to one 
 
78 EQUIPOTENTIAL SURFACES [98. 
 
 centre, and /, that due to the other, the potential due to both will be 
 V,+V,.=V. Hence, since at every intersection of the equipotential 
 surfaces belonging to the two series we know both V, and /,, we 
 also know the value of VY. If therefore we draw a surface which 
 passes through all those intersections for which the value of V is 
 the same, this surface will coincide with a true equipotential surface 
 at all these intersections, and if the original systems of surfaces 
 be drawn sufficiently close, the new surface may be drawn with 
 any required degree of accuracy. The equipotential surfaces due to 
 two points whose charges are equal and opposite are represented by 
 the continuous lines on the right hand side of Fig. 21. 
 
 This method may be applied to the drawing of any system of 
 equipotential surfaces when the potential is the sum of two po- 
 tentials, for which we have already drawn the equipotential surfaces. 
 
 The lines of force due to a single centre of force are straight 
 lines radiating from that centre. If we wish to indicate by these 
 lines the intensity as well as the direction of the force at any point, 
 we must draw them so that they mark out on the equipotential 
 surfaces portions over which the surface-integral of induction has 
 definite values. The best way of doing this is to suppose our 
 plane figure to be the section of a figure in space formed by the 
 revolution of the plane figure about an axis passing through the 
 centre of force. Any straight line radiating from the centre and 
 making an angle 6 with the axis will then trace out a cone, 
 and the surface-integral of the induction through that part of any 
 surface which is cut off by this cone on the side next the positive 
 direction of the axis, is 27 H(1—cos6). 
 
 If we further suppose this surface to be bounded by its inter- 
 section with two planes passing through the axis, and inclined at 
 the angle whose are is equal to half the radius, then the induction 
 through the surface so bounded is 
 
 H(1—cos0) = 2, say ; 
 . 
 and 6 = cos-1(1—2 +) . 
 
 If we now give to ¥ a series of values 1, 2, 3... #, we shall find 
 a corresponding series of values of 0, and if H be an integer, the 
 number of corresponding lines of force, including the axis, will be 
 equal to Z. 
 
 We have therefore a method of drawing lines of force so that 
 the charge of any centre is indicated by the number of lines which 
 converge to it, and the induction through any surface cut off in the 
 
To face page 78. 
 
 Fig. 21. 
 
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 Lines of Force. Equipotential Surfaces. 
 
 Method of drawing 
 Lines of Force and Equipotential Surfaces. 
 
 University Press, Oxford. 
 
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98. | AND LINES OF INDUCTION. 79 
 
 way described is measured by the number of lines of force which 
 pass through it. The dotted straight lines on the left hand side 
 of Fig. 21 represent the lines of force due to each of two electrified 
 points whose charges are 10 and — 10 respectively. 
 
 If there are two centres of force on the axis of the figure we 
 may draw the lines of force for each axis corresponding to values 
 of ¥, and ¥,, and then, by drawing lines through the consecutive 
 intersections of these lines, for which the value of ¥, + ¥, is the 
 same, we may find the lines of force due to both centres, and in 
 the same way we may combine any two systems of lines of force 
 which are symmetrically situated about the same axis. The con- 
 tinuous curves on the left hand side of Fig. 21 represent the lines 
 of force due to the electrified points acting at once. 
 
 After the equipotential surfaces and lines of force have been 
 constructed by this method, the accuracy of the drawing may be 
 tested by observing whether the two systems of lines are every- 
 where orthogonal, and whether the distance between consecutive 
 equipotential surfaces is to the distance between consecutive lines 
 of force as half the distance from the axis is to the assumed unit of 
 length. 
 
 In the case of any such system of finite dimensions the line of 
 foree whose index number is ¥ has an asymptote which passes 
 through the centre of gravity of the system, and is inclined to the 
 axis at an angle whose cosine is hy where / is the total 
 electrification of the system, provided ¥ is less than #. Lines of 
 force whose index is greater than / are finite lines. 
 
 The lines of force corresponding to a field of uniform force parallel 
 to the axis are lines parallel to the axis, the distances from the 
 axis being the square roots of an arithmetical series, 
 
CHAPTER VII. 
 
 THEORY OF ELECTRICAL IMAGES. 
 
 99.| Tue calculation of the distribution of electrification on the 
 surface of a conductor when electrified bodies are placed near it is in 
 general an operation beyond the powers of existing mathematical 
 methods. 
 
 When the conductor is a sphere, and when the distribution of 
 electricity on external bodies is given, a solution, depending on 
 an infinite series was obtained by Poisson. This solution agrees 
 with that which was afterwards given in a far simpler form by 
 Sir W. Thomson, and which is the foundation of his method of 
 Electric Images. 
 
 By this method he has solved problems in electricity which 
 have never been attempted by any other method, and which, even 
 after the solution has been pointed out, no other method seems 
 capable of attacking. This method has the great advantage of 
 being intelligible by the aid of the most elementary mathematical 
 reasoning, especially when it is considered in connection with the 
 diagrams of equipotential surfaces described in Arts. 93-96. 
 
 100.| The idea of an image is most easily acquired by considering 
 the optical phenomena on account of which the term i was 
 first introduced into science. 
 
 We are accustomed to make use of the visual impressions we 
 receive through our eyes in order to ascertain the positions of 
 distant objects. We are doing this all day long in a manner 
 sufficiently accurate for ordinary purposes. Surveyors and astro- 
 nomers by means of artificial instruments and mathematical de- 
 ductions do the same thing with greater exactness. In whatever 
 way, however, we make our deductions, we find that they are 
 consistent with the hypothesis that an object exists in a certain 
 position in space, from which it emits light which travels to our 
 eyes or to our instruments in straight lines. 
 
101.] ELECTRICAL IMAGES. 81 
 
 But if we stand in front of a plane mirror and make observations 
 on the apparent direction of the objects reflected therein, we find 
 that these observations are consistent with the hypothesis that 
 there is no mirror, but that certain objects exist in the region 
 beyond the plane of the mirror. These hypothetical objects are 
 eeometrically related to certain real objects in front of the plane of 
 the mirror, and they are called the zmages of these objects. 
 
 We are not provided with a special sense for enabling us to 
 ascertain the presence and the position of distant bodies by means 
 of their electrical effects, but we have instrumental methods by 
 which the distribution of potential and of electric force in any part 
 of the field may be ascertained, and from these data we obtain a 
 certain amount of evidence as to the position and electrification of 
 the distant body. 
 
 If an astronomer, for instance, could ascertain the direction and 
 magnitude of the force of gravitation at any desired point in the 
 heavenly spaces, he could deduce the positions and masses of the 
 bodies to which the force is due. When Adams and Leverrier 
 discovered the hitherto unknown planet Neptune, they did so by 
 ascertaining the direction and magnitude of the gravitating force 
 due to the unseen planet at certain points of space. In the elec- 
 trical problem we employed an electrified pith ball, which we 
 moved about in the field at pleasure. The astronomers employed 
 for a similar purpose the planet Uranus, over which, indeed, they 
 had no control, but which moved of itself into such positions that 
 the alterations of the elements of its orbit served to indicate the 
 position of the unknown disturbing planet. 
 
 101.] In one of the electrified systems which we have already 
 investigated, that of a spherical conductor 4 within a concentric 
 spherical conducting vessel B, we have one of the simplest cases of 
 the principle of electric images. 
 
 The electric field is in this case the region which les between 
 the two concentric spherical surfaces. The electric force at any 
 point P within this region is in the direction of the radius OP 
 and numerically equal to the charge of the inner sphere, 4, divided 
 by the square of the distance, OP, of the point from the common 
 centre. It is evident, therefore, that the force within this region 
 will be the same if we substitute for the electrified spherical sur- 
 faces, A and B, any other two concentric spherical surfaces, ( and 
 D, one of them, C, lying within the smaller sphere, 4, and the 
 other, D, lying outside of B, the charge of C being equal to that 
 
 G 
 
82 CONCENTRIC SPHERES. [ 102. 
 
 of A in the former case. The electric phenomena in the region 
 between 4 and B are therefore the same as before, the only differ- 
 ence between the cases is that in the region between 4 and C and 
 also in the region between B and D we now find electric forces 
 acting according to the same law 
 as in the region between 4 and 
 B, whereas when the region was 
 bounded by the conducting sur- 
 faces A and B there was no elec- 
 trical force whatever in the regions 
 beyond these surfaces. We may 
 even, for mathematical purposes, 
 suppose the inner sphere C to be 
 reduced to a physical point at 0, 
 and the outer sphere D to expand 
 Fig. 22. to an infinite size, and thus we 
 assimilate the electric action in 
 the region between 4 and # to that due to an electrified point at 
 O placed in an infinite region. 
 
 It appears, therefore, that when a spherical surface is uniformly 
 electrified, the electric phenomena in the region outside the sphere 
 are exactly the same as if the spherical surface had been removed, 
 and a very small body placed at the centre of the sphere, having 
 the same electric charge as the sphere. 
 
 This is a simple instance in which the phenomena in a certain 
 region are consistent with a false hypothesis as to what exists 
 beyond that region. The action of a uniformly electrified spherical 
 surface in the region outside that surface is such that the phenomena 
 may be attributed to an imaginary electrified point at the centre of 
 the sphere. 
 
 The potential, y, of a sphere of radius a, placed in infinite space 
 
 
 
 and charged with a quantity e of electricity, is . Hence if y is 
 
 the potential of the sphere, the imaginary charge at its centre 
 is wa. 
 
 102.] Now let us calculate the potential at a point P (Fig. 23.) 
 in a spherical surface whose centre is C and radius CP, due to two 
 electrified points 4 and / in the same radius produced, and such 
 that the product of their distances from the centre is equal to the 
 square of the radius. Points thus related to one another are called 
 averse points with respect to the sphere. 
 
102. | IMAGE OF A POINT. 83 
 
 Let a = CP be the radius of the sphere. Let CA = ma, then CB 
 : a 
 will be — - 
 m 
 Also the triangle 4PC is similar to PCB, and 
 AP:PB:: AC: PG, 
 
 or AP =mBP. See Euclid vi. prop. E. 
 Now let a charge of electricity equal to e be placed at A and a 
 
 charge ¢ = — of the opposite kind be placed at B. The poten- 
 tial due to these charges at P will be 
 
 / 
 
 
 
 = é if ec 
 
 pee Rp. 
 
 a6 Gi Cc 
 eae ae 
 
 = Whe 
 
 or the potential due to the charges at 4 and B at any point P of the 
 spherical surface is zero. 
 
 We may nowsuppose the spherical 
 surface to be a thin shell of metal. 
 Its potential is already zero at every 
 point, so that if we connect it by a 
 fine wire with the earth there will 
 be no alteration of its potential, 
 and therefore the potential at every 
 point, whether within or without 
 the surface, will remain unaltered, and will be that due to the two 
 electrified points 4 and S. 
 
 If we now keep the metallic shell in connection with the earth 
 and remove the electrified point B, the potential at every point 
 within the sphere will become zero, but outside it will remain as 
 before. or the surface of the sphere still remains of the same 
 potential, and no change has been made in the distribution of 
 electrified bodies in the region outside the sphere. 
 
 Hence, if an electrified point 4 be placed outside a spherical con- 
 ductor which is at potential zero, the electrical action at all points 
 outside the sphere will be equivalent to that due to the point 4 
 together with another point, B, within the sphere, which is the 
 inverse point to 4, and whose charge is to that of 4 as —1 is to m. 
 The point B with its imaginary charge is called the electric image of A. 
 
 In the same way by removing 4 and retaining 6, we may shew 
 
 G 2 
 
 
 
 Fig. 23 
 
84 ELECTRICAL IMAGES. iro. 
 
 that if an electrified point B be placed inside a hollow conductor 
 having its inner surface spherical, the electrical action within the 
 hollow is equivalent to that of the point 5, together with an 
 imaginary point, 4, outside the sphere, whose charge is to that 
 of Bas m is to —1. 3 | 
 
 If the sphere, instead of being in connection with the earth, and 
 therefore at potential zero, is at potential w, the electrical effects 
 outside the sphere will be the same as if, in addition to the image 
 of the electrified point, another imaginary charge equal to ya were 
 placed at the centre of the sphere. 
 
 Within the sphere the potential will simply be increased by y. 
 
 103.] As an example of the method of electric images let us 
 caleulate the electric state of two spheres whose radii are a and 6 
 respectively, and whose potentials are P, and P,, the distance be- 
 tween their centres being ¢: We shall suppose 0 to be small com- 
 pared with c. 
 
 
 
 Fig. 24. 
 
 We may consider the actual electrical effects at any point out- 
 side the two spheres as due to a series of electric images. 
 
 In the first place, since the potential of the sphere 4 is P, we 
 must place an image at the centre 4 with a charge aP,. 
 
 Similarly at B, the centre of the other sphere, we must place a 
 charge bP,. 
 
 Each of these images will have an image of the second order in 
 
 the other sphere. The image of B in the sphere a will be at D, 
 where a? a 
 Wh 7 and the charge D = — --)P,. 
 c 
 The image of 4 in the sphere 4 will be at H, where 
 2 
 
 b 
 Bias ap and the charge # = — oa ee 
 
104. | _ ' TWO SPHERES. 85 
 
 Each of these will have an image of the third order. That of # 
 in a will be at /, where 
 
 2 ae a? b 
 
 
 
 
 
 a i 
 Fa = ae 5h and / = A opt Whee 
 That of D in @ will be at G, where 
 b? & c ab? 
 bG = 75 = @aga? and 7 = ep Po 
 The images of the eesti order will be, 
 of Gin a at A, where 
 a> a® (c*—a*) 3 a? 6? 
 eMac (eas 78) Cie gt 8) 
 of Fin B at J, where 
 fg G2 (c?—2?) a b2 
 SS aa CE) aa (a 
 
 We might go on with a series of images for ever, but if ) is small 
 compared with c, the images will rapidly become smaller and may 
 be neglected after the fourth order. 
 
 If we now write 
 a®h 
 
 EN: 
 i a* | 
 eb a 
 
 ati: (c?— a? — 0?) 
 
 Jaa = 4+ = + &e., 
 
 — &e., 
 
 
 
 the whole charge of the sphere a will be 
 Ea = Yaa Pat Yan Pr: 
 and that of the sphere 4 will be 
 Ly = Yan Pat In Pro- 
 104.] From these results we may calculate the potentials of the 
 
 two spheres when their charges are given, and if we neglect 
 terms involving J? we find 
 
 ft _ E, + - ~E,, 
 1 ae 
 nals fa at al B 
 The electric energy of the system is 
 
 1 ee al tk (1 at a 
 g FaPat Ly Pr) = 5 7 be +5 BB+ 315 — aaa L,’. 
 
86 TWO SPHERES. [105. 
 
 The repulsion, 2, between the two spheres is measured by the 
 rate at which the energy diminishes as ¢ increases ; therefore, 
 
 Ef, a® (2c?—a?*) 
 k= — ahh —L, oa c(c?— e(2—a?)? t. 
 
 In order that the are may be repulsive it is necessary that the 
 
 charges of the spheres should be of the same sign, and 
 2 A 
 f,, must be greater than L, a 
 Hence the force 1s always attractive, 
 
 1. When either sphere is uninsulated ; 
 
 2. When either sphere has no charge ; 
 
 3. When the spheres are very nearly in contact, if their poten- 
 tials are different. 
 
 When the potentials of the two spheres are equal the force is 
 always repulsive. 
 
 105.] To determine the electrie foree at any point just outside of 
 the surface of a conducting sphere connected with the earth arising 
 from the presence of an electrified point 4 outside the sphere. 
 
 The electrical conditions at all points outside the sphere are equi- 
 valent, as we have seen, to those due to the point 4 together with its 
 image at 6. If eis the charge of the point 4 (Fig. 23), the force 
 due to it at P is Tp in the direction 4P. Resolving this force in 
 al oes parallel to 4C and along the ar its components are 
 
 ¥ im AC in the direction parallel to 4C and 7 a CP in the direc- 
 
 tion CP. The charge of the image . A at Bis —e a and the 
 
 CP 
 force due to the image at P is e — CA api 
 solving this force in the same direction as the other, its components 
 oe e SSS in a direction parallel to Cd, and — 
 
 CA” BP? i ’ 
 ; Gre 
 CARLES 
 If a is the radius of the sphere and if CA = f= ma and AP =7, 
 
 in the direction PB. Re- 
 
 in the direction PC. 
 
 1 niga . 
 then CB = a and BP = Pate and if e is the charge of the point 
 
 ee : 1 
 A, the charge of its image at B is — mate 
 
 The force at P due to the charge e at A is 4 in the direction AP, 
 
106. | DENSITY OF INDUCED CHARGE. 87 
 
 Resolving this force in the direction of the radius and a direction 
 parallel to AC, its components are 
 
 ae = in the direction AC, and 
 
 sas) 
 
 ive ca. ere: 
 bey I the direction CP. 
 
 ] mM 
 ——— or ¢— 
 BL a 
 in the direction P&. Resolving this in the same directions as the 
 other force, its components are 
 
 : 1 ; 
 The force at P due to the image — oon: at B is - e 
 
 
 
 
 
 m BC ema, maar: ; 
 
 € 2 Bp > @ the direction Cd, and 
 m.CP ema , ; ; 
 
 € 2 BP OY 3 in the direction PC. 
 
 The components in the direction parallel to AC are equal but in 
 opposite directions. The resultant force is therefore in the direc- 
 tion of the radius, which confirms what we have already proved, 
 that the sphere is an equipotential surface to which the resultant 
 force is everywhere normal. ‘The resultant force is therefore in the 
 
 direction PC, and is equal to 5 (m*—1) in the direction PC, that is 
 
 to say, towards the centre of the sphere. 
 
 From this we may ascertain the surface density of the electrifica- 
 tion at any point of the sphere, for, by Coulomb’s law, if o is the 
 _ surface density, 
 
 4no = fF, where & is the resultant force acting outwards. 
 
 Hence, as the resultant force in this ease acts inwards, the surface 
 density is everywhere negative, and is 
 
 a 
 c= — - “5 (m?—1). 
 
 Hence the surface density is inversely as the cube of the distance 
 from the inducing point A. 
 
 106.] In the case of the two spheres 4 and B (Fig. 24), whose 
 radii are w and 6 and potentials P, and P,, the distance between 
 their centres being c, we may determine the surface density at any 
 point of the sphere 4 by considering it as due to the action of a 
 charge aP, at A, together with that due to the pairs of points B, 
 D and FL, F &c., the successive pairs of images. 
 
88 DENSITY OF INDUCED CHARGE. [ 106. 
 
 Putting r= PBS Tea ais <tr eeee 
 we find 
 PY PAN tad ye 
 r=7-P,|7 + DAN Peg i hetia + &e. | 
 es ac 2 
 
 47 a 
 
 1 b b? ¢2 c? —q2—§7 2% “as 
 =e BP =; (c?—a*) aa 73 (a?) =a -S} + ke, | 
 
 If we call B the inducing body and 4 the induced body, then we 
 may consider the electrification induced on 4 as consisting of two 
 parts, one depending on the potential of £ and the other on its 
 own potential. 
 
 -The part depending on /P, is called by some writers on electricity 
 the induced electrification of the first species. When A is not in- 
 sulated it constitutes the whole electrification, and if P, is positive 
 it is negative over every part of the surface, but greatest in 
 numerical nature at the point nearest to B. 
 
 The part depending on P, is called the enduced electrification of 
 the second species. It can only exist when 4 is insulated, and it 
 is everywhere of the same sign as P,. If A is insulated and with- 
 out charge, then the induced electrifications of the first and second 
 species must be equal and opposite. The surface-density is negative 
 on the side next to B and positive on the side furthest from B, but 
 though the total quantities of positive and negative electrification 
 are equal, the negative electrification is more concentrated than the 
 positive, so that the neutral line which separates the positive from 
 the negative electrification is not the equator of the sphere, but les 
 nearer to B. 
 
 The condition that there shall be both positive and negative elec- 
 trification on the sphere is that the value of o at the points nearest 
 to B and farthest from B shall have opposite signs. Ifa and # are 
 small compared with c, we may neglect all the terms of the co- 
 efficients of P, and P, after the first. The values of 7 lie between 
 b(c—a) andes d(c+a) 
 
 (e+a) (e—a)? 
 there will be both positive and negative electrification on A, divided 
 by a neutral line, but if P, is beyond these limits, the electrification 
 of every part of the surface will be of one kind; negative if P, is 
 below the lower limit, and positive if it is above the higher limit. 
 
 c+aandc—a. Hence, if P, is between P 
 
 
 
 
 
CHAPTER VIII. 
 ON ELECTROSTATIC CAPACITY. 
 
 107.| Tu capacity of a conductor is measured by the charge of 
 electricity which will raise its potential to the value unity, the 
 potential of all other conductors in the field being kept at zero. 
 The capacity of a conductor depends not only on its own form and 
 size, but on the form and position of the other conductors in the 
 field. The nearer the uninsulated conductors are placed the greater 
 is the capacity of the charged conductor. 
 
 An apparatus consisting of two insulated conductors, each pre- 
 senting a large surface to the other with a small distance between 
 them, is called a condenser, because a small electromotive force is 
 able to charge such an apparatus with a large quantity of elec- 
 tricity. 
 
 The simplest form of condenser, that to which the name is most 
 commonly applied, consists of two disks placed parallel to each 
 other, the medium between them being air. When one of these 
 disks 1s connected to the zinc and the other to the copper electrode 
 of a voltaic battery, the disks become charged with negative and 
 positive electricity respectively, and the amount of the charge is 
 the greater the nearer the disks are placed to each other, being 
 approximately inversely as the distance between them. Hence by 
 bringing the disks very close to each other, connecting them with 
 the electrodes of the battery and then disconnecting them from the 
 battery, we have two large charges of opposite kinds insulated on 
 the disks. If we now remove one of the disks from the other we 
 do work against the electric attraction which draws them together, 
 and we may thus increase the energy of the system so much that, 
 though the original electromotive force was only that of a single 
 voltaic cell, either of the disks when separated may be raised to so 
 
90 DISCHARGE BY ALTERNATE CONTACTS, [ 108. 
 
 high a potential that the gold leaves of an electrometer connected 
 with it are deflected. 
 
 It was in this way that Volta demonstrated that the electrifica- 
 tion due to a voltaic cell is of the same kind as that due to friction, 
 the copper electrode being positive with respect to the zine elec- 
 trode. In this condenser the capacity of each disk depends prin- 
 cipally on the distance between it and the other disk, but it also 
 depends in a smaller degree on the nature of the electric field at 
 the back of the disk. 
 
 There are other forms of condensers, however, in which one of 
 the conductors is almost or altogether surrounded by the other. 
 In this case the capacity of the inner conductor is almost or alto- 
 eether independent of everything but the outer conductor. This is 
 the case in the Leyden jar, and in a cable with a copper core sur- 
 rounded by an insulator the outside of which is protected by a 
 sheathing of iron wires. 
 
 108.] But in most cases the charge of each conductor depends 
 not only on the difference between its potential and that of the 
 other conductor, but also in part on the difference between its 
 potential and that of some other body, such as the earth, or the 
 walls of the room where the experiment is made. The charges of 
 the two conductors may, therefore, in the simpler cases be written 
 
 Q = K(P—p)+HP, ........00 (1) 
 
 q= K(p—P)+%p, ...... (2) 
 
 where P and p are the potentials, that of the walls of the room 
 being zero, @ and g the charges of the two conductors respectively, 
 K is the capacity of the condenser in so far as 1t depends on the 
 mutual relation of the two conductors, and // and / represent those 
 parts of the capacity of each conductor which depend on their rela- 
 tion to external objects, such as the walls of the room. 
 
 If we connect the second conductor with the earth we make p 
 zero while Q remains the same, and we get for the new values of 
 
 Pe) anus 
 
 dt a Q,=(K+H)P,, 9, = — hee eee 
 
 K+” 
 If we now insulate the second conductor and connect the first 
 with the earth we make P zero, and 
 
 % 
 Sieenr yam yl Q. = — Kp,, 9, =(K+4)p,. +a 
 
T10. | DISCHARGE BY ALTERNATE CONTACTS. 91 
 
 If we again nsulate the first conductor and put the second to 
 earth, 
 K : 
 P3=— Ky? Q3=(K+H)P;, 9,= —KPs. ...... (5) 
 From this it appears that if we connect first the one and then 
 the other conductor with the earth the values of the potentials and 
 K2 
 
 charges will be diminished in the ratio of (K+H) (K+) to 
 
 unity. 
 
 Comparison of two condensers. 
 
 109.| Let us suppose the condensers to be Leyden jars having 
 an inner and an outer coating. 
 
 Let the inner coating of the first jar and the outer coating of 
 the second be connected with a source of electricity and brought to 
 the potential P, while the outer coating of the first and the inner 
 coating of the second are connected with the earth. 
 
 Then if Q, and Q, are the charges of the inner coatings of the 
 two jars, AY eS OTT IA WOE 1 Od errr eer tie (7) 
 
 Now let the outer coatings of both jars be connected with the 
 earth, and let the inner coatings be connected with each other. 
 Required the common potential of the inner coatings. 
 
 Hence we have pl = pl = 0, 
 Q, te 0. = QO + On eels eleleuerais: ¢ eletereie he ater eratelers (8) 
 YER eed 15 fe Pe ePeReae dace ticserbcvauevivcntarwe'e (9) 
 
 and we have to find P’. 
 
 Equation (8) becomes, in virtue of (9), 
 
 (K,+H,—K,) P = (K,+H,+ K,+H,) P’. 
 
 If K,+H,= K, the discharge is complete. 
 
 110.| The following method, by which the existence of a deter- 
 minate relation between the capacities of four condensers may be 
 verified, has been employed by Sir W. Thomson.* It corresponds 
 in electrostatics to Wheatstone’s Bridge in current electricity. 
 
 In Fig. 25 the condensers are represented as Leyden jars. Two 
 of these, P and Q, are plaeed with their external coatings in contact 
 with an insulating stand 8; the other two, & and 4, have their 
 
 * Gibson and Barclay. 
 
92 COMPARISON OF TWO CONDENSERS. [ 110. 
 
 external coatings connected to the earth. The inner coatings of P 
 and # are permanently connected; so are those of Q and 8. In 
 performing the experiment the internal coatings of P and & are 
 first charged to a potential, 4, while those of Q and S are charged 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 “a 
 
 ees 
 aloe A 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 to a different potential, C. During this process the stand f is 
 connected to the earth. The stand £ is then disconnected from the 
 earth and connected to one electrode of an electrometer, the other 
 
 ¥ B Q 
 
 x 
 
 aoe 
 
 R 
 
 T! 
 
 Fig. 26. 
 
 electrode being connected to earth. Since f is already reduced to 
 potential zero by connection with the earth, there will be no dis- 
 turbance of the electrometer unless there is leakage in some of the 
 jars. We shall assume, however, that there is no leakage, and 
 that the electrometer remains at zero. 
 
Fi. COMPARISON OF TWO CONDENSERS. 93 
 
 The inner coatings of the four jars are now made to communicate 
 with each other by dropping the small insulated wire w so as to 
 fall on the two hooks connected with a and y. Since the potentials 
 of a and y are different a discharge will occur, and the potential of 
 8 will in general be affected, and this will be indicated by the 
 electrometer. If, however, there is a certain relation among the 
 capacities of the jars the potential of 8 will remain zero. 
 
 111.] Let us ascertain what this relation must be. In Fig. 26 
 the same electrical arrangement is represented under a simpler 
 form, in which the condensers consist each of a pair of disks. 
 Under this form the analogy with Wheatstone’s Bridge becomes 
 apparent to the eye. We have to consider the potentials and 
 charges of four conductors. The first consists of the inner coatings 
 of P and £&, together with the connecting wire. We shall call this 
 conductor a, its charge a, and its potential 4. The second consists 
 of the outer coatings of P and Q, together with the insulating stand 
 8. We shall call this conductor £, its charge J, and its potential 
 4, The third consists of the inner coatings of Q and S and the 
 connecting wire y. We shall call this y, its charge c, and its 
 potential C. The fourth consists of the outer coatings of @ and S 
 and of the earth with which they are kept connected. We might 
 use the letters 6, d, and D with reference to this conductor, but as 
 its potential is always zero and its charge equal and opposite to 
 that of the other conductors we shall not require to consider it. 
 
 The charge of any one of the conductors depends on its own 
 potential together with the potentials of the two adjacent con- 
 ductors, and also, but in a very slight degree, on that of the oppo- 
 site conductor. 
 
 Let the coefficients of induction between the different pairs of 
 the four conductors be as in the following scheme,— 
 
 P B Q 
 
 
 
 Fig. 27. 
 
 in which € and n are very small compared with P, Q, R, and 8. 
 The coefficient of capacity of any one of the conductors will exceed 
 
94 COMPARISON OF CONDENSERS. Bae 
 
 the sum of its three coefficients of induction by a quantity which 
 will be small if the capacity of the knobs of the jars and their 
 connecting wires are small compared with the whole capacities of 
 the jars. Let us denote this excess by the symbols a, 8, y, 6, which 
 belong to the conductors. The capacities therefore will be, 
 
 P+h+a+n, 
 P+Q+B+& 
 Q+S+y+n, 
 R+S+38 +6, 
 and the charges will therefore be, 
 for a, a=(P+Rh+a+n) A—PB-RD—-7C, 
 for B, 6=(P+Q+4+8+4+6&) B-PA—QC- ED, 
 for y, c=(Q4+8+y+n)C—-QB-SD—7 A, 
 for 6, d=(R+8 +8+£)D—RA—SC—EB. 
 
 In the first part of the experiment the potentials of a and y are 
 A and C respectively, while those of 8 and 6 are zero. Hence, at 
 first, a=(P+Rh+a+n)A—-7C, 
 
 b= —PA—QC, 
 e=(Q4+S+y+7)C —nA. 
 We need not determine the charge of 6. 
 
 Now let a communication be made between a and y, and let us 
 denote the charges and potentials of the conductors after the dis- 
 charge by accented letters. The potentials of a and y will become 
 equal; let us call their common potential y, then 
 
 ea ee 
 
 The sum of their charges remains the same, or 
 
 a+dé=ate. 
 
 The charge of 8 remains the same as before, or 
 
 C0, 
 
 but its potential is no longer zero, but B’, and we have to deter- 
 mine the value of 5” in terms of 4 and C by eliminating the other 
 quantities entering into the equations. 
 After discharge, 
 a= (P+Rht+a)y—PP, 
 = (P+Q+B+€)B (P+ Q)y, 
 f= (Q+S+y)y—QR. 
 
1II.| COMPARISON OF CONDENSERS, 95 
 
 Hence, the equation a’ +¢’ = a+c becomes 
 (P+Rh+Q+Sta+y)y—(P+ Q)B=(P+Rh+a)\At+(Q+St+y)C, 
 and J’= 6 becomes 
 
 (P+Q+B+6&) B-(P4+Q)y = —PA-~QC. 
 
 Eliminating ¥ from these equations, we find 
 
 Bi(P+ Q)(R+S8)+(P+Q)(a+b+y+é)+(R+St+a+y) (6+ €)} 
 = (Q(R-+a)—P(8+y)} (4—C). 
 If, therefore, the electrometer is not disturbed by the discharge, 
 
 B= O, and 
 P:Q::h+a:8+y. 
 
CHAPTER IX. 
 
 THE ELECTRIC CURRENT. 
 
 112.] Ler 4 and B be two metal bodies connected respectively 
 with the inner and outer coatings of a Leyden jar, the inner 
 coating of the jar being positive, so that the 
 
 | potential of 4 is higher than that of B. 
 
 ) € Let C be a gilt pith ball suspended by a silk 
 thread. If C is brought into contact with 4 and 
 B alternately, it will receive a small charge of 
 positive electricity from A every time it touches 
 it, and will communicate positive electricity to B 
 when. it touches B, 
 
 l = There will thus be a transference of positive 
 Fig. 28. electricity from A to B along the path travelled 
 over by the pith ball, and this is what occurs 
 in every electric current, namely, the passage of electricity along a 
 definite direction. During the motion of the pith ball from 4 to 
 4% it is charged positively, and the electric force between 4 and 
 B tends to move it in the direction from 4 to B. After touching 
 B, it becomes charged negatively, so that the electric force, during 
 its return journey, acts from £6 to 4. Hence the ball is acted 
 on by the electric force always in the direction in which it is 
 moving at the time, so that if it is properly suspended the electric 
 force will not only keep up the backward and forward motion, but 
 will communicate to the moving ball an amount of energy which it 
 will expend in a series of rattling blows against the balls 4 and B. 
 The current of positive electricity from 4 to Bis thus kept up by 
 means of the electromotive force from 4 to S. 
 113.] The phenomenon we have been describing may be called 
 a current of Convection. The motion of the electrification takes 
 
 
 
 
 
 
 
 
 
 
 
113.] CONVECTION AND CONDUCTION CURRENTS. 97 
 
 place in virtue of the motion of the electrified body which conveys 
 or carries the electricity as it moves from one place to another. 
 But if, instead of the pith ball, we take a metal wire carried 
 by an insulating handle, and cause the two ends of the wire 
 to touch 4 and B respectively, there will be a transference of 
 electricity from 4 to B along the wire, though the wire itself does 
 not move. 
 
 What takes place in the wire is called a current of Conduction. 
 The effects of the current of conduction on the electrical state of 
 A and B are of precisely the same kind as those of the current of 
 convection. In both cases there is a transference of electrification 
 from one place to another along a continuous path. 
 
 In the case of the convection of the charge on the pith ball we 
 may observe the actual motion of the ball, and therefore in this 
 case we may distinguish between the act of carrying a positive 
 charge from 4 to B and that of carrying a negative charge from 
 & to A, though the Electrical effects of these two operations are 
 identical. We may also distinguish between the act of carrying a 
 number of small charges from 4 to # in rapid succession and 
 with great velocity, and the act of carrying a single great 
 charge, equivalent to the sum of these charges, slowly from A to 
 B in the time occupied by the whole series of journeys in the other 
 case. 
 
 But in the case of the current conduction through a wire we 
 have no reason to suppose that the mode of transference of the 
 charge resembles one of these methods rather than another. All 
 that we know is that a charge of so much electricity is conveyed 
 from A to B in a certain time, but whether this is effected by 
 carrying positive electricity from A to B, or by carrying negative 
 electricity from B to A, or by a combination of both processes, is a 
 question which we have no means of determining. We are equally 
 unable to determine whether the ‘ velocity of electricity’ in the 
 wire is great or small. If there be a substance pervading bodies, 
 the motion of which constitutes an electric current, then the excess 
 of this substance in connexion with a body above a certain normal 
 value constitutes the observed charge of that body, and is a 
 quantity capable of measurement. But we have no means of 
 estimating the normal charge itself. The only evidence we possess 
 is deduced from experiments on the quantity of electricity evolved 
 during the decomposition of one grain of an electrolyte, and this 
 quantity is enormous when compared with any positive or negative 
 
 H 
 
98 MEASURE OF CURRENT. [114. 
 
 charge which we can accumulate within the space occupied by the 
 electrolyte. If, then, the normal charge of a portion of the wire 
 the millionth of an inch in length is equal to the total charge 
 transferred from A to B, the transference may be effected by the 
 displacement of the electricity in the wire whose linear extent is 
 only the millionth of an inch. 
 
 ‘It is therefore quite possible that the velocity of electricity in a 
 telegraph wire may be exceedingly small, less, say, than the 
 hundredth of an inch in an hour, though signals, that is to say, 
 changes in the state of the current, may be propagated along the 
 wire many thousands of miles in a second. 
 
 Since, therefore, we are ignorant of-the true linear velocity of an 
 electric current, we must measure the strength of the current by the 
 quantity of electricity discharged through any section of the con- 
 ductor in the unit of time, just as engineers measure the discharge 
 of water and gas through pipes, not by the velocity of the water or 
 gas, but by the quantity which passes in a minute. 
 
 114.] In many cases we have to consider the whole quantity of 
 electricity which passes rather than the rate at which it passes. 
 This is especially the case when the current lasts only a very short 
 time, or when the current is considered merely as a transition 
 from one permanent state of the system to another. In these cases 
 it is convenient to speak of the total current as the Electric 
 Displacement, the word displacement indicating the final result 
 of a motion without reference to the rate at which it takes place. 
 The passage of a given quantity of electricity along a given path is 
 called an Electric Discharge. 
 
 Classification of bodies according to their relation to the 
 transference of electricity, 
 
 115.] For the sake of distinction we shall consider a portion of 
 matter whose ends are formed by two equipotential surfaces having 
 different potentials, and whose sides are formed by lines of electric 
 current or displacement. 
 
 The ends of the body are called its Electrodes, that at which 
 electricity enters is called the Anode, and that at which it leaves 
 the body is called the Cathode. 
 
 The excess of the potential of the anode over that of the cathode 
 is called the External Electromotive Force. 
 
 The Form of the body may vary from that of a long wire sur- 
 
117.| OHMS LAW. 99 
 
 rounded by air or other insulating matter to that of a thin sheet of 
 the substance, the electricity passing through the thickness of the 
 sheet. 
 
 Bodies may be divided into three great classes according to the 
 mode in which they are acted on by electromotive force,—Metals, 
 Electrolytes, and Dielectrics. 
 
 First Crass.—Metats, &c. 
 
 116.| The first class includes all the metals, whether in the solid 
 or liquid state, together with some other substances not regarded 
 by chemists as metals. In these the smallest external electromotive 
 force is capable of producing an electric current, and this current 
 continues to flow as long as the electromotive force continues to 
 act, without producing any change in the chemical properties of the 
 substance. The strength of the permanent current is proportional 
 to the electromotive force. The ratio of the numerical value of the 
 electromotive force to that of the current is called the Resistance 
 of the conductor. The same thing may be otherwise stated by 
 saying that the flow of the current is opposed by an internal 
 electromotive force, proportional to the strength of the current, 
 and to a quantity called the Resistance of the conductor, depending 
 on its form and nature. When the strength of the current is such 
 that this internal electromotive force balances the external electro- 
 motive force the current neither increases nor diminishes in strength. 
 It is then said to be a steady current 
 
 These relations were first established by Dr. G. S. Ohm, in a 
 work published in 1827, They are expressed by the formula, 
 
 Electromotive foree = Current x Resistance, 
 which is called Ohm’s Law. 
 
 Generation of Heat by the current. 
 
 117.] During the flow of a steady current through a conductor 
 of uniform material of the first class heat is generated in the 
 conductor, but the substance of the conductor will not be affected 
 in any way, for if the heat is allowed to escape as fast as it is 
 generated, the temperature and every other physical condition of 
 the conductor remains the same. 
 
 The whole work done by the external electromotive force in 
 urging electricity through the body is therefore spent in generating 
 
 H 2 
 
100 JOULES LAW. [118. 
 
 heat. The dynamical equivalent of the heat generated is therefore 
 equal to the electrical work spent, that is, to the product of the 
 electromotive force into the quantity of electricity transmitted by 
 the current. 
 
 Now, the electromotive force is, by Ohm’s law the product of 
 the strength of the current into the resistance, and the quantity 
 of electricity is, by the definition of a current, the product of the 
 current into the time during which it flows, so that we find, 
 
 Heat generated measured in dynamical units 
 = Square of Current x Resistance x Time. 
 
 This relation was first established by Dr. Joule, and is therefore 
 called Joule’s law. It was also established independently by Lenz. 
 
 SECOND CiAass.—ELECTROLYTES. 
 
 118.] The second class of substances consists of compound bodies, 
 generally in the liquid form, called Electrolytes. 
 
 When an electric current passes through fused chloride of silver, 
 which is an electrolyte, chlorine appears at the anode where the 
 current enters, and silver at the cathode where the current leaves 
 the electrolyte. The quantities of these two substances are such 
 that if combined they would form chloride of silver. The com- 
 position of those portions of the electrolyte which lie between the 
 electrodes remains unaltered. Hence, if we fix our attention upon 
 a portion of the electrolyte between two fixed planes perpendicular 
 to the direction of the current, the quantity of silver or of chlorine 
 which enters the portion through one plane must be equal to the 
 quantity which leaves it through the other plane. It follows from 
 this that in every part of the electrolyte the silver is moving in the 
 direction of the current, and the chlorine in the opposite direction. 
 
 This operation, in which a compound body is decomposed by an 
 electric current, is called Electrolysis, and the mode in which the 
 current is transmitted is called Electrolytic Conduction. The 
 compound body is called an Electrolyte, and the components into 
 which it is separated are called Ions. That which appears at the 
 anode is called the Anion, and that which appears at the cathode is 
 called the Cation. 
 
 The quantity of the substance which is decomposed is propor- 
 tional to the total quantity of electricity which passes through it, 
 and is independent of the time during which the electricity is. 
 
120. | FARADAY S LAWS OF ELECTROLYSIS. 101 
 
 passing. The quantity corresponding to the passage of one unit 
 of electricity is called the Electrochemical Equivalent of the sub- 
 stance. Thus, when one unit of electricity is passed through fused 
 chloride of silver, one electrochemical equivalent of silver appears 
 at the cathode and one electrochemical equivalent of chlorine at 
 the anode, and one electrochemical equivalent of chloride of silver 
 disappears. 
 
 119.] The electrochemical equivalents of the same substance, as 
 deduced from experiments on different electrolytes which contain 
 it, are consistent with each other. Thus the electrochemical 
 equivalent of chlorine is the same, whether we deduce it from 
 experiments on chloride of silver, or from experiments on hydro- 
 ehlorie acid, and that of silver is the same, whether we deduce 
 it from experiments on chloride of silver, or from experiments on 
 nitrate of silver. ‘These laws of electrolysis were established by 
 Faraday*. If they are strictly true the conduction of electricity 
 through an electrolyte is always electrolytic conduction, that is to 
 say, the electric current is always associated with a flow of the 
 components of the electrolyte in opposite directions. 
 
 Such a flow of the components necessarily involves their appear- 
 ance In a separate form at the anode and the cathode. ‘To effect 
 this separation a certain electromotive force is required depending 
 on the energy of combination of the electrolyte. Thus the electro- 
 motive force of one of Daniell’s cells is not sufficient to decompose 
 dilute sulphuric acid. 
 
 If, therefore, an electrolytic cell, consisting of a vessel of 
 acidulated water, in which two platinum plates are placed as 
 electrodes, is inserted in the circuit of a single Daniell’s cell, along 
 with a galvanometer to measure the current, it will be found that 
 though there is a transient current at the instant the circuit is 
 closed, this current rapidly diminishes in intensity, so as to become 
 in a very short time too weak to be measured except by a very 
 sensitive galvanometer. 
 
 Neither oxygen nor hydrogen, the chemical components of water, 
 appear in a gaseous form at the electrodes, but the electrodes them- 
 selves acquire new properties, shewing that a chemical action has 
 taken place at the surface of the platinum plates. 
 
 120.] If the Daniell’s cell is taken out of the circuit, and the 
 circuit again closed, the galvanometer indicates a current passing 
 through the electrolytic cell in the opposite direction to the original 
 
 * Exp. Res., series vii and viii. 
 
102 ELECTROLYTIC POLARIZATION. [121. 
 
 current. This current rapidly diminishes in strength and soon 
 vanishes, so that the whole quantity of electricity which is trans- 
 mitted by it is never greater than that of the primitive current. 
 This reverse current indicates that the platinum plates have ac- 
 quired a difference of properties by being used as electrodes. They 
 are said to be polarized. The cathode is polarized positively and 
 the anode negatively, so that an electromotive force is exerted in 
 the circuit opposite to that of the Daniell’s cell. This electromotive 
 force, which is called the electromotive force of polarization, is the 
 cause of the rapid diminution in the strength of the original current, 
 and of its final cessation. 
 
 A chemical examination of the platinum plates shews that a 
 certain quantity of hydrogen has been deposited on the cathode. 
 This hydrogen is not in the ordinary gaseous form, but adheres to 
 the surface of the platinum so firmly that it is not easy to remove 
 the last traces of it. 
 
 121.] Faraday’s law, that conduction takes place in electrolytes 
 only by electrolysis, was long supposed not to be strictly true. 
 In the experiment in which a single Daniell’s cell furnishes the 
 electromotive force in a cireuit containing an electrolyte and a 
 galvanometer, it is found that the current soon becomes very ‘feeble 
 but never entirely vanishes, so that if the electromotive force is 
 maintained long enough, a very considerable quantity of electricity 
 may be passed through the electrolyte without any visible de- 
 composition. — 
 
 Hence it was argued that electrolytes conduct electricity in two 
 different ways, by electrolysis in a very conspicuous manner and 
 also, but in a very slight degree, in the manner of metals, without 
 decomposition. Helmholtz has recently* shewn that the feeble 
 permanent current can be explained in a different manner, and that 
 we have no evidence that an electrolyte can conduct electricity 
 without electrolysis. . 
 
 122.| In the case of platinum plates immersed in dilute sulphuric 
 acid, if the liquid is carefully freed from all trace of oxygen or 
 of hydrogen in solution, and if the surfaces of the platinum plates 
 are also freed from adhering oxygen or hydrogen, the current con- 
 tinues only till the platinum plates have become polarized and no 
 permanent current can be detected, even by means of a sensitive 
 galvanometer. When the experiment is made without these pre- 
 
 * Ueber galvanische Polarisation in gusfreien Fliissigkeiten. Monatsbericht d. K. 
 Akad. d. Berlin, July 1873, p. 587. 
 
123.| ELECTROLYTIC CONVECTION. 103 
 
 cautions, there is generally a certain amount of oxygen or of hydrogen 
 ‘in solution in the liquid, and this, when it comes in contact with 
 the hydrogen or the oxygen adhering to the platinum surface, 
 combines slowly with it, as even the free gases do in presence 
 of platinum. The polarization is thus diminished, and the electro- 
 motive force is consequently enabled to keep up a permanent 
 current, by what Helmholtz has called electrolytic convection. 
 Besides this, it is probable that the molecular motion of the liquid 
 may be able occasionally to dislodge molecules of oxygen or of 
 hydrogen adhering to the platinum plates. These molecules when 
 thus absorbed into the liquid will travel according to the ordinary 
 laws of diffusion, for it is only when in chemical combination that 
 their motions are governed by the electromotive foree. They will 
 therefore tend to diffuse themselves uniformly through the liquid, 
 and will thus in time reach the opposite electrode, where, in contact 
 with a platinum surface, they combine with and neutralize part of 
 the other constituent adhering to that surface. In this way a 
 constant circulation is kept up, each of the constituents travelling 
 in one direction by electrolysis, and back again by diffusion, so that 
 a permanent current may exist without any visible accumulation 
 of the products of decomposition. We may therefore conclude that 
 the supposed inaccuracy of Iaraday’s law has not yet been confirmed 
 by experiment. 
 
 123.| The verification of Ohm’s law as applied to electrolytic 
 conduction is attended with considerable difficulty, because the 
 varying polarization of the electrodes introduces a variable electro- 
 motive force, and renders it difficult to ascertain the true electro- 
 motive force at any instant. By using electrodes in the form of 
 plates, having an area large compared with the section of the 
 electrolyte, and employing currents alternately in opposite direc- 
 tions, the effect of polarization may be diminished relatively to 
 that of true resistance. It appears from experiments conducted 
 in this way that Ohm’s law is true for electrolytes as well as 
 for metals, that is to say, that the current is always proportional 
 to the electromotive force, whatever be the amount of that force. 
 _ The reason that the external resistance of an electrolyte appears 
 greater for small than for large electromotive forces is that the 
 external electromotive force between the metallic electrodes is not 
 the true electromotive force acting on the electrolyte. There is, 
 in general, a force of polaiization acting in the opposite direction 
 to the external electromotive force, and it is only the excess of 
 
104 CLAUSIUS’ THEORY OF [124. 
 
 the external force above the force of polarization that really acts on 
 the electrolyte. 
 
 It appears, therefore, that the very smallest electromotive force, 
 if it really acts on the electrolyte, is able to produce conduction by 
 electrolysis. How, then, is this to be reconciled with the fact that 
 in order to produce complete decomposition a very considerable 
 electromotive force is required ? 
 
 124.| Clausius* has pointed out that on the old theory of 
 electrolysis, according to which the electromotive force was sup- 
 posed to be the sole agent in tearing asunder the components 
 of the molecules of the electrolyte, there ought to be no decom- 
 position and no current as long as the electromotive force is 
 below a certain value, but that as soon as it has reached this 
 value a vigorous decomposition ought to commence, accompanied 
 by a strong current. This, however, is by no means the case, 
 for the current is strictly proportional to the electromotive force for 
 all values of that force. 
 
 Clausius explains this in the following way :— 
 
 According to the theory of molecular motion of which he has 
 himself been the chief founder, every molecule of the fluid is 
 moving in an exceedingly irregular manner, being driven first 
 one way and then another by the impacts of other molecules which 
 are also in a state of agitation. 
 
 This molecular agitation goes on at all times independently of 
 the action of electromotive force. The diffusion of one fluid through 
 another is brought about by this molecular agitation, which in- 
 creases in velocity as the temperature rises. The agitation being 
 exceedingly irregular, the encounters of the molecules take place 
 with various degrees of violence, and it is probable that even at 
 low temperatures some of the encounters are so violent that one 
 or both of the compound molecules are split up into their con- 
 stituents. Each of these constituent molecules then knocks about 
 among the rest till it meets with another molecule of the opposite 
 kind and unites with it to form a new molecule of the compound. 
 In every compound, therefore, a certain proportion of the mole- 
 cules at any instant are broken up into their constituent atoms. 
 At high temperatures the proportion becomes so large as to 
 produce the phenomenon of dissociation studied by M. St. Claire 
 Deville f. 
 
 * Pogg. Ann. CI. 338 (1857). 
 
 + (Legons sur la Dissociation, professées devant la Société Chimique. L. Hachette 
 et Ci*, 1866. ] 
 
126, ELECTROLYSIS. 105 
 
 125.] Now Clausius supposes that it is on the constituent mole- 
 cules in their intervals of freedom that the electromotive force acts, 
 deflecting them slightly from the paths they would otherwise have 
 followed, and causing the positive constituents to travel, on the 
 whole, more in the positive than in the negative direction, and 
 the negative constituents more in the negative direction than in 
 the positive. The electromotive force, therefore, does not produce 
 the disruptions and reunions of the molecules, but finding these 
 disruptions and reunions already going on, it influences the motion 
 of the constituents during their intervals of freedom. The amount 
 of this influence is proportional to the electromotive force when the 
 temperature is given. The higher the temperature, however, the 
 greater the molecular agitation, and the more numerous are the free 
 constituents. Hence the conductivity of electrolytes increases as 
 the temperature rises. 
 
 This effect of temperature is the opposite of that which takes 
 place in metals, the resistance of which increases as the temperature 
 rises. This difference of the effect of temperature is sometimes 
 used as a test whether a conductor is of the metallic or the 
 electrolytic kind. The best test, however, is the existence of 
 polarization, for even when the quantity of the free ions is too 
 small to be observed or measured, their presence may be indicated 
 by the electromotive force which they excite. 
 
 126.] Kohlrausch* finds that if the electromotive force is one 
 volt per centimetre in length of the electrolyte, then if the electro- 
 lyte differs but slightly from pure water at 18°C the velocity of 
 hydrogen is about 0-0029 centimetres per second, and that the 
 actual force on a gramme of hydrogen in the solution required 
 to make it move at the rate of one centimetre per second 
 through the solution is equal to the weight of 330,000,000 kilo- 
 grammes. 
 
 The velocities of the components of unibasic acids and their salts 
 were found by Kohlrausch to be in the following proportion :— 
 
 TaB.eE I. 
 H K NH, Na Li 5Ba 4Sr 4Ca 4Mg 
 273 48 46 30 19 31 28 24 21 
 J Br Cl F NO; ClO; C,H;0, 
 55 53 50 29 47 36 22 
 
 * Gottingen Nachrichten, 5 Aug., 1874, 17 May, 1876, and 4 April, 1877, 
 
106 WATER NOT AN ELECTROLYTE. inevs 
 
 127.| The specific molecular conductivity (7) of an electrolyte 
 is the sum of the velocities of its components*, and the actual 
 conductivity of any weak solution is found by multiplying the 
 number Z by the number of grammes of the substance in a litre 
 and dividing by the molecular weight of the substance, that of 
 hydrogen being: 1. 
 
 128.] We have reason to believe that water is not an electrolyte, 
 and that it is not a conductor of the electric current. It is 
 exceedingly difficult to obtain water free from foreign matter. 
 Kohlrauschf, however, has obtained water so pure that its resistance 
 was enormous compared with ordinary distilled water. When 
 exposed to the air for [4-3 hours its conductivity rose 70 per 
 cent.], and [in 1060 hours it was increased nearly fortyfold. After 
 long exposure to the air the conductivity was more than doubled in 
 4-5 hours by the action of tobacco smoke.] Water kept in glass 
 
 vessels very soon dissolves enough of foreign matter to enable it 
 to conduct-freely. 
 
 Kohlrausch t has determined the resistance of water containing a 
 very small percentage of different electrolytes, and he finds that 
 the results agree very well with the hypothesis that the velocity 
 with which each ion travels through the liquid is proportional 
 to the electromotive force, the velocity corresponding to unit of 
 eloctromotive force being different for different ions, but the same 
 for the same ion, whatever the other ion may be with which it is 
 combined. The velocities of different ions in centimetres per 
 second, corresponding to an electromotive force of one volt, are 
 given in Table II. 
 
 
 
 - Taste IT. ai 
 ial K NH, Na Li Ba Sr Ca Meg 
 ‘0029 -00051 -00049 -00082 -00020 -.00083 .00030 -00025 -00022 
 | I Br Cl F NO, ClO; C,H,0, - 
 
 
 
 00058 -00056 -00053 .00031 -00050 00088 -00023 ot 
 
 When the water contains a large percentage of foreign matter 
 the velocities of the ions are no longer the same, as it is no longer 
 through water, but through a liquid of quite different physical 
 properties that they have to make their way. It appears from 
 
 * [Compare Cavendish Papers, pp. 446, 447.] 
 
 + | Poggendorff, Ergdnzungsband, VIII (1876), pp. 7, 9, 11.] 
 
 t [Pogg. Ann. Vol. CLIV (1875), p. 215; Vol. CLIX (1876), p. 242; Phil. Mag. 
 June 1875.] 
 
PROPERTIES OF DIELECTRICS. 107 
 
 Table III* that though for small percentages of sulphuric acid in 
 water the conducting power is proportional to the percentage of 
 acid, yet as the proportion of acid increases the conducting power 
 increases more slowly till a maximum conducting power is reached, 
 after which the addition of acid diminishes the conducting power fT. 
 
 Taste III. 
 
 Conductivity of Sulphuric Acid at 18°C referred to that 
 of Mercury at 0°C as unity. 
 
 Percentage of 108K Percentage of 10°K Percentage of 108K 
 
 
 
 He2S04 H2804 HeSO4 
 1 429 60 3487 87 944 
 2.5 1020 65 2722 88 965 
 5 1952 70 2016 89 986 
 10 3665 75 1421 90 1005 
 15 5084 78 1158 91 1022 
 20 6108 80 1032 92 1030 
 25 6710 81 985 93 1024 
 30 6912 82 947 94 1001 
 35 6776 83 924 95 958 
 40 6361 || . 84 915 96 885 
 45 5766 85 916 97 750 
 50 5055 86 926 99.4 80 
 a 55 4280 | 
 
 129.| The oxygen and hydrogen which are given off at the 
 electrodes in so many experiments on water containing foreign 
 in¢redients are, therefore, not the ions of water separated by strict 
 electrolysis, but secondary products of the electrolysis of the matter 
 in solution. Thus, if the cation is a metal which decomposes water, 
 it unites with an equivalent of oxygen and allows the two equiva- 
 lents of hydrogen to escape in the form of gas. The anion may 
 be a [compound radicle] which cannot exist in a separate state, 
 [but which exists in the nascent condition, and] contains one equi- 
 valent [or more] of [some electronegative element which reacts 
 upon water and liberates oxygen. | 
 
 Tuirp CLAss.— DIELECTRICS. 
 
 130.] The third class of bodies has an electric resistance so much 
 ereater than that of metals, or even of electrolytes, that they are 
 often called insulators of electricity. All the gases, many liquids 
 which are not electrolytes, such as spirit of turpentine, naphtha, &c., 
 and many solid bodies, such as gutta-percha, caoutchouc in its 
 various forms, amber and resins, crystallized electrolytes, glass 
 when cold, &c., are insulators. 
 
 * [See also p. 201.] 
 + [A similar result was found with nitric acid and some viscous saline solution. ] 
 
108 SPECIFIC INDUCTIVE CAPACITY. [13I. 
 
 They are called insulators because they do not allow a current 
 of electricity to pass through them. They are called dielectrics 
 because certain electrical actions can be transmitted through them. 
 According to the theory adopted in this book, when an electro- 
 motive force acts on a dielectric it causes the electricity to be 
 displaced within it in the direction of the electromotive force, the 
 amount of the displacement being proportional to the electromotive 
 force, but depending also on the nature of the dielectric, the dis- 
 placement due to equal electromotive forces being greater in solid 
 and liquid dielectrics than in air or other gases. 
 
 When the electromotive force is increasing, the increase of 
 electric displacement is equivalent to an electric current in the 
 same direction as the electromotive force. When the electromotive 
 force is constant there is still displacement, but no current. When 
 the electromotive force is diminishing, the diminution of the electric 
 displacement is equivalent to a current in the opposite direction. 
 
 131.] Ina dielectric, electric displacement calls into action an 
 internal electromotive force in a direction opposite to that of the 
 displacement, and tending to reduce the displacement to zero. 
 The seat of this internal force is in every part of the dielectric 
 where displacement exists. 
 
 To produce electric displacement in a dielectric requires an 
 expenditure of work measured by half the product of the electro- 
 motive force into the electric displacement. This work is stored 
 up as energy within the dielectric, and is the source of the 
 energy of an electrified system which renders it capable of doing 
 mechanical work. 
 
 The amount of displacement produced by a given electromotive 
 force is different in different dielectrics. The ratio of the displace- 
 ment in any dielectric to the displacement in a vacuum due to the 
 same electromotive force is called the Specific Inductive Capacity 
 of the dielectric, or more briefly, the Dielectric Constant. This 
 quantity is greater in dense bodies than in a so-called vacuum, and 
 is approximately equal to the square of the index of refraction. 
 Thus Dr. L. Boltzmann® finds for various substances, 
 
 o Index of 
 
 D. VD. refraction. 
 Sulphur (cast) 3-84 1-960 2-040 
 Colophonium 2-55 1-597 1.543 
 Paraffin 2-32 1-523 1-536 
 
 Ebonite (Hartgummi) 3-15 1.775 
 * [Pogg. Ann. CLI. (1874), p. 482.] 
 
T33-| PROPERTIES OF A DIELECTRIC. 109 
 
 For a sphere cut from a crystal of sulphur Boltzmann finds D 
 by electrical experiments for the three principal axes, and compares 
 them with the results as calculated from the three indices of 
 refraction. 
 
 D, = 3-970 
 D, = 3-886 
 
 D; = 3-811 
 D; = 3-591 
 
 By electrical experiments D, = 4-773 
 By optical measurements D, = 4-596 
 {Sitzungsb. (Vienna), 9 Jan., 1873.} 
 132.] Schiller (Pogg. Aun. CLIT. 535) ascertained the time of 
 the electrical vibrations when a condenser is discharged through an 
 electromagnet. He finds in this way the following values of the 
 dielectric coefficients of various substances, and compares them with 
 those found by Siemens by the method of a rapid commutator. 
 
 Schiller. Siemens. i p 
 Ebonite (Hartgummi) 2-21 2-76 
 Pure rubber 2-12 2-34 2-25 1.50 
 Vulcanized grey, do. 2-69 2-94 
 Paraffin, quick cooled, clear 1.68 
 slow cooled, milk white 1-81 © 1.92 2-19 1-48 
 * another specimen 1.89 2-47 2-34 1-53 
 Straw coloured glass 2-96 4.12 
 ” ; ”? 3-66 
 White mirror glass 5-83 6.34 
 P. Silow {Pogg. Ann. CLVI (1875), [p. 395]!* finds for oil of 
 turpentine 
 D = 2.21 WD = 1.490 Hoo = 1.456. 
 
 Faraday did not succeed in detecting any difference in the 
 dielectric constants of different gases. Dr. Boltzmann +} however 
 has succeeded by a very ingenious method in determining it for 
 various gases at 0°“C, and at one atmosphere pressure. 
 
 D: A/D. H. 
 Air 1.000590 1.000295 1.000294 
 Carbonic Acid 1.000946 1.000473 1.000449 
 Hydrogen 1.000264 1.000182 1.000188 
 Carbonic Oxide 1.000690 1-000345 1.000340 
 Nitrous Oxide 1.000994 1.000497 1.000503 
 Olefiant Gas 1-001312 1-000656 1.000678 
 Marsh Gas 1.000944 1.000472 1.000443 
 
 Disruptive DiscHaRGE. 
 133.] If the electromotive force acting at any point of a dielectric 
 is gradually increased, a limit is at length reached at which there 
 
 * [See also CLVIII. (1876), pp. 306 et sqq.] 
 + |Pogg. Ann. CLI. (1875), p. 403.] 
 
110 MECHANICAL ILLUSTRATIONS OF [134. 
 
 is a sudden electrical discharge through the dielectric, generally 
 accompanied with light and sound. The dielectric, if solid, is often 
 pierced, cracked, or broken, and portions of it are often dispersed in 
 the form of vapour. This phenomenon appears to be analogous 
 to the rupture of a solid body when exposed to a continually 
 increasing stress. This analogy is so complete that we may make 
 use of the same terms in describing the behaviour of media under 
 the action of electromotive force as we apply to bodies under the 
 action of stress. Thus electromotive force and electric displacement 
 correspond to ordinary force and ordinary displacement ; the electro- 
 motive force which produces disruptive discharge corresponds to 
 the breaking stress. Conduction, or the transmission of electricity, 
 corresponds to permanent bending. 
 
 Thus if we consider the twisting of a wire on the one hand, and 
 the transmission of electricity through a body on the other, the 
 moment of the couple which twists the wire will correspond to 
 the electromotive force acting on the body, and the angle through 
 which the wire is twisted will correspond to the electric displace- 
 ment. If the wire, when the force is removed, returns to its 
 former shape and becomes completely untwisted it is said to be 
 elastic. Such a wire corresponds to a dielectric which acts as a 
 perfect insulator with respect to the electromotive force employed. 
 If the twisting couple is increased up to a certain limit the wire 
 gives way and is broken. This corresponds to disruptive discharge, 
 and the ultimate strength of the wire corresponds to the greatest 
 electromotive force which the dielectric can support, which we may 
 eall its electric strength. | 
 
 If before rupture takes place the wire yields so that it will no 
 longer completely untwist itself when the force is removed it is 
 said to be plastic. It corresponds to a dielectric which conducts 
 electricity to a certain extent. 
 
 If no such permanent twist can be given to the wire by a force 
 which is not sufficient to break it, the wire is called brittle. In 
 like manner we may speak of those dielectrics such as air, which 
 will not transmit electricity except by the disruptive discharge, as 
 electrically brittle. 
 
 134.]| Many wires after being kept twisted for some time and 
 then set free immediately untwist themselves, but through a smaller 
 angle than they were twisted. In the course of time, however, they 
 go on untwisting themselves, but very slowly, the process some- 
 times going on for days or weeks. In like manner many dielectrics 
 
135. | THE PROPERTIES OF A DIELECTRIC. 111 
 
 such as the glass of a Leyden jar or the gutta percha of a submarine 
 cable, after being subjected for some time to electromotive force 
 and then placed in a closed circuit give an instantaneous discharge 
 which is less than the original charge. After this discharge, how- 
 ever, they are capable of giving residual discharges which become 
 more and more feeble, and if the circuit is kept closed a quantity of 
 electricity will slowly ooze out through the circuit, the current 
 becoming feebler and feebler as the charge is more nearly 
 exhausted, 
 
 Mechanical Itlustration of the Properties of a Dielectric. 
 
 135*.] Five tubes of equal sectional area 4, B, C, D and P are 
 arranged in circuit as in the figure. 4, B, C and D are vertical 
 and equal, and / is horizontal. 
 
 The lower halves of 4, B, C, D 
 are filled with mercury, their upper 
 halves and the horizontal tube P are 
 filled with water. 
 
 A tube with a stopeock Q con- 
 nects the lower part of 4 and B 
 with that of C and J, and a piston 
 P is made to slide in the horizontal 
 tube. 
 
 Let us begin by supposing that 
 the level of the mercury in the four 
 tubes is the same, and that it is in- 
 dicated by 4,, B,, Cy), Dy, that the 
 piston is at P,, and that the stop- 
 eock @ is shut. 
 
 Now let the piston be moved from 
 P, to P,, a distance a. Then, since 
 the sections of all the tubes are equal, the level of the mercury 
 in A and C will rise a distance a, or to 4, and C,, and the mercury 
 in B and D will sink an equal distance a, or to B, and D,. 
 
 The difference of pressure on the two sides of the piston will 
 be represented by 4a. 
 
 This arrangement may serve to represent the state of a dielectric 
 acted on by an electromotive force 4a. 
 
 The excess of water in the tube D may be taken to represent a 
 positive charge of electricity on one side of the dielectric, and the 
 
 
 
 Q 
 Fig. 29. 
 
112 RESIDUAL DISCHARGE. [F35.- 
 
 excess of mercury in the tube 4 may represent the negative charge 
 on the other side. The excess of pressure in the tube P on the 
 side of the piston next D will then represent the excess of potential 
 on the positive side of the dielectric. 
 
 If the piston is free to move it will move back to P, and be in 
 equilibrium there. This represents the complete discharge of the 
 dielectric. 
 
 During the discharge there is reversed motion of the liquids 
 throughout the whole tube, and this represents that change of 
 electric displacement which we have supposed to take place in a 
 dielectric. 
 
 I have supposed every part of the system of tubes filled with 
 incompressible liquids, in order to represent the property of all 
 electric displacement that there is no real accumulation of elec- 
 tricity at any place. 
 
 Let us now consider the effect of opening the stopcock Q while 
 the piston P is at P,. 
 
 The level of 4, and D, will remain unchanged, but that of B and 
 C will become the same, and will coincide with B, and C,. 
 
 The opening of the stopcock Q corresponds to the existence of 
 a part of the dielectric which has a slight conducting power, but 
 which does not extend through the whole dielectric so as to form 
 an open channel. 
 
 The charges on the opposite sides of the dielectric remain in- 
 sulated, but their difference of potential diminishes. 
 
 In fact, the difference of pressure on the two sides of the piston 
 sinks from 4a to 2a during the passage of the fluid through Q. 
 
 If we now shut the stopcock Q and allow the piston P to move 
 freely, it will come to equilibrium at a point P,, and the discharge 
 will be apparently only half of the charge. 
 
 The level of the mereury in A and B will be 4a above its 
 original level, and the level in the tubes C and D will be 2a 
 below its original level. This is indicated by the levels 4,, B,, 
 Che Rs. 
 
 If the piston is now fixed and the stopcock opened, mercury will 
 flow from B to C till the level in the two tubes is again at B, and 
 C,. There will then be a difference of pressure = a on the two 
 sides of the piston P. If the stopcock is then closed and the piston 
 P \eft free to move, it will again come to equilibrium at a point P,, 
 half way between P, and P,. ‘This corresponds to the residual 
 charge which is observed when a charged dielectric is first dis- 
 
136.] ELECTRIC STRENGTH OF AIR. 113 
 
 charged and then left to itself. It gradually recovers part of its 
 charge, and if this is again discharge a third charge is formed, the 
 successive charges diminishing in quantity. In the case of the 
 illustrative experiment each charge is half of the preceding, and the 
 discharges, which are $, 4, &c. of the original charge, form a series 
 whose sum is equai to the original charge. 
 
 If, instead of opening and closing the stopcock, we had allowed it 
 to remain nearly, but not quite, closed during the whole experiment, 
 ‘we should have had a case resembling that of the electrification of a 
 dielectric which is a perfect insulator and yet exhibits the phe- 
 nomenon called ‘ electric absorptic n.’ 
 
 To represent the case in which there is true conduction through 
 the dielectric we must either make the piston leaky, or we must 
 establish a communication between the top of the tube 4 and the 
 top of the tube D. 
 
 In this way we may construct a mechanical illustration of the 
 properties of a dielectric of any kind, in which the two electricities 
 are represented by two real fluids, and the electric potential is 
 represented by fluid pressure. Charge and discharge are repre- 
 sented by the motion of the piston /, and electromotive force by 
 the resultant force on the piston. 
 
 186.| The electric strength of a dielectric medium depends on the 
 nature of the medium and its density and temperature. Thus the 
 electromotive force required to produce a disruptive discharge is 
 greater in glass or ebonite than in air. 
 
 The electric strength of air or any other gas may be tested by 
 causing sparks. to pass through a portion of the gas between two 
 balls of metal. If the experiment is conducted in a glass vessel 
 from which the air may be exhausted by an air pump, it is found 
 that the electromotive force necessary to produce the discharge 
 diminishes, while the pressure is reduced from that of the atmo- 
 sphere to that of about 3 millimetres of mercury. If the supply of 
 electricity is kept up at a constant rate, the sparks become smaller 
 and more frequent, till at last there appears to be a continuous flow. 
 If, however, the exhaustion be carried further, the electric strength 
 again increases, till in the most perfect vacuum hitherto made the 
 electromotive force required to produce a spark between electrodes 
 -6 centimetres apart is so great that the discharge does not take 
 place between the electrodes, but passes round the outside of the 
 vessel through a distance of 20 centimetres of air at the ordinary 
 pressure. It would therefore seem as if a perfect vacuum would 
 
 : , 
 
114 ELECTRIC STRENGTH OF AIR. eye 
 
 present an almost insuperable resistance to the passage of electricity. 
 A small quantity of gas, however, introduced into the empty space 
 renders it incapable of withstanding even a small electromotive 
 force. This diminution of the electric strength, however, does not 
 go on when the density of the gas is still further increased, but for 
 pressures of a centimetre and upwards the electric strength in- 
 creases as the density increases. 
 
 137.| The electric strength of air diminishes rapidly as the tem- 
 perature rises. The heated air which rises from a flame conducts 
 electricity freely. The best way of discharging the electrification 
 of the surface of a solid dielectric is to pass the electrified body 
 over a flame. In most experiments with heated air the air is in 
 motion. It is therefore desirable that experiments should be made 
 on the conductivity of air at various temperatures, contained in a 
 closed vessel and free from currents. 
 
 138.] In order to test the insulating properties of air and other 
 gases I made the following experiment :— 
 
 A tube half an inch in diameter, CD, is supported on an insulated 
 stand c. <A rod dS, a quarter of an inch in diameter, is supported 
 by the insulating stand a@ so that about 6 inches of the rod is within 
 the tube with a cylindrical shell of air about an eighth of an inch 
 thick between it and the inside of the tube. The tube is connected 
 with one electrode of a battery of 50 Leclanché cells, the other 
 electrode being connected to earth. The rod is connected to one 
 electrode of Thomson’s quadrant electrometer, the other electrode 
 being connected to earth. A tube, /, which is fixed so as not to 
 touch the tube CD, is used for sending a current of hot air or steam 
 through the tube CD. ‘The part of the tube CD which contains the 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 rod AB is surrounded by a wider tube # of thick brass which may 
 be heated by a gas furnace so as to keep the inner tube and rod hot 
 
139. | CONDUCTIVITY OF GASES. 115 
 
 without exposing them to the current of the products of combustion 
 of the burner. 
 
 The sensitiveness of this apparatus was shewn by the effect of 
 communicating a small charge to the tube #. The electrometer 
 was immediately deflected on account of induction between the 
 tube and the rod 44. The rod 4B was then discharged to earth 
 so that the electrometer indicated zero, the tube remaining at a 
 higher potential. If any conduction were now to take place 
 through the air between the tube and the rod it would be indicated 
 by the electrometer. No conduction however could be observed 
 even after the lapse of a quarter of an hour, and when hot air and 
 steam were blown through the tube. At the end of the experi- 
 ment the tube was discharged to earth, when a negative deflection 
 of the electrometer was observed, shewing that the tube had re- 
 mained charged during the whole experiment. 
 
 139.] Other experiments were afterwards made in which mercury 
 and sodium were made to boil in a bent glass tube while raised to 
 a high potential by a battery of 50 Leclanché cells. A thick 
 copper wire (Fig. 31) was placed on an insulating stand so that the 
 end of the wire was within the glass tube and surrounded by the 
 vapour of the metal. It was necessary that the wire should not 
 be allowed to touch the tube, because glass at a high tempera- 
 ture is a good conductor. It was also necessary to see that the 
 products of combustion from the Bunsen burner did not come 
 in contact with the wire after becoming electrified by the hot 
 tube. 
 
 
 
 Fig. 31. 
 
 The wire was connected with the electrometer, but no evidence of 
 conduction of electricity could be observed, even when the mercury 
 was boiling briskly, and its vapour was being condensed on the 
 
 12 
 
116 CONDUCTIVITY OF GASES. [ 140. 
 
 wire. But whenever so much mercury had collected on the wire 
 that a drop fell off at the end of the wire, there was a deflection 
 of the electrometer because the drop had become charged by in- 
 duction from the tube and the removal of this charge affected 
 the electrometer. This however was no evidence of conduction 
 through the metallic vapour, but only indicated that the apparatus 
 was in such a state of electrification that any conduction, if it took 
 place, would produce a sensible indication at the electrometer. 
 
 It is difficult to reconcile these experiments on the insulating 
 power of hot gases and vapours with the well-known phenomena 
 of the communication of electricity along the stream of heated 
 matter rising from a flame or even from red-hot metal. ‘This 
 stream acts as a powerful conductor of electricity between the flame 
 and bodies placed at a foot or a yard above it where the temperature 
 of the ascending current is much lower than it was in the experi- 
 ment of the tube and rod. | 
 
 140.] The whole theory of the electric properties of gases is in a 
 very imperfect state. According to the kinetic theory of gases, 
 their molecules are in a state of agitation so that they are con- 
 tinually striking against each other. The velocity of this agitation 
 is greater the higher the temperature. It would appear, therefore, 
 that the electric conduction of gases is of the nature of convection. 
 At every collision the whole charge of two of the molecules would 
 be equally divided between them, and thus the tendency of the 
 agitation would be to equalize the charges of all the molecules. 
 
 But we can hardly admit a theory of this kind when we consider ~ 
 that we have hitherto obtained no evidence of the conduction of 
 electricity through air at the ordinary pressure and temperature 
 under a feeble electromotive force. 
 
 Whenever a body free from projecting points and sharp edges 
 and charged to a low potential is found to lose its charge, the 
 result can always he traced to conduction through the substance 
 or along the surface of the apparatus which is required to support 
 it. The more perfectly insulating we make this apparatus the 
 more slowly does the electrified body lose its charge, so that it 
 is probable that 1f we could support an electrified body on a per- 
 feetly insulating stand so that it could lose its charge only by 
 conduction through the air, 1t would never lose its charge. 
 
T4t.] PYRO-ELECTRIC PHENOMENA. iy 
 
 Iilectrie Phenomena of Tourmaline. 
 
 141.] Certain crystals of tourmaline and of other minerals 
 possess what may be called Electric Polarity. Suppose a crystal 
 of tourmaline to be at a uniform temperature, and apparently 
 free from electrification on its surface. Let its temperature be 
 now raised, the crystal remaining insulated. One end will be 
 found positively and the other end negatively electrified. Let the 
 surface be deprived of this apparent electrification by means of a 
 flame or otherwise ; then if the crystal be made still hotter, electrifi- 
 cation of the same kind as before will appear, but if the crystal be 
 cooled the end which was positive when the crystal was heated will 
 become negative. | 
 
 These electrifications are observed at the extremities of the crys- 
 tallographie axis. Some crystals are terminated by a six-sided 
 pyramid at one end and by a three-sided pyramid at the other. 
 In these the end having the six-sided pyramid becomes positive 
 when the erystal is heated. | 
 
 Sir W. Thomson supposes every portion of these and other hemi- 
 hedral crystals to have a definite electric polarity, the intensity 
 of which depends on the temperature. When the surface is passed 
 through a flame, every part of the surface becomes electrified to 
 such an extent as to exactly neutralize, for all external points, 
 the effect of the internal polarity. The crystal then has no ex- 
 ternal electrical action, nor any tendency to change its mode of 
 electrification. But if it be heated or cooled the interior polariza- 
 tion of each particle of the erystal is altered, and can no longer 
 be balanced by the superficial electrification, so that there is a 
 resultant external action. 
 
 In tourmaline and other pyroelectric crystals it is probable 
 that a state of electric polarization exists, which depends upon 
 temperature, and does not require an external electromotive force 
 to produce it. If the interior of a body were in a state of 
 permanent electric polarisation, the outside would gradually become 
 charged in such a manner as to neutralize the action of the internal 
 electrification for all points outside the body. This external super- 
 ficial charge could not be detected by any of the ordinary tests, 
 and could not be removed by any of the ordinary methods for 
 discharging superficial electrification. The internal polarization 
 of the substance would therefore never be discovered unless by 
 some means, such as change of temperature, the amount of the 
 
118 THE ELECTRIC GLOW. [ra2: 
 
 internal polarization could be increased or diminished. The ex- 
 ternal electrification would then be no longer capable of neutralizing 
 the external effect of the internal polarization, and an apparent 
 electrification would be observed, as in the case of tourmaline. 
 
 The Electric Glow. 
 
 142.] It can be proved by the mathematical theory of electricity 
 that if a conductor having on its surface a sharp conical point 
 is placed in a perfectly insulating medium and electrified, the 
 surface-density of the electricity will increase without limit for 
 points nearer and nearer to the conical point, so that at the conical 
 point itself the surface-density, and therefore the electromotive 
 force, would be infinite. But this result depends on the hypothesis 
 that the air or other surrounding dielectric has an invincible 
 insulating power, which is not the case, and therefore as soon 
 as the electromotive force at the conical point reaches a certain 
 limiting value the insulating power of the air gives way, and 
 there is a disruptive discharge of electricity into the air. <A small 
 portion of air close to the conical point thus becomes electrified. 
 The electrified system now consists of the metal conductor with 
 its conical point, together with a rounded mass of electrified air, 
 which covers the point and acts as a sort of sheath to it, so that 
 the boundary of the electrified system is no longer pointed. 
 
 This electrified system, if it were solid, would retain its charge, 
 for the electromotive force is not great enough at any place to 
 produce disruptive discharge, but since the air is fluid, and since 
 the electromotive force is greatest in the line of prolongation of 
 the conical point, the electrified particles of air move off in that 
 direction. When they are gone other unelectrified particles take 
 their place round the point, and the point being no longer 
 protected by electrified air, there is another discharge, as at first. 
 
 Thus there is continually kept up an influx of uncharged air 
 to the point, a luminous discharge of electricity from the point, 
 called the Electric Glow, and a stream of charged air in the 
 direction of the prolongation of the axis of the cone called the 
 Electric Wind. By checking the influx of air behind the point 
 we may weaken the glow and by increasing the current of air by 
 blowing we may make the glow stronger. 
 
 143.] The electric wind which blows from the conical point 
 may be made to drive a little windmill, or if the conductor is 
 
145.] THE ELECTRIC WIND. 119 
 
 made of two wires crossed and having their sharpened ends bent 
 backwards, as in Fig, 32, and supported so as to be capable of 
 rotating, the reaction of the electric 
 
 wind will make the cross rotate in 
 the direction of the arrows. ¥ 
 
 It is only close to the electrified ) 
 point that the motion of the elec- a . 
 trified air is in any degree influenced 
 by its electrification. At a short dis- 
 
 tance from the point the electrified Fig. 32. 
 
 alr becomes mixed with other air, and 
 
 is carried about by the ordinary currents of the atmosphere as an 
 invisible electric cloud 
 
 If we calculate the force due to the electrification of a large 
 body of air at a considerable distance from other eleetrified bodies, 
 we shall find that it is not capable of producing effects on the 
 motion of so large a mass which are at all comparable to the 
 effects of the slight variations of density and other causes which 
 produce the movements of the atmosphere. Hence the motion of 
 thunder clouds is due almost entirely to atmospheric currents and 
 is not sensibly affected by their electrification. 
 
 144,] When an electrified portion of air comes near the surface 
 of a conductor, it induces on that surface an electrification opposite 
 to its own and is attracted towards the surface, but since the 
 electromotive force is small the electrified particles may remain for 
 a long time in the neighbourhood of the conductor without being 
 drawn into contact with it and discharged. 
 
 To detect the presence of this electrified atmosphere clinging 
 to a conductor we have only to insulate the conductor and connect 
 it with an electrometer. If we now blow away the electrified air 
 from its surface, the-electrometer will indicate the electrification 
 of the conductor itself, which is of course of the opposite kind 
 to that of the electrified air. Thus, if we hold in the hand a 
 hollow metal cylinder over an electrified point, we may electrify 
 the air within it. If we then place it on an insulated stand in 
 connexion with the electrometer, the electrometer will remain at 
 zero till the electrified air is removed, which may be done by 
 blowing air through the cylinder. The electrometer will then 
 indicate the electrification of the cylinder, which is of the opposite 
 kind from that of the electrified air. 
 
 145.] The glow is more easily formed in rare air than in dense 
 
120 ACTION OF POINTS. (146. 
 
 air, and more easily when the point is positive than when it is 
 negative. This and many other differences between positive and 
 negative electrification seem to depend upon a condition analogous 
 to electrolytic polarization in the stratum of air in contact with 
 the electrode. It appears that the electromotive force required 
 to cause an electric discharge to take place is somewhat smaller 
 where the electrode at which the discharge begins is negative, but 
 that the quantity of electricity in each discharge is greatest when 
 _ the ele «de at which the discharge begins is positive. 
 
 146.] A fine point may be used instead of a proof plane to deter- 
 mine the nature of the electrification of any part of the surface of 
 a conductor when, electricity is induced upon it in presence of 
 anoth re ectrified body. Jor this purpose the point is fixed to the 
 conductor so as to project a few millimetres from its surface. If 
 the part of the surface to which it is fixed is electrified positively 
 the point gives off positive electricity to the air, and the conductor 
 loses positive electricity or gains negative electricity. This may 
 be ascertained either by removing or discharging the inductor and 
 ascertainin the character of the charge of the induced body, or by 
 connecting the induced body with the electrometer and observing 
 the change of potential as the point throws off its electricity. 
 
 It has been found that some vegetable thorns, prickles, or spines 
 act more perfectly in throwing off electricity than the finest pomted 
 needles which can be procured. 
 
 The action of the point may be assisted by blowing air from 
 a blowpipe over the point, and in this way we may prevent the 
 electrified air from discharging itself on the surface of the inductor. 
 
 The Electric Brush. 
 
 147.| The electric brush is a phenomenon which may be pro- 
 duced by ele:’rifying a blunt point or a small ball in air so as to 
 produce an electric field in which the tension diminishes as the 
 distance from the ball increases, but not so rapidly as in the case of 
 a sharp point. ‘The brush consists of a succession of discharges, 
 ramifying as they diverge from the ball into the air, and termin- 
 ating either by charging portions of air or by reaching some other 
 conductor. The brush produces a sound, the pitch of which depends 
 on the interval between the successive discharges, and there is no 
 current of air as in the case of the glow. 
 
149. | DISRUPTIVE DISCHARGE. 121 
 
 The Electrie Spark. 
 
 148.| When the tension in the space between the two electrodes 
 is considerable all the way between them, as in the case of two balls 
 whose distance is not very great compared with their radii, the 
 discharge, when it occurs, usually takes the form of a spark, by 
 which nearly the whole electrification is discharged at once. 
 
 In this case, when any part of the dielectric has given way, the 
 part next to it in the direction of the electric force is put into 
 a state of greater tension, so that 1t also gives way, and so the 
 discharge proceeds right through the dielectric. We may compare 
 this breaking down of the dielectric to what occurs when we make 
 a little rent perpendicular to the edge of a piece of paper and then 
 apply tension to the paper in the direction of the edge. The paper 
 is torn through, the disruption beginning at the little rent, but 
 diverging occasionally so as to take in weak places in the paper. 
 The electric spark in the same way begins at the point where the 
 electric tension first overcomes the ‘electric strength’ of the dielec- 
 tric, and proceeds from that point, in an apparently irregular path, 
 so as to take in other weak points, such as particles of dust floating 
 in the air. 
 
 149.] The investigation of the phenomena of the luminous 
 electric discharge has been greatly assisted by the use of the spec- 
 troscope. The light of the spark or other discharge is made to fall 
 on the slit of the collimator of the spectroscope, and after being 
 analysed by the prisms is observed through the telescope. The 
 light as thus analysed is found to consist of a great number of 
 bright lines and bands forming what is called the spectrum of the 
 light. By comparing light from different sources it is found that 
 these bright lines may be divided into groups, each group being 
 due to the presence of a particular substance in the medium through 
 which the discharge takes place. 
 
 By using the method introduced by Mr. Lockyer of forming 
 an image of the spark upon the slit by means of a lens, we may 
 obtain at one view a comparison of the constituents of the medium 
 which are rendered luminous by the dielectric discharge at the 
 
 ifferent points of its path. Close to either electrode the lines are 
 principally due to the metal or metals of which that electrode 
 consists. At greater distances these lines become fainter, thinner, 
 and less numerous, but the spectrum belonging to the gas through 
 which the discharge takes place remains visible, 
 
122 SPECTRUM OF THE ELECTRIC SPARK. [150*: 
 
 Some of the lines due to the metals appear longer than others, 
 shewing that they can be formed in regions of the spark where 
 others are no longer visible, owing either to a deficiency in the 
 amount of the metallic vapour or to a want of vigour in the electric 
 disturbance. It thus appears that the electric discharge separates 
 an appreciable amount of matter even from the hardest metals and 
 carries the particles through the air to a distance of several milli- 
 metres from the surface of the metal. It also appears by a com- 
 parison of sparks from different electrodes and through different 
 gases that no part of the light is emitted by any substance common 
 to all the different cases, but that every line is due to one or other 
 of the chemical substances present. 
 
 It follows from this that neither the electric fluid, if there be 
 such a substance, nor any ethereal medium such as is supposed 
 to pervade all ordinary matter, is rendered luminous during the 
 discharge, for if 1t were so its spectrum would be visible in all 
 discharges. 
 
 On Steady Currents. 
 
 150*.] In the case of the current between two insulated con- 
 ductors at different potentials the operation is soon brought to an 
 end by the equalization of the potentials of the two bodies, and the 
 current is therefore essentially a Transient current. 
 
 But there are methods by which the difference of potential of 
 the conductors may be maintained constant, in which case the 
 current will continue to flow with uniform strength as a Steady 
 Current. 
 
 The Voltaic Battery. 
 
 The most convenient method of producing a steady current is by 
 means of the Voltaic Battery. | 
 
 For the sake of distinctness we shall describe Daniell’s Constant 
 Battery :— 
 
 A solution of sulphate of zinc is placed in a cell of porous earthen- 
 ware, and this cell is placed in a vessel containing a saturated 
 solution of sulphate of copper. A piece of zinc is dipped into the 
 sulphate of zine, and a piece of copper is dipped into the sulphate 
 of copper. Wires are soldered to the zine and to the copper above 
 the surface of the liquid. ‘This combination is called a cell or 
 element of Daniell’s battery. See Art. 193.* 
 
Fao: | DANIELL'S BATTERY. 123 
 
 151*.] If the cell is insulated by being placed on a non-con- 
 ducting stand, and if the wire connected with the copper is put 
 in contact with an insulated conductor A, and the wire connected 
 - with the zine is put in contact with 4, another insulated conductor 
 of the same metal as 4, then it may be shewn by means of a deli-: 
 eate electrometer that the potential of 4 exceeds that of B by a 
 certain quantity. ‘This difference of potentials is called the Elec- 
 tromotive Force of Daniell’s Cell. 
 
 If 4 and £& are now disconnected from the cell and put in 
 communication .by means of a wire, a transient current passes 
 through the wire from 4 to &, and the potentials of 4 and B 
 become equal. 4 and 4 may then be charged again by the cell, 
 and the process repeated as long as the cell will work. But if 
 A and B be connected by means of the wire C, and at the same 
 time connected with the battery as before, then the cell will main- 
 tain a constant current through C, and also a constant difference 
 of potentials between A and 6. This difference will not, as we 
 shall see, be equal to the whole electromotive force of the cell, for 
 part of this force is spent In maintaining the current through the 
 cell itself. 
 
 A number of cells placed in series so that the zine of the first 
 cell is connected by metal with the copper of the second, and so 
 on, is called a Voltaic Battery. The electromotive force of such a 
 battery is the sum of the electromotive forces of the cells of which 
 it is composed. If the battery is insulated it may be charged with 
 electricity as a whole, but the potential of the copper end will 
 always exceed that of the zine end by the electromotive force of 
 the battery, whatever the absolute value of either of these potentials 
 may be. ‘The cells of the battery may be of very various construc- 
 tion, containing different chemical substances and different metals, 
 provided-they are such that chemical action does not go on when 
 no current passes. 
 
 152*.]| Let us now consider a voltaic battery with its ends in- 
 sulated from each other. ‘he copper end will be positively or 
 vitreously electrified, and the zine end will be negatively or resin- 
 ously electrified. 
 
 Let the two ends of the battery be now connected by means 
 of a wire. An electric current will commence, and will in a very 
 short time attain a constant value. It is then said to be a Steady 
 Current. 
 
124 OERSTED’S DISCOVERY. Kiera 
 
 Magnetic Action of the Current. 
 
 153*.| Oersted discovered that a magnet placed near a straight 
 electric current tends to place itself at right angles to the plane — 
 passing through the magnet and the current. 
 
 If a man were to place his body in the line of the current so 
 that the current from copper through the wire to zine should flow 
 from his head to his feet, and if he were to direct his face towards 
 the centre of the magnet, then that end of the magnet which tends 
 to point to the north would, when the current flows, tend to point 
 towards the man’s right hand. Thus we see that the electric 
 current has a magnetic action which is exerted outside the current, 
 and by which its existence can be ascertained and its intensity 
 measured without breaking the cireuit or introducing anything 
 into the current itself. 
 
 The amount of the magnetic action has been ascertained to be 
 strictly proportional to the strength of the current as measured 
 by the products of electrolysis in the voltameter, and to be quite 
 independent of the nature of the conductor in which the current 
 is flowing, whether it be a metal or an electrolyte. 
 
 154**,] An instrument which indicates the strength of an electric 
 current by its magnetic effects is called a Galvanometer. 
 
 Galvanometers in general consist of one or more coils of silk- 
 covered wire within which a magnet is suspended with its axis 
 horizontal. When a current is passed through the wire the magnet 
 tends to set itself with its axis perpendicular to the plane of the 
 coils. If we suppose the plane of the coils to be placed parallel 
 to the plane of the earth’s equator, and the current to flow round 
 the coil from east to west in the direction of the apparent motion 
 of the sun, then the magnet within will tend to set itself with 
 its magnetization in the same direction as that of the earth con- 
 sidered as a great magnet, the north pole of the earth being similar 
 to that end of the compass needle which points south. 
 
 The galvanometer is the most convenient instrument for measur- 
 ing the strength of electric currents. We shall therefore assume 
 the possibility of constructing such an instrument in studying the 
 laws of these currents, and when we say that an electric current is 
 of a certain strength we suppose that the measurement is effected 
 by the galvanometer. 
 
Peon OHMS LAW. 125 
 
 On Systems of Linear Conductors, 
 
 155*,| Any conductor may be treated as a linear conductor if it 
 is arranged so that the current must always pass in the same manner 
 between two portions of its surface which are called its electrodes. 
 For instance, a mass of metal of any form, the surface of which is 
 entirely covered with insulating material except at two places, at 
 which the exposed surface of the conductor is in metallic contact 
 with electrodes formed of a perfectly conducting material, may be 
 treated as a linear conductor. For if the current be made to enter 
 at one of these electrodes and escape at the other the lines of flow 
 will be determinate, and the relation between electromotive force, 
 current, and resistance will be expressed by Ohm’s Law, for the 
 current In every part of the mass will be a linear function of Z. 
 But if there be more possible electrodes than two, the conductor 
 may have more than one independent current through it. 
 
 Ohm's Law, 
 
 156*.] Let # be the electromotive force in a linear conductor 
 from the electrode dA, to the electrode A,. (See Art. 5.) Let 
 C be the strength of the electric current along the conductor, that 
 is to say, let C units of electricity pass across every section in 
 the direction .4, 4, in unit of time, and let A be the resistance of 
 the conductor, then the expression of Ohm’s Law is 
 
 TGR Let cB 9 tes tac cic oe c ca ate (1) 
 
 The Resistance of a conductor is defined to be the ratio of 
 the electromotive force to the strength of the current which it 
 produces. The introduction of this term would have been of no 
 scientific value unless Ohm had shewn, as he did experimentally, 
 that it corresponds to a real physical quantity, that is, that it has 
 a definite value which is altered only when the nature of the con- 
 ductor is altered. 
 
 In the first place, then, the resistance of a conductor is indepen- 
 dent of the strength of the current flowing through it. 
 
 In the second place the resistance is independent of the electric 
 potential at which the conductor is maintained, and of the density 
 of the distribution of electricity on the surface of the conductor. 
 
 It depends entirely on the nature of the material of which the 
 conductor is composed, the state of aggregation of its parts and its 
 temperature. 
 
126 RESISTANCE OF CONDUCTORS IN SERIES. [157*. 
 
 The resistance of a conductor may be measured to within one 
 ten thousandth or even one hundred thousandth part of its value, 
 and so many conductors have been tested that our assurance of the 
 truth of Ohm’s Law is now very high*. 
 
 Innear Conductors arranged in Series. 
 157*.| Let 4,, 4, be the electrodes of the first conductor and let 
 the second conductor be placed with one of its electrodes in contact 
 with A,, so that the second conductor has for its electrodes 4,, 43. 
 The electrodes of the third conductor may be denoted by 4d, 
 and A,. 
 Let the electromotive force along each of these conductors be 
 denoted by /,,, £3, 4, and so on for the other conductors. 
 Let the resistance of the conductors be 
 UP BE Mts Seo 
 Then, since the conductors are arranged in series so that the same 
 current C flows through each, we have by Ohm’s Law, 
 HiA= Ch,, Ho, = Ch,,, T2,—= Cha. nn (2) 
 If # is the resultant electromotive force, and / the resultant 
 resistance of the system, we must have by Ohm’s Law, 
 1 OR ( 3) 
 Now EE = Fy + Bog Lg, whisasvavesy ovis see teee een (4) 
 the sum of the separate electromotive forces, 
 = C(R,,+R,3+R,,) by equations (2). 
 Comparing this result with (3), we find 
 Be = Byy + Bog + Bigg ecseecsseccsecewoeszers (5) 
 Or, the resistance of a series of conductors is the sum of the resistances 
 of the conductors taken separately. 
 
 Potential at any Point of the Series. 
 
 Let A and C be the electrodes of the series, 6 a point between 
 them, a, ¢, and 6 the potentials of these points respectively. Let 
 R, be the resistance of the part from 4 to 5, &, that of the part 
 from B to C, and & that of the whole from A to C, then, since 
 
 a—b=khC, b—c=C, and a—c= RC, 
 the potential at 5 is ay R,atR,e 
 
 * [See Report of British Association, 1876. ] 
 
158*.] RESISTANCE OF CONDUCTORS IN MULTIPLE ARC. 127 
 
 which determines the potential at B when those at 4 and C are 
 given. 
 
 Resistance of a Multiple Conductor. 
 
 158*.| Let a number of conductors 4BZ, ACZ, ADZ be arranged 
 side by side with their extremities in contact with the same two 
 points 4 and Z. They are then said to be arranged in multiple 
 are. 
 
 Let the resistances of these conductors be 2,, R,, , respect- 
 ively, and the currents C,, C,, C,, and let the resistance of the 
 multiple conductor be #, and the total current C. Then, since the 
 potentials at 4 and Z are the same for all the conductors, they have 
 the same difference, which we may call #. We then have 
 
 B= (0,R, =C,R, = C,R, = CR, 
 
 but C=C,4+06,+C,, 
 1 1 1 1 
 whence Ramee Te ve jt ia rT ee nS Tc neee (7) 
 
 Or, the reciprocal of the resistance of a multiple conductor is the sum 
 of the reciprocals of the component conductors. 
 
 If we call the reciprocal of the resistance of a conductor the 
 conductivity of the conductor, then we may say that the con- 
 ductivity of a multiple conductor is the sum of the conductivities of 
 the component conductors. 
 
 Current in any Branch of a Multiple Conductor. 
 
 From the equations of the preceding article, it appears that if 
 C, is the current in any branch of the multiple conductor, and 
 f, the resistance of that branch, 
 
 where C is the total current, and £& is the resistance of the multiple 
 conductor as previously determined. 
 
 Kirchhoff has stated the conditions of a linear system in’ the 
 following manner, in which the consideration of the potential is 
 avoided. 
 
 (1) (Condition of ‘continuity.’) At any point of the system the 
 sum of all the currents which flow towards that point is zero. 
 
 (2) In any complete circuit formed by the conductors the sum 
 of the electromotive forces taken round the circuit is equal to the 
 
128 RESISTANCE OF A ‘WIRE. . [rsg7. 
 
 sum of the products of the currents in each conductor multiplied 
 by the resistance of that conductor. 
 
 Longitudinal Resistance of Conductors of Uniform Section. 
 
 159*,| Let the resistance of a cube of a given material to a 
 current parallel to one of its edges be p, the side of the cube being 
 unit of length, p is called the ‘ specific resistance of that material 
 for unit of volume.’ 
 
 Consider next a prismatic conductor of the same material whose 
 length is 7, and whose section is unity. This is equivalent to / 
 cubes arranged in series. The resistance of the conductor is there- 
 fore Zp. 
 
 Finally, consider a conductor of length / and uniform section s. 
 This is equivalent to s conductors similar to the last arranged in 
 multiple are. The resistance of this conductor is therefore 
 
 tt 
 8 
 
 When we know the resistance of a uniform wire we can determine 
 the specific resistance of the material of which it is made if we can 
 measure its length and its section. 
 
 The sectional area of small wires is most accurately determined 
 by calculation from the length, weight, and specific gravity of the 
 specimen. The determination of the specific gravity is sometimes 
 inconvenient, and in such cases the resistance of a wire of unit 
 length and unit mass is used as the ‘specific resistance per unit of 
 weight.’ 
 
 If vis this resistance, / the length, and m the mass of a wire, then 
 
CHAPTER X. 
 
 PHENOMENA OF AN ELECTRIC CURRENT WHICH FLOWS 
 THROUGH HETEROGENEOUS MEDIA. 
 
 1. Lhermo-electric phenomena. 
 
 160.] Seebeck, in 1822, discovered that if a circuit is formed of 
 two different metals, and if the two junctions of the metals are 
 kept at different temperatures, an electric current tends to flow 
 round the cireuit. If the metals are iron and copper at tempera- 
 tures below 280°C, the current flows from copper to iron through 
 the hotter junction. There is therefore, in general, an electro- 
 motive force acting in a definite direction round the circuit, whenever 
 the two junctions are at different temperatures. 
 
 In a cireuit- formed of any number of metals all at the same 
 temperature, there can be no current, for if there were a current 
 it might be constantly employed to work a machine or to generate 
 heat in a conductor, and this without any energy being supplied 
 to the system from without, for in order to keep the circuit at 
 a constant temperature nothing is required except to prevent heat 
 from entering or leaving it. Hence at any given temperature 
 the electromotive force in a cireuit of three metals, 4, B, C must 
 be zero for the whole circuit. Hence if the electromotive force 
 from C to 4 is a, and that from C to B is 4, and that from B to 4 
 x, then in the cireuit 4, B, C, the total electromotive force 1s 
 a—b—xz=0, so that x, the electromotive force from B to A is 
 represented by a—0, where a and @ are quantities determined by 
 observation of the electromotive force from any third metal C 
 to the metals 4 and &. We may express this by saying that 
 the quantities a and J are the potentials of the metals 4 and B 
 with respect to a third metal C at the given temperature. The 
 potential of 4 with respect to Bis a—06. The actual determina- 
 tion of the relative potentials of the metals will be explained in 
 Art. 182. 
 
 K 
 
130 LAW OF MAGNUS. ioe 
 
 161.] It has been shewn by Magnus* that if a cireuit be formed 
 of a single metal, no current will be formed in it, however the 
 temperature and the section of the conducting circuit may vary 
 in different parts. Since in this case there is necessarily con- 
 duction of heat, and consequently dissipation of energy, we canrot, 
 as in the former case, consider the result as self-evident. The 
 electromotive force, for instance, between two portions of the 
 circuit at given temperatures might depend on the length or 
 the mode of variation of the section of the intermediate portion 
 of the circuit. In fact the experiments of Le Roux and others 
 have shewn that the law of Magnus is no longer applicable in 
 a circuit in which there is a very abrupt variation of temperature, 
 as at the instant when the circuit is closed by a hot wire coming 
 in contact with a cold wire of the same metal. Even without 
 
 os | 
 coy bk) — ia 
 
 Fig. 33. 
 
 any physical discontinuity of the ciremt such as is implied in 
 the contact of two separate pieces of wire, a sufficiently abrupt 
 variation of temperature may be produced by taking a thick 
 wire and filing down a certain length of it till it is very thin. 
 If the junction of the thick and the thin portions is placed in 
 a flame, the thin portion will be heated so much more rapidly 
 than the thick portion, that the variation of temperature will be 
 so abrupt that the law of Magnus fails, and we obtain a current 
 in a circuit of one metal; we must therefore modify the statement 
 of the law of Magnus as follows :— 
 
 The electromotive force from one point of a conductor of homogeneous 
 metal to another depends only on the temperature of these points unless 
 at any part of the conductor a sensible variation of temperature occurs 
 between points whose distance is within the limits of molecular action. 
 
 Thermo-electric power of a metal at a given temperature. 
 
 162.| Let us now consider a linear circuit made up of alternate 
 pieces of two metals, say lead and iron. We shall assume lead 
 to be the standard metal, and study the properties of iron in 
 relation to lead. 
 In the figure the pieces of iron are distinguished by shading. 
 Let the temperatures of the junctions be those indicated in the 
 * [Pogg. Ann. 1851.] 
 
163.| THERMOELECTRIC POWER. 131 
 
 figure, in which the temperatures of the extremities of each piece 
 of iron differ by one degree, but the temperatures of the extremities 
 of each of the intermediate pieces of lead are equal. The total 
 electromotive force round the circuit is the sum of the electromotive 
 
 
 
 Fig. 34. 
 
 forces due to the thermo-electric action of the different pairs of 
 junctions. Now if we consider the pairs 4 and B, C and D, # 
 and /' belonging to the pieces of iron we find that the temperature 
 rises one degree in each piece, but if we take the pairs Bb and C, 
 D and £ belonging to the pieces of lead, the temperature in 
 each piece is uniform and therefore there is no electromotive force 
 in these pieces. We may therefore leave the intermediate pieces 
 of lead out of account, and consider the electromotive force due 
 to the junctions 4 and F as equivalent to the sum of the electro- 
 motive forces of the three pairs of junctions 4 and B, C and D, 
 E and F. 
 
 Hence if a diagram is constructed in which the axis OZ is 
 marked with the degrees of the thermometric scale and in which 
 the area 0°PQ1° represents the electromotive force when the 
 junctions are at 0° and 1° and so on, then the electromotive force 
 
 
 
 0) 0° Lp 2° a3 4° Z 
 Fig. 35. 
 
 when the junctions are at any given temperatures will be re- 
 presented by the area included between the axis, the ordinates at 
 the given temperatures and the line PQAST. 
 
 163.] Any ordinate such as O°P, 1° Q, &c., is called the Thermo- 
 electric Power of iron with respect to lead at O°, 1°, &c., and is 
 
 K 2 
 
132 THERMOELECTRIC INVERSION. [ 164. 
 
 reckoned positive when, for a small difference of temperature, the 
 current is from lead to iron through the hot junction. 
 
 We may also on the same diagram construct other lines, the 
 ordinates of which represent the thermo-electric powers of any 
 other metals with respect to lead, being reckoned positive and 
 measured upwards when for a small difference of temperatures 
 the current sets from lead to that metal through the hot junction. 
 Such a diagram is called a thermo-electric diagram, and from it 
 we can deduce the electromotive force due to any pair of metals 
 with their junctions at any given temperatures. 
 
 Thus if a A is the line representing the metal 4, and 4 B another 
 
 representing the metal B, and 7, ¢ the 
 b temperatures of the junctions, the 
 electromotive force of the circuit is 
 B represented by the area A Blad and 
 it acts in the direction indicated, 
 namely, from the metal 4 to the 
 metal B through the hot junction. 
 
 - _ If, instead of lead, we had assumed 
 any other metal as the standard 
 metal, the diagram would have been 
 
 : i altered in form, but all areas measured 
 Fig. 36. on the diagram would have remained 
 
 the same, the change of form being 
 due to a shearing strain in which the slipping is along vertical 
 lines. 
 
 Thermo-electric Inversion. 
 
 164.| Cumming in 1823 discovered several cases in which the 
 thermo-electric order of two metals as observed at ordinary tempera- 
 tures becomes inverted at high temperatures. The lines corre- 
 sponding to these metals on the thermo-electric diagram must 
 therefore cross one another at some intermediate temperature, called 
 the Neutral Temperature for these metals. 
 
 Tait has recently investigated the lines which represent a con- 
 siderable number of metals in the thermo-electric diagram, and he 
 finds that for most metals they are nearly if not exactly straight 
 lines. The lines for iron and nickel however have considerable 
 sinuosities, so that they may intersect the straight lines belonging 
 to another metal in several different points corresponding to several 
 different neutral temperatures. 
 
166. | PELTIER EFFECT. 133 
 
 Thermal effects of the Current. 
 
 165.| By applying the principle of the conservation of energy to 
 the case of a thermo-electric current, it is easy to see that certain 
 thermal effects must accompany the electric current. 
 
 Let us consider what takes place while one unit of electricity 
 is transmitted across any section of the circuit. The work done 
 on the electric current is the product of the electromotive force 
 into the quantity of electricity transmitted, and since this latter 
 quantity is unity, the work is numerically equal to the electro- 
 motive force, and is represented by the area Ada in the thermo- 
 electric diagram. If the current is allowed to flow without 
 anything to impede it except the resistance of the circuit, the 
 whole of the work will be converted into heat, but if the resistance 
 of any part of the circuit such as a long and fine wire greatly 
 exceeds that of the thermo-electric couple, the heat generated in 
 that part of the circuit will greatly exceed that generated in the 
 thermo-electric couple itself. Instead of allowing the current to 
 generate heat, we may make it drive a magneto-electric engine, 
 and so convert any given proportion of the work into mechanical 
 work, | 
 
 Thus for every unit of electricity which is transmitted, a certain 
 amount of work is done by the thermo-electric forces on the current. 
 The only source of this work is the heat of the thermo-electric 
 couple, and therefore, by the principle of the conservation of energy, 
 we conclude that an amount of heat, dynamically equivalent to this 
 work, must have disappeared in some part of the circuit. 
 
 166.| Now Peltier* in 1834 found that when an electric currnet 
 is made to pass from one metal to another which has a higher 
 thermo-electric power, the junction is cooled, or, since there is no 
 permanent change in the metals, there is a disappearance of heat. 
 When the current is made to flow in the opposite direction the 
 junction is heated, indicating a generation of heat. - 
 
 This thermal effect of the current at the junction is of quite 
 a different kind from the ordinary generation of heat by the current 
 while it overcomes the resistance of a conductor. The latter, 
 which we may call with Thomson the frictional generation of heat, 
 is the same when the direction of the current is reversed, and 
 varies as the square of the strength of the current. The former 
 
 * Annales de Chimie et de Physique, lvi. p. 371 (1834), 
 
134 THOMSON EFFECT. [167. 
 
 which we shall call the Peltier effect, is reversed when the current 
 is reversed, and depends simply on the strength of the current. 
 
 167.] But Thomson has shewn that besides the Peltier effect, 
 there must in certain metals be another reversible thermal effect 
 of the current. The current must generate or absorb heat when 
 it passes from hotter to colder or from colder to hotter parts of 
 the came metal. Thus, let a thermo-electric couple of copper and 
 iron be kept with one junction AB at the neutral temperature 
 which is about 280°C, and the other, ad, at some lower temperature. 
 The thermo-electric current is from copper to iron at the hot 
 junction 4B and from iron to copper at the cold junction ab, 
 "Now the Peltier effect at the hot junction, 4B, is zero, for that 
 junction is at the neutral temperature, and the Peltier effect at the 
 cold junction, ad, 1s a generation of heat, for the current is there 
 passing from the metal of 
 higher to the metal of lower 
 thermo-electric power. Hence 
 the absorption of heat which 
 must exist in order to account 
 for the work done by the cur- 
 rent must take place in some 
 
 Fig. 37. other part of the circuit, either 
 in the copper where the cur- 
 rent is flowing from cold to hot, or in the iron where it is flowing 
 from hot to cold, or in both metals. This thermal effect of the 
 current was predicted by Thomson as the result of reasoning 
 similar to that here given. He afterwards verified this pre- 
 diction experimentally, and found that in iron unequally heated 
 a current from hot to cold cools the metal, while a current from 
 cold to hot heats it, and that the reverse thermal effect takes 
 place in copper. We shall refer to this thermal effect as the 
 Thomson effect. . 
 
 168.] Thomson has shewn that a very close analogy subsists be- 
 tween these thermo-electric phenomena and those of a fluid cireu- 
 lating in a tube consisting of two vertical branches connected by 
 two horizontal branches. A fluid, heated in one part of the circuit, 
 and passing on into cooler parts of the system, will give out heat, 
 and when it passes from colder to warmer parts will absorb heat, 
 the amount of heat emitted or absorbed depending on the specific 
 heat of the fluid. According to this analogy, positive or vitreous 
 electricity carries heat with it in copper as if it were a real fluid, 
 
 
 
171.| SPECIFIC HEAT OF ELECTRICITY. 135 
 
 but in iron it behaves as if its specific heat were a negative quan- 
 tity, which cannot be the case in a real fluid. Hence Thomson 
 expresses the fact by saying that negative or resinous electricity 
 earries heat with it in iron. Neither kind of electricity, therefore, 
 can be regarded in this respect as a real fluid. We may therefore 
 adhere to the usual convention, and speaking of the positive elec- 
 tricity only, we may say that in copper it behaves as if its specific 
 heat were positive, and in iron as if it were negative. 
 
 169.| M. Le Roux*, who has made some very careful experiments 
 on the Thomson effect, finds that in lead the specifie heat of 
 electricity is either zero or very small indeed. Professor Tait has 
 therefore adopted lead as the standard metal in his thermo-electric 
 measurements. 
 
 170.| We may express both the Peltier and the Thomson effects 
 by stating that when an electric current is flowing from places of 
 smaller to places of greater thermo-electric power, heat is absorbed, 
 and when it is flowing in the reverse direction heat is generated, 
 and this, whether the difference of thermo-electric power in the two 
 places arises from a difference in the nature of the metal or from a 
 difference of temperature in the same metal. 
 
 171.] The amount of heat absorbed corresponding to a given 
 
 
 
 Fig. 38. 
 
 increase of thermo-electric power, must depend on the temperature 
 as well as on the amount of that increase. For consider a circuit 
 consisting of two metals, neither of which exhibits the Thomson 
 effect. Such a circuit would be represented in the thermo-electric 
 diagram by the parallelogram AabB with horizontal and vertical 
 sides. If the current flows in the direction AabB heat is absorbed 
 in BA and generated on ad, and no reversible thermal effect occurs 
 elsewhere. Also the heat absorbed in BA exceeds that generated 
 
 * Annales de Chimie et de Physique (4), x. p. 243 (1867). 
 
136 SECOND LAW OF THERMODYNAMICS. pe. 
 
 in ab by a quantity represented by the parallelogram BdAab. Hence 
 if we produce Aa and Bd and draw the vertical line af at such a 
 distance that the heat absorbed at the junction AB is represented 
 by the parallelogram BAaB, the heat generated at the junction ad, 
 which, as we have seen, is less than this by the parallelogram 5.daé, 
 will be represented by the parallelogram aba. The Peltier effect. 
 therefore is measured by the product of the increase of thermo- 
 electric power in passing from the first metal to the second into 
 the temperature reckoned from some point lower than any observed 
 temperature, and is of the form (¢,—¢,) (¢—7¢,), when the current 
 flows from a metal in which the thermo-electric power is ¢, to 
 a metal in which it is ¢,, and ¢ is the thermometer reading, and 
 ¢, 1s a constant, the value of which can be ascertained only by 
 experiment. 
 
 172.] Thus far we are led by the principle of the Conservation 
 of Energy. It is a consequence, however, of the Second Law of 
 Thermodynamics, that in all strictly reversible operations in which 
 heat is transformed into work or work into heat, the amount of heat 
 absorbed or emitted at the higher temperature is to that emitted or 
 absorbed at the lower temperature as the higher temperature is to 
 the lower temperature, both being reckoned from absolute zero of 
 the thermodynamic scale. It follows that the line a8 must be 
 drawn in the position corresponding to the absolute zero of the 
 thermodynamic scale, and that the expression for the heat absorbed 
 may be written (¢,—¢,)0, where 0 is the temperature reckoned 
 from absolute zero. It is true that the thermo-electric operations 
 cannot be made completely reversible, as the conduction of heat, 
 which is an irreversible operation, 1s always going on, and cannot 
 be prevented. We must therefore consider the application of the 
 Second Law of Thermodynamics to the reversible part of the 
 phenomena as a very probable conjecture consistent with other 
 parts of the theory of heat, and verified approximately by the 
 measurements of the Peltier and Thomson effects by Le Roux. 
 
 173.] We are now able to express all the thermal and electro- 
 motive effects in terms of the areas in the thermo-electric diagram. 
 Let /7 be the line for one metal, say iron, Ce that for another, say 
 copper. Let 7’ be the higher temperature and ¢ the lower, and let 
 O represent the position of absolute zero. Let the current flow in 
 the direction Clic till one unit of electricity has passed. Then the 
 heat absorbed at the hot junction will be represented by the area 
 CIQk. This is the Peltier effect. 
 
174. | ENTROPY. 137 
 
 The heat absorbed in the iron is represented by PQ... Thomson 
 
 effect. 
 The heat generated in the cold junction, by acSP... Peltier 
 
 effect. 
 The heat absorbed in the copper, by cCRS,,, Thomson 
 
 effect, 
 
 
 
 Fig. 39. 
 
 The whole heat absorbed is therefore represented by CliPSc, and 
 the heat generated by icSP, leaving Clic for the heat absorbed as 
 the result of the whole operation. This heat is converted into the 
 work done on the electric current. 
 
 174,| Entropy*, in Thermodynamics, is a quantity relating to 
 a body such that its increase or diminution implies that heat has 
 entered or left the body. The amount of heat which enters or 
 leaves the body is measured by the product of the increase or 
 diminution of entropy into the temperature at which it takes 
 place. 
 
 In this treatise we have avoided making any assumption that 
 electricity is a body or that it is not a body, and we must also avoid 
 any statement which might suggest that, like a body, electricity 
 may receive or emit heat. 
 
 We may, however, without any such assumption, make use of 
 the idea of entropy, introduced by Clausius and Rankine into the 
 theory of heat, and extend it to certain thermo-electric phenomena, 
 always remembering that entropy is not a thing but a mere instru- 
 ment of scientific thought, by which we are enabled to express in a 
 
 * [Arts. 174-181 consist principally of a repetition of Arts. 167-173, but expressed 
 in the language of the doctrine of Entropy. It was probably the intention of Pro- 
 fessor Clerk Maxwell to insert them or sume modification of them in place of the 
 foregoing Articles, but it has been thought best not to alter the continuous MS., but 
 simply to insert the separate Articles here as representing a slightly different method 
 of applying the Second Law of Thermodynamics to thermo-electric phenomena. ] 
 
138 ELECTRIC ENTROPY. [175: 
 
 compact and convenient manner the conditions under which heat 
 is emitted or absorbed. 
 
 175.] When an electric current passes from one metal to another 
 heat is emitted or absorbed at the junction of the metals. We shall 
 therefore suppose that the electric entropy has diminished or in- 
 creased when the electricity passes from the one metal to the other, 
 the electric entropy being different according to the nature of the 
 medium in which the electricity is, and being affected by its 
 temperature, stress, strain, &e. It is only, however, during the 
 motion of electricity that any thermo-electric phenomena are pro- 
 duced. 
 
 176.] It is proved in treatises on thermodynamics that in all re- 
 versible thermal operations, what is called the entropy of the system 
 remains the same. (Maxwell’s Theory of Heat, 5th ed. p. 190.) 
 
 The entropy of a body is a quantity which when the body re- 
 ceives (or emits) a quantity of heat, H, increases (or diminishes) by 
 
 y dal 
 a quantity oa where @ is the temperature reckoned on the ther- 
 
 modynamic scale. The entropy of a material system is the sum of 
 the entropies of its parts. 
 
 177.| The thermal effects of electric currents are in part re- 
 versible and in part irreversible, but the reversible effects, such as 
 those discovered by Peltier and Thomson, are always small com- 
 pared with the irreversible effects—the frictional generation of heat 
 and the diffusion of heat by conduction. Hence we cannot extend 
 the demonstration of the theorem, which applies to completely re- 
 versible thermal operations, to thermo-electric phenomena. 
 
 But, as Sir Wm. Thomson has pointed out, we have great 
 reason to conjecture that the reversible portion of the thermo-electric 
 effects are subject to the same condition as other reversible thermal 
 operations. This conjecture has not hitherto been disproved by any 
 experiments, and it may hereafter be verified by careful electric and 
 calorimetric measurements. In the meantime the consequences which 
 flow from this conjecture may be conveniently described by an ex- 
 tension of the term entropy to electric phenomena. 
 
 The term Electric Entropy, as we shall use it, corresponds to the 
 term Thermo-electrie Power, as defined by Sir W. Thomson in his 
 fifth paper on the Dynamical Theory of Heat (Trans. R. 8S. E. 
 Ist May, 1854; Art. 140, p. 151). 
 
oo 
 
 178. | THERMO-ELECTRIC DIAGRAM. 159 
 
 Thermo-electric Diagram. 
 
 178.| The most convenient method of studying the theory of 
 thermo-electric phenomena is by means of a diagram in which the 
 temperature and electric entropy of a metal at any instant are 
 represented by the horizontal and vertical coordinates of a point 
 on the diagram. Thus, if OC represents the temperature, reckoned 
 from absolute zero on the thermodynamic scale, of a piece of a 
 certain metal, and if Cd represents the electric entropy corre- 
 sponding to the same piece of metal, then the point 4 will indicate 
 by its position in the diagram the thermo-electric state of the piece 
 of metal. In the same way we may find other points in the 
 diagram corresponding to the same metal under other conditions or 
 to other metals. 
 
 If in the path of an electric current electricity passes from one 
 metal to another or from one portion of a metal to another at 
 a different temperature, the different points of the electric circuit 
 
 id 
 
 
 
 Fig. 40. 
 
 will be represented by corresponding points on the thermo-electric 
 diagram. The path of the current will thus be represented by 
 a line or path on the thermo-electric diagram. When the current 
 flows in a single metal, 4, from a point at a temperature OC to 
 another at a temperature Oc, the path is represented by the line da, 
 the points of which represent the state of the metal at intermediate 
 temperatures. The form of the path depends on the nature of 
 the metal and on the other influences which act on it besides 
 temperature, such as stress and strain. Professor Tait, however, 
 
140 SPECIFIC HEAT OF ELECTRICITY. [179. 
 
 finds that for most of the metals except iron and nickel, the path 
 on the thermo-electric diagram is a straight line. 
 
 When the current flows from the metal 4 to another metal B 
 at the same temperature, the path is represented by 4B, a vertical 
 straight line. The circuit traversed by the electric current will 
 thus be represented by a circuit on the thermo-electric diagram. 
 
 The heat generated while a unit of electricity moves along the 
 path Aa is represented by the area of the figure 4aQPA, bounded 
 by the path da, the horizontal ordinate at a, the line of zero tem- 
 perature and the horizontal ordinate at 4. If this area les on the 
 right of the path, it represents heat generated ; if it lies to the left 
 of the path it represents heat absorbed. 
 
 179.| If electricity were a fluid, running through the conductor 
 as water does through a tube, and always giving out or absorbing 
 heat till its temperature is that of the conductor, then in passing 
 from hot to cold it would give out heat and in passing from cold to 
 hot it would absorb heat, and the amount of this heat would depend 
 on the specific heat of the fluid. 
 
 In the diagram the specific heat of the fluid at 4 would be 
 represented by the line af where a is the point where the tan- 
 gent to the path at 4 cuts the line of zero temperature, and P 
 is the intersection with the same line of the horizontal ordinate 
 through 4. 
 
 The line Aaa in the diagram is such that the electric entropy 
 increases as the temperature rises. This is the case with copper, 
 and therefore we may assert that the specific heat of electricity in- 
 copper is positive. 
 
 In other metals, as for instance iron, the electric entropy 
 diminishes as the temperature rises, as is represented by the line 
 BOB. The specific heat of electricity in such metals is negative, 
 and at B is represented by the line B7. 
 
 180.| Thomson, who discovered first from theory and then by 
 experimental verification the thermal effect of an electric current in 
 an unequally heated metal, expresses the fact by saying that 
 vitreous electricity carries heat with it in copper, while resinous 
 electricity carries heat with it in iron. 
 
 We must remember, however, that these phrases are not in- 
 tended by Thomson, and must not be understood by us, to imply 
 that electricity either positive or negative is a fluid which can 
 be heated or cooled and which has a definite specific heat. Since, 
 therefore, the whole set of phrases are merely analogical we shall 
 
181. | ELECTROMOTIVE FORCE. 141 
 
 adhere to the ordinary convention according to which vitreous 
 electricity is reckoned positive, and we shall say that the specific 
 heat of electricity is positive in copper but negative in iron. 
 
 The obvious fact that no real fluid can have a negative specific 
 heat need not disturb us, for we do not assert that electricity is a 
 real fluid. 
 
 181.] Let us next consider a circuit consisting of two linear 
 conductors of the metals 4 and & respectively, the two junctions 
 being kept at different temperatures, represented in the diagram 
 by OC and Oc. This electric circuit will be represented in the 
 diagram by the circuit 4adBA. If the current flows in the 
 direction 4adB till one unit of electricity has been transmitted, 
 the following thermal effects will take place. 
 
 (1) In the metal 4 heat will be generated as the electricity flows 
 from the hot junction to the cold junction. The amount of this 
 heat is represented by the area 4aQPA. 
 
 (2) At the cold junction, where the electricity passes from the 
 metal 4A to the metal 8, heat will be generated. The amount of 
 this heat is represented by the area abSQa. 
 
 (3) In the metal B heat will be generated as the electricity flows 
 from the cold junction to the hot junction. The amount of this 
 heat is represented by the area L780. 
 
 (4) At the hot junction, where the electricity passes from the 
 metal B to the metal 4, heat will be absorbed. The amount of 
 this heat is represented by the area LAPT'B. The reverse order of 
 the letters shews that this area is to be taken negatively. 
 
 The whole heat generated is therefore represented by the area 
 AabBTPA, and the whole heat absorbed by LAPTB. The total 
 effect is therefore an absorption of heat represented by the area 
 AabBa. 
 
 The energy corresponding to this heat cannot be lost. It is 
 transformed into electrical work spent upon the current by an 
 electromotive force acting in the direction of the current. Since 
 the quantity of electricity transmitted by the current is supposed 
 to be unity, the energy, which is the product of the electromotive 
 force into the quantity of electricity transmitted, must be equal to 
 the electromotive force itself. 
 
 Hence the electromotive force is represented by the area dabB4A, 
 and it acts in the direction represented by the order of the letters— 
 that is, 
 
 Hot, metal 4, cold, metal B, hot. 
 
142 MEASUREMENT OF ELECTROMOTIVE FORCE. [182. 
 
 This electromotive force will, if the resistance of the circuit is 
 finite, produce an actual current*. It was by means of such 
 currents that the thermo-electric properties of metallic circuits 
 were first discovered by Seebeck in 1822. 
 
 182.] The electrical effects due to heat were discovered before 
 the thermal effects due to the electric current, but the application 
 of the thermal effects of the current to determine the electromotive 
 forces acting along different portions of the circuit is due to Sir 
 W. Thomsony. It is manifest that in a heterogeneous circuit 
 we cannot determine the electromotive force acting from the point 
 A to the point B by simply connecting these points by wires to the 
 electrodes of a galvanometer or electrometer, for we are ignorant of 
 the electromotive forces acting at the junctions of these wires with 
 the matter of the circuit at A and JB. 
 
 But if we cause a current of known strength to flow from 4 to B, 
 and if this current causes the generation of a quantity of heat equal 
 to H in that portion of the circuit, and if no chemical, magnetic or 
 other permanent effect takes place in the matter of the conductor 
 between 4 and B, then we know that if Q is the total quantity of 
 electricity which has been transmitted from 4 to B, and # the 
 electromotive force in the direction from £ to A which the current 
 has to overcome, then the work done by the current is Q#. This 
 work is done within a definite region, namely the portion AB of 
 the conductor, and it is entirely expended in generating heat within 
 that region. Hence, if the quantity of heat generated in the 
 portion AZ is H, as expressed in dynamical measure, we have the 
 equation 
 
 QH= H, 
 and since Q and JZ are capable of being measured we can determine 
 the electromotive force # acting against the current. When the 
 electromotive force acts in the same direction as the current is 
 flowing, the quantity of heat generated is negative; or, in other 
 words, there is an absorption of heat. 
 
 In this investigation we must remember that / represents the 
 whole electromotive force acting against the current. Now part of 
 this electromotive force arises from the electric resistance of the 
 
 * [The energy expended in driving the current will, if not otherwise employed, be 
 ultimately converted into heat through the frictional resistance of the metals. The 
 heat produced by this irreversible action must be distinguished from the Thomson 
 and Peltier effects, and is represented on the Thermo-electric diagram by the area 
 A Bba A.) 
 
 + Zrans. fh. S. Edin. 1854. 
 
184.] MEASUREMENT OF ELECTROMOTIVE FORCE. 143 
 
 conductor. ‘This part always acts against the current, and is pro- 
 portional to the current according to Ohm’s law. 
 
 The other part of the electromotive force acts in a definite direc- 
 tion, either from 4 to B or from B to A, and is independent of the 
 direction of the current. It is generally this latter part of the 
 electromotive force which is referred to as the electromotive force 
 from A to B. 
 
 It is easy to eliminate the part due to resistance by making two 
 
 experiments in which currents of equal strength are made to flow 
 in one case from 4 to B and in the other from B to 4. The excess 
 of the heat generated in the second case over that generated in the 
 first case, per unit of electricity transmitted, is numerically equal 
 to twice the electromotive force from 4 to B. 
 
 183.| The total electromotive force round any circuit is easily 
 measured by breaking the cireuit in a place where it is homo- 
 geneous, and determining: the difference of potentials of the two ends. 
 This may be done by any of the ordinary methods for determining 
 electromotive force or difference of potentials, because in this case 
 the two ends are of the same substance and at the same tempera- 
 ture. But we cannot by this method determine how much of this 
 electromotive force has its seat in a particular part of the circuit, 
 as for instance, between 4 and 2, where 4 and # are of different 
 substances or at different temperatures. The only method by which 
 we can determine where the electromotive force acts is that of 
 measuring the heats generated or absorbed during the transmission 
 of a unit of electricity from 4 to B. 
 
 184.| In the cases we have hitherto considered the only per- 
 manent effect of the current has been the generation or absorption 
 of heat, for metals are not altered in any respect by the continuous 
 flow of a current through them. But when the current flows from ‘ 
 a metal to an electrolyte or from an electrolyte to a metal, there 
 are chemical changes, and in applying the principle of the conserva- 
 tion of energy we must take account of these as well as of the 
 thermal effects. 
 
 We shall consider the current as flowing through an electrolyte 
 from the anode to the cathode. The fundamental phenomenon of 
 electrolysis is the liberation of the components or ions of the electro- 
 lyte, the anion at the anode and the cation at the cathode. This is 
 the only purely electrolytic effect; the subsequent phenomena 
 depend on the nature of the ions, the electrodes and the electrolyte, 
 and take place according to chemical and physical laws in a manner 
 
144 ELECTROMOTIVE FORCE BETWEEN [185. 
 
 apparently independent of the electric current. Thus the ion, when 
 liberated at the electrode, may behave in several different ways, 
 according to the conditions in which it finds itself. It may be in 
 such a condition that it acts neither on the electrode nor on the 
 electrolyte, as when it is a gas which escapes in bubbles, or sub- 
 stance insoluble in the electrolyte, which is precipitated. It may 
 be deposited on the surface of the electrode, as hydrogen is on 
 platinum, and may adhere to it with various degrees of tenacity, 
 from mere juxtaposition up to chemical combination. If it is 
 soluble in the electrolyte, it will diffuse through the electrolyte 
 according to the ordinary law of diffusion, and the rate of this 
 diffusion is not, so far as we know, affected by the existence of the 
 electric current through the electrolyte, for it is only when in com- 
 bination, and not when in mere solution, that the current produces 
 the electrolytic transfer of the ions. Thus when hydrogen is an 
 ion, part of it may escape in bubbles, part of it may be condensed 
 on the electrode, and part of it may be absorbed into the electro- 
 lyte without combination, and travel through it by ordinary 
 diffusion. 
 
 185.| The liberated ion may also act chemically on the electrode 
 or on the electrolyte. The results of such action are called 
 secondary products of electrolysis, and these secondary products may 
 remain at the surface of the electrodes, or may become diffused 
 through the electrolyte. ‘Thus, when the same current is passed, 
 first through a solution of sulphate of soda between platinum elec- 
 trodes, and then through sulphuric acid, equal volumes of oxygen 
 are given off at the anodes of the two electrolytes, and equal 
 volumes of hydrogen, each equal to double the volume of oxygen, 
 are given off at the cathodes, 
 
 But if the electrolysis is conducted in suitable vessels, such as 
 U-shaped tubes or vessels with a porous diaphragm, so that the 
 substance surrounding each electrode may be examined, it is found 
 that at the anode of the sulphate of soda there is an equiva- 
 lent of sulphuric acid as well as an equivalent of oxygen, and at 
 the cathode there is an equivalent of soda as well as two equivalents 
 ‘ of hydrogen. It would at first sight appear as if (according to the 
 old theory of the constitution of salts) the sulphate of soda were 
 electrolysed into its constituents, sulphuric acid and soda, while the 
 water of the solution is electrolysed at the same time into oxygen 
 and hydrogen. But this explanation would involve the assumption 
 that the same current which passing through dilute sulphuric acid 
 
186. A METAL AND AN ELECTROLYTE. 145 
 
 electrolyses one equivalent of water, when it passes through so- 
 lution of sulphate of soda electrolyses two equivalents, one of the 
 salt and one of water, and this would be contrary to the law of 
 electrochemical equivalents. But if we suppose that the com- 
 ponents of sulphate of soda are not SO, and Na,O, but SO, and Na, 
 —not sulphuric acid and soda but sulphion and sodium—then an 
 equivalent of sulphion travels to the anode and is set free, but being 
 unable to exist in a free state, it breaks up into sulphuric anhydride 
 and oxygen, one equivalent of each. At the same time [two] equiva- 
 lents of sodium are set free at the cathode, and then decompose the 
 water of the solution, forming two equivalents of soda [NaHO] 
 and two of hydrogen. 
 
 In the dilute sulphuric acid, the gases collected at the elec- 
 trodes are the constituents of water, namely one volume of oxy- 
 gen and two volumes of hydrogen. There is also an increase 
 of sulphuric acid at the anode, but its amount is less than one 
 equivalent. 
 
 186.| It follows from these considerations that in order to ascer- 
 tain the electromotive force acting from a metal to an electrolyte, 
 we must take account of the whole permanent effects of the passage 
 of one unit of electricity from the metal to the electrolyte. Thus, 
 if the electrolyte is sulphate of zine, with zinc electrodes, a certain 
 amount of heat is generated at the anode for every unit of elec- 
 tricity and at the same time one equivalent of zine combines with 
 one equivalent of sulphion and forms sulphate of zinc. Now the 
 quantity of heat generated when one equivalent of zinc combines 
 with oxygen is known from the experiments of Andrews and others, 
 and also the heat generated when an equivalent of oxide of zine 
 combines with sulphuric acid, and is dissolved in water so as to 
 form a solution of sulphate of zine of the same strength as that 
 which surrounds the electrode. The sum of these quantities of 
 heat, which we may call //, is equivalent to the total work done by 
 the chemical action at the anode, which is therefore JH | where 
 J represents Joule’s equivalent, or the mechanical equivalent of 
 heat]. Let 4 be the quantity of heat generated at the anode during 
 the passage of one unit of electricity, and let / be the electromotive 
 force acting from the zinc to the electrolyte, that is, in the direction 
 of the current. Then the work done in generating heat is J/, and 
 the work done in driving the current is # so that the equation of 
 work is JH=Jh+H#H 
 
 or E=J(H— kh). 
 L 
 
146 MEASUREMENT OF ELECTROMOTIVE FORCE. [187. 
 
 Of these quantities H is known very accurately but it is some- 
 what difficult to measure /, the quantity of heat generated at the 
 electrode, because the electrode must be in contact with the electro- 
 lyte, and therefore a large and unknown fraction of the heat 
 generated will be carried away by conduction and convection 
 through the electrolyte. The only method which seems likely 
 to succeed is to compare the stationary temperature at a certain 
 distance from the electrode with the temperature at the same 
 point when in the place of the electrode we put a fine wire of 
 known resistance through which we pass a known current so as 
 to generate heat at a known rate. If the temperatures are equal 
 in the two cases we may conclude that the heat is generated at the 
 same rate in the zine electrode and in the wire. But if the current 
 is a strong one a very sensible portion of the whole heat generated 
 will be due to the work done by the current in overcoming the 
 ordinary resistance of the electrode and the electrolyte. As the elec- 
 trode is generally made of a metal whose resistance is very small 
 compared with that of the electrolyte, this frictional generation of 
 heat will take place principally in the electrolyte. This frictional 
 generation of heat may be made very small compared with the 
 reversible part by diminishing the strength of the current, but then 
 the rate of generation of heat becomes so small that it is difficult 
 to measure it in the presence of unavoidable thermal disturbances, 
 such as arise from changes in the temperature of the air, &e. The 
 experimental investigation is therefore one of considerable difficulty, 
 and I am not aware that the electromotive force from a metal to an 
 electrolyte has as yet been measured even approximately*. If, how- 
 ever, we assume that the electromotive forces from the metals 4 
 and B to the electrolyte C are 4 and B respectively, and that the 
 thermo-electric powers of these metals at the temperature 6 are a 
 and 6 respectively, then the electromotive force from - to B at 
 their junction is (4—a) 6. 
 
 The total electromotive force round the circuit in the cyclical 
 direction ABC is (6—a)6 +B—A. 
 
 On the Conservation of Energy in Electrolysis. 
 
 187.] Consider an electric current flowing in a circuit consisting 
 partly of metals and partly of electrolytes placed in series. 
 During the passage of one unit of electricity through any section 
 
 * [See Art. 192 and last two paragraphs of note, p. 150.] 
 
roa: JOULE’S EXPERIMENTS, 147 
 
 of the circuit one electrochemical equivalent of each of the electro- 
 lytes is electrolysed, There is therefore a definite amount of 
 chemical action corresponding to a definite quantity of electricity 
 passed through the circuit. The energy equivalent to any chemical 
 process can be ascertained either directly or indirectly. When the 
 process is such that it will go on of itself, and if the only effect 
 external to the system is the giving off of heat generated during 
 the process, then the intrinsic energy of the system must be 
 diminished during the process by a quantity of energy equivalent 
 to the heat given out. Ifa material system consisting of definite 
 quantities of so many chemical substances is capable of existing in 
 several different states, and if the system will not of itself pass 
 from one of these states (4) to another (4) we can still find the 
 relative energy of the state (4) with respect to the state (3) 
 provided we can cause both the state (4) and the state (2) to 
 pass into the state (C) which we may suppose to be the state 
 in which all the energies of combination of the system have been 
 exhausted. 
 
 Thus if the substances in the system are oxygen, hydrogen and 
 earbon, and if the states (4) and (J) consist of two different 
 hydrocarbons with free hydrogen and oxygen, we cannot in general 
 cause the state (4) to pass into the state (2), but we can cause 
 either (4) or (B) to pass into the state (C) in which all the 
 hydrogen is combined with oxygen as water and all the carbon 
 is combined with oxygen as carbonic acid. In this way the 
 energy of the state (4) relatively to the state (B) can be determined 
 by measurements of heat. 
 
 188.] It has been proved experimentally by Joule that the heat 
 developed throughout the whole electric circuit is the same for the 
 same amount of chemical action whatever be the resistance of the 
 circuit provided no other form of energy than heat is given off by 
 the system. 
 
 Thus in a battery the electrodes of which are connected by a 
 short thick wire the current is very strong and the heat is gener- 
 ated principally in the cells of the battery and to a much smaller 
 extent in the wire; but if the wire is long and thin, the heat 
 generated in the wire is far greater than that generated in the 
 cells, but if we take into account the heat generated in the wire 
 as well as that generated in the cells, we find that the whole 
 heat generated for each grain of zine dissolved is the same in 
 both cases. 
 
 L 2 
 
148 ELECTROMOTIVE FORCE IN [189. 
 
 189.] If, however, the circuit includes a cell in which dilute 
 acid is electrolysed into oxygen and hydrogen the heat generated 
 in the circuit, per grain of zinc dissolved, is less than before, by the 
 quantity of heat which would be generated if the oxygen and 
 hydrogen evolved in the electrolytic cell were made to combine. 
 
 Or if the circuit includes an electromagnetic engine which is 
 employed to do work, the heat generated in the circuit is less than 
 that corresponding to the zine consumed by an amount equal to 
 the heat which would be generated if the work done by the engine 
 were entirely expended in friction. 
 
 190.| If the arrangement is such that the amount of chemical 
 action depends entirely on the quantity of electricity transmitted 
 we can determine the electromotive force of the circuit by the 
 following method, first given by Thomson (PAz/. Mag., Dec. 1851). 
 Let the resistance of the circuit be made so great that the heat 
 generated by the current in the electrolytes may be neglected. 
 Let # be the electromotive force of the circuit; then the work 
 done in driving one unit of electricity through the circuit is 
 numerically equal to #. But during this process one electro- 
 chemical equivalent of the electrolyte undergoes the chemical 
 process which goes on in the cell. Hence, if the energy given 
 out during this process is entirely expended in maintaining the 
 current, the dynamical value of the process must be numerically 
 equal to H, the electromotive force of the circuit, or, as Thomson 
 stated it, 
 
 ‘The electromotive force of an electrochemical apparatus is in 
 absolute measure equal to the mechanical equivalent of the chemical 
 action on one electrochemical equivalent of the substance.’ 
 
 EXAMPLES. 
 
 191.] If the action in the cell consists in part of irreversible 
 processes, such as 
 1. The frictional generation of heat by resistance in the elec- 
 trolyte, 
 2. Diffusion of the primary or secondary products of electrolysis 
 through the electrolyte, or, 
 3. Any other action which is not reversed when the direction of 
 the current 1s reversed, 
 there will be a certain amount of dissipation of energy and the 
 electromotive force of the circuit will be less than the loss of 
 
192.] A VOLTAIC CIRCUIT. 149 
 
 intrinsic energy corresponding to the electrolysis of one electro- 
 chemical equivalent. 
 
 It is only the strictly reversible processes that must be taken 
 into account in calculating the electromotive force of the circuit. 
 
 192.] The determination of the total electromotive force in an 
 electrochemical circuit is therefore always possible. If, however, 
 we wish to determine the precise points in the circuit where the 
 different portions of this electromotive force are exerted, we find 
 the investigation much more difficult than in the case of a purely 
 metallic circuit. 
 
 For the chemical action at the junction of a metal with an 
 electrolyte is generally of such a kind that it cannot take place 
 by itself, that is to say, without an action equivalent to that 
 which takes place at the other electrode. Thus, when a current 
 passes between silver electrodes through fused chloride of silver, 
 chlorine is liberated at the anode which immediately acts on the 
 electrode so as to form chloride of silver and silver is deposited on 
 the cathode. 
 
 Now we know the amount of heat given out when an equivalent 
 of free chlorine combines with an equivalent of silver, and this is 
 equivalent to the energy which must be spent in electrolysing 
 chloride of silver into free chlorine and free silver, but the process 
 that takes place at the anode is the combination of silver, not with 
 free chlorine, but with chlorine in the act of being electrolysed out 
 of chloride of silver*. 
 
 * [The following note is an extract from Professor Maxwell’s letter on Potential 
 published in the Electrician, April 26th, 1879.] 
 
 In a voltaic circuit the sum of the electromotive forces from zinc to the electrolyte, 
 from the electrolyte to copper, and from copper to zinc, is not zero but is what is called 
 the electromotive force of the circuit—a measurable quantity. Of these three electro- 
 motive forces only one can be separately measured by a legitimate process, that, 
 namely, from copper to zinc. 
 
 Now it is found by thermoelectric experiments that this electromotive force is ex- 
 ceedingly small at ordinary temperatures, being less than a microvolt, and that it is 
 from zinc to copper. 
 
 Hence the statement deduced from experiments in which air is the third medium, 
 that the electromotive force from copper to zinc is -75 volts, cannot be correct. In 
 fact, what is really measured is the difference between the potential in air near the 
 surface of copper, and the potential in air near the surface of zine, the zinc and copper 
 being in contact. The number -75 is therefore the electromotive force, in volts of 
 the circuit copper, zinc, air, copper, and is the sum of three electromotive forces, only 
 one of which has as yet been measured. 
 
 Mr. J. Brown has shown (Phil. Mag., Aug. 1878, p. 142), by the divided ring method 
 of Sir W. Thomson, that whereas copper is negative with respect to iron in air it is 
 positive with respect to iron in hydrogen sulphide. ; 
 
 It would appear, therefore, that the reason why the results of the comparison of 
 metals by the ordinary ‘ contact force’ experiments harmonise so well with the com- 
 parison by dipping both metals in water or an oxidizing electrolyte is not because the 
 electromotive force between a metal and a gas or an electrolyte is small, but because 
 
150 CONSTANT BATTERIES. [193*. 
 
 On Constant Voltaic Elements. 
 
 -193*.] When a series of experiments is made with a voltaic 
 battery in which polarization occurs, the polarization diminishes 
 during the time that the current is not flowing, so that when 
 it begins to flow again the current is stronger than after it has 
 flowed for some time. If, on the other hand, the resistance of the 
 circuit is diminished by allowing the current to flow through a 
 short shunt, then, when the current is again made to flow through 
 the ordinary circuit, it is at first weaker than its normal strength 
 on account of the great polarization produced by the use of the 
 short circuit. 
 
 To get rid of these irregularities in the current, which are 
 exceedingly troublesome in experiments involving exact measure- 
 ments, it is neeessary to get rid of the polarization, or at least 
 to reduce it as much as possible. 
 
 It does not appear that there is much polarization at the surface 
 of the zinc plate when immersed in a solution of sulphate of zine 
 or in dilute sulphuric acid. The principal seat of polarization is 
 at the surface of the negative metal. When the fluid in which 
 the negative metal is immersed is dilute sulphurie acid, it is seen 
 to become covered with bubbles of hydrogen gas, arising from the 
 electrolytic decomposition of the fluid. Of course these bubbles, 
 by preventing the fluid from touching the metal, diminish the 
 surface of contact and increase the resistance of the cireuit. But 
 besides the visible bubbles it is certain that there is a thin coating 
 of hydrogen, probably not in a free state, adhering to the metal, 
 and as we have seen that this coating is able to produce an elec- 
 tromotive force in the reverse direction, it must necessarily diminish 
 the electromotive force of the battery. 
 
 Various plans have been adopted to get rid of this coating of 
 hydrogen. It may be diminished to some extent by mechanical 
 
 the properties of air agree, to a.certain extent, with those of oxidising electrolytes. 
 For, if the active component of the electrolyte is sulphur, the results are quite different, 
 and the same kind of difference occurs when hydrogen sulphide is substituted for air. 
 
 We know so little about the nature of the ions as they exist in an electrolyte that, 
 even if we could measure the quantity of heat generated or absorbed when unit of 
 electricity passes from a metal to an electrolyte, or from an electrolyte to a metal, we 
 could not determine from this the value of the electromotive force from the metal to 
 the electrolyte. i 
 
 If this is the case with liquid electrolytes, we have still less hope of determining the 
 electromotive force from a metal to a gas, for we cannot produce a current from the 
 one to the other without tumultuary and non-reversible effects, such as disintegration 
 of the metal and violent disturbance of the gas by the discontinuous discharge. 
 
193%: CONSTANT BATTERIES. 151 
 
 means, such as stirring the liquid, or rvbbing the surface of the 
 negative plate. In Smee’s battery the negative plates are ver- 
 tical, and covered with finely divided platinum from which the 
 bubbles of hydrogen easily escape, and in their ascent produce a 
 current of liquid which helps to brush off other bubbles as they 
 are formed. 
 
 A far more efficacious method, however, is to employ chemical 
 means. These are of two kinds. In the batteries of Grove and 
 Bunsen the negative plate is immersed in a fluid rich in oxygen, 
 and the hydrogen, instead of forming a ceating on the plate, 
 combines with this substance. In Grove’s battery the plate is 
 of platinum immersed in strong nitric acid. In Bunsen’s first 
 battery it is of carbon in the same acid. Chromic acid is also used 
 for the same purpose, and has the advantage of being free from the 
 acid fumes produced by the reduction of nitric acid. 
 
 A different mode of getting rid of the hydrogen is by using — 
 copper as the negative metal, and covering the surface with a coat 
 of oxide. This, however, rapidly disappears when it is used as 
 the negative electrode. ‘To renew it Joule has proposed to make 
 the copper plates in the form of disks, half immersed in the liquid, 
 and to rotate them slowly, so that the air may act on the parts 
 exposed to it in turn. 
 
 The other methed is by using as the liquid an electrolyte, the 
 cation of which is a metal hiehly negative to zine. 
 
 In Daniell’s battery a copper plate is immersed in a saturated 
 solution of sulphate of copper. When the current flows through 
 the solution from the zine to the copper no hydrogen appears on 
 the copper plate, but copper is deposited on it. When the solution 
 is saturated, and the current is not too strong, the copper appears 
 to act as a true cation, the anion SO, travelling towards the zine. 
 
 When these conditions are not fulfilled hydrogen is evolved at 
 the cathode, but immediately acts on the solution, throwing down 
 copper, and uniting with SO, to form oil of vitriol. When this 
 is the case, the sulphate of copper next the copper plate is replaced 
 by oil of vitriol, the liquid becomes colourless, and polarization by 
 hydrogen gas again takes place. The copper deposited in this way 
 is of a looser and more friable structure than that deposited by true 
 electrolysis. 
 
 To ensure that the liquid in contact with the copper shall be 
 saturated with sulphate of copper, crystals of this substance must 
 be placed in the liquid close to the copper, so that when the solution 
 
152 DANIELL'S BATTERY. [193*. 
 
 is made weak by the deposition of the copper, more of the crystals 
 may be dissolved. 
 
 We have seen that it is necessary that the liquid next the copper 
 should be saturated with sulphate of copper. It is still more 
 necessary that the liquid in which the zine is immersed should be. 
 free from sulphate of copper. If any of this salt makes its way 
 to the surface of the zine it is reduced, and copper is deposited 
 on the zinc. The zinc, copper, and fluid then form a little cireuit 
 in which rapid electrolytic action goes on, and the zine is eaten 
 away by an action which contributes nothing to the useful effect 
 of the battery. 
 
 To prevent this, the zinc is immersed either in dilute sulphuric 
 acid or in a solution of sulphate of zine, and to prevent the solution 
 of sulphate of copper from mixing with this liquid, the two liquids 
 are separated by a division consisting of bladder or porous earthen- 
 ware, which allows electrolysis to take place through it, but 
 effectually prevents mixture of the fluids by visible currents. 
 
 In some batteries sawdust is used to prevent currents. The 
 experiments of Graham, however, shew that the process of diffusion 
 goes on nearly as rapidly when two liquids are separated by a 
 division of this kind as when they are in direct contact, provided 
 there are no visible currents, and it is probable that if a septum 
 is employed which diminishes the diffusion, it will inerease in 
 exactly the same ratio the resistance of the element, because elec- 
 trolytic conduction is a process the mathematical laws of which 
 have the same form as those of diffusion, and whatever interferes 
 with one must interfere equally with the other. The only differ- 
 ence 1s that diffusion is always going on, while the current flows 
 only when the battery is in action. 
 
 In all forms of Daniell’s battery the final result is that the 
 sulphate of copper finds its way to the zine and spoils the battery. 
 To retard this result indefinitely, Sir W. Thomson * has constructed 
 Daniell’s battery in the form shewn in Fig. 41. 
 
 In each cell the copper plate is placed horizontally at the bottom, 
 and a saturated solution of sulphate of zinc is poured over it. The 
 zinc is in the form of a grating and is placed horizontally near the 
 surface of the solution. A glass tube is placed vertically in the 
 solution with its lower end just above the surface of the copper 
 plate. Crystals of sulphate of copper are dropped down this tube, 
 and, dissolving in the liquid, form a solution of greater density - 
 
 * Proc. R. S., Jan. 19, 1871, 
 
13". | DANIELL'S BATTERY. 153 
 
 than that of sulphate of zine alone, so that it cannot get to the 
 zine except by diffusion. To retard this process of diffusion, a 
 siphon, consisting of a glass tube stuffed with cotton wick, is 
 placed with one extremity midway between the zine and copper, 
 and the other in a vessel outside the cell, so that the liquid is 
 
 eS 
 
 ee ELECTRODES 
 
 oD anes 
 
 
 
 
 
 Zine 
 
 Zn S04 
 
 LEVEL oF SIPHON 
 
 22 S047 Cu 504 
 4 COPPER 
 
 Fig. 41. 
 
 very slowly drawn off near the middle of its depth. To supply 
 its place, water, or a weak solution of sulphate of zine, is added 
 above when required. In this way the greater part of the sulphate 
 of copper rising through the liquid by diffusion is drawn off by the 
 siphon before it reaches the zinc, and the zine is surrounded by 
 liquid nearly free from sulphate of copper, and having a very slow 
 downward motion in the cell, which still further retards the upward 
 motion of the sulphate of copper. During the action of the battery 
 copper is deposited on the copper plate, and SO, travels slowly 
 through the liquid to the zinc with which it combines, forming 
 sulphate of zinc. ‘Thus the liquid at the bottom becomes less dense 
 by the deposition of the copper, and the liquid at the top becomes 
 more dense by the addition of the zinc. To prevent this action 
 from changing the order of density of the strata, and so producing 
 instability and visible currents in the vessel, care must be taken to 
 keep the tube well supplied with crystals of sulphate of copper, 
 and to feed the cell above with a solution of sulphate of zine suffi- 
 ciently dilute to be lighter than any other stratum of the liquid 
 in the cell. 
 
 Daniell’s battery is by no means the most powerful in common 
 -use. The electromotive force of Grove’s cell is 192,000,000, of 
 Daniell’s 107,900,000, and that of Bunsen’s 188,000,000. 
 
154 ELECTROMOTIVE FORCE OF BATTERIES.  [193*. 
 
 The resistance of Daniell’s cell is in general greater than that of 
 Grove’s or Bunsen’s of the same size. 
 
 These defects, however, are more than counterbalanced in all 
 cases where exact measurements are required, by the fact that 
 Daniell’s cell exceeds every other known arrangement in constancy 
 of electromotive force. It has also the advantage of continuing 
 in working order for a long time, and of emitting no gas. 
 
CHAPTER XI. 
 
 METHODS OF MAINTAINING AN ELECTRIC CURRENT. 
 
 194.| THE principal methods of maintaining a steady electric 
 current are— 
 (1) The Frictional Machine. 
 (2) The Voltaic Battery. 
 (3) The Thermo-electric Battery. 
 (4) The Magneto-electric Machine. 
 
 (1) The Frictional Electric Machine. 
 
 195.| The electrification is here produced between the surfaces of 
 two different substances, such as glass and amalgam or ebonite and 
 fur. By the motion of the machine one of these electrified surfaces 
 is carried away from the other, and both are made to discharge 
 their electrification into the electrodes of the machine, from which 
 the current is conveyed along any required circuit. - 
 
 In the ordinary form of the machine a circular plate or a cylinder 
 of glass is made to revolve about its axis. Let us suppose that the 
 revolving part is a plate of glass. The rubber is fixed so that it 
 presses against the surface of the plate as it rotates. The surface 
 of the rubber is of leather, on which is spread an amalgam of zine 
 and mercury. By the friction between the glass and the amalgam 
 the surface of the glass becomes electrified positively, and that of 
 the rubber negatively. As the plate revolves the electrified surface 
 of the glass is carried away from under the rubber, and another 
 part of the surface of the glass, previously unelectrified, is brought 
 under the rubber to be electrified. As long as the oppositely 
 electrified surfaces of the glass and the rubber remain in contact, 
 the electrical effects in the neighbourhood are very small, but when 
 
156 FRICTIONAL ELECTRIC MACHINE. [196. 
 
 the glass is removed from the rubber, strong electrical forces are 
 developed. The potential of the rubber becomes negative, and as, 
 on account of the amalgam, it conducts freely its electrification is 
 at once carried off to the negative electrode. At the same time the 
 potential of the electrified glass becomes highly positive, but as the 
 glass is an insulating substance it does not so readily part with its 
 electrification. The positive electrode of the machine is therefore 
 furnished with a comb, consisting of a number of sharp pointed 
 wires terminating near the electrified surface of the glass. As the 
 potential at the surface of the glass is much higher than that of 
 the comb there is a great accumulation of negative electrification 
 at the point of the comb, and this breaks into a negative electric 
 glow accompanied by an electric wind blowing from the comb to 
 the glass. The negatively electrified particles of air spread them- 
 selves over the positively electrified surface of the glass, and cause 
 the electrification of the glass to be discharged. It is possible, 
 however, that part of them may be carried round with the glass till 
 they are wiped off by the rubber, though I have not been able to 
 obtain experimental evidence of this. 
 
 Thus the rotation of the machine carries the positive electri- 
 fication of the surface of the glass from the rubber to the comb, 
 and the negative electric wind of the comb either neutralizes the 
 positively electrified surface, or is carried round with it to the 
 rubber, so that there is a continual current of positive electricity 
 kept up from the rubber to the comb, or, what is the same thing, 
 of negative electricity from the comb to the rubber, or, since the 
 mode of expressing the fact is indifferent, we may, if we please, 
 describe it as consisting of a positive current in the one direction 
 combined with a negative current in the other the arithmetical 
 sum of these two imaginary currents being the actual current 
 observed. The action of the machine thus depends on the electri- 
 fication of the surface of the glass by the rubber, the convection of 
 this electrification, by the motion of the machine, to the comb and 
 the discharge of the electrification by the comb. 
 
 196.] The strength of the current produced depends on the 
 surface-density. of the electrification, the area of the electrified 
 surface and the number of turns in a minute. 
 
 The electromotive force of the machine is the excess of the 
 potential of the comb above that of the rubber. The most con- 
 venient test of the electromotive force of an electrical machine is 
 the length of the sparks which it will give. 
 
196. | ACTION OF THE SILK FLAPS. 157 
 
 During the passage of the electrified surface from the rubber to 
 the comb it is passing from places of low to places of high potential, 
 and is therefore acted on by a force in the direction opposite to 
 that of its motion. The work done in turning the machine there- 
 fore exceeds that necessary to overcome the friction of the rubber, 
 the axle, and other mechanical resistances by the electrical work 
 done in carrying the electricity from the rubber to the comb. 
 
 At every point of its course the electricity on the surface of the 
 glass plate is acted on by a force the value of which is measured 
 by the rate at which the potential varies from one point to another 
 of the surface. If this force exceeds a certain value it will cause 
 the electrification to slide along the surface of the plate, and this 
 will take place under the action of a much smaller force than that 
 which is required to remove the electricity from the surface. This 
 discharge along the surface of the plate may be seen when the 
 electric machine is worked in a dark room, and it is evident that 
 the electricity which thus flashes back is so much lost from the 
 principal current of the machine. 
 
 In order that the machine may work to the best advantage 
 this slipping back of the electricity must be prevented. The 
 slipping takes place whenever the rate of variation of the potential 
 from point to point of the surface exceeds a certain value. If by 
 any distribution of the electrification the rate of variation of the 
 potential can be kept just below this value all the way from the 
 rubber to the comb the electromotive force of the machine will have 
 its highest possible value. 
 
 In most electrical machines flaps of oiled silk are attached to 
 the rubber so that as the plate revolves the electrified surface as 
 it leaves the rubber is covered with the silk flap which extends 
 from the rubber nearly up to the comb. ‘These silk flaps become 
 negatively electrified and therefore adhere of themselves to the 
 surface of the glass. If in any part of the revolution of the plate, 
 the rate of increase of the potential is so great that a slipping 
 back of the electrification occurs, the positive electricity which so 
 slips back neutralizes part of the negative electrification of the 
 silk flap and so raises the electric potential just behind the place 
 where the slipping occurred. In this way the slope of the electric 
 potential is equalized and the electromotive force of the machine is 
 raised to its highest possible value, so as to give the longest sparks 
 which a machine of given dimensions can furnish, 
 
 When the silk flaps are removed the slope of the potential 
 
158 THE REVOLVING DOUBLER. [197% 
 
 becomes much greater close to the rubber than at any other place, 
 the electricity slips back on the glass just as it leaves the rubber 
 and very little electricity, and that at a comparatively low potential, 
 reaches the comb. 
 
 In the best machines, in which the slope of the potential is 
 uniform from the rubber to the comb, the length of the spark 
 must depend principally on the distance between the rubber and 
 the comb. Hence a machine which, like Winter’s, has the rubber 
 and the comb at opposite extremities of a diameter of the plate will 
 give a longer spark than one from a machine whose plate has the 
 same diameter but which like Cuthbertson’s has two rubbers and 
 two combs, the distance between each rubber and its comb being a 
 quadrant. 
 
 On Machines producing Electrification by Mechanical Work. 
 
 197*.] In the ordinary frictional electrical machine the work done 
 in overcoming friction is far greater than that done in increasing 
 the electrification. Hence any arrangement by which the elec- — 
 trification may be produced entirely by mechanical work against 
 the electrical forces is of scientific importance if not of practical 
 value. The first machine of this kind seems to have been Nicholson’s 
 Revolving Doubler, described in the PAzlosophical Transactions for 
 1788 as ‘an instrument which by the turning of a Winch produces 
 the two states of Electricity without friction or communication 
 with the Earth.’ 
 
 198*.] It was by means of the revolving doubler that Volta 
 succeeded in developing from the electrification of the pile an 
 electrification capable of affecting his electrometer. Instruments 
 on the same principle have been invented independently by Mr. 
 C. F. Varley*, and Sir W. Thomson. 
 
 These instruments consist essentially of insulated conductors of 
 various forms, some fixed and others moveable. ‘The moveable 
 conductors are called Carriers, and the fixed ones may be called 
 Inductors, Receivers, and Regenerators. The inductors and receivers 
 are so formed that when the carriers arrive at certain points in 
 their revolution they are almost completely surrounded by a con- 
 ducting body. As the inductors and receivers cannot completely 
 surround the carrier and at the same time allow it to move freely 
 in and out without a complicated arrangement of moveable pieces, 
 the instrument is not theoretically perfect without a pair of re- 
 
 * Specification of Patent, Jan, 27, 1860, No. 206. 
 
198*.] THE REVOLVING DOUBLER. 159 
 
 generators, which store up the small amount of electricity which 
 the carriers retain when they emerge from the receivers. 
 
 For the present, however, we may suppose the inductors and 
 receivers to surround the carrier completely when it is within them, 
 in which case the theory is much simplified. 
 
 We shall suppose the machine to consist of two inductors 4 and 
 
 —C, and of two receivers B and D, with two carriers / and G. 
 
 Suppose the inductor dA to be positively electrified so that its 
 potential is 4A, and that the carrier / is within it and is at 
 potential / Then, if Q is the coefficient of induction (taken 
 positive) between 4 and JS, the quantity of electricity on the carrier 
 will be Q (7— A). 
 
 If the carrier, while within the inductor, is put in connexion with 
 the earth, then / = 0, and the charge on the carrier will be — Q4, 
 a negative quantity. Let the carrier be carried round till it is 
 within the receiver 4, and let it then come in contact with a spring 
 so as to be in electrical connexion with &. It will then, as was 
 shewn in Art. 20, become completely discharged, and will com- 
 municate its whole negative charge to the receiver J. 
 
 The carrier will next enter the inductor C, which we shall suppose 
 charged negatively. While within C it is put in connexion with 
 the earth and thus acquires a positive charge, which it carries off 
 and communicates to the receiver D, and so on. 
 
 In this way, if the potentials of the inductors remain always 
 constant, the receivers B and D receive successive charges, which 
 are the same for every revolution of the carrier, and thus every 
 revolution: produces an equal increment of electricity in the re- 
 celvers. 
 
 But by putting the inductor 4 in communication with the re- 
 ceiver D, and the inductor C with the receiver 2B, the potentials 
 of the inductors will be continually increased, and the quantity 
 of electricity communicated to the receivers in each revolution will 
 continually increase. 
 
 For instance, let the potential of 4 and D be U, and that of B 
 and C, VY, and when the carrier is within 4 let the charge on 4 
 and C be w, and that on the carrier z, then, since the potential 
 of the carrier is zero, being in contact with earth, its charge is 
 z=—QU. The carrier enters B with this charge and communicates 
 it to B. If the capacity of B and C is B, their potential will be 
 
 changed from / to /. = | 
 
160 THOMSONS REPLENISHER. [198*. 
 
 If the other carrier has at the same time carried a charge —QV 
 from C to D, it will change the potential of 4 and O from U to 
 
 bon V, if Q’ is the coefficient of induction between the carrier 
 
 and C,and A the capacity of 4 and D. If, therefore, U, and /,, 
 be the potentials of the two inductors after ~ half revolutions, and 
 On414 %+1 half revolutions, 
 
 Q’ 
 Us — Ue SP ah” 
 
 Q 
 Past = lu — Bun 
 
 lA 
 
 If we write p? = = and gq? = ee » we find 
 
 DM art Ql nay = (DUn+ Qn) (1—29) = (U0 + 97%) (1—pq)"*3, 
 DM 41 — Vong = (DU n—-Gln) (1 +09) = (pUp—97,) (1 +.9)"*1. 
 Hence 
 
 eee (tpg) + (1 tpg +2711 — Pq)" — (1 +79)" 
 
 Fa = UY pay (1 +20)" + To Yl pa) + (pf 
 
 It appears from these equations that the quantity pU+qV/ con- 
 tinually diminishes, so that whatever be the initial state of elec- 
 trification the receivers are ultimately oppositely electrified, so that 
 the potentials of 4 and B are in the ratio of g to —p. 
 
 On the other hand, the quantity »U—qV continually increases, 
 so that, however little »U may exceed or fall short of gV at first,. 
 the difference will be increased in a geometrical ratio in each 
 revolution till the electromotive forces become so great that the 
 insulation of the apparatus is overcome. 
 
 Instruments of this kind may be used for various purposes. 
 
 For producing a copious supply of electricity at a high potential, 
 as is done by means of Mr. Varley’s large machine. 
 
 For adjusting the charge of a condenser, as in the case of 
 Thomson’s electrometer, the charge of which can be increased or 
 diminished by a few turns of a very small machine of this kind, 
 which is then called a Replenisher. 
 
 For multiplying small differences of potential. The inductors 
 may be charged at first to an exceedingly small potential, as, for 
 instance, that due to a thermo-electric pair, then, by turning the 
 machine, the difference of potentials may be continually multiplied 
 
200. | WATER DROPPING ACCUMULATOR. 161 
 
 ‘till it becomes capable of measurement by an ordinary electrometer. 
 By determining by experiment the ratio of increase of this difference 
 due to each turn of the machine, the original electromotive force 
 with which the inductors were charged may be deduced from the 
 number of turns and the final electrification. 
 
 In most of these instruments the:carriers are made to revolve 
 about an axis and to come into the proper positions with respect, 
 to the inductors by turning an axle. The connexions are made by 
 means of springs so placed that the carriers come in contact with 
 them at the proper instants. 
 
 199*.| Sir W. Thomson, however, has constructed a machine 
 for multiplying electrical charges in which the carriers are drops of 
 water falling out of the inside of an inductor into an insulated 
 receiver. The receiver is thus continually supplied with electricity 
 of opposite sign to that of the inductor. If the inductor is electrified 
 positively, the receiver will receive a continually increasing: charge 
 of negative electricity. 
 
 The water is made to escape from the receiver by means of a 
 funnel, the nozzle of which is almost surrounded by the metal of 
 the receiver. The drops falling from this nozzle are therefore 
 nearly free from electrification. Another inductor and receiver of 
 the same construction are arranged so that the inductor of the 
 one system is in connexion with the receiver of the other. The 
 rate of increase of charge of the receivers is thus no longer constant, 
 but increases in a geometrical progression with the time, the 
 charges of the two receivers being of opposite signs. This increase 
 goes on till the falling drops are so diverted from their course by 
 the electrical action that they fall outside of the receiver or even 
 strike the inductor. 
 
 In this instrument the energy of the electrification is drawn 
 from that of the falling drops. 
 
 200.| In Holtz’s ‘ Influence-Machine’ a plate of varnished glass 
 is made to rotate in front of a fixed plate of varnished glass. The 
 inductors consist of two pointed pieces of card sometimes covered 
 with tin foil and placed on the further side of the fixed plate so 
 that their points are at opposite extremities of a diameter. Holes 
 are cut in the fixed plate opposite the points of the inductors. The 
 electrodes are first put in connexion with each other and the 
 machine is set in rotation. One of the inductors is then electrified, 
 either by an ordinary machine or by an excited piece of ebonite. 
 
 * Proc. R. S., June 20, 1867. 
 M 
 
162 HOLTZS MACHINE. [zor 
 
 _ Let us suppose it electrified positively. The comb in front of the 
 charged inductor immediately begins to glow and discharges nega- 
 tive electricity against the rotating disk. This negative electrifica- 
 tion is carried round by the disk to the other side where it is free 
 i Coulee inductor. The other inductor 
 
 gricity from its point and becomes 
 itself ee el he comb of the negative electrode 
 al positive grep sits wit 
 
 
 
 
 
 
 
 
 t h is carried round-the disk on 
 the othen side back to! ositiye electrode. In this way there 
 
 is kept upan/electric current fy yom the positive to the negative 
 electrode. A rushing noise is“h Reard and in the dark a glow is 
 seen extending itself fromthe positive comb over the surface of the 
 rotating disk in the direction opposite to its motion. If the elec- 
 trodes are now gradually separated a succession of sparks will pass 
 between them. 
 
 Influence Machine. 
 
 1865. Holtz exhibited his machine to the Berlin Academy, April 
 1865. 8 to 10 cm. diam. 
 
 1866. Topler, metal inductors, two metal carriers on a glass 
 disk, 
 
 1867. Tdpler’s multiple machine, 8 rotating disks, 32 em. diam. 
 sparks 6 to 9 cm. 
 
 1867. Holtz with two disks rotating oppositely. 
 1868. Kundt. 
 Carré, inductor disk 38 em. induced 49, spark 15 to 18. 
 
 201*.| In the electrical machines already described sparks occur 
 whenever the carrier comes in contact with a conductor at a 
 different potential from its own. 
 
 Now we have shewn that whenever this occurs there is a loss 
 of energy, and therefore the whole work employed in turning the 
 machine is not converted into electrification in an available form, 
 but part is spent in producing the heat and noise of electric 
 sparks. 
 
 I have therefore thought it desirable to shew how an electrical 
 machine may be constructed which is not subject to this loss of 
 efficiency. I do not propose it as a useful form of machine, but 
 as an example of the method by which the contrivance called in 
 heat-engines a regenerator may be applied to an electrical machine 
 to prevent loss of work. 
 
201*.] MACHINE WITHOUT SPARKS, 163 
 
 In the figure let 4, B, C, A’, B’, C’ represent hollow fixed 
 conductors, so arranged that the carrier P passes in succession 
 within each of them. Of these 4, 4’ and B, B’ nearly surround the 
 ' earrier when it is at the middle point of its passage, but C, C’ do not 
 cover it so much. 
 
 We shall suppose 4A, B, C to be -¢onnected with a Leyden jar 
 of great capacity at potential /, wor Bb’, & to be connected with 
 another jar at potential —/’. § 
 
 P is one of the carriers movifig in a “circle from 4 to C’, &e. 
 and touching in its course cer- 
 tain springs, of which @ and 
 a’ are connected with 4 and 4’ 
 respectively, and ¢, e’ are con- 
 nected with the earth. 
 
 Let us suppose that when 
 the carrier P is in the middle 
 of A the coefficient of induction 
 between P and 4 is— A. The 
 capacity of P in this position 
 is greater than A, since it is not 
 completely surrounded by the 
 receiver 4. Let it be 4-+a. 
 
 Then if the potential of P is U, and that of 4, V, the charge 
 on P will be (4+a) U—AYP. 
 
 Now let P be in contact with the spring @ when in the middle 
 of the receiver 4, then the potential of P is V, the same as that 
 of A, and its charge is therefore a/. 
 
 If P now leaves the spring @ it carries with it the charge a/. 
 As P leaves A its potential diminishes, and it diminishes still more 
 when it comes within the influence of C’, which is negatively 
 electrified. 
 
 If when P comes within C its coefficient of induction on C is 
 —C’, and its capacity is C’+c’, then, if U is the potential of P 
 the charge on / is 
 
 
 
 (1 4+6¢)U+CV=a?P. 
 
 If CV’=a V, 
 then at this point U the potential of P will be reduced to zero. 
 
 Let P at this point come in contact with the spring e which is 
 connected with the earth. Since the potential of P is equal to that 
 of the spring there will be no spark at contact. 
 
 This conductor C’, by which the carrier is enabled to be connected 
 
 M 2 
 
164 MACHINE WITHOUT SPARKS, [201*. 
 
 to earth without a spark, answers to the contrivance called a 
 regenerator in heat-engines. We shall therefore call it a Re- 
 generator. 
 
 Now let P move on, still in contact with the earth-spring é’, till 
 if comes into the middle of the inductor B, the potential of which 
 is V. If —B is the coefficient of induction between P and B at 
 this point, then, since U=0 the charge on P will be —BY/. 
 
 When P moves away from the earth-spring it carries this charge 
 with it. As it moves out of the positive inductor B towards the 
 
 negative receiver 4’ its potential will be increasingly negative. At 
 the middle of 4’, if it retained its charge, its be would be 
 AV + BV 
 ae A a a’ 2 
 and if BV is greater than a’V’ its numerical value will be greater 
 than that of V’. Hence there is some point before P reaches the 
 middle of A’ where its potential is —/V’. At this point let it come 
 in contact with the negative receiver-spring a’. There will be no 
 spark since the two bodies are at the same potential. Let P move 
 on to the middle of 4’, still in contact with the spring, and therefore 
 at the same potential with 4’. During this motion it communicates 
 a negative charge to 4’. At the middle of 4’ it leaves the spring 
 and carries away a charge —a’/” towards the positive regenerator 
 C, where its potential is reduced to zero and it touches the earth- 
 spring e. It then slides along the earth-spring into the negative 
 inductor B’, during which motion it acquires a positive charge BV” 
 which it finally communicates to the positive receiver 4, and the 
 cycle of operations is repeated. 
 
 During this cycle the positive receiver has lost a charge aV and 
 gained a charge B’V’. Hence the total gain of positive electricity 
 is BV’ —aV. 
 
 Similarly the total gain of negative cle is BV—ad'y’, 
 
 By making the inductors so as to be as close to the surface of 
 the carrier as is consistent with insulation, B and B’ may be made 
 large, and by making the receivers so as nearly to.surround the 
 carrier when it is within them, « and a’ may be made very small, 
 and then the charges of both the Leyden jars will be increased in 
 every revolution. 
 
 The conditions to be fulfilled by the regenerators are 
 
 CV'=aV¥, and CV=a' lV’. 
 
 Since a and a’ are small the regenerators do not require to be 
 
 either large or very close to the carriers. 
 
po2".| TORSION BALANCE, 165 
 
 Coulomb's Torsion Balance. 
 
 202*.| A great number of the experiments by which Coulomb 
 established the fundamental laws of electricity were made by mea- 
 suring the force between two small spheres charged with electricity, 
 one of which was fixed while the other was held in equilibrium by 
 two forces, the electrical action between the spheres, and the 
 torsional elasticity of a glass fibre or metal wire. 
 
 The balance of torsion consists of a horizontal arm of gum-lac, 
 suspended by a fine wire or glass fibre, and carrying at one end a 
 little sphere of elder pith, smoothly gilt. The suspension wire is 
 fastened above to the vertical axis of an arm which can be moved 
 round a horizontal graduated circle, so as to twist the upper end 
 of the wire about its own axis any number of degrees. 
 
 The whole of this apparatus is enclosed in a case. Another little 
 sphere is so mounted on an insulating stem that it can be charged 
 and introduced into the case through a hole, and brought so that 
 its centre coincides with a definite point in the horizontal circle 
 described by the suspended sphere. ‘The position of the suspended 
 sphere is ascertained by means of a graduated circle engraved on 
 the cylindrical glass case of the instrument. 
 
 Now suppose both spheres charged, and the suspended sphere 
 in equilibrium in a known position such that the torsion-arm makes 
 an angle 6 with the radius through the centre of the fixed sphere. 
 The distance of the centres is then 2 sin $0, where a is the radius 
 of the torsion-arm, and if / is the force between the spheres the 
 moment of this force about the axis of torsion is Ya cos 46. 
 
 Let both spheres be completely discharged, and let the torsion- 
 arm now bein equilibrium at an angle @ with the radius through 
 the fixed sphere. 
 
 Then the angle through which the electrical force twisted the 
 torsion-arm must have been 0—q¢, and if I is the moment of 
 the torsional elasticity of the fibre, we shall have the equation 
 
 Facos$ 0 = M(d—¢). 
 
 Hence, if we can ascertain W/, we can determine J; the actual 
 force between the spheres at the distance 2¢ sin 30. 
 
 To find 1/, the moment of torsion, let J be the moment of inertia 
 of the torgion-arm, and 7’ the time of a double vibration of the arm 
 under the action of the torsional elasticity, then 
 
 4n?J 
 Ma te. 
 
 
 
166 INFLUENCE OF THE CASE. [202*. 
 
 In all electrometers it is of the greatest importance to know 
 what force we are measuring. The force acting on the suspended 
 sphere is due partly to the direct action of the fixed sphere, but 
 partly also to the electrification, if any, of the sides of the case. 
 
 If the case is made of glass it is impossible to determine the 
 electrification of its surface otherwise than by very difficult mea- 
 surements at every point. If, however, either the case is made 
 of metal, or if a metallic case which almost completely encloses the 
 apparatus is placed as a screen between the spheres and the glass 
 case, the electrification of the inside of the metal screen will depend 
 entirely on that of the spheres, and the electrification of the glass 
 case will have no influence on the spheres. In this way we may 
 avoid any indefiniteness due to the action of the case. 
 
 To illustrate this by an example in which we can calculate all 
 the effects, let us suppose that the case is a sphere of radius 4, 
 that the centre of motion of the torsion-arm coincides with the 
 centre of the sphere and that its radius is a; that the charges on 
 the two spheres are /, and /, and that the angle between their 
 positions is 6; that the fixed sphere is at a distance a, from the 
 centre, and that 7 is the distance between the two small spheres. 
 
 Neglecting for the present the effect of induction on the dis- 
 tribution of electricity on the small spheres, the force between 
 them will be a repulsion 
 
 
 
 and the moment of this force round a vertical axis through the 
 centre will be 
 
 HE, aa, sin 0 
 
 Seal 
 
 The image of /, due to the spherical surface of the case is a point 
 : : boeke 
 in the same radius at a distance a with a charge Ssh , and the 
 1 1 
 moment of the attraction between H and this image about the axis 
 of suspension is 
 32 
 
 j aS sin 0 
 EE, ——_____7———77— 
 a {a*—2 7 eos6 +t 
 ay ay 
 — ER, aa, sin 0 
 
 
 
 ; aa 0242.) 
 }1 2 cos. 0 48 7a ae 
 
203*.| ATTRACTED DISK ELECTROMETERS. 167 
 
 If J, the radius of the spherical case, is large compared with a 
 and a,, the distances of the spheres from the centre, we may neglect 
 the second and third terms of the factor in the denominator. The 
 whole moment tending to turn the torsion-arm may then be written 
 
 ‘ 1 1 
 EE, aa, sin 0 = ~ a = M(d—}¢). 
 
 : EHlectrometers for the Measurement of Potentials. 
 
 203*.| In all electrometers the moveable part is a body charged 
 with electricity, and its potential is different from that of certain 
 of the fixed parts round it. When, as in Coulomb’s method, an 
 insulated body having a certain charge is used, it is the charge 
 which is the direct object of measurement. We may, however, 
 connect the balls of Coulomb’s electrometer, by means of fine wires, 
 with different conductors. The charges of the balls will then 
 depend on the values of the potentials of these conductors and on 
 the potential of the case of the instrument. The charge on each 
 ball will be approximately equal to its radius multiplied by the 
 excess of its potential over that of the case of the instrument, 
 provided the radii of the balls are small compared with their 
 distances from each other and from the sides or opening of the 
 case. 
 
 Coulomb’s form of apparatus, however, is not well adapted for 
 measurements of this kind, owing to the smallness of the force 
 between spheres at the proper distances when the difference of 
 potentials is small. A more convenient form is that of the 
 Attracted Disk Electrometer. The first electrometers on this 
 principle were constructed by Sir W. Snow Harris*. They have 
 since been brought to great perfection, both in theory and con- 
 struction, by Sir W. Thomson f. 
 
 When two disks at different potentials are brought face to face 
 with a small interval between them there will be a nearly uniform 
 electrification on the opposite faces and very little electrification 
 on the backs of the disks, provided there are no other conductors 
 or electrified bodies in the neighbourhood. ‘The charge on the 
 positive disk will be approximately proportional to its area, and to 
 the difference of potentials of the disks, and inversely as the distance 
 
 * Phil. Trans. 1834. 
 + See an excellent report on Electrometers by Sir W. Thomson. Report of the 
 British Association, Dundee, 1867. 
 
168 PRINCIPLE OF THE GUARD RING. ‘[204*. 
 
 between them. Hence, by making the areas of the disks large 
 and the distance between them small, a small difference of potential 
 may give rise to a measurable force of attraction. 
 
 204.*.| The addition of the guard-ring to the attracted disk 3 is 
 one of the chief improvements which Sir W. Thomson has made 
 on the apparatus. 
 
 Instead of suspending the whole of one of the disks and deter- 
 mining the force acting upon it, a central portion of the disk is 
 separated from the rest to form the attracted disk, and the outer 
 ring forming the remainder of the disk is fixed. In this way the 
 
 COUNTERPOISE 
 
 
 
 
 
 TWO BLACK OOTS 
 
 
 
 iN LENS 
 
 Ale 
 rte SUSPENDED 
 if 
 
 be 
 \7 D/ a 
 \ GUARD RING Y, 
 
 INSULATED STEM 
 
 WORKED BY 
 MICROMETER SCREW 
 
 Fig. 43. 
 
 force is measured only on that part of the disk where it is most 
 regular, and the want of uniformity of the electrification near the 
 edge is of no importance, as it occurs on the guard-ring and not 
 on the suspended part of the disk. 
 
 Besides this, by connecting the guard-ring with a metal case 
 surrounding the back of the attracted disk and all its suspending 
 apparatus, the electrification of the back of the disk is rendered 
 impossible, for it is part of the inner surface of a closed hollow 
 conductor all at the same potential. 
 
 Thomson's Asbolute Electrometer therefore consists essentially © 
 
204". | ABSOLUTE ELECTROMETER. 169 
 
 of two parallel plates at different potentials, one of which is made 
 so that a certain area, no part of which is near the edge of the 
 plate, is moveable under the action of electric force. To fix our 
 ideas we may suppose the attracted disk and guard-ring uppermost. 
 The fixed disk is horizontal, and is mounted on an insulating stem 
 which has a measurable vertical motion given to it by means of 
 a micrometer screw. The guard-ring is at least as large as the 
 fixed disk; its lower surface is truly plane and parallel to the fixed 
 disk. A delicate balance is erected on the guard-ring to which 
 is suspended a light moveable disk which almost fills the circular 
 aperture in the guard-ring without rubbing against its sides, The 
 lower surface of the suspended disk must be truly plane, and we 
 must have the means of knowing when its plane coincides with that 
 of the lower surface of the guard-ring, so as to form a single plane 
 interrupted only by the narrow interval between the disk and its 
 guard-ring. 
 
 For this purpose the lower disk is screwed up till it is in contact 
 with the guard-ring, and the suspended disk is allowed to rest 
 upon the lower disk, so that its lower surface is in the same plane 
 as that of the guard-ring. Its position with respect to the guard- 
 ring is then ascertained by means of a system of fiducial marks. 
 Sir W. Thomson generally uses for this purpose a black hair 
 attached to the moveable part. This hair moves up or down just 
 in front of two black dots on a white enamelled ground and is 
 viewed along with these dots by means of a plano convex lens with 
 the plane side next the eye. If the hair as seen through the lens 
 appears straight and bisects the interval between the black dots 
 it is said to be in its sighted position, and indicates that the sus- 
 pended disk with which it moves is in its proper position as regards 
 height. The horizontality of the suspended disk may be tested by 
 comparing the reflexion of part of any object from its upper surface 
 with that of the remainder of the same object from the upper 
 surface of the guard-ring. 
 
 The balance is then arranged so that when a known weight 1s 
 placed on the centre of the suspended disk it is in equilibrium 
 in its sighted position, the whole apparatus being freed from 
 electrification by putting every part in metallic communication. 
 A metal case is placed over the guard-ring so as to enclose the 
 balance and suspended disk, sufficient apertures being left to see 
 the fiducial marks. 
 
 The guard-ring, case, and suspended disk are all in metallic 
 
170 ABSOLUTE ELECTROMETER. [204*. 
 
 communication with each other, but are insulated from the other 
 parts of the apparatus. 
 
 Now let it be required to measure the difference of potentials 
 of two conductors. The conductors are put In communication with 
 the upper and lower disks respectively by means of wires, the 
 weight is taken off the suspended disk, and the lower disk is 
 moved up by means of the micrometer screw till the electrical 
 attraction brings the suspended disk down to its sighted position. 
 We then know that the attraction between the disks is equal to 
 the weight which brought the disk to its sighted position. 
 
 If V be the numerical value of the weight, and g the force of 
 eravity, the force is Wg, and if A is the area of the suspended 
 disk, D the distance between the disks, and V the difference of the 
 potentials of the disks, 
 
 wy 24 
 PSn D4 
 
 or V=D WY) ele. 
 
 If the suspended disk is circular, of radius #, and if the radius of 
 the aperture of the guard-ring is #’, then 
 
 A=}n(R24+R)*, and V = 4D \/ 
 
 * Let us denote the radius of the suspended disk by &, and that of the aperture of 
 the guard-ring by #’, then the breadth of the annular interval between the disk and 
 the ring will be B = R’—R. 
 
 If the distance between the suspended disk and the large fixed disk is D, and the 
 difference of potentials between these disks is V, then (see Hlectricity and Magnetism, 
 Art. 201) the quantity of electricity on the suspended disk will be 
 
 
 
 R+R? RIK a 
 Py fe fee Wiehe SB a TE 2 
 @ 8D 8D D+a 
 where a= B a r a= 0.220635 (R’—R). 
 
 If the surface of the guard-ring is not exactly in the plane of the surface of 
 the suspended disk, let us suppose that the distance between the fixed disk and 
 the guard-ring is not D but D + 2= D’, then (see Electricity and Magnetism, Art. 225) 
 there will be an additional charge of electricity near the edge of the disk on 
 account of its height 2 above the general surface of the guard-ring. The whole 
 charge in this case is therefore 
 
 +24 AR? Rt. va aa 4n(R+ Rf’) 
 v i= sepa at, oe 
 oF. | S80 aay a DRT ae) a) cone 
 and in the expression for the attraction we must substitute for A, the area of the disk, 
 the corrected quantity 
 
 A 
 
 fu } RY+ R?_(R2_B) -"_ +8(R+ BY) (D'—D) logy ee 
 
205*.] SMALL ELECTROMOTIVE FORCES MEASURED. Lm 
 
 205*.| Since there is always some uncertainty in determining the 
 micrometer reading corresponding to D=0, and since any error 
 in the position of the suspended disk is most important when JD - 
 is small, Sir W. Thomson prefers to make all his measurements 
 depend on differences of the electromotive force ’. Thus, if V and 
 V’ are two potentials, and D and J” the corresponding distances, 
 
 : a Y =(D-D) nf BEY. 
 
 For instance, in order to measure the electromotive force of a 
 galvanic battery, two electrometers are used. 
 
 By means of a condenser, kept charged if necessary by a re- 
 plenisher, the lower disk of the principal electrometer is maintained 
 at a constant potential. This is tested by connecting the lower 
 disk of the principal electrometer with the lower disk of a secondary 
 electrometer, the suspended disk of which is connected with the 
 earth. The distance between the disks of the secondary elec- 
 trometer and the force required to bring the suspended disk to 
 its sighted position being constant, if we raise the potential of the 
 condenser till the secondary electrometer is in its sighted position, 
 we know that the potential of the lower disk of the principal 
 electrometer exceeds that of the earth by a constant quantity which 
 we may call /. 
 
 If we now connect the positive electrode of the battery to earth, 
 and connect the suspended disk of the principal electrometer to the 
 negative electrode, the difference of potentials between the disks 
 will be V + v, if v is the electromotive force of the battery. Let 
 D be the reading of the micrometer in this case, and let D’ be the 
 reading when the suspended disk is connected with earth, then 
 
 v = (D—D’ jy aL, 
 
 In this way a small electromotive force v may be measured 
 by the electrometer with the disks at conveniently measurable 
 distances. When the distance is too small a small change of 
 absolute distance makes a great change in the force, since the 
 
 
 
 
 
 where & = radius of suspended disk, 
 R’ = radius of aperture in the guard-ring, 
 D = distance between fixed and suspended disks, 
 D’ = distance between fixed disk and guard-ring, 
 a = 0.220635 (R’— BR). 
 When a is small compared with D we may neglect the second term, and when 
 D'—D is small we may neglect the last term. 
 
ie ABSOLUTE ELECTROMETER. [205*. 
 
 force varies inversely as the square of the distance, so that any 
 error in the absolute distance introduces a large error in the result 
 unless the distance is large compared with the limits of error of 
 the micrometer screw. 
 
 The effects of small irregularities of form im the surface of the 
 disks and of the interval between them diminish according to the 
 inverse cube and higher inverse powers of the distance, and what- 
 ever be the form of a corrugated surface, the emimences of which 
 just reach a plane surface, the electrical effect at any distance 
 which is considerable compared to the breadth of the corrugations, 
 is the same as that of a plane at a certain small distance behind 
 the plane of the tops of the eminences. 
 
 By means of the auxiliary electrification, tested by the auxiliary 
 electrometer, a proper interval between the disks is secured. 
 
 The auxiliary electrometer may be of a simpler construction, in 
 which there is no provision for the determination of the force 
 of attraction in absolute measure, since all that is wanted is to 
 secure a constant electrification. Such an electrometer may be 
 ealled a gauge electrometer. 
 
 This method of using an auxiliary electrification besides the elec- 
 trification to be measured is called the Heterostatic method of 
 electrometry, in opposition to the Idiostatie method in which the 
 whole effect is produced by the electrification to be measured. 
 
 In several forms of the attracted disk electrometer, the attracted 
 disk is placed at one end of an arm which is supported by being 
 attached to a platinum wire passing through its centre of gravity 
 and kept stretched by means of a spring. The other end of the 
 arm carries the hair which is brought to a sighted position: by 
 altering the distance between the disks, and so adjusting the force 
 of the electric attraction to a constant value. In these electro- 
 meters this force is not in general determined in absolute measure, 
 but is known to be constant, provided the torsional elasticity of 
 the platinum wire does not change. 
 
 The whole apparatus is placed in a Leyden jar, of which the inner 
 surface is charged and connected with the attracted disk and 
 guard-ring. The other disk is worked by a micrometer screw and 
 is connected first with the earth and then with the conductor whose 
 potential is to be measured. The difference of readings multiplied 
 by a constant to be determined for each electrometer gives the 
 potential required. 
 
eM) 
 
 206%. | MEASUREMENT OF POTENTIAL. 17 
 
 On the Measurement of Electric Potential. 
 
 206*.| In order to determine large differences of potential in ab- 
 solute measure we may employ the attracted disk electrometer, and 
 compare the attraction with the effect of a weight. Ifat the same 
 time we measure the difference of potential of the same conductors 
 by means of the quadrant electrometer, we shall ascertain the 
 absolute value of certain readings of the scale of the quadrant 
 electrometer, and in this way we may deduce the value of the scale 
 readings of the quadrant electrometer in terms of the potential 
 of the suspended part, and the moment of torsion of the suspension 
 apparatus. 
 
 To ascertain the potential of a charged conductor of finite size 
 we may connect the conductor with one electrode of the electro- 
 meter, while the other is connected to earth or to a body of 
 constant potential. The electrometer reading will give the potential 
 ‘of the conductor after the division of its electricity between it 
 and the part of the electrometer with which it is put in contact. 
 If K denote the capacity of the conductor, and KA’ that of this part 
 of the electrometer, and if VY, V” denote the potentials of these 
 bodies before making contact, then their common potential after 
 making contact will be 
 
 7 KV+KV’ 
 a ae 
 Hence the original potential of the conductor was 
 Be ERS A 
 [i aa ays Sd 7 
 P= Ti+ K (V—T’). 
 
 If the conductor is not large compared with the electrometer, 
 K’ will be comparable with XK, and unless we can ascertain the 
 values of K and XK” the second term of the expression will have 
 a doubtful value. But if we can make the potential of the electrode 
 of the electrometer very nearly equal to that of the body before 
 making contact, then the uncertainty of the values of K and K’ 
 will be of little consequence. 
 
 If we know the value of the potential of the body approximately, 
 we may charge the electrode by means of a ‘replenisher’ or other- 
 wise to this approximate potential, and the next experiment wiil 
 give a closer approximation. In this way we may measure the 
 potential of a conductor whose capacity is small compared with that 
 of the electrometer. | 
 
174 POTENTIAL AT ANY POINT IN THE AIR. [207%*. 
 
 To Measure the Potential at any Point in the Air. 
 
 207*.| Hirst Method. Place a sphere, whose radius is small com- 
 pared with the distance of electrified conductors, with its centre 
 at the given point. Connect it by means of a fine wire with the 
 earth, then insulate it, and carry it to an electrometer and ascertain 
 the total charge on the sphere. 
 
 Then, if VY be the potential at the given point, and a@ the 
 radius of the sphere, the charge of the sphere will be —Va= Q, 
 and if V’ be the potential of the sphere as measured by an 
 electrometer when placed in a room whose walls are connected 
 with the earth, then Q= Wa, 
 
 whence V+V'=0, 
 
 or the potential of the air at the point where the centre of the 
 sphere was placed is equal but of opposite sign to the potential of 
 the sphere after being connected to earth, then insulated, and 
 brought into a room. 
 
 This method has been employed by M. Delmann of Creuznach in 
 measuring the potential at a certain height above the earth's 
 surface *. , 
 
 Second Method. We have supposed the sphere placed at the 
 given point and first connected to earth, and then insulated, and 
 carried into a space surrounded with conducting matter at potential 
 ZeYO. 
 
 Now let us suppose a fine insulated wire carried from the elec- 
 trode of the electrometer to the place where the potential is to 
 be measured, Let the sphere be first discharged completely. This 
 may be done by putting it into the inside of a vessel of the same 
 metal which nearly surrounds it and making it touch the vessel. 
 Now let the sphere thus discharged be carried to the end of the 
 wire and made to touch it. Since the sphere is not eleetrified it 
 will be at the potential of the air at the place. If the electrode 
 wire is at the same potential it will not be affected by the contact, 
 but if the electrode is at a different potential it will by contact 
 with the sphere be made nearer to that of the air than it was 
 before. By a succession of such operations, the sphere being 
 alternately discharged and made to touch the electrode, the poten- 
 tial of the electrode of the electrometer will continually approach 
 that of the air at the given point. 
 
 * [Compare Art, 50.] 
 
208". | POTENTIAL OF A CONDUCTOR. 175 
 
 208*.| To measure the potential of a conductor without touching 
 it, we may measure the potential of the air at any point in the 
 neighbourhood of the conductor, and calculate that of the conductor 
 from the result. If there be a hollow nearly surrounded by the 
 conductor, then the potential at any point of the air in this hollow 
 will be very nearly that of the conductor. 
 
 In this way it has been ascertained by Sir W. Thomson that if 
 two hollow conductors, one of copper and the other of zine, are 
 in metallic contact, then the potential of the air in the hollow 
 surrounded by zine is positive with reference to that of the air in 
 the hollow surrounded by copper. 
 
 Third Method. If by any means we can cause a succession of 
 small bodies to detach themselves from the end of the electrode, 
 the potential of the electrode will approximate to that of the sur- 
 rounding air. This may be done by causing shot, filings, sand, or 
 water to drop out of a funnel or pipe connected with the electrode. 
 The point at which the potential is measured is that at which 
 the stream ceases to be continuous and breaks into separate parts 
 or drops. 
 
CHAPTER XII. 
 
 THE MEASUREMENT OF ELECTRIC RESISTANCE. 
 
 209*.| In the present state of electrical science, the determi- 
 nation of the electric resistance of a conductor may be considered 
 as the cardinal operation in electricity, in the same sense that the 
 determination of weight is the cardinal operation in chemistry. 
 
 The reason of this is that the determination in absolute measure 
 of other electrical magnitudes, such as quantities of electricity, 
 electromotive forces, currents, &c., requires in each case a com- 
 plicated series of operations, involving generally observations of 
 time, measurements of distances, and determinations of moments 
 of inertia, and these operations, or at least some of them, must be 
 repeated for every new determination, because it is impossible to pre- 
 serve a unit of electricity, or of electromotive force, or of current, in 
 an unchangeable state, so as to be available for direct comparison. 
 
 But when the electric resistance of a properly shaped conductor 
 of a properly chosen material has been once determined, it is found 
 that it always remains the same for the same temperature, so that 
 the conductor may be used as a standard of resistance, with which 
 that of other conductors can be compared, and the comparison of 
 two resistances is an operation which admits of extreme accuracy. 
 
 When the unit of electrical resistance has been fixed on, material 
 copies of this unit, in the form of ‘ Resistance Coils,’ are prepared 
 for the use of electricians, so that in every part of the world 
 electrical resistances, may be expressed in terms of the same unit. 
 These unit resistance coils are at present the only examples of 
 material electric standards which can be preserved, copied, and used 
 for the purpose of measurement. Measures of electrical capacity, 
 which are also of great importance, are still defective, on account 
 of the disturbing influence of electric absorption. 
 
 210*.| The unit of resistance may be an entirely arbitrary one, 
 as in the case of Jacobi’s Etalon, which was a certain coppals 
 
 * [Recent observations have shewn that it is far from easy to find a material 
 
 satisfying this condition. | 
 
ZER*, | UNIT OF RESISTANCE. Lit 
 
 wire of 22-4932 grammes weight, 7-61975 metres length, and 0-667 
 millimetres diameter. Copies of this have been made by Leyser of 
 Leipsig, and are to be found in different places. 
 
 According to another method the unit may be defined as the 
 resistance of a portion of a definite substance of definite dimensions. 
 Thus, Siemens’ unit is defined as the resistance of a column of 
 mercury of one metre long, and one square millimetre section, at 
 the temperature 0°C. 
 
 211*.| Finally, the unit may be defined with reference to the elec- 
 trostatic or the electromagnetic system of units. In practice the elec- 
 tromagnetic system is used in all telegraphic operations, and therefore 
 the only systematic units actually in use are those of this system. 
 
 In the electromagnetic system a resistance is a quantity homoge- 
 _ neous with a velocity, and may therefore be expressed as a velocitiy. 
 
 212*.] The first actual measurements on this system were made 
 by Weber, who employed as his unit one millimetre per second. 
 Sir W. Thomson afterwards used one foot per second as a unit, 
 but a large number of electricians have now agreed to use the 
 unit of the British Association, which professes to represent a 
 resistance which, expressed as a velocity, is ten millions of metres 
 per second. The magnitude of this unit is more convenient than 
 that of Weber’s unit, which is too small. It is sometimes referred 
 to as the B.A. unit, but in order to connect it with the name of 
 the discoverer of the laws of resistance, it is called the Ohm. 
 
 213*.] To recollect its value in absolute measure it is useful 
 to know that ten millions of metres is professedly the distance 
 from the pole to the equator, measured along the meridian of Paris. 
 A body, therefore, which in one second travels along a meridian 
 from the pole to the equator would have a velocity which, on the 
 electromagnetic system, is professedly represented by an Ohm. 
 
 I say professedly, because, if more accurate researches should 
 prove that the Ohm, as constructed from the British Association’s 
 material standards, is not really represented by this velocity, elec- 
 tricians would not alter their standards*, but would apply a cor- 
 rection. In the same way the metre is professedly one ten-millionth 
 of a certain quadrantal are, but though this is found not to be 
 exactly true, the length of the metre has not been altered, but the 
 dimensions of the earth are expressed by a less simple number. 
 
 According to the system of the British Association, the absolute 
 walue of the unit is originally chosen so as to represent as nearly 
 
 * [Electricians have scarcely acted up to this principle in the reform of the Ohm.] 
 
 N 
 
178 STANDARD RESISTANCE COILS. epedig 
 
 as possible a quantity derived from the electromagnetic absolute 
 system. 
 
 214*.| When a material unit representing this abstract quantity 
 has been made, other standards are constructed by copying this unit, 
 a process capable of extreme accuracy—of much greater accuracy 
 than, for instance, the copying of foot-rules from a standard foot. 
 
 These copies, made of the most permanent materials, are dis- 
 tributed over all parts of the world, so that it is not lkely that 
 any difficulty will be found in obtaining copies of them if the 
 original standards should he lost. 
 
 But such units as that of Siemens can without very great 
 labour be reconstructed with considerable accuracy, so that as the 
 relation of the Ohm to Siemens unit is known, the Ohm can be 
 reproduced even without having a standard to copy, though the 
 labour is much greater and the accuracy 
 much less than by the method of copying. 
 
 Finally, the Ohm may be reproduced by 
 the electromagnetic method by which it 
 was originally determined. This method, 
 which is considerably more laborious than 
 the determination of a foot from the seconds 
 pendulum, is probably inferior in accuracy 
 to that last mentioned. On the other hand, 
 the determination of the electromagnetic 
 unit in terms of the Ohm with an amount 
 of accuracy corresponding to the progress 
 of electrical science, is a most important 
 physical research and well worthy of being 
 repeated. 
 
 The actual resistance coils constructed 
 to represent the Ohm were made of an 
 alloy of two parts of silver and one of pla- 
 tinum in the form of wires from -5 milli- 
 metres to -8 millimetres diameter, and from 
 one to two metres in length. These wires 
 were soldered to stout copper electrodes. The wire itself was 
 covered with two layers of silk, imbedded in solid paraffin, and 
 enclosed in a thin brass case, so that it can be easily brought to 
 a temperature at which its resistance is accurately one Ohm. 
 This temperature is marked on the insulating support of the coil. 
 (See Fig. 44.) 
 
 
 
 
 
 
 
 
 
 
 
215*.] FORMS OF RESISTANCE COILS. 179 
 
 On the Forms of Resistance Coils. 
 
 215*.| A Resistance Coil is a conductor capable of being easily 
 placed in the voltaic circuit, so as to introduce into the circuit 
 a known resistance. | 
 
 The electrodes or ends of the coil must be such that no appre- 
 ciable error may arise from the mode of making the connexions 
 For resistances of considerable magnitude it is sufficient that the 
 electrodes should be made of stout copper wire or rod well amal- 
 gamated with mercury at the ends, and that the ends should be 
 made to press on flat amalgamated copper surfaces placed in 
 mereury cups. 
 
 For very great resistances it is sufficient that the electrodes 
 should be thick pieces of brass, and that the connexions should 
 be made by inserting a wedge of brass or copper into the interval 
 between them. This method is found very convenient. 
 
 The resistance coil itself consists of a wire well covered with 
 silk, the ends of which are soldered permanently to the elec- 
 trodes. 
 
 The coil must be so arranged that its temperature may be easily 
 observed. or this purpose the wire is coiled on a tube and 
 covered with another tube, so that it may be placed in a vessel 
 of water, and that the water may have access to the inside and the 
 outside of the coil. 
 
 To avoid the electromagnetic effects of the current in the coil 
 the wire is first doubled back on itself and then coiled on the tube, 
 so that at every part of the coil there are equal and opposite 
 currents in the adjacent parts of the wire. 
 
 When it is desired to keep two coils at the same temperature the 
 wires are sometimes placed side by side and coiled up together. 
 This method is especially useful when it is more important to 
 secure equality of resistance than to know the absolute value of 
 the resistance, as in the case of the equal arms of Wheatstone’s 
 Bridge (Art. 221). 
 
 When measurements of resistance were first attempted, a resist- 
 ance coil, consisting of.an uncovered wire coiled in a spiral groove 
 round a cylinder of insulating material, was much used. It was 
 called a Rheostat. The accuracy with which it was found possible 
 to compare resistances was soon found to be inconsistent with the 
 use of any instrument in which the contacts are not more perfect 
 than can be obtained in the rheostat. The rheostat, however, is 
 
 N 2 
 
180 RESISTANCE BOXES. Pens 
 
 still used for adjusting the resistance where accurate measurement 
 is not required. 
 
 Resistance coils are generally made of those metals whose resist- 
 ance is greatest and which vary least with temperature. German 
 silver fulfils these conditions very well, but some specimens are 
 found to change their properties during the lapse of years. Hence 
 for standard coils, several pure metals, and also an alloy of platinum 
 and silver, have been employed, and the relative resistance of these 
 during several years has been found constant up to the limits of 
 modern accuracy. 
 
 216*.] For very great resistances, such as several millions of 
 Ohms, the wire must be either very long or very thin, and the 
 construction of the coil is expensive and difficult. Hence tellurium 
 
 and selenium have been proposed as materials for constructing — 
 
 standards of great resistance. A very Ingenious and easy method 
 of construction has been lately proposed by Phillipst. On a piece 
 of ebonite or ground glass a fine pencil-line is drawn. The ends 
 of this filament of plumbago are connected to metallic electrodes, 
 and the whole is then covered with insulating varnish. If it 
 should be found that the resistance of such a pencil-line remains 
 constant, this will be the best method of obtaining a resistance of 
 several millions of Ohms. 
 
 217*.] There are various arrangements by which resistance coils 
 may be easily introduced into a circuit. 
 
 For instance, a series of coils of which the resistances are 1, 2, 
 4, 8, 16, &c., arranged according to the powers of 2, may be placed 
 
 PO ag 
 Dob 
 aOuu ue 
 
 G+ 32 1G 
 
 
 
 Fig. 45. 
 The electrodes consist of stout brass plates, so arranged on the 
 
 * [More recent experiments indicate a small change in resistance in course of time. ] 
 + Phil. Mag., July, 1870, 
 
248", | RESISTANCE BOXES. 181 
 
 outside of the box that by inserting a brass plug or wedge between 
 two of them as a shunt, the resistance of the corresponding coil 
 may be put out of the circuit. This arrangement was introduced 
 by Siemens. 
 
 Each interval between the electrodes is marked with the resist- 
 ance of the corresponding coil, so that if we wish to make the 
 resistance box equal to 107 we express 107 in the binary scale as 
 644+32+8+2+41 or 1101011. We then take the plugs out of 
 the holes corresponding to 64, 32, 8, 2 and 1, and leave the plugs 
 in 16 and 4. 
 
 This method, founded on the binary scale, is that in which the 
 smallest number of separate coils is needed, and it is also that 
 which can be most readily tested. For if we have another coil 
 equal to 1 we can test the equality of 1 and 1’, then that of 141’ 
 and 2, then that of 1+ 1’+2 and 4, and so on. 
 
 The only disadvantage of the arrangement is that it requires 
 a familiarity with the binary scale of notation, which is not 
 generally possessed by those accustomed to express every number 
 in the decimal scale. 
 
 218*.] A box of resistance coils may be arranged in a different 
 way for the purpose of mea- 
 suring conductivities instead of 
 resistances. 
 
 The coils are placed so that 
 one end of each is connected 
 with a long thick piece of 
 metal which forms one elec- 
 trode of the box, and the other 
 end is connected with a stout piece of brass plate as in the former 
 case. ) 
 The other electrode of the box is a long brass plate, such that 
 by inserting brass plugs between it and the electrodes of the coils 
 it may be connected to the first electrode through any given set of 
 coils. The conductivity of the box is then the sum of the con- 
 ductivities of the coils. 
 
 In the figure, in which the resistances of the coils are 1, 2, 4, &e., 
 and the plugs are inserted at 2 and 8, the conductivity of the box 
 is }+4 = 2, and the resistance of the box is therefore = or 1-6. 
 
 This method of combining resistance coils for the measurement 
 of fractional resistances was introduced by Sir W. Thomson under 
 the name of the method of multiple arcs. (See Art. 158.) 
 
 
 
182 THE COMPARISON OF RESISTANCES. Beioy: 
 
 On the Comparison of Resistances. 
 
 219*.] If # is the electromotive force of a battery, and & the 
 resistance of the battery and its connexions, including the galvan- 
 ometer used in measuring the current, and if the strength of the 
 current is J when the battery connexions are closed, and 4, J, 
 when additional resistances 7,, 7, are introduced into the circuit, 
 then, by Ohm’s Law, 
 
 H=IR= if (2 +7) I, (R+7,). 
 
 Eliminating /, the electromotive force of the battery, and R 
 the resistance of the battery and its connexions, we get Ohm’s 
 formula a (I-L)Z 
 
 Pom PTE i 
 This method requires a measurement of the ratios of J, Z and J,, 
 and this implies a galvanometer graduated for absolute mea- 
 surements. 
 
 If the resistances 7, and 7, are equal, then J, and J, are equal, 
 and we can test the equality of currents by a galvanometer which 
 is not capable of determining their ratios. 
 
 But this is rather to be taken as an example of a faulty method 
 than as a practical method of determining resistance. The electro- 
 motive force # cannot be maintained rigorously constant, and the 
 internal resistance of the battery is also exceedingly variable, so 
 that any methods in which these are assumed to be even for a short 
 time constant are not to be depended on. 
 
 220*.| The comparison of resistances can be made with extreme 
 
 
 
 Fig. 47. 
 
 accuracy by either of two methods, in which the result is in- 
 dependent of variations of R and £. 
 
220", | THE COMPARISON OF RESISTANCES. 183 
 
 The first of these methods depends on the use of the differential 
 galvanometer, an instrument in which there are two coils, the 
 currents in which are independent of each other, so that when 
 the currents are made to flow in opposite directions they act in 
 opposite directions on the needle, and when the ratio of these 
 currents is that of m to 2 they have no resultant effect on the 
 galvanometer needle. 
 
 Let /,, J, be the currents through the two coils of the galvan- 
 ometer, then the deflexion of the needle may be written 
 
 6= mI,—nI,. 
 
 Now let the battery current J be divided between the coils of 
 the galvanometer, and let resistances 4 and B be introduced into 
 the first and second coils respectively. Let the remainder of the 
 resistance of the coils and their connexions be a and § respect- 
 ively, and let the resistance of the battery and its connexions 
 between C and D be 7, and its electromotive force £. 
 
 Then we find, by Ohm’s Law, for the difference of potentials 
 between C and J, 
 
 C—D=T,(A+a)=1,(B+8) = #_-F, 
 
 
 
 
 
 and since lima 
 B+B A+a A+a+B+8 
 Ieee sh D ? gels D b) LEY Perey; ee 
 where D= (A+a)(B+6)+7(44+a+ B4+68). 
 
 The deflexion of the galvanometer needle is therefore 
 E ' 
 — D {m(B+B)—n(A+a)}, 
 
 and if there is no observable deflexion, then we know that the 
 quantity enclosed in brackets cannot differ from zero by more than 
 a certain small quantity, depending on the power of the battery, 
 the suitableness of the arrangement, the delicacy of the galvan- 
 ometer, and the accuracy of the observer. 
 
 Suppose that B has been adjusted so that there is no apparent 
 deflexion. 
 
 Now let another conductor 4’ be substituted for A, and let A’ be 
 adjusted till there is no apparent deflexion. Then evidently to a 
 first approximation d’= 4. 
 
 To ascertain the degree of accuracy of this estimate, let the 
 altered quantities in the second observation be accented, then 
 
184 MEASUREMENT OF RESISTANCE. eexs 
 
 m(B+pB)—2(d+a) = i }, 
 
 m(B+B)—n(dA’ +a) = wih 
 Hence n(d’— A) = Cte: a o’. 
 
 If 6 and 3’, instead of being both apparently zero, had been only 
 observed to be equal, then, unless we also could assert that 4 = L’, 
 the right-hand side of the equation might not be zero. In fact, the 
 method would be a mere modification of that already deseribed. 
 
 The merit of the method consists in the fact that the thing 
 observed in the absence of any deflexion, or in other words, the ~ 
 method is a Null method, one in which the non-existence of a force 
 is asserted from an observation in which the force, if it had been 
 different from zero by more than a certain small amount, would 
 have produced an observable effect. 
 
 Null methods are of great value where they can be employed, 
 but they can only be employed where we can cause two equal and 
 opposite quantities of the same kind to enter into the experiment 
 together. 
 
 In the case before us both 6 and & are quantities too small to be 
 observed, and therefore any change in the value of / will not affect 
 the accuracy of the result. 
 
 The actual degree of accuracy of this method might be ascer- 
 tained by taking a number of observations in each of which 4’ 
 is separately adjusted, and comparing the result of each observation 
 with the mean of the whole series. 
 
 But by putting 4’ out of adjustment by a known quantity, as, 
 for instance, by inserting at 4 or at B an additional resistance 
 equal to a hundredth part of 4 or of #B, and then observing 
 the resulting deviation of the galvanometer needle, we can estimate 
 the number of degrees corresponding to an error of one per cent. 
 To find the actual degree of precision we must estimate the smallest 
 deflexion which could not escape observation, and compare it with 
 the deflexion due to an error of one per cent. 
 
 *If the comparison is to be made between 4 and B, and if the 
 positions of d and & are exchanged, then the second equation 
 becomes 
 
 * This investigation is taken from Weber’s treatise on Galvanometry. Géttingen 
 Transactions, x. p. 65. 
 
220".] DIFFERENTIAL GALVANOMETER. 185 
 
 m(A+8)—x (B+a) = ety 
 JN ky 
 whence (m+n) (B—A) = =, wana EF o. 
 
 If m and n, A and a a and 6 are peas + equal, then 
 
 i At ae — (4 +a) (Ad +a+4 27) (8-0). 
 Here 6—8’ may be taken to be the smallest observable deflexion 
 of the galvanometer. 
 
 If the galvanometer wire be made longer and thinner, retaining 
 the same total mass, then z will vary as the length of the wire 
 and a as the square of the length. Hence there will be a minimum 
 A A 27 
 peed A eae, = eez!) when 
 
 Sees 
 at 7 | _—_—_ —_—- —_—_—_———— e 
 Mae) A/F ae 
 
 If we suppose 7, the battery resistance, small compared with 4, 
 this gives 
 
 value of 
 
 a= 34; 
 or, the resistance of each coil of the galvanometer should be one-third 
 of the resistance to be measured. 
 
 We then find RB 4=24 0 %’), 
 
 If we allow the current to flow through one only of the coils 
 of the galvanometer, and if the deflexion thereby produced is A 
 (supposing the deflexion strictly proportional to the deflecting 
 force), then 
 VEER AY 
 — Atatr 44d 
 
 Ba Ae 20 0. 
 Pe oer SK 
 
 In the differential galvanometer two currents are made to 
 produce equal and opposite effects on the suspended needle. The 
 force with which either current acts on the needle depends not 
 only on the strength of the current, but on the position of the 
 windings of the wire with respect to the needle. Hence, unless 
 the coil is very carefully wound, the ratio of m to ~ may change 
 when the position of the needle is changed, and therefore it is 
 necessary to determine this ratio by proper methods during each 
 
 
 
 if y= 0 and a=5A. 
 
 Hence 
 
 
 
186 MEASUREMENT OF RESISTANCE. Bega’ 
 
 course of experiments if any alteration of the position of the needle 
 is suspected. 
 
 The other null method, in which Wheatstone’s Bridge is used, 
 requires only an ordinary galvanometer, and the observed zero 
 deflexion of the needle is due, not to the opposing action of two 
 currents, but to the non-existence of a current in the wire. Hence 
 we have not merely a null deflexion, but a null current as the 
 phenomenon observed, and no etrors can arise from want of 
 regularity or change of any kind in the coils of the galvanometer. 
 The galvanometer is only required to be sensitive enough to detect 
 the existence and direction of a current, without in any way 
 determining its value or comparing its value with that of another 
 current. 
 
 221*,| Wheatstone’s Bridge consists essentially of six conductors 
 connecting four points. An electromotive 
 force / is made to act between two of the 
 points by means of a voltaic battery in- 
 troduced between £6 and C. The current 
 between the other two points O and JA is 
 measured by a galvanometer. 
 
 Under certain circumstances this current 
 becomes zero. The conductors LC and OA 
 are then said to be conjugate to each other, 
 which implies a certain relation between the resistances of the 
 other four conductors, and this relation is made use of in measuring 
 resistances. 
 
 If the current in OA is zero, the potential at O must be equal 
 to that at 4. Now when we know the potentials at B and C we 
 can determine those at O and A by the rule given at Art. 157, 
 provided there is no current in OA, 
 
 
 
 Fig. 48. 
 
 
 
 o — Pyt+e8 _ bb+Ce 
 = sie ey ’ maar Bite ’ 
 whence the condition is Coie 
 where 4, c, B, y are the resistances in CA, AB, BO and OC re- 
 
 spectively. 
 
 To determine the degree of accuracy attainable by this method 
 we must ascertain the strength of the current in O4 when this 
 condition is not fulfilled exactly. 
 
 Let A, 6, C and O be the four points. Let the currents along 
 BC, CA and AB be x, y and z, and the resistances of these 
 
222". | WHEATSTONE’S BRIDGE. 187 
 
 conductors a, dandc. Let the currents along OA, OB and OC be 
 €, n, ¢ and the resistances a, 8 and y. Let an electromotive force 
 Lact along LC. Required the current € alone OA. 
 
 Let the potentials at the points 4, 5, C and O be denoted by 
 the symbols 4, 6, Cand O. The equations of conduction are 
 
 ax= B—C+H, af = O—A, 
 by=C-—A Bn = O-B, 
 eoz= A—B y(= O-C; 
 with the equations of continuity 
 ft+y—z= 0, 
 H+ 2—x@ = 0, 
 C+27—y= 0. 
 
 By considering the system as made up of three cireuits OBC, 
 OCA and OAB in which the currents are 2, y, 2 respectively, and 
 applying Kirchhoff’s rule [ Art. 158] to each cycle, we eliminate the 
 values of the potentials O, 4, 6, C, and the currents & 7, ¢ and 
 obtain the following equations for z, y and z, 
 
 (@+B+y)e-yy —B2 = L, 
 —y@ +(b+y+a)y—az =a 
 —Bue —ay +(cta+f)z= 0. 
 Hence, if we put 
 D=|a+prty ay —p 
 —~y b+yta oh ead be 
 —p —a e+at+ 
 we find a= Ba (4B—cy), 
 D 
 and v= {(b+7) (+B) +a(b+e4B8+y)}. 
 
 222*.| The value of D may be expressed in the symmetrical form, 
 D=alhe+be(B+y)+ca(y+a)+ab(a+ B)+(a+4 +e) (By+ya+t af) 
 or, since we suppose the battery in the conductor a and the 
 galvanometer in a, we may put B the battery resistance for a and 
 G the galvanometer resistance for a. We then find 
 
 D= BG b+e+B+y)+B(d+y)(c+8) 
 + G(b+0)(B+y)+ be (B+) + By (4 +e). 
 
 If the electromotive force # were made to act along OA, the 
 
 resistance of OA being still a, and if the galvanometer were placed 
 
188 MEASUREMENT OF RESISTANCE. (239% 
 
 in BC, the resistance of BC being still a, then the value of D would » 
 remain the same, and the current in BC due to the electromotive 
 foree # acting along OA would be equal to the current in OA4 due 
 to the electromotive force # acting in BC. 
 
 But if we simply disconnect the battery and the galvanometer, 
 and without altering their respective resistances connect the battery 
 to O and 4 and the galvanometer to B and C, then in the value of 
 D we must exchange the values of Band G. If D’ be the value of 
 D after this exchange, we find 
 
 D—D = (G—B) {(b+e)(B+y)—+y) (B+e)}, 
 = (B—@) {(0—8) (e—7)}. 
 
 Let us suppose that the resistance of the galvanometer is greater 
 than that of the battery. 
 
 Let us also suppose that in its original position the galvanometer 
 connects the junction of the two conductors of least resistance 8, y 
 with the junction of the two conductors of greatest resistance J, ¢, 
 or, in other words, we shall suppose that if the quantities 4, c, y, 8 
 are arranged in order of magnitude, 4 and ¢ stand together, and 
 y and 6 stand together. Hence the quantities 6—8 and c—y are 
 of the same sign, so that their product is positive, and therefore 
 D—T is of the same sign as B— G. 
 
 If therefore the galvanometer is made to connect the junction of 
 the two greatest resistances with that of the two least, and if 
 the galvanometer resistance is greater than that of the battery, 
 then the value of D will be less, and the value of the deflexion of 
 the galvanometer greater, than if the connexions are exchanged. 
 
 The rule therefore for obtaining the greatest galvanometer de- 
 _ flexion in a given system is as follows: 
 
 Of the two resistances, that of the battery and that of the 
  galvanometer, connect the greater resistance so as to join the two 
 \. greatest to the two least of the four other resistances. 
 
 223*.] We shall suppose that we have to determine the ratio of 
 the resistances of the conductors 4B and AC, and that this is to be 
 done by finding a point O on the conductor LOC, such that when 
 the points 4 and O are connected by a wire, in the course of which 
 a galvanometer is inserted, no sensible deflexion of the galvano- 
 meter needle occurs when the battery is made to act between B 
 and C. 
 
 The conductor BOC may be supposed to be a wire of uniform 
 resistance divided into equal parts, so that the ratio of the resist- 
 ances of BO and OC may be read off at once. 
 
223%. | WHEATSTONES BRIDGE. 189 
 
 Instead of the whole conductor being a uniform wire, we may 
 make the part near O of such a wire, and the parts on each side 
 may be coils of any form, the resistance of which is accurately 
 known. ‘ 
 
 We shall now use a different notation instead of the symmetrical 
 notation with which we commeneed. 
 
 Let the whole resistance of BAC be R, 
 
 Let ¢ = mF and 6 = (1—m) R. 
 
 Let the whole resistanee of BOC be S. 
 
 Let B = 2S and y = (1—2)S. 
 
 The value of ~ is read off directly, and that of m is deduced from 
 it when there is no sensible deviation of the galvanometer. 
 
 Let the resistanee of the battery and its connexions be PB, and 
 that of the galvanometer and its connexions G. 
 
 We find as before 
 D= GSBR+ BS+ RS} +m (1—m) BR? (B+ 8)+n(1—n) 8S? (B+ R) 
 +(m+n—2mn) BRS, 
 and if € is the current in the galvanometer wire 
 
 gé= = S (n—m). 
 
 
 
 In order to obtain the most accurate results we must make the 
 deviation of the needle as great as possible compared with the 
 value of (x—m). This may be done by properly choosing the 
 dimensions of the galvanometer and the standard resistance wire. 
 
 It may be shewn that when the form of a galvanometer wire 
 is changed while its mass remains constant, the deviation of the 
 needle for unit} current is proportional to the length, but the 
 resistance increases as the square of the length. Hence the 
 maximum deflexion is shewn to occur when the resistance of the 
 galvanometer wire is equal to the constant resistance of the rest 
 of the circuit. 
 
 In the present case, if 6 is the deviation, 
 
 8=CVGE | 
 where C is some constant, and G is the galvanometer resistance 
 which varies as the square of the length of the wire. Hence we 
 find that in the value of D, when 6 is a maximum, the part involv- | 
 ing G must be made equal to the rest of the expression. 
 
 If we also put m=, as is the case if we have made a correct 
 observation, we find the best value of G to be 
 
 G=n(1—2)(R+S). 
 
190 MEASUREMENT OF RESISTANCE. [224*. 
 
 This result is easily obtained by considering the resistance from 
 A to O through the system, remembering that BC, being conjugate 
 to AO, has no effect on this resistance. 
 
 In the same way we should find that if the total area of the 
 acting: surfaces of the battery is given, the most advantageous ar- 
 rangement of the battery is when 
 
 RS 
 R+8 
 
 Finally, we shall determine the value of S such that a given 
 change in the value of z may produce the greatest galvanometer 
 deflexion. By differentiating the expression for € we find 
 
 Je 
 
 §2= 
 
 = $75 @ aaeiery)) 
 
 If we have a great many determinations of resistance to make 
 in which the actual resistance has nearly the same value, then it 
 may be worth while to prepare a galvanometer and a battery for 
 this purpose. In this case we find that the best arrangement is 
 
 eas ie Daan G = 2n(1—2) R, 
 angeles eG ee 
 
 On the Use of Wheatstone’s Bridge. 
 
 224*,| We have already explained the general theory of Wheat- 
 stone’s Bridge, we shall now consider some of its applications. 
 
 
 
 The comparison which can be effected with the greatest exactness 
 is that of two equal resistances. 
 
g24".| USE OF WHEATSTONE’S BRIDGE. 191 
 
 Let us suppose that 8 is a standard resistance coil, and that we 
 wish to adjust y to be equal in resistance to B. 
 
 Two other coils, d and ¢, are prepared which are equal or nearly 
 equal to each other, and the four coils are placed with their electrodes 
 in mereury cups so that the current of the battery is divided 
 between two branches, one consisting of 8B and y and the other 
 of 6 and ec. The coils 4 and e¢ are connected by a wire PR, as 
 uniform in its resistance as possible, and furnished with a scale of 
 equal parts. 
 
 The galvanometer wire connects the junction of 8 and y with 
 a point Q of the wire PA, and the point of contact at Q@ is made 
 to vary till on closing first the battery circuit and then the 
 galvanometer circuit, no deflexion of the galvanometer needle is 
 observed. 
 
 The coils B and y are then made to change places, and a new 
 position is found for Q. If this new position is the same as the 
 old one, then we know that the exchange of 8 and y has produced 
 no change in the proportions of the resistances, and therefore y 
 is rightly adjusted. If Q has to be moved, the direction and 
 amount of the change will indicate the nature and amount of the 
 alteration of the length of the wire of y, which will make its resist- 
 ance equal to that of £. 
 
 If the resistances of the coils 4 and ¢, each including part of the 
 wire PR up to its zero reading, are equal to that of 4 and ¢ divisions 
 of the wire respectively, then, if # is the scale reading of Q in the 
 first case, and y that in the second, 
 
 
 
 
 
 gamete CSET LY 
 
 Peay by. 8 
 
 whence pie (2+¢)(y—#) 
 pe (¢+#) (6-9) 
 
 Since 4—y is nearly equal to c+, and both are great with 
 respect to x or ¥, we may write this 
 
 
 
 
 
 2 
 y y—% 
 1 Sey Se 
 ek te 
 and i YR 
 y= B(1 eh) 
 
 When y is adjusted as well as we can, we substitute for 4 and ¢ 
 other coils of (say) ten times greater resistance. 
 
 The remaining difference between 6 and y will now produce 
 a ten times greater difference in the position of Q than with the 
 
192 MEASUREMENT OF RESISTANCE. Feces 
 
 original coils 4 and c, and in this way we can continually increase 
 the accuracy of the comparison. 
 
 The adjustment by means of the wire with sliding contact piece 
 is more quickly made than by means of a resistance box, and it is 
 capable of continuous variation. 
 
 The battery must never be introduced instead of the galvano- 
 meter into the wire with a sliding contact, for the passage of a 
 powerful current at the point of contact would injure the surface 
 of the wire. Hence this arrangement is adapted for the case in 
 which the resistance of the galvanometer is greater than that of the 
 battery. 
 
 When y, the resistance to be measured, a, the resistance of the 
 battery, and a, the resistance of the galvanometer, are given, the 
 best values of the other resistances have been shewn by Mr. Oliver 
 Heaviside (Phil. Mag., Feb. 1873) to be 
 
 peere b= n/ayStt, te r/o! ree, 
 
 Thomson’s* Method for the Determination of the Resistance of 
 the Galvanometer. 
 
 
 
 
 
 225*.| An arrangement similar to Wheatstone’s Bridge has been 
 employed with advantage by 
 
 A Gatvanometer f 4 y 
 : Sir W. Thomson in determin- 
 C) ing the resistance of the gal- 
 ; Z IE ag vanometer when in actual use. 
 
 Zz Key y : 
 
 ‘ o uae > it was suggested to Sir W. 
 Ww Sa Thomson by Mance’s Method. 
 y a fr" (See Art. 226.) i 
 Let the battery be placed, 
 e i as before, between / and C 
 me in the figure of Article 221, 
 > but Jet the galvanometer be 
 
 placed in CA instead of in 
 OA. If b8—cy is zero, then 
 the conductor O4 is conjugate 
 to BC, and, as there is no cur- 
 rent produced in OA by the battery in BC, the strength of the 
 current in any other conductor is independent of the resistance 
 
 
 
 Fig. 50. 
 
 * Proc. R. 8., Jan. 19, 1871. 
 
256°; | MANCE S METHOD. 193 
 
 in OA. Hence, if the galvanometer is placed in CA its deflexion 
 will remain the same whether the resistance of OA is small or 
 ereat. We therefore observe whether the deflexion of the galvano- 
 meter remains the same when O and 4 are joined by a conductor 
 of small resistance, as when this connexion is broken, and if, by 
 properly adjusting the resistances of the conductors, we obtain this 
 result, we know that the resistance of the galvanometer is 
 
 iro at 
 
 os 
 
 B 
 where ¢, y, and # are resistance coils of known resistance. 
 
 It will be observed that though this is not a null method, in the 
 sense of there being no current in the galvanometer, it is so in 
 the sense of the fact observed being the negative one, that the 
 deflexion of the galvanometer is not changed when a certain con- 
 tact is made. An observation of this kind is of greater value 
 than an observation of the equality of two different deflexions of 
 the same galvanometer, for in the latter case there is time for 
 alteration in the strength of the battery or the sensitiveness of 
 the galvanometer, whereas when the deflexion remains constant, 
 in spite of certain changes which we can repeat at pleasure, we are 
 sure that the current is quite independent of these changes. 
 
 The determination of the resistance of the coil of a galvanometer 
 ean easily be effected in the ordinary way of using Wheatstone’s 
 Bridge by placing another galvanometer in OA. By the method 
 now described the galvanometer itself is employed to measure its 
 own resistance. 
 
 Mance’'s* Method of determining the Resistance of the Battery. 
 
 226*.| The measurement of the resistance of a battery when in 
 action is of a much higher order of difficulty, since the resistance 
 of the battery is found to change considerably for some time after 
 the strength of the current through it ischanged. In many of the 
 methods commonly used to measure the resistance of a battery such 
 alterations of the strength of the current through it occur in the 
 course of the operations, and therefore the results are rendered 
 doubtful. 
 
 In Manee’s method, which is free from this objection, the battery 
 is placed in BC and the galvanometer in Cd. The connexion 
 between O and B is then alternately made and broken. - 
 
 * Proc, R. 8., Jan. 19, 1871. 
 O 
 
194 MEASUREMENT OF RESISTANCE. [226*. 
 
 If the deflexion of the galvanometer remains unaltered, we know 
 that OB is conjugate to CA, whence cy = aa, and a, the resistance 
 of the battery, is obtained in terms of known resistances ¢, y, a. 
 
 When the condition cy = aa is fulfilled, then the current through 
 the galvanometer is 
 
 < Ta 
 
 = bate(b+a+y) 
 
 and this is independent of the resistance 8B between O and B. To 
 test the sensibility of the method let us suppose that the condition 
 cy = aa is nearly, but not accurately, fulfilled, and that y is the 
 current through the galvanometer when O and B are connected 
 
 us 
 
 
 
 Fig. 51. 
 
 by a conductor of no sensible resistance, and y, the current when 
 O and B are completely disconnected. 
 
 To find these values we must make 8 equal to 0 and to o in the 
 general formula for y, and compare the results. 
 
 In this way we find 
 
 of Sad 1) eee heme arene eee 
 y — y(e+a)(a+y)’ 
 where y, and 7, are supposed to be so nearly equal that we may, 
 when their difference is not in question, put either of them equal 
 to y, the value of the current when the adjustment is perfect. 
 
 The resistance, c, of the conductor 4B should be equal to a, 
 that of the battery, a and y, should be equal and as small as 
 possible, and 4 should be equal to a+ y. 
 
 Since a galvanometer is most sensitive when its deflexion is 
 small, we should brin» the needle nearly to zero by means of fixed 
 magnets before making contact between O and BL. 
 
 In this method of measuring the resistance of the battery, the 
 current in the battery is not in any way interfered with during the 
 operation, so that we may ascertain its resistance for any given 
 
 
 
227*.] COMPARISON OF ELECTROMOTIVE FORCES. 195 
 
 strength of current, so as to determine how the strength of current 
 affects the resistance. 
 
 If y is the current in the galvanometer, the actual current through 
 the battery is a, with the key down and 2, with the key up, where 
 
 b b ac 
 marry  saa(ist git) 
 the resistance of the battery is 
 a=—, 
 a 
 
 .and the electromotive force of the battery is 
 
 B=y(b+e4+£+y))- 
 
 The method of Art. 225 for finding the resistance of the galva- 
 nometer differs from this only in making and breaking contact 
 between O and 4 instead of between O and B, and by exchanging 
 a and 6 we obtain for this case 
 
 Jo—-"N aay B cy—b B : 
 y y(¢+8) (8+y) 
 
 On the Comparison of Electromotive Forces. 
 227*.| The following method of comparing the electromotive 
 forces of voltaic and thermoelectric arrangements, when no current 
 passes through them, requires only a set of resistance coils and a 
 constant battery. 
 Let the electromotive force / of the battery be greater than that 
 of either of the electromotors to be compared, then, if a sufficient 
 
 
 
 
 
 * (This method, as has been pointed out by Professor Oliver Lodge, is not free from 
 error on account of the variation of the E.M.F. of the battery, as the current through 
 it is diminished or increased by raising or depressing the key.] 
 
 O 2 
 
196 POGGENDORFF'S COMPENSATION METHOD. [227*. 
 
 resistance, &,, be interposed between the points 4,, B, of the 
 primary circuit 1 B,A,H, the electromotive force from B, to A, 
 may be made equal to that of the electromotor Z;. If the elec- 
 trodes of this electromotor are now connected with the points 
 A,, B, no current will flow through the electromotor. By placing 
 a galvanometer G, in the circuit of the electromotor £,, and 
 adjusting the resistance between 4, and J,, till the galvanometer 
 G, indicates no current, we obtain the equation 
 EB, = RC, 
 
 where /, is the resistance between 4, and B,, and C is the strength 
 of the current in the primary circuit. 
 
 In the same way, by taking a second electromotor /, and placing 
 its electrodes at 4, and B,, so that no current is indicated by the 
 galvanometer G,, 
 
 EL, = h, C, 
 where F, is the resistance between 4, and B,. If the observations 
 of the galvanometers G, and G, are simultaneous, the value of C, 
 the current in the primary circuit, is the same in both equations, 
 and we find 
 yO OREO fh oct 
 
 In this way the electromotive force of two electromotors may be 
 compared*. The absolute electromotive force of an electromotor 
 may be measured either electrostatically by means of the electro-— 
 meter, or electromagnetically by means of an absolute galvano- 
 meter, 
 
 This method, in which, at the time of the comparison, there 
 is no current through either of the electromotors, is a modification 
 of Poggendorff’s method, and is due to Mr. Latimer Clark, who 
 has deduced the following values of electromotive forces : 
 
 Concentrated Volts 
 solution of ; 
 Daniell I. Amalgamated Zinc H,SO,+ 4aq. CuSO, Copper =1.079 
 Il. re H,80,+ 12 aq. CuSO, Copper =0.978 
 ioe a H,S0O,+12 aq. Cu2(NO;) Copper =1.00 
 Bunsen I. ir A Ae HNO; Carbon =1.964 
 Lie oy ¥ » sp. g. 1.38 Carbon =1.888 
 Grove 7M H,80,+ 4 aq. HNO, Platinum = 1,956 
 
 A Volt is an electromotive force equal to 100,000,000 units of the centimetre-gramme- 
 second system. 
 
 * [Any number of batteries may be compared by the help of only one galvanometer 
 if one pole of each battery is connected with the same electrode of the galvanometer, 
 the other poles being connected through separate keys to points A,, A,, &c. upon 
 the wire and the keys being depressed one at a time but in rapid succession. | 
 
CHAPTER XIII. 
 
 ON THE ELECTRIC RESISTANCE OF SUBSTANCES. 
 
 228*,| TueErz are three classes in which we may place different 
 substances in relation to the passage of electricity through them. 
 
 The first class contains all the metals and their alloys, some 
 sulphurets, and other compounds containing metals, to which we 
 must add carbon in the form of gas-coke, and selenium in the 
 crystalline form. 
 
 In all these substances conduction takes place without any 
 decomposition, or alteration of the chemical nature of the substance, 
 either in its interior or where the current enters and leaves the 
 body. In all of them the resistance increases as the temperature 
 rises. 
 
 The second class consists of substances which are called electro- 
 lytes, because the current is associated with a decomposition of 
 the substance into two components which appear at the electrodes. 
 As a rule a substance is an electrolyte only when in the liquid 
 form, though certain colloid substances, such as glass at 100°C, 
 which are apparently solid, are electrolytes. It would appear from 
 the experiments of Sir B. C. Brodie that certain gases are capable 
 of electrolysis by a powerful electromotive force. 
 
 In all substances which conduct by electrolysis the resistance 
 diminishes as the temperature rises. 
 
 The third class consists of substances the resistance of which is 
 so great that it is only by the most refined methods that the 
 passage of electricity through them can be detected. These are 
 ealled Dielectrics. To this class belong a considerable number 
 of solid bodies, many of which are electrolytes when melted, some 
 liquids, such as turpentine, naphtha, melted paraffin, &c., and all 
 gases and vapours. Carbon in the form of diamond, and selenium 
 in the amorphous form, belong to this class. 
 
 The resistance of this class of bodies is enormous compared with 
 that of the metals. It diminishes as the temperature rises. It 
 
198 RESISTANCE OF METALS. [229*. 
 
 is difficult, on account of the great resistance of these substances, 
 to determine whether the feeble current which we can force through 
 them is or is not associated with electrolysis. 
 
 On the LHlectric Resistance of Metals. 
 
 229%*.] There is no part of electrical research in which more 
 numerous or more accurate experiments have been made than in 
 the determination of the resistance of metals. It is of the utmost 
 importance in the electric telegraph that the metal of which the 
 wires are made should have the smallest attainable resistance. 
 Measurements of resistance must therefore be made before selecting 
 the materials. When any fault occurs in the line, its position is 
 at once ascertained by measurements of resistance, and these mea- 
 surements, in which so many persons are now employed, require 
 the use of resistance coils, made of metal the electrical properties 
 of which have been carefully tested. 
 
 The electrical properties of metals and their alloys have been 
 studied with great care by MM. Matthiessen, Voet, and Hockin, 
 and by MM. Siemens, who have done so much to introduce exact 
 electrical measurements into practical work. 
 
 It appears from the researches of Dr. Matthiessen, that the effect 
 of temperature on the resistance is nearly the same for a considerable 
 number of the pure metals, the resistance at 100°C being to that 
 at 0°C in the ratio of 1.414 to 1, or of 1 to .707. For pure iron 
 the ratio is 1.645, and for pure thallium 1.458. 
 
 The resistance of metals has been observed by Dr. C. W. Siemens* 
 through a much wider range of temperature, extending from the 
 freezing point to 350°C, and in certain cases to 1000°C. He finds 
 that the resistance increases as the temperature rises, but that the 
 rate of increase diminishes as the temperature rises, The formula, 
 which he finds to agree very closely both with the resistances 
 observed at low temperatures by Dr. Matthiessen and with his 
 own observations through a range of 1000°C, is 
 
 r=al?+PT+y, 
 where 7’ is the absolute temperature reckoned from — 273°C, and 
 a, 8, y are constants. Thus, for 
 
 Platine r = 0.0393697 + 0.002164077'— 0.2413, 
 
 Copper.......+. r= 0.0265777'2 + 0.00314437 —0.22751, 
 
 Tron ee r= 0.0725457* + 0.00381337 —1.23971. 
 * Proc. R, S., April 27, 1871. 
 
230. _ RESISTANCE OF METALS. | 199 
 
 From data of this kind the temperature of a furnace may be 
 determined by means of an observation of the resistance of a 
 platinum wire placed in the furnace. 
 
 Dr. Matthiessen found that when two metals are combined to 
 form an alloy, the resistance of the alloy is in most cases greater 
 than that calculated from the resistance of the component metals 
 and their proportions. In the case of alloys of gold and silver, the 
 resistance of the alloy is greater than that of either pure gold or 
 pure silver, and, within certain limiting proportions of the con- 
 stituents, 1t varies very little with a slight alteration of the pro- 
 portions. For this reason Dr. Matthiessen recommended an alloy 
 of two parts by weight of gold and one of silver as a material 
 for reproducing the unit of resistance. 
 
 The effect of change of temperature on electric resistance is 
 generally less in alloys than in pure metals. 
 
 Hence ordinary resistance coils are made of German silver, on 
 account of its great resistance, and its small variation with tem- 
 perature. 
 
 An alloy of silver and platinum is also used for standard coils. 
 
 230*.| In the following table & is the resistance in Ohms of a 
 column one metre long and one gramme weight at 0°C, and 7 is 
 the resistance in centimetres per second of a cube of one centi- 
 metre, according to the experiments of Matthiessen*. 
 
 Percentage 
 increment of 
 Specific resistance for 
 gravity R r 1°C at 20°C, 
 PENCE rates viaccess ese 10-50 hard drawn 0-1689 1609 0.377 
 SPOR Peleg tex ts cates ss 8-95 hard drawn 0-1469 1642 0-388 
 Son Et 2 ae eee 19.27 hard drawn 0-4150 2154 0-365 
 LS eae 11-391 pressed 2-257 19847 0-387 
 BE OLCUrY tics cts. sce 13-595 liquid 13-071 96146 0-072 
 Gold 2, Silver 1... .15-218 hard or annealed 1-668 10988 0-065 
 Selenium at 100°C Crystalline form 6x108 ~=1.00 
 
 It appears from the researches of Matthiessen and Hockin that 
 the resistance of a uniform column of mercury of one metre in 
 length, and weighing one gramme at 0°C, is 13.071 Ohms, whence 
 it follows that if the specific gravity of mercury is 13-595, the 
 resistance of a column of one metre in length and one square 
 millimetre in section is 0.96146 Ohms. 
 
 * Phil. Mag., May, 1865. 
 
200 RESISTANCE OF ELECTROLYTES. Eee 
 
 On the Electric Resistance of Electrolytes. 
 
 231*.| The measurement of the electric resistance of electrolytes 
 is rendered difficult on account of the polarization of the electrodes, 
 which causes the observed difference of potentials of the metallic 
 electrodes to be greater than the electromotive force which actually 
 produces the current. 
 
 This difficulty can be overcome in various ways. In certain 
 cases we can get rid of polarization by using electrodes of proper 
 material, as, for instance, zinc electrodes in a solution of sulphate 
 of zinc. By making the surface of the electrodes very large com- 
 pared with the section of the part of the electrolyte whose resist- 
 ance is to be measured, and by using only currents of short duration 
 in opposite directions alternately, we can make the measurements 
 before any considerable intensity of polarization has been excited 
 by the passage of the current. 
 
 Finally, by making two different experiments, in one of which 
 the path-of the current through the electrolyte is much longer than 
 in the other, and so adjusting the electromotive force that the 
 actual current, and the time during which it flows, are nearly the 
 same in each case, we can eliminate the effect of polarization 
 altogether. 
 
 232*.| In the experiments of Dr. Paalzow* the electrodes were 
 in the form of large disks placed in separate flat vessels filled with 
 the electrolyte, and the connexion was made by means of a long | 
 siphon filled with the electrolyte and dipping into both vessels. 
 Two such siphons of different lengths were used. 
 
 The observed resistances of the electrolyte in these siphons 
 being #, and &,, the siphons were next filled with mercury, and 
 their resistances when filled with mercury were found to be 2,’ 
 and Ry’. 
 
 The ratio of the resistance of the electrolyte to that of a mass 
 of mercury at 0°C of the same form was then found from the 
 formula 
 
 h,—R, 
 fiat Ry 
 
 To deduce from the values of p the resistance of a centimetre in 
 
 * Berlin Monatsbericht, July, 1868. 
 
233°.] RESISTANCE OF ELECTROLYTES, 201 
 
 length having a section of a square centimetre, we must multiply 
 them by the value of 7 for mercury at 0°C. (See Art. 230.) 
 The results given by Paalzow are as follow: 
 
 Mixtures of Sulphuric Acid and Water. 
 
 Resistance compared 
 
 Temp, with mercury. 
 
 (eee iy Senay 15°C 96950 
 
 mepOp ey 14-0 .,2..2... 19°C 14157 
 
 pape rete lord Oo. yon 22°C 13310 
 
 H,SO, + 499 HO ......... 22°C 184773 
 Sulphate of Zine and Water. 
 
 2n 50; + 23°H°O) 42... 23°C 194400 
 
 2050; + 24°H"*Oe.. 3... 23°C 191000 
 
 ZnSO, + 105 HO .,........ 23°C 354000 
 Sulphate of Copper and Water. 
 
 CuSO, + 45 HO ,,.!..... 22°C 202410 
 
 Cu S80, 4+ 205 H?O* 7... .:. 22°C 339341 
 
 Sulphate of Magnesium and Water. 
 
 MgSO,+ 34 HO ......... 22°C 199180 
 MgSO, + 107 HO ......... 22°C 324600 
 
 Hydrochloric Acid and Water. 
 
 Clee tO), as 23°C 13626 
 Peet OHIO ET) 2 oe 23°C 86679 
 
 2338*.| MM. F. Kohlrausch and W. A. Nippoldt* have de- 
 termined the resistance of mixtures of sulphuric acid and water. 
 They used alternating magneto-electric currents, the electromotive 
 force of which varied from } to 7; of that of a Grove’s cell, and — 
 by means of a thermoelectric copper-iron pair they reduced the 
 electromotive force to zzayoz Of that of a Grove’s cell. They found 
 that Ohm’s law was applicable to this electrolyte throughout the 
 range of these electromotive forces. 
 
 The resistance is a minimum in a mixture containing about one- 
 third of sulphuric acid. 
 
 The resistance of electrolytes diminishes as the temperature 
 increases. 'The percentage increment of conductivity for a rise of 
 1-C is given in the following table: 
 
 * Pogg. Ann. cxxxviii, p. 286, Oct. 1869. 
 
202 RESISTANCE OF DIELECTRICS. [234*. 
 
 Resistance of Mixtures of Sulphuric Acid and Water at 22°C in terms 
 of Mercury at 0°C. MM. Kohlrausch and Nippoldt. 
 
 i : Resistance Percentage 
 Specific gravity Percentage at2o°C increment of 
 at 18°5 of H2SO4 (Hg=1) conductivity 
 for 1°C 
 0-9985 0-0 746300 0-47 
 1-00 0.2 465100 0-47 
 1.0504 8-3 34530 0-653 
 1-0989 14.2 18946 0-646 
 1.1431 20-2 14990 0-799 
 1.2045 28-0 131383 1-317 
 1.2631 35-2 13132 1.259 
 1.3163 41.5 14286 1-410 
 1.8547 46-0 15762 1.674 
 1.3994 50-4 17726 1.582 
 1.4482 55-2 20796 1.417 
 1.5026 60-3 25574 1-794 
 
 On the Electrical Resistance of Dielectrics. 
 
 234*.] A great number of determinations of the resistance of 
 gutta-percha, and other materials used as insulating media, in the 
 manufacture of telegraphic cables, have been made in order to 
 ascertain the value of these materials as insulators. 
 
 The tests are generally applied to the material after it has been 
 used to cover the conducting wire, the wire being used as one 
 electrode, and the water of a tank, in which the cable is plunged, 
 as the other. Thus the current is made to pass through a eylin- 
 drical coating of the insulator of greater area and small thickness. 
 
 It is found that when the electromotive force begins to act, the 
 current, as indicated by the galvanometer, is by no means constant. 
 The first effect is of course a transient current of considerable 
 intensity, the total quantity of electricity being that required to 
 change the surfaces of the insulator with the superficial distribution 
 of electricity corresponding to the electromotive force. This first 
 current therefore is a measure not of the conductivity, but of the 
 capacity of the insulating layer. 
 
 But even after this current has been allowed to subside the 
 residual current is not constant, and does not indicate the true 
 conductivity of the substance. It is found that the current con- 
 tinues to decrease for at least half an hour, so that a determination 
 of the resistance deduced from the current will give a greater value 
 if a certain time is allowed to elapse than if taken immediately 
 after applying the battery. 
 
Rae | RESISTANCE OF DIELECTRICS, 203 
 
 Thus, with Hooper’s insulating material the apparent resistance 
 at the end of ten minutes was four times, and at the end of 
 nineteen hours twenty-three times that observed at the end of 
 one minute. When the direction of the electromotive force is 
 reversed, the resistance falls as low or lower than at first and 
 then gradually rises. 
 
 These phenomena seem to be due to a condition of the gutta- 
 percha, which, for want of a better name, we may call polarization, 
 and which we may compare on the one hand with that of a series 
 of Leyden jars charged by cascade, and, on the other, with Ritter’s 
 secondary pile, 
 
 If a number of Leyden jars of great capacity are connected in 
 series by means of conductors of great resistance (such as wet 
 cotton threads in the experiments of M. Gaugain), then an electro- 
 motive force acting on the series will produce a current, as 
 indicated by a galvanometer, which will gradually diminish till 
 the jars are fully charged. 
 
 The apparent resistance of such a series will increase, and if the 
 dielectric of the jars is a perfect insulator it will increase without 
 limit. If the electromotive force be removed and connexion made 
 between the ends of the series, a reverse current will be observed, 
 the total quantity of which, in the case of perfect insulation, will be 
 the same as that of the direct current. Similar effects are observed 
 in the case of the secondary pile, with the difference that the final 
 insulation is not so good, and that the capacity per unit of surface 
 is immensely greater. 
 
 In the case of the cable covered with gutta-percha, &c., it is 
 found that after applying the battery for half an hour, and then 
 connecting the wire with the external electrode, a reverse current 
 takes place, which goes on for some time, and gradually reduces 
 the system to its original state. 
 
 These phenomena are of the same kind with those indicated 
 by the ‘residual discharge’ of the Leyden jar, except that the 
 amount of the polarization is much greater in gutta-percha, &c. 
 than in glass. 
 
 This state of polarization seems to be a directed property of the 
 material, which requires for its production not only electromotive 
 force, but the passage, by displacement or otherwise, of a con- 
 siderable quantity of electricity, and this passage requires a con- 
 siderable time. When the polarized state has been set up, there 
 is an internal electromotive force acting in the substance in the 
 
204 RESISTANCE OF DIELECTRICS. [235% 
 
 reverse direction, which will continue till it has either produced 
 a reversed current equal in total quantity to the first, or till the 
 state of polarization has quietly subsided by means of true con- 
 duction through the substance. 
 
 The whole theory of what has been called residual discharge, 
 absorption of electricity, electrification, or polarization, deserves 
 a careful investigation, and will probably lead to important dis- 
 coveries relating to the internal structure of bodies. 
 
 235*.| The resistance of the greater number of dielectrics di- 
 minishes as the temperature rises. 
 
 Thus the resistance of gutta-percha is about twenty times as great 
 at O°C as at 24°C. Messrs. Bright and Clark have found that the 
 following formula gives results agreeing with their experiments. 
 If ris the resistance of gutta-percha at temperature 7’ centigrade, 
 then the resistance at temperature 7’+¢ will be 
 
 R=r x 0.887 8%, 
 the number varies between 0.8878 and 0.9. 
 
 _ Mr, Hockin has verified the curious fact that it is not until some 
 hours after the gutta-percha has taken its temperature that the 
 resistance reaches its corresponding: value. 
 
 The effect of temperature on the resistance of india-rubber is not 
 so great as on that of gutta-percha. 
 
 The resistance of gutta-percha increases considerably on the ap- 
 plication of pressure. 
 
 The resistance, in Ohms, of a cubic metre of various specimens of 
 gutta-percha used in different cables is as follows*. 
 
 Name of Cable. 
 
 Réd *Sé6aiee eae .267-x 1012 to. .362 5 = 
 Malta-Alexandria............00: 23810 
 Persian Gulls (ees cree 180 %4.0) 
 peconmd vAtlantic Ae... ee 34 Dexa 
 Hooper's Persian Gulf Core...74.7 x 10” 
 Gutta-percha at 24°C ......... 3.53 x 10™ 
 
 236*.| The following table, calculated from the experiments of 
 M. Buff't, shews the resistance of a cubic metre of glass in Ohms 
 at different temperatures : 
 
 * Jenkin’s Cantor Lectures. 
 + [Annalen der Chemie und Pharmacie, bd. xc. 257 (1854).] 
 
238°%| RESISTANCE OF DIELECTRICS. 205 
 
 Temperature. Resistance. 
 200°C 227000 
 250° 13900 
 300° 1480 
 350° 1035 
 400° “00 
 
 237*.|] Mr. C. F. Varley * has recently investigated the conditions 
 of the current through rarefied gases, and finds that the electro- 
 motive force / is equal to a constant H, together with a part 
 depending on the current according to Ohm’s Law, thus 
 
  #H == ik RC, 
 
 For instance, the electromotive force required to cause the 
 
 current to begin in a certain tube was that of 323 Daniell’s cells, 
 but an electromotive force of 304 cells was just sufficient to 
 maintain the current. The intensity of the current, as measured 
 by the galvanometer, was proportional to the number of cells above 
 304. Thus for 305 cells the deflexion was 2, for 306 it was 4, 
 for 307 it was 6, and so on up to 380, or 304+ 76 for which the 
 deflexion was 150, or 76 x 1.97. 
 _ From these experiments it appears that there is a kind of 
 polarization of the electrodes, the electromotive force of which 
 is equal to that of 304 Daniell’s cells, and that up to this electro- 
 motive force the battery is occupied in establishing this state of 
 polarization. When the maximum polarization is established, the 
 excess of electromotive force above that of 304 cells is devoted to 
 maintaining the current according to Ohm’s Law. 
 
 The Law of the current in a rarefied gas is therefore very similar 
 to the law of the current through an electrolyte in which we have 
 to take account of the polarization of the electrodes. 
 
 In connexion with this subject we should study Thomson’s re- 
 sults}, in which the electromotive force required to produce a spark 
 in air was found to be proportional not to the distance, but to the 
 distance together with a constant quantity. The electromotive 
 force corresponding to this constant quantity may be regarded as 
 the intensity of polarization of the electrodes. 
 
 238*.| MM. Wiedemann and Ruhlmann have recently { investi- 
 gated the passage of electricity through gases. The electric current 
 was produced by Holtz’s machine, and the discharge took place 
 
 * Proc. R. S., Jan. 12, 1871. 
 
 + (Proc. R. 8. 1860, or Reprint, chap. xix.] 
 t Berichte der Kénigl. Sdchs. Gesellschaft, Oct. 20, 1871. 
 
206 RESISTANCE OF DIELECTRICS. [238*. 
 
 between spherical electrodes within a metallic vessel containing 
 rarefied gas. The discharge was in general discontinuous, and the 
 interval of time between successive discharges was measured by 
 means of a mirror revolving along with the axis of Holtz’s machine. 
 The images of the series of discharges were observed by means of 
 a heliometer with a divided object-glass, which was adjusted till 
 one image of each discharge coincided with the other image of 
 the next discharge. By this method very consistent results were 
 obtained. It was found that the quantity of electricity in each 
 discharge is independent of the strength of the current and of 
 the material of the electrodes, and that 1t depends on the nature 
 and density of the gas, and on the distance and form of the. 
 electrodes. 
 
 These researches confirm the statement of Faraday* that the 
 electric tension (see Art. 46) required to cause a disruptive discharge 
 to begin at the electrified surface of a conductor is a little less 
 when the electrification is negative than when it is positive, but 
 that when a discharge does take place, much more electricity passes 
 at each discharge when it begins at a positive surface. They also 
 tend to support the hypothesis, that the stratum of gas condensed 
 on the surface of the electrode plays an important part in the 
 phenomenon, and they indicate that this condensation is greatest 
 at the positive electrode. 
 
 Note on Wheatstone’s Bridge. 
 
 [The following method of determining the current in the Gal- 
 vanometer of Wheatstone’s Bridge was given by Professor Maxwell 
 in his last course of lectures, and is a good illustration of the method 
 of treating a system of linear conductors. It has been communicated 
 to the present editor by Professor J. A. Fleming of University 
 College, Nottingham. The method simply assumes Ohm's Law for 
 each conductor, and that the whole electromotive force around a 
 linear circuit is the sum of the electromotive forces in the several 
 conductors forming the cireuits, and therefore equal to the sum of 
 the products of the resistance of each conductor and the current 
 flowing in it, the currents being taken in cyclic order. 
 
 Let P, Q, 8, Rk, Gand B (Fig. 53) denote the resistances in the 
 several conductors forming the bridge, and let them be arranged as 
 indicated in the figure. Now the six conductors may be considered 
 
 * Exp. Res., 1501. 
 
WHEATSTONES BRIDGE. 207 
 
 as forming three independent circuits viz.:—PGQ, RSG, and QSB. 
 Let x+y, y and z denote the currents in these circuits respectively, 
 each current being considered as flowing in the directions indicated 
 by the arrows. Then the actual current in Q is z—#—y, that in 
 Sis z—y and that in G, is x, and the electromotive force between 
 
 
 
 Fig. 53. 
 
 the ends of Q is Q (e—y—z) and so on for the other conductors. 
 Of the three circuits specified above the E. M. F. in the first two is 
 zero while that in the third is /, the electromotive force of the 
 battery. Hence, applying Ohm’s Law to each cireuit in order we 
 have 
 
 (P+G4+Q)e+y—Gy—-Qz =0 
 
 (R+S+@)y —Sz2—Gut+yH=O Pp ceveeeees (I) 
 
 (Q+S8+B)e —Sy—Qu+y=P 
 
 
 
 
 
 or 
 
 (P+ @+ Q)at+(P+ Q)y—Qz = 0 
 
 
 
 
 
 
 
 —Gx +(R+8)y—Sz = 0 Peel) 
 —Qx © —(S+Q)yt+(Q+S+B)z=F 
 Solving for w we obtain 
 Leta, = @ 
 nes fo Pres Se 8 
 A 
 _ £(QR-PS) 
 = z ; 
 
 where A is the determinant of the system of equations (II). 
 The condition for no current in the galvanometer is # = 0, or 
 
 ya 
 QR—PS=0,0r 5 = 3. 
 
208 WHEATSTONE’S BRIDGE. 
 
 To obtain the current equations, (I), the rule is— 
 
 ‘Multiply each cycle sign (i.e. current) by the sum of all the 
 resistances which bound that cycle, and subtract from it the sign 
 of each neighbouring cycle multiplied by the resistance separating 
 the cycles, and equate the result to the E. M. F. in the cycle.’ 
 
 It will be seen that. the method is a simple application of 
 Kirchhoff’s second law, but the above rule is very convenient 
 in its application. | | | 
 
 THE END. 
 
Pik d\ EIS Sy 
 

 
Prats I, 
 
 Art. 93. 
 
 
 
 
 
 
 
 
 
 x 
 
 Lines of Force and Equipotential Surfuces. 
 
 A = 20. B= 5. P. Point of Equilibrium. AP = xAB. 
 
 University Press, Oxford, 
 

 
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Prats II, 
 
 Art. 94. 
 
 
 
 Lines of Force and Equipotential Surfaces. 
 
 - A= 20. = -54. P. Point of Equilibrium. AP = 2AB. 
 Q. Spherical surface of Zero potential. 
 M. Point of Maximum Force along the axis. 
 
 The dotted line is the Line of Force ¥ = 0.1 thus _._._._.- 
 
 University Press, Oxford, 
 
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Priate III. 
 
 Art. 95. 
 
 
 
 
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Prats LV. 
 
 Art. 96. 
 
 
 
 Lines of Force and Equipotential Surfaces. 
 
 C= 20; 
 
 B= —72. 
 
 University Press, Oxford. 
 

 
PLATE V. 
 
 Art. 193. 
 
 
 
 eee ite 
 Et ae a Soe 
 
 
 
 se cerech er erosle 
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 Lines of Force near the edge of a Plate. 
 

 
Pirate VI, 
 
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 Lines of Force near a Grating 
 
 University Press, Oxford. 
 

 

 
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