OF THE
UNIVERSITY
OF
L i G *7 A R Y
OF T;;E
ASTRONOMICAL SOCIETY
OF THE PACIFIC
ODlarentron
27
A TREATISE
ON
ELECTRICITY AND MAGNETISM
MAXWELL
VOL. I.
Hontron
HENRY FROWDE
OXFORD UNIVERSITY PRESS WAREHOUSE
7 PATERNOSTER ROW
Clarentron r
A TREATISE
ELECTRICITY AND MAGNETISM
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
VOL. I
SECOND EDITION
©.Tfotlr
AT THE CLARENDON PRESS
1881
[ AH rights reserved ]
AS/flONOMY
M3.2
V.I
ASTRONOMY
fc JJBRAK* ,
PREFACE TO THE FIRST EDITION,
THE fact that certain bodies, after being rubbed,
appear to attract other bodies, was known to the
ancients. In modern times, a great variety of other
phenomena have been observed, and have been found
to be related to these phenomena of attraction. They
have been classed under the name of Electric phe
nomena, amber, fa&crpov, 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 phe
nomena 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, vayvi?,
being found in the Thessalian Magnesia.
These two classes of phenomena have since been
found to be related to each other, and the relations
between the various phenomena of both classes, so
far as they are known, constitute the science of Elec-
tromagnetism.
In the following Treatise I propose to describe the
M877187
VI
PREFACE.
most important of these phenomena, to shew how they
may be subjected to measurement, and to trace the
mathematical connexions of the quantities measured.
Having thus obtained the data for a mathematical
theory of electromagnetism, and having shewn how
this theory may be applied to the calculation of phe
nomena, I shall endeavour to place in as clear a light
as I can the relations between the mathematical form
of this theory and that of the fundamental science of
Dynamics, in order that we may be in some degree
prepared to determine the kind of dynamical pheno
mena among which we are to look for illustrations or
explanations of the electromagnetic phenomena.
In describing the phenomena, I shall select those
which most clearly illustrate the fundamental ideas of
the theory, omitting others, or reserving them till the
reader is more advanced.
The most important aspect of any phenomenon from
a mathematical point of view is that of a measurable
quantity. I shall therefore consider electrical pheno
mena chiefly with a view to their measurement, de
scribing the methods of measurement, and defining
the standards on which they depend.
In the application of mathematics to the calculation
of electrical quantities, I shall endeavour in the first
place to deduce the most general conclusions from the
data at our disposal, and in the next place to apply
the results to the simplest cases that can be chosen.
I shall avoid, as much as I can, those questions which,
though they have elicited the skill of mathematicians,
have not enlarged our knowledge of science.
PREFACE. vii
The internal relations of the different branches of
the science which we have to study are more numerous
arid complex than those of any other science hitherto
developed. Its external relations, on the one hand to
dynamics, and on the other to heat, light, chemical
action, and the constitution of bodies, seem to indicate
the special importance of electrical science as an aid
to the interpretation of nature.
It appears to me, therefore, that the study of electro-
magnetism in all its extent has now become of the
first importance as a means of promoting the progress
of science.
The mathematical laws of the different classes of
phenomena have been to a great extent satisfactorily
made out.
The connexions between the different classes of phe
nomena have also been investigated, and the proba
bility of the rigorous exactness of the experimental
laws has been greatly strengthened by a more extended
knowledge of their relations to each other.
Finally, some progress has been made in the re
duction of electromagnetism to a dynamical science,
by shewing that no electromagnetic phenomenon is
contradictory to the supposition that it depends on
purely dynamical action.
What has been hitherto done, however, has by no
means exhausted the field of electrical research. It
has rather opened up that field, by pointing out sub
jects of enquiry, and furnishing us with means of
investigation.
It is hardly necessary to enlarge upon the beneficial
Vlll
PREFACE.
results of magnetic research on navigation, and the
importance of a knowledge of the true direction of
the compass, and of the effect of the iron in a ship.
But the labours of those who have endeavoured to
render navigation more secure by means of magnetic
observations have at the same time greatly advanced
the progress of pure science.
Gauss, as a member of the German Magnetic Union,
brought his powerful intellect to bear on the theory
of magnetism, and on the methods of observing it,
and he not only added greatly to our knowledge of
the theory of attractions, but reconstructed the whole
of magnetic science as regards the instruments used,
the methods of observation, and the calculation of the
results, so that his memoirs on Terrestrial Magnetism
may be taken as models of physical research by all
those who are engaged in the measurement of any
of the forces in nature.
The important applications of electromagnetism to
telegraphy have also reacted on pure science by giving
a commercial value to accurate electrical measure
ments, and by affording to electricians the use of
apparatus on a scale which greatly transcends that
of any ordinary laboratory. The consequences of this
demand for electrical knowledge, and of these experi
mental opportunities for acquiring it, have been already
very great, both in stimulating the energies of ad
vanced electricians, and in diffusing among practical
men a degree of accurate knowledge which is likely
to conduce to the general scientific progress of the
whole engineering profession.
PREFACE. ix
There are several treatises in which electrical and
magnetic phenomena are described in a popular way.
These, however, are not what is wanted by those who
have been brought face to face with quantities to be
measured, and whose minds do not rest satisfied with
lecture-room experiments.
There is also a considerable mass of mathematical
memoirs which are of great importance in electrical
science, but they lie concealed in the bulky Trans
actions of learned societies ; they do not form a con
nected system; they are of very unequal merit, and
they are for the most part beyond the comprehension
of any but professed mathematicians.
I have therefore thought that a treatise would be
useful which should have for its principal object to
take up the whole subject in a methodical manner,
and which should also indicate how each part of the
subject is brought within the reach of methods of
verification by actual measurement.
The general complexion of the treatise differs con
siderably from that of several excellent electrical
works, published, most of them, in Germany, and it
may appear that scant justice is done to the specu
lations of several eminent electricians and mathema
ticians. One reason of this is that before I began
the study of electricity I resolved to read no mathe
matics on the subject till I had first read through
Faraday's Experimental Researches on Electricity. I
was aware that there was supposed to be a difference
between Faraday's way of conceiving phenomena and
that of the mathematicians, so that neither he nor
x PREFACE.
they were satisfied with each other's language. I had
also the conviction that this discrepancy did not arise
from either party being wrong. I was first convinced
of this by Sir William Thomson *, to whose advice and
assistance, as well as to his published papers, I owe
most of what I have learned on the subject.
As I proceeded with the study of Faraday, I per
ceived that his method of conceiving the phenomena
was also a mathematical one, though not exhibited
in the conventional form of mathematical symbols. I
also found that these methods were capable of being-
expressed in the ordinary mathematical forms, and
thus compared with those of the professed mathema
ticians.
For instance, Faraday, in his mind's eye, saw lines
of force traversing all space where the mathematicians
saw centres of force attracting at a distance : Faraday
saw a medium where they saw nothing but distance :
Faraday sought the seat of the phenomena in real
actions going on in the medium, they were satisfied
that they had found it in a power of action at a
distance impressed on the electric fluids.
When I had translated what I considered to be
Faraday's ideas into a mathematical form, I found
that in general the results of the two methods coin
cided, so that the same phenomena were accounted
for, and the same laws of action deduced by both
methods, but that Faraday's methods resembled those
* I take this opportunity of acknowledging my obligations to Sir
W. Thomson and to Professor Tait for many valuable suggestions made
during the printing of this work.
PREFACE. xi
in which we begin with the whole and arrive at the
parts by anlaysis, while the ordinary mathematical
methods were founded on the principle of beginning
with the parts and building up the whole by syn
thesis.
I also found that several of the most fertile methods
of research discovered by the mathematicians could be
expressed much better in terms of ideas derived from
Faraday than in their original form.
The whole theory, for instance, of the potential, con
sidered as a quantity which satisfies a certain partial
differential equation, belongs essentially to the method
which I have called that of Faraday. According to
the other method, the potential, if it is to be considered
at all, must be regarded as the result of a summation
of the electrified particles divided each by its distance
from a given point. Hence many of the mathematical
discoveries of Laplace, Poisson, Green and Gauss find
their proper place in this treatise, and their appropriate
expression in terms of conceptions mainly derived from
Faraday.
Great progress has been made in electrical science,
chiefly in Germany, by cultivators of the theory of
action at a distance. The valuable electrical measure
ments of W. Weber are interpreted by him according
to this theory, and the electromagnetic speculation
which was originated by Gauss, and carried on by
Weber, Eiemann, J. and C. Neumann, Lorenz, &c. is
founded on the theory of action at a distance, but
depending either directly on the relative velocity of the
particles, or on the gradual propagation of something,
xii PREFACE.
whether potential or force, from the one particle to
the other. The great success which these eminent
men have attained in the application of mathematics
to electrical phenomena, gives, as is natural, addi
tional weight to their theoretical speculations, so that
those who, as students of electricity, turn to them as
the greatest authorities in mathematical electricity,
would probably imbibe, along with their mathematical
methods, their physical hypotheses.
These physical hypotheses, however, are entirely
alien from the way of looking at things which I
adopt, and one object which I have in view is that
some of those who wish to study electricity may, by
reading this treatise, come to see that there is another
way of treating the subject, which is no less fitted to
explain the phenomena, and which, though in some
parts it may appear less definite, corresponds, as I
think, more faithfully with our actual knowledge, both
in what it affirms and in what it leaves undecided.
In a philosophical point of view, moreover, it is
exceedingly important that two methods should be
compared, both of which have succeeded in explaining
the principal electromagnetic phenomena, and both of
which have attempted to explain the propagation of
light as an electromagnetic phenomenon, and have
actually calculated its velocity, while at the same time
the fundamental conceptions of what actually takes
place, as well as most of the secondary conceptions of
the quantities concerned, are radically different.
I have therefore taken the part of an advocate rather
than that of a judge, and have rather exemplified one
PREFACE. xiii
method than attempted to give an impartial description
of both. I have no doubt that the method which I
have called the German one will also find its sup
porters, and will be expounded with a skill worthy
of its ingenuity.
I have not attempted an exhaustive account of elec
trical phenomena, experiments, and apparatus. The
student who desires to read all that is known on these
subjects will find great assistance from the Traite
d' Electricite of Professor A. de la Rive, and from several
German treatises, such as Wiedemann's Galvanismus,
Biess' Beibungseleldricitat, Beer's Einleitung in die Elek-
trostatik, &c.
I have confined myself almost entirely to the ma
thematical treatment of the subject, but I would
recommend the student, after he has learned, experi
mentally if possible, what are the phenomena to be
observed, to read carefully Faraday's Experimental
Researches in Electricity. He will there find a strictly
contemporary historical account of some of the greatest
electrical discoveries and investigations, carried on in
an order and succession which could hardly have been
improved if the results had been known from the
first, and expressed in the language of a man who
devoted much of his attention to the methods of
accurately describing scientific operations and their
results *.
It is of great advantage to the student of any
subject to read the original memoirs on that subject,
for science is always most completely assimilated when
* Life and Letters of Faraday, vol. i. p. 395.
xiv PREFACE.
it is in the nascent state, and in the case of Faraday's
Researches this is comparatively easy, as they are
published in a separate form, and may be read con
secutively. If by anything I have here written I
may assist any student in understanding Faraday's
modes of thought and expression, I shall regard it as
the accomplishment of one of my principal aims — to
communicate to others the same 'delight which I have
found myself in reading Faraday's Researches.
The description of the phenomena, and the ele
mentary parts of the theory of each subject, will be
found in the earlier chapters of each of the four Parts
into which this treatise is divided. The student will
find in these chapters enough to give him an elementary
acquaintance with the whole science.
The remaining chapters of each Part are occupied
with the higher parts of the theory, the processes of
numerical calculation, and the instruments and methods
of experimental research.
The relations between electromagnetic phenomena
and those of radiation, the theory of molecular electric
currents, and the results of speculation on the nature
of action at a distance, are treated of in the last four
chapters of the second volume.
Feb. 1, 1873.
PREFACE TO THE SECOND EDITION.
WHEN I was asked to read the proof-sheets of the
second edition of the Electricity and Magnetism the
work of printing had already reached the ninth chapter,
the greater part of which had been revised by the
author.
Those who are familiar with the first edition will see
from a comparison with the present how extensive were
the changes intended by Professor Maxwell both in the
substance and in the treatment of the subject, and how
much this edition has suffered from his premature death.
The first nine chapters were in some cases entirely re
written, much new matter being added and the former
contents rearranged and simplified.
From the ninth chapter onwards the present edition
is little more than a reprint. The only liberties I have
taken have been in the insertion here and there of a
step in the mathematical reasoning where it seemed to
be an advantage to the reader, and of a few foot-notes
on parts of the subject which my own experience or that
of pupils attending my classes shewed to require further
elucidation. These footnotes are in square brackets.
There were two parts of the subject in the treatment
xvi PREFACE.
of which it was known to me that the Professor con
templated considerable changes : viz. the mathematical
theory of the conduction of electricity in a network of
wires, and the determination of coefficients of induction
in coils of wire. In these subjects I have not found
myself in a position to add, from the Professor's notes,
anything substantial to the work as it stood in the
former edition, with the exception of a numerical table,
printed in vol. ii, pp. 317-319. This table will be found
very useful in calculating coefficients of induction in
circular coils of wire.
In a work so original, and containing so many details
of new results, it was impossible but that there should
be a few errors in the first edition. I trust that in
the present edition most of these will be found to have
been corrected. I have the greater confidence in ex
pressing this hope as, in reading some of the proofs, I
have had the assistance of various friends conversant
with the work, among whom I may mention particularly
my brother Professor Charles Niven, and Mr. J. J. Thom
son, Fellow of Trinity College, Cambridge.
W. D. NIVEN.
TRINITY COLLEGE, CAMBBIDGE,
Oct. i, 1881.
CONTENTS,
PRELIMINARY.
ON THE MEASUKEMENT OP QUANTITIES.
Art. Page
1. The expression of a quantity consists of two factors, the nu
merical value, and the name of the concrete unit 1
2. Dimensions of derived units 1
3-5. The three fundamental units — Length, Time and Mass . . 2, 3
6. Derived units 5
7. Physical continuity and discontinuity 6
8. Discontinuity of a function of more than one variable . . . . 7
9. Periodic and multiple functions 8
10. Relation of physical quantities to directions in space . . . . 8
11. Meaning of the words Scalar and Vector 9
12. Division of physical vectors into two classes, Forces and Fluxes 10
13. Relation between corresponding vectors of the two classes . . 11
14. Line-integration appropriate to forces, surface-integration to
fluxes 12
15. Longitudinal and rotational vectors 13
16. Line-integrals and potentials 13
17. Hamilton's expression for the relation between a force and its
potential 15
18. Cyclic regions and geometry of position 16
19. The potential in an acyclic region is single valued 17
20. System of values of the potential in a cyclic region 18
21. Surface-integrals 19
22. Surfaces, tubes, and lines of flow .. 21
23. Right-handed and left-handed relations in space 24
24. Transformation of a line-integral into a surface-integral . . . . 25
25. Effect of Hamilton's operation V on a vector function . . . . 28
26. Nature of the operation V2 29
VOL. I. b
xvm CONTENTS.
PAET I.
ELECTROSTATICS.
CHAPTER I.
DESCRIPTION OF PHENOMENA.
Art. page
27. Electrification by friction. Electrification is of two kinds, to
which the names of Vitreous and Resinous, or Positive and
Negative, have been given 31
28. Electrification by induction 32
29. Electrification by conduction. Conductors and insulators . . 33
30. In electrification by friction the quantity of the positive elec
trification is equal to that of the negative electrification . . 34
31. To charge a vessel with a quantity of electricity equal and
opposite to that of an excited body 34
32. To discharge a conductor completely into a metallic vessel . . 35
33. Test of electrification- by gold-leaf electroscope 35
34. Electrification, considered as a measurable quantity, may be
called Electricity 36
35. Electricity may be treated as a physical quantity 37
36. Theory of Two fluids 38
37. Theory of One fluid 40
38. Measurement of the force between electrified bodies 41
39. Relation between this force and the quantities of electricity . . 42
40. Variation of the force with the distance 43
41,42. Definition of the electrostatic unit of electricity. — Its
dimensions 4^, 4
43. Proof of the law of electric force 44
44. Electric field 45
45. Electromotive force and potential 46
46. Equipotential surfaces. Example of their use in reasoning
about electricity 47
47. Lines of force
48. Electric tension
49. Electromotive force
50. Capacity of a conductor. Electric Accumulators 49
51. Properties of bodies. — Resistance 50
52. Specific Inductive capacity of a dielectric 52
53. ' Absorption' of electricity 53
CONTENTS. xix
Art. Page
54. Impossibility of an absolute charge 54
55. Disruptive discharge. — Glow 54
56. Brush 57
57. Spark 57
58. Electrical phenomena of Tourmaline 58
59. Plan of the treatise, and sketch of its results 59
60. Electric polarization and displacement 61
61. The motion of electricity analogous to that of an incompressible
fluid _ 64
62. Peculiarities of the theory of this treatise 65
CHAPTER II.
ELEMENTARY MATHEMATICAL THEOEY OF ELECTRICITY.
63. Definition of electricity as a mathematical quantity 68
64. Volume-density, surface-density, and line-density 68
65. Definition of the electrostatic unit of electricity 70
66. Law of force between electrified bodies 70
67. Resultant force between two bodies 71
68. Resultant intensity at a point 71
69. Line-integral of electric intensity ; electromotive force . . . . 72
70. Electric potential 73
71. Resultant intensity in terms of the potential . . . . . . . . 74
72. The potential of all points of a conductor is the same . . . . 75
73. Potential due to an electrified system 76
74 a. Proof of the law of the inverse square. Cavendish's experiments 76
74 b. Cavendish's experiments repeated in a modified form . . . . 77
74 c, d, e. Theory of the experiments 79-81
75. Surface-integral of electric induction 82
76. Induction through a closed surface due to a single centre of
force 83
77. Poisson's extension of Laplace's equation 84
78 a, b, c. Conditions to be fulfilled at an electrified surface . . 85-88
79. Resultant force on an electrified surface 88
80. The electrification of a conductor is entirely on the surface . . 90
81. A distribution of electricity on lines or points is physically
impossible 91
82. Lines of electric induction 92
83 a. Specific inductive capacity 94
83 b. Apparent distribution of electricity 94
xx CONTENTS.
CHAPTER III.
ON ELECTRICAL WORK AND ENERGY IN A SYSTEM OF CONDUCTORS.
Art. Page
84. On the superposition of electrified systems. Expression for the
energy of a system of conductors 96
85 a. Change of the energy in passing from one state to another . . 97
856. Relations between the potentials and the charges 98
86. Theorems of reciprocity 98
87. Theory of a system of conductors. Coefficients of potential. Ca
pacity. Coefficients of induction 100
88. Dimensions of the coefficients 103
89 a. Necessary relations among the coefficients of potential . . . . 103
896. Relations derived from physical considerations 104
89 c. Relations among coefficients of capacity and induction . . . . 105
89 d. Approximation to capacity of one conductor 105
89 e. The coefficients of potential changed by a second conductor . . 106
90 a. Approximate determination of the coefficients of capacity and
induction of two conductors 107
90 b. Similar determination for two condensers 107
91. Relative magnitudes of coefficients of potential 109
92. And of induction 110
93 a. Mechanical force on a conductor expressed in terms of the
charges of the different conductors of the system 110
936. Theorem in quadratic functions Ill
93 c. Work done by the electric forces during the displacement of a
system when the potentials are maintained constant . . . . Ill
94. Comparison of electrified systems 112
CHAPTER IV.
GENERAL THEOREMS.
95 a, 6. Two opposite methods of treating electrical questions 115, 116
96 a. Green's Theorem 118
96 6. "When one of the functions is many valued 120
96 c. When the region is multiply connected 120
96 d. When one of the functions becomes infinite in the region . . 121
97 a, 6. Applications of Green's method 123, 124
98. Green's Function 125
99 a. Energy of a system expressed as a volume integral . . . . 126
CONTENTS. xxi
Art.
99 b. Proof of unique solution for the potential when its value is
given at every point of a closed surface ........ 127
100 a-e. Thomson's Theorem .............. 129-132
101 a-h. Expression for the energy when the dielectric constants
are different in different directions. Extension of Green's
Theorem to a heterogeneous medium ........ 133-137
102 a. Method of finding limiting values of electrical coefficients . . 138
102 b. Approximation to the solution of problems of the distribution
of electricity on conductors at given potentials ...... 140
102 c. Application to the case of a condenser with slightly curved
plates ...................... 142
CHAPTER V.
MECHANICAL ACTION BETWEEN TWO ELECTRICAL SYSTEMS.
103. Expression for the force at any point of the medium in terms
of the potentials arising from the presence of the two systems 144
104. In terms of the potential arising from both systems . . . . 145
105. Nature of the stress in the medium which would produce the
same force
106. Further determination of the type of stress 148
107. Modification of the expressions at the surface of a conductor. . 149
108. Discussion of the integral of Art. 104 expressing the force
when taken over all space 151
109. Statements of Faraday relative to the longitudinal tension and
lateral pressure of the lines of force 153
110. Objections to stress in a fluid considered 153
111. Statement of the theory of electric polarization 154
CHAPTER VI.
POINTS AND LINES OF EQUILIBRIUM.
112. Conditions for a point of equilibrium 157
113. Number of points of equilibrium 158
114. At a point or line of equilibrium there is a conical point or a
line of self-intersection of the equipotential surface . . . . 159
115. Angles at which an equipotential surface intersects itself . . 160
116. The equilibrium of an electrified body cannot be stable . . . . 161
xxn CONTENTS.
CHAPTER VII.
FOEMS OP EQUIPOTENTIAL SUEFACES AND LINES OF FLOW.
Art. page
117. Practical importance of a knowledge of these forms in simple
cases 164
118. Two electrified points, ratio 4:1. (Fig. I) 165
119. Two electrified points, ratio 4 : — 1. (Fig. II) 166
1 20. An electrified point in a uniform field of force (Fig. Ill) . . 167
121. Three electrified points. Two spherical equipotential sur
faces. (Fig. IV) 167
122. Faraday's use of the conception of lines of force 168
123. Method employed in drawing the diagrams 169
CHAPTER VIII.
SIMPLE CASES OF ELECTEIFICATION.
124. Two parallel planes 172
125. Two concentric spherical surfaces 174
126. Two coaxal cylindric surfaces 176
127. Longitudinal force on a cylinder, the ends of which are sur
rounded by cylinders at different potentials 177
CHAPTER IX.
SPHEEICAL HAEMONICS.
128. Heine, Todhunter, Ferrers 179
129 a. Singular points 179
1296. Definition of an axis 180
129 c. Construction of points of different orders 181
129 d. Potential of such points. Surface harmonics Yn .. ..182
130 a. Solid harmonics. Hn = rnYn 182
130 b. There are 2^+1 independent constants in a solid harmonic
of the nth order 183
131 a. Potential due to a spherical shell 184
1 3 1 b. Expressed in harmonics 184
131 c. Mutual potential of shell and external system 185
132. Value of J'fYmYnds 186
133. Trigonometrical expressions for Yn 187
134. Value of ffYmYnds, when m — n 189
135 a. Special case when Ym is a zonal harmonic 190
135 b. Laplace's expansion of a surface harmonic 190
136. Conjugate harmonics 192
CONTENTS. xxiii
Art. Page
137. Standard harmonics of any order 192
138. Zonal harmonics 193
139. Laplace's coefficient or Biaxal harmonic 194
140 a. Tesseral harmonics. Their trigonometrical expansion .. 194
140 b. Notations used by various authors .. 197
140 c. Forms of the tesseral and sectorial harmonics 197
141. Surface integral of the square of a tesseral harmonic . . . . 198
142 a. Determination of a given tesseral harmonic in the expansion
of a function 199
142 b. The same in terms of differential coefficients of the function. . 199
143. Figures of various harmonics 200
144 a. Spherical conductor in a given field of force 201
144 b. Spherical conductor in a field for which Green's function is
known 201-
145 a. Distribution of electricity on a nearly spherical conductor . . 204
145 b. When acted on by external electrical force . . . . . . . . 206
145 c. When enclosed in a nearly spherical and nearly concentric
vessel 207
146. Equilibrium of electricity on two spherical conductors . . . . 208
CHAPTER X.
CONFOCAL SURFACES OF THE SECOND DEGREE.
147. The lines of intersection of two systems and their intercepts
by the third system 215
148. The characteristic equation of V in terms of ellipsoidal co
ordinates 216
149. Expression of a, /3, y in terms of elliptic functions 217
150. Particular solutions of electrical distribution on the confocal
surfaces and their limiting forms 218
151. Continuous transformation into a figure of revolution about
the axis of z 221
152. Transformation into a figure of revolution about the axis of x. . 222
153. Transformation into a system of cones and spheres 223
154. Confocal paraboloids 223
CHAPTER XI.
THEORY OF ELECTRIC IMAGES.
155. Thomson's method of electric images 226
156. When two points are oppositely and unequally electrified, the
surface for which the potential is zero is a sphere . . . . 227
xxiv CONTENTS,
Art. Page
157. Electric images 228
158. Distribution of electricity on the surface of the sphere . . . . 230
159. Image of any given distribution of electricity 231
160. Resultant force between an electrified point and sphere . . . . 232
161. Images in an infinite plane conducting surface 234
162. Electric inversion 235
163. Geometrical theorems about inversion 236
164. Application of the method to the problem of Art. 158 . . . . 237
165. Finite systems of successive images 238
166. Case of two spherical surfaces intersecting at an angle - . . 240
n
167. Enumeration of the cases in which the number of images is
finite 241
168. Case of two spheres intersecting orthogonally 242
169. Case of three spheres intersecting orthogonally 245
170. Case of four spheres intersecting orthogonally 246
171. Infinite series of images. Case of two concentric spheres . . 247
172. Any two spheres not intersecting each other 249
173. Calculation of the coefficients of capacity and induction , . . . 251
174. Calculation of the charges of the spheres, and of the force
between them 253
175. Distribution of electricity on two spheres in contact. Proof
sphere 255
176. Thomson's investigation of an electrified spherical bowl. . . . 257
177. Distribution on an ellipsoid, and on a circular disk at po
tential V 257
178. Induction on an uninsulated disk or bowl by an electrified
point in the continuation of the plane or spherical surface . . 258
179. The rest of the sphere supposed uniformly electrified . . . . 259
180. The bowl maintained at potential V and uninfluenced . . . . 259
181. Induction on the bowl due to a point placed anywhere . . . . 260
CHAPTER XII.
CONJUGATE FUNCTIONS IN TWO DIMENSIONS.
182. Cases in which the quantities are functions of x and y only . . 262
183. Conjugate functions 263
184. Conjugate functions may be added or subtracted 264
185. Conjugate functions of conjugate functions are themselves
conjugate 265
186. Transformation of Poisson's equation 267
187. Additional theorems on conjugate functions 268
CONTENTS. XXV
Art. Page
188. Inversion in two dimensions 268
189. Electric images in two dimensions 269
190. Neumann's transformation of this case 270
191. Distribution of electricity near the edge of a conductor formed
by two plane surfaces 272
192. Ellipses and hyperbolas. (Fig. X) 273
193. Transformation of this case. (Fig. XI) 274
194. Application to two cases of the flow of electricity in a con
ducting sheet 276
195. Application to two cases of electrical induction 276
196. Capacity of a condenser consisting of a circular disk between
two infinite planes 277
197. Case of a series of equidistant planes cut off by a plane at right
angles to them 279
198. Case of a furrowed surface 280
199. Case of a single straight groove 281
200. Modification of the results when the groove is circular . . . . 281
201. Application to Sir W. Thomson's guard-ring 284
202. Case of two parallel plates cut off by a perpendicular plane.
(Fig. XII) 285
203. Case of a grating of parallel wires. (Fig. XIII) 286
204. Case of a single electrified wire transformed into that of the
grating 287
205. The grating used as a shield to protect a body from electrical
influence 288
206. Method of approximation applied to the case of the grating . . 289
CHAPTER XIII.
ELECTROSTATIC INSTRUMENTS.
207. The frictional electrical machine 292
208. The electrophorus of Volta 293
209. Production of electrification by mechanical work. — Nicholson's
Revolving Doubler 294
210. Principle of Varley's and Thomson's electrical machines. . . . 294
211. Thomson's water-dropping machine. . „ 297
212. Holtz's electrical machine 298
213. Theory of regenerators applied to electrical machines . . . . 298
214. On electrometers and electroscopes. Indicating instruments
and null methods. Difference between registration and mea
surement 300
215. Coulomb's Torsion Balance for measuring charges 301
XXVI CONTENTS.
Art. Page
216. Electrometers for measuring potentials. Snow-Harris's and
Thomson's 304
217. Principle of the guard-ring. Thomson's Absolute Electrometer 305
218. Heterostatic method 308
219. Self-acting electrometers. — Thomson's Quadrant Electrometer 309
220. Measurement of the electric potential of a small body . . . . 312
221. Measurement of the potential at a point in the air 313
222. Measurement of the potential of a conductor without touching it 314
223. Measurement of the 'superficial density of electrification. The
proof plane 315
224. A hemisphere used as a test 316
225. A circular disk 317
226. On electric accumulators. The Leyden jar 319
227. Accumulators of measurable capacity 320
228. The guard-ring accumulator 321
229. Comparison of the capacities of accumulators 323
PAET II
ELECTRO KINEMATICS.
CHAPTER I.
THE ELECTBIC CUKEENT.
-\
230. Current produced when conductors are discharged 326
231. Transference of electrification 326
232. Description of the voltaic battery 327
233. Electromotive force 328
234. Production of a steady current 328
235. Properties of the current 329
236. Electrolytic action 329
237. Explanation of terms connected with electrolysis 330
238. Different modes of passage of the current 330
239. Magnetic action of the current 331
240. The Galvanometer 332
CONTENTS. xxvn
CHAPTER II.
CONDUCTION AND RESISTANCE.
Art. Page
241. Ohm's Law 333
242. Generation of heat by the current. Joule's Law 334
243. Analogy between the conduction of electricity and that of heat 335
244. Differences between the two classes of phenomena 335
245. Faraday's doctrine of the impossibility of an absolute charge. . 336
CHAPTER III.
ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT.
246. Volta's law of the contact force between different metals at the
same temperature 337
247. Effect of electrolytes . . , 338
248. Thomson's voltaic current in which gravity performs the part
of chemical action 338
249. Peltier's phenomenon. Deduction of the thermoelectric elec
tromotive force at a junction 338
250. Seebeck's discovery of thermoelectric currents 340
251. Magnus's law of a circuit of one metal 340
252. Cumming's discovery of thermoelectric inversions 342
253. Thomson's deductions from these facts, and discovery of the
reversible thermal effects of electric currents in copper and
in iron 342
254. Tait's law of the electromotive force of a thermoelectric pair. . 343
CHAPTER IV.
ELECTROLYSIS.
255. Faraday's law of electrochemical equivalents 345
256. Clausius's theory of molecular agitation 347
257. Electrolytic polarization 347
258. Test of an electrolyte by polarization 348
259. Difficulties in the theory of electrolysis 348
260. Molecular charges 349
261. Secondary actions observed at the electrodes 351
262. Conservation of energy in electrolysis 353
263. Measurement of chemical affinity as an electromotive force . . 354
xxviu CONTENTS.
CHAPTER V.
ELECTROLYTIC POLARIZATION.
Art. Page
264. Difficulties of applying Ohm's law to electrolytes 356
265. Ohm's law nevertheless applicable .. , 356
266. The effect of polarization distinguished from that of resistance 356
267. Polarization due to the presence of the ions at the electrodes.
The ions not in a free state 357
268. Relation between the electromotive force of polarization and
the state of the ions at the electrodes 358
269. Dissipation of the ions and loss of polarization 359
270. Limit of polarization 359
271. Ritter's secondary pile compared with the Leyden jar . . . . 360
272. Constant voltaic elements. — Daniell's cell . 363
CHAPTER VI.
MATHEMATICAL THEORY OF THE DISTRIBUTION OF ELECTRIC CURRENTS.
273. Linear conductors •• 367
274. Ohm's Law 367
275. Linear conductors in series 367
276. Linear conductors in multiple arc 368
277. Resistance of conductors of uniform section 369
278. Dimensions of the quantities involved in Ohm's law . . . . 370
279. Specific resistance and conductivity in electromagnetic measure 371
280. Linear systems of conductors in general 371
281. Reciprocal property of any two conductors of the system . . 373
282 a, b. Conjugate conductors 373, 374
283. Heat generated in the system 374
284. The heat is a minimum when the current is distributed ac
cording to Ohm's law 375
CHAPTER VII.
CONDUCTION IN THREE DIMENSIONS.
285. Notation 376
286. Composition and resolution of electric currents 376
287. Determination of the quantity which flows through any surface 377
288. Equation of a surface of flow 378
CONTENTS. xxix
Art. Page
289. Eelation between any three systems of surfaces of flow . . . . 378
290. Tubes of flow 378
291. Expression for the components of the flow in terms of surfaces
of flow 379
292. Simplification of this expression by a proper choice of para
meters .. .. 379
293. Unit tubes of flow used as a complete method of determining
the current 379
294. Current-sheets and current-functions 380
295. Equation of ' continuity' 380
296. Quantity of electricity which flows through a given surface . . 382
CHAPTER VIII.
EESISTANCE AND CONDUCTIVITY IN THEEE DIMENSIONS.
297. Equations of resistance 383
298. Equations of conduction , 384
299. Rate of generation of heat 384
300. Conditions of stability 385
301. Equation of continuity in a homogeneous medium 386
302. Solution of the equation 386
303. Theory of the coefficient T. It probably does not exist .. 387
304. Generalized form of Thomson's theorem 388
305. Proof without symbols 389
306. Strutt's method applied to a wire of variable section. — Lower
limit of the value of the resistance 390
307. Higher limit 393
308. Lower limit for the correction for the ends of the wire .. .. 395
309. Higher limit 396
CHAPTER IX.
CONDUCTION THEOUGH HETEEOGENEOUS MEDIA.
310. Surface-conditions 398
311. Spherical surface 400
312. Spherical shell 401
313. Spherical shell placed in a field of uniform flow 402
314. Medium in which small spheres are uniformly disseminated .. 403
315. Images in a plane surface 404
316. Method of inversion not applicable in three dimensions .. .. 405
317. Case of conduction through a stratum bounded by parallel
planes 405
xxx CONTENTS.
Art. Page
318. Infinite series of images. Application to magnetic induction .. 406
319. On stratified conductors. Coefficients of conductivity of a con
ductor consisting of alternate strata of two different substances 407
320. If neither of the substances has the rotatory property denoted
by T the compound conductor is free from it 408
321. If the substances are isotropic the direction of greatest resist
ance is normal to the strata .. .. 408
322. Medium containing parallelepipeds of another medium .. .. 409
323. The rotatory property cannot be introduced by means of con
ducting channels 410
324. Construction of an artificial solid having given coefficients of
longitudinal and transverse conductivity 411
CHAPTER X.
CONDUCTION IN DIELECTRICS.
325. In a strictly homogeneous medium there can be no internal
charge 412
326. Theory of a condenser in which the dielectric is not a perfect
insulator 413
327. No residual charge due to simple conduction 414
328. Theory of a composite accumulator 414
329. Residual charge and electrical absorption 416
330. Total discharge 418
331. Comparison with the conduction of heat 419
332. Theory of telegraph cables and comparison of the equations
with those of the conduction of heat 421
333. Opinion of Ohm on this subject 422
334. Mechanical illustration of the properties of a dielectric .. .. 423
CHAPTER XI.
MEASUREMENT OF THE ELECTRIC RESISTANCE OF CONDUCTORS.
335. Advantage of using material standards of resistance in electrical
measurements 426
336. Different standards which have been used and different systems
which have been proposed 426
337. The electromagnetic system of units 427
338. Weber's unit, and the British Association unit or Ohm .. .. 427
339. Professed value of the Ohm 10,000,000 metres per second .. 427
CONTENTS. xxxi
Art. Page
340. Reproduction of standards 428
341. Forms of resistance coils 429
342. Coils of great resistance 430
343. Arrangement of coils in series 430
344. Arrangement in multiple arc 431
345. On the comparison of resistances. (1) Ohm's method .. .. 432
346. (2) By the differential galvanometer .. 432
347. (3) By Wheatstone's Bridge 436
348. Estimation of limits of error in the determination 437
349. Best arrangement of the conductors to be compared .. .. 438
350. On the use of Wheatstone's Bridge 440
351. Thomson's method for small resistances 442
352. Matthiessen and Hockin's method for small resistances .. .. 444
353. Comparison of great resistances by the electrometer .. .. 446
354. By accumulation in a condenser 447
355. Direct electrostatic method 447
356. Thomson's method for the resistance of a galvanometer .. .. 448
357. Mance's method of determining the resistance of a battery ,. 449
358. Comparison of electromotive forces 452
CHAPTER XII.
ELECTKIC RESISTANCE OF SUBSTANCES.
359. Metals, electrolytes, and dielectrics 454
360. Resistance of metals 455
361. Resistance of mercury 456
362. Table of resistance of metals 457
363. Resistance of electrolytes 458
364. Experiments of Paalzow 458
365. Experiments of Kohlrausch and Nippoldt 459
366. Resistance of dielectrics • •• 460
367. Gutta-percha 462
368. Glass 462
369. Gases 463
370. Experiments of Wiedemann and Riihlmann 463
ELECTEICITY AND MAGNETISM,
PEELIMINARY.
ON THE MEASUREMENT OF QUANTITIES.
1.] EVEEY expression of a Quantity consists of two factors or
components. One of these is the name of a certain known quan
tity of the same kind as the quantity to be expressed, which is
taken as a standard of reference. The other component is the
number of times the standard is to be taken in order to make up
the required quantity. The standard quantity is technically called
the Unit, and the number is called the Numerical Value of the
quantity.
There must be as many different units as there are different
kinds of quantities to be measured, but in all dynamical sciences
it is possible to define these units in terms of the three funda
mental units of Length, Time, and Mass. Thus the units of area
and of volume are defined respectively as the square and the cube
whose sides are the unit of length.
Sometimes, however, we find several units of the same kind
founded on independent considerations. Thus the gallon, or the
volume of ten pounds of water, is used as a unit of capacity as well
as the cubic foot. The gallon may be a convenient measure in
some cases, but it is not a systematic one, since its numerical re
lation to the cubic foot is not a round integral number.
2.] In framing a mathematical system we suppose the funda
mental units of length, time, and mass to be given, and deduce
all the derivative units from these by the simplest attainable de
finitions.
The formulae at which we arrive must be such that a person
VOL. I. B
2 PRELIMINARY. [3.
V
of any nation, by substituting for the different symbols the nu
merical values of the quantities as measured by his own national
units, would arrive at a true result.
Hence, in all scientific studies it is of the greatest importance
to employ units belonging to a properly denned system, and to
know the relations of these units to the fundamental units, so that
we may be able at once to transform our results from one system to
another.
This is most conveniently done by ascertaining the dimensions
of every unit in terms of the three fundamental units. When a
given unit varies as the ^th power of one of these units, it is said
to be of n dimensions as regards that unit.
For instance, the scientific unit of volume is always the cube
whose side is the unit of length. If the unit of length varies,
the unit of volume will vary as its third power, and the unit of
volume is said to be of three dimensions with respect to the unit of
length.
A knowledge of the dimensions of units furnishes a test which
ought to be applied to the equations resulting from any lengthened
investigation. The dimensions of every term of such an equa
tion, with respect to each of the three fundamental units, must
be the same. If not, the equation is absurd, and contains some
error, as its interpretation would be different according to the arbi
trary system of units which we adopt *.
The Three Fundamental Units.
3.] (1) Length. The standard of length for scientific purposes
in this country is one foot, which is the third part of the standard
yard preserved in the Exchequer Chambers.
In France, and other countries which have adopted the metric
system, it is the metre. The metre is theoretically the ten mil
lionth part of the length of a meridian of the earth measured
from the pole to the equator ; but practically it is the length of
a standard preserved in Paris, which was constructed by Borda
to correspond, when at the temperature of melting ice, with the
value of the preceding length as measured by Delambre. The metre
has not been altered to correspond with new and more accurate
measurements of the earth, but the arc of the meridian is estimated
in terms of the original metre.
* The theory of dimensions was first stated by Fourier, Theorie de Chaleur, § 160.
5-] THE THREE FUNDAMENTAL UNITS. 3
In astronomy the mean distance of the earth from the sun is
sometimes taken as a unit of length.
In the present state of science the most universal standard of
length which we could assume would be the wave length in vacuum
of a particular kind of light, emitted by some widely diffused sub
stance such as sodium, which has well-defined lines in its spectrum.
Such a standard would be independent of any changes in the di
mensions of the earth, and should be adopted by those who expect
their writings to be more permanent than that body.
In treating of the dimensions of units we shall call the unit of
length [Z/]. If I is the numerical value of a length, it is under
stood to be expressed in terms of the concrete unit [Z], so that
the actual length would be fully expressed by I [It].
4.] (2) Time. The standard unit of time in all civilized coun
tries is deduced from the time of rotation of the earth about its
axis. The sidereal day, or the true period of rotation of the earth,
can be ascertained with great exactness by the ordinary observa
tions of astronomers ; and the mean solar day can be deduced
from this by our knowledge of the length of the year.
The unit of time adopted in all physical researches is one second
of mean solar time.
In astronomy a year is sometimes used as a unit of time. A
more universal unit of time might be found by taking the periodic
time of vibration of the particular kind of light whose wave length
is the unit of length.
We shall call the concrete unit of time [T7], and the numerical
measure of time t.
5.] (3) Mass. The standard unit of mass is in this country the
avoirdupois pound preserved in the Exchequer Chambers. The
grain, which is often used as a unit, is defined to be the 7000th
part of this pound.
In the metrical system it is the gramme, which is theoretically
the mass of a cubic centimetre of distilled water at standard tem
perature and pressure, but practically it is the thousandth part
of the standard kilogramme preserved in Paris.
The accuracy with which the masses of bodies can be com
pared by weighing is far greater than that hitherto attained in
the measurement of lengths, so that all masses ought, if possible,
to be compared directly with the standard, and not deduced from
experiments on water.
In descriptive astronomy the mass of the sun or that of the
4 PRELIMINARY. [5.
earth is sometimes taken as a unit, but in the dynamical theory
of astronomy the unit of mass is deduced from the units of time
and length, combined with the fact of universal gravitation. The
astronomical unit of mass is that mass which attracts another
body placed at the unit of distance so as to produce in that body
the unit of acceleration.
In framing a universal system of units we may either deduce
the unit of mass in this way from those of length and time
already defined, and this we can do to a rough approximation in
the present state of science ; or, if we expect* soon to be able to
determine the mass of a single molecule of a standard substance,
we may wait for this determination before fixing a universal
standard of mass.
We shall denote the concrete unit of mass by the symbol [M]
in treating of the dimensions of other units. The unit of mass
will be taken as one of the three fundamental units. When, as
in the French system, a particular substance, water, is taken as
a standard of density, then the unit of mass is no longer inde
pendent, but varies as the unit of volume, or as [I/3].
If, as in the astronomical system, the unit of mass is defined
with respect to its attractive power, the dimensions of [M] are
For the acceleration due to the attraction of a mass m at a
ay*
distance r is by the Newtonian Law -j . Suppose this attraction
to act for a very small time t on a body originally at rest, and to
cause it to describe a space s, then by the formula of Galileo,
O
whence m = 2^-. Since r and s are both lengths, and t is a
t
time, this equation cannot be true unless the dimensions of m, are
[L*T-2]. The same can be shewn from any astronomical equa
tion in which the mass of a body appears in some but not in all
of the terms f.
* See Prof. J. Loschmidt, ' Zur Grosse der Luftmolecule,' Academy of Vienna,
Oct. 12, 1865 ; G. J. Stoney on 'The Internal Motions of Gases,' Phil. Mag., Aug.
1868 ; and Sir W. Thomson on ' The Size of Atoms,5 Nature, March 31, 1870.
t If a centimetre and a second are taken as units, the astronomical unit of mass
would be about 1-537 x 107 grammes, or 15'37 tonnes according to Baily's repetition
of Cavendish's experiment. Baily adopts 5'6604 as the result of all his experiments
t* the mean density of the earth, and this, with the values used by Baily for the
dimensions of the earth and the intensity of gravitation at its surface, gives the above
>-alue as the direct result of his experiments.
*f '
6.] DERIVED UNITS. 5
Derived Units.
6.] The unit of Velocity is that velocity in which unit of length
is described in unit of time. Its dimensions are [X27"1].
If we adopt the units of length and time derived from the
vibrations of light, then the unit of velocity is the velocity of
light.
The unit of Acceleration is that acceleration in which the velo
city increases by unity in unit of time. Its dimensions are [Z27"2].
The unit of Density is the density of a substance which contains
unit of mass in unit of volume. Its dimensions are [JfZ/~3],
The unit of Momentum is the momentum of unit of mass moving
with unit of velocity. Its dimensions are \_MLT~l~\.
The unit of Force is the force which produces unit of momentum
in unit of time. Its dimensions are [MLT~2~].
This is the absolute unit of force, and this definition of it is
implied in every equation in Dynamics. Nevertheless, in many
text books in which these equations are given, a different unit of
force is adopted, namely, the weight of the national unit of mass;
and then, in order to satisfy the equations, the national unit of mass
is itself abandoned, and an artificial unit is adopted as the dynamical
unit, equal to the national unit divided by the numerical value of
the intensity of gravity at the place. In this way both the unit of
force and the unit of mass are made to depend on the value of the
intensity of gravity, which varies from place to place, so that state
ments involving these quantities are not complete without a know
ledge of the intensity of gravity in the places where these statements
were found to be true.
The abolition, for all scientific purposes, of this method of measur
ing forces is mainly due to the introduction by Gauss of a general
system of making observations of magnetic force in countries in
which the intensity of gravity is different. All such forces are
now measured according to the strictly dynamical method deduced
from our definitions, and the numerical results are the same in
whatever country the experiments are made.
The unit of Work is the work done by the unit of force acting
through the unit of length measured in its own direction. Its
dimensions are [MIPT"*].
The Energy of a system, being its capacity of performing work,
is measured by the work which the system is capable of performing
by the expenditure of its whole energy.
6 PRELIMINAKY. [7.
The definitions of other quantities, and of the units to which
they are referred, will be given when we require them.
In transforming the values of physical quantities determined in
terms of one unit, so as to express them in terms of any other unit
of the same kind, we have only to remember that every expres
sion for the quantity consists of two factors, the unit and the nu
merical part which expresses how often the unit is to be taken.
Hence the numerical part of the expression varies inversely as the
magnitude of the unit, that is, inversely as the various powers of
the fundamental units which are indicated by the dimensions of the
derived unit.
On Physical Continuity and Discontinuity.
7.] A quantity is said to vary continuously if, when it passes
from one value to another, it assumes all the intermediate values.
We may obtain the conception of continuity from a consideration
of the continuous existence of a particle of matter in time and space.
Such a particle cannot pass from one position to another without
describing a continuous line in space, and the coordinates of its
position must be continuous functions of the time.
In the so-called ' equation of continuity,' as given in treatises
on Hydrodynamics, the fact expressed is that matter cannot appear
in or disappear from an element of volume without passing in or out
through the sides of that element.
A quantity is said to be a continuous function of its variables
if, when the variables alter continuously, the quantity itself alters
continuously.
Thus, if u is a function of a?, and if, while x passes continuously
from #0 to a?!, u passes continuously from UQ to u19 but when sc
passes from ^ to #2, u passes from u{ to uz, u{ being different from
%, then u is said to have a discontinuity in its variation with
respect to x for the value x = xl9 because it passes abruptly from ^
to u{ while x passes continuously through #r
If we consider the differential coefficient of u with respect to x for
the value x=ce1 as the limit of the fraction
when x.2 and XQ are both made to approach ^ without limit, then,
if a?0 and #2 are always on opposite sides of aslt the ultimate value of
the numerator will be u^—u^ and that of the denominator will
be zero. If u is a quantity physically continuous, the discontinuity
8.] CONTINUITY AND DISCONTINUITY. 7
can exist only with respect to the particular variable x. We must
in this case admit that it has an infinite differential coefficient
when 0? = ^. If u is not physically continuous, it cannot be dif
ferentiated at all.
It is possible in physical questions to get rid of the idea of
discontinuity without sensibly altering the conditions of the case.
If OCQ is a very little less than x^ , and #2 a very little greater than
#!, then u0 will be very nearly equal to u^ and n2 to u{ . We
may now suppose u to vary in any arbitrary but continuous manner
from u0 to u2 between the limits #0 and a?2. In many physical
questions we may begin with a hypothesis of this kind, and then
investigate the result when the values of #0 and #2 are made to
approach that of x± and ultimately to reach it. If the result is
independent of the arbitrary manner in which we have supposed
u to vary between the limits, we may assume that it is true when it,
is discontinuous.
Discontinuity of a Function of more than One Variable.
8.] If we suppose the values of all the variables except x to be
constant, the discontinuity of the function will occur for particular
values of #, and these will be connected with the values of the
other variables by an equation which we may write
4> = <J>(0,y,s,&c.) = 0.
The discontinuity will occur when $ = 0. When <£ is positive the
function will have the form F2 (x, yt z, &c.). When <£ is negative
it will have the form F1 (x,y9 z, &c.). There need be no necessary
relation between the forms Fl and F2.
To express this discontinuity in a mathematical form, let one of
the variables, say #, be expressed as a function of $ and the other
variables, and let F1 and F2 be expressed as functions of $,y, z, &c.
We may now express the general form of the function by any
formula which is sensibly equal to F2 when $ is positive, and to
F1 when $ is negative. Such a formula is the following —
F
As long as n is a finite quantity, however great, F will be a
continuous function, but if we make n infinite F will be equal to
F2 when <£ is positive, and equal to F^ when <p is negative.
8 PRELIMINARY. [9.
Discontinuity of the Derivatives of a Continuous Function.
The first derivatives of a continuous function may be discon
tinuous. Let the values of the variables for which the discon
tinuity of the derivatives occurs be connected by the equation
<£ = $(x,y,z...) = 0,
and let Fl and F2 be expressed in terms of <£ and ^—1 other
variables, say (y, z . . .).
Then, when <£ is negative, Fl is to be taken, and when <£ is
positive F2 is to be taken, and, since F is itself continuous, when
<£ is zero, Fl = F2.
Hence, when d> is zero, the derivatives -=-1 and -~ may be
u<(p a<p
different, but the derivatives with respect to any of the other
variables, such as — =-^ and — =-^ , must be the same. The discon-
dy dy
tinuity is therefore confined to the derivative with respect to (/>, all
the other derivatives being continuous.
Periodic and Multiple Functions.
9.] If u is a function of x such that its value is the same for
x, iv + at x-\-na, and all values of x differing by a, u is called a
periodic function of a?, and a is called its period.
If x is considered as a function of u, then, for a given value of
n, there must be an infinite series of values of x differing by-
multiples of a. In this case x is called a multiple function of u}
and a is called its cyclic constant.
S] SY*
The differential coefficient -=- has only a finite number of values
du
corresponding to a given value of u.
On the Relation of Physical Quantities to Directions In Space.
10.] In distinguishing the kinds of physical quantities, it is of
great importance to know how they are related to the directions
of those coordinate axes which we usually employ in defining the
positions of things. The introduction of coordinate axes into geo
metry by Des Cartes was one of the greatest steps in mathematical
progress, for it reduced the methods of geometry to calculations
performed on numerical quantities. The position of a point is made
to depend on the length of three lines which are always drawn in
determinate directions, and the line joining two points is in like
manner considered as the resultant of three lines.
II.] VECTORS, OR DIRECTED QUANTITIES. 9
But for many purposes of physical reasoning, as distinguished
from calculation, it is desirable to avoid explicitly introducing the
Cartesian coordinates, and to fix the mind at once on a point of
space instead of its three coordinates, and on the magnitude and
direction of a force instead of its three components. This mode
of contemplating geometrical and physical quantities is more prim
itive and more natural than the other, although the ideas connected
with it did not receive their full development till Hamilton made
the next great step in dealing with space, by the invention of his
Calculus of Quaternions.
As the methods of Des Cartes are still the most familiar to
students of science, and as they are really the most useful for
purposes of calculation, we shall express all our results in the
Cartesian form. I am convinced, however, that the introduction
of the ideas, as distinguished from the operations and methods of
Quaternions, will be of great use to us in the study of all parts
of our subject, and especially in electrodynamics, where we have to
deal with a number of physical quantities, the relations of which
to each other can be expressed far more simply by a few expressions
of Hamilton's, than by the ordinary equations.
11.] One of the most important features of Hamilton's method is
the division of quantities into Scalars and Vectors.
A Scalar quantity is capable of being completely defined by a
single numerical specification. Its numerical value does not in
any way depend on the directions we assume for the coordinate
axes.
A Vector, or Directed quantity, requires for its definition three
numerical specifications, and these may most simply be understood
as having reference to the directions of the coordinate axes.
Scalar quantities do not involve direction. The volume of a
geometrical figure, the mass and the energy of a material body,
the hydrostatical pressure at a point in a fluid, and the potential
at a point in space, are examples of scalar quantities.
A vector quantity has direction as well as magnitude, and is
such that a reversal of its direction reverses its sign. The dis
placement of a point, represented by a straight line drawn from
its original to its final position, may be taken as the typical vector
quantity, from which indeed the name of Vector is derived.
The velocity of a body, its momentum, the force acting on it,
an electric current, the magnetization of a particle of iron, are
instances of vector quantities.
10 PRELIMINARY. [l2.
There are. physical quantities of another kind which are related
to directions in space, but which are not vectors. Stresses and
strains in solid bodies are examples of these, and so are some of
the properties of bodies considered in the theory of elasticity and
in the theory of double refraction. Quantities of this class require
for their definition nine numerical specifications. They are ex
pressed in the language of Quaternions by linear and vector
functions of a vector.
The addition of one vector quantity to another of the same kind
is performed according to the rule given in Statics for the com
position of forces. In fact, the proof which Poisson gives of the
' parallelogram of forces ' is applicable to the composition of any
quantities such that turning them end for end is equivalent to a
reversal of their sign.
When we wish to denote a vector quantity by a single symbol,
and to call attention to the fact that it is a vector, so that we must
consider its direction as well as its magnitude, we shall denote
it by a German capital letter, as §1, S3, &c.
In the calculus of Quaternions, the position of a point in space
is defined by the vector drawn from a fixed point, called the origin,
to that point. If we have to consider any physical quantity whose
value depends on the position of the point, that quantity is treated
as a function of the vector drawn from the origin. The function
may be itself either scalar or vector. The density of a body, its
temperature, its hydrostatic pressure, the potential at a point,
are examples of scalar functions. The resultant force at a point,
the velocity of a fluid at a point, the velocity of rotation of
an element of the fluid, and the couple producing rotation, are
examples of vector functions.
12.] Physical vector quantities may be divided into two classes,
in one of which the quantity is defined with reference to a line,
while in the other the quantity is defined with reference to an
area.
For instance, the resultant of an attractive force in any direction
may be measured by finding the work which it would do on a
body if the body were moved a short distance in that direction
and dividing it by that short distance. Here the attractive force
is defined with reference to a line.
On the other hand, the flux of heat in any direction at any
point of a solid body may be defined as the quantity of heat which
crosses a small area drawn perpendicular to that direction divided
1 3.] INTENSITIES AND FLUXES. 11
by that area and by the time. Here the flux is defined with
reference to an area.
There are certain cases in which a quantity may be measured
with reference to a line as well as with reference to an area.
Thus, in treating of the displacements of elastic solids, we may
direct our attention either to the original and the actual position
of a particle, in which case the displacement of the particle is
measured by the line drawn from the first position to the second,
or we may consider a small area fixed in space, and determine
what quantity of the solid passes across that area during the dis
placement.
In the same way the velocity of a fluid may be investigated
either with respect to the actual velocity of the individual particles,
or with respect to the quantity of the fluid which flows through
any fixed area.
But in these cases we require to know separately the density of
the body as well as the displacement or velocity, in order to apply
the first method, and whenever we attempt to form a molecular
theory we have to use the second method.
In the case of the flow of electricity we do not know anything
of its density or its velocity in the conductor, we only know the
value of what, on the fluid theory, would correspond to the product
of the density and the velocity. Hence in all such cases we must
apply the more general method of measurement of the flux across
an area.
In electrical science, electromotive and magnetic intensity
belong to the first class, being defined with reference to lines.
When we wish to indicate this fact, we may refer to them as
Intensities.
On the other hand, electric and magnetic induction, and electric
currents, belong to the second class, being defined with reference
to areas. When we wish to indicate this fact, we shall refer to them
as Fluxes.
Each of these forces may be considered as producing, or tending
to produce, its corresponding flux. Thus, electromotive intensity
produces electric currents in conductors, and tends to produce them
in dielectrics. It produces electric induction in dielectrics, and pro
bably in conductors also. In the same sense, magnetic intensity
produces magnetic induction.
13.] In some cases the flux is simply proportional to the force
and in the same direction, but in other cases we can only affirm
12 PRELIMINARY. [14.
that the direction and magnitude of the flux are functions of the
direction and magnitude of the force.
The case in which the components of the flux are linear functions
of those of the force is discussed in the chapter on the Equations
of Conduction, Art. 297. There are in general nine coefficients
which determine the relation between the force and the flux. In
certain cases we have reason to believe that six of these coefficients
form three pairs of equal quantities. In such cases the relation be
tween the line of direction of the force and the normal plane of the
flux is of the same kind as that between a diameter of an ellipsoid
and its conjugate diametral plane. In Quaternion language, the
one vector is said to be a linear and vector function of the other, and
when there are three pairs of equal coefficients the function is said
to be self-conjugate.
In the case of magnetic induction in iron, the flux, (the mag
netization of the iron,) is not a linear function of the magnetizing
force. In all cases, however, the product of the force and the
flux resolved in its direction, give a result of scientific import
ance, and this is always a scalar quantity.
14.] There are two mathematical operations of frequent occur
rence which are appropriate to these two classes of vectors, or
directed quantities.
In the case of forces, we have to take the integral along a line
of the product of an element of the line, and the resolved part of
the force along that element. The result of this operation is
called the Line-integral of the force. It represents the work
done on a body carried along the line. In certain cases in which
the line-integral does not depend on the form of the line, but
only on the positions of its extremities, the line-integral is called
the Potential.
In the case of fluxes, we have to take the integral, over a surface,
of the flux through every element of the surface. The result of
this operation is called the Surface-integral of the flux. It repre
sents the quantity which passes through the surface.
There are certain surfaces across which there is no flux. If
two of these surfaces intersect, their line of intersection is a line
of flux. In those cases in which the flux is in the same direction
as the force, lines of this kind are often called Lines of Force. It
would be more correct, however, to speak of them in electrostatics
and magnetics as Lines of Induction, and in electrokinematics as
Lines of Flow.
1 6.] LINE-INTEGRALS. 13
15.] There is another distinction between different kinds of
directed quantities, which, though very important in a physical
point of view, is not so necessary to be observed for the sake of
the mathematical methods. This is the distinction between longi
tudinal and rotational properties.
The direction and magnitude of a quantity may depend upon
some action or effect which takes place entirely along a certain
line, or it may depend upon something of the nature of rota
tion about that line as an axis. The laws of combination of
directed quantities are the same whether they are longitudinal or
rotational, so that there is no difference in the mathematical treat
ment of the two classes, but there may be physical circumstances
which indicate to which class we must refer a particular pheno
menon. Thus, electrolysis consists of the transfer of certain sub
stances along a line in one direction, and of certain other sub
stances in the opposite direction, which is evidently a longitudinal
phenomenon, and there is no evidence of any rotational effect
about the direction of the force. Hence we infer that the electric
current which causes or accompanies electrolysis is a longitudinal,
and not a rotational phenomenon.
On the other hand, the north and south poles of a magnet do
not differ as oxygen and hydrogen do, which appear at opposite
places during electrolysis, so that we have no evidence that mag
netism is a longitudinal phenomenon, while the effect of magnetism
in rotating the plane of polarized light distinctly shews that mag
netism is a rotational phenomenon.
On Line-integrals.
16.] The operation of integration of the resolved part of a vector
quantity along a line is important in physical science generally,
and should be clearly understood.
Let x, y, z be the coordinates of a point P on a line whose
length, measured from a certain point A, is s. These coordinates
will be functions of a single variable s.
Let R be the numerical value of the vector quantity at P, and
let the tangent to the curve at P make with the direction of R the
angle e, then R cos e is the resolved part of R along the line, and the
integral r»
L — \ Rcostds
*J 0
is called the line-integral of R along the line s.
14 PEELIMINAEY. [l6.
We may write this expression
o ds d
where X, 7, Z are the components of E parallel to x, y, z respect
ively.
This quantity is, in general, different for different lines drawn
between A and P. When, however, within a certain region, the
quantity Xd®+7dy + Zdz=—D3f,
that is, when it is an exact differential within that region, the
value of L becomes
L = VA-*P,
and is the same for any two forms of the path between A and P,
provided the one form can be changed into the other by continuous
motion without passing out of this region.
On Potentials.
The quantity ^ is a scalar function of the position of the point,
and is therefore independent of the directions of reference. It is
called the Potential Function, and the vector quantity whose com
ponents are J, J, Z is said to have a potential #, if
When a potential function exists, surfaces for which the potential
is constant are called Equipotential surfaces. The direction of B at
anv point of such a surface coincides with the normal to the surface,
d<V
and if n be a normal at the point P, then E = — -^ •
The method of considering the components of a vector as the
first derivatives of a certain function of the coordinates with re
spect to these coordinates was invented by Laplace* in his treat
ment of the theory of attractions. The name of Potential was first
given to this function by Green f, who made it the basis of his
treatment of electricity. Green's essay was neglected by mathe
maticians till 1846, and before that time most of its important
theorems had been rediscovered by Gauss, Chasles, Sturm, anc
Thomson {.
^ Esty°of ttipVpJLion of Mathematical Analysis to tte Theories of! Electncity
and Magnetism, Nottingham, 1828. Reprinted in CrdUs Journal, and m Mr. Ferrers
edition of Green's Works.
J Thomson and Tait, Natural Philosophy, § 483.
1 7.] RELATION BETWEEN FORCE AND POTENTIAL. 15
In the theory of gravitation the potential is taken with the
opposite sign to that which is here used, and the resultant force
in any direction is then measured by the rate of increase of the
potential function in that direction. In electrical and magnetic
investigations the potential is denned so that the resultant force
in any direction is measured by the decrease of the potential in
that direction. This method of using the expression makes it
correspond in sign with potential energy, which always decreases
when the bodies are moved in the direction of the forces acting
on them.
17.] The geometrical nature of the relation between the poten
tial and the vector thus derived from it receives great light from
Hamilton's discovery of the form of the operator by which the vector
is derived from the potential.
The resolved part of the vector in any direction is, as we have
seen, the first derivative of the potential with respect to a co
ordinate drawn in that direction, the sign being reversed.
Now if i, j, Jc are three unit vectors at right angles to each
other, and if X, Y, Z are the components of the vector g resolved
parallel to these vectors, then
% = iX+jY+kZ\ (1)
and by what we have said above, if # is the potential,
If we now write V for the operator,
. d . d . d
i -j — -\- 1 -=— + K -j— i (3)
ax ay dz
g=-V*. (4)
The symbol of operation V may be interpreted as directing us
to measure, in each of three rectangular directions, the rate of
increase of *£, and then, considering the quantities thus found as
vectors, to compound them into one. This is what we are directed
to do by the expression (3). But we may also consider it as directing
us first to find out in what direction ^ increases fastest, and then
to lay off in that direction a vector representing this rate of
increase.
M. Lame, in his Traite des Fonctlons Inverses, uses the term
Differential Parameter to express the magnitude of this greatest
rate of increase, but neither the term itself, nor the mode in which
16 PRELIMINARY. [l8.
Lame* uses it, indicates that the quantity referred to has direction
as well as magnitude. On those rare occasions in which I shall have
to refer to this relation as a purely geometrical one, I shall call the
vector S ^ne space-variation of the scalar function ^, using the
phrase to indicate the direction, as well as the magnitude, of the
most rapid decrease of #.
18.] There are cases, however, in which the conditions
dZ dY dX dZ dY dX
— = 0, -, =- = 0, and -= — = 0,
dy dz dz dso dx dy
which are those of Xdx + Ydy + Zdz being a complete differential,
are satisfied throughout a certain region of space, and yet the line-
integral from A to P may be different for two lines, each of
which lies wholly within that region. This may be the case if
the region is in the form of a ring, and if the two lines from A
to P pass through opposite segments of the ring. In this case,
the one path cannot be transformed into the other by continuous
motion without passing out of the region.
We are here led to considerations belonging to the Geometry
of Position, a subject which, though its importance was pointed
out by Leibnitz and illustrated by Gauss, has been little studied.
The most complete treatment of this subject has been given by
J. B. Listing*.
Let there be p points in space, and let I lines of any form be
drawn joining these points so that no two lines intersect each
other, and no point is left isolated. We shall call a figure com
posed of lines in this way a Diagram. Of these lines, p— 1 are
sufficient to join the p points so as to form a connected system.
Every new line completes a loop or closed path, or, as we shall
call it, a Cycle. The number of independent cycles in the diagram
is therefore K = I— _p+ 1.
Any closed path drawn along the lines of the diagram is com
posed of these independent cycles, each being taken any number of
times and in either direction.
The existence of cycles is called Cyclosis, and the number of
cycles in a diagram is called its Cyclomatic number.
Cyclosis in Surfaces and Regions.
Surfaces are either complete or bounded. Complete surfaces are
either infinite or closed. Bounded surfaces are limited by one or
* Der Census RailmlicTier Complete, Gott. Abh., Bd. x. S. 97 (1861).
1 9-] CYCLIC REGIONS. 17
more closed lines, which may in the limiting cases become double
finite lines or points.
A finite region of space is bounded by one or more closed
surfaces. Of these one is the external surface, the others are
included in it and exclude each other, and are called internal
surfaces.
If the region has one bounding surface, we may suppose that
surface to contract inwards without breaking its continuity or
cutting itself. If the region is one of simple continuity, such as
a sphere, this process may be continued till it is reduced to a
point ; but if the region is like a ring, the result will be a closed
curve; and if the region has multiple connexions, the result will
be a diagram of lines, and the cyclomatic number of the diagram
will be that of the region. The space outside the region has the
same cyclomatic number as the region itself. Hence, if the region
is bounded by internal as well as external surfaces, its cyclomatic
number is the sum of those due to all the surfaces.
When a region encloses within itself other regions, it is called a
Periphractic region.
The number of internal bounding surfaces of a region is called
its periphractic number. A closed surface is also periphractic, its
periphractic number being unity.
The cyclomatic number of a closed surface is twice that of either
of the regions which it bounds. To find the cyclomatic number of
a bounded surface, suppose all the boundaries to contract inwards,
without breaking continuity, till they meet. The surface will then
be reduced to a point in the case of an acyclic surface, or to a linear
diagram in the case of cyclic surfaces. The cyclomatic number of
the diagram is that of the surface.
19.] THEOREM I. If throughout any acyclic region
the value of the line-integral from a point A to a point P taken
along any path within the region will be the same.
We shall first shew that the line-integral taken round any closed
path within the region is zero.
Suppose the equipotential surfaces drawn. They are all either
closed surfaces or are bounded entirely by the surface of the re
gion, so that a closed line within the region, if it cuts any of the
surfaces at one part of its path, must cut the same surface in
the opposite direction at some other part of its path, and the
VOL. i. c
V
18 PRELIMINARY. [20.
corresponding portions of the line-integral being equal and opposite,
the total value is zero.
Hence if AQP and AQ'P are two paths from A to P, the line-
integral for AQ'P is the sum of that for AQP and the closed path
AQ'PQA. But the line-integral of the closed path is zero, there
fore those of the two paths are equal.
Hence if the potential is given at any one point of such a
region, that at any other point is determinate.
20.] THEOREM II. In a cyclic region in which the equation
Xdx + Ydy+Zdz = -DV
is everywhere satisfied, the line-integral from A to P, along a
line drawn within the region, will not in general be determinate
unless the channel of communication between A and P be specified.
Let K be the cyclomatic number of the region, then K sections
of the region may be made by surfaces which we may call Dia
phragms, so as to close up K of the channels of communication,
and reduce the region to an acyclic condition without destroying
its continuity.
The line-integral from A to any point P taken along a line
which does not cut any of these diaphragms will be, by the last
theorem, determinate in value.
Now let A and P be taken indefinitely near to each other, but
on opposite sides of a diaphragm, and let K be the line-integral
from A to P.
Let A' and P' be two other points on opposite sides of the same
diaphragm and indefinitely near to each other, and let K' be the
line-integral from A to P'. Then K'= K.
For if we draw AA' and PP', nearly coincident, but on opposite
sides of the diaphragm, the line-integrals along these lines will
be equal. Suppose each equal to L, then K', the line-integral of
AT, is equal to that of A'A + AP + PP^ -l+K+L=K= that
ofAP.
Hence the line-integral round a closed curve which passes through
one diaphragm of the system in a given direction is a constant
quantity K. This quantity is called the Cyclic constant corre
sponding to the given cycle.
Let any closed curve be drawn within the region, and let it cut
the diaphragm of the first cycle p times in the positive direction
and/ times in the negative direction, and let p— / = %. Then
the line-integral of the closed curve will be n^K^
21.] SURFACE-INTEGRALS. 19
Similarly the line-integral of any closed curve will be
where nK represents the excess of the number of positive passages
of the curve through the diaphragm of the cycle K over the
number of negative passages.
If two curves are such that one of them may be transformed
into the other by continuous motion without at any time passing
through any part of space for which the condition of having a
potential is not fulfilled, these two curves are called Reconcileable
curves. Curves for which this transformation cannot be effected
are called Irreconcileable curves *.
The condition that Xdx + Ydy+Zdz is a complete differential
of some function ^ for all points within a certain region, occurs in
several physical investigations in which the directed quantity and
the potential have different physical interpretations,
In pure kinematics we may suppose X, T3 Z to be the com
ponents of the displacement of a point of a continuous body whose
original coordinates area?, ^, z\ the condition then expresses that
these displacements constitute a non-rotational strain f.
If X, Y, Z represent the components of the velocity of a fluid at
the point #, y> z, then the condition expresses that the motion of the
fluid is irrotational.
If X, Y, Z represent the components of the force at the point
a?, y, z, then the condition expresses that the work done on a
particle passing from one point to another is the difference of the
potentials at these points, and the value of this difference is the
same for all reconcileable paths between the two points.
On Surface-Integrals.
21.] Let dS be the element of a surface, and € the angle which
a normal to the surface drawn towards the positive side of the
surface makes with the direction of the vector quantity R, then
RcosedS is called the surface-integral of R over ike surface 8.
THEOREM III. The surface-integral of the flux inwards through a
closed surface may be expressed as the volume-integral of its con
vergence taken within the surface. (See Art. 25.)
Let X, Y, Z be the components of R, and let /, m, n be the
* See Sir W. Thomson ' On Vortex Motion,' Trans. R. S. Edin., 1867-8.
t See Thomson and Tait's Natural Philosophy, § 190 (*).
C 2,
20 PRELIMINARY. [21.
direction-cosines of the normal to S measured inwards. Then the
surface-integral of R over S is
JJR coavdS =j'j'xid8 +f/Ym dS+ffzndS; (1)
the values of X, Y, Z being those at a point in the surface, and
the integrations being extended over the whole surface.
If the surface is a closed one, then, when y and z are given,
the coordinate x must have an even number of values, since a line
parallel to x must enter and leave the enclosed space an equal
number of times provided it meets the surface at all.
At each entrance
IdS = dydzt
and at each exit 7 7 Q 7 ,
IdS =.—dydz.
Let a point travelling from # = — oo to # = + oo first enter
the space when x = selt then leave it when x = a?2, and so on ;
and let the values of X at these points be X^X^, &c., then
. (2)
If X is a quantity which is continuous, and has no infinite values
between xl and #2, then
where the integration is extended from the first to the second
intersection, that is, along the first segment of x which is within
the closed surface. Taking into account all the segments which lie
within the closed surface, we find
—ff /%*+*, (4)
the double integration being confined to the closed surface, but
the triple integration being extended to the whole enclosed space.
Hence, if X, Y, Z are continuous and finite within a closed surface
$, the total surface-integral of R over that surface will be
COS € dS = -
the triple integration being extended over the whole space within S.
Let us next suppose that X, Y, Z are not continuous within the
closed surface, but that at a certain surface F(x, y, z) = 0 the
values of Z, Y, Z alter abruptly from X, Y, Z on the negative side
of the surface to X', Y't Z' on the positive side.
22.] SOLENOIDAL DISTRIBUTION. 21
If this discontinuity occurs, say, between ssl and #2? ^ne value
where in the expression under the integral sign only the finite
values of the derivative of X are to be considered.
In this case therefore the total surface-integral of R over the
closed surface will be expressed by
(7)
or, if V, m', n' are the direction-cosines of the normal to the surface
of discontinuity, and d$' an element of that surface,
//
*— «--
-Z)} dS', (8)
where the integration of the last term is to be extended over the
surface of discontinuity.
If at every point where X, Y, Z are continuous
and at every surface where they are discontinuous
VX' + m'T+n'Z'= I'X+m'Y+n'Z, (10)
then the surface-integral over every closed surface is zero, and the
distribution of the vector quantity is said to be Solenoidal.
We shall refer to equation (9) as the General solenoidal con
dition, and to equation (10) as the Superficial solenoidal condition.
22.] Let us now consider the case in which at every point
within the surface S the equation
dX dY dZ
is satisfied. We have as a consequence of this the surface-integral
over the closed surface equal to zero.
Now let the closed surface 8 consist of three parts $1} $0, and
$2. Let Sl be a surface of any form bounded by a closed line L^.
Let $0 be formed by drawing lines from every point of L^ always
22 PRELIMINARY. [22.
coinciding with the direction of E. If J, m, n are the direction-
cosines of the normal at any point of the surface Sot we have
£cose = Xl+Ym + Zn= 0. (12)
Hence this part of the surface contributes nothing towards the
value of the surface-integral.
Let S2 be another surface of any form bounded by the closed
curve L2 in which it meets the surface SQ.
Let Qlt Q0, Q2 be the surface-integrals of the surfaces Slt SQ, S2)
and let Q be the surface-integral of the closed surface S. Then
Q= <21+go+Q2 = 0; (13)
and we know that Q0 = 0 ; (14)
therefore Q*=-Qi'> (15)
or, in other words, the surface-integral over the surface S2 is equal
and opposite to that over Sl whatever be the form and position
of £2, provided that the intermediate surface SQ is one for which E
is always tangential.
If we suppose L± a closed curve of small area, £0 will be a
tubular surface having the property that the surface-integral over
every complete section of the tube is the same.
Since the whole space can be divided into tubes of this kind
provided d X dY dZ_ __ /16\
~5 — "T" ""7 ~r 7 •"-" ) V /
das dy dz
a distribution of a vector quantity consistent with this equation is
called a Solenoidal Distribution.
On Tubes and Lines of Flow.
If the space is so divided into tubes that the surface-integral
for every tube is unity, the tubes are called Unit tubes, and the
surface-integral over any finite surface 8 bounded by a closed
curve L is equal to the number of such tubes which pass through
S in the positive direction, or, what is the same thing, the number
which pass through the closed curve L.
Hence the surface-integral of 8 depends only on the form of
its boundary L, and not on the form of the surface within its
boundary.
On Periphrastic Eeglons.
If, throughout the whole region bounded externally by the single
closed surface /S, the solenoidal condition
dX dY + ^=0
dos + dy dz ~
22.] PERIPHRACTIC REGIONS. 23
is satisfied, then the surface-integral taken over any closed surface
drawn within this region will be zero, and the surface-integral
taken over a bounded surface within the region will depend only
on the form of the closed curve which forms its boundary.
It is not, however, generally true that the same results follow
if the region within which the solenoidal condition is satisfied is
bounded otherwise than by a single surface.
For if it is bounded by more than one continuous surface, one of
these is the external surface and the others are internal surfaces,
and the region S is a periphractic region, having within it other
regions which it completely encloses.
If within one of these enclosed regions, say, that bounded by the
closed surface S19 the solenoidal condition is not satisfied, let
A =
be the surface-integral for the surface enclosing this region, and
let Q2, Q3, &c. be the corresponding quantities for the other en
closed regions S2, $3, &c.
Then, if a closed surface $' is drawn within the region S, the
value of its surface-integral will be zero only when this surface
S' does not include any of the enclosed regions Slt S2t &c* -^ ^
includes any of these, the surface-integral is the sum of the surface-
integrals of the different enclosed regions which lie within it.
For the same reason, the surface-integral taken over a surface
bounded by a closed curve is the same for such surfaces only, bounded
by the closed curve, as are reconcileable with the given surface by
continuous motion of the surface within the region S.
When we have to deal with a periphractic region, the first thing
to be done is to reduce it to an aperiphractic region by drawing
lines 2/u .Z/2, &c. joining the internal surfaces Slt $2, &c. to the
external surface S. Each of these lines, provided it joins surfaces
which were not already in continuous connexion, reduces the
periphractic number by unity, so that the whole number of lines
to be drawn to remove the periphraxy is equal to the periphractic
number, or the number of internal surfaces. In drawing these lines
we must remember that any line joining surfaces which are already
connected does not diminish the periphraxy, but introduces cyclosis.
When these lines have been drawn we may assert that if the
solenoidal condition is satisfied in the region S, any closed surface
drawn entirely within S, and not cutting any of the lines, has its
surface-integral zero. If it cuts any line, say L^ , once or any odd
24: PRELIMINARY. [23.
number of times, it encloses the surface S1 and the surface-integral
The most familiar example of a periphractic region within which
the solenoidal condition is satisfied is the region surrounding a mass
attracting or repelling inversely as the square of the distance.
In this case we have
A«O A«O A«O
where m is the mass, supposed to be at the origin of coordinates.
At any point where r is finite
dX dY dZ_0
dx dy dz
but at the origin these quantities become infinite. For any closed
surface not including the origin, the surface-integral is zero. If a
closed surface includes the origin, its surface-integral is 4 urn.
If, for any reason, we wish to treat the region round m as if it
were not periphractic, we must draw a line from m to an infinite
distance, and in taking surface-integrals we must remember to add
4 Tim whenever this line crosses from the negative to the positive
side of the surface.
On Right-handed and Left-handed Relations in Space.
23.] In this treatise the motions of translation along any axis
and of rotation about that axis will be assumed to be of the same
sign when their directions correspond to those of the translation
and rotation of an ordinary or right-handed screw *.
For instance, if the actual rotation of the earth from west to east
is taken positive, the direction of the earth's axis from south to
north will be taken positive, and if a man walks forward in the
positive direction, the positive rotation is in the order, head, right-
hand, feet, left-hand.
* The combined action of the muscles of the arm when we turn the upper side of
the right-hand outwards, and at the same time thrust the hand forwards, will
impress the right-handed screw motion on the memory more firmly than any verbal
definition. A common corkscrew may be used as a material symbol of the same
relation.
Professor W. H. Miller has suggested to me that as the tendrils of the vine are
right-handed screws and those of the hop left-handed, the two systems of relations in
space might be called those of the vine and the hop respectively.
The system of the vine, which we adopt, is that of Linnaeus, and of screw-makers
in all civilized countries except Japan. De Candolle was the first who called the
hop-tendril right-handed, and in this he is followed by Listing, and by most writers
on the circular polarization of light. Screws like the hop-tendril are made for the
couplings of railway-carriages, and for the fittings of wheels on the left side of or
dinary carriages, but they are always called left-handed screws by those who use
them.
24.] LINE-INTEGRAL AND SURFACE-INTEGRAL. 25
If we place ourselves on the positive side of a surface, the positive
direction along its bounding curve will be opposite to the motion
of the hands of a watch with its face towards us.
This is the right-handed system which is adopted in Thomson
and Tait's Natural Philosophy -, § 243, and in Tait's Quaternions.
The opposite, or left-handed system, is adopted in Hamilton's
Quaternions (Lectures, p. 76, and Elements, p. 108, and p. 117 note).
The operation of passing from the one system to the other is called,
by Listing, Perversion.
The reflexion of an object in a mirror is a perverted image of the
object.
When we use the Cartesian axes of oc, y, z, we shall draw them
so that the ordinary conventions about the cyclic order of the
symbols lead to a right-handed system of directions in space. Thus,
if x is drawn eastward and y northward, z must be drawn upward.
The areas of surfaces will be taken positive when the order of
integration coincides with the cyclic order of the symbols. Thus,
the area of a closed curve in the plane of xy may be written either
/ x dy or — \yAx\
the order of integration being SB, y in the first expression, and y, x
in the second.
This relation between the two products dx dy and dy dx may
be compared with the rule for the product of two perpendicular
vectors in the method of Quaternions, the sign of which depends
on the order of multiplication ; and with the reversal of the sign
of a determinant when the adjoining rows or columns are ex
changed.
For similar reasons a volume-integral is to be taken positive when
the order of integration is in the cyclic order of the variables x, y, z,
and negative when the cyclic order is reversed.
We now proceed to prove a theorem which is useful as esta
blishing a connexion between the surface-integral taken over a
finite surface and a line-integral taken round its boundary,
24.] THEOREM IV. A line-integral taken round a closed curve
may be expressed in terms of a surface-integral taken over a
surface bounded by the curve.
Let Z, 7, Z be the components of a vector quantity 51 whose line-
integral is to be taken round a closed curve s.
Let S be any continuous finite surface bounded entirely by the
26 PRELIMINARY. [24.
closed curve s, and let £, 77, f be the components of another vector
quantity 33, related to X, 7, Z by the equations
t:_d^_clY _dX dZ dY dX
-dy"dz' r7-^™^' ^-~fa"~dy'
Then the surface-integral of 33 taken over the surface 8 is equal to
the line-integral of 51 taken round the curve s. It is manifest that
f, 77, f satisfy of themselves the solenoidal condition
Let /, m, n be the direction- cosines of the normal to an element
of the surface dS, reckoned in the positive direction. Then the
value of the surface-integral of 33 may be written
8. (2)
In order to form a definite idea of the meaning of the element
dS, we shall suppose that the values of the coordinates #, y> z for
every point of the surface are given as functions of two inde
pendent variables a and p. If p is constant and a varies, the point
(#, y, z) will describe a curve on the surface, and if a series of values
is given to j3, a series of such curves will be traced, all lying on
the surface 8. In the same way, by giving a series of constant
values to a, a second series of curves may be traced, cutting the
first series, and dividing the whole surface into elementary portions,
any one of which may be taken as the element dS.
The projection of this element on the plane of y z is, by the
ordinary formula,
dp dp da
The expressions for m dS and n dS are obtained from this by sub
stituting a?, y, z in cyclic order.
The surface-integral which we have to find is
(4)
or, substituting the values of £, 77, f in terms of X, Yy Z>
fff dX dX dY 7dY , 7dZ dZ\1Q (^
(m-= -- n-j- +n-7 -- l-j- +l-= -- m — )dS. (5)
JJ \ dz dy dx dz dy dx'
The part of this which depends on X may be written
dX ,dz dx dz dx\ dX ,dx dy dx dy^i, , , ,
dz (la d~p * dp To) ~~ ~dy (da dp " dp TV)] *fti
fft
JJ i
24-]
LINE INTEGRAL AND SURFACE INTEGRAL. 27
... _ ,. dXdx dx , . .
adding and subtracting ~r~ ~r~ ~ry this becomes
//!
dx da
dx ,dXdx dXdy dX dz\
dfi ^dx da dy da dz da)
dx fdX dx dX dy
, v
Let us now suppose that the curves for which a is constant form
a series of closed curves surrounding a point on the surface for
which a has its minimum value, a0, and let the last curve of the
series, for which a = ax , coincide with the closed curve s.
Let us also suppose that the curves for which j3 is constant form
a series of lines drawn from the point at which a = a0 to the closed
curve s, the first, /30, and the last, ft, being identical.
Integrating (8) by parts, the first term with respect to a and the
second with respect to /3, the double integrals destroy each other
and the expression becomes
) rfa. (9)
V ^ Vft)
Since the point (a, ft) is identical with the point (a, /30), the
third and fourth terms destroy each other ; and since there is
but one value of x at the point where a = a0, the second term is
zero, and the expression is reduced to the first term :
Since the curve a = ax is identical with the closed curve 5, we
may write the expression in the form
where the integration is to be performed round the curve 5. We
may treat in the same way the parts of the surface-integral which
depend upon T and Z, so that we get finally,
where the first integral is extended over the surface $, and the
second round the bounding curve 5 *.
* This theorem was given by Professor Stokes, Smith's Prize Examination, 1854,
question 8. It is proved in Thomson and Tait's Natural Philosophy, § 190 (j).
'
28 PKELIMINAEY. [25.
On the effect of the operator V on a vector function.
25.] We have seen that the operation denoted by V is that by
which a vector quantity is deduced from its potential. The same
operation, however, when applied to a vector function, produces
results which enter into the two theorems we have just proved
(III and IV). The extension of this operator to vector displace
ments, and most of its further development, is due to Professor
Tait*.
Let a be a vector function of /o, the vector of a variable point.
Let us suppose, as usual, that
p = iso+jy + kz,
and o- = iX+jY + JcZ\
where X, Y, Z are the components of <r in the directions of the
axes.
We have to perform on a- the operation
. d .d . d
V = *T- +J-J- + &T"
dx dy dz
Performing this operation, and remembering the rules for the
multiplication of i, j, k, we find that Vo- consists of two parts,
one scalar and the other vector.
The scalar part is
/dX dY dZ\ rj,, TTT
tfVo- = — (-=- + -T- + -i-}, see Theorem III,
W# dy dz'
and the vector part is
.dZ dY .,dX dZ . ,dY dX
If the relation between X, Y} Z and £ 77, f is that given by
equation (1) of the last theorem, we may write
YVff = i{+jri + &{. See Theorem IV.
It appears therefore that the functions of X, Y, Z which occur
in the two theorems are both obtained by the operation V on the
vector whose components are X, J, Z. The theorems themselves
may be written
III8 V " ds = f/S ' * Uv dS) (m)
and fstrdp = (( S .V<rUvds\ (IV)
* See Proc. R. S. Edin., April 28, 1862. ' On Green's and other allied Theorems,'
Trans. R. S. Edin., 1869-70, a very valuable paper; and 'On some Quaternion
Integrals,' Proc. R. S. Edin., 1870-71.
26.] HAMILTON'S OPERATOR V. 29
where ds is an element of a volume, da of a surface, dp of a curve,
and Uv a unit- vector in the direction of the normal.
To understand the meaning1 of these functions of a vector, let us
suppose that a0 is the value of a- at a point P, and let us examine
the value of cr— o-Q in the neighbourhood of P.
If we draw a closed surface round P, then, if the I
surface-integral of <r over this surface is directed \^ » S
inwards, /SVo- will be positive, and the vector
<r— (70 near the point P will be on the whole p
directed towards P, as in the figure (l). d . \.
I propose therefore to call the scalar part of
V<r the convergence of <r at the point P. Tig. i.
To interpret the vector part of V<r, let us
suppose ourselves to be looking in the direction of the vector
whose components are £ rj, £ and let us examine
the vector <r—cr0 near the point P. It will appear •* —
as in the figure (2), this vector being arranged on
the whole tangentially in the direction opposite to
the hands of a watch. j,. 2
I propose (with great diffidence) to call the vector
part of V<r the rotation of <7 at the point P.
In Fig. 3 we have an illustration of rotation com- /
bined with convergence. \
Let us now consider the meaning of the equation \
TV o-=0. /
This implies that Vo- is a scalar, or that the vector rig. 3.
<r is the space-variation of some scalar function ^.
26.] One of the most remarkable properties of the operator V is
that when repeated it becomes
an operator occurring in all parts of Physics, which we may refer to
as Laplace's Operator.
This operator is itself essentially scalar. When it acts on a
scalar function the result is scalar, when it acts on a vector function
the result is a vector.
If, with any point P as centre, we draw a small sphere whose
radius is r, then if q0 is the value of q at the centre, and q the
mean value of q for all points within the sphere,
30 PRELIMINARY. [26.
so that the value at the centre exceeds or falls short of the mean
value according as V2 q is positive or negative.
I propose therefore to call V2 q the concentration of q at the
point P, because it indicates the excess of the value of q at that
point over its mean value in the neighbourhood of the point.
If q is a scalar function, the method of finding its mean value is
well known. If it is a vector function, we must find its mean
value by the rules for integrating vector functions. The result
of course is a vector.
PART I.
ELECTROSTATIC S.
CHAPTEK I.
DESCRIPTION OF PHENOMENA.
Electrification by Friction.
27.] EXPERIMENT I*. Let a piece of glass and a piece of resin,
neither of which exhibits any electrical properties, be rubbed to
gether and left with the rubbed surfaces in contact. They will
still exhibit no electrical properties. Let them be separated. They
will now attract each other.
If a second piece of glass be rubbed with a second piece of
resin, and if the pieces be then separated and suspended in the
neighbourhood of the former pieces of glass and resin, it may be
observed —
(1) That the two pieces of glass repel each other.
(2) That each piece of glass attracts each piece of resin.
(3) That the two pieces of resin repel each other.
These phenomena of attraction and repulsion are called Elec
trical phenomena, and the bodies which exhibit them are said to
be electrified, or to be charged with electricity.
Bodies may be electrified in many other ways, as well as by
friction.
The electrical properties of the two pieces of glass are similar
to each other but opposite to those of the two pieces of resin :
the glass attracts what the resin repels and repels what the resin
attracts.
* See Sir W. Thomson ' On the Mathematical Theory of Electricity,' Cambridge
and Dublin Mathematical Journal) March, 1848.
32 ELECTROSTATIC PHENOMENA. [28.
If a body electrified in any manner whatever behaves as the
glass does, that is, if it repels the glass and attracts the resin, the
body is said to be vitreously electrified, and if it attracts the glass
and repels the resin it is said to be resinously electrified. All
electrified bodies are found to be either vitreously or resinously
electrified.
It is the established practice of men of science to call the vitreous
electrification positive, and the resinous electrification negative.
The exactly opposite properties of the two kinds of electrification
justify us in indicating them by opposite signs, but the applica
tion of the positive sign to one rather than to the other kind must
be considered as a matter of arbitrary convention, just as it is a
matter of convention in mathematical diagrams to reckon positive
distances towards the right hand.
No force, either of attraction or of repulsion, can be observed
between an electrified body and a body not electrified. When, in
any case, bodies not previously electrified are observed to be acted
on by an electrified body, it is because they have become electrified
by induction.
Electrification by Induction.
28.] EXPERIMENT II*. Let a hollow vessel of metal be hung
up by white silk threads, and let a similar thread
be attached to the lid of the vessel so that the vessel
may be opened or closed without touching it.
Let the pieces of glass and resin be similarly sus
pended and electrified as before.
Let the vessel be originally unelectrified, then if
an electrified piece of glass is hung up within it by
its thread without touching the vessel, and the lid
closed, the outside of the vessel will be found to
be vitreously electrified, and it may be shewn that
the electrification outside of the vessel is exactly the
Fig. 4. same in whatever part of the interior space the glass
is suspended.
If the glass is now taken out of the vessel without touching it,
the electrification of the glass will be the same as before it was
put in, and that of the vessel will have disappeared.
This electrification of the vessel, which depends on the glass
* This, and several experiments which follow, are due to Faraday, 'On Static
Electrical Inductive Action,' Phil. Mag., 1843, or Exp. Res., vol. ii. p. 279.
29-] ELECTRIFICATION. 33
being within it, and which vanishes when the glass is removed, is
called electrification by Induction.
Similar effects would be produced if the glass were suspended
near the vessel on the outside, but in that case we should find
an electrification, vitreous in one part of the outside of the vessel
and resinous in another. When the glass is inside the vessel
the whole of the outside is vitreously and the whole of the inside
resinously electrified.
Electrification by Conduction.
29.] EXPERIMENT III. Let the metal vessel be electrified by
induction, as in the last experiment, let a second metallic body
be suspended by wliite silk threads near it, and let a metal wire,
similarly suspended, be brought so as to touch simultaneously the
electrified vessel and the second body.
The second body will now be found to be vitreously electrified,
and the vitreous electrification of the vessel will have diminished.
The electrical condition has been transferred from the vessel to
the second body by means of the wire. The wire is called a con
ductor of electricity, and the second body is said to be electrified
ly conduction.
Conductors and Insulators.
EXPERIMENT IV. If a glass rod, a stick of resin or gutta-percha,
or a white silk thread, had been used instead of the metal wire, no
transfer of electricity would have taken place. Hence these latter
substances are called Non-conductors of electricity. Non-conduc
tors are used in electrical experiments to support electrified bodies
without carrying off their electricity. They are then called In
sulators.
The metals are good conductors ; air, glass, resins, gutta-percha,
vulcanite, paraffin, &c. are good insulators; but, as we shall see
afterwards, all substances resist the passage of electricity, and all
substances allow it to pass, though in exceedingly different degrees.
This subject will be considered when we come to treat of the
motion of electricity. For the present we shall consider only two
classes of bodies, good conductors, and good insulators.
In Experiment II an electrified body produced electrification in
the metal vessel while separated from it by air, a non-conducting
medium. Such a medium, considered as transmitting these electrical
effects without conduction, has been called by Faraday a Dielectric
VOL. I. D
34 ELECTROSTATIC PHENOMENA. [30.
medium,, and the action which takes place through it is called
Induction.
In Experiment III the electrified vessel produced electrification
in the second metallic body through the medium of the wire. Let
us suppose the wire removed, and the electrified piece of glass taken
out of the vessel without touching it, and removed to a sufficient
distance. The second body will still exhibit vitreous electrifica
tion, but the vessel, when the glass is removed, will have resinous
electrification. If we now bring the wire into contact with both
bodies, conduction will take place along the wire, and all electri
fication will disappear from both bodies, shewing that the elec
trification of the two bodies was equal and opposite.
30.] EXPERIMENT V. In Experiment II it was shewn that if
a piece of glass, electrified by rubbing it with resin, is hung up in
an insulated metal vessel, the electrification observed outside does
not depend on the position of the glass. If we now introduce the
piece of resin with which the glass was rubbed into the same vessel,
without touching it or the vessel, it will be found that there is
no electrification outside the vessel. From this we conclude that
the electrification of the resin is exactly equal and opposite to that
of the glass. By putting in any number of bodies, electrified in
any way, it may be shewn that the electrification of the outside of
the vessel is that due to the algebraic sum of all the electrifica
tions, those being reckoned negative which are resinous. We have
thus a practical method of adding the electrical effects of several
bodies without altering the electrification of each.
31.] EXPERIMENT VI. Let a second insulated metallic vessel, B,
be provided, and let the electrified piece of glass be put into the
first vessel A, and the electrified piece of resin into the second vessel
B. Let the two vessels be then put in communication by the metal
wire, as in Experiment III. All signs of electrification will dis
appear.
Next, let the wire be removed, and let the pieces of glass and of
resin be taken out of the vessels without touching them. It will
be found that A is electrified resinously and B vitreously.
If now the glass and the vessel A be introduced together into a
larger insulated vessel C, it will be found that there is no elec
trification outside C. This shews that the electrification of A is
exactly equal and opposite to that of the piece of glass, and that
of B may be shewn in the same way to be equal and opposite to that
of the piece of resin.
33-] SUMMATION OF ELECTRIC EFFECTS. 35
We have thus obtained a method of charging a vessel with a
quantity of electricity exactly equal and opposite to that of an
electrified body without altering the electrification of the latter,
and we may in this way charge any number of vessels with exactly
equal quantities of electricity of either kind, which we may take
for provisional units.
32.] EXPERIMENT VII. Let the vessel B, charged with a quan
tity of positive electricity, which we shall call, for the present,
unity, be introduced into the larger insulated vessel C without
touching it. It will produce a positive electrification on the out
side of C. Now let B be made to touch the inside of C. No change
of the external electrification will be observed. If B is now taken
out of C without touching it, and removed to a sufficient distance,
it will be found that B is completely discharged, and that C has
become charged with a unit of positive electricity.
We have thus a method of transferring the charge of B to C.
Let B be now recharged with a unit of electricity, introduced
into C already charged, made to touch the inside of C, and re
moved. It will be found that B is again completely discharged,
so that the charge of C is doubled.
If this process is repeated, it will be found that however highly
C is previously charged, and in whatever way B is charged, when
B is first entirely enclosed in C, then made to touch C, and finally
removed without touching C, the charge of B is completely trans
ferred to C, and B is entirely free from electrification.
This experiment indicates a method of charging a body with
any number of units of electricity. We shall find, when we come
to the mathematical theory of electricity, that the result of this
experiment affords an accurate test of the truth of the theory.
33.] Before we proceed to the investigation of the law of
electrical force, let us enumerate the facts we have already estab
lished.
By placing any electrified system inside an insulated hollow con
ducting vessel, and examining the resultant effect on the outside
of the vessel, we ascertain the character of the total electrification
of the system placed inside, without any communication of elec
tricity between the different bodies of the system.
The electrification of the outside of the vessel may be tested
with great delicacy by putting it in communication with an elec
troscope.
We may suppose the electroscope to consist of a strip of gold
36 ELECTROSTATIC PHENOMENA. [34.
leaf hanging between two bodies charged, one positively, and the
other negatively. If the gold leaf becomes electrified it will incline
towards the body whose electrification is opposite to its own. By
increasing the electrification of the two bodies and the delicacy of
the suspension, an exceedingly small electrification of the gold leaf
may be detected.
When we come to describe electrometers and multipliers we
shall find that there are still more delicate methods of detecting
electrification and of testing the accuracy of our theories, but at
present we shall suppose the testing to be made by connecting the
hollow vessel with a gold leaf electroscope.
This method was used by Faraday in his very admirable de
monstration of the laws of electrical phenomena *.
34.] I. The total electrification of a body, or system of bodies,
remains always the same, except in so far as it receives electrifi
cation 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 is due to want
of perfect 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 placed in a perfectly in
sulating medium would remain perfectly 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 two bodies are enclosed in the hollow vessel, no change
of the total electrification is observed.
III. When electrification is produced by friction, or by any
other known method, equal quantities of positive and negative elec
trification are produced.
For the electrification of the whole system may be tested in
the hollow vessel, or the process of electrification may be carried
on within the vessel itself, and however intense the electrification of
the parts of the system may be, the electrification of the whole,
as indicated by the gold leaf electroscope, is invariably zero.
The electrification of a body is therefore a physical quantity
capable of measurement, and two or more electrifications can be
combined experimentally with a result of the same kind as when
* 'On Static Electrical Inductive Action,' Phil. May., 1843, or Exp. Res., vol. ii.
p. 249.
35-] ELECTRICITY AS A QUANTITY. 37
two quantities are added algebraically. We therefore are entitled
to use language fitted to deal with electrification as a quantity as
well as a quality, and to speak of any electrified body as c charged
with a certain quantity of positive or negative electricity.'
35.] While admitting electricity, as we have now done, to the
rank of a physical quantity, we must not too hastily assume that
it is, or is not, a substance, or that it is, or is not, a form of
energy, or that it belongs to any known category of physical
quantities. All that we have hitherto proved is that it cannot
be created or annihilated, so that if the total quantity of elec
tricity within a closed surface is increased or diminished, the in
crease or diminution must have passed in or out through the closed
surface.
This is true of matter, and is expressed by the equation known as
the Equation of Continuity in Hydrodynamics.
It is not true of heat, for heat may be increased or diminished
within a closed surface, without passing in or out through the
surface, by the transformation of some other form of energy into
heat, or of heat into some other form of energy.
It is not true even of energy in general if we admit the imme
diate action of bodies at a distance. For a body outside the closed
surface may make an exchange of energy with a body within
the surface. But if all apparent action at a distance is the
result of the action between the parts of an intervening medium,
it is conceivable that in all cases of the increase or diminution
of the energy within a closed surface we may be able, when the
nature of this action of the parts of the medium is clearly under
stood, to trace the passage of the energy in or out through that
surface.
There is, however, another reason which warrants us in asserting
that electricity, as a physical quantity, synonymous with the total
electrification of a body, is not, like heat, a form of energy. An
electrified system has a certain amount of energy, and this energy
can be calculated by multiplying the quantity of electricity in
each of its parts by another physical quantity, called the Potential
of that part, and taking half the sum of the products. The quan
tities ' Electricity ' and ' Potential,' when multiplied together,
produce the quantity ' Energy.' It is impossible, therefore, that
electricity and energy should be quantities of the same category, for
electricity is only one of the factors of energy, the other factor
being ' Potential.'
38 ELECTROSTATIC PHENOMENA. [36.
Energy, which is the product of these factors, may also be con
sidered as the product of several other pairs of factors, such as
A Force x A distance through which the force is to act.
A Mass x Gravitation acting through a certain height.
A Mass x Half the square of its velocity.
A Pressure x A volume of fluid introduced into a vessel at
that pressure.
A Chemical Affinity x A chemical change, measured by the number
of electro-chemical equivalents which enter
into combination.
If we ever should obtain distinct mechanical ideas of the nature of
electric potential, we may combine these with the idea of energy
to determine the physical category in which ' Electricity ' is to be
placed.
36.] In most theories on the subject, Electricity is treated as
a substance, but inasmuch as there are two kinds of electrification
which, being combined, annul each other, and since we cannot
conceive of two substances annulling each other, a distinction has
been drawn between Free Electricity and Combined Electricity.
Theory of Two Fluids.
In what is called the Theory of Two Fluids, all bodies, in their
unelectrified state, are supposed to be charged with equal quan
tities of positive and negative electricity. These quantities are
supposed to be so great that no process of electrification has ever
yet deprived a body of all the electricity of either kind. The pro
cess of electrification, according to this theory, consists in taking
a certain quantity P of positive electricity from the body A and
communicating it to .5, or in taking a quantity N of negative
electricity from B and communicating it to A, or in some com
bination of these processes.
The result will be that A will have P + N units of negative
electricity over and above its remaining positive electricity, which
is supposed to be in a state of combination with an equal quantity
of negative electricity. This quantity P-f N is called the Free elec
tricity, the rest is called the Combined, Latent, or Fixed electricity.
In most expositions of this theory the two electricities are called
'Fluids,' because they are capable of being transferred from one
body to another, and are, within conducting bodies, extremely
mobile. The other properties of fluids, such as their inertia,
36.] THEOEY OF TWO FLUIDS. 39
weight, and elasticity, are not attributed to them by those who
have used the theory for merely mathematical purposes; but the
use of the word Fluid has been apt to mislead the vulgar, including
many men of science who are not natural philosophers, and who
have seized on the word Fluid as the only term in the statement
of the theory which seemed intelligible to them.
We shall see that the mathematical treatment of the subject has
been greatly developed by writers who express themselves in terms
of the ' Two Fluids ' theory. Their results, however, have been
deduced entirely from data which can be proved by experiment,
and which must therefore be true, whether we adopt the theory of
two fluids or not. The experimental verification of the mathe
matical results therefore is no evidence for or against the peculiar
doctrines of this theory.
The introduction of two fluids permits us to consider the negative
electrification of A and the positive electrification of B as the effect
of any one of three different processes which would lead to the same
result. We have already supposed it produced by the transfer of
P units of positive electricity from A to B, together with the
transfer of N units of negative electricity from £ to A. But if
P + N units of positive electricity had been transferred from A
to B, or if P + N units of negative electricity had been transferred
from B to A, the resulting ' free electricity ' on A and on B would
have been the same as before, but the quantity of 'combined
electricity' in A would have been less in the second case and greater
in the third than it was in the first.
It would appear therefore, according to this theory, that it is
possible to alter not only the amount of free electricity in a body,
but the amount of combined electricity. But no phenomena have
ever been observed in electrified bodies which can be traced to the
varying amount of their combined electricities. Hence either the
combined electricities have no observable properties, or the amount
of the combined electricities is incapable of variation. The first
of these alternatives presents no difficulty to the mere mathema
tician, who attributes no properties to the fluids except those of
attraction and repulsion, for he conceives the two fluids simply to
annul one another, like +e and — e, and their combination to be a
true mathematical zero. But to those who cannot use the word
Fluid without thinking of a substance it is difficult to conceive how
the combination of the two fluids can have no properties at all, so
that the addition of more or less of the combination to a body shall
40 ELECTROSTATIC PHENOMENA. [37.
not in any way affect it, either by increasing its mass or its weight,
or altering some of its other properties. Hence it has been supposed
by some, that in every process of electrification exactly equal quan
tities of the two fluids are transferred in opposite directions, so
that the total quantity of the two fluids in any body taken to
gether remains always the same. By this new law they ' contrive
to save appearances,' forgetting that there would have been no need
of the law except to reconcile the ' two fluids ' theory with facts,
and to prevent it from predicting non-existent phenomena.
Theory of One Fluid.
37.] In the theory of One Fluid everything is the same as in
the theory of Two Fluids except that, instead of supposing the two
substances equal and opposite in all respects, one of them, gene
rally the negative one, has been endowed with the properties and
name of Ordinary Matter, while the other retains the name of The
Electric Fluid. The particles of the fluid are supposed to repel
one another according to the law of the inverse square of the
distance, and to attract those of matter according to the same
law. Those of matter are supposed to repel each other and attract
those of electricity.
If the quantity of the electric fluid in a body is such that a
particle of the electric fluid outside the body is as much repelled
by the electric fluid in the body as it is attracted by the matter
of the body, the body is said to be Saturated. If the quantity of
fluid in the body is greater than that required for saturation, the
excess is called the Redundant fluid, and the body is said to be
Overcharged. If it is less, the body is said to be Undercharged,
and the quantity of fluid which would be required to saturate it
is sometimes called the Deficient fluid. The number of units of
electricity required to saturate one gramme of ordinary matter
must be very great, because a gramme of gold may be beaten out
to an area of a square metre, and when in this form may have a
negative charge of at least 60,000 units of electricity. In order to
saturate the gold leaf, this quantity of electric fluid must be
communicated to it, so that the whole quantity required to saturate
it must be greater than this. The attraction between the matter
and the fluid in two saturated bodies is supposed to be a very little
greater than the repulsion between the two portions of matter and
that between the two portions of fluid. This residual force is sup
posed to account for the attraction of gravitation.
38.] THEORY OF ONE FLUID. 41
This theory does not, like the Two-Fluid theory, explain too
much. It requires us, however, to suppose the mass of the electric
fluid so small that no attainable positive or negative electrification
has yet perceptibly increased or diminished either the mass or the
weight of a body, and it has not yet been able to assign sufficient
reasons why the vitreous rather than the resinous electrification
should be supposed due to an excess of electricity.
One objection has sometimes been urged against this theory by
men who ought to have reasoned better. It has been said that
the doctrine that the particles of matter uncombined with elec
tricity repel one another, is in direct antagonism with the well-
established fact that every particle of matter attracts every other
particle throughout the universe. If the theory of One Fluid were
true we should have the heavenly bodies repelling one another.
But it is manifest that the heavenly bodies, according to this
theory, if they consisted of matter uncombined with electricity,
would be in the highest state of negative electrification, and would
repel each other. We have no reason to believe that they are in
such a highly electrified state, or could be maintained in that
state. The earth and all the bodies whose attraction has been
observed are rather in an unelectrified state, that is, they contain
the normal charge of electricity, and the only action between them
is the residual force lately mentioned. The artificial manner, how
ever, in which this residual force is introduced is a much more
valid objection to the theory.
In the present treatise I propose, at different stages of the in
vestigation, to test the different theories in the light of additional
classes of phenomena. For my own part, I look for additional
light on the nature of electricity from a study of what takes place
in the space intervening between the electrified bodies. Such is the
essential character of the mode of investigation pursued by Faraday
in his Experimental Researches, and as we go on I intend to exhibit
the results, as developed by Faraday, W. Thomson, &c., in a con
nected and mathematical form, so that we may perceive what
phenomena are explained equally well by all the theories, and what
phenomena indicate the peculiar difficulties of each theory.
Measurement of the Force between Electrified Bodies.
38.] Forces may be measured in various ways. For instance,
one of the bodies may be suspended from one arm of a delicate
balance, and weights suspended from the other arm, till the body,
42 ELECTROSTATIC PHENOMENA. [39.
when unelectrified, is in equilibrium. The other body may then
be placed at a known distance beneath the first, so that the
attraction or repulsion of the bodies when electrified may increase
or diminish the apparent weight of the first. The weight which
must be added to or taken from the other arm, when expressed
in dynamical measure, will measure the force between the bodies.
This arrangement was used by Sir W. Snow Plan-is, and is that
adopted in Sir W. Thomson's absolute electrometers. See Art. 217.
It is sometimes more convenient to use a torsion-balance, in
which a horizontal arm is suspended by a fine wire or fibre, so as
to be capable of vibrating about the vertical wire as an axis, and
the body is attached to one end of the arm and acted on by the
force in the tangential direction, so as to turn the arm round the
vertical axis, and so twist the suspension wire through a certain
angle. The torsional rigidity of the wire is found by observing
the time of oscillation of the arm, the moment of inertia of the
arm being otherwise known, and from the angle of torsion and
the torsional rigidity the force of attraction or repulsion can be
deduced. The torsion-balance was devised by Michell for the de
termination of the force of gravitation between small bodies, and
was used by Cavendish for this purpose. Coulomb, working in
dependently of these philosophers, reinvented it, thoroughly studied
its action, and successfully applied it to discover the laws of electric
and magnetic forces ; and the torsion-balance has ever since been
used in all researches where small forces have to be measured. See
Art. 215.
39.] Let us suppose that by either of these methods we can
measure the force between two electrified bodies. We shall suppose
the dimensions of the bodies small compared with the distance
between them, so that the result may not be much altered by
any inequality of distribution of the electrification on either body,
and we shall suppose that both bodies are so suspended in air as
to be at a considerable distance from other bodies on which they
might induce electrification.
It is then found that if the bodies are placed at a fixed distance
and charged respectively with e and e of our provisional units of
electricity, they will repel each other with a force proportional
to the product of e and /. If either e or / is negative, that is,
if one of the charges is vitreous and the other resinous, the force
will be attractive, but if both e and / are negative the force is again
repulsive.
41.] MEASUREMENT OF ELECTRIC FORCES. 43
We may suppose the first body, A, charged with m units of
vitreous and n units of resinous electricity, which may be con
ceived separately placed within the body, as in Experiment V.
Let the second body, B, be charged with m units of positive
and ri units of negative electricity.
Then each of the m positive units in A will repel each of the m'
positive units in B with a certain force, say/; making a total effect
equal to m m f.
Since the effect of negative electricity is exactly equal and
opposite to that of positive electricity, each of the m positive units
in A will attract each of the n negative units in B with the same
force/ making a total effect equal to mnf.
Similarly the n negative units in A will attract the mf positive
units in B with a force nm'f, and will repel the n' negative units
in B with a force nn'f.
The total repulsion will therefore be (mm'+ nn)f\ and the total
attraction will be (mn' + m'n)f.
The resultant repulsion will be
(mm'-{- nnf — mnf — nm'}f or (m — n) (in — n')f.
Now m — n = e is the algebraical value of the charge on A, and
m' — n'= e is that of the charge on B, so that the resultant re
pulsion may be written eef, the quantities e and e' being always
understood to be taken with their proper signs.
Variation of the Force with the Distance.
40.] Having established the law of force at a fixed distance,
we may measure the force between bodies charged in a constant
manner and placed at different distances. It is found by direct
measurement that the force, whether of attraction or repulsion,
varies inversely as the square of the distance, so that if,/ is the
repulsion between two units at unit distance, the repulsion at dis
tance r will be/>~2, and the general expression for the repulsion
between e units and ef units at distance r will be
Definition of the Electrostatic Unit of Electricity.
41.] We have hitherto used a wholly arbitrary standard for our
unit of electricity, namely, the electrification of a certain piece of
glass as it happened to be electrified at the commencement of our
experiments. We are now able to select a unit on a definite
44 ELECTEOSTATIC PHENOMENA. [42.
principle, and in order that this unit may belong to a general
system we define it so thatymay be unity, or in other words —
The electrostatic unit of electricity is that quantity of positive elec
tricity which) when placed at unit of distance from an equal quantity,
repels it with unit of force.
This unit is called the Electrostatic unit to distinguish it from
the Electromagnetic unit, to be afterwards defined.
We may now write the general law of electrical action in the
simple form F=ee' r~2 ; or,
The repulsion between two small bodies charged respectively with e and
ef units of electricity is numerically equal to the product of the charges
divided ~by the square of the distance.
Dimensions of the Electrostatic Unit of Quantity.
42.] If [Q] is the concrete electrostatic unit of quantity itself,
and <?, e' the numerical values of particular quantities; if \_L~\ is
the unit of length, and r the numerical value of the distance ; and
if [F] is the unit of force, and F the numerical value of the force,
then the equation becomes
whence [Q] =
This unit is called the Electrostatic Unit of electricity. Other
units may be employed for practical purposes, and in other depart
ments of electrical science, but in the equations of electrostatics
quantities of electricity are understood to be estimated in electro
static units, just as in physical astronomy we employ a unit of
mass which is founded on the phenomena of gravitation, and which
differs from the units of mass in common use.
Proof of the Law of Electrical Force.
43.] The experiments of Coulomb with the torsion-balance may
be considered to have established the law of force with a certain
approximation to accuracy. Experiments of this kind, however,
are rendered difficult, and in some degree uncertain, by several
disturbing causes, which must be carefully traced and corrected for.
In the first place, the two electrified bodies must be of sensible
dimensions relative to the distance between them, in order to be
capable of carrying charges sufficient to produce measurable forces.
44-] LAW OF ELECTRIC FORCE. 45
The action of each, body will then produce an effect on the dis
tribution of electricity on the other, so that the charge cannot be
considered as evenly distributed over the surface, or collected at
the centre of gravity ; but its effect must be calculated by an
intricate investigation. This, however, has been done as regards
two spheres by Poisson in an extremely able manner, and the
investigation has been greatly simplified by Sir W. Thomson in
his Theory of Electrical Images. See Arts. 172-175.
Another difficulty arises from the action of the electricity
induced on the sides of the case containing the instrument. By
making the inner surface of the instrument of metal, this effect
can be rendered definite and measurable.
An independent difficulty arises from the imperfect insulation
of the bodies, on account of which the charge continually de
creases. Coulomb investigated the law of dissipation, and made
corrections for it in his experiments.
The methods of insulating charged conductors, and of measuring
electrical effects, have been greatly improved since the time of
Coulomb, particularly by Sir W. Thomson ; but the perfect ac
curacy of Coulomb's law of force is established, not by any direct
experiments and measurements (which may be used as illustrations
of the law), but by a mathematical consideration of the pheno
menon described as Experiment VII, namely, that an electrified
conductor J5, if made to touch the inside of a hollow closed con
ductor C and then withdrawn without touching C, is perfectly dis
charged, in whatever manner the outside of C may be electrified.
By means of delicate electroscopes it is easy to shew that no
electricity remains on B after the operation, and by the mathe
matical theory given at Art. 74, this can only be the case if the
force varies inversely as the square of the distance, for if the law
were of any different form B would be electrified.
The Electric Field.
44.] The Electric Field is the portion of space in the neigh
bourhood of electrified bodies, considered with reference to electric
phenomena. It may be occupied by air or other bodies, or it may
be a so-called vacuum, from which we have withdrawn every sub
stance which we can act upon with the means at our disposal.
If an electrified body be placed at any part of the electric field
it will, in general, produce a sensible disturbance in the electri
fication of the other bodies.
46 ELECTROSTATIC PHENOMENA. [45.
But if the body is very small, and its charge also very small,
the electrification of the other bodies will not be sensibly disturbed,
and we may consider the position of the body as determined by
its centre of mass. The force acting on the body will then be
proportional to its charge, and will be reversed when the charge
is reversed.
Let e be the charge of the body, and F the force acting on the
body in a certain direction, then when e is very small F is propor
tional to e, or F — Rey
where R depends on the distribution of electricity on the other
bodies in the field. If the charge e could be made equal to
unity without disturbing the electrification of other bodies we
should have F = R.
We shall call R the Resultant Electromotive Intensity at the
given point of the field. When we wish to express the fact that
this quantity is a vector we shall denote it by the German letter (£.
Electromotive Force and Potential.
45.1 If the small body carrying the small charge e be moved
from one given point, A, to another B, along a given path, it
will experience at each point of its course a force Re, where R
varies from point to point of the course. Let the whole work
done on the body by the electrical force be Ee, then E is called
the Total Electromotive Force along the path A E. If the path
forms a complete circuit, and if the total electromotive force round
the circuit does not vanish, the electricity cannot be in equi
librium but a current will be produced. Hence in Electrostatics
the electromotive force round any closed circuit must be zero, so
that if A and B are two points on the circuit, the electromotive
force from A to B is the same along either of the two paths into
which the circuit is broken, and since either of these can be altered
independently of the other, the electromotive force from A to B is
the same for all paths from A to B.
If B is taken as a point of reference for all other points, then
the electromotive force from A to B is called the Potential of A.
It depends only on the position of A. In mathematical investi
gations, B is generally taken at an infinite distance from the
electrified bodies.
A body charged positively tends to move from places of greater
positive potential to places of smaller positive, or of negative,
46.] ELECTEIC POTENTIAL. 47
potential, and a body charged negatively tends to move in the
opposite direction.
In a conductor the electrification is free to move relatively to
the conductor. If therefore two parts of a conductor have different
potentials, positive electricity will move from the part having
greater potential to the part having less potential as long as that
difference continues. A conductor therefore cannot be in electrical
equilibrium unless every point in it has the same potential. This
potential is called the Potential of the Conductor.
Equip otential Surfaces.
46.] If a surface described or supposed to be described in the
electric field is such that the electric potential is the same at every
point of the surface it is called an Equipotential surface.
An electrified particle constrained to rest upon such a surface
will have no tendency to move from one part of the surface to
another, because the potential is the same at every point. An
equi potential surface is therefore a surface of equilibrium or a level
surface.
The resultant force at any point of the surface is in the direction
of the normal to the surface, and the magnitude of the force is such
that the work done on an electrical unit in passing from the surface
Fto the surface V is V— V .
No two equipotential surfaces having different potentials can
meet one another, because the same point cannot have more than
one potential, but one equipotential surface may meet itself, and
this takes place at all points and along all lines of equilibrium.
The surface of a conductor in electrical equilibrium is necessarily
an equipotential surface. If the electrification of the conductor is
positive over the whole surface, then the potential will diminish as
we move away from the surface on every side, and the conductor
will be surrounded by a series of surfaces of lower potential.
But if (owing to the action of external electrified bodies) some
regions of the conductor are charged positively and others ne
gatively, the complete equipotential surface will consist of the
surface of the conductor itself together with a system of other
surfaces, meeting the surface of the conductor in the lines which
divide the positive from the negative regions. These lines will
be lines of equilibrium, and an electrified particle placed on one
of these lines will experience no force in any direction.
When the surface of a conductor is charged positively in some
^W-a Crt
48 ELECTROSTATIC PHENOMENA. [47.
parts and negatively in others, there must be some other electrified
body in the field besides itself. For if we allow a positively
electrified particle, starting from a positively charged part of the
surface, to move always in the direction of the resultant force
upon it, the potential at the point will continually diminish till
the point reaches either a negatively charged surface at a potential
less than that of the first conductor, or moves off to an infinite
distance. Since the potential at an infinite distance is zero, the
latter case can only occur when the potential of the conductor is
positive.
In the same way a negatively electrified particle, moving off
from a negatively charged part of the surface, must either reach
a positively charged surface, or pass off to infinity, and the latter
case can only happen when the potential of the conductor is
negative.
Therefore, if both positive and negative charge exist on
a conductor, there must be some other body in the field whose
potential has the same sign as that of the conductor but a greater
numerical value, and if a conductor of any form is alone in the
field the charge of every part is of the same sign as the potential
of the conductor.
The interior surface of a hollow conducting vessel containing
no charged bodies is entirely free from charge. For if any part of
the surface were charged positively, a positively electrified particle
moving in the direction of the force upon it, must reach a nega
tively charged surface at a lower potential. But the whole in
terior surface has the same potential. Hence it can have no
charge.
A conductor placed inside the vessel and communicating with
it, may be considered as bounded by the interior surface. Hence
such a conductor has no charge.
Lines of Force.
47.] The line described by a point moving always in the direc
tion of the resultant intensity is called a Line of force. It cuts the
equipotential surfaces at right angles. The properties of lines of
force will be more fully explained afterwards, because Faraday has
expressed many of the laws of electrical action in terms of his
conception of lines of force drawn in the electric field, and in
dicating both the direction and the intensity at every point.
50.]
ELECTRIC TENSION.
49
Electric Tension.
48.] Since the surface of a conductor is an equipotential surface,
the resultant force is normal to the surface, and it will be shewn
in Art. 78 that it is proportional to the superficial density of the
electrification. Hence the electricity on any small area of the
surface will be acted on by a force tending from the conductor
and proportional to the product of the resultant force and the
density, that is, proportional to the square of the resultant force.
This force, which acts outwards as a tension on every part of
the conductor, will be called electric Tension. It is measured like
ordinary mechanical tension, by the force exerted on unit of area.
The word Tension has been used by electricians in several vague
senses, and it has been attempted to adopt it in mathematical
language as a synonym for Potential ; but on examining the cases
in which the word has been used, I think it will be more con
sistent with usage and with mechanical analogy to understand by
tension a pulling force of so many pounds weight per square inch
exerted on the surface of a conductor or elsewhere. We shall find
that the conception of Faraday, that this electric tension exists not
only at the electrified surface but all along the lines of force, leads
to a theory of electric action as a phenomenon of stress in a
medium.
Electromotive Force.
49.] When two conductors at different potentials are connected
by a thin conducting wire, the tendency of electricity to flow
along the wire is measured by the difference of the potentials of
the two bodies. The difference of potentials between two con
ductors or two points is therefore called the Electromotive force
between them.
Electromotive force cannot in all cases be expressed in the
form of a difference of potentials. These cases, however, are not
treated of in Electrostatics. We shall consider them when we
come to heterogeneous circuits, chemical actions, motions of mag
nets, inequalities of temperature, &c.
Capacity of a Conductor.
50.] If one conductor is insulated while all the surrounding con
ductors are kept at the zero potential by being put in commu
nication with the earth, and if the conductor, when charged with
VOL. i. E
50 ELECTROSTATIC PHENOMENA. [51.
a quantity E of electricity, has a potential F, the ratio of E to V
is called the Capacity of the conductor. If the conductor is com
pletely enclosed within a conducting vessel without touching- it,
then the charge on the inner conductor will be equal and op
posite to the charge on the inner surface of the outer conductor,
and will be equal to the capacity of the inner conductor multiplied
by the difference of the potentials of the two conductors.
Electric Accumulators.
A system consisting of two conductors whose opposed surfaces
are separated from each other by a thin stratum of an insulating
medium is called an electric Accumulator. The two conductors are
called the Electrodes and the insulating medium is called the
Dielectric. The capacity of the accumulator is directly propor
tional to the area of the opposed surfaces and inversely proportional
to the thickness of the stratum between them. A Leyden jar is an
accumulator in which glass is the insulating medium. Accumu
lators are sometimes called Condensers, but I prefer to restrict
the term ' condenser ' to an instrument which is used not to hold
electricity but to increase its superficial density.
PROPERTIES OP BODIES IN RELATION TO STATICAL ELECTRICITY.
Resistance to the Passage of Electricity through a Body.
51.] When a charge of electricity is communicated to any part
of a mass of metal the electricity is rapidly transferred from places
of high to places of low potential till the potential of the whole
mass becomes the same. In the case of pieces of metal used in
ordinary experiments this process is completed in a time too short
to be observed, but in the case of very long and thin wires, such
as those used in telegraphs, the potential does not become uniform
till after a sensible time, on account of the resistance of the wire
to the passage of electricity through it.
The resistance to the passage of electricity is exceedingly dif
ferent in different substances, as may be seen from the tables at
Arts. 36 2, 366, and 369, which will be explained in treating of
Electric Currents.
All the metals are good conductors, though the resistance of lead
is 1 2 times that of copper or silver, that of iron 6 times, and that
of mercury 60 times that of copper. The resistance of all metals
increases as their temperature rises.
51.] ELECTRIC RESISTANCE. 51
Many liquids conduct electricity by electrolysis. This mode of
conduction will be considered in Part II. For the present, we may
regard all liquids containing water and all damp bodies as con
ductors, far inferior to the metals, but incapable of insulating a
charge of electricity for a sufficient time to be observed. The re
sistance of electrolytes diminishes as the temperature rises.
On the other hand, the gases at the atmospheric pressure, whether
dry or moist, are insulators so nearly perfect when the electric tension
is small that we have as yet obtained no evidence of electricity
passing through them by ordinary conduction. The gradual loss of
charge by electrified bodies may in every case be traced to imperfect
insulation in the supports, the electricity either passing through the
substance of the support or creeping over its surface. Hence, when
two charged bodies are hung up near each other, they will preserve
their charges longer if they are electrified in opposite ways, than if
they are electrified in the same way. For though the electromotive
force tending to make the electricity pass through the air between
them is much greater when they are oppositely electrified, no per
ceptible loss occurs in this way. The actual loss takes place through
the supports, and the electromotive force through the supports is
greatest when the bodies are electrified in the same way. The result
appears anomalous only when we expect the loss to occur by the
passage of electricity through the air between the bodies. The
passage of electricity through gases takes place, in general, by dis
ruptive discharge, and does not begin till the electromotive force
has reached a certain value. The value of the electromotive force
which can exist in a dielectric without a discharge taking place
is called the Electric Strength of the dielectric. The electric
strength of air diminishes as the pressure is reduced from the atmo
spheric pressure to that of about three millimetres of mercury.
When the pressure is still further reduced, the electric strength
rapidly increases ; and when the exhaustion is carried to the highest
degree hitherto attained, the electromotive force required to produce
a spark of a quarter of an inch is greater than that which will give
a spark of eight inches in air at the ordinary pressure.
A vacuum, that is to say, that which remains in a vessel after
we have removed everything which we can remove from it, is there
fore an insulator of very great electric strength.
The electric strength of hydrogen is much less than that of air.
Certain kinds of glass when cold are marvellously perfect in
sulators, and Sir W. Thomson has preserved charges of electricity
52 ELECTROSTATIC PHENOMENA. [52.
for years in bulbs hermetically sealed. The same glass, however,
becomes a conductor at a temperature below that of boiling water.
Gutta-percha, caoutchouc, vulcanite, paraffin, and resins are good
insulators, the resistance of gutta-percha at 75° F. being about
6 x 1 019 times that of copper.
Ice, crystals, and solidified electrolytes, are also insulators.
Certain liquids, such as naphtha, turpentine, and some oils, are
insulators, but inferior to the best solid insulators.
DIELECTllICS.
Specific Inductive Capacity.
52.] All bodies whose insulating power is such that when they
are placed between two conductors at different potentials the elec
tromotive force acting on them does not immediately distribute
their electricity so as to reduce the potential to a constant value, are
called by Faraday Dielectrics.
It appears from the hitherto unpublished researches of Cavendish
that he had, before 1773, measured the capacity of plates of glass,
rosin, beeswax, and shellac, and had determined the ratio in which
their capacity exceeded that of plates of air of the same dimensions.
Faraday, to whom these researches were unknown, discovered
that the capacity of an accumulator depends on the nature of the
insulating medium between the two conductors, as well as on the
dimensions and relative position of the conductors themselves.
By substituting other insulating media for air as the dielectric of
the accumulator, without altering it in any other respect, he found
that when air and other gases were employed as the insulating
medium the capacity of the accumulator remained sensibly the
same, but that when shellac, sulphur, glass, &c. were substituted
for air, the capacity was increased in a ratio which was different
for each substance.
By a more delicate method of measurement Boltzmann succeeded
in observing the variation of the inductive capacity of gases at
different pressures.
This property of dielectrics, which Faraday called Specific In
ductive Capacity, is also called the Dielectric Constant of the sub
stance. It is defined as the ratio of the capacity of an accumulator
when its dielectric is the given substance, to its capacity when the
dielectric is a vacuum.
If the dielectric is not a good insulator, it is difficult to measure
53-] ELECTRIC ABSORPTION-. 53
its inductive capacity, because the accumulator will not hold a
charge for a sufficient time to allow it to be measured ; but it is
certain that inductive capacity is a property not confined to good
insulators, and it is probable that it exists in all bodies.
Absorption of Electricity.
53.] It is found that when an accumulator is formed of certain
dielectrics, the following phenomena occur.
When the accumulator has been for some time electrified and is
then suddenly discharged and again insulated, it becomes recharged
in the same sense as at first, but to a smaller degree, so that it may
be discharged again several times in succession, these discharges
always diminishing. This phenomenon is called that of the Re
sidual Discharge.
The instantaneous discharge appears always to be proportional
to the difference of potentials at the instant of discharge, and the
ratio of these quantities is the true capacity of the accumulator;
but if the contact of the discharger is prolonged so as to include
some of the residual discharge, the apparent capacity of the accu
mulator, calculated from such a discharge, will be too great.
The accumulator if charged and left insulated appears to lose its
charge by conduction, but it is found that the proportionate rate
of loss is much greater at first than it is afterwards, so that the
measure of conductivity, if deduced from what takes place at first,
would be too great. Thus, when the insulation of a submarine
cable is tested, the insulation appears to improve as the electrifi
cation continues.
Thermal phenomena of a kind at first sight analogous take place
in the case of the conduction of heat when the opposite sides of a
body are kept at different temperatures. In the case of heat we
know that they depend on the heat taken in and given out by the
body itself. Hence, in the case of the electrical phenomena, it
has been supposed that electricity is absorbed and emitted by the
parts of the body. We shall see, however, in Art. 329, that the
phenomena can be explained without the hypothesis of absorp
tion of electricity, by supposing the dielectric in some degree
heterogeneous.
That the phenomenon called Electric Absorption is not an
actual absorption of electricity by the substance may be shewn by
charging the substance in any manner with electricity while it is
surrounded by a closed metallic insulated vessel. If, when the
54: ELECTROSTATIC PHENOMENA. [54.
substance is charged and insulated, the vessel be instantaneously
discharged and then left insulated, no charge is ever communicated
to the vessel by the gradual dissipation of the electrification of the
charged substance within it.
54.] This fact is expressed by the statement of Faraday that
it is impossible to charge matter with an absolute and independent
charge of one kind of electricity *.
In fact it appears from the result of every experiment which
has been tried that in whatever way electrical actions may take
place among a system of bodies surrounded by a metallic vessel, the
charge on the outside of that vessel is not altered.
Now if any portion of electricity could be forced into a body
so as to be absorbed in it, or to become latent, or in any way
to exist in it, without being connected with an equal portion of
the opposite electricity by lines of induction, or if, after having
being absorbed, it could gradually emerge and return to its ordi
nary mode of action, we should find some change of electrification
in the surrounding vessel.
As this is never found to be the case, Faraday concluded that
it is impossible to communicate an absolute charge to matter, and
that no portion of matter can by any change of state evolve or
render latent one kind of electricity or the other. He therefore
regarded induction as ' the essential function both in the first
development and the consequent phenomena of electricity.3 His
'induction' is (1298) a polarized state of the particles of the
dielectric, each particle being positive on one side and negative
on the other, the positive and the negative electrification of each
particle being always exactly equal.
Disruptive DiscJiarge f.
55.] If the electromotive intensity at any point of a dielectric
is gradually increased, a limit is at length reached at which there
is a sudden electrical discharge through the dielectric, generally
accompanied with light and sound, and with a temporary or per
manent rupture of the dielectric.
The intensity of the electromotive force when this takes place
is a measure of what we may call the electngjitrength of the di
electric. It depends on the nature of the dielectric, and is greater
in dense air than in rare air, and greater in glass than in air, but
* Exp. Res., vol. i. series xi. H ii. ' On the Absolute Charge of Matter,' and (1244).
f See Faraday, Exp. Res., vol. i., series xii. and xiii.
55-] ELECTRIC GLOW. 55
in every case, if the electromotive force be made great enough,
the dielectric gives way and its insulating power is destroyed, so
that a current of electricity takes place through it. It is for this
reason that distributions of electricity for which the electromotive
intensity becomes anywhere infinite cannot exist.
The Electric Glow.
Thus, when a conductor having a sharp point is electrified, the
theory, based on the hypothesis that it retains its charge, leads
to the conclusion that as we approach the point the superficial
density of the electricity increases without limit, so that at the
point itself the surface-density, and therefore the resultant electrical
force, would be infinite. If the air, or other surrounding dielectric,
had an invincible insulating power, this result would actually occur ;
but the fact is, that as soon as the resultant force in the neigh
bourhood of the point has reached a certain limit, the insulating
power of the air gives way, so that the air close to the point
becomes a conductor. At a certain distance from the point the
resultant force is not sufficient to break through the insulation
of the air, so that the electric current is checked, and the electricity
accumulates in the air round the point.
The point is thus surrounded by particles of air charged with
electricity of the same kind with its own. The effect of this charged
air round the point is to relieve the air at the point itself from
part of the enormous electromotive force which it would have ex
perienced if the conductor alone had been electrified. In fact the
surface of the electrified body is no longer pointed, because the
point is enveloped by a rounded mass of charged air, the surface
of which, rather than that of the solid conductor, may be regarded
as the outer electrified surface.
If this portion of charged air could be kept still, the electrified
body would retain its charge, if not on itself at least in its
neighbourhood, but the charged particles of air being free to move
under the action of electrical force, tend to move away from the
electrified body because it is charged with the same kind of elec
tricity. The charged particles of air therefore tend to move off
in the direction of the lines of force and to approach those sur
rounding bodies which are oppositely electrified. When they are
gone, other uncharged particles take their place round the point,
and since these cannot shield those next the point itself from the
excessive electric tension, a new discharge takes place, after which
56 ELECTROSTATIC PHENOMENA. [55.
the newly charged particles move off, and so on as long as the body
remains electrified.
In this way the following phenomena are produced : — At and
close to the point there is a steady glow, arising from the con
stant discharges which are taking place between the point and the
air very near it.
The charged particles of air tend to move off in the same general
direction, and thus produce a current of air from the point, con
sisting of the charged particles, and probably of others carried along
by them. By artificially aiding this current we may increase the
glow, and by checking the formation of the current we may pre
vent the continuance of the glow *.
The electric wind in the neighbourhood of the point is sometimes
very rapid, but it soon loses its velocity, and the air with its charged
particles is carried about with the general motions of the atmo
sphere, and constitutes an invisible electric cloud. When the
charged particles come near to any conducting surface, such as a
wall, they induce on that surface a charge opposite to their own,
and are then attracted towards the wall, but since the electro
motive force is small they may remain for a long time near the
wall without being drawn up to the surface and discharged. They
thus form an electrified atmosphere clinging to conductors, the
presence of which may sometimes be detected by the electrometer.
The electrical forces, however, acting between large masses of
charged air and other bodies are exceedingly feeble compared with
the ordinary forces which produce winds, and which depend on
inequalities of density due to differences of temperature, so that it is
very improbable that any observable part of the motion of ordinary
thunder clouds arises from electrical causes.
The passage of electricity from one place to another by the
motion of charged particles is called Electrical Convection or Con-
vective Discharge.
The electrical glow is therefore produced by the constant passage
of electricity through a small portion of air in which the tension
is very high, so as to charge the surrounding particles of air which
are continually swept off by the electric wind, which is an essential
part of the phenomenon.
The glow is more easily formed in rare air than in dense air,
and more easily when the point is positive than when it is negative.
* See Priestley's History of Electricity, pp. 117 and 591 ; and Cavendish's 'Elec
trical Researches,' Phil. Trans., 1771, § 4, or Art. 125 of Reprint of Cavendish.
57-] ELECTRIC SPARK. 57
This and many other differences between positive and negative elec
trification must be studied by those who desire to discover some
thing- about the nature of electricity. They have not, however,
been satisfactorily brought to bear upon any existing theory.
The Electric Brush.
56.] The electric brush is a phenomenon which may be pro
duced by electrifying a blunt point or small ball so as to produce
an electric field in which the tension diminishes as the distance
increases, but in a less rapid manner than when a sharp point is
used. It consists of a succession of discharges, ramifying as they
diverge from the ball into the air, and terminating either by
charging portions of air or by reaching some other conductor. It
is accompanied by 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.
The Electric Spark.
57.] When the tension in the space between two conductors is
considerable all the way between them, as in the case of two balls
whose distance is not 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 parts on either side of it in the direction of the electric force
are put into a state of greater tension so that they also give way,
and so the discharge proceeds right through the dielectric, just as
when a little rent is made in the edge of a piece of paper a tension
applied to the paper in the direction of the edge causes the paper to
be torn through, beginning at the rent, but diverging occasionally
where there are weak places in the paper. The electric spark in
the same way begins at the point where the electric tension first
overcomes the insulation of the dielectric, 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 air.
All these phenomena differ considerably in different gases, and in
the same gas at different densities. Some of the forms of electrical
discharge through rare gases are exceedingly remarkable. In some
cases there is a regular alternation of luminous and dark strata, so
that if the electricity, for example, is passing along a tube contain
ing a very small quantity of gas, a number of luminous disks will
58
ELECTEOSTATIC PHENOMENA. [58.
be seen arranged transversely at nearly equal intervals along1 the
axis of the tube and separated by dark strata. If the strength of
the current be increased a new disk will start into existence, and
it and the old disks will arrange themselves in closer order. In
a tube described by Mr. Gassiot* the light of each of the disks
is bluish on the negative and reddish on the positive side, and
bright red in the central stratum.
These, and many other phenomena of electrical discharge, are
exceedingly important, and when they are better understood they
will probably throw great light on the nature of electricity as well
as on the nature of gases and of the medium pervading space. At
present, however, they must be considered as outside the domain of
the mathematical theory of electricity.
Electric Phenomena of Tourmaline.
58.] Certain crystals of tourmaline, and of other minerals, possess
what may be called Electric Polarity. Suppose a crystal of tour
maline 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 de
prived of this apparent electrification by means of a flame or other
wise, then if the crystal be made still hotter,, electrification 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-
tallographic 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 crystal 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-
* Intellectual Observer, March, 1866.
59-] PLAN OF THIS TREATISE. 59
tion of each particle of the crystal is altered,, and can no longer
be balanced by the superficial electrification, so that there is a
resultant external action.
Plan of this Treatise.
59.] In the following1 treatise I propose first to explain the
ordinary theory of electrical action, which considers it as depending
only on the electrified bodies and on their relative position, with
out taking account of any phenomena which may take place in the
intervening media. In this way we shall establish the law of the
inverse square, the theory of the potential, and the equations of
Laplace and Poisson. We shall next consider the charges and
potentials of a system of electrified conductors as connected by
a system of equations, the coefficients of which may be supposed
to be determined by experiment in those cases in which our present
mathematical methods are not applicable, and from these we shall
determine the mechanical forces acting between the different elec
trified bodies.
We shall then investigate certain general theorems by which
Green, Gauss, and Thomson have indicated the conditions of so
lution of problems in the distribution of electricity. One result
of these theorems is, that if Poisson's equation is satisfied by any
function, and if at the surface of every conductor the function
has the value of the potential of that conductor, then the func
tion expresses the actual potential of the system at every point.
We also deduce a method of finding problems capable of exact
solution.
In Thomson's theorem, the total energy of the system is ex
pressed in the form of the integral of a certain quantity extended
over the whole space between the electrified bodies, and also in
the form of an integral extended over the electrified surfaces
only. The equality of these two expressions may be thus inter
preted physically. We may conceive the physical relation between
the electrified bodies, either as the result of the state of the
intervening medium, or as the result of a direct action between
the electrified bodies at a distance. If we adopt the latter con
ception, we may determine the law of the action, but we can go
no further in speculating on its cause. If, on the other hand,
we adopt the conception of action through a medium, we are
led to enquire into the nature of that action in each part of the
medium.
60 ELECTROSTATIC PHENOMENA. [59.
It appears from the theorem, that if we are to look for the seat
of the electric energy in the different parts of the dielectric me
dium, the amount of energy in any small part must depend on
the square of the resultant electromotive intensity at that place
multiplied by a coefficient called the specific inductive capacity of
the medium.
It is better, however, in considering the theory of dielectrics
from the most general point of view, to distinguish between the
electromotive intensity at any point and the electric polarization of
the medium at that point, since these directed quantities, though
related to one another, are not, in some solid substances, in the
same direction. The most general expression for the electric
energy of the medium per unit of volume is half the product of
the electromotive intensity and the electric polarization multiplied
by the cosine of the angle between their directions. In all fluid
dielectrics the electromotive intensity and the electric polarization
are in the same direction and in a constant ratio.
If we calculate on this hypothesis the total energy residing
in the medium, we shall find it equal to the energy due to the
electrification of the conductors on the hypothesis of direct action
at a distance. Hence the two hypotheses are mathematically
equivalent.
If we now proceed to investigate the mechanical state of the
medium on the hypothesis that the mechanical action observed
between electrified bodies is exerted through and by means of the
medium, as in the familiar instances of the action of one body
on another by means of the tension of a rope or the pressure of
a rod, we find that the medium must be in a state of mechanical
stress.
The nature of this stress is, as Faraday pointed out *, a tension
along the lines of force combined with an equal pressure in all
directions at right angles to these lines. The magnitude of these
stresses is proportional to the energy of the electrification per unit
of volume, or, in other words, to the square of the resultant electro
motive intensity multiplied by the specific inductive capacity of the
medium.
This distribution of stress is the only one consistent with the
observed mechanical action on the electrified bodies, and also with
the observed equilibrium of the fluid dielectric which surrounds
them. I have therefore thought it a warrantable step in scientific
* Exp. Res., series xi. 1297.
60.] STRESS IN DIELECTEICS. 61
procedure to assume the actual existence of this state of stress, and
to follow the assumption into its consequences. Finding the phrase
electric tension used in several vague senses, I have attempted to
confine it to what I conceive to have been in the mind of some
of those who have used it, namely, the state of stress in the
dielectric medium which causes motion of the electrified bodies,
and leads, when continually augmented, to disruptive discharge.
Electric tension, in this sense, is a tension of exactly the same
kind, and measured in the same way, as the tension of a rope,
and the dielectric medium, which can support a certain tension
and no more, may be said to have a certain strength in exactly
the same sense as the rope is said to have a certain strength.
Thus, for example, Thomson has found that air at the ordinary
pressure and temperature can support an electric tension of 9600
grains weight per square foot before a spark passes.
60.] From the hypothesis that electric action is not a direct
action between bodies at a distance, but is exerted by means of
the medium between the bodies, we have deduced that this medium
must be in a state of stress. We have also ascertained the cha
racter of the stress, and compared it with the stresses which may
occur in solid bodies. Along the lines of force there is tension,
and perpendicular to them there is pressure, the numerical mag
nitude of these forces being equal, and each proportional to the
square of the resultant intensity at the point. Having established
these results, we are prepared to take another step, and to form
an idea of the nature of the electric polarization of the dielectric
medium.
An elementary portion of a body may be said to be polarized
when it acquires equal and opposite properties on two opposite
sides. The idea of internal polarity may be studied to the greatest
advantage as exemplified in permanent magnets, and it will be
explained at greater length when we come to treat of magnetism.
The electric polarization of an elementary portion of a dielectric
is a forced state into which the medium is thrown by the action
of electromotive force, and which disappears when that force is
removed. We may conceive it to consist in what we may call
an electrical displacement, produced by the electromotive intensity.
When the electromotive force acts on a conducting medium it
produces a current through it, but if the medium is a non-con
ductor or dielectric, the current cannot flow through the medium,
but the electricity is displaced within the medium in the direction
62 ELECTROSTATIC PHENOMENA. [60.
of the electromotive intensity, the extent of this displacement
depending- on the magnitude of the electromotive intensity, so that
if the electromotive intensity increases or diminishes, the electric
displacement increases and diminishes in the same ratio.
The amount of the displacement is measured hy the quantity
of electricity which crosses unit of area, while the displacement
increases from zero to its actual amount. This, therefore, is the
measure of the electric polarization.
The analogy between the action of electromotive force in pro
ducing electric displacement and of ordinary mechanical force in
producing the displacement of an elastic body is so obvious that
I have ventured to call the ratio of the electromotive intensity to
the corresponding electric displacement the coefficient of electric
elasticity of the medium. This coefficient is different in different
media, and varies inversely as the specific inductive capacity of each
medium.
The variations of electric displacement evidently constitute electric
currents. These currents, however, can only exist during the
variation of the displacement, and therefore, since the displace
ment cannot exceed a certain value without causing disruptive
discharge, they cannot be continued indefinitely in the same direc
tion, like the currents through conductors.
In tourmaline, and other pyro-electric crystals, it is probable that
a state of electric polarization exists, which depends upon tem
perature, and does not require an external electromotive force to
produce it. If the interior of a body were in a state of permanent
electric polarization, the outside would gradually become charged
in such a manner as to neutralize the action of the internal
polarization 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 internal
polarization could be increased or diminished. The external elec
trification would then be no longer capable of neutralizing the
external effect of the internal polarization, and an apparent elec
trification would be observed, as in the case of tourmaline.
If a charge e is uniformly distributed over the surface of a sphere,
the resultant force at any point of the medium surrounding the
sphere is numerically equal to the charge e divided by the square of
60.] ELECTRIC DISPLACEMENT. 63
the distance from the centre of the sphere. This resultant force,
according to our theory, is accompanied by a displacement of elec
tricity in a direction outwards from the sphere.
If we now draw a concentric spherical surface of radius r, the
whole displacement, E9 through this surface will be proportional to
the resultant force multiplied by the area of the spherical surface. -
But the resultant force is directly as the charge e and inversely as
the square of the radius, while the area of the surface is directly ^^utP*t
as the square of the radius. * '**+**
Hence the whole displacement, E, is proportional to the charge e,
and is independent of the radius.
To determine the ratio between the charge e, and the quantity
of electricity, E, displaced outwards through any one of the
spherical surfaces, let us consider the work done upon the medium
in the region between two concentric spherical surfaces, while the
displacement is increased from E to E + §E. If 71 and F2 denote
the potentials at the inner and the outer of these surfaces respect
ively, the electromotive force by which the additional displacement
is produced is V± — F2, so that the work spent in augmenting the
displacement is (Vl — F~2) 8 E.
If we now make the inner surface coincide with that of the
electrified sphere, and make the radius of the other infinite, V±
becomes F", the potential of the sphere, and F"2 becomes zero, so
that the whole work done in the surrounding medium is FbE.
But by the ordinary theory, the work done in augmenting the
charge is Tbe, and if this is spent, as we suppose, in augmenting
the displacement, bE = be, and since E and e vanish together,
E = e, or —
The displacement outwards through any spherical surface concentric
with the sphere is equal to the charge on the sphere.
To fix our ideas of electric displacement, let us consider an accu
mulator formed of two conducting plates A and B, separated by a
stratum of a dielectric C. Let W be a conducting wire joining
A and B, and let us suppose that by the action of an electromotive
force a quantity Q of positive electricity is transferred along the
wire from B to A. The positive electrification of A and the
negative electrification of B will produce a certain electromotive
force acting from A towards B in the dielectric stratum, and this
will produce an electric displacement from A towards B within the
dielectric. The amount of this displacement, as measured by the
quantity of electricity forced across an imaginary section of the
64 ELECTROSTATIC PHENOMENA. [6 1.
dielectric dividing it into two strata, will be, according to our
theory, exactly Q. See Arts. 75, 76, 111.
It appears, therefore, that at the same time that a quantity
Q of electricity is being transferred along the wire by the electro
motive force from J5 towards A, so as to cross every section of
the wire, the same quantity of electricity crosses every section
of the dielectric from A towards £ by reason of the electric dis
placement.
The displacements of electricity during the discharge of the accu
mulator will be the reverse of these. In the wire the discharge
will be Q from A to B> and in the dielectric the displacement will
subside, and a quantity of electricity Q will cross every section
from B towards A.
Every case of charge or discharge may therefore be considered
as a motion in a closed circuit, such that at every section of
the circuit the same quantity of electricity crosses in the same
time, and this is the case, not only in the voltaic circuit where
it has always been recognised, but in those cases in which elec
tricity has been generally supposed to be accumulated in certain
places.
61.] We are thus led to a very remarkable consequence of the
theory which we are examining, namely, that the motions of elec
tricity are like those of an incompressible fluid, so that the total
quantity within an imaginary fixed closed surface remains always
the same. This result appears at first sight in direct contradiction
to the fact that we can charge a conductor and then introduce
it into the closed space, and so alter the quantity of electricity
within that space. But we must remember that the ordinary
theory takes no account of the electric displacement in the sub
stance of dielectrics which we have been investigating, but confines
its attention to the electrification at the bounding surfaces of the
conductors and dielectrics. In the case of the charged conductor
let us suppose the charge to be positive, then if the surrounding
dielectric extends on all sides beyond the closed surface there will
be electric polarization, accompanied with displacement from within
outwards all over the closed surface, and the surface-integral of the
displacement taken over the surface will be equal to the charge on
the conductor within.
Thus when the charged conductor is introduced into the closed
space there is immediately a displacement of a quantity of elec
tricity equal to the charge through the surface from within out-
62.] THEOEY PROPOSED. 65
wards, and the whole quantity within the surface remains the
same.
The theory of electric polarization will be discussed at greater
length in Chapter V, and a mechanical illustration of it will be
given in Art. 334, but its importance cannot be fully understood
till we arrive at the study of electromagnetic phenomena.
62.] The peculiar features of the theory are : —
That the energy of electrification resides in the dielectric medium,
whether that medium be solid, liquid, or gaseous, dense or rare,
or even what is called a vacuum, provided it be still capable of
transmitting electrical action.
That the energy in any part of the medium is stored up in
the form of a state of constraint called electric polarization, the
amount of which depends on the resultant electromotive intensity
at the place.
That electromotive force acting on a dielectric produces what
we have called electric displacement, the relation between the in
tensity and the displacement being in the most general case of a
kind to be afterwards investigated in treating of conduction, but in
the most important cases the displacement is in the same direc-
tion as the force, and is numerically equal to the intensity mul-
tiplied by — — K, where K is the specific inductive capacity of the
That the energy per unit of volume of the dielectric arising from
the electric polarization is half the product of the electromotive
intensity and the electric displacement, multiplied, if necessary, by
the cosine of the angle between their directions.
That in fluid dielectrics the electric polarization is accompanied
by a tension in the direction of the lines of induction, combined
with an equal pressure in all directions at right angles to the
lines of induction, the tension or pressure per unit of area being
numerically equal to the energy per unit of volume at the same
place.
That the surface of any elementary portion into which we may
conceive the volume of the dielectric divided must be conceived
to be charged so that the surface-density at any point of the
surface is equal in magnitude to the displacement through that
point of the surface reckoned inwards. If the displacement is in
the positive direction, the surface of the element will be charged
negatively on the positive side of the element, and positively on
VOL. i. F
66 ELECTROSTATIC PHENOMENA. [62.
the negative side. These superficial charges will in general destroy
one another when consecutive elements are considered, except
where the dielectric has an internal charge, or at the surface of
the dielectric.
That whatever electricity may be, and whatever we may under
stand by the movement of electricity, the phenomenon which we
have called electric displacement is a movement of electricity in the
same sense as the transference of a definite quantity of electricity
through a wire is a movement of electricity, the only difference
being that in the dielectric there is a force which we have called
electric elasticity which acts against the electric displacement, and
forces the electricity back when the electromotive force is removed;
whereas in the conducting wire the electric elasticity is continually
giving way, so that a current of true conduction is set up, and
the resistance depends, not on the total quantity of electricity dis
placed from its position of equilibrium, but on the quantity which
crosses a section of the conductor in a given time.
That in every case the motion of electricity is subject to the
same condition as that of an incompressible fluid, namely, that
at every instant as much must flow out of any given closed surface
as flows into it.
It follows from this that every electric current must form a
closed circuit. The importance of this result will be seen when we
investigate the laws of electro-magnetism.
Since, as we have seen, the theory of direct action at a distance
is mathematically identical with that of action by means of a
medium, the actual phenomena may be explained by the one
theory as well as by the other, provided suitable hypotheses be
introduced when any difficulty occurs. Thus, Mossotti has deduced
the mathematical theory of dielectrics from the ordinary theory
of attraction merely by giving an electric instead of a magnetic
interpretation to the symbols in the investigation by which Poisson
has deduced the theory of magnetic induction from the theory of
magnetic fluids. He assumes the existence within the dielectric of
small conducting elements, capable of having their opposite surfaces
oppositely electrified by induction, but not capable of losing or
gaining electricity on the whole, owing to their being insulated
from each other by a non-conducting medium. This theory of
dielectrics is consistent with the laws of electricity, and may be
actually true. If it is true, the specific inductive capacity of
a dielectric may be greater> but cannot be less, than that of a
62.] METHOD OF THIS WORK. 67
vacuum. No instance has yet been found of a dielectric having
an inductive capacity less than that of a vacuum, but if such should
be discovered, Mossotti's physical theory must be abandoned,
although his formulae would all remain exact, and would only
require us to alter the sign of a coefficient.
In many parts of physical science, equations of the same form
are found applicable to phenomena which are certainly of quite
different natures, as, for instance, electric induction through di
electrics, conduction through conductors, and magnetic induction.
In all these cases the relation between the force and the effect
produced is expressed by a set of equations of the same kind,
so that when a problem in one of these subjects is solved, the
problem and its solution may be translated into the language
of the other subjects and the results in their new form will still
be true.
CHAPTEE II.
ELEMENTARY MATHEMATICAL THEORY OF STATICAL
ELECTRICITY.
Definition of Electricity as a Mathematical
63.] We have seen that the properties of charged bodies are
such that the charge of one body may be equal to that of an
other, or to the sum of the charges of two bodies, and that when
two bodies are equally and oppositely charged they have no elec
trical effect on external bodies when placed together within a closed
insulated conducting vessel. We may express all these results in
a concise and consistent manner by describing an electrified body as
charged with a certain quantity of electricity, which we may denote
by e. When the charge is positive, that is, according to the usual
convention, vitreous, e will be a positive quantity. When the
charge is negative or resinous, e will be negative, and the quantity
— -e may be interpreted either as a negative quantity of vitreous
electricity or as a positive quantity of resinous electricity.
The effect of adding together two equal and opposite charges of
electricity, +<? and — e, is to produce a state of no charge expressed
by zero. We may therefore regard a body not charged as virtually
charged with equal and opposite charges of indefinite magnitude,
and a charged body as virtually charged with unequal quantities of
positive and negative electricity, the algebraic sum of these charges
constituting the observed electrification. It is manifest, however,
that this way of regarding an electrified body is entirely artificial,
and may be compared to the conception of the velocity of a body as
compounded of two or more different velocities, no one of which
is the actual velocity of the body.
ON ELECTRIC DENSITY.
Distribution in Three Dimensions.
64] Definition. The electric volume-density at a given point
in space is the limiting ratio of the quantity of electricity within
64.] ELECTRIC DENSITY. 69
a sphere whose centre is the given point to the volume of the
sphere, when its radius is diminished without limit.
We shall denote this ratio by the symbol p, which may be posi
tive or negative.
Distribution over a Surface.
It is a result alike of theory and of experiment, that, in certain
cases, the charge of a body is entirely on the surface. The density
at a point on the surface, if defined according to the method given
above, would be infinite. We therefore adopt a different method
for the measurement of surface-density.
Definition. The electric density at a given point on a surface is
the limiting ratio of the quantity of electricity within a sphere
whose centre is the given point to the area of the surface contained
within the sphere, when its radius is diminished without limit.
We shall denote the surface-density by the symbol o-.
Those writers who supposed electricity to be a material fluid
or a collection of particles, were obliged in this case to suppose
the electricity distributed on the surface in the form of a stratum
of a certain thickness 0, its density being />0, or that value of p
which would result from the particles having the closest contact
of which they are capable. It is manifest that on this theory
P0 6 = (T.
When o- is negative, according to this theory, a certain stratum
of thickness 0 is left entirely devoid of positive electricity, and
filled entirely with negative electricity, or, on the theory of one
fluid, with matter.
There is, however, no experimental evidence either of the elec
tric stratum having any thickness, or of electricity being a fluid
or a collection of particles. We therefore prefer to do without the
symbol for the thickness of the stratum, and to use a special symbol
for surface-density.
Distribution on a Line.
It is sometimes convenient to suppose electricity distributed
on a line, that is, a long narrow body of which we neglect the
thickness. In this case we may define the line-density at any point
to be the limiting ratio of the charge on an element of the
line to the length of that element when the element is diminished
without limit.
70 ELECTROSTATICS. [65.
If A denotes the line-density, then the whole quantity of elec
tricity on a curve is e — I \ds, where ds is the element of the curve.
Similarly, if cr is the surface-density, the whole quantity of elec
tricity on the surface is
where dS is the element of surface.
If p is the volume-density at any point of space, then the whole
electricity within a certain volume is
e = / / / p dx dy dz.
where dx dy dz is the element of volume. The limits of integration
in each case are those of the curve, the surface, or the portion of
space considered.
It is manifest that 0, A, o- and p are quantities differing in kind,
each being one dimension in space lower than the preceding, so that
if I be a line, the quantities #, IX, I2 a; and I3 p will be all of the
same kind, and if [Z] be the unit of length, and [A], [o-], [p] the
units of the different kinds of density, [Y], [£A], [X2<r], and [^3/o]
will each denote one unit of electricity.
Definition of the Unit of Electricity.
65.] Let A and B be two points the distance between which
is the unit of length. Let two bodies, whose dimensions are small
compared with the distance AS, be charged with equal quantities
of positive electricity and placed at A and B respectively, and
let the charges be such that the force with which they repel each
other is the unit of force, measured as in Art. 6. Then the charge
of either body is said to be the unit of electricity.
If the charge of the body at B were a unit of negative electricity,
then, since the action between the bodies would be reversed, we
should have an attraction equal to the unit of force. If the charge
of A were also negative, and equal to unity, the force would be
repulsive, and equal to unity.
Since the action between any two portions of electricity is not
affected by the presence of other portions, the repulsion between
e units of electricity at A and e' units at B is ee\ the distance
AB being unity. See Art?: 39. Asvti 4/
Law of Force between Charged Bodies.
66.] Coulomb shewed by experiment that the force between
68.] LAW OF ELECTRIC FORCE. 71
charged bodies whose dimensions are small compared with the
distance between them, varies inversely as the square of the dis
tance. Hence the repulsion between two such bodies charged with
quantities e and / and placed at a distance r is
72-*
We shall prove in Art. 74 that this law is the only one con
sistent with the observed fact that a conductor, placed in the inside
of a closed hollow conductor and in contact with it, is deprived of
all electrical charge. Our conviction of the accuracy of the law
of the inverse square of the distance may be considered to rest
on experiments of this kind, rather than on the direct measure
ments of Coulomb.
Resultant Force between Two Bodies.
67.] In order to calculate the resultant force between two bodies
we might divide each of them into its elements of volume, and
consider the repulsion between the electricity in each of the elements
of the first body and the electricity in each of the elements of the
second body. We should thus get a system of forces equal in
number to the product of the numbers of the elements into which
we have divided each body, and we should have to combine the
effects of these forces by the rules of Statics. Thus, to find the
component in the direction of x we should have to find the value
of the sextuple integral
/Y/Y/Y PP'(X—~X') dx dy dz dx'dy' dz
JJJJJJ { (# _ a/)2 _j_ (y — /)2 + (z— /)2 } % '
where #, y, z are the coordinates of a point in the first body at
which the electrical density is /o, and #', y', /, and p' are the
corresponding quantities for the second body, and the integration
is extended first over the one body and then over the other.
Eesultant Intensity at a Point.
68.] In order to simplify the mathematical process, it is con
venient to consider the action of an electrified body, not on another
body of any form, but on an indefinitely small body, charged with
an indefinitely small amount of electricity, and placed at any point
of the space to which the electrical action extends. By making
the charge of this body indefinitely small we render insensible its
disturbing action on the charge of the first body.
72 ELECTROSTATICS. [69.
Let e be the charge of the small body, and let the force acting
on it when placed at the point (a?,y, z) be Re, and let the direction-
cosines of the force be I, m, n, then we may call R the resultant
electrical Intensity at the point (x, y, z).
If X, Y, Z denote the components of R, then
X=Rl, Y=Rm, Z=Rn.
In speaking of the resultant electrical intensity at a point, we
do not necessarily imply that any force is actually exerted there,
but only that if an electrified body were placed there it would be
acted on by a force Re, where e is the charge of the body*.
Definition. The Resultant electric Intensity at any point is the
force which would be exerted on a small body charged with the
unit of positive electricity, if it were placed there without disturbing
the actual distribution of electricity.
This force not only tends to move a body charged with
electricity, but to move the electricity within the body, so that
the positive electricity tends to move in the direction of R
and the negative electricity in the opposite direction. Hence
the quantity R is also called the Electromotive Intensity at the point
0», v> 4
When we wish to express the fact that the resultant intensity is
a vector, we shall denote it by the German letter (£. If the body
is a dielectric, then, according to the theory adopted in this
treatise, the electricity is displaced within it, so that the quantity
of electricity which is forced in the direction of & across unit of
area fixed perpendicular to (£ is
where 2) is the displacement, (£ the resultant intensity, and K the
specific inductive capacity of the dielectric.
If the body is a conductor, the state of constraint is continually
giving way, so that a current of conduction is produced and main
tained as long as (£ acts on the medium.
Line-Integral of Electric Intensity > or Electromotive Force along
an Arc of a Curve.
f 69.] The Electromotive force along a given arc AP of a curve is
-I numerically measured by the work which would be done by the
* The Electric and Magnetic Intensity correspond, in electricity and mag
netism, to the intensity of gravity, commonly denoted by g, in the theory of heavy
bodies.
70.] ELECTROMOTIVE FORCE. 73
electric force on a unit of positive electricity carried along the curve
from A, the beginning", to P, the end of the arc.
If s is the length of the arc, measured from A, and if the re
sultant intensity R at any point of the curve makes an angle e with
the tangent drawn in the positive direction, then the work done
on unit of electricity in moving along the element of the curve
ds will be R cos € dS)
and the total electromotive force D will be
E = I R cos e ds,
J
the integration being extended from the beginning to the end
of the arc.
If we make use of the components of the intensity, the expres
sion becomes
o
If X, Y, and Z are such that Xdse+ Ydy + Zdz is the complete
differential of — F, a function of x, y, z, then
E = [P(Xdx + Ydy + Zdz} = - f*dF = VA- VP\
JA J A
where the integration is performed in any way from the point A
to the point P, whether along the given curve or along any other
line between A and P.
In this case Fis a scalar function of the position of a point in
space, that is, when we know the coordinates of the point, the value
of Fis determinate, and this value is independent of the position
and direction of the axes of reference. See Art. 16.
On Functions of the Position of a Point.
In what follows, when we describe a quantity as a function of
the position of a point, we mean that for every position of the point
the function has a determinate value. We do not imply that this
value can always be expressed by the same formula for all points of
space, for it may be expressed by one formula on one side of a
given surface and by another formula on the other side.
On Potential Functions.
70.] The quantity Xdx+Ydy + Zdz is an exact differential
whenever the force arises from attractions or repulsions whose in
tensity is a function of the distances from any number of points.
74 ELECTROSTATICS. [71.
For if ^ be the distance of one of the points from the point (#, y, 2),
and if R± be the repulsion, then
Y - 7? *""*i - 7? ^i
•*i — **1 - = **1 ~T~ '
/! l<?#
with similar expressions for T^ and Zlt so that
Tl dy + Z± dz = R± dr± ;
and since Rl is a function of r± only, ^ dr^ is an exact differential
of some function of rlt say — V^ .
Similarly for any other force E^ acting from a centre at dis
tance r2, Xzdx + Y2dy + Z2dz = R2dr2 = -dV2.
But X = X1 -j- X2 + &c. and T and Z are compounded in the same
way, therefore
The integral of this quantity, under the condition that it vanishes
at an infinite distance, is called the Potential Function.
The use of this function in the theory of attractions was intro
duced by Laplace in the calculation of the attraction of the earth.
Green, in his essay ' On the Application of Mathematical Analysis
to Electricity/ gave it the name of the Potential Function. Gauss,
working independently of Green, also used the word Potential.
Clausius and others have applied the term Potential to the work
which would be done if two bodies or systems were removed to
an infinite distance from one another. We shall follow the use of
the word in recent English works, and avoid ambiguity by adopting
the following definition due to Sir W. Thomson.
Definition of Potential. The Potential at a Point is the work
which would be done on a unit of positive electricity by the elec
tric forces if it were placed at that point without disturbing the
electric distribution, and carried from that point to an infinite
distance : or, what comes to the same thing, the work which
must be done by an external agent in order to bring the unit
of positive electricity from an infinite distance (or from any place
where the potential is zero) to the given point.
71.] Expressions for the Resultant Intensity and its components in
terms of the Potential.
Since the total electromotive force along any arc AB is
72.] POTENTIAL** 75
if we put ds for the arc AB we shall have for the force resolved
in the direction of ds,
T> dY
H cos e = =- ;
ds
whence, by assuming ds parallel to each of the axes in succession,
we get
Y dV _ dV dV
JL=— — • I = — , Z— r- J
dx dy dz
2 dV
+ -5-
dV
+ -T-
dz
We shall denote the intensity itself, whose magnitude, or tensor,
is E and whose components are X, Y, Z, by the German letter (£, as
in Arts. 17 and 68.
The Potential at all Points within a Conductor is the same.
72.] A conductor is a body which allows the electricity within
it to move from one part of the body to any other when acted on
by electromotive force. When the electricity is in equilibrium
there can be no electromotive force -acting within the conductor.
Hence R = 0 throughout the whole space occupied by the con
ductor. From this it follows that
dV dV dV
-7- = °> :r- = °> -7-=°;
dx dy dz
and therefore for every point of the conductor
F=C,
where C is a constant quantity.
Since the potential at all points within the substance of the
conductor is C, the quantity C is called the Potential of the con
ductor. C may be defined as the work which must be done by
external agency in order to bring a unit of electricity from an
infinite distance to the conductor, the distribution of electricity
being supposed not to be disturbed by the presence of the unit.
It will be shewn at Art. 246 that in general when two bodies
of different kinds are in contact, an electromotive force acts from
one to the other through the surface of contact, so that when they
are in equilibrium the potential of the latter is higher than that
of the former. For the present, therefore, we shall suppose all our
conductors made of the same metal, and at the same temperature.
If the potentials of the conductors A and B be VA and ~PB re
spectively, then the electromotive force along a wire joining A and
B will be YA — V*
76 ELECTROSTATICS. [73.
in the direction AS, that is, positive electricity will tend to pass
from the conductor of higher potential to the other.
Potential, in electrical science, has the same relation to Elec
tricity that Pressure, in Hydrostatics, has to Fluid, or that Tem
perature, in Thermodynamics, has to Heat. Electricity, Fluids,
and Heat all tend to pass from one place to another, if the Poten
tial, Pressure, or Temperature is greater in the first place than in
the second. A fluid is certainly a substance, heat is as certainly
not a substance, so that though we may find assistance from ana
logies of this kind in forming clear ideas of formal relations of
electrical quantities, we must be careful not to let the one or the
other analogy suggest to us that electricity is either a substance
like water, or a state of agitation like heat.
Potential due to any Electrical System.
73.] Let there be a single electrified point charged with a quantity
e of electricity, and let r be the distance of the point #', y' , / from
it, then f*> r°° ,, ^
V = \ Edr = ~dr = -•
Jr Jr r2 r
Let there be any number of electrified points whose coordinates
are (xl9 ylt zj, (x2, y^ z^ &c. and their charges elt <?2, &c., and
let their distances from the point (#', /, /) be rlt r2, &c., then the
potential of the system at (of, y ', /) will be
Let the electric density at any point (a?, y> z) within an elec
trified body be pt then the potential due to the body is
where r = {(x-xj + (y-yj + (*-/)2}*,
the integration being extended throughout the body.
On the Proof of the Law of the Inverse Square.
74 «.] The fact that the force between electrified bodies is inversely
as the square of the distance may be considered to be established by
Coulomb's direct experiments with the torsion-balance. The results,
however, which we derive from such experiments must be regarded
as affected by an error depending on the probable error of each
experiment, and unless the skill of the operator be very great,
74 fr-] PKOOF OF THE LAW OF FOECE. 77
the probable error of an experiment with the torsion-balance is
considerable.
A far more accurate verification of the law of force may be
deduced from an experiment similar to that described at Art 32
(Exp. VII).
Cavendish, in his hitherto unpublished work on electricity, makes
the evidence of the law of force depend on an experiment of this
kind.
He fixed a globe on an insulating- support, and fastened two
hemispheres by glass rods to two wooden frames hinged to an axis
so that the hemispheres, when the frames were brought together,
formed an insulated spherical shell concentric with the globe.
The globe could then be made to communicate with the hemispheres
by means of a short wire, to which a silk string was fastened so
that the wire could be removed without discharging the apparatus.
The globe being in communication with the hemispheres, he
charged the hemispheres by means of a Leyden jar, the potential
of which had been previously measured by an electrometer, and
immediately drew out the communicating wire by means of the
silk string, removed and discharged the hemispheres, and tested
the electrical condition of the globe by means of a pith ball electro
meter.
No indication of any charge of the globe could be detected by
the pith ball electrometer, which at that time (1773) was considered
the most delicate electroscope.
Cavendish next communicated to the globe a known fraction of
the charge formerly communicated to the hemispheres, and tested
the globe again with his electrometer.
He thus found that the charge of the globe in the original
experiment must have been less than -fa- of the charge of the whole
apparatus, for if it had been greater it would have been detected by
the electrometer.
He then calculated the ratio of the charge of the globe to that of
the hemispheres on the hypothesis that the repulsion is inversely as
a power of the distance differing slightly from 2, and found that if
this difference was -^ there would have been a charge on the globe
equal to -gV of that of the whole apparatus, and therefore capable of
being detected by the electrometer.
74 #.] The experiment has recently been repeated at the Cavendish
Laboratory in a somewhat different manner.
The hemispheres were fixed on an insulating stand, and the globe
78 ELECTROSTATICS. [74 b.
fixed in its proper position within them by means of an ebonite
ring-. By this arrangement the insulating support of the globe
was never exposed to the action of any sensible electric force, and
therefore never became charged, so that the disturbing effect of
electricity creeping along the surface of the insulators was entirely
removed.
Instead of removing the hemispheres before testing the potential
of the globe, they were left in their position, but discharged to
earth. The effect of a given charge of the globe on the electro
meter was not so great as if the hemispheres had been removed,
but this disadvantage was more than compensated by the perfect
security afforded by the conducting vessel against all external
electric disturbances.
The short wire which made the connexion between the shell and
the globe was fastened to a small metal disk which acted as a lid to
a small hole in the shell, so that when the wire and the lid were
lifted up by a silk string, the electrode of the electrometer could be
made to dip into the hole and rest on the globe within.
The electrometer was Thomson's Quadrant Electrometer described
in Art. 219. The case of the electrometer and one of the electrodes
were always connected to earth, and the testing electrode was con
nected to earth till the electricity of the shell had been discharged.
To estimate the original charge of the shell, a small brass ball
was placed on an insulating support at a considerable distance from
the shell.
The operations were conducted as follows : —
The shell was charged by communication with a Leyden jar.
The small ball was connected to earth so as to give it a negative
charge by induction, and was then left insulated.
The communicating wire between the globe and the shell was
removed by a silk string.
The shell was then discharged, and kept connected to earth.
The testing electrode was disconnected from earth, and made
to touch the globe, passing through the hole in the shell.
Not the slightest effect on the electrometer could be observed.
To test the sensitiveness of the apparatus the shell was discon
nected from earth and the small ball was discharged to earth. The
electrometer then showed a positive deflection, D.
The negative charge of the brass ball was about -f^ of the ori
ginal charge of the shell, and the positive charge induced by the
ball when the shell was put to earth was about | of that of the ball.
74 c-~\ PROOF OF THE LAW OF FORCE. 79
Hence when the ball was put to earth the potential of the shell, as
indicated by the electrometer, was about T|F of its original potential.
But if the repulsion had been as rq~2, the potential of the globe
would have been —0-1478 q of that of the shell by equation 22, p. 81.
Hence if + d be the greatest deflexion of the electrometer which
could escape observation, and D the deflexion observed in the second
part of the experiment, q cannot exceed
+ J-£.
- 72 D
Now even in a rough experiment D was more than 300^ so that
q cannot exceed 1
- 21600
Theory of the Experiment.
74 cl\ To find the potential at any point due to a uniform spherical
shell, the repulsion between two units of matter being any given
function of the distance.
Let 0 (/*) be the repulsion between two units at distance r, and
let/(r) be such that
(=/'(')) = 'j* w*. 0)
Let the radius of the shell be a, and its surface density cr, then, if
a denotes the whole mass of the shell,
a = 47T02o-. (2)
Let b denote the distance of the given point from the centre
of the shell, and let r denote its distance from any given point
of the shell.
If we refer the point on the shell to spherical coordinates, the
pole being the centre of the shell, and the axis the line drawn
to the given point, then
r2 = a2 + b2-2abcos0. (3)
The mass of the element of the shell is
o- a2 sin 6 d$ dd, (4)
and the potential due to this element at the given point is
<ja2siuO'^-^ded<t>; (5)
and this has to be integrated with respect to $ from $ = 0 to
<J> = 2 IT, which gives
27ro-fl2sin<9^-^0, (6)
which has to be integrated from 0 — 0 to B = TT.
80 ELECTROSTATICS. [74 C.
Differentiating (3) we find
(7)
Substituting the value of dO in (6) we obtain
2 * erf
the integral of which is
-/r2}, (9)
when rx is the greatest value of r, which is always a + b, and ?\
is the least value of r, which is b— a when the given point is out
side the shell and a—b when it is within the shell.
If we write a for the whole charge of the shell, and V for its
potential at the given point, then for a point outside the shell
r=^L {/(*+«)-/(*-«)}. (10)
For a point on the shell itself
and for a point inside the shell
We have next to determine the potentials of two concentric
spherical shells, the radii of the outer and inner shells being a and b,
and their charges a and /3.
Calling the potential of the outer shell A, and that of the inner
J5, we have by what precedes
2 a'2
~
In the first part of the experiment the shells communicate by the
short wire and are both raised to the same potential, say V.
By putting A = B = V, and solving the equations (13) and (14)
for /3, we find the charge of the inner shell
a/(2 a) -«
In the experiment of Cavendish, the hemispheres forming the
outer shell were removed to a distance which we may suppose in-
74 eJ] PROOF OF THE LAW OF FORCE. 81
finite, and discharged. The potential of the inner shell (or globe)
would then become
S^-^AZt). • (16)
In the form of the experiment as repeated at the Cavendish
Laboratory the outer shell was left in its place, but connected to
earth, so that A — 0. In this case we find for the potential of the
inner shell in terms of V
.] Let us now assume, with Cavendish, that the law offeree
is some inverse power of the distance, not differing much from the
inverse square, and let us put
$(*•) = /•'-»; (18)
then /W==_l_f*+i. (19)
If we suppose q to be small, we may expand this by the ex
ponential theorem in the form
+?lo&?'+(?logr)2"F&c-; (20)
and if we neglect terms involving q2, equations (16) and (17) be
come
from which we , may determine q in terms of the results of the
experiment.
740.] Laplace gave the first demonstration that no function of
the distance except the inverse square satisfies the condition that a
uniform spherical shell exerts no force on a particle within it *.
If we suppose that /3 in equation (15) is always zero, we may
apply the method of Laplace to determine the form of /(r). We
have by (15),
a/(2 *)-*/(*+*)+«/(« -a) = o.
Differentiating twice with respect to b, and dividing by #, we find
f"(a + b) =f"(a-l).
If this equation is generally true
f" (r) = C0, a constant.
* Mec. Cel, I. 2.
VOL. I. G
82 ELECTROSTATICS. [75.
Hence, f (r) = (70r + C\;
and by (1) f* <f>(r)dr= £^-=(70+^-,
J f T T
We may observe, however, that though the assumption of
Cavendish, that the force varies as some power of the distance, may
appear less general than that of Laplace, who supposes it to be any
function of the distance, it is the only one consistent with the fact
that similar figures can be electrified so as to have similar electrical
properties.
For if the force were any function of the distance except a power
of the distance, the ratio of the force at two different distances
would not be a function of the ratio of the distances, but would
depend on the absolute value of the distances, and would therefore
involve the ratios of these distances to an absolutely fixed length.
Indeed Cavendish himself points out that on his own hypothesis
as to the constitution of the electric fluid, it is impossible for the
distribution of electricity to be accurately similar in two conductors
geometrically similar, unless the charges are proportional to the
volumes. For he supposes the particles of the electric fluid to be
closely pressed together near the surface of the body, and this is
equivalent to supposing that the law of repulsion is no longer the
inverse square, but that as soon as the particles come into contact,
their repulsion begins to increase at a much greater rate with any
further diminution of their distance.
Surface-Integral of Electric Induction, and Electric Displacement
through a surface.
75.] Let R be the resultant intensity at any point of the surface,
and e the angle which E makes with the normal drawn towards
the positive side of the surface, then R cos e is the component of
the intensity normal to the surface, and if dS is the element of the
surface, the electric displacement through dS will be, by Art. 68,
since we do not at present consider any dielectric except air, K=l.
We may, however, avoid introducing at this staye the theory of
electric displacement, by calling RcosedS the Induction through
the element dS. This quantity is well known in mathematical
76.] ELECTRIC INDUCTION. 83
physics, but the name of induction is borrowed from Faraday.
The surface-integral of induction is
R cos € dS,
and it appears by Art. 21, that if X, Y, Z are the components of R,
and if these quantities are continuous within a region bounded by a
closed surface S, the induction reckoned from within outwards is
/Yr, fff/dX dY dz\ •
EcosedS = / / / (-y- + -y- + -J-) dxdydz,
JJ JJJ \dx dy dz'
the integration being extended through the whole space within the
surface.
Induction through a Closed Surface due to a Single Centre of Force.
76.] Let a quantity e of electricity be supposed to be placed at a
point 0, and let r be the distance of any point P from 0, the force
at that point is R = er~2 in the direction OP.
Let a line be drawn from 0 in any direction to an infinite dis
tance. If 0 is without the closed surface this line will either not
cut the surface at all, or it will issue from the surface as many
times as it enters. If 0 is within the surface the line must first
issue from the surface, and then it may enter and issue any number
of times alternately, ending by issuing from it.
Let € be the angle between OP and the normal to the surface
drawn outwards where OP cuts it, then where the line issues from
the surface, cos e will be positive, and where it enters, cos e will
be negative.
Now let a sphere be described with centre 0 and radius unity,
and let the line OP describe a conical surface of small angular
aperture about 0 as vertex.
This cone will cut off a small element d<& from the surface of the
sphere, and small elements dSlt dS2, &c. from the closed surface at
the different places where the line OP intersects it.
Then, since any one of these elements dS intersects the cone at a
distance r from the vertex and at an obliquity e,
dS — r2 sec e du> ;
and, since R = er~2, we shall have
R cos € dS = ±edu>;
the positive sign being taken when r issues from the surface, and
the negative where it enters it.
If the point 0 is without the closed surface, the positive values
G 2
84 ELECTROSTATICS. [77.
are equal in number to the negative ones, so that for any direction
and therefore / / R cos e dS = 0,
the integration being extended over the whole closed surface.
If the point 0 is within the closed surface the radius vector OP
first issues from the closed surface, giving a positive value of e da,
and then has an equal number of entrances and issues, so that in
this case 2 R Cos e dS = e dv.
Extending the integration over the whole closed surface, we shall
include the whole of the spherical surface, the area of which is 4 TT,
so that rr rr
I I R cos e dS = e / / da = 47i<?.
Hence we conclude that the total induction outwards through a
closed surface due to a centre of force e placed at a point 0 is
zero when 0 is without the surface, and 4 tie when 0 is within
the surface.
' Since in air the displacement is equal to the induction divided
by 4-77, the displacement through a closed surface, reckoned out-
v wards, is equal to the electricity within the surface.
Corollary. It also follows that if the surface is not closed but
is bounded by a given closed curve, the total induction through
it is we, where o> is the solid angle subtended by the closed curve
at 0. This quantity, therefore, depends only on the closed curve,
and the form of the surface of which it is the boundary may be
changed in any way, provided it does not pass from one side to the
other of the centre of force.
On tJie Equations of Laplace and Poisson.
77.] Since the value of the total induction of a single centre
of force through a closed surface depends only on whether the
centre is within the surface or not, and does not depend on its
position in any other way, if there are a number of such centres
elt e2, &c. within the surface, and */, ez', &c. without the surface,
we shall have rr
/ / R cose dS = 4ne ;
where e denotes the algebraical sum of the quantities of electricity
at all the centres of force within the closed surface, that is, the
total electricity within the surface, resinous electricity being reck
oned negative.
78 a.] EQUATIONS OP LAPLACE AND POISSON. 85
If the electricity is so distributed within the surface that the
density is nowhere infinite, we shall have by Art. 64,
4 TT e = 4 TT / / / p dx dy dz,
and by Art. 75,
/Y» ff[fdx dY dz\ i
I RcostdS = I I /(— + — + —}dacdydz.
J J JJJ ^dx dy dz'
If we take as the closed surface that of the element of volume
dx dy dzj we shall have, by equating these expressions,
dX dY dZ
and if a potential V exists, we find by Art. 7 1 ,
d27
This equation, in the case in which the density is zero, is called
Laplace's Equation. In its more general form it was first given by
Poisson. It enables us, when we know the potential at every point,
to determine the distribution of electricity.
We shall denote, as in Art. 26, the quantity
d27 .
and we may express Poisson's equation in words by saying that
the electric density multiplied by 4?r is the concentration of the
potential. Where there is no electrification, the potential has no
concentration, and this is the interpretation of Laplace's equation.
By Art. 72, V is constant within a conductor. Hence within a
conductor the volume-density is zero, and the whole charge must
be on the surface.
If we suppose that in the superficial and linear distributions of
electricity the volume-density p remains finite, and that the elec
tricity exists in the form of a thin stratum or a narrow fibre, then,
by increasing p and diminishing the depth of the stratum or the
section of the fibre, we may approach the limit of true superficial
or linear distribution, and the equation being true throughout the
process will remain true at the limit, if interpreted in accordance
with the actual circumstances.
Variation of the Potential at a Charged Surface.
78 #.] The potential function, F, must be physically continuous
in the sense defined in Art. 7, except at the bounding surface of
86 ELECTROSTATICS. [78 a.
two different media, in which case, as we shall see in Art. 246,
there may be a difference of potential between the substances,
so that when the electricity is in equilibrium, the potential at
a point in one substance is higher than the potential at the
contiguous point in the other substance by a constant quantity,
C, depending on the natures of the two substances and on their
temperatures.
But the first derivatives of V with respect to #, y, or z may be
discontinuous, and, by Art. 8, the points at which this discontinuity
occurs must lie in a surface, the equation of which may be expressed
in the form ^ _ $ fa ^ zj = 0. (l)
This surface separates the region in which (/> is negative from the
region in which <p is positive.
Let T[ denote the potential at any given point in the negative
region, and V% that at any given point in the positive region, then
at any point in the surface at which $ = 0, and which may be
said to belong to both regions,
r^c=rt, (2)
where C is the constant excess of potential, if any, in the substance
on the positive side of the surface.
Let /, m, n be the direction-cosines of the normal v2 drawn from
a given point of the surface into the positive region. Those of the
normal vl drawn from the same point into the negative region will
be — /, — m, and — n.
The rates of variation of V along the normals are
dV, .dK dV, dV,
-_!=-/— 1-^-^-fc—l, (3)
di\ else ay dz
d72 7dK dK dK
-T- = l-T~ + m-r- +n-j±-- (4)
dv% ace ay dz
Let any line be drawn on the surface, and let its length, measured
from a fixed point in it, be <?, then at every point of the surface,
and therefore at every point of this line, V^— T[ = C. Differentiating
this equation with respect to s, we get
v dx dx ' ds v dy dy ' ds V dz dz ' ds
m(o)
and since the normal is perpendicular to this line
, dx dy dz
l~ + m-f + n-r = 0. (6)
ds ds ds
-.-*
c ^ ^5 ^o; A ^ ip ^
» -* I '*f . *'*T " j -^" r^-xx
78 b.] POTENTIAL NEAR A CHARGED SURFACE. 87
From (3), (4), (5), (6) we find
dr, W__l(WvW}y (7)
W dx ' ^ h dvj*
£-£_.('5+<£), (8)
ay ay ^avl dvz'
dK dK fdV^ dV^ ...
-=-*- -- r1 = n ( -T± + -j-M • (9)
dz dz ^dv-L dv%'
If we consider the variation of the electromotive intensity at a
point in passing through the surface, that component of the in
tensity which is normal to the surface may change abruptly at the
surface, but the other two components parallel to the tangent plane
remain continuous in passing through the surface.
783.] To determine the charge of the surface, let us consider a
closed surface which is partly in the positive region and partly in
the negative region, and which therefore encloses a portion of the
surface of discontinuity.
The surface integral,
extended over this surface, is equal to lire, where e is the quantity
of electricity within the closed surface.
Proceeding as in Art. 2 1 , we find
dY
S) (2)
where the triple integral is extended throughout the closed surface,
and the double integral over the surface of discontinuity.
Substituting for the terms of this equation their values from
(7), (8), (9),
\ /' v y \ /'
But by the definition of the volume-density, p, and the surface-
density, ,, ^ = ^ j j f pdxdydz + ^fjads_ (12)
Hence, comparing the last terms of these two equations,
0. (13)
This equation is called the characteristic equation of V at an elec
trified surface of which the surface-density is <r.
88 ELECTROSTATICS.
78<?.] If V is a function of x,y, z which, throughout a given con
tinuous region of space, satisfies Laplace's equation
dtf df
and if throughout a finite portion of this region T is constant and
equal to C, then V must be constant and equal to C throughout the
whole region in which Laplace's equation is satisfied.
If V is not equal to C throughout the whole region, let 8 be the
surface which bounds the finite portion within which V — C.
At the surface 8, V = C.
Let v be a normal drawn outwards from the surface 8. Since
8 is the boundary of the continuous region for which V — C, the
value of Fas we travel from the surface along the normal begins
dV
to differ from C. Hence -=— just outside the surface may be posi-
wV
tive or negative, but cannot be zero except for normals drawn from
the boundary line between a positive and a negative area.
But if v is the normal drawn inwards from the surface S, V — C
j d?'
and -j-r = 0.
civ
Hence, at every point of the surface except certain boundary lines,
dv dr.
— + -—-(=— 47TO-)
dv dv ^
is a finite quantity, positive or negative, and therefore the surface
8 has a continuous distribution of electricity over all parts of it
except certain boundary lines which separate positively from nega
tively charged areas.
Laplace's equation is not satisfied at the surface 8 except at
points lying on certain lines on the surface. The surface 8 there
fore, within which V — C, includes the whole of the continuous
region within which Laplace's equation is satisfied.
Force Acting on a Charged Surface.
79.] The general expression for the components of the force
acting on a charged body parallel to the three axes are of the form
A =f[fp %dx dy dz, (14)
with similar expressions for B and C, the components parallel to y
and z.
But at a charged surface p is infinite, and X is discontinuous, so
.79-] FORCE ACTING ON A CHARGED SURFACE. 89
that we cannot calculate the force directly from expressions of this
form.
We have proved, however, that the discontinuity affects only
that component of the intensity which is normal to the charged
surface, the other two components being continuous.
Let us therefore assume the axis of x normal to the surface at
the given point, and let us also assume, at least in the first part
of our investigation, that X is not really discontinuous; but that
it changes continuously from X1 to X2 while x changes from xl
to #?2. If the result of our calculation gives a definite limiting
value for the force when x^—x^ is diminished without limit, we
may consider it correct when x2 = x± , and the charged surface has
no thickness.
Substituting for p its value as found in Art. 77,
A i ffffdx t AY ciz.vl 1 . , .
--+ + Xd*d*dz'
Integrating this expression with respect to as from x = a?x to x = os
it becomes
This is the value of A for a stratum parallel to yz of which the
thickness is x^—x^
Since Y and Z are continuous, — + —- is finite, and since X
dy dz
is also finite,
,dY d
where C is the greatest value of (-j- -\--j-jX between x — x^ and
x —
Hence when x.2 — x-^ is diminished without limit this term must
ultimately vanish, leaving
(17)
where X1 is the value of X on the negative and X2 on the positive
side of the surface.
But by Art. 78, Xj-^i = - = ***> (18)
dx dx
so that we may write
A =jj\(Xz + Xl}(rdydz. (19)
Here dydz is the element of the surface, <r is the surface-density,
90 ELECTROSTATICS. [80.
and J (X2 -f Xj) is the arithmetical mean of the electromotive in
tensity on the two sides of the surface.
Hence an element of a charged surface is acted on by a force,
the component of which normal to the surface is equal to the charge
of the element into the arithmetical mean of the normal electro
motive intensities on the two sides of the surface.
Since the other two components of the electromagnetic intensity
are not discontinuous, there can be no ambiguity in estimating the
corresponding components of the force acting on the surface.
We may now suppose the direction of the normal to the surface to be
in any direction with respect to the axes, and write the general expres
sions for the components of the force on the element of surface dS,
A = ±(Xl + X2)<rdS,
J9 = i(rl+ra)cr^, (20)
C =
Charged Surface of a Conductor.
80.] We have already shewn (Art. 72) that throughout the sub
stance of a conductor in electric equilibrium X = Y = Z— 0, and
therefore V is constant.
dX dY dZ
Hence -z — \- —7 — |- -=— = 4?rp = 0,
ax ay dz
and therefore p must be zero throughout the substance of the
conductor, or there can be no electricity in the interior of the con
ductor.
Hence a superficial distribution of electricity is the only possible
distribution in a conductor in equilibrium.
A distribution throughout the mass of a body can exist only
when the body is a non-conductor.
Since the resultant intensity within the conductor is zero, the
resultant intensity just outside the conductor must be in the direc
tion of the normal and equal to 47T0-, acting outwards from th>%
conductor.
This relation between the surface-density and the resultant in
tensity close to the surface of a conductor is known as Coulomb's
Law, Coulomb having ascertained by experiment that the intensity
of the electric force near a given point of the surface of a conductor
is normal to the surface and proportional to the surface-density at
the given point. The numerical relation
R = 4 77 (7
was established by Poisson.
8 1.] CHARGED WIRE. 91
The force acting on an element, dS, of the charged surface of
a conductor is, by Art. 79, (since the intensity is zero on the inner
side of the surface,)
8 77
This force acts outwards from the conductor, whether the charge
of the surface is positive or negative.
Its value in dynes per square centimetre is
\R<r = 2770-2 = — R2,
Sir
acting as a tension outwards from the surface of the conductor.
81.] If we now suppose an elongated body to be electrified, we
may, by diminishing its lateral dimensions, arrive at the conception
of an electrified line.
Let ds be the length of a small portion of the elongated body,
and let c be its circumference, and a the surface density of the
electricity on its surface; then, if A. is the charge per unit of
length, A = ca, and the resultant electrical intensity close to the
surface will be X
4 TTO- = 47T--
c
If, while A remains finite, c be diminished indefinitely, the in
tensity at the surface will be increased indefinitely. Now in every
dielectric there is a limit beyond which the intensity cannot be
increased without a disruptive discharge. Hence a distribution of
electricity in which a finite quantity is placed on a finite portion
of a line is inconsistent with the conditions existing in nature.
Even if an insulator could be found such that no discharge could
be driven through it by an infinite force, it would be impossible
to charge a linear conductor with a finite quantity of electricity,
for an infinite electromotive force would be required to bring the
electricity to the linear conductor.
In the same way it may be shewn that a point charged with
a finite quantity of electricity cannot exist in nature. It is con
venient, however, in certain cases, to speak of electrified lines and
points, and we may suppose these represented by electrified wires,
and by small bodies of which the dimensions are negligible com
pared with the principal distances concerned.
Since the quantity of electricity on any given portion of a wire
at a given potential diminishes indefinitely when the diameter of
the wire is indefinitely diminished, the distribution of electricity on
bodies of considerable dimensions will not be sensibly affected by
92 ELECTROSTATICS. [82.
the introduction of very fine metallic wires into the field, such as
are used to form electrical connexions between these bodies and the
earth, an electrical machine, or an electrometer.
On Lines of Force.
82.] If a line be drawn whose direction at every point of its
course coincides with that of the resultant intensity at that point,
the line is called a Line of Force.
In every part of the course of a line of force, it is proceeding
from a place of higher potential to a place of lower potential.
Hence a line of force cannot return into itself, but must have a
beginning and an end. The beginning of a line of force must be
in a positively charged surface, and the end of a line of force must
be in a negatively charged surface.
The beginning and the end of the line are called corresponding
points on the positive and negative surface respectively.
If the line of force moves so that its beginning traces a closed
curve on the positive surface, its end will trace a corresponding
closed curve on the negative surface, and the line of force itself
will generate a tubular surface called a tube of induction. Such a
tube is called a Solenoid *.
Since the force at any point of the tubular surface is in the
tangent plane, there is no induction across the surface. Hence
if the tube does not contain any electrified matter, by Art. 77
the total induction through the closed surface formed by the
tubular surface and the two ends is zero, and the values of
U cos e dS for the two ends must be equal in magnitude
but opposite in sign.
If these surfaces arc the surfaces of conductors
e=0 and R=
and / / R cos e dS becomes — 4 IT / / a dS, or the charge of the sur
face multiplied by 4 TT.
Hence the positive charge of the surface enclosed within the
closed curve at the beginning of the tube is numerically equal to
the negative charge enclosed within the corresponding closed curve
at the end of the tube.
* From aw\T]vt a tube. Faraday uses (3271) the term ' Sphondyloid ' in the same
sense.
82.]
LINES OF FOECE,
93
Several important results may be deduced from the properties of
lines of force.
The interior surface of a closed conducting vessel is entirely
free from charge, and the potential at every point within it is
the same as that of the conductor, provided there is no insulated
and charged body within the vessel.
For since a line of force must begin at a positively charged
surface and end at a negatively charged surface, and since no
charged body is within the vessel, a line of force, if it exists
within the vessel, must begin and end on the interior surface of
the vessel itself.
But the potential must be higher at the beginning of a line
of force than at the end of the line, whereas we have proved that
the potential at all points of a conductor is the same.
Hence no line of force can exist in the space within a hollow
vessel, provided no charged body be placed inside it.
If a conductor within a closed hollow vessel is placed in com
munication with the vessel, its potential becomes the same as
that of the vessel, and its surface becomes continuous with the
inner surface of the vessel. The conductor is therefore free from
charge.
If we suppose any charged surface divided into elementary por
tions such that the charge of each element is unity, and if solenoids
having these elements for their bases are drawn through the field of
force, then the surface-integral for any other surface will be re
presented by the number of solenoids which it cuts. It is in this
sense that Faraday uses his conception of lines of force to indicate
not only the direction but the amount of the force at any place in
the field.
We have used the phrase Lines of Force because it has been used
by Faraday and others. In strictness, however, these lines should
be called Lines of Electric Induction.
In the ordinary cases the lines of induction indicate the direction
and magnitude of the resultant electromotive intensity at every
point, because the intensity and the induction are in the same
direction and in a constant ratio. There are other cases, how
ever, in which it is important to remember that these lines indi
cate primarily the induction, and that the intensity is directly
indicated by the equipotential surfaces, being normal to these
surfaces and inversely proportional to the distances of consecutive
surfaces.
94 ELECTROSTATICS. [83 a.
On Specific Inductive Capacity.
83tf .] In the preceding investigation of surface-integrals we have
adopted the ordinary conception of direct action ajt*. a distance, and
have not taken into consideration any effects Depending on the
nature of the dielectric medium in which the forces are observed.
But Faraday has observed that the quantity of electricity in
duced by a given electromotive force on the surface of a conductor
which bounds a dielectric is not the same for all dielectrics. The
induced electricity is greater for most solid and liquid dielectrics
than for air and gases. Hence these bodies are said to have a
greater specific inductive capacity than air, which he adopted as
the standard medium.
We may express the theory of Faraday in mathematical language
by saying that in a dielectric medium the induction across any
surface is the product of the normal electric force into the coefficient
of specific inductive capacity of that medium. If we denote this
coefficient by Kt then in every part of the investigation of sur
face-integrals we must multiply X, Y, and Z by K, so that the
equation of Poisson will become
,.o. (i)
a x dy dy dz dz
At the surface of separation of two media whose inductive capa
cities are K-^ and K2, and in which the potentials are ^ and ^2, the
characteristic equation may be written
KW+fW+t** = 0., (2)
1 dvi dv%
where vlt v2 are the normals drawn in the two media, and <r is
the true surface-density on the surface of separation; that is to
say, the quantity of electricity which is actually on the surface
in the form of a charge, and which can be altered only by con
veying electricity to or from the spot.
Apparent distribution of Electricity.
835.] If we begin with the actual distribution of the potential and
deduce from it the volume density />' and the surface density a-' on
the hypothesis that K is everywhere equal to unity, we may call p'
the apparent volume density and </ the apparent surface density,
because a distribution of electricity thus defined would account for
the actual distribution of potential, on the hypothesis that the law
83 &.] SPECIFIC INDUCTIVE CAPACITY. 95
of electric force as given in Art. 66 requires no modification on
account of the different properties of dielectrics.
The apparent charge of electricity within a given region may
increase or diminish without any passage of electricity through the
bounding surface of the region. We must therefore distinguish it
from the true charge, which satisfies the equation of continuity.
In a heterogeneous dielectric in which K varies continuously, if
p' be the apparent volume-density,
+ -TT+ -TV +4w/= 0. (3)
dy2 dz2
Comparing this with the equation above, we find
dKdV dKdV dKdV
47r(p—Kp) +_—-+__--- + __— - = 0. (4)
r ' dx dx dy dy dz dz v '
«/ «7
The true electrification, indicated by p, in the dielectric whose
variable inductive capacity is denoted by K, will produce the same
potential at every point as the apparent electrification, denoted by
/>', would produce in a dielectric whose inductive capacity is every
where equal to unity.
The apparent surface charge, o-', is that deduced from the electrical
forces in the neighbourhood of the surface, using the ordinary
characteristic equation
dE dK
-r^+rrj-+4irc/=0. (5)
di\ dvz
If a solid dielectric of any form is a perfect insulator, and if
its surface receives no charge, then the true electrification remains
zero, whatever be the electrical forces acting on it.
r , T 2
Hence JT ^-1- + jr _2- = 0.
1 2
The surface-density o-' is that of the apparent electrification
produced at the surface of the solid dielectric by induction. It
disappears entirely when the inducing force is removed, but if
during the action of the inducing force the apparent electrification
of the surface is discharged by passing a flame over the surface,
then, when the inducing force is taken away, there will appear a
true electrification opposite to a' *.
* See Faraday's ' Remarks on Static Induction,' Proceedings of the Royal In
stitution, Feb. 12, 1858.
CHAPTEE III.
ON ELECTKICAL WORK AND ENERGY IN A SYSTEM
OF CONDUCTORS.
84.] On the IVorJc which must be done ~by an external agent in order
to charge an electrified system in a given manner.
The work spent in bringing a quantity of electricity be from an
infinite distance (or from any place where the potential is zero) to a
given part of the system where the potential is F, is, by the defi
nition of potential (Art. 70), 7be.
The effect of this operation is to increase the charge of the given
part of the system by be, so that if it was e before, it will become
e + be after the operation.
We may therefore express the work done in producing a given
alteration in the charges of the system by the integral
; 0)
where the summation, (2), is to be extended to all parts of the
electrified system.
It appears from the expression for the potential in Art. 73,
that the potential at a given point may be considered as the sum
of a number of parts, each of these parts being the potential due
to a corresponding part of the charge of the system.
Hence if 7 is the potential at a given point due to a system
of charges which we may call 2 (e\ and V the potential at the
same point due to another system of charges which we may call
2 (/), the potential at the same point due to both systems of
charges existing together would be 7 + V .
If, therefore, every one of the charges of the system is altered in
the ratio of n to 1, the potential at any given point in the system
will also be altered in the ratio of n to 1 .
85 a.] WORK DONE IN CHARGING A SYSTEM. 97
Let us, therefore, suppose that the operation of charging the
system is conducted in the following manner. Let the system
be originally free from charge and at potential zero, and let the
different portions of the system be charged simultaneously, each
at a rate proportional to its final charge.
Thus if e is the final charge, and V the final potential of any
part of the system, then, if at any stage of the operation the
charge is ne, the potential will be nF, and we may represent
the process of charging by supposing n to increase continuously
from 0 to 1.
While n increases from n to n + bn, any portion of the system
whose final charge is e, and whose final potential is F, receives
an increment of charge e bn, its potential being n7, so that the
work done on it during this operation is eVnbn.
Hence the whole work done in charging the system is
(2)
or half the sum of the products of the charges of the different
portions of the system into their respective potentials.
This is the work which must be done by an external agent in
order to charge the system in the manner described, but since
the system is a conservative system, the work required to bring
the system into the same state by any other process must be the
same.
We may therefore call
W=\-S.(e7) (3)
the electric energy of the system, expressed in terms of the charges
of the different parts of the system and their potentials.
85 «.] Let us next suppose that the system passes from the state
(e, 7) to the state (/, 7') by a process in which the different
charges increase simultaneously at rates proportional for each to
its total increment e' — e.
If at any instant the charge of a given portion of the system
is e+n(i—e)j its potential will be V+n(V'—V}, and the work
done in altering the charge of this portion will be
(S-e)[7+n(7'-7)]dn = \(e'-e) (7+ 7');
so that if we denote by W the energy of the system in the state
(*-, n
w-w=^(s-e)(7'+7). (4)
VOL. I. H
98 SYSTEM OF CONDUCTORS. [856.
But W=\^(eV\
and r'=iS(4T).
Substituting these values in equation (4) we find
S(*F') = S(*T). (5)
Hence if, in the same fixed system of electrified conductors, we
consider two different states of electrification, the sum of the
products of the charges in the first state into the potentials of
the corresponding portions of the conductors in the second state,
is equal to the sum of the products of the charges in the second
state into the potentials of the corresponding conductors in the
first state.
This result corresponds, in the elementary theory of electricity,
to Green's Theorem in the analytical theory. By properly choosing
the initial and final state of the system, we may deduce a number
of useful results.
85 b.~\ From (4) and (5) we find another expression for the in
crement of the energy, in which it is expressed in terms of the
increments of potential,
w-w=\v(<f+e)(r'-r). (6)
If the increments are infinitesimal, we may write (4) and (6)
and if we denote by We and Wv the expressions for W in terms
of the charges and the potentials of the system respectively, and
by Ar, er, and Vr a particular conductor of the system, its charge,
and its potential, then
r =
(9)
86.] If in any fixed system of conductors, any one of them,
which we may denote by At, is without charge, both in the initial
and final state, then for that conductor ei = 0, and e{ = 0, so
that the terms depending on At vanish from both members of
equation (5).
If another conductor, say Att, is at potential zero in both states
of the system, then Tu = 0 and 7U' = 0, so that the terms depending
on Au vanish from both members of equation (5).
If, therefore, all the conductors except two, Ar and As, are either
86] RECIPROCAL RELATIONS. 99
insulated and without charge, or else connected to the earth,
equation (5) is reduced to the form
<TV + etf = er'rr + e.'ff (10)
If in the initial state
er = 1 and es — 0,
and in the final state
<?/= 0 and e'= 1,
equation (10) becomes Yf=Jrs; (11)
or if a unit charge communicated to Ar raises As to a potential V,
then a unit charge communicated to As will raise Ar to the same
potential T7, provided that every one of the other conductors of
the system is either insulated and without charge, or else connected
to earth so that its potential is zero.
This is the first instance we have met with in electricity of a
reciprocal relation. Such reciprocal relations occur in every branch
of science, and often enable us to deduce the solution of new
problems from those of simpler problems already solved.
Thus from the fact that at a point outside a conducting sphere
whose charge is 1 the potential is r~l, where r is the distance
from the centre, we conclude that if a small body whose charge
is 1 is placed at a distance r from the centre of a conducting sphere
without charge, it will raise the potential of the sphere to r~l.
Let us next suppose that in the initial state
Tr = 1 and V& = 0,
and in the final state
rr'= 0 and 77= 1,
equation (10) becomes e8 = er'\ (12)
or if, when Ar is raised to unit potential, a charge e is induced
on As , then if A8 is raised to unit potential, an equal charge e will
be induced on Ar.
Let us suppose in the third place, that in the initial state
Pr = 1 and es = 0,
and that in the final state
7? = 0 and */= 1,
equation (10) becomes in this case
«/+7. = 0. (13)
Hence if when A8 is without charge, the operation of charging
Ar to potential unity raises As to potential F9 then if Ar is kept
H 2,
100 SYSTEM OF CONDUCTOKS. [87.
at potential zero, a unit charge communicated to As will induce
on Ar a negative charge, the numerical value of which is V.
In all these cases we may suppose some of the other conductors
to be insulated and without charge, and the rest to be connected to
earth.
The third case is an elementary form of one of Green's theorems.
As an example of its use let us suppose that we have ascertained
the distribution of electric charge on the different elements of a
conducting system at potential zero, induced by a charge unity
communicated to a given body A8 of the system.
Let rjr be the charge of Ar under these circumstances. Then
if we suppose As without charge, and the other bodies raised each
to a different potential, the potential of A8 will be
^=-2(1,,^. (14)
Thus if we have ascertained the surface density at any given
point of a hollow conducting vessel due to a unit charge placed at
a given point within it, then, if we know the value of the potential
at every point of a surface of the same size and form as the interior
surface of the vessel, we can deduce the potential at a point within
it the position of which corresponds to that of the unit charge.
Hence if the potential is known for all points of a closed surface
it may be determined for any point within the surface, if there be
no electrified body within it, and for any point outside, if there
be no electrified body outside.
Tfaory of a system of conductors.
87.] Let Alt A.2, ... An be n conductors of any form; let elf e2,
... en be their charges; and V^ 72, ...7n their potentials.
Let us suppose that the dielectric medium which separates the
conductors remains the same, and does not become charged with
electricity during the operations to be considered.
We have shown in Art. 84 that the potential of each conductor
is a homogeneous linear function of the n charges.
Hence since the electric energy of the system is half the sum
of the products of the potential of each conductor into its charge,
the electric energy must be a homogeneous quadratic function of
the n charges, of the form
The suffix e indicates that W is to be expressed as a function
87.] COEFFICIENTS OF POTENTIAL AND OF INDUCTION. 101
of the charges. When W is written without a suffix it denotes
the expression (3), in which both charges and potentials occur.
From this expression we can deduce the potential of any one
of the conductors. For since the potential is defined as the work
which must be done to bring a unit of electricity from potential -.y
zero to the given potential, and since this work is spent in -
increasing W, we have only to differentiate We with respect to the
charge of the given conductor to obtain its potential. We thus
obtain
(16)
n= /?1M^... + prner... +pnnen,
a system of n linear equations which express the n potentials in
terms of the n charges.
The coefficients prs &c., are called coefficients of potential. Each
has two suffixes, the first corresponding with that of the charge,
and the second with that of the potential.
The coefficient prr, in which the two suffixes are the same,
denotes the potential of Ar when its charge is unity, that of all
the other conductors being zero. There are n coefficients of this
kind, one for each conductor.
The coefficient jorg, in which the two suffixes are different, denotes
the potential of A8 when Ar receives a charge unity, the charge of
each of the other conductors, except Ar , being zero.
We have already proved in Art. 86 thatj?rs = psr, but we may
prove it more briefly by considering that
_ .-- r,7i
lrs ~ der ~ der des ~ des der ~ dee ~ Ar*
The number of different coefficients with double suffix is there
fore \n(n—\\ being one for each pair of conductors.
By solving the equations (16) for elt e2 &c., we obtain n equations
giving the charges in terms of the potentials
(18)
102 SYSTEM OF CONDUCTORS. [87.
We have in this case also qrs = qsr, for
de d dWv _ ^dWy __des _ , }
^*-W8-Ws~Wr~~- dvr dVK ~dvr-qsr
By substituting the values of the charges in the equation for
the electric energy
r=i[^+ ... +errr...+e»K]t (20)
we obtain an expression for the energy in terms of the potentials
A coefficient in which the two suffixes are the same is called the
Electric Capacity of the conductor to which it belongs.
Definition. The Capacity of a conductor is its charge when its
own potential is unity, and that of all the other conductors is
zero.
.
This is the proper definition of the capacity of a conductor when
no further specification is made. But it is sometimes convenient
to specify the condition of some or all of the other conductors in
a different manner, as for instance to suppose that the charge of
certain of them is zero, and we may then define the capacity of the
conductor under these conditions as its charge when its potential is
unity.
The other coefficients are called coefficients of induction. Any
one of them, as qrs denotes the charge of Ar when As is raised to
potential unity, the potential of all the conductors except As being
zero.
The mathematical calculation of the coefficients of potential and
of capacity is in general difficult. We shall afterwards prove that
they have always determinate values, and in certain special cases
we shall calculate these values. We shall also shew how they may
be determined by experiment.
When the capacity of a conductor is spoken of without specifying
the form and position of any other conductor in the same system,
it is to be interpreted as the capacity of the conductor when no
other conductor or electrified body is within a finite distance of the
conductor referred to.
It is sometimes convenient, when we are dealing with capacities
and coefficients of induction only, to write them in the form [A . P],
this symbol being understood to denote the charge on A when P is
raised to unit potential.
In like manner [(A + B) . (P+ Q)] would denote the charge on
89 a.] PROPERTIES OF THE COEFFICIENTS. 103
A + B when P and Q are both raised to potential 1, and it is
manifest that since
\_(A+B) (P+ Q)] = [AP] + [AQ] + [SP] + [SQ]
the compound symbols may be combined by addition and multipli
cation as if they were symbols of quantity.
The symbol [A . A~\ denotes the charge on A when the potential
of A is 1, that is to say, the capacity of A.
In like manner [(A + B) (A + Q)] denotes the sum of the charges
on A and B when A and Q are raised to potential 1, the potential
of all the conductors except A and Q, being zero.
It may be decomposed into
[A.A] + [A.S] + [A.Q-] + [S.Q].
The coefficients of potential cannot be dealt with in this way.
The coefficients of induction represent charges, and these charges
can be combined by addition, but the coefficients of potential
represent potentials, and if the potential of A is \ and that of
B is ?2, the sum ^4-^ has no physical meaning bearing on the
phenomena, though 7J— ?2 represents the electromotive force from
AtoB.
The coefficients of induction between two conductors may be
expressed in terms of the capacities of the conductors and that of
the two conductors together, thus :
Dimensions of the coefficients.
£
88.] Since the potential of a charge e at a distance r is - ,
the dimensions of a charge of electricity are equal to those of the
product of a potential into a line.
The coefficients of capacity and induction have therefore the
same dimensions as a line, and each of them may be represented
by a straight line, the length of which is independent of the
system of units which we employ.
For the same reason, any coefficient of potential may be repre
sented as the reciprocal of a line.
On certain conditions which the coefficients must
89 a.] In the first place, since the electric energy of a system
is an essentially positive quantity, its expression as a quadratic
104
SYSTEM OF CONDUCTORS.
function of the charges or of the potentials must be positive,
whatever values, positive or negative, are given to the charges
or the potentials.
Now the conditions that a homogeneous quadratic function of n
variables shall be always positive are n in number, and may be
written
Ai > 0,
Pl2
> o,
Pin
- - > 0.
r
(22)
Pnl'-Pnn
These n conditions are necessary and sufficient to ensure that
W shall be essentially positive *.
But since in equation (16) we may arrange the conductors in any
order, every determinant must be positive which is formed sym
metrically from the coefficients belonging to any combination of the
n conductors, and the number of these combinations is 2n— 1.
Only n, however, of the conditions so found can be independent.
The coefficients of capacity and induction are subject to con
ditions of the same form.
89 £.] The coefficients of potential are all positive, lut none of the
coefficients prs is greater than prr or pss.
For let a charge unity be communicated to Ar, the other con
ductors being uncharged. A system of equipotential surfaces will
be formed. Of these one will be the surface of Ar, and its potential
will be prr. If A8 is placed in a hollow excavated in Ar so as to be
completely enclosed by it, then the potential of As will also be prr.
If, however, As is outside of Ar its potential prs will lie between
prr and zero.
For consider the lines of force issuing from the charged con
ductor Ar. The charge is measured by the excess of the number
of lines which issue from it over those which terminate in it.
Hence, if the conductor has no charge, the number of lines which
enter the conductor must be equal to the number which issue from
it. The lines which enter the conductor come from places of greater
potential, and those which issue from it go to places of less poten-
* See Williamson's Differential Calculus, 3rd edition, p. 407.
89 d.] PEOPERTIES OF THE COEFFICIENTS. 105
tial. Hence the potential of an uncharged conductor must be
intermediate between the highest and lowest potentials in the field,
and therefore the highest and lowest potentials cannot belong to
any of the uncharged bodies.
The highest potential must therefore be prr, that of the charged
body Ar, the lowest must be that of space at an infinite distance,
which is zero, and all the other potentials such as prs must lie
between prr and zero.
If As completely surrounds At, i\ienprs = prt.
89 <?.] None of the coefficients of induction are positive, and the sum
of all those belonging to a single conductor is not numerically
greater than the coefficient of capacity of that conductor ', which
is always positive.
For let Ar be maintained at potential unity while all the other
conductors are kept at potential zero, then the charge on Ar is qrr,
and that on any other conductor As is qrs.
The number of lines of force which issue from Ar is qrr. Of these
some terminate in the other conductors, and some may proceed to
infinity, but no lines of force can pass between any of the other
conductors or from them to infinity, because they are all at poten
tial zero.
No line of force can issue from any of the other conductors such
as AS9 because no part of the field has a lower potential than As.
If As is completely cut off from Ar by the closed surface of one
of the conductors, then qrs is zero. If As is not thus cut off, qr8 is a
negative quantity.
If one of the conductors At completely surrounds Ar) then all
the lines of force from Ar fall on At and the conductors within it,
and the sum of the coefficients of induction of these conductors with
respect to Ar will be equal to qrr with its sign changed. But if
Ar is not completely surrounded by a conductor the arithmetical
sum of the coefficients of induction qrs, &c. will be less than qrr.
We have deduced these two theorems independently by means
of electrical considerations. We may leave it to the mathematical
student to determine whether one is a mathematical consequence
of the other,
89 d.~\ When there is only one conductor in the field its coefficient
of potential on itself is the reciprocal of its capacity.
The centre of mass of the electricity when there are no external
forces is called the electric centre of the conductor. If the conductor
106 SYSTEM OF CONDUCTORS. [89 e.
is symmetrical about a centre of figure, this point is the electric
centre. If the dimensions of the conductor are small compared with
the distances considered, the position of the electric centre may be
estimated sufficiently nearly by conjecture.
The potential at a distance c from the electric centre must be
between P , ni , P /-,2 x
where e is the charge, and a is the greatest distance of any part of
the surface of the body from the electric centre.
For if the charge be concentrated in two points at distances a on
opposite sides of the electric centre, the first of these expressions
.is the potential at a point in the line joining the charges, and the
second at a point in a line perpendicular to the line joining the
charges. For all other distributions within the sphere whose radius
is a the potential is intermediate between those values.
If there are two conductors in the field, their mutual coefficient
of potential is - . where c' cannot differ from c, the distance between
c
a2 _j_ £2
the electric centres, by more than - — ; a and b being the greatest
cC c
distances of any part of the surfaces of the bodies from their re
spective electric centres.
89 £.] If a new conductor is brought into the field the coefficient
of potential of any one of the others on itself is diminished.
For let the new body, B, be supposed at first to be a non-conductor
free from charge in any part, then when one of the conductors, A^
receives a charge elt the distribution of the electricity on the con
ductors of the system will not be disturbed by B, as B is still
without charge in any part, and the electric energy of the system
will be simply i^i = 4*i2/>u-
3 /(A
Now let B become a conductor. Electricity will flow from
places of higher to places of lower potential, and in so doing will
diminish the electric energy of the system, so that the quantity
2 ei2Pi\ must diminish.
But el remains constant, therefore p1L must diminish.
Also if B increases by another body b being placed in contact
with it, pll will be further diminished.
For let us first suppose that there is no electric communication
between B and b ; the introduction of the new body b will
diminish j?11. Now let a communication be opened between B
90&.] APPEOXIMATE VALUES OF THE COEFFICIENTS. 107
and I. If any electricity flows through it, it flows from a place
of higher to a place of lower potential, and therefore, as we have
shown, still further diminishes pn.
Hence the diminution of j?n by the body B is greater than
that which would be produced by any body the surface of which
can be inscribed in B, and less than that produced by any body the
surface of which can be described about B.
We shall shew in Chapter XI, that a sphere of diameter b at a
distance r diminishes the value of pn by a quantity which is
#3
approximately ^ —^ •
Hence if the body B is of any other figure, and if & is its
greatest diameter, the diminution of the value of pn must be less
£3
than % — .
Hence if the greatest diameter of B is so small compared with
its distance from A1 that we may neglect quantities of the order
#3
i -4- , we may consider the reciprocal of the capacity of Al when
alone in the field as a sufficient approximation to pllt
90 #.] Let us therefore suppose that the capacity of A1 when alone
in the field is Klt and that of A2, K2, and let the mean distance
between A± and A2 be r, where r is very great compared with the
dimensions of A1 and A2) then we may write
1 1 1
Ai-^. A*--, ^-T2;
^^-i + ^r-1,
Hence
Of these coefficients qn and q22 are the capacities of Ai and A2
when, instead of being each alone at an infinite distance from any
other body, they are brought so as to be at a distance r from each
other.
90 #.] When two conductors are placed so near together that
their coefficient of mutual induction is large, the combination is
called a Condenser.
Let A and B be the two conductors or electrodes of a con
denser.
~, <*.
SYSTEM OF CONDUCTORS. [906.
Let L be the capacity of A, JVthat of 5, and if the coefficient
of mutual induction. (We must remember that M is essentially
negative, so that the numerical value of L + M and M+N is less
than L or N.)
Let us suppose that a and £ are the electrodes of another con
denser at a distance R from the first, R being very great com
pared with the dimensions of either condenser, and let the
coefficients of capacity and induction of the condenser al when
alone be I, m, n. Let us calculate the effect of one of the
condensers on the coefficients of the other.
Let D = LN-M* and d = ln-m2;
then the coefficients of potential for each condenser by itself are
PAR = —D-lM, pab=—d~l m,
PBB = D-*L, pbb = d~ll.
The values of these coefficients will not be sensibly altered when
the two condensers are at a distance R.
The coefficient of potential of any two conductors at distance R
is R'1, so that
PAa = PAb = PSa = J?Bb = R~l •
The equations of potential are therefore
V = D-iNe-I)
Va =
Solving these equations for the charges, we find
r T
-W-
" fi*
where L't M', N' are what L, M, N become when the second con
denser is brought into the field.
91.] APPROXIMATE VALUES OF THE COEFFICIENTS. 109
If only one conductor, a, is brought into the field, m=n=0, and
q_AA = L' = L + ™-
(M+N)l
El(L+M)
If there are only the two simple conductors, A and a,
M=N=m=.n— 0,
L2l ELI
qAA = L + ~E^rr ^=-W^LI>
expressions which are the same as those found in Art. 90#.
The quantity L + 2 M + N is the total charge of the condenser
when its electrodes are at potential 1. It cannot exceed half the
greatest diameter of the condenser.
L + M is the charge of the first electrode, and M + N that of the
second when both are at potential 1. These quantities must be
each of them positive and less than the capacity of the electrode by
itself. Hence the corrections to be applied to the coefficients of
capacity of a condenser are much smaller than those for a simple
conductor of equal capacity.
Approximations of this kind are often useful in estimating the
capacities of conductors of irregular form placed at a finite distance
from other conductors.
91.] When a round conductor, Ast of small size compared with
the distances between the conductors, is brought into the field, the
coefficient of potential of Al on A2 will be increased when A^ is
inside and diminished when A3 is outside of a sphere whose
diameter is the straight line ALA2.
For if A1 receives a unit charge there will be a distribution of
electricity on ABi -\-e being on the side furthest from Alt and — e on
the side nearest Alt The potential at A2 due to this distribution
on A3 will be positive or negative as +e or -~e is nearest to A2,
and if the form of A3 is not very elongated this will depend on
whether the angle Al A3 A2 is obtuse or acute, and therefore on
whether Az is inside or outside the sphere described on A1 A2 as
diameter.
$/* A. <
If A3 is of an elongated form it is easy to see that if it is placed
with its longest axis in the direction of the tangent to the circle
110 SYSTEM OF CONDUCTORS. [92.
g^ drawn through the points Alt Aa, A2 it may increase the potential
of AZ3 even when it is entirely outside the sphere, and how by
placing it with its longest axis in the direction of the radius of
.the sphere, it may diminish the potential of A2, even when entirely
within the sphere. But this proposition is only intended for
forming a rough estimate of the phenomena to be expected in
a given arrangement of apparatus.
92.] If a new conductor, A^ is introduced into the field, the
capacities of all the conductors already there are increased, and the
numerical values of the coefficients of induction between every pair
.
of them are diminished.
Let us suppose that A1 is at potential unity and all the rest at
potential zero. Since the charge of the new conductor is negative
it will induce a positive charge on every other conductor, and
will therefore increase the positive charge of Al and diminish the
negative charge of each of the other conductors.
93 «.] Work done ~by the electric forces during the displacement of
a system of insulated charged conductors.
Since the conductors are insulated, their charges remain_constant
during the displacement. Let their potentials be 7^ V^ . . . ~Pn before
and JJ', 72', ...7n' after the displacement. The electrical energy is
before the displacement, and
after the displacement.
The work done by the electric forces during the displacement is
the excess of the initial energy W over the final energy W, or
~nr ~ttff — l 5 I P ( r — 7 1 1
- 2 L V AT
This expression gives the work done during any displacement,
small or large, of an insulated system.
To find the force tending to produce a particular kind of dis
placement, let 0 be the variable whose variation corresponds to the
kind of displacement, and let 4> be the corresponding force, reckoned
positive when the electric force tends to increase <£, then
dW
or 4> = TT
where We denotes the expression for the electric energy as a
quadratic function of the charges.
93 C.] MECHANICAL FOECES. Ill
93 *.] To prove that - + = °-
d<p d(f>
We have three different expressions for the energy of the system,
(i) r=is(«n,
a definite function of the n charges and n potentials
(2) ^=j2S(«r«.A.),
where r and s may be the same or different,, and both rs and sr are
to be included in the summation.
This is a function of the n charges and of the variables which
define the configuration. Let $ be one of these.
(3) rr=iss(^.?r<),
where the summation is to be taken as before. This is a function
of the n potentials and of the variables which define the configura
tion of which <j> is one.
Since W=We=Wv,
-2W= 0.
Now let the n charges, the n potentials, and $ vary in any con
sistent manner, and we must have
Now the n charges, the n potentials, and $ are not all independent
of each other, for in fact only n + 1 of them can be independent.
But we have already proved that
so that the first sum of terms vanishes identically, and it follows
from this, even if we had not already proved it that
dWv
~~dJ^ '' 6s>
and that lastly,
^ , dWY = 0_
Work done by the electric forces during the displacement of a
whose potentials are maintained constant.
AW
93 <?.] It follows from the last equation t^at the force 4> = —^
112 SYSTEM OF CONDUCTORS. [94.
and if the system is displaced under the condition that all the
potentials remain constant, the work done by the electric forces is
r r '
or the work done by the electric forces in this case is equal to the
increment of the electric energy.
Here, then, we have an increase of energy together with a quan
tity of work done by the system. The system must therefore be
supplied with energy from some external source, such as a voltaic
battery, in order to maintain the potentials constant during the
displacement.
The work done by the battery is therefore equal to the sum of
the work done by the system and the increment of energy, or,
since these are equal, the work done by the battery is twice the
work done by the system of conductors during the displacement.
On the comparison of similar electrified systems.
94.] If two electrified systems are similar in a geometrical sense,
so that the lengths of corresponding lines in the two systems are
as L to L', then if the dielectric which separates the conducting
bodies is the same in both systems, the coefficients of induction
and of capacity will be in the proportion of L to If. For if we
consider corresponding portions, A and A\ of the two systems, and
suppose the quantity of electricity on A to be e, and that on A'
to be /, then the potentials 7 and 7' at corresponding points
B and B', due to this electrification, will be
But AB is to A'B' as L to L't so that we must have
e-.Snir: L'V.
But if the inductive capacity of the dielectric is different in the
two systems, being K in the first and K' in the second, then if the
potential at any point of the first system is to that at the cor
responding point of the second as V to V , and if the quantities
of electricity on corresponding parts are as E to E', we shall have
By this proportion we may find the relation between the total
charges of corresponding parts of two systems, which are
in the first place geometrically similar, in the second place com
posed of dielectric media of which the specific inductive capacity
94-] SIMILAR SYSTEMS. 113
at corresponding points is in the proportion of K to K', and in
the third place so electrified that the potentials of corresponding
points are as V to V .
From this it appears that if q be any coefficient of capacity or
induction in the first system, and %' the corresponding one in the
second, q'.cf'.-.LK-.L'K';
and if p and pf denote corresponding coefficients of potential in
the two systems, \ j
P:J3'::^K:VTC'
If one of the bodies be displaced in the first system, and the
corresponding body in the second system receive a similar dis
placement, then these displacements are in the proportion of L
to L\ and if the forces acting on the two bodies are as F to /"',
then the work done in the two systems will be as FL to I"J/.
But the total electrical energy is half the sum of the charges
of electricity multiplied each by the potential of the charged
body, so that in the similar systems, if W and W be the total
electrical energy in the two systems respectively,
W: W \ : eV \ e'V,
and the difference of energy after similar displacements in the two
systems will be in the same proportion. Hence, since FL is pro
portional to the electrical work done during the displacement,
FLiF'L' nerie'Y'.
Combining these proportions, we find that the ratio of the
resultant force on any body of the first system to that on the
corresponding body of the second system is
f>2 f/2
or F - F' • •
'
The first of these proportions shews that in similar systems the
force is proportional to the square of the electromotive force and
to the inductive capacity of the dielectric, but is independent of the
actual dimensions of the system.
Hence two conductors placed in a liquid whose inductive capacity
is greater than that of air, and electrified to given potentials, will
attract each other more than if they had been electrified to the
same potentials in air.
The second proportion shews that if the quantity of electricity
on each body is given, the forces are proportional to the squares
VOL. I. !
114 SYSTEM OF CONDUCTORS. [94.
of the charges and inversely to the squares of the distances, and
also inversely to the inductive capacities of the media.
Hence, if two conductors with given charges are placed in a
liquid whose inductive capacity is greater than that of air, they
will attract each other less than if they had been surrounded with
air and charged with the same quantities of electricity.
CHAPTEE IV.
GENERAL THEOREMS.
95 a.~\ IN the second chapter we have calculated the potential
function and investigated some of its properties on the hypothesis
that there is a direct action at a distance between electrified bodies,
which is the resultant of the direct actions between the various
electrified parts of the bodies.
If we call this the direct method of investigation, the inverse
method will consist in assuming that the potential is a function
characterised by properties the same as those which we have already
established, and investigating the form of the function.
In the direct method the potential is calculated from the dis
tribution of electricity by a process of integration, and is found
to satisfy certain partial differential equations. In the inverse
method the partial differential equations are supposed given, and
we have to find the potential and the distribution of electricity.
It is only in problems in which the distribution of electricity
is given that the direct method can be used. When we have to
find the distribution on a conductor we must make use of the
inverse method.
We have now to shew that the inverse method leads iu every
case to a determinate result, and to establish certain general
theorems deduced from Poisson's partial differential equation
The mathematical ideas expressed by this equation are of a
different kind from those expressed by the definite integral
r+x r+oo r + oo n
r= I tu
J — ao J — ao J —<x> '
In the differential equation we express that the sum of the second
derivatives of Y in the neighbourhood of any point is related to
116 GENERAL THEOREMS. [95 &•
the density at that point in a certain manner, and no relation
is expressed between the value of V at that point and the value
of p at any point at a finite distance from it.
In the definite integral, on the other hand, the distance of
the point (of, /, z'\ at which p exists, from the point (x, y, z\ at
which V exists, is denoted by r, and is distinctly recognised in the
expression to be integrated.
The integral, therefore, is the appropriate mathematical expression
for a theory of action between particles at a distance, whereas the
differential equation is the appropriate expression for a theory of
action exerted between contiguous parts of a medium.
We have seen that the result of the integration satisfies the
differential equation. We have now to shew that it is the only
solution of that equation satisfying certain conditions.
We shall in this way not only establish the mathematical equi
valence of the two expressions, but prepare our minds to pass from
the theory of direct action at a distance to that of action between
contiguous parts of a medium.
955.] The theorems considered in this chapter relate to the
properties of certain volume-integrals taken throughout a finite
region of space which we may refer to as the electric field.
The element of these integrals, that is to say, the quantity
under the integral sign, is either the square of a certain vector
quantity whose direction and magnitude varies from point to point
in the field, or the product of one vector into the resolved part of
another in its own direction.
Of the different modes in which a vector quantity may be dis
tributed in space, two are of special importance.
The first is that in which the vector may be represented
as the space-variation [Art. 17] of a scalar function called the
Potential.
Such a distribution may be called an Irrotational distribution.
The resultant force arising from the attraction or repulsion of any
combination of centres of force, the law of each being any given
function of the distance, is distributed irrotationally.
The second mode of distribution is that in which the convergence
[Art. 25] is zero at every point. Such a distribution may be
called a Solenoidal distribution. The velocity of an incompressible
fluid is distributed in a solenoidal manner.
When the central forces which, as we have said, give rise to an
irrotatioaal distribution of the resultant force, vary according to
95 &•] IRROTATIONAL AND SOLENOIDAL DISTRIBUTIONS. 117
the inverse square of the distance, then, if these centres are outside
the field, the distribution within the field will be solenoidal as well
as irrotational.
When the motion of an incompressible fluid which, as we have
said, is solenoidal, arises from the action of central forces depending1
on the distance, or of surface pressures, on a frictionless fluid
originally at rest, the distribution of velocity is irrotational as well
as solenoidal.
When we have to specify a distribution which is at once irrota
tional and solenoidal, we shall call it a Laplacian distribution;
Laplace having- pointed out some of the most important properties
of such a distribution.
The volume integrals discussed in this chapter are, as we shall
see, expressions for the energy of the electric field. In the first
group of theorems, beginning with Green's Theorem, the energy is
expressed in terms of the electromotive intensity, a vector which is
distributed irrotationally in all cases of electric equilibrium. It is
shewn that if the surface-potential be given, then of all irrotational
distributions, that which is also solenoidal has the least energy;
whence it also follows that there can be only one Laplacian distri
bution consistent with the surface potentials.
In the second group of theorems, including Thomson's Theorem,!' i
the energy is expressed in terms of the electric displacement,^ •
vector of which the distribution is solenoidal. It is shewn that
if the surface-charges are given, then of all solenoidal distributions
that has least energy which is also irrotational, whence it also
follows that there can be only one Laplacian distribution consistent
with the given surface-charges.
The demonstration of all these theorems is conducted in the same
way. In order to avoid the repetition in every case of the steps
of a surface integration conducted with reference to rectangular
axes, we make use in each case of the result of Theorem III, Art.
21,* where the relation between a volume-integral and the corre
sponding surface-integral is fully worked out. All that we have to
do, therefore, is to substitute for X, 7, and Z in that Theorem the
components of the vector on which the particular theorem depends.
In the first edition of this book the statement of each theorem
was cumbered with a multitude of alternative conditions which
* This theorem seems to have been first given by Ostrogradsky in a paper read in
1828, but published in 1831 in the Mem. de VAcad. de St. Petersbourg, T. I. p. 39. It
may be regarded, however, as a form of the equation of continuity.
118 GENERAL THEOREMS. [96 a.
were intended to shew the generality of the theorem and the variety
of cases to which it might be applied, but which tended rather to
confuse in the mind of the reader what was assumed with what was
to be proved.
In the present edition each theorem is at first stated in a more
definite, if more restricted, form, and it is afterwards shewn what
further degree of generality the theorem admits of.
We have hitherto used the symbol V for the potential, and we
shall continue to do so whenever we are dealing with electrostatics
only. In this chapter, however, and in those parts of the second
volume in which the electric potential occurs in electro-magnetic
investigations, we shall use ^ as a special symbol for the electric
potential.
f
Green's Theorem.
96 a.~\ The following important theorem was given by George
Green, in his ' Essay on the Application of Mathematics to Elec
tricity and Magnetism.'
The theorem relates to the space bounded by the closed surface
s. We may refer to this finite space as the Field. Let v be a
normal drawn from the surface 8 into the field, and let I, m, n be
the direction cosines of this normal, then
7d^ d$ dV d^
£-=- -f m— +n-j- = -j- (1)
dx dy dz dv
will be the rate of variation of the function ^ in passing along
dy
the normal v. Let it be understood that the value of —r- is to be
dv
taken at the surface itself, where v = 0.
Let us also write, as in Arts. 26 and 77,
( '
dx* df dz* ~
and when there are two functions, y and <£, let us write
********** ** = _&vvf,v<l). (3)
dx dx dy dy dz dz
The reader who is not acquainted with the method of Quater
nions may, if it pleases him, regard the expressions V2x£ and
tf.V^V^ as mere conventional abbreviations for the quantities to
which they are equated above, and as in what follows we shall
employ ordinary Cartesian methods, it will not be necessary to
remember the Quaternion interpretation of these expressions. The
96 a.] GREEN'S THEOREM. 119
reason, however, why we use as our abbreviations these expressions
and not single letters arbitrarily chosen, is, that in the language
of Quaternions they represent fully the quantities to which they
are equated. The operator V applied to the scalar function y
gives the space-variation of that function, and the expression
— xS.V^V^ is the scalar part of the product of two space- variations,
or the product of either space-variation into the resolved part of the
dy
other in its own direction. The expression -=- is usually written
in Quaternions S.UvVy, Uv being a unit- vector in the direction
of the normal. There does not seem much advantage in using
this notation here, but we shall find the advantage of doing so
when we come to deal with anisotropic media.
Statement of Green's Theorem.
Let y and 3> be two functions of a?, y, z, which, with their first
derivatives, are finite and continuous within the acyclic region s,
bounded by the closed surface 5, then
ds—
(4)
where the double integrals are to be extended over the whole
closed surface <?, and the triple integrals throughout the field, s,
enclosed by that surface.
To prove this, let us write, in Art. 21, Theorem III,
,
dx dy
_ , 7d<b d$>
then TZcos e = *- +
(6)
, dX dY dZ , sd2® d2® d2®
and h --r- + ^r- =
*
dx dy dz ^ dx*
, by (2) and (3). (7)
dx dx ~dy dy dz dz
But by Theorem III
dY
120 GENERAL THEOREMS. [966.
or by (6) and (?)
(8)
= jjj S.
Since in the second member of this equation ^ and <£ may be
interchanged, we may do so in the first, and we thus obtain the
complete statement of Green's Theorem, as given in equation (4).
96 £.] We have next to shew that Green's Theorem is true when
one of the functions, say ^, is a many-valued one, provided that
its first derivatives are single -valued, and do not become infinite
within the acyclic region s.
Since V^ and V^ are single-valued, the second member of equa
tion (4) is single-valued ; but since ^ is many-valued, any one
element of the first member, as ^ V2 $>, is many -valued. If,
however, we select one of the many values of ^j as tyQ , at the point
A within the region s, then the value of # at any other point, P,
will be definite. For, since the selected value of ^ is continuous
within the region, the value of ^ at P must be that which is '
arrived at by continuous variation along any path from A to P,
beginning with the value ^0 at A. If the value at P were different
for two paths between A and P, then these two paths must embrace
between them a closed curve at which the first derivatives of ^
become infinite. Now this is contrary to the specification, for
since the first derivatives do not become infinite within the region
s, the closed curve must be entirely without the region ; and since
the region is acyclic, two paths within the region cannot embrace ^
anything outside the region.
Hence, if ^0 is given as the value of ^ at the point A, the value
at P is definite.
If any other value of *, say ^0 -f HK, had been chosen as the
value at A, then the value at P would have been ^ + UK.. But the
value of the first member of equation (4) would be the same as before,
for the change amounts to increasing the first member by
[//£"-///-«}
and this, by Theorem III, is zero.
96 c.~\ If the region s is doubly or multiply connected, we may
reduce it to an acyclic region by closing each of its circuits
with a diaphragm.
Let <?! be one of these diaphragms, and ^ the corresponding
cyclic constant, that is to say, the increment of ^ in going once
g6d.~\ GREEN'S THEOREM. 121
round the circuit in the positive direction. Since the region s lies
on both sides of the diaphragm slf every element of s1 will occur
twice in the surface integral.
If we suppose the normal vl drawn towards the positive side of
ds1) and i\ drawn towards the negative side,
at*! avL
and ^ = v^ + K,
so that the element of the surface-integral arising from ds-^ will be
= — K
l -j- i i T7-
1 dvl l dv\
Hence if the region 9 is multiply connected, the first term of equa
tion (4) must be written
//* lll *-•>// *-*-*// *.-///**-* ; w
where the first surface-integral is to be taken over the bounding
surface, and the others over the different diaphragms, each element
of surface of a diaphragm being taken once only, and the normal
being drawn in the positive direction of the circuit.
This modification of the theorem in the case of multiply-
connected regions was first shewn to be necessary by Helmholtz "*,
and was first applied to the theorem by Thomson f.
96 d~\ Let us now suppose, with Green, that one of the functions,
say 4>, does not satisfy the condition that it and its first derivatives
do not become infinite within the given region, but that it becomes
infinite at the point P, and at that point only, in that region, and
that very near to P the value of <J> is <J>0-f e/r%t where 4>0 is a finite
and continuous quantity, and r is the distance from P. This will be
the case if 3> is the potential of a quantity of electricity e concen
trated at the point P, together with any distribution of electricity
the volume density of which is nowhere infinite within the region
considered.
Let us now suppose a very small sphere whose radius is a to
be described about P as centre ; then since in the region outside
this sphere, but within the surface s, 4> presents no singularity, we
* ' Ueber Integrate der hydrodynamischen Gleichungen welche den Wirbelbewe-
gungen entsprechen,' Crelle, 1858. Translated by Prof. Tait, Phil. Mag., 1867 (I).
t ' On Vortex Motion,' Trans. R. S. Edin. xxv. part i. p. 241 (1867).
% The mark / separates the numerator from the denominator of a fraction.
122 GENERAL THEOREMS. [96 d.
may apply Green's Theorem to this region, remembering that the
surface of the small sphere is to be taken account of in forming
the surface-integral.
In forming the volume-integrals we have to subtract from the
volume-integral arising from the whole region that arising from
the small sphere.
Now / / / 4>V2 ^idxdydz for the sphere cannot be numerically
greater than
or g
where the suffix, gt attached to any quantity, indicates that the
greatest numerical value of that quantity within the sphere is to be
taken.
This volume-integral, therefore, is of the order a2, and may be
neglected when a diminishes and ultimately vanishes.
The other volume-integral
cannot be numerically greater than
and is of the order a3, and may be neglected when a vanishes.
The surface-integral / / $> -=- ds cannot be numerically greater
Now by Theorem III
//
* --*• ****
dv
and this cannot be numerically greater than (V2 ^i^tf3, and <£«,
at the surface is approximately -, so that / / <I> -j- ds cannot be nu
merically greater than
and is therefore of the order a2, and may be neglected when a
vanishes.
But the surface-integral on the other side of the equation, namely
*£"*-
97 &•] GREEK'S THEOREM. 123
does not vanish, for / / — - ds = — 4 ire ;
and if ^0 be the value of ^ at the point P,
/Y d&
I I ^ -y- ds — — 4 ne %.
J J dv
Equation (4) therefore becomes in this case
97 «.] We may illustrate this case of Green's Theorem by em
ploying it as Green does to determine the surface-density of a
distribution which will produce a potential whose values inside and
outside a given closed surface are given. These values must
coincide at the surface, also within the surface V2 4* = 0, and outside
V2#' = 0.
Green begins with the direct process, that is to say, the distribu
tion of the surface density, cr, being given, the potentials at an
internal point P and an external point P/ are found by integrating
the expressions
(9)
//
where r and / are measured from the points P and P/ respectively.
Now let 4> = 1/r, then applying Green's Theorem to the space
within the surface, and remembering that V24> = 0 and V2 * = 0,
we find _ 1
where tyP is the value of ^ at P.
Again, if we apply the theorem to the space between the surface s
and a surface surrounding it at an infinite distance a, the part of the
surface-integral belonging to the latter surface will be of the order
I/a and may be neglected, and we have
1/
*-
Now at the surface, * = ^, and since the normals v and v are
drawn in opposite directions,
.1 ,1
d- d-
r r
dv dv
124: GENERAL THEOREMS. [97 &•
Hence on adding equations (10) and (ll), the left-hand members
destroy each other, and we have
-«•*
-//><£+£)"
97 £.] Green also proves that if the value of the potential at
every point of a closed surface s be given arbitrarily, the potential
at any point inside or outside the surface may be determined.
For this purpose he supposes the function <J> to be such that
near the point P its value is sensibly 1/r, while at the surface s its
value is zero, and at every point within the surface V2 4> = 0.
That such a function must exist, Green proves from the physical
consideration that if s is a conducting surface connected to the
earth, and if a unit of electricity is placed at the point P, the
potential within s must satisfy the above conditions. For since
s is connected to the earth the potential must be zero at every
point of 5, and since the potential arises from the electricity at P
and the electricity induced on «?, V24> — 0 at every point within
the surface.
Applying Green's Theorem to this case, we find
rr d$> _
4Typ=zJJy—d8, (13)
where, in the surface-integral, ^ is the given value of the potential
at the element of surface ds ; and since, if o> is the density of the
electricity induced on s by unit of electricity at P,
we may write equation (13)
4^ + ^=0, (14)
dv
(15)
where a is the surface-density of the electricity induced on ds by
a charge equal to unity at the point P.
Hence if the value of <r is known at every point of the surface
for a particular position of P, then we can calculate by ordinary
integration the potential at the point P, supposing the potential
at every point of the surface to be given, and the potential
within the surface to be subject to the condition
V2^ = 0.
We shall afterwards prove that if we have obtained a value of
^ which satisfies these conditions, it is the only value of *
which satisfies them.
98.] GREEN'S FUNCTION. 125
y"
Green's Function.
S
98.] Let a closed surface s be maintained at potential zero. Let
P and Q be two points on the positive side of the surface s (we may
suppose either the inside or the outside positive), and let a small
body charged with unit of electricity be placed at P ; the potential
at. the point Q will consist of two parts, of which one is due to the
direct action of the electricity at P, while the other is due to the
action of the electricity induced on s by P. The latter part of the
potential is called Green's Function, and is denoted by Gpq.
This quantity is a function of the positions of the two points P
and Q) the form of the function depending on the surface s. It
has been calculated for the case in which $ is a sphere, and for a
very few other cases. It denotes the potential at Q due to the
electricity induced on s by unit of electricity at P.
The actual potential at any point Q due to the electricity at P
and to the electricity induced on s is l/rpq -f Gpq, where rpq denotes
the distance between P and Q.
At the surface s, and at all points on the negative side of s, the
potential is zero, therefore
pa
where the suffix a indicates that a point A on the surface s is taken
instead of Q.
Let vpa' denote the surface-density induced by P at a point A'
of the surface s, then, since Gpq is the potential at Q due to the
superficial distribution,
<?„ =
where ds' is an element of the surface s at A ', and the integration
is to be extended over the whole surface s.
But if unit of electricity had been placed at Q, we should have
had by equation (l),
where a-qa is the density at A of the electricity induced by Q, els is
an element of surface, and ratf is the distance between A and A'.
126 GENERAL THEOREMS. [99 a.
Substituting this value of l/rqci in the expression for Gpq, we find
Since this expression is not altered by changing p into q and
into p, we find that
^a = Gw 5 (6)
a result which we have already shewn to be necessary in Art. 87,
but which we now see to be deducible from the mathematical process
by which Green's function may be calculated.
If we assume any distribution of electricity whatever, and place
in the field a point charged with unit of electricity, and if the
surface of potential zero completely separates the point from the
assumed distribution, then if we take this surface for the surface s,
and the point for P, Green's function, for any point on the same
side of the surface as P, will be the potential of the assumed dis
tribution on the other side of the surface. In this way we may
construct any number of cases in which Green's function can be
found for a particular position of P. To find the form of the
function when the form of the surface is given and the position
of P is arbitrary, is a problem of far greater difficulty, though,
as we have proved, it is mathematically possible.
Let us suppose the problem solved, and that the point P is
taken within the surface. Then for all external points the potential
of the superficial distribution is equal and opposite to that of P.
The superficial distribution is therefore centrobaric*, and its action
on all external points is the same as that of a unit of negative
electricity placed at P.
99 a.] If in Green's Theorem we make ^=4>, we find
If # is the potential of a distribution of electricity in space with a
volume-density p and on conductors whose surfaces are sv s.2, &c.,
and whose potentials are *1}*2' &c"> w^n surface- densities c^, cr2, &c.,
then V2* =477,0, (17)
where e} is the charge of the surface sv
* Thomson and Tail's Natural Philosophy, § 526.
&-] UNIQUE MINIMUM OF Wy. 127
Dividing (16) by — Sir, we find
(*1 6l + *2*2 + &C.) +
The first term is the electric energy of the system arising from the
surface-distributions, and the second is that arising from the distri
bution of electricity through the field, if such a distribution exists.
Hence the second member of the equation expresses the whole
electric energy of the system, the potential * being a given function
of #, yt z.
As we shall often have occasion to employ this volume-integral,
we shall denote it by the abbreviation W^ so that
'••-£///[<£> + (£ >'*£)>**• <«»
If the only charges are those on the surfaces of the conductors,
p = 0, and the second term of the first member of equation (20)
disappears.
The first term is the expression for the energy of the charged
system expressed, as in Art. 84, in terms of the charges and the
potentials of the conductors, and this expression for the energy we
denote by W.
99 #.] Let ^ be a function of x, y, zt subject to the condition that
its value at the closed surface s is ty, a known quantity for every
point of the surface. The value of *P at points not on the surface
s is perfectly arbitrary.
Let us also write
the integration being extended throughout the space within the
surface; then we shall prove that if ^ is a particular form of ty
which satisfies the surface condition and also satisfies Laplace's
Equation ^2 ^ __ 0 (23)
at every point within the surface, then W^ the value of W corre
sponding to ^j, is less than that corresponding to any function which
differs from ^ at any point within the surface.
For let ^ be any function coinciding with ^ at the surface but
not at every point within it, and let us write
* = 4'1 + *2; (24)
then ^2 is a function which is zero at every point of the surface.
128 GENERAL THEOREMS. [99 6.
The value of W for ^ will be evidently
By Green's Theorem the last term may be written
The volume-integral vanishes because V2 4^ = 0 within the
surface, and the surface-integral vanishes because at the surface
4>2 = 0. Hence equation (25) is reduced to the form
Now the elements of the integral W^ being sums of three squares,
are incapable of negative values, so that the integral itself can only
be positive or zero. Hence if W2 is not zero it must be positive,
and therefore W greater than Wr But if W2 is zero, every one of
its elements must be zero, and therefore
~^f = °' ^ = °' W2 = °
at every point within the surface, and *2 must be a constant within
the surface. But at the surface ^2 = 0, therefore ^2 = 0 at every
point within the surface, and ^ = 3*v so that if W is not greater
than W^ V must be identical with ^ at every point within the
surface.
It follows from this that ^ is the only function of #, y, z which
becomes equal to 5 at the surface, and which satisfies Laplace's
Equation at every point within the surface.
For if these conditions are satisfied by any other function #3,
then W9 must be less than any other value of W. But we have
already proved that W^ is less than any other value, and therefore
than Wy Hence no function different from ^ can satisfy the
conditions.
The case which we shall find most useful is that in which the
'' field is bounded by one exterior surface, s, and any number of
f interior surfaces, SL, s2, &c., and when the conditions are that the
v*/j6 ' value of * shall be zero at s, ^ at sv *2 at *2, and so on, where
*,. ^o, &c. are constant for each surface, as in a system of conductors,
V 2'
the potentials of which are given.
Of all values of ^ satisfying these conditions, that gives the
minimum value of W^ for which V2# = 0 at every point in the
field.
ti
^0^-
1 00 6.] LEMMA. 129
Thomson's Theorem.
Lemma.
100 a.] Let ^ be any function of x, y, z which is finite and
continuous within the closed surface s, and which at certain closed
surfaces, slt szt sp9 &cv has the values ^, ^2, Vpt &c. constant for
each surface.
Let M, v, w be functions of #, y, z, which we may consider as the
components of a vector (£ subject to the solenoidal condition
and let us put in Theorem III
X=*«, Y=Vv, Z=Vw\ (29)
we find as the result of these substitutions
/Y/V d* d* dy^
+JJJ (U-^ +^ + ^^-)^y^ = o5 (so)
the surface-integrals being extended over the different surfaces and
the volume-integrals being taken throughout the whole field.
Now the first volume-integral vanishes in virtue of the solenoidal
condition for u, v, w, and the surface-integrals vanish in the follow
ing cases : —
(1) When at every point of the surface * =.0.
(2) When at every point of the surface lu + mv + nw = 0.
(3) When the surface is entirely made up of parts which satisfy
either (l) or (2).
(4) When ^ is constant over the whole closed surface, and
nw) ds = 0.
Hence in these four cases the volume-interal
1003.] Now consider a field bounded by the external closed
surface s, and the internal closed surfaces slt s2, &c.
Let * be a function of a?, y, z, which within the field is finite
and continuous and satisfies Laplace's Equation
V2xP=0, (32)
and has the constant, but not given, values V19 #2, &c. at the
surfaces slt s2, &c. respectively, and is zero at the external
surface s.
VOL. i. K
130 GENERAL THEOREMS. [lOOC.
The charge of any of the conducting surfaces, as sl} is given
by the surface-integral
(33)
the normal v1 being drawn from the surface s: into the electric
field.
100 c.] Now let /, y, h be functions of x, y, z, which we may
consider as the components of a vector £>, subject only to the
conditions that at every point of the field they must satisfy the
solenoidal equation
df da dJi /0.x
— -I- — £• -I- — = 0 (34)
dx dy dz l '
and that at any one of the internal closed surfaces, as slt the surface-
integral
(35)
where I, m, n are the direction cosines of the normal ^ drawn
outwards from the surface s1 into the electric field, and el is the
same quantity as in equation (33), being, in fact, the electric charge
of the conductor whose surface is s1 .
We have to consider the value of the volume-integral
Wv = **///(/* +?+**) dxdydz, (36)
extended throughout the whole of the field within s and without
«?13 s2) &c., and to compare it with
the limits of integration being the same.
Let us write
d* 1 d*
--- j- ~* -J-
4.77 dx 4-n- dy 477 dz
and W* = 2 T///V + v2 + w*}dxdydz; (39)
then since
ioo c.] THOMSON'S THEOEEM. 131
Now in the first place, n, v, w satisfy the solenoidal condition at
every point of the field, for by equations (38)
du dv dw df dg dli 1
and by the conditions expressed in equations (34) and (32), both
parts of the second member of (41) are zero.
In the second place, the surface-integral
Jj
*k, (42)
but by (35) the first term of the second member is e, and by (33)
the second term is — 0, so that
JJ
w)dsl = 0. (43)
Hence, since ^ is constant, the fourth condition of Art. 1 00 a is
satisfied, and the last term of equation (40) is zero, so that the
equation is reduced to the form
S. (44)
Now since the element of the integral W® is the sum of three
squares, uz+v2+wz, it must be either positive or zero. If at any
point within the field u, v, and w are not each of them equal to zero,
the integral #6 must have a positive value, and #J must therefore
be greater than W*. But the values u = v = w = 0 at every point
satisfy the conditions.
Hence, if at every point
f_ I d* id* 1 d*
~^<¥' -4^' *=-n5-J ^
tlien ^ = 0J, (46)
and the value of W^ corresponding to these values of f, g, kt is less
than the value corresponding to any values of /, g, h, differing
from these.
Hence the problem of determining the displacement and po
tential, at every point of the field, when the charge on each
conductor is given, has one and only one solution.
This theorem in one of its more general forms was first stated
by Sir W. Thomson*. We shall afterwards show of what gene
ralization it is capable.
* Cambridge md Dublin Mathematical Journal, February, 1848.
132 GENERAL THEOREMS. [lOO d.
This theorem may be modified by supposing that the
vector 2), instead of satisfying the solenoidal condition at every
point of the field, satisfies the condition
df da dk ,._»
^+!+<fo="' <47)
where p is a finite quantity, whose value is given at every point in
the field, and may be positive or negative, continuous or discon
tinuous, its volume-integral within a finite region being, however,
finite.
We may also suppose that at certain surfaces in the field
lf+ mg + nh + I'f + m'g' + ri V = cr, (48)
when I, m, n and l\ m', n' are the direction cosines of the normals
drawn from a point of the surface towards those regions in which
the components of the displacement are f, g, li and /", /, V re
spectively, and a- is a quantity given at all points of the surface,
the surface-integral of which, over a finite surface, is finite.
100 <?.] We may also alter the condition at the bounding surfaces
by supposing that at every point of these surfaces
lf+mff+nfi = <r, (49)
where o- is given for every point.
(In the original statement we supposed only the value of the
integral of a- over each of the surfaces to be given. Here we
suppose its value given for every element of surface, which comes
to the same thing as if, in the original statement, we had considered
every element as a separate surface.)
None of these modifications will affect the truth of the theorem
provided we remember that ^ must satisfy the corresponding
conditions, namely, the general condition,
d2* d*V d*y f .
TT + TT + TT-+47rP = °> V50)
dx* dy* dz*
and the surface condition
£+£+«—«• (51)
For if, as before,
du dv dw _
+ += J
then u, v, w will satisfy the general solenoidal condition
du
H^
and the surface condition
1 01 J.] INTENSITY AND DISPLACEMENT. 133
and at the bounding- surface
lu+mv+nw = 0,
whence we find as before that
and that W^—
Hence as before it is shewn that W^ is a unique minimum when
W§ — 0, which implies that (£ is everywhere zero, and therefore
1 d^_ _!_<?# 1 dV
~~4^n~dx' ~~4^~dy' ~ ~4^ ~dz '
101 a.~\ In our statement of these theorems we have hitherto
confined ourselves to that theory of electricity which assumes that
the properties of an electric system depend on the form and relative
position of the conductors, and on their charges, but takes no
account of the nature of the dielectric medium between the
conductors.
According- to that theory , for example, there is an invariable
relation between the surface density of a conductor and the electro
motive intensity just outside it, as expressed in the law of Coulomb
R = 47TCT.
But this is true only in the standard medium, which we may
take to be air. In other media the relation is different, as was
proved experimentally, though not published, by Cavendish, and
afterwards rediscovered independently by Faraday.
In order to express the phenomenon completely, we find it
necessary to consider two vector quantities, the relation between
which is different in different media. One of these is the electro
motive intensity, the other is the electric displacement. The
electromotive intensity is connected by equations of invariable
form with the potential, and the electric displacement is connected
by equations of invariable form with the distribution of electricity,
but the relation between the electromotive intensity and the electric
displacement depends on the nature of the dielectric medium, and
must be expressed by equations, the most general form of which
is as yet not fully determined, and can be determined only by ex
periments on dielectrics.
101 £.] The electromotive intensity is a vector defined in Art. 68,
as the mechanical force on a small quantity e of electricity divided
by e. We shall denote its components by the letters P9 Q, It,
and the vector itself by (£.
In electrostatics, the line integral of (£' is always independent
134 GENERAL THEOREMS. [lOI C.
of the path of integration, or in other words (£ is the space- variation
of a potential. Hence
^ ^7? ^
f — -- 7— » (/ = -- =— > zt = -- =— >
dx dy dz
or more briefly, in the language of Quaternions
101 <?.] The electric displacement in any direction is defined
in Art. 68, as the quantity of electricity carried through a small
area A, the plane of which is normal to that direction, divided
by A. We shall denote the rectangular components of the electric
displacement by the letters /, g^ k, and the vector itself by 2).
The volume-density at any point is determined by the equation
df da dJi
P = -T- + -T + J-9
dx dy dz
or in the language of Quaternions
P= -&V2).
The surface-density at any point of a charged surface is deter
mined by the equation
<r = lf+mg + nh + I'f + m'tf + n'lt,
where f, g, Ji are the components of the displacement on one side
of the surface, the direction cosines of the normal drawn from the
surface on that side being /, m, n, and /", /, h' and I', m', n' are the
components of the displacements, and the direction cosines of the
normal on the other side.
This is expressed in Quaternions by the equation
<r= -[S.UvQ + S.Uv®'],
where Uv, Uv are unit normals on the two sides of the surface,
and 8 indicates that the scalar part of the product is to be taken.
When the surface is that of a conductor, v being the normal
drawn outwards, then since/', /, h' and £)' are zero, the equation is
reduced to the form
o- = (lf+ mg 4- nli) ',
= -S.Uv®.
The whole charge of the conductor is therefore
=
8.
101 d.] The electric energy of the system is, as was shown in
Art. 84, half the sum of the products of the charges into their
respective potentials. Calling this energy W9
IOT <3.] PROPERTIES OF A DIELECTRIC. 135
where the volume-integral is to be taken throughout the electric
field, and the surface-integral over the surfaces of the conductors.
Writing in Theorem III, Art. 21,
we find
rr rrr ,df dg dii
\ l^ttf +mq -\-nli) ds — — / / V '( — + -~ + -j-
JJ vyn JJJ \lx dy dz
rrr, d-% dy
— ( f \-q— +
JJJ \J dx y dy
Substituting this value for the surface-integral in W we find
or
W = \jjj(fp + ^ 5
101 <?.] We now come to the relation between & and (£.
The unit of electricity is usually defined with reference to
experiments conducted in air. We now know from the experiments
of Boltzmann that the dielectric constant of air is somewhat greater
than that of a vacuum, and that it varies with the density. Hence,
strictly speaking, all measurements of electric quantity require to
be corrected to reduce them either to air of standard pressure and
temperature, or, what would be more scientific, to a vacuum, just
as indices of refraction measured in air require a similar correction,
the correction in both cases being so small that it is sensible only
in measurements of extreme accuracy.
In the standard medium
477$ = (g,
Or 47T/=P, 47H7 =Q, ±Tl7l = R.
In an isotropic medium whose dielectric constant is K
477,7 rr
There are some media, however, of which glass has been the most
carefully investigated, in which the relation between 2) and ($
136 GENERAL THEOREMS. [ I Q I /.
is more complicated, and involves the time variation of one or
both of these quantities, so that the relation must be of the form
^(£>, @, 2), @, 5), §, &c.) = 0.
We shall not attempt to discuss relations of this more general kind
at present, but shall confine ourselves to the case in which 3) is
a linear and vector function of (£.
The most general form of such a relation may be written
where $ during the present investigation always denotes a linear
and vector function. The components of 2) are therefore homo
geneous linear functions of those of (£, and may be written in
the form
where the first suffix of each coefficient K indicates the direction
of the displacement, and the second, that of the electromotive
intensity.
The most general form of a linear and vector function involves
nine independent coefficients. When the coefficients which have
the same pair, of suffixes are equal, the function is said to be
self-conjugate.
If we express (£ in terms of 3) we shall have
R = 4 TT (kxnf+ Jcyzg + kzs k).
101 /.] The work done by the electromotive intensity whose
components are P, Q, R, in producing a displacement whose com
ponents are dft dg, and dk, in unit of volume of the medium, is
dW=Pdf+Qdff+Rdh.
Since a dielectric under electric displacement is a conservative
system, W must be a function of f, g, h, and since f, gy k may vary
independently, wre have
aw aw aw
Hence
T* f dP
But -=- = <
dg
. dQ
and -7-^= 47i/<
df
~ df' ~ dg' ~ dh
dP d*W d*W dQ
p,
dg "~ dfdg " dgdf~df
l7T&yX) the coefficient of g in the expression for
'exv, the coefficient of /"in the expression for Q,
101 /L] EXTENSION OF GREEN'S THEOREM. 137
Hence if a dielectric is a conservative system, (and we know that
it is so, because it can retain its energy for an indefinite time),
and (f)~l is a self-conjugate function.
Hence it follows that $ also is self-conjugate, and
101 g.~\ The expression for the energy may therefore be written
in either of the forms
R2 + 2KyzQR
Kzx RP+2 Kxy PQ}dxdy dz,
+ 2 Jczx hf+ 2 7cxy fg\ dx dy ch,
where the suffix denotes the vector in terms of which TFis to be
expressed. When there is no suffix, the energy is understood to be
expressed in terms of both vectors.
We have thus, in all, six different expressions for the energy
of the electric field. Three of these involve the charges and poten
tials of the surfaces of conductors, and are given in Art. 87.
The other three are volume-integrals taken throughout the
electric field, and involve the components of electromotive intensity
or of electric displacement, or of both.
The first three therefore belong to the theory of action at a
distance, and the last three to the theory of action by means of the
intervening medium.
These three expressions for W may be written,
101 ^.] To extend Green's Theorem to the case of a hetero
geneous anisotropic medium, we have only to write in Theorem III,
138 GENERAL THEOREMS. [lO2 a.
and we obtain (remembering that the order of the suffixes of the
coefficients is indifferent),
rll
d
jjj [Kxx dz~fa + K™~d^ ~tiu + K** dz dz
dx d® vv dy dy
,dy
1-
.,,
v\ else dy dy
^4>Nl 7 , 7
5 -- 5— ) \dxdydz
d -dx /J
,7
K-
Using quaternion notation the result may be written more briefly,
//* -S. ZTi; 4» (V 4>) ^ - jj^ ^. ( V* V) 4^ ^r
= _ IJJ8.
Limits between which the electric capacity of a conductor must lie.
102 a.] The capacity of a conductor or system of conductors
has been already defined as the charge of that conductor or system
102 a.] LIMITING VALUES OF CAPACITY. 139
of conductors when raised to potential unity, all the other con
ductors in the field being at potential zero.
The following method of determining limiting values between
which the capacity must lie, was suggested by a paper ' On the
Theory of Resonance/ by the Hon. J. W. Strutt, Phil. Trans. 1871.
See Art. 308.
Let <?! denote the surface of the conductor, or system of con
ductors, whose capacity is to be determined, and s0 the surface of
all other conductors. Let the potential of ^ be ^15 and that of sQ3
^0. Let the charge of s1 be <%. That of s0 will be — elf
Then if q is the capacity of s1 ,
* = ^r> 0
and if W is the energy of the system with its actual distribution of
electricity W = i e1 (^ - *0), (2)
2W e2
*=(+-ig5=2F-
To find an upper limit of the value of the capacity. Assume any
value of ^ which is equal to 1 at s1 and equal to zero at <?0, and
calculate the value of the volume-integral
extended over the whole field.
Then as we have proved (Art. 99$) that W cannot be greater
than %, the capacity, q, cannot be greater than 2%.
To find a lower limit of the value of the capacity. Assume any
system of values of f> g, h, which satisfies the equation
and let it make
Calculate the value of the volume-integral
/ / (^f+ m^g + n1h)dsl = e-^. (6)
extended over the whole field ; then as we have proved (Art. 100 c)
that W cannot be greater than #$>, the capacity, ^, cannot be less
than e^
"?!
The simplest method of obtaining a system of values of ft g, /i,
which will satisfy the solenoidal condition, is to assume a distribu
tion of electricity on the surface of <?15 and another on <?0, the sum
140 GENERAL THEOREMS.
of the charges being zero, then to calculate the potential,, #, due
to this distribution, and the electric energy of the system thus
arranged, which we may call Wff.
If we then make
,_ 1 d^ id* 1 d*
"47aT ~I7^' '77^'
these values off, g, k will satisfy the solenoidal condition.
But in this case we can determine W^ without going through
the process of finding the volume-integral. For since this solution
makes V2x£ = 0 at all points in the field, we can obtain W$> in the
form of the surface-integrals,
^o *o, (9)
where the first integral is extended over the surface s1 and the
second over the surface *0.
If the surface s0 is at an infinite distance from s1, the potential
at s0 is zero and the second term vanishes.
102 $.] An approximation to the solution of any problem of the
distribution of electricity on conductors whose potentials are given
may be made in the following manner : —
Let s1 be the surface of a conductor or system of conductors
maintained at potential 1, and let SQ be the surface of all the other
conductors, including the hollow conductor which surrounds the
rest, which last, however, may in certain cases be at an infinite
distance from the others.
Begin by drawing a set of lines, straight or curved, from
*j to s0.
Along each of these lines, assume SP so that it is equal to 1 at s13
and equal to 0 at s0 . Then if P is a point on one of these lines we
Ps
may take 4^ = - — as a first approximation.
siso
We shall thus obtain a first approximation to ^ which satisfies
the condition of being equal to unity at s1 and equal to zero at s0 .
The value of W* calculated from ^l would be greater than W.
Let us next assume as a second approximation to the lines of
force
The vector whose components are a, b, c is normal to the surfaces
for which ^ is constant. Let us determine p so as to make a, 5, c
satisfy the solenoidal condition. We thus get
I02J!] CALCULATION OF CAPACITY. 141
_
dtf dy* dz* dxdx " dy dy "*" <fe <fe " V '
If we draw a line from SL to s0 whose direction is always normal
to the surfaces for which ^is constant, and if we denote the length
of this line measured from s0 by s, then
dx d^ dy d*! dz d^
tt — — -- •=— ) ti-j-— -- j— > .#-=-= -- — -} (12)
as dx ds ay as dz v '
dty
where E is the resultant intensity = — -y-'» so that
dpd'V, dpd^f. dpdty __ dp
dx dx dy dy dz dz ds'
and equation (11) becomes
^V2^=H2-~j (14-}
d^i \"/
whence p—Cexp.\ 1 d^, , (15)
J0 Jxr
the integral being a line integral taken along the line s.
Let us next assume that along the line s,
d^o dx , dy dz
then *2 = <7o exp.-j^dd^, (17)
the integration being always understood to be performed along the
line s.
The constant C is now to be determined from the condition that
= 1 at SL when also ^ = 1 , so that
'* 2
l. (18)
ri r<
? exp.
JQ ^o
This gives a second approximation to *, and the process may
be repeated.
The results obtained from calculating W#Li Tf^2, ^J2, &c., give
capacities alternately above and below the true capacity and con
tinually approximating thereto.
The process as indicated above involves the calculation of the
form of the line s and integration along this line, operations which
are in general too difficult for practical purposes.
If/ft
142 GENERAL THEOREMS. [lO2 C.
In certain cases however we may obtain an approximation by a
simpler process.
102 c.] As an illustration of this method, let us apply it to
obtain successive approximations to the equipotential surfaces and
lines of induction in the electric field between two surfaces which
are nearly but not exactly plane and parallel, one of which is
maintained at potential zero, and the other at potential unity.
Let the equations of the two surfaces be
'i=/i(»^) = « (19)
for the surface whose potential is zero, and
zz =/2 (x>y) — * (20)
for the surface whose potential is unity, a and b being given
functions of x and y, of which b is always greater than a. The
first derivatives of a and b with respect to x and y are small quan
tities of which we may neglect powers and products of more than
two dimensions.
We shall begin by supposing that the lines of induction are
parallel to the axis of z, in which case
dh
/=°> ff=\te=°~ (21)
Hence Ji is constant along each individual line of induction, and
^ = — 477 / Jidz = — ±Tili(z— a). (22)
* a,
When z = I, * = 1, hence
1
AW/,_,A ' (23)
(24)
\ /
and \j/ _ j
b — a
which gives a first approximation to the potential, and indicates a
series of equipotential surfaces the intervals between which,
measured parallel to z9 are equal.
To obtain a second approximation to the lines of induction, let us
assume that they are everywhere normal to the equipotential
surfaces as given by equation (24).
This is equivalent to the conditions
dx* y ~ dy dz
*s
where A is to be determined so that at every point of the field
df da dk , .
/ + -f + — = 0, (26)
dx dy dz
102 C.] POTENTIAL BETWEEN TWO NEARLY FLAT SURFACES. 143
and also so that the line-integral
taken along any line of induction from the surface a to the surface
b, shall be equal to — 1 .
Let us assume
A= l+A + B(z-a) + C(z-a)2, (28)
and let us neglect powers and products of A, B, C, and at this stage
of our work powers and products of the first derivatives of a and b.
The solenoidal condition then gives ( ~~ """^^*W/<wZ £,
If instead of taking the line-integral along the new line of
induction, we take it along the old line of induction, parallel to
z, the second condition gives
Hence
and
We thus find for the second approximation to the components of
displacement,
X rda d(b—a) z—a^
•T-^ ~~
(33)
- - ,
I— a
and for the second approximation to the potential,
Z — a
^ ,j
—a — a *+"-ji*~£^j /{ <*
If o-0 and o-6 are the surface- densities and ^0 and *6the potentials
of the surfaces a and b respectively,
CHAPTEK V.
MECHANICAL ACTION BETWEEN TWO ELECTRICAL SYSTEMS.
103.] Let E1 and E2 be two electrical systems, the mutual action
between which we propose to investigate. Let the distribution of
electricity in E± be defined by the volume-density, pl5 of the
element whose coordinates are x^y^z-^. Let p2 be the volume-
density of the element of E2 , whose coordinates are oc2 , y2 , z2 .
Then the ^-component of the force acting on the element of El
on account of the repulsion of the element of E2 will be
Pi Pz l 3 2 ^i fy\ dzi d®z dy2 dz2 ,
where r* = (tf1-*2
and if A denotes the x component of the whole force acting on El
on account of the presence of E2
A =ffffff'!*=pplpt<l*1dy1d*1d*tdysd*,, (1)
where the integration with respect to al9 yl} z^ is extended
throughout the region occupied by E1 , and the integration with
respect to x^y^z^ is extended throughout the region occupied
by E2.
Since, however, pl is zero except in the system Elt and p2 is zero
except in the system E2, the value of the integral will not be
altered by extending the limits of the integrations, so that we may
suppose the limits of every integration to be + oo.
This expression for the force is a literal translation into mathe
matical symbols of the theory which supposes the electric force
to act directly between bodies at a distance, no attention being
bestowed on the intervening medium.
If we now define ^2, the potential at the point x^y^z^ arising
from the presence of the system E2, by the equation
*2=fff^dx2dy2dz2, (2)
^2 will vanish at an infinite distance, and will everywhere satisfy
the equation V 2 *2 = 4 7r/32 . (3)
1 04.] MECHANICAL ACTIOJT. 145
We may now express A in the form of a triple integral
A = jiiv pl do°l d/1 dZl ' ^
Here the potential #2 is supposed to have a definite value at
every point of the field, and in terms of this, together with the
distribution, />15 of electricity in the first system E^ the force A is
expressed, no explicit mention being made of the distribution of
electricity in the second system E%.
Now let *j be the potential arising from the first system,
expressed as a function of a?,y, z9 and defined by the equation
*! will vanish at an infinite distance, and will everywhere satisfy
the equation
V2*l== 41^. (6)
We may now eliminate px from A and obtain
in which the force is expressed in terms of the two potentials only.
104.] In all the integrations hitherto considered, it is indifferent
what limits are prescribed, provided they include the whole of the
system Elt In what follows we shall suppose the systems El and
E2 to be such that a certain closed surface s contains within it the
whole of El but no part of U2 .
Let us also write
p = Pl+P2, * = *1+*a, (8)
then within s, p2 = 0, p =Pi>
and without s p1 = 0, p = p2. (9)
Now Au = - Pi *»i «& &!
represents the resultant force, in the direction %} on the system El
arising from the electricity in the system itself. But on the theory
of direct action this must be zero, for the action of any particle P on
another Q is equal and opposite to that of Q on P. and since the
components of both actions enter into the integral, they will
destroy each other.
We may therefore write
VOL. I.
146 MECHANICAL ACTION. [IO5-
where ^ is the potential arising from both systems, the integration
being now limited to the space within the closed surface s, which
includes the whole of the system E± but none of E2.
105.] If the action of E.2 on E^ is effected, not by direct action
at a distance, but by means of a distribution of stress in a medium
extending continuously from Ez to El , it is manifest that if we
know the stress at every point of any closed surface s which
completely separates El from JE2, we shall be able to determine
completely the mechanical action of E2 on E±. For if the force on
El is not completely accounted for by the stress through $, there
must be direct action between something outside of s and some
thing inside of s.
Hence if it is possible to account for the action of E2 on El by
means of a distribution of stress in the intervening medium, it
must be possible to express this action in the form of a surface-
integral extended over any surface s which completely separates
E2 from El.
Let us therefore endeavour to express
1 rrfdV
= - - -j
4TTJJJ doo
j j
doo \_ dx2 dy* dz
in the form of a surface integral.
By Theorem III we may do so if we can determine X, Fand Z,
so that
dX_ dY_ dZ_
' ~ * 2 )~ ~~ dx " d + ~d^ '
Taking the terms separately,
_
dx dx2 ~~ 2 dx
_ d ,dV d*^ d
dx dy2 dy ^dx dy ' dy dxdy
^ _ _
dy dx dy~^ dx
. d sd^d^ I d
Similarly to d^ = dz (jx~ -r) - 2 dx
If, therefore, we write
STRESS IN A MEDIUM.
147
(14)
then A =
dy dz
dz dx
dx dy
dp™ dp*
dx dy
= *1tPx»>
(15)
the integration being extended throughout the space within «,
Transforming the volume-integral by Theorem III, Art. 2.1,
A =
(16)
where ds is an element of any closed surface including the whole
of EI but none of U2, and Imn are the direction cosines of the
normal drawn from ds outwards.
For the components of the force on E^ in the directions of y and
2, we obtain in the same way
\ds, (17)
(18)
If the action of the system E2 on E1 does in reality take place
by direct action at a distance, without the intervention of any
medium, we must consider the quantities pxx &c. as mere abbreviated
forms for certain symbolical expressions, and as having no physical
significance.
But if we suppose that the mutual action between U2 and E^ is
kept up by means of stress in the medium between them, then since
the equations (16), (17), (18) give the components of the resultant
force arising from the action, on the outside of the surface <$, of
the stress whose six components are pxx &c., we must consider
pxx &c. as the components of a stress actually existing in the
medium.
148 MECHANICAL ACTION. [106.
106.] To obtain a clearer view of the nature of this stress let
us alter the form of part of the surface s so that the element ds
may become part of an equipotential surface. (This alteration of
the surface is legitimate provided we do not thereby exclude any
part of EI or include any part of E2).
Let v be a normal to ds drawn outwards.
dy
Let E — — — be the intensity of the electromotive force in
the direction of v, then
dy p dy
= — El, —=- = — Em, — — = —En.
, , —
dx dy dz
Hence the six components of stress are
8TT 47T
If a, b, c are the components of the force on ds per unit of area
1
= ~STT '
Hence the force exerted by the part of the medium outside ds
on the part of the medium inside ds is normal to the element and
directed outwards, that is to say, it is a tension like that of a rope,
and its value per unit of area is - E2.
Sir
Let us next suppose that the element ds is at right angles
to the equipotential surfaces which cut it, in which case
7dV d$ dty
l-T-+m-T- + n— — = 0. (19)
dx dy dz
jf/^*\2 fdV\* fd^\2~}
Now S fj = I --) - (-) - (-^) J
..
-= -- ,-• (20)
—7 =-
dx dy dx dz
dy
Multiplying (19) by 2-y— and subtracting from (20), we find
I07-] COMPONENTS OF STRESS. 149
_
Hence the components of the tension per unit of area of ds are
» = -— ~RH,
8 7T
Hence if the element ^ is at right angles to an equipotential
surface, the force which acts on it is normal to the surface, and its
numerical value per unit of area is the same as in the former case,
but the direction of the force is different, for it is a pressure instead
of a tension.
We have thus completely determined the type of the stress at
any given point of the medium.
The direction of the electromotive intensity at the point is a
principal axis of stress, and the stress in this direction is a tension
whose numerical value is
where R is the electromotive intensity.
Any direction at right angles to this is also a principal axis of
stress, and the stress along this axis is a pressure whose numerical
magnitude is also p.
The stress as thus denned is not of the most general type, for
it has two of its principal stresses equal to each other, and the
third has the same value with the sign reversed.
These conditions reduce the number of independent variables
which determine the stress from six to three, and accordingly it is
completely determined by the three components of the electro
motive intensity
dx dy dz
The three relations between the six components of stress are
9 f \ I \ v
P yz ~ (Pxx + Pn) (P**+Pxx)> }
P2zx = (Pyy + Pzz) (Pxx+Pyy\ ( (23)
+^«)« )
107.] Let us now examine whether the results we have obtained /2<5S1
y £n>«_ < ot ( - <x c
150 MECHANICAL ACTION. [lO/.
will require modification when a finite quantity of electricity is
collected on a finite surface so that the volume-density becomes
infinite at the surface.
In this case, as we have shown in Art. 78, the components
of the electromotive intensity are discontinuous at the surface.
Hence the components of stress will also be discontinuous at the
surface.
Let I m n be the direction cosines of the normal to ds. Let
P, Q, R be the components of the electromotive intensity on the
side on which the normal is drawn, and Pr Q' R' their values
on the other side.
Then by Art. (78 a) if <r is the surface-density
P-P'= 4770-/,
(24)
Let a be the ^-component of the resultant force acting- on
the surface per unit of area, arising1 from the stress on both sides,
then
-Lm(PQ-P'Q') + ±-
-L
O1T
(25)
Hence, assuming that the stress at any point is given by
equations (14), we find that the resultant force in the direction
of a? on a charged surface per unit of volume is equal to the
surface-density multiplied into the arithmetical mean of the x-
components of the electromotive intensity on the two sides of the
surface.
I08.] FORCE OX A CHARGED SURFACE. 151
This is the same result as we obtained in Art. 79 by a process
essentially similar.
Hence the hypothesis of stress in the surrounding- medium is
applicable to the case in which a finite quantity of electricity is
collected on a finite surface.
The resultant force on an element of surface is usually deduced
from the theory of action at a distance by considering a portion
of the surface, the dimensions of which are very small compared
with the radii of curvature of the surface."*
On the normal to the middle point of this portion of the surface
take a point P whose distance from the surface is very small com
pared with the dimensions of the portion of the surface. The
electromotive intensity at this point, due to the small portion of the
surface, will be approximately the same as if the surface had been
an infinite plane, that is to say 2-Tro- in the direction of the normal
drawn from the surface. For a point P' just on the other side of
the surface the intensity will be the same, but in the opposite
direction.
Now consider the part of the electromotive intensity arising from
the rest of the surface and from other electrified bodies at a finite
distance from the element of surface. Since the points P and £'
are infinitely near one another, the components of the electromotive
intensity arising from electricity at a finite distance will be the
same for both points.
Let P0 be the a?-component of the electromotive intensity on
A or A' arising from electricity at a finite distance, then the total
value of the ^-component for A will be
and for A' P= P0-2*<rl.
Hence P0= i (P-f P').
Now the resultant mechanical force on the element of surface
must arise entirely from the action of electricity at a finite distance,
since the action of the element on itself must have a resultant zero.
Hence the ^-component of this force per unit of area must be
a — o-P0,
108.] If we define the potential (as in equation (2)) in terms
of a distribution of electricity supposed to be given, then it follows
* This method is due to Laplace. See Poisson, ' Sur la Distribution cle 1'electricit^
&c.' Mem. de I'lmtitut, 1811, p. 30.
152 MECHANICAL ACTION. [108.
from the fact that the action and reaction between any pair of
electric particles are equal and opposite, that the ^-component of
the force arising from the action of a system on itself must be
zero, and we may write this in the form
But if we define ^ as a function of #, y, z which satisfies the
equation V2^ = 0
at every point outside the closed surface s, and is zero at an infinite
distance, the fact, that the volume-integral extended throughout
any space including $ is zero, would seem to require proof.
One method of proof is founded on the theorem (Art. 100 a), that
if V2v£ is given at every point, and ^ = 0 at an infinite distance,
then the value of V at every point is determinate and equal to
(27)
where r is the distance between the element dx dy clz at which the
concentration of ^ is given = V2^ and the point af if / at which
*' is to be found.
This reduces the theorem to what we deduced from the first
definition of 3>.
But when we consider ^ as the primary function of no, y, z, from
which the others are derived, it is more appropriate to reduce (26)
to the form of a surface-integral,
dS> (28)
and if we suppose the surface S to be everywhere at a great
distance a from the surface s, which includes every point where
V2vP differs from zero, then we know that ^ cannot be numerically
greater than e/a, where 4: ire is the volume-integral of V2v£, and that
R cannot be greater than d^/da or — e/a2, and that the quantities
PxxiPxy>Px* can none of them be greater than p or IP/Sir or
£2/8T7«4. Hence the surface-integral taken over a sphere whose
radius is very great and equal to a cannot exceed <?2/2 a2, and
when a is increased without limit, the surface-integral must become
ultimately zero.
But this surface-integral is equal to the volume-integral (26),
and the value of this volume-integral is the same whatever be the
size of the space enclosed within St provided S encloses every point
at which V2s£ differs from zero. Hence, since the integral is zero
no.] FAEADAY'S THEORY. 153
when a is infinite, it must also be zero when the limits of integra
tion are denned by any surface which includes every point at
which V2^ differs from zero.
109.] The distribution of stress considered in this chapter is pre
cisely that to which Faraday was led in his investigation of induc
tion through dielectrics. He sums up in the following words : —
'(1297) The direct inductive force, which may be conceived to
be exerted in lines between the two limiting and charged con
ducting surfaces, is accompanied by a lateral or transverse force
equivalent to a dilatation or repulsion of these representative lines
(1224.); or the attracting force which exists amongst the par
ticles of the dielectric in the direction of the induction is ac
companied by a repulsive or a diverging force in the transverse
direction.
' (1298) Induction appears to consist in a certain polarized state
of the particles, into which they are thrown by the electrified body
sustaining the action, the particles assuming positive and negative
points or parts, which are symmetrically arranged with respect
to each other and the inducting surfaces or particles. The state
must be a forced one, for it is originated and sustained only by
force, and sinks to the normal or quiescent state when that force
is removed. It can be continued only in insulators_by the same
portion of electricity, because they only can retain this state of the
particles.'
This is an exact account of the conclusions to which we have
been conducted by our mathematical investigation. At every point
of the medium there is a state of stress such that there is tension
along the lines of force and pressure in all directions at right angles
to these lines, the numerical magnitude of the pressure being equal
to that of the tension, and both varying as the square of the
resultant force at the point.
The expression ' electric tension ' has been used in various senses
by different writers. I shall always use it to denote the tension
along the lines of force, which, as we have seen, varies from point
to point, and is always proportional to the square of the resultant
force at the point.
110.] The hypothesis that a state of stress of this kind exists
in a fluid dielectric, such as air or turpentine, may at first sight
appear at variance with the established principle that at any point
in a fluid the pressures in all directions are equal. But in the
deduction of this principle from a consideration of the mobility
154: MECHANICAL ACTION. [ill.
and equilibrium of the parts of the fluid it is taken for granted
that no action such as that which we here suppose to take place
along the lines of force exists in the fluid. The state of stress
which we have been studying is perfectly consistent with the
mobility and equilibrium of the fluid, for we have seen that, if
any portion of the fluid is devoid of electric charge, it experi
ences no resultant force from the stresses on its surface,, however
intense these may be. It is only when a portion of the fluid
becomes charged that its equilibrium is disturbed by the stresses
on its surface, and we know that in this case it actually tends to
move. Hence the supposed state of stress is not inconsistent with
the equilibrium of a fluid dielectric.
The quantity W, which was investigated in Chapter IV, Art. 99,
may be interpreted as the energy in the medium due to the
distribution of stress. It appears from the theorems of that
chapter that the distribution of stress which satisfies the conditions
there given also makes W an absolute minimum. Now when the
energy is a minimum for any configuration, that configuration is
one of equilibrium, and the equilibrium is stable. Hence the
dielectric, when subjected to the inductive action of electrified
bodies, will of itself take up a state of stress distributed in the
way we have described.
It must be carefully borne in mind that we have made only one
step in the theory of the action of the medium. We have supposed
it to be in a state of stress, but we have not in any way accounted
for this stress, or explained how it is maintained. This step,
however, seems to me to be an important one, as it explains, by
the action of the consecutive parts of the medium, phenomena which
were formerly supposed to be explicable only by direct action at
a distance.
111.] I have not been able to make the next step, namely, to
account by mechanical considerations for these stresses in the
dielectric. I therefore leave the theory at this point, merely
stating what are the other parts of the phenomenon of induction
in dielectrics.
I. Electric Displacement. When induction is transmitted through
a dielectric, there is in the first place a displacement of electricity
in the direction of the induction. For instance, in a Ley den jar,
of which the inner coating is charged positively and the outer
coating negatively, the displacement of positive electricity in the
substance of the glass is from within outwards.
III.] ELECTRIC POLARIZATION. 155
Any increase of this displacement is equivalent, during the time
of increase, to a current of positive electricity from within outwards,
and any diminution of the displacement is equivalent to a current
in the opposite direction.
The whole quantity of electricity displaced through any area
of a surface fixed in the dielectric is measured by the quantity which
we have already investigated (Art. 75) as the surface-integral of
induction through that area, multiplied by K/lir, where K is the
specific inductive capacity of the dielectric.
II. Surface charge of the particles of the dielectric. Conceive any
portion of the dielectric, large or small, to be separated (in imagi
nation) from the rest by a closed surface, then we must suppose
that on every elementary portion of this surface there is a charge
measured by the total displacement of electricity through that
element of surface reckoned inwards.
In the case of the Leyden jar of which the inner coating is
charged positively, any portion of the glass will have its inner
side charged positively and its outer side negatively. If this
portion be entirely in the interior of the glass, its surface charge
will be neutralized by the opposite charge of the parts in contact
with it, but if it be in contact with a conducting body, which
is incapable of maintaining in itself the inductive state, the
surface charge will not be neutralized, but will constitute that
apparent charge which is commonly called the Charge of the
Conductor.
The charge therefore at the bounding surface of a conductor and
the surrounding dielectric, which on the old theory was called the
charge of the conductor, must be called in the theory of induction
the surface charge of the surrounding dielectric.
According to this theory, all charge is the residual effect of the
polarization of the dielectric. This polarization exists throughout
the interior of the substance, but it is there neutralized by the
juxtaposition of oppositely charged parts, so that it is only at the
surface of the dielectric that the effects of the charge become
apparent.
The theory completely accounts for the theorem of Art. 77, that
the total induction through a closed surface is equal to the total
quantity of electricity within the surface multiplied by 4-rr. For
what we have called the induction through the surface is simply the
electric displacement multiplied by 47r, and the total displacement
outwards is necessarily equal to the total charge within the surface.
156 MECHANICAL ACTION. [ill.
The theory also accounts for the impossibility of communicating1
an ' absolute charge ' to matter. For every particle of the dielectric
has equal and opposite charges on its opposite sides, if it would not
be more correct to say that these charges are only the manifestations
of a single phenomenon, which we may call Electric Polarization.
A dielectric medium, when thus polarized, is the seat of electrical
energy, and the energy in unit of volume of the medium is nu
merically equal to the electric tension on unit of area, both quan
tities being equal to half the product of the displacement and the
resultant electromotive intensity, or
where p is the electric tension, 3) the displacement, (£ the electro
motive intensity, and K the specific inductive capacity.
If the medium is not a perfect insulator, the state of constraint,
which we call electric polarization, is continually giving way. The
medium yields to the electromotive force, the electric stress is
relaxed, and the potential energy of the state of constraint is con
verted into heat. The rate at which this decay of the state of
polarization takes place depends on the nature of the medium.
In some kinds of glass, days or years may elapse before the polar
ization sinks to half its original value. In copper, a similar change
is effected in less than the billionth of a second.
We have supposed the medium after being polarized to be simply
left to itself. In the phenomenon called the electric current the
constant passage of electricity through the medium tends to restore
the state of polarization as fast as the conductivity of the medium
allows it to decay. Thus the external agency which maintains the
current is always doing work in restoring the polarization of the
medium, which is continually becoming relaxed, and the potential
energy of this polarization is continually becoming transformed
into heat, so that the final result of the energy expended in main
taining the current is to gradually raise the temperature of the
conductor, till as much heat is lost by conduction and radiation
from its surface as is generated in the same time by the electric
current.
CHAPTER VI.
ON POINTS AND LINES OF EQUILIBRIUM.
112.] IF at any point of the electric field the resultant force is
zero, the point is called a Point of equilibrium.
If every point on a certain line is a point of equilibrium, the line
is called a Line of equilibrium.
The conditions that a point shall be a point of equilibrium are
that at that point
dV cW dV
-J- = 0, -T- = 0, -y- = 0.
dx dy dz
At such a point, therefore, the value of V is a maximum, or
a minimum, or is stationary, with respect to variations of the
coordinates. The potential, however, can have a maximum or a
minimum value only at a point charged with positive or with
negative electricity, or throughout a finite space bounded by a
surface charged positively or negatively. If, therefore, a point
of equilibrium occurs in an uncharged part of the field it must
be a stationary point, and not a maximum or a minimum.
In fact, the first condition of a maximum or minimum is that
~dtf' Hf' and 3?
must be all negative or all positive, if they have finite values.
Now, by Laplace's equation, at a point where there is no charge,
the sum of these three quantities is zero, and therefore this condition
cannot be satisfied.
Instead of investigating the analytical conditions for the cases
in which the components of the force simultaneously vanish, we
shall give a general proof by means of the equipotential surfaces.
If at any point, P, there is a true maximum value of 7, then, at
all other points in the immediate neighbourhood of P, the value
of V is less than at P. Hence P will be surrounded by a series of
closed equipotential surfaces, each outside the one before it, and at
all points of any one of these surfaces the electrical force will be
158 POINTS AND LINES OF EQUILIBRIUM. [IJ3»
directed outwards. But we have proved, in Art. 76, that the surface-
integral of the electromotive intensity taken over any closed surface
gives the total charge within that surface multiplied by 4 IT. Now,
in this case the force is everywhere outwards, so that the surface-
integral is necessarily positive, and therefore there is positive charge
within the surface, and, since we may take the surface as near
to P as we please, there is positive charge at the point P.
In the same way we may prove that if V is a minimum at P,
then P is negatively charged.
Next, let P be a point of equilibrium in a region devoid of charge,
and let us describe a sphere of very small radius round P, then,
as we have seen, the potential at this surface cannot be everywhere
greater or everywhere less than at P. It must therefore be greater
at some parts of the surface and less at others. These portions
of the surface are bounded by lines in which the potential is equal
to that at P. Along lines drawn from P to points at which
the potential is less than that at P the electrical force is from P,
and along lines drawn to points of greater potential the force
is towards P. Hence the point P is a point of stable equilibrium
for some displacements, and of unstable equilibrium for other
displacements.
113.] To determine the number of the points and lines of equi
librium, let us consider the surface or surfaces for which the
potential is equal to (7, a given quantity. Let us call the regions
in which the potential is less than C the negative regions, and
those in which it is greater than C the positive regions. Let
VQ be the lowest, and V^ the highest potential existing in the
electric field. If we make C =^, the negative region will in
clude only the point or conductor of lowest potential, and this
is necessarily charged negatively. The positive region consists
of the rest of space, and since it surrounds the negative region
it is periphractic. See Art. 18.
If we now increase the value of C, the negative region will
expand, and new negative regions will be formed round negatively
charged bodies. For every negative region thus formed the sur
rounding positive region acquires one degree of periphraxy.
As the different negative regions expand, two or more of them
may meet in a point or a line. If n+l negative regions meet,
the positive region loses n degrees of periphraxy, and the point
or the line in which they meet is a point or line of equilibrium
of the wth degree.
1I4-] THEIR NUMBER, 159
When C becomes equal to 7£ the positive region is reduced to
the point or the conductor of highest potential, and has therefore
lost all its periphraxy. Hence, if each point or line of equilibrium
counts for one, two, or n, according to its degree, the number so
made up by the points or lines now considered will be less by one
than the number of negatively charged bodies.
There are other points or lines of equilibrium which occur
where the positive regions become separated from each other,
and the negative region acquires periphraxy. The number of
these, reckoned according to their degrees, is less by one than
the number of positively charged bodies.
If we call a point or line of equilibrium positive when it is the
meeting-place of two or more positive regions, and negative when
the regions which unite there are negative, then, if there are p
bodies positively and n bodies negatively charged, the sum of
the degrees of the positive points and lines of equilibrium will be
p—l, and that of the negative ones n—l. The surface which sur
rounds the electrical system at an infinite distance from it is to be
reckoned as a body whose charge is equal and opposite to the sum
of the charges of the system.
But, besides this definite number of points and lines of equi
librium arising from the junction of different regions, there may
be others, of which we can only affirm that their number must be
even. For if, as any one of the negative regions expands, it meets
itself, it becomes a cyclic region, and it may acquire, by repeatedly
meeting itself, any number of degrees of cyclosis, each of which
corresponds to the point or line of equilibrium at which the cyclosis
was established. As the negative region continues to expand till
it fills all space, it loses every degree of cyclosis it has acquired,
and becomes at last acyclic. Hence there is a set of points or
lines of equilibrium at which cyclosis is lost, and these are equal in
number of degrees to those at which it is acquired.
If the form of the charged bodies or conductors is arbitrary, we
can only assert that the number of these additional points or lines
is even, but if they are charged points or spherical conductors, the
number arising in this way cannot exceed (n— l) (ft — 2), where n
is the number of bodies.
114.] The potential close to any point P may be expanded in
the series y= ^ + J5r1 + J22+&c.;
where fflf H2, &c. are homogeneous functions of #, y, z, whose
dimensions are 1, 2, &c. respectively.
160 POINTS AND LINES OF EQUILIBRIUM. [IT5-
Since the first derivatives of V vanish at a point of equilibrium,
H-L = 0, if P be a point of equilibrium.
Let Hn be the first function which does not vanish, then close to
the point P we may neglect all functions of higher degrees as
compared with Hn.
Now Hn = 0
is the equation of a cone of the degree n, and this cone is the cone
of closest contact with the equipotential surface at P.
It appears, therefore, that the equipotential surface passing
through P has, at that point, a conical point touched by a cone
of the second or of a higher degree. The intersection of this cone
with a sphere whose centre is the vertex is called the Nodal line.
If the point P is not on a line of equilibrium the nodal line
does not intersect itself, but consists of n or some smaller number
of closed curves.
If the nodal line intersects itself, then the point P is on a line
of equilibrium, and the equipotential surface through P cuts itself
in that line.
If there are intersections of the nodal line not on opposite points
of the sphere, then P is at the intersection of three or more lines
of equilibrium. For the equipotential surface through P must cut
itself in each line of equilibrium.
115.] If two sheets of the same equipotential surface intersect,
they must intersect at right angles.
For let the tangent to the line of intersection be taken as the
axis of z, then d*7/dz* = 0. Also let the axis of x be a tangent
to one of the sheets, then d*7/da? = 0. It follows from this, by
Laplace's equation, that cl^V/df = 0, or the axis of y is a tangent
to the other sheet.
This investigation assumes that H2 is finite. If H2 vanishes, let
the tangent to the line of intersection be taken as the axis of z, and
let SB = r cos 0, and y = r sin 0, then, since
Id7. I d*7 _
'~~ ~~
the solution of which equation in ascending powers of r is
At a point of equilibrium Al is zero. If the first term that does
not vanish is that in rnt then
V— 70 = An rn cos (n 6 + an) -} terms in higher powers of r.
Il6.] THEIR PROPERTIES. 161
This equation shews that n sheets of the equipotential surface
^= VQ intersect at angles each equal to ir/n. This theorem was
given by Rankine*.
It is only under certain conditions that a line of equilibrium can
exist in free space, but there must be a line of equilibrium on the
surface of a conductor whenever the surface density of the conductor
is positive in one portion and negative in another.
In order that a conductor may be charged oppositely on different
portions of its surface, there must be in the field some places where
the potential is higher than that of the body and others where it is
lower.
Let us begin with two conductors electrified positively to the
same potential. There will be a point of equilibrium between the
two bodies. Let the potential of the first body be gradually
diminished. The point of equilibrium will approach it, and, at a
certain stage of the process, will coincide with a point on its
surface. During the next stage of the process, the equipotential
surface round the second body which has the same potential as the
first body will cut the surface of the second body at right angles
in a closed curve, which is a line of equilibrium. This closed
curve, after sweeping over the entire surface of the conductor,
will again contract to a point ; and then the point of equilibrium
will move off on the other side of the first body, and will be at an
infinite distance when the charges of the two bodies are equal and
opposite.
Earnshaw's Theorem.
116.] A charged body placed in a field of electric force cannot
be in stable equilibrium.
First, let us suppose the electricity of the moveable body (A), and
also that of the system of surrounding bodies (.5), to be fixed in
those bodies.
Let V be the potential at any point of the moveable body due to
the action of the surrounding bodies (B), and let e be the electricity
on a small portion of the moveable body A surrounding this point.
Then the potential energy of A with respect to B will be
M = 2(7e),
where the summation is to be extended to every charged portion of A*
* 'Summary of the Properties of certain Stream Lines/ Phil. Mag., Oct. 1864.
See also, Thomson and Tait's Natural Philosophy, § 780 ; and Kankine and Stokes,
in the Proc. R. S., 1867, p. 468 ; also W. K. Smith, Proc. £. S. Edin., 1869-70, p. 79.
VOL. I. M
162 POINTS AND LINES OF EQUILIBRIUM. [ll6.
Let a, b} c be the coordinates of any charged part of A with
respect to axes fixed in A, and parallel to those of #,y, #. Let the
absolute coordinates of the origin of these axes be f, rj, f
Let us suppose for the present that the body A is constrained to
move parallel to itself, then the absolute coordinates of the point
&, b, e will be
The potential of the body A with respect to B may now be
expressed as the sum of a number of terms, in each of which V
is expressed in terms of «, b, c and £ TJ, £ and the sum of these
terms is a function of the quantities a, b, c, which are constant for
each point of the body, and of f, r\, £ which vary when the body is
moved.
Since Laplace's equation is satisfied by each of these terms it is
satisfied by their sum, or
d*M d*M d*M
cie " drf " df2
Now let a small displacement be given to A, so that
and let dM be the increment of the potential of A with respect to
the surrounding system B.
If this be positive, work will have to be done to increase r, and
there will be a force E = dM/dr tending to diminish/ and to restore
A to its former position, and for this displacement therefore the
equilibrium will be stable. If, on the other hand, this quantity is
negative, the force will tend to increase r, and the equilibrium will
be unstable.
Now consider a sphere whose centre is the origin and whose
radius is /•, and so small that when the point fixed in the body
lies within this sphere no part of the moveable body A can coincide
with any part of the external system B. Then, since within the
sphere V2 JT = 0, the surface-integral
taken over the surface of the sphere, is zero.
Hence, if at any part of the surface of the sphere dM/dr is
positive, there must be some other part of the surface where it is
negative, and if the body A be displaced in a direction in which
dM/dr is negative, it will tend to move from its original position,
and its equilibrium is therefore necessarily unstable.
The body therefore is unstable even when constrained to move
1 1 6.] EQUILIBRIUM ALWAYS UNSTABLE. 163
parallel to itself, and a fortiori it is unstable when altogether
free.
Now let us suppose that the body A is a conductor. We might
treat this as a case of equilibrium of a system of bodies, the move-
able electricity being considered as part of that system, and we
might argue that as the system is unstable when deprived of so
many degrees of freedom by the fixture of its electricity, it must
a fortiori be unstable when this freedom is restored to it.
But we may consider this case in a more particular way, thus —
First, let the electricity be fixed in A, and let A move through
the small distance dr. The increment of the potential of A due to
this cause has been already considered.
Next, let the electricity be allowed to move within A into its
position of equilibrium, which is always stable. During this motion
the potential will necessarily be diminished by a quantity which we
may call Cdr.
Hence the total increment of the potential when the electricity
is free to move will be
,dM ,
(-f-C)dr'
and the force tending to bring A back towards its original position
will be
AM
W'^'
where C is always positive.
Now we have shewn that dM/dr is negative for certain directions
of r, hence when the electricity is free to move the instability in
these directions will be increased.
CHAPTER VII.
FORMS OF THE EQUIPOTENTIAL SURFACES AND LINES OF
INDUCTION IN SIMPLE CASES.
117.] WE have seen that the determination of the distribution
of electricity on the surface of conductors may be made to depend
on the solution of Laplace's equation
V being a function of os, y, and z, which is always finite and con
tinuous, which vanishes at an infinite distance, and which has
a given constant value at the surface of each conductor.
It is not in general possible by known mathematical methods
to solve this equation so as to fulfil arbitrarily given conditions,
but it is easy to write down any number of expressions for the
function V which shall satisfy the equation, and to determine in
each case the forms of the conducting surfaces, so that the function
V shall be the true solution.
It appears, therefore, that what we should naturally call the
inverse problem of determining the forms of the conductors when
the expression for the potential is given is more manageable than
the direct problem of determining the potential when the form of
the conductors is given.
In fact, every electrical problem of which we know the solution
has been constructed by this inverse process. 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 con
structed 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
1 1 8.] USE OP DIAGRAMS. 165
whose forms are given, we may begin with some case in which one
of the equipotential surfaces takes a form somewhat resembling the
given form, and then by a tentative method we may modify the pro
blem till it more nearly corresponds to the given case. 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 cal
culation 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, there are known mathematical methods by
which we may proceed. In other cases we cannot afford to despise
the humbler method of actually drawing tentative figures on paper,
and selecting 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 equi
potential surfaces and lines of induction, so that the student may
make himself familiar with the forms of the lines. The methods by
which such diagrams may be drawn will be explained in Art. 123.
118.] In the first figure at the end of this volume we have the
sections of the equipotential surfaces surrounding two points
charged with quantities 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,
though none of them are accurately spheres. If two of these sur
faces, one surrounding each point, 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 each somewhat more
remote than the middle point from the other body.
The same diagram enables us to see what will be the distribution
166 EQUIPOTENTIAL SURFACES [^19-
of electricity on one of the oval figures, larger at one end than
the other, which surround both centres. Such a body, if charged
with 25 units of electricity 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.
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.
119.] In the second figure we have again two points whose
charges are as 20 to 5, but the one positive and the other negative.
In this case one of the equipotential surfaces, that, namely, corre
sponding 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 equilibrium.
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 surface-density at any point. That at the bottom of
the dimple will be zero.
Surrounding this surface we have others having a rounded1
dimple which flattens and finally disappears in the equipotential
surface passing through the point marked M.
The lines of force in this diagram form two systems divided by a
surface which passes through the point of equilibrium.
121.] AM) LINES OF INDUCTION. 167
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. It then changes sign, and reaches a maximum
at M, after which it continually diminishes.
This maximum, however, is only a maximum relatively to other
points on the axis, for if we consider a surface through M per
pendicular to the axis, M is a point of minimum force relatively to
neighbouring points on that surface.
120.] Figure III represents the equipotential surfaces and lines
of induction due to a point whose charge is 10 placed at A, and
surrounded by a field of force, which, before the introduction of the
charged point, was uniform in direction and magnitude at every
part.
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 A. Those below this surface
have one sheet with a depression near the axis. Those above have
a closed portion surrounding A 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 A 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
prominent according to the particular equipotential surface we
choose.
121.] Figure IV represents the equipotential surfaces and lines
of induction due to three points A, B and C, the charge of A being
15 units of positive electricity, that of .3 12 units of negative
electricity, and that of C 20 units of positive electricity. These
points are placed in one straight line, so that
AB = 9, BC=16, AC =25.
In this case, the surface for which the potential is zero consists
of two spheres whose centres are A and C and their radii 15 and 20.
These spheres intersect in the circle which cuts the plane of the
paper at right angles in D and 2/t so that B is the centre of this
circle and its radius is 12. This circle is an example of a line
168 EQUIPOTENTIAL SURFACES [l22.
of equilibrium, for the resultant force vanishes at every point of
this line.
If we suppose the sphere whose centre is A 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 A. The part of this spherical surface within
the small circle Dl/ will be negatively charged by the influence
of C. All the rest of the sphere will be positively charged, and
the small circle Dlf itself will be a line of no charge.
We may also consider the diagram to represent the sphere whose
centre is (7, charged with 8 units of positive electricity, and in
fluenced by 1 5 units of positive electricity placed at A.
The diagram may also be taken to represent a conductor whose
surface consists of the larger segments of the two spheres meeting
in DD', charged with 23 units of positive electricity.
We shall return to the consideration of this diagram as an
illustration of Thomson's Theory of Electrical Images. See Art. 168.
122.] 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.
The word Force denotes a restricted aspect of that action between
two material bodies by which their motions are rendered different
from what they would have been in the absence of that action.
The whole phenomenon, when both bodies are contemplated at
once, is called Stress, and may be described as a transference of
momentum from one body to the other. When we restrict our
attention to the first of the two bodies, we call the stress acting
on it the Moving Force, or simply the Force on that body, and
it is measured by the momentum which that body is receiving per
unit of time.
The mechanical action between two charged bodies is a stress^
and that on one of them is a force. The force on a small charged
body is proportional to its own charge, and the force per unit of
charge is called the Intensity of the force.
The word Induction was employed by Faraday to denote the
mode in which the charges of electrified bodies are related to
each other, every unit of positive charge being connected with
a unit of negative charge by a line, the direction of which,
in fluid dielectrics, coincides at every part of its course with
that of the electric intensity. Such a line is often called a
I23-] AND LINES OF INDUCTION. 169
line of Force, but it is more correct to call it a line of In
duction.
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 results 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.
123.] 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 e. The potential at a distance r is V— e/r ;
hence, if we make r = e/F, we shall find r, the radius of the sphere
for which the potential is V. 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, each of which we may mark with the number denoting its
potential. These are indicated by the dotted semi-circles on the
right hand of Fig. 6.
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 T[ be the potential due to one
centre, and 7J that due to the other, the potential due to both will be
7f + J%= V. Hence, since at every intersection of the equipotential
surfaces belonging to the two series we know both 7^ and /£, we
also know the value of V. 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
170 EQUIPOTENTIAL SURFACES. [123.
are 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. 6.
This method may be applied to the drawing of any system
of equipotential surfaces when the potential is the sum of two
potentials, 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 0 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 2 ire (1 —cos 6).
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 arc is equal to half the radius, then the induction
through the surface so bounded is
e ( I — cos 6) — 2 <J>, say ;
and 9 = cos-1 (l — 2 — V
V e '
If we now give to 4> a series of values 1, 2, 3 ... e, we shall find
a corresponding series of values of 0, and if e be an integer, the
number of corresponding lines of force, including the axis, will be
equal to e.
We have thus a method of drawing lines of force so that the
charge of any centre is indicated by the number of lines which
diverge from it, and the induction through any surface cut off in the
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. 6 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
TofactPffO.
Fig: 6.
lines of force
' Surfaces
of
Lines of Force and Equipolenlial Surfaces.
l~br Ikz Delegates of the Clcur&ndon Press.
1 2 3.] AND LINES OF INDUCTION. 171
of 4>! and <J>2 , and then, by drawing lines through the consecutive
intersections of these lines for which the value of <J>j + ^2 ^s ^e
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. 6 represent the lines of
force due to the two charged 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 mean distance from the axis is to the assumed
unit of length.
In the case of any such system of finite dimensions the line of force
whose index number is 4> has an asymptote which passes through
the electric centre (Art. 89 d) of the system, and is inclined to the
axis at an angle whose cosine is 1 — 2 <£/<?, where e is the total elec
trification of the system, provided <J> is less than e. Lines of force
whose index is greater than e are finite lines. If e is zero, they are
all finite.
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.
The theory of equipotential surfaces and lines of force in two
dimensions will be given when we come to the theory of conjugate
functions*.
* See a paper 'On the Flow of Electricity in Conducting Surfaces,' by Prof. W. K.
Smith, Proc. B.S.Edin., 1869-70, p. 79.
CHAPTEE VIII.
SIMPLE CASES OF ELECTRIFICATION.
Two Parallel Planes.
124.] WE shall consider, in the first place, two parallel plane
conducting surfaces of infinite extent, at a distance e from each
other, maintained respectively at potentials A and B.
It is manifest that in this case the potential V will be a function
of the distance z from the plane A, and will be the same for all
points of any parallel plane between A and B, except near the
boundaries of the electrified surfaces, which by the supposition
are at an infinitely great distance from the point considered.
Hence, Laplace's equation becomes reduced to
the integral of which is
and since when z = 0, F= A, and when z = c, 7= B,
%
c
For all points between the planes, the resultant intensity is
normal to the planes, and its magnitude is
***%> "B
In the substance of the conductors themselves, R — 0. Hence
the distribution of 'electricity on the first plane has a surface-
density cr, where yrr
*
c
On the other surface, where the potential is B, the surface-
density or' will be equal and opposite to o-, and
124.] SIMPLE CASES. PARALLEL PLANES. 173
Let us next consider a portion of the first surface whose area
is S, taken so that no part of S is near the boundary of the
surface.
The quantity of electricity on this surface is el = So; and, by
Art. 79, the force acting on every unit of electricity is \R, so that
the whole force acting on the area S, and attracting it towards
the other plane, is
8 77 8 77 C2
Here the attraction is expressed in terms of the area S, the
difference of potentials of the two surfaces (A — B), and the distance
between them c. The attraction, expressed in terms of the charge
el , on the area S, is „ _ 2 TT 2
~S~61'
The electrical energy due to the distribution of electricity on the
area S, and that on the corresponding area S' on the surface B
defined by projecting 8 on the surface B by a system of lines of
force, which in this case are normals to the planes, is
8 A-
STT
27T
The first of these expressions is the general expression of elec
trical energy (Art. 84).
The second gives the energy in terms of the area, the distance,
and the difference of potentials.
The third gives it in terms of the resultant force R, and the
volume So included between the areas S and $', and shews that the
energy in unit of volume is p where 8 TTJ? = Rz.
The attraction between the planes isjpS, or in other words, there
is an electrical tension (or negative pressure) equal to p on every
unit of area.
The fourth expression gives the energy in terms of the charge.
The fifth shews that the electrical energy is equal to the work
which would be done by the electric force if the two surfaces were
to be brought together, moving parallel to themselves, with their
electric charges constant.
174 SIMPLE CASES. [125.
To express the charge in terms of the difference of potentials,
we have 1 S , . .
The coefficient q represents the charge due to a difference of
potentials equal to unity. This coefficient is called the Capacity
of the surface S, due to its position relatively to the opposite
surface.
Let us now suppose that the medium between the two surfaces
is no longer air hut some other dielectric substance whose specific
inductive capacity is K, then the charge due to a given difference
of potentials will be K times as great as when the dielectric is air,
e
The total energy will be
.
~ KS 1
The force between the surfaces will be
2 ir
Hence the force between two surfaces kept at given potentials
varies directly as K, the specific capacity of the dielectric, but the
force between two surfaces charged with given quantities of elec
tricity varies inversely as K.
Two Concentric Spherical Surfaces.
125.] Let two concentric spherical surfaces of radii a and b, of
which b is the greater, be maintained at potentials A and B
respectively, then it is manifest that the potential V is a function
of r the distance from the centre. In this case, Laplace's equation
becomes
dr2 r dr ~~
The solution of this is
F=<?1+£2r-i;
and the condition that 7= A when r = a, and V=B when r = b,
gives for the space between the spherical surfaces,
1 2 5.] CONCENTRIC SPHERICAL SURFACES. 175
Aa—Bb A-B _,
If o-15 o-2 are the surface-densities on the opposed surfaces of a
solid sphere of radius a, and a spherical hollow of radius b, then
1 A-B 1 B-A
If ^ and 6?2 are the whole charges of electricity on these surfaces,
A-B
The capacity of the enclosed sphere is therefore »
b-a
If the outer surface of the shell be also spherical and of radius c,
then, if there are no other conductors in the neighbourhood, the
charge on the outer surface is
e3 = Be.
Hence the whole charge on the inner sphere is
e1 = jL(A-S)}
and that of the outer shell
If we put b = oo, we have the case of a sphere in an infinite
space. The electric capacity of such a sphere is a, or it is numeri
cally equal to its radius.
The electric tension on the inner sphere per unit of area is
i PV-S?
f=-s^?-^syr-
The resultant of this tension over a hemisphere is va2_p = F
normal to the base of the hemisphere, and if this is balanced by a
surface tension exerted across the circular boundary of the hemi
sphere, the tension on unit of length being T, we have
F = 2vaT.
6* (A-B)* e*
Hence F— — ^-r- - L-—-L-,
8 b-a2 8a2
(A-B)
IGva (b-a)
176 SIMPLE CASES. [126.
If a spherical soap bubble is electrified to a potential A, then, if
its radius is a, the charge will be Aa, and the surface- density
will be \ A
<j = •
47T a
The resultant intensity just outside the surface will be 4770-,
and inside the bubble it is zero, so that by Art. 79 the electrical
force on unit of area of the surface will be 27rcr2, acting outwards.
Hence the electrification will dimmish the pressure of the air
within the bubble by 2 TT o-2, or
1 A2
877 o2"
But it may be shewn that if T0 is the tension which the liquid
film exerts across a line of unit length, then the pressure from
within required to keep the bubble from collapsing is 2 TJa. If the
electrical force is just sufficient to keep the bubble in equilibrium
when the air within and without is at the same pressure,
Two Infinite Coaxal Cylindric Surfaces.
126.] Let the radius of the outer surface of a conducting cylinder
be a, and let the radius of the inner surface of a hollow cylinder,
having the same axis with the first, be b. Let their potentials
be A and B respectively. Then, since the potential Fis in this
case a function of r, the distance from the axis, Laplace's equation
becomes j^y \dV
I — o,
whence V= Ci -f (72 log r.
Since V= A when r = a, and F"= B when r = #,
»
L
If o-1} o-2 are the surface-densities on the inner and outer
surfaces,
A-B 4^ B-A
a log- Hog-
1 2 7.] COAXAL CYLINDERS. 177
If e1 and ez are the charges on the portions of the two cylinders
between two sections transverse to the axis at a distance I from
each other, A-B _
e1== 2-7r#/o-1= J- j-l = —e^
i ^
los«
The capacity of a length I of the interior cylinder is therefore
'
If the space between the cylinders is occupied by a dielectric of
specific capacity K instead of air, then the capacity of the inner
cylinder is IK
*• "' '
7
log-
The energy of the electrical distribution on the part of the infinite
cylinder which we have considered is
Ef.
1
Fig. 5.
127.] Let there be two hollow cylindric conductors A and B>
Fig. 5, of indefinite length, having the axis of x for their common
axis, one on the positive and the other on the negative side of the
origin, and separated by a short interval near the origin of co
ordinates.
Let a hollow cylinder C of length 2 1 be placed with its middle
point at a distance x on the positive side of the -origin, so as to
extend into both the hollow cylinders.
Let the potential of the positive hollow cylinder be A, that of
the negative one £, and that of the internal one C, and let us put
a for the capacity per unit of length of C with respect to A, and
/3 for the same quantity with respect to B.
The surface densities of the parts of the cylinders at fixed
points near the origin and at points at given small distances
from the ends of the inner cylinder will not be affected by the
VOL. i. N
178 SIMPLE CASES. [127.
value of x provided a considerable length of the inner cylinder
enters each of the hollow cylinders. Near the ends of the hollow
cylinders, and near the ends of the inner cylinder, there will be
distributions of electricity which we are not yet able to calculate,
but the distribution near the origin will not be altered by the
motion of the inner cylinder provided neither of its ends comes
near the origin, and the distributions at the ends of the inner
cylinder will move with it, so that the only effect of the motion
will be to increase or diminish the length of those parts of the
inner cylinder where the distribution is similar to that on an
infinite cylinder.
Hence the whole energy of the system will be, so far as it depends
on x,
Q = \a(l+x) (C-Af + \p(l-x) (C-.5)3 + quantities
independent of oo ;
and the resultant force parallel to the axis of the cylinders will be
If the cylinders A and B are of equal section, a = /3, and
It appears, therefore, that there is a constant force acting 011
the inner cylinder tending to draw it into that one of the outer
cylinders from which its potential differs most.
If C be numerically large and A + B comparatively small, then
the force is approximately X= a (B— A) C;
so that the difference of the potentials of the two cylinders can be
measured if we can measure X, and the delicacy of the measurement
will be increased by raising C, the potential of the inner cylinder.
This principle in a modified form is adopted in Thomson's
Quadrant Electrometer, Art. 219.
The same arrangement of three cylinders may be used as a
measure of capacity by connecting B and C. If the potential of
A is zero, and that of B and C is 7, then the quantity of electricity
on A will be E3 = (^13 + a (I +00)} V;
so that by moving C to the right till a becomes x + £ the capacity of
the cylinder C becomes increased by the definite quantity of, where
1
a and b being the radii of the opposed cylindric surfaces.
CHAPTEK IX.
SPHEEICAL HAKMONICS.
128.] The mathematical theory of spherical harmonics has been
made the subject of several special treatises. The Handbuch der
Kugelfunctionen of Dr. E. Heine, which is the most elaborate work
on the subject, has now (1878) reached a second edition in two
volumes, and Dr. F. Neumann has published his Beitrcige zur
Theorie der Kugelfunctionen (Leipzig, Teubner, 1878). The treat
ment of the subject in Thomson and Tait's Natural Philosophy is
considerably improved in the second edition (1879), and Mr. Tod-
hunter's Elementary Treatise on Laplace's Functions, Lame's Func
tions, and Vessel's Functions, together with Mr. Ferrers' Elementary
Treatise on Spherical Harmonics and subjects connected with them,
have rendered it unnecessary to devote much space in a book on
electricity to the purely mathematical development of the subject.
I have retained however the specification of a spherical harmonic
in terms of its poles.
On Singular Points at which the Potential becomes Infinite.
129 a] If a charge, AQ) of electricity is uniformly spread over
the surface of a sphere the coordinates of whose centre are (a, d, c)
the potential at any point (#, y, z) outside the sphere is, by Art. 1 25,
r=4- (i)
where r2 = (x-a)2 + (y-b)2 + (z-c)2. (2)
As the expression for V is independent of the radius of the
sphere, the form of the expression will be the same if we suppose
the radius infinitely small. The physical interpretation of the
expression would be that the charge A0 is placed on the surface
of an infinitely small sphere, which is sensibly the same as a
N 2,
180 SPHERICAL HARMONICS. [1296.
mathematical point. We have already (Arts. 55, 81) shewn that
there is a limit to the surface-density of electricity, so that it is
physically impossible to place a finite charge of electricity on a
sphere of less than a certain radius.
Nevertheless as the equation (l) represents a possible distri
bution of potential in the space surrounding- a sphere, we may
for mathematical purposes treat it as if it arose from a charge A0
condensed at the mathematical point (a, b, <?) and we may call
the point an infinite point of order zero.
There are other kinds of singular points, the properties of which
we shall now investigate, but before doing so we must define
certain expressions which we shall find useful in dealing with
directions in space, and with the points on a sphere which cor
respond to them.
1295.] An axis is any definite direction in space. We may
suppose it defined by a mark made on the surface of a sphere at the
point where the radius drawn from the centre in the direction
of the axis meets the surface. This point is called the Pole of
the axis. An axis has therefore one pole only, not two.
If ju is the cosine of the angle between the axis h and any vector
r, and if ^ = ^ (3)
p is the resolved part of r in the direction of the axis Ji.
Different axes are distinguished by different suffixes, and the
cosine of the angle between two axes is denoted by Amn, where m
and n are the suffixes specifying the axes.
Differentiation with respect to an axis, Ji, whose direction cosines
are L, M, N, is denoted by
4 = J* +Jf£+tf£. (4)
dh dx dy dz
.
From these definitions it is evident that
' (6)
Amn—
If we now suppose that the potential at the point (#, y, z] due to
a singular point of any order placed at the origin is
I2QC.] INFINITE POINTS. 181
then if such a point be placed at the extremity of the axis /£,
the potential at (#, y, z] will be
4f[(*-Ll), (9- MX), (t-NXft,
and if a point in all respects the same, except that the sign of A is
reversed, be placed at the origin, the potential due to the pair
of points will be
= — Ah ~f(x, y} z) + terms containing h*.
Cvnt
If we now diminish Ji and increase A without limit, their pro
duct continuing finite and equal to A', the ultimate value of the
potential of the pair of points will be
V'=-A'^f(x,y,z). (8)
If/(#, y, z) satisfies Laplace's equation, then, since this equation
is linear, 7', which is the difference of two functions, each of which
separately satisfies the equation, must itself satisfy it.
129 61.] Now the potential due to an infinite point of order zero
V, = A«\, (9)
satisfies Laplace's equation, therefore every function formed from
this by differentiation with respect to any number of axes in suc
cession must also satisfy that equation.
A point of the first order may be formed by taking two points
of order zero, having equal and opposite charges — AQ and AQ} and
placing the first at the origin and the second at the extremity
of the axis h^ . The value of 7^ is then diminished and that of AQ
increased indefinitely, but so that the product A0 h^ is always equal
to AL. The ultimate result of this process, when the two points
coincide, is a point of the first order whose moment is Al and
whose axis is \. A point of the first order is therefore a double
point. Its potential is
-4$- Co)
By placing a point of the first order at the origin, whose moment
is — Aly and another at the extremity of the axis hz whose moment
is Alt and then diminishing ^2 and increasing Alt so that
Alk2= \Ay,, (ll)
182 SPHERICAL HARMONICS. [129 d.
we obtain a point of the second order, whose potential is
V - i li d V
' V*~ -2/^
A 3
We may call a point of the second order a quadruple point
because it is constructed by making four points of order zero ap
proach each other. It has two axes h^ and ?i2 and a moment A2.
The directions of these axes and the magnitude of the moment
completely define the nature of the point.
By differentiating with respect to n axes in succession we obtain
the potential due to a point of the nih order. It will be the
product of three factors, a constant, a certain combination of
cosines, and /•-(n+1). It is convenient, for reasons which will appear
as we go on, to make the numerical value of the constant such
that when all the axes coincide with the vector, the coefficient of
the moment is f-(n+^. We therefore divide by n when we differ
entiate with respect to hn.
In this way we obtain a definite numerical value for a particular
potential, to which we restrict the name of The Solid Harmonic of
degree — (n + 1), namely
F-f-y*--1 _ — •— — •-• (is)
} 1.2.3...?* dhi dk2" dhn r
If this quantity is multiplied by a constant it is still the poten
tial due to a certain point of the nih order.
129 d.~\ The result of the operation (13) is of the form
F= rnr-("+1>, ' (14)
where Tn is a function of the n cosines ^ . . . jun of the angles
between r and the n axes, and of the \n(n — 1) cosines A12, &c. of
the angles between pairs of the axes.
If we consider the directions of r and the n axes as determined
by points on a spherical surface, we may regard Tn as a quantity
varying from point to point on that surface, being a function of the
\n(n+\) distances between the n poles of the axes and the pole
of the vector. We therefore call Yn the Surface Harmonic of
order n.
130&.] We have next to shew that to every surface-harmonic
of order n there corresponds not only a solid harmonic of degree
— (n+1) but another of degree n, or that
Hn= Yurn = rnr**+l (15)
satisfies Laplace's equation.
I3O?).] SOLID HARMONIC OF POSITIVE DEGREE. 183
For
/ — • — •»!—.*» w ' -w, I * 7 *
ax dx
' dx
Hence
Now, since ^ is a homogeneous function of a?, ^, and 2, of
negative degree # + 1 ,
The first two terms therefore of the right-hand member of
equation (16) destroy each other, and, since Vn satisfies Laplace's
equation, the third term is zero, so that Hn also satisfies Laplace's"
equation, and is therefore a solid harmonic of degree n.
This is a particular case of the more general theorem of electrical
inversion, which asserts that if F (a?, y, z) is a function of #, ^,
and z which satisfies Laplace's equation, then there exists another
function, a a*x a2y a2~
- Jf (— H-> — s-> — 2~)>
/ \ p )• f* '
which also satisfies Laplace's equation. See Art. 162.
130#.] The surface harmonic Tn contains 2n arbitrary variables,
for it is defined by the positions of its n poles on the sphere, and
each of these is defined by two coordinates.
Hence the solid harmonics Vn and Hn also contain In arbitrary
variables. Each of these quantities, however, when multiplied by
a constant, will still satisfy Laplace's equation.
To prove that AHn is the most general rational homogeneous
function of degree n which can satisfy Laplace's equation, we
observe that K, the general rational homogeneous function of
degree n, contains $(n+l)(n+2) terms. But VZK is a homo
geneous function of degree n — 2, and therefore contains \n(n — 1)
terms, and the condition VZK= 0 requires that each of these must
vanish. There are therefore \n(n — 1) equations between the
184 SPHERICAL HARMONICS. l3I«-
coefficients of the J (ti+ 1) (n+2) terms of the function K, leaving
2 ft + 1 independent constants in the most general form of the homo
geneous function of degree n which satisfies Laplace's equation.
But Nn, when multiplied by an arbitrary constant, satisfies the
required conditions, and has 2n+l arbitrary constants. It is
therefore of the most general form.
131 aJ\ We are now able to form a distribution of potential such
that neither the potential itself nor its first derivatives become
infinite at any point.
The function 7n = Ynr~(n+l"> satisfies the condition of vanishing
at infinity, but becomes infinite at the origin.
The function ffn=Ynrn is finite and continuous at finite dis
tances from the origin, but does not vanish at an infinite distance.
But if we make a"Ynr-(n+V the potential at all points outside
a sphere whose centre is the origin, and whose radius is a, and
a-(n+i)Ynrn the potential at all points within the sphere, and if
on the sphere itself we suppose electricity spread with a surface
density o- such that
n, (18)
then all the conditions will be satisfied for the potential due to a
shell charged in this manner.
For the potential is everywhere finite and continuous, and
vanishes at an infinite distance ; its first derivatives are everywhere
finite and are continuous except at the charged surface, where they
satisfy the equation
and Laplace's equation is satisfied at all points both inside and
outside of the sphere.
This, therefore, is a distribution of potential which satisfies the
conditions, and by Art. 100 a it is the only distribution which can
satisfy them.
131 #.] The potential due to a sphere of radius a whose surface
density is given by the equation
47Ttf2<r = (2ft+l)rn, (20)
is, at all points external to the sphere, identical with that due to
the corresponding singular point of order n.
Let us now suppose that there is an electrical system which
we may call E, external to the sphere, and that * is the potential
due to this system, and let us find the value of 2(*e?) for the
131 C.] SINGULAR POINT EQUIVALENT TO CHARGED SHELL. 185
singular point. This is the part of the electric energy depending
on the action of the external system on the singular point.
If AQ is the charge of a single point of order zero, then the
potential energy in question is
^ = 4>*. (21)
If there are two such points, a negative one at the origin and a
positive one of equal numerical value at the extremity of the axis
kl} then the potential energy will be
/7vl; /72 q/
-A*+^0(*^1_ + 4V — + &c.),
and when AQ increases and h^ diminishes indefinitely, but so that
AQ&! = A.lf the value of the potential energy for a point of the first
order will be
* = 4jf (22)
Similarly for a point of order n the potential energy will be
* = i^^aB*;* (23)
131 <?.] If we suppose the external system to be made up of
parts, any one of which is denoted by dEt and the singular point
to be made up of parts any one of which is de, then
* = 2(1^). (24)
But if Va is the potential due to the singular point,
j; = S(i<fc), (25)
and the potential energy due to the action of E on e is
JT=2(l>de) = 22 (-dEde) = ^VndE, (26)
the last expression being the potential energy due to the action of
e on E.
Similarly, if crds is an element of electricity on the shell, since
the potential due to the shell at the external system E is Fn,
we have
2(*<r<fo). (27)
The last term contains a summation to be extended over the
* We shall find it convenient, in what follows, to denote the product of the positive
integral numbers 1.2.3...nbyw!
vr *
186 SPHERICAL HABMONICS. [132.
surface of the sphere. Equating it to the first expression for F,
we have
-—A — (28)
~ n\ n dl^...dliu
If we remember that 4 TTO- a2 = (2 n+ 1) 7n, and that An = an, this
becomes
T 7 ^7r "+2 ^ (29)
JL .j ff5 ^^ , • ~~r tt IT j L \ /
This equation reduces the operation of taking the surface integral
o?VYnds over every element of the surface of a sphere of radius a,
to that of differentiating * with respect to the n axes of the
harmonic and taking the value of the differential coefficient at
the centre of the sphere, provided that * satisfies Laplace's equa
tion at all points within the sphere, and Yn is a surface harmonic of
order n.
132.] Let us now suppose that * is a solid harmonic of positive
degree m of the form
* = a-m Ymrm. (30)
At the spherical surface, r = a, and * = Jm, so that equation
(29) becomes in this case
ffrr* 4w «*-+•£&£*, (si)
JJ T™T«d* -n\(2n+l)a dk,..Mn
where the value of the differential coefficient is to be taken at the
centre of the sphere.
When n is less than m, the result of the differentiation is a
homogeneous function of a?, y and z of degree m-n, the value of
which at the centre of the sphere is zero. If n is equal to m the
result of the differentiation is a constant, the value of which we
shall determine in Art. 134 b. If the differentiation is carried
further, the result is zero. Hence the surface-integral JJ Ym Yn ds
vanishes whenever m and n are different.
The steps by which we have arrived at this result are all of
them purely mathematical, for though we have made use of terms
having a physical meaning, such as electrical energy, each of these
terms is regarded not as a physical phenomenon to be investigated,
but as a definite mathematical expression. A mathematician has
as much right to make use of these as of any other mathematical
functions which he may find useful, and a physicist, when he has
1 33-] TRIGONOMETRICAL EXPRESSION. 187
to follow a mathematical calculation, will understand it all the
better if each of the steps of the calculation admits of a physical
interpretation.
133.] We shall now determine the form of the surface harmonic
Tn as a function of the position of a point P on the sphere with
respect to the n poles of the harmonic.
We have
3 l
and so on.
Every term of Tn therefore consists of products of cosines, those
of the form /u, with a single suffix, being" cosines of the angles
between P and the different poles, and those of the form A, with
double suffixes, being cosines of the angles between the poles.
Since each axis is introduced by one of the n differentiations, the
symbol of that axis must occur once and only once among the
suffixes of the cosines of each term.
Hence if in any term there are s cosines with double suffixes,
there must be n — 2s cosines with single suffixes.
Let the sum of all products of cosines in which s of them have
double suffixes be written in the abbreviated form
2(MM-28AS).
In every one of the products all the suffixes occur once, and none
is repeated.
If we wish to express that a particular suffix, m, occurs among
the //'s only or among the A's only, we write it as a suffix to the /*
or the A. Thus the equation
2 (^* A') = 2 (^-a- As) + 2 (p«-*'\w>) (33)
expresses that the whole set of products may be divided into two
parts, in one of which the suffix m occurs among the direction
cosines of the variable point P, and in the other among the cosines
of the angles between the poles.
Let us now assume that for a particular value of n
Yn = An 1 0 2 (p*) -Mn, i s O"-2 A1) + &c.
+ A.82<yi-2sAs) + &c., (34)
when the A9s are numerical coefficients. We may write the series
in the abbreviated form
Yn = S[Ant.2(n«-*'\')], (35)
when S indicates a summation in which all values of s, including
zero, not greater than J», are to be taken.
188 SPHERICAL HARMONICS. [l
OO-
To obtain the corresponding solid harmonic of negative degree
(n+ 1) and order n, we multiply by ?-(n+i)? and obtain
^ = 5[^...»a-il"-1 2 (!>"-»• V)]; (36)
putting rp — p, as in equation (3).
If we differentiate Vn with respect to a new axis hm we obtain
and therefore
If we wish to obtain the terms containing s cosines with double
suffixes, we must diminish s by unity in the last term, and we find
-A.s-i2(^-2s+1Aw*)}]. (38)
Now the two classes of products are not distinguished from each
other in any way except that the suffix m occurs among the p's
in one and among the A's in the other. Hence their coefficients
must be the same, and since we ought to be able to obtain the
same result by putting n 4- 1 for n in the expression for Vn and
multiplying by n+l} we obtain the following equations,
(n+1) An+lt8 = (2»-2«-f 1)4,,. = -4...-1' (39)
If we put s = 0, we obtain
(n+l)An+L = (2n+l)An-t (40)
and therefore, since Al 0 = 1,
A ^ • (41)
^•0-^pTp
and from this we obtain the general value of the coefficient
and finally the trigonometrical expression for the surface harmonic,
as
Yn = S[(-)s njin7,2^\. S (jutn-2s As)1. (43)
This expression gives the value of the surface harmonic at any
point P of the spherical surface in terms of the cosines of the
distances of P from the different poles and of the distances of the
poles from each other.
It is easy to see that if any one of the poles be removed to
the opposite point of the spherical surface, the value of the
harmonic will have its sign reversed. For any cosine involving
YnJs. 189
the index of this pole will have its sign reversed, and in each
term of the harmonic the index of the pole occurs once and only
once.
Hence if two or any even number of poles are removed to the
points respectively opposite to them, the value of the harmonic
will be unaltered.
Professor Sylvester, however,, has shewn (Phil. Mag., Oct. 1876)
that when the harmonic is given, the problem of finding the n
lines which coincide with the axes has one and only one solution,
though, as we have just seen, the directions to be reckoned positive
along these axes may be reversed in pairs.
134.] We are now able to determine the value of the surface
integral / / Ym Yn ds when the order of the two surface harmonics
is the same, though the directions of their axes may be in general
different.
For this purpose we have to form the solid harmonic Tmrn and
to differentiate it with respect to each of the n axes of Tn .
Any term of Tmrf of the form rm^m~28X.8 may be written
r28_pmm~28\mm8. Differentiating this n times in succession with
respect to the n axes of J"n, we find that in differentiating r2*
with respect to s of these axes we introduce s of the j?n's, and
the numerical factor
2*(2*— 2)...2, or 2°s\
In continuing the differentiation with respect to the next s axes,
the jt?n's become converted into Xwn's, but no numerical factor is
introduced, and in differentiating with respect to the remaining
n — 2s axes, the pm's become converted into Amn's, so that the
result is 2s* ! \nn* \mms Xmnm-28.
We have therefore, by equation (31),
. -«!(2»+l)
and by equation (43),
Hence performing the differentiations and remembering that
= n, we find
190 SPHERICAL HARMONICS.
135 #.] The expression (46) for the surface-integral of the
product of two surface-harmonics assumes a remarkable form if
we suppose all the axes of one of the harmonics, Ym, to coincide
with each other, so that Ym becomes what we shall afterwards
define as the zonal harmonic of order m, denoted by the symbol Pm .
In this case all the cosines of the form \nm may be written fj,n,
where pn denotes the cosine of the angle between the common
axis of Pm and one of the axes of Yn. The cosines of the form
\mm will all become equal to unity, so that for 2ASTOTO we must
put the number of combinations of s symbols, each of which is
distinguished by two suffixes out of n, no suffix being repeated.
Hence
The number of permutations of the remaining n—2s indices of
the axes of Pm is (n — 2 s) I Hence
SW£") = («-2.)!M"-»'. (48)
Equation (46) therefore becomes, when all the axes of Ym coincide
with each other,
4Wa y«(»), by equation (43), (50)
;^>vr 2K+1
where 7n(m) denotes the value of Yn at the pole of Pm.
We may obtain the same result by the following shorter pro
cess : —
Let a system of rectangular coordinates be taken so that the
axis of z coincides with the axis of Pm, and let Ynrn be expanded
as a homogeneous function of #, y, z of degree n.
At the pole of Pmi x =y = 0 and z = r, so that if Czn is the
term not involving x or y, C is the value of Yn at the pole of Pm.
Equation (31) becomes in this case
n
If m is equal to n, the result of differentiating Czn is n \ C, and
is zero for the oth«r terms. Hence
P ds - C
fm<t* ~^
C being the value of Yn at the pole of Pm.
135 £.] This result is- a very important one in the theory of
135&-] EXPANSION IN SPHERICAL HARMONICS. 191
spherical harmonics, as it shews how to determine a series of
spherical harmonics which expresses the value of a quantity having
any arbitrarily assigned finite and continuous value at each point
of a spherical surface.
For let F be the value of the quantity and ds the element of
surface at a point Q of the spherical surface, then if we multiply
Fds by Pn, the zonal harmonic whose pole is the point P of the
same surface, and integrate over the surface, the result, since
it depends on the position of the point P, may be considered as
a function of the position of P.
But since the value at P of the zonal harmonic whose pole is Q
is equal to the value at Q of the zonal harmonic of the same order
whose pole is P, we may suppose that for every element ds of the
surface a zonal harmonic is constructed having its pole at Q and
having a coefficient Fds.
We shall thus have a system of zonal harmonics superposed on
each other with their poles at every point of the sphere where F
has a value. Since each of these is a multiple of a surface harmonic
of order ny their sum is a multiple of a surface harmonic (not
necessarily zonal) of order n.
The surface integral / / FPnds considered as a function of the
point P is therefore a multiple of a surface harmonic Yn ; so that
is also that particular surface harmonic of the nih order which
belongs to the series of harmonics which expresses F, provided F
can be so expressed.
For if F can be expressed in the form
then if we multiply by Pnds and take the surface integral over the
whole sphere, all terms involving products of harmonics of different
orders will vanish, leaving
Hence the only possible expansion of F in spherical harmonics is
f= si / U<'JJ,.as4-&c. + (2n4-l) I I 4'P..d*4.Rr.(*. I. (51)
192 SPHERICAL HARMONICS. [137.
Conjugate Harmonics.
136.] We have seen that the surface integral of the product of
two harmonics of different orders is always zero. But even when
the two harmonics are of the same order, the surface integral of
their product may be zero. The two harmonics are then said to
be conjugate to each other. The condition of two harmonics of the
same order being conjugate to each other is expressed in terms of
equation (46) by making its members equal to zero.
If one of the harmonics is zonal, the condition of conjugacy is
that the value of the other harmonic at the pole of the zonal
harmonic must be zero.
If we begin with a given harmonic of the ni}l order, then, in
order that a second harmonic may be conjugate to it, its 2n
variables must satisfy one condition.
If a third harmonic is to be conjugate to both, its 2 n variables
must satisfy two conditions. If we go on constructing harmonics,
each of which is conjugate to all those before it, the number of
conditions for each will be equal to the number of harmonics
already in existence, so that the (2rc+l)th harmonic will have 2n
conditions to satisfy by means of its 2 n variables, and will therefore
be completely determined.
Any multiple A Tn of a surface harmonic of the nih order can
be expressed as the sum of multiples of any set of 2 n + 1 conjugate
harmonics of the same order, for the coefficients of the 2n+l
conjugate harmonics are a set of disposable quantities equal in
number to the 2 n variables of Tn and the coefficient A.
In order to find the coefficient of any one of the conjugate
harmonics, say Yn°, suppose that
Multiply by Yn*ds and find the surface integral over the sphere.
All the terms involving products of harmonics conjugate to each
other will vanish, leaving
2 », (52)
an equation which determines Aff.
Hence if we suppose a set of 2n+l conjugate harmonics given,
any other harmonic of the nih order can be expressed in terms of
them, and this only in one way. Hence no other harmonic can be
conjugate to all of them.
137.] We have seen that if a complete system of 2^+1 har-
I38-] ZONAL HARMONICS. 193
monies of the nih order, all conjugate to each other, be given,
any other harmonic of that order can be expressed in terms of
these. In such a system of 2 n -f 1 harmonics there are 2n(2n+l)
variables connected by n(2n+l) equations, n(2n+l) of the
variables may therefore be regarded as arbitrary.
We might, as Thomson and Tait have suggested, select as a
system of conjugate harmonics one in which each harmonic has
its n poles distributed so that j of them coincide at the pole of the
axis of x, k at the pole of y, and l(= n—j—Jc) at the p'ole of z.
The n -f 1 distributions for which I = 0 and the n distributions
for which 1=1 being given, all the others may be expressed in
terms of these.
The system which has been actually adopted by all mathe
maticians (including Thomson and Tait) is that in which n — o- of
the poles are made to coincide at a point which we may call the
Positive Pole of the sphere, and the remaining <r poles are placed at
equal distances round the equator when their number is odd, or
at equal distances round one half of the equator when their number
is even.
In this case j/ls //2 , . . . /*„_, are each of them equal to cos 0, which
we shall denote by /u. If we also write v for sin 0, nn-v+l...i*.n are
of the form v cos (0—0), where 0 is the azimuth of one of the poles
on the equator.
Also the value of Xpq is unity, if_p and q are both less than n — o-,
zero when one is greater and the other less than this number, and
cos 7-77/0- when both are greater, r being an integral number less
than a:
138.] When all the poles coincide at the pole of the sphere,
o- = 0, and the harmonic is called a Zonal harmonic. As the
zonal harmonic is of great importance we shall reserve for it the
symbol Pn.
We may obtain its value either from the trigonometrical ex
pression (43) or more directly by differentiation, thus
«-<->•££(!)> («)
1 Q £ /O/M 1\ r
£ =
[— / 4 \
n_ *\n—L)
2.(2rc-ir
1.2.3...* 2.(2n-l)
n(n-l)(n-2) («-3)
W*
= S|f-V... . v — u"-2*
VOL. I.
194: SPHERICAL HARMQNICS. [139.
where we must give to p every integral value from zero to the
greatest integer which does not exceed \n.
It is sometimes convenient to express Pn as a homogeneous
function of cos 0 and sin 0, or, as we write them, /u, and v,
P = ._ --
It is shewn in the mathematical treatises on this subject that
Pn (ju) is the coefficient of hn in the expansion of (l — 2^h + h2)"^.
The surface integral of the square of the zonal harmonic, or
(p. oo)« ^ = ^ - (55)
Hence (P. (M))2 *M = • (56)
139.] If we consider a zonal harmonic simply as a function of /u,
and without any explicit reference to a spherical surface, it may be
called a Legendre's Coefficient.
If we consider the zonal harmonic as existing on a spherical
surface the points of which are defined by the coordinates 6 and </>,
and if we suppose the pole of the zonal harmonic to be at the point
(0', $'), then the value of the zonal harmonic at the point (0, <£)
is a function of the four angles 0', $', 0, $, and because it is a
function of ju, the cosine of the arc joining the points (0, $) and
(Of, <p'), it will be unchanged in value if 6 and 0', and also $ and $',
are made to change places. The zonal harmonic so expressed has
been called Laplace's Coefficient. Thomson and Tait call it the
Biaxal Harmonic.
Any homogeneous function of a?, y, z- which satisfies Laplace's equa
tion may be called a Solid harmonic, and the value of a solid
harmonic at the surface of a sphere whose centre is the origin may
be called a Surface harmonic, In this book we have defined a
surface harmonic by means of its n poles, so that it has only 2n
variables. The more general surface harmonic, which has 2#-M
variables, is the more restricted surface harmonic multiplied by an
arbitrary constant. The more general surface harmonic, when
expressed in terms of 0 and $, is called a Laplace's Function.
140 #.] To obtain the other harmonics of the symmetrical system,
we have to differentiate with respect to <r axes in the plane of xy
inclined to each other at angles equal to ir/cr. This may be most
1 40 a.] TESSERAL HARMONICS. 195
conveniently done by means of the system of imaginary coordinates
given in Thomson and Tait's Natural Philosophy, vol. I, p. 148 (or
p. 185 of 2nd edition).
If we write
( = x + iy9 n = x — iy9 (57)
where i denotes \A— 1, the operation of differentiating with respect
to the or axes may be written
J ££*. ^^ ,
if one of the axes coincides with the axis ^
if the axis of^ bisects the angle between two of the axes/~ l'* ' ^
We shall find it convenient to express these operations by the
abbreviated symbols of operation Ds and DC, respectively. They
are, of course, real operations, and may be expressed without the
use of imaginary symbols thus —
^^ 1.2
We shall also write
so that Ds and DC denote the operations of differentiating with
n n
respect to n axes, n — o- of which coincide with the axis of #, while
the remaining o- make angles TT/CT with each other in the plane of
xy, Ds being used when the axis of y coincides with one of the
n
axes,. and Do when the axis of y bisects the angle between two
n
of the axes.
The two tesseral surface harmonics of order n and type o- may
now be written
(63)
(64)
196 SPHERICAL HARMONICS. [140 a.
Writing fx = cos 0, v = sin 0, p2 = #2 +/,
so that 2 = pr, p = vr, x — p cos <£, y = p sin $,
we have J? I = (- 1)'
in which we may write
(67)
We have now only to differentiate with respect to z, which we
may do either so as to obtain the result in terms of r and z, or as a
homogeneous function of z and p divided by a power of r,
n\ (2o-)\
(*-*) (n-°- l) zn-*-* r* + &c.1 , (68)
2(2^—1)
r ^ (^-^-.Q.-I) ^ + &c l ^
L 4 ((7+1)
If we write
2. 4. (2»-l) (2»-3)
and
2. 2^-1
I (7())
(K-g) (^-(7-1) ^-(7- 2) fc-(7-3) -(r_4 _
0-3) .^ ,r 1
(<r) 2M-^^!fc + q-)! .(*)
then 0« : (2»)IcrI -- 3« '
so tli at these two functions differ only by a constant factor.
We may now write the expressions for the two tesseral harmonics
of order n and type o- in terms either of 0 or ^,
- (74)
Wre must remember that when o- = 0, sin 0-$ = 0 and cos o-<f> = 1
140 C.] TESSERAL HARMONICS. 197
For every value of a from 1 to n inclusive there is a pair of
(0) (0)
harmonics, but when a = 0, Ts = 0 and Yc = 2Ptt, the zonal bar-
n n
monic. The whole number of harmonics of order n is therefore
2n+l, as it ought to be.
1400.] The numerical value of Y adopted in this treatise is that
which we find by differentiating- r~l with respect to the n axes and
dividing by n \ It is the product of four factors, the sine or cosine
of 0-0, v*9 a function of ^ (or of //, and v), and a numerical co
efficient.
The product of the second and third factors, that is to say, the
part depending on 0, has been expressed in terms of three different
symbols which differ from each other only by their numerical
factors. When it is expressed as the product of v° into a series of
descending powers of /u, the first term being /xw~^ it is the function
which we, following Thomson and Tait, denote by 0.
The function which Heine (Ilandbuch der Kugelfunctionen, § 47)
denotes by P^ and calls eine zugeordnete Function erster Art, or, as
Todhunter translates it, an ' Associated Function of the First Kind,'
is related to ©^ by the equation
0W=(_/p(»). (75)
The series of descending powers of ju, beginning with ju""17, is
expressed by Heine by the symbol *$£\ and by Todhunter by the
symbol or (a-, n).
This series may also be expressed in two other forms,
_-
(2n)l dp* *' (n
The last of these, in which the series is obtained by differentiating
the zonal harmonic with respect to jn, seems to have suggested the
symbol adopted by Ferrers, who defines it thus
When the same quantity is expressed as a homogeneous function
of /u and v, and divided by the coefficient of /^n-<r vv, it is what we
have already denoted by 3^ •
140 <?.] The harmonics of the symmetrical system have been
classified by Thomson and Tait with reference to the form of the
spherical curves at which they become zero.
198 SPHERICAL HARMONICS.
The value of the zonal harmonic at any point of the sphere is
a function of the cosine of the polar distance, which if equated
to zero gives an equation of the nib degree, all whose roots lie
between — 1 and -f 1 , and therefore correspond to n parallels of
latitude on the sphere.
The zones included between these parallels are alternately positive
and negative, the circle surrounding the pole being always positive.
The zonal harmonic is therefore suitable for expressing a function
which becomes zero at certain parallels of latitude on the sphere,
or at certain conical surfaces in space.
The other harmonics of the symmetrical system occur in pairs,
one involving the cosine and the other the sine of <r$. They
therefore become zero at a- meridian circles on the sphere and also
at n — a- parallels of latitude, so that the spherical surface is divided
into 2o-(n — o-— l) quadrilaterals or tesserae, together with 40-
triangles at the poles. They are therefore useful in investigations
relating to quadrilaterals or tesserae on the sphere bounded by
meridian circles and parallels of latitude.
They are all called Tesseral harmonics except the last pair, which
becomes zero at n meridian circles only, which divide the spherical
surface into 2n sectors. This pair are therefore called Sectorial
harmonics.
141.] We have next to find the surface integral of the square of
any tesseral harmonic taken over the sphere. This we may do by
the method of Art. 134. We convert the surface harmonic Y^} into
a solid harmonic of positive degree by multiplying it by rn, we
differentiate this solid harmonic with respect to the n axes of the
harmonic itself, and then make x = y = z = 0, and we multiply the
<">
,, ,
result by .
* »!(2»+l)
These operations are indicated in our notation by
Writing the solid harmonic in the form of a homogeneous func
tion of z and f, rj, viz.,
rnls =
; (79)
we find that on performing the differentiations with respect to z,
all the terms of the series except the first disappear, and the factor
(n — a)l is introduced.
142 &.] SURFACE INTEGRALS. 199
Continuing the differentiation with respect to £ and 77 we get rid
also of these variables and introduce the factor a-!, so that the final
result is
)! («-*)! . ,
We shall denote the second member of this equation by the
abbreviated symbol [n, <r].
This expression is correct for all values of cr from 1 to n inclusive,
but there is no harmonic in sin o-</> corresponding to o- = 0.
In the same way we can shew that
zdl (8!)
*
for all values of a from 1 to n inclusive.
When o- = 0, the harmonic becomes the zonal harmonic, and
-//«>•*-
a result which may be obtained directly from equation (50) by
putting Tn = Pm and remembering that the value of the zonal
harmonic at its pole is unity.
142 #.] We can now apply the method of Art. 136 to determine
the coefficient of any given tesseral surface harmonic in the
expansion of any arbitrary function of the position of a point on
a sphere. For let F be the arbitrary function, and let A* be the
coefficient of Y^ in the expansion of this function in surface
harmonics of the symmetrical system
FJ? ds = 4? T?1* = ^ & *], (83)
JJ
where [n, a] is the abbreviation for the value of the surface integral
given in equation (80).
142 b.~\ Let ^ be any function which satisfies Laplace's equation,
and which has no singular values within a distance a of a point 0,
which we may take as the origin of coordinates. It is always
possible to expand such a function in a series of solid harmonics
of positive degree, having their origin at 0.
One way of doing this is to describe a sphere about 0 as centre
with a radius less than a, and to expand the value of the potential
at the surface of the sphere in a series of surface harmonics.
Multiplying each of these harmonics by r/a raised to a power
equal to the order of the surface harmonic, we obtain the solid
harmonics of which the given function is the sum.
200 SPHERICAL HARMONICS. [143.
But a more convenient method, and one which does not involve
integration, is by differentation with respect to the axes of the
harmonics of the symmetrical system.
For instance, let us suppose that in the expression of ^, there is
(<>•) (<>•)
a term of the form Ac Yc rn.
n n
If we perform on ^ and on its expansion the operation
d"-* ,d* d9 s
and put #, y, z equal to zero after differentiating1, all the terms
((7)
of the expansion vanish except that containing1 Ac.
n
Expressing the operator on ^ in terms of differentiators with
respect to the real axes, we obtain the equation
1.2
from which we can determine the coefficient of any harmonic of the
series in terms of the differential coefficients of ^ with respect to
x, y> z at the origin.
143.] It appears from equation (50) that it is always possible
to express a harmonic as the sum of a system of zonal harmonics
of the same order, having their poles distributed over the surface
of the sphere. The simplification of this system, however, does not
appear easy. I have, however, for the sake of exhibiting to the
eye some of the features of spherical harmonics, calculated the zonal
harmonics of the third and fourth orders, and drawn, by the
method already described for the addition of functions, the equi-
potential lines on the sphere for harmonics which are the sums of
two zonal harmonics. See Figures VI to IX at the end of this
volume.
Fig. VI represents the difference of two zonal harmonics of the
third order whose axes are inclined 120° in the plane of the paper,
and this difference is the harmonic of the second type in which o- = 1 ,
the axis being perpendicular to the paper.
In Fig. VII the harmonic is also of the third order, but the
axes of the zonal harmonics of which it is the sum are inclined
90°, and the result is not of any type of the symmetrical system.
One of the nodal lines is a great circle, but the other two which are
intersected by it are not circles.
Fig. VIII represents the difference of two zonal harmonics of
144 &•] DIAGRAMS OF SPHERICAL HARMONICS. 201
the fourth order whose axes are at right angles. The result is a
tesseral harmonic for which n = 4, a = 2.
Fig. IX represents the sum of the same zonal harmonics. The
result gives some notion of one type of the more general har
monic of the fourth order. In this type the nodal line on the
sphere consists of six ovals not intersecting each other. Within
these ovals the harmonic is positive, and in the sextuply connected
part of the spherical surface which lies outside the ovals, the har
monic is negative.
All these figures are orthogonal projections of the spherical
surface.
I have also drawn in Fig. V a plane section through the axis
of a sphere, to shew the equipotential surfaces and lines of force
due to a spherical surface electrified according to the values of a
spherical harmonic of the first order.
Within the sphere the equipotential surfaces are equidistant
planes, and the lines of force are straight lines parallel to the axis,
their distances from the axis being as the square roots of the
natural numbers. The lines outside the sphere may be taken as a
representation of those which would be due to the earth's magnetism
if it were distributed according to the most simple type.
144 #.] We are now able to determine the distribution of
electricity on a spherical conductor under the action of electric
forces whose potential is given.
By the methods already given we expand ^ the potential due
to the given forces, in a series of solid harmonics of positive
degree having their origin at the centre of the sphere.
Let AnrnTn be one of these, then since within the conducting
sphere the potential is uniform, there must be a term —AnrnYn
arising from the distribution of electricity on the surface of the
sphere, and therefore in the expansion of 4^0- there must be a term
In this way we can determine the coefficients of the harmonics of
all orders except zero in the expression for the surface density.
The coefficient corresponding to order zero depends on the charge,
e, of the sphere, and is given by 47rcr0 = a~2e.
The potential of the sphere is
144 #.] Let us next suppose that the sphere is placed in the
neighbourhood of conductors connected with the earth, and that
202 SFHEKICAL HAKMONICS.
Green's Function, G, has been determined in terms of x, y, z and
#', y, /, the coordinates of any two points in the region in which
the sphere is placed.
If the surface density on the sphere is expressed in a series
of spherical harmonics, then the electrical phenomena outside the
sphere, arising1 from this charge on the sphere, are identical with
those arising from an imaginary series of singular points all
at the centre of the sphere, the first of which is a single point
having a charge equal to that of the sphere and the others are
multiple points of different orders corresponding to the harmonics
which express the surface density.
Let Green's function be denoted by Gpt/, where p indicates the
point whose coordinates are x, y, #, and pf the point whose co
ordinates are #', y', /.
If a charge AQ is placed at the point p', then, considering
x ', y\ z as constants, Gpp> becomes a function of x, y, z and the
potential arising from the electricity induced an surrounding bodies
by 4, is * = A0Gpp,. (1)
If, instead of placing the charge A0 at the point y, it were
distributed uniformly over a sphere of radius a having its centre
at y, the value of ^ at points outside the sphere would be the
same.
If the charge on the sphere is not uniformly distributed, let
its surface density be expressed,, as it always can, in a series of
spherical harmonics, thus
47T«2n- = J0 + 3Jiri-f&C. + (2«+ l)AnYn. (2)
The potential arising from any term of this distribution, say
7n, (3)
will be -^-AnYn for points inside the sphere, and -—^ AnYn for
points outside the sphere.
Now the latter expression, by equations (13), (14), Art. 129, is
equal to , ,. . £_ dn I.
(~L) *nn\dhr..dhnr'
or the potential outside the sphere, due to the charge on the
surface of the sphere, is equivalent to that due to a certain
multiple point whose axes are Ji^..Jin and whose moment is
Anan.
Hence the distribution of electricity on the surrounding con
ductors and the potential due to this distribution is the same as
that which would be due to such a multiple point.
144 &•] GREEN'S FUNCTION. 203
The potential, therefore, at the point p, or (#, y, z\ due to the
induced electrification of surrounding- bodies, is
flU J'll
*« = A"^. d'!h...d'k,G' W
where the accent over the d's indicates that the differentiations are
to be performed with respect to x', yf, z' . These coordinates are
afterwards to be made equal to those of the centre of the sphere.
It is convenient to suppose Tn broken up into its 2n+l con
stituents of the symmetrical system. Let A(^ Y^ be one of these,
then d" iy<'>
rf'V.,rf'A. "
It is unnecessary here to supply the affix s or c9 which indicates
whether sino-0 or coso-0 occurs in the harmonic
We may now write the complete expression for $>,
(6)
But within the sphere the potential is constant, or
= constant- 7
Now perform on this expression the operation D^\ where the
differentiations are to be with respect to x y, z, and the values
of % and o-j are independent of those of n and o-. All the terms of
(7) will disappear except that in Y^\ and we find
_ 2 fa+Q-iVK-^)! 1 M
22<ri»1! ani+l «i
= A<1)G + 2S[<^^X^]. (8)
We thus obtain a set of equations, the first member of each of
which contains one of the coefficients which we wish to determine.
The first term of the second member contains A0, the charge of
the sphere, and we may regard this as the principal term.
Neglecting, for the present, the other terms, we obtain as a
first approximation
If the shortest distance from the centre of the sphere to the
nearest of the surrounding conductors is denoted by d,
204 SPHERICAL HARMONICS. [145 a.
If, therefore, I is large compared with a, the radius of the sphere,
the coefficients of the other spherical harmonics are very small
compared with AQ . The terms after the first on the right-hand
side of equation (8) will therefore be of an order of magnitude
a 2n + n1+l
similar to i-j\
We may therefore neglect them in a first approximation, and in
a second approximation we may insert in these terms the values
of the coefficients obtained by the first approximation, and so on
till we arrive at the degree of approximation required.
Distribution of electricity on a nearly spherical conductor.
145 a.] Let the equation of the surface of the conductor be
r = a(l+F), (1)
where F is a function of the direction of r, that is to say of 0 and $,
and is a quantity the square of which may be neglected in this
investigation.
Let F be expanded in the form of a series of surface harmonics
F=/0+/iri+/272 + &C.+/,1I';, (2)
Of these terms, the first depends on the excess of the mean
radius above a. If therefore we assume that a is the mean radius,
that is to say, approximately the radius of a sphere whose volume
is equal to that of the given conductor, the coefficient /0 will
disappear.
The second term, that in /x , depends on the distance of the
centre of mass of the conductor, supposed of uniform density, from
the origin. If therefore we take that centre for origin, the
coefficient /i will also disappear.
We shall begin by supposing that the conductor has a charge J0,
and that no external electrical force acts on it. The potential
outside the conductor must therefore be of the form
'~+&C. + A,tYa'-, (3)
where the surface harmonics are not assumed to be of the same
types as in the expansion of F.
At the surface of the conductor the potential is that of the
conductor, namely, the constant quantity a.
Hence, expanding the powers of r in terms of a and F, and
neglecting the square and higher powers of F9 we have
145 «•] NEARLY SPHERICAL CONDUCTORS. 205
(4)
Since the coefficients Alt &c. are evidently small compared with
A0, we may begin by neglecting products of these coefficients
into F.
If we then write for F in its first term its expansion in spherical
harmonics, and 'equate to zero the terms involving harmonics of
the same order, we find
« = 4>5« ' (5)
AJl^AtafiY^O, (6)
4,7.' =4, a"/.*".. (7)
It follows from these equations that the Y"s must be of the
same type as the Y's, and therefore identical with them, and that
Al = 0 and An = A0anfn.
To determine the density at any point of the surface, we have
the equation fif $Y
where v is the normal and e is the angle which the normal makes
with the radius. Since in this investigation we suppose F and its
first differential coefficients with respect to 6 and 0 to be small,
we may put cos e = 1 , so that
.- (9)
Expanding the powers of r in terms of a and Ft and neglecting
products of F into An , we find
^Yn. (10)
Expanding F in spherical harmonics and giving An its value
as already found, we obtain
Hence, if the surface differs from that of a sphere by a thin
stratum whose depth varies according to the values of a spherical
harmonic of order n, the ratio of the difference of the surface
densities at any two points to their sum will be n—1 times,
206 SPHERICAL HARMONICS. [145 I.
the ratio of the difference of the radii at the same two points to
their sum.
145 #.] If a nearly spherical conductor is acted on by external
electric forces, let the potential, U, arising from these forces be
expanded in a series of spherical harmonics of positive degree,
having their origin. at the centre of volume of the conductor
U^S. + S.r T/ + J32 r« Y2' + &c. + S, r*Yu', (12)
where the accent over Y indicates that this harmonic is not
necessarily of the same type as the harmonic of the same order
in the expansion of F.
If the conductor had been accurately spherical, the potential
arising from its surface charge at a point outside the conductor
would have been
v = A. I - A £ ti- &c. - sn ££ r.'. (is)
Let the actual potential arising from the surface charge be
^f_l_ W^ where
^+...i (14)
the harmonics with a double accent being different from those
occurring either in F or in U, and the coefficients C being small
because Fis small.
The condition to be fulfilled is that, when r = a(l+F),
= constant = AQ -
a
the potential of the conductor.
Expanding the powers of r in terms of a and Ft and retaining
the first power of F when it is multiplied by A or B, but neglecting
it when it is multiplied by the small quantity C, we find
7=0. (15)
To determine the coefiicients C, we must perform the multipli
cation indicated in the first term, and express the result in a series
of spherical harmonics. This series, with the signs reversed, will be
the series for W at the surface of the conductor.
The product of two spherical harmonics of orders n and m, is
a rational function of degree n + m in x/r, y/r, and z/r, and can
therefore be expanded in a series of spherical harmonics of orders
not exceeding m+n. If, therefore, F can be expanded in spherical
145 c-] NEARLY SPHERICAL VESSEL. 207
harmonics of orders not exceeding m, and if the potential due to
external forces can be expanded in spherical harmonics of orders
not exceeding n, the potential arising from the surface charge will
involve spherical harmonies of orders not exceeding m -f n.
This surface density can then be found from the potential by
the equation ,
(16)
1456?.] A nearly spherical conductor enclosed in a nearly spherical
and nearly concentric
Let the equation of the surface of the conductor be
r = a(l+F)9 (17)
where F =/> T, + &c. +tf> Y?\ (18)
Let the equation of the inner surface of the vessel be
r = 6(l + G), (,19)
where Q = ffl Yl + &c. +£> j£\ (20)
the /'s and /s being small compared with unity, and r(<7) being
the surface harmonic of order n and type cr.
Let the potential of the conductor be a, and that of the vessel j3.
Let the potential at any point between the conductor and the
vessel be expanded in spherical harmonics, thus
*£>, (21)
then we have to determine the constants of the forms k and k so
that when r = a (1 +F), V = a, and when r = I ( 1 + G) , * = ft.
It is manifest, from our former investigation, that all the /&'s
and /fc's except hQ and £0 will be small quantities, the products of
which into jPmay be neglected. We may, therefore, write
' « = ^ + ^oi(l-^)+&e.+ (4'V + ^-lI)rr! (22)
ft = A0 + (l-G) + &c.+ (&" + ?l)Y^. (23)
We have therefore 1
> (24)
-, (25)
208 SPHERICAL HARMONICS. [146.
whence we find for the charge of the inner conductor
(27)
and for the coefficients of the harmonics of order n
(30)
X /
where we must remember that the coefficients /"„, ^n, /&„, ^n are
those belonging to the same type as well as order.
The surface density on the inner conductor is given by the
equation
where A—
146.] As an example of the application of zonal harmonics,
let us investigate the equilibrium of electricity on two spherical
conductors.
Let a and b be the radii of the spheres, and c the distance
between their centres. We shall also, for the sake of brevity,
write a = ex, and I = cy> so that x and y are numerical quantities
less than unity.
Let the line joining the centres of the spheres be taken as
the axis of the zonal harmonics, and let the pole of the zonal
harmonics belonging to either sphere be the point of that sphere
nearest to the other.
Let r be the distance of any point from the centre of the first
sphere, and s the distance of the same point from that of the second
sphere.
Let the surface density, <rlf of the first sphere be given by the
equation
4770-^2 = ^4 J1P1+34,P2 + &c. + (2^ + l)^TOPm, (1)
so that A is the total charge of the sphere, and Alt &c. are the
coefficients of the zonal harmonics Pl , &c.
TWO SPHERICAL CONDUCTORS. 209
The potential due to this distribution of charge may be repre
sented by
for points inside the sphere, and by
1a-+A2P^+&c.+AmPm^] (3)
for points outside.
Similarly, if the surface density on the second sphere is given
by the equation
n, (4)
the potential inside and outside this sphere may be represented
by equations of the form
(6)
where the general harmonics are related to the second sphere.
The charges of the sphere are A and B respectively.
The potential at every point within the first sphere is constant
and equal to a, the potential of that sphere, so that within the
first sphere Uf + F= a. (7)
Similarly, if the potential of the second sphere is /3, for points
within that sphere, U+ 7'= (3. (8)
For points outside both spheres the potential is #, where
U+7=V. (9)
On the axis, between the centres of the spheres,
r + s= c. (10)
Hence, differentiating with respect to r, and after differentiation
making r = 0, and remembering that at the pole each of the
zonal harmonics is unity, we find
1 dV
where, after differentiation, s is to be made equal to c.
VOL. i. p
210 SPHERICAL HARMONICS. [146.
If we perform the differentiations, and write a/c - x and b/c — y,
these equations become
0 =
0 =
m"'''*
0 =
By the corresponding operations for the second sphere we find;
-(13)
!
To determine the potentials, a and 0, of the two spheres we have
the equations (7) and (8), which we may now write
If, therefore, we confine our attention to the coefficients Al to Am
and Bl to .Sw, we have m + n equations from which to determine
these quantities in terms of A and .#, the charges of the two
spheres, and by inserting the values of these coefficients in (14)
and (15) we may express the potentials of the spheres in terms of
their charges.
These operations may be expressed in the form of determinants,
but for purposes of calculation it is more convenient to proceed as
follows.
Inserting in equations (12) the values of Bl...Bn from equa
tions (13), we find
(16)
TWO SPHERICAL CONDUCTORS. 211
1 -f 1 0 . ly2 + 20 .
2 + 10. 32
(18)
V5.2. (19)
By substituting in the second members of these equations the
approximate values of A1 &c., and repeating the process for further
approximations, we may cany the approximation to the coefficient
to any extent in ascending powers and products of x and y If
we write
we find
30/+ 75/+154/ + 280
+ 288/+735/ + &C.
+ 780/ -f &c.
[144 + &c.
(20)
4- 9/+ 16jf*+ 25/+ 36/+
+18/+ 40/+ 75/4-126/4-
+30/+ 80/+175/ + 336/4-&C
212 SPHERICAL HARMONICS. [146.
16+' 72/ + 209/+488/ + &C.
+ a?10/[ 60+ 342/+1222/ + &C.
+ #12/ [150 + 1 050/ + &C.
64 + &C. (21)
It will be more convenient in subsequent operations to write
these coefficients in terms of a, b, and c, and to arrange the terms
according to their dimensions in c. This will make it easier to
differentiate with respect to c. We thus find
19. (22)
+ (6 a7 tf + 9 a5 65) c"10
(23)
(24)
(25)
*I46.] TWO SPHERICAL CONDUCTORS. 213
(26)
-f 525tf969 + 336«70n)c-18. (27)
)c-u
(28)
:i7. (29)
(30)
(31)
(32)
15. (33)
(34)
(35)
(36)
£8=*9<r-9. (37)
The values of the r's and *'s may be written down by exchanging'
a and b in the ^'s and jo's respectively.
If we now calculate the potentials of the two spheres in terms
of these coefficients in the form
(38)
(39)
then I, m, n are the coefficients of potential (Art. 87), and of these
a2 c~3 + &c. , (40)
~3— &c., (41)
214 SPHERICAL HARMONICS. [146.
or, expanding in terms of a, d, c,
-f a
5750*$° + 209«3^10 +
(42)
(43)
The value of / can be obtained from that of n by exchanging a
and 6.
The potential energy of the system is, by Art. 87,
W=\lA*+mAB+\nB*, (44)
and the repulsion between the two spheres is, by Art. 9 3 a,
-rtJ.* (45)
dc d
The surface density at any point of either sphere is given by
equations (l) and (4) in terms of the coefficients An and _Z?n.
CHAPTEE X.
CONFOCAL QUADRIC SURFACES*.
147.] LET the general equation of a confocal system be
where A. is a variable parameter, which we shall distinguish by a
suffix for the species of quadric, viz. we shall take Ax for the hyper-
boloids of two sheets, A2 for the hyperboloids of one sheet, and A3
for the ellipsoids. The quantities
#, A15 6, A2, c, A3
are in ascending order of magnitude. The quantity a is introduced
for the sake of symmetry, but in our results we shall always suppose
0 = 0.
If we consider the three surfaces whose parameters are A1? A2, A3,
we find, by elimination between their equations, that the value of
x2 at their point of intersection satisfies the equation
a*)(c*-a*) = (A12-«2)(A22-«2)(A32-«2). (2)
The values of y2 and z2 may be found by transposing a, &, c
symmetrically.
Differentiating this equation with respect to A1? we find
If dsl is the length of the intercept of the curve of intersection of
A2 and A3 cut off between the surfaces Ax and A1 + ^A1, then
doc dy
dz
1 d^
* This investigation is chiefly borrowed from a very interesting work, — Leqons sur
les Fonctions Inverses dcs Tramcendantea et les Surfaces Isothermes. Par G. ~
Puris, 1857.
216 CONFOCAL QUADRIC SURFACES. [148.
The denominator of this fraction is the product of the squares of
the semi-axes of the surface AJ .
If we put
IV = A32-A22, A2 = V-V» and Aa=V-V» (5)
and if we make a = 0, then
It is easy to see that D2 and D3 are the semi-axes of the central
section of A: which is conjugate to the diameter passing through
the given point, and that D2 is parallel to ds2, and D3 to ds3.
If we also substitute for the three parameters Al5 A2, A3 their
values in terms of three functions a, ft, y, defined by the equations
(7)
-/'
Jr.
A2\ Ifi x 2^
o ) (c — A.2 ;
^3
then ^ = -D2D3cla, ds2 = -D^dfi, ds3 --D^dy. (8)
c c c
148.] Now let V be the potential at any point a, /3, y, then the
resultant force in the direction of dsl is
CIL __
l~ " ^ = ^a^ " da ^$1
Since ^, ds2, and ^«s3 are at right angles to each other, the
surface-integral over the element of area ds2 ds3 is
<••)
Now consider the element of volume intercepted between the
surfaces a, /3, y, and a + da, fi + dfi, y + dy. There will be eight
such elements, one in each octant of space.
We have found the surface-integral of the normal component of
the force (measured inwards) for the element of surface intercepted
from the surface a by the surfaces /3 and ft -f dj3, y and y -f dy.
I49-] TRANSFORMATION OF POISSON's EQUATION. 217
The surface-integral for the corresponding element of the surface
a-f da will be
,n, . 7
-f 3 -- -dfidy + -j-^ -±- da dj3 dy
da c da2 c
since D^ is independent of a. The surface-integral for the two
opposite faces of the element of volume will be the sum of these
quantities, or
Similarly the surface-integrals for the other two pairs of faces
will be
_
and —-J-
These six faces enclose an element whose volume is
7) 2 7) 2 7) 2
i -i -i -*-'~\ J-/9 -*-''\ til
ds^ ds2 ds3 = — - — | - - da d(3 dy,
and if p is the volume-density within that element, we find by
Art. 77 that the total surface-integral of the element, together with
the quantity of electricity within it, multiplied by 4 IT is zero, or,
dividing by da dfi dy,
which is the form of Poisson's extension of Laplace's equation re
ferred to ellipsoidal coordinates.
If p = 0 the fourth term vanishes, and the equation is equivalent
to that of Laplace.
For the general discussion of this equation the reader is referred
to the work of Lame already mentioned.
149.] To determine the quantities a, /3, y, we may put them in
the form of ordinary elliptic integrals by introducing the auxiliary
angles 0, $, and \j/, where
(12)
A2 = v/c2sin20-f £2cos20, (13)
A3 = csee\jr. (14)
If we put b = kc> and /fi + Jc'2 = 1, we may call k and k' the two
complementary moduli of the confocal system, and we find
[Q do
a = . => (15)
218 CONFOCAL QUADRIC SURFACES. [l5O.
an elliptic integral of the first kind, which we may write according
to the usual notation F(k,Q).
In the same way we find
^
-0 Vl—y?;'2 COS2(/>
where F(k") is the complete function for modulus Jc,
y = = F(t)-F(t, *). (17)
JQ VI—^COS2^
Here a is represented as a function of the angle 0, which is ac
cordingly a function of the parameter \19 /3 as a function of <£ and
thence of A2, and y as a function of \j/ and thence of A3.
But these angles and parameters may be considered as functions
of a, /3, y. The properties of such inverse functions, and of those
connected with them, are explained in the treatise of M. Lame on
that subject.
It is easy to see that since the parameters are periodic functions
of the auxiliary angles, they will be periodic functions of the
quantities a, /3, y : the periods of Xl and A3 are ±F(K), and that of A2
is 2F(k').
•f y- ,.:c^- ^ <*-*"" «
Particular Solutions.
150.] If F is a linear function of a, /3, or y, the equation is
satisfied. Hence we may deduce from the equation the distribution
of electricity on any two confocal surfaces of the same family
maintained at given potentials, and the potential at any point
between them.
The Hyperboloids of Two Sheets.
When a is constant the corresponding surface is a hyperboloid
of two sheets. Let us make the sign of a the same as that of x in
the sheet under consideration. We shall thus be able to study one
of these sheets at a time.
Let alt a2 be the values of a corresponding to two single sheets,
whether of different hyperboloids or of the same one, and let ^, J£
be the potentials at which they are maintained. Then, if we make
the conditions will be satisfied at the two surfaces and throughout
the space between them. If we make 7 constant and equal to h
in the space beyond the surface alf and constant and equal to ^
150.] DISTRIBUTION OF ELECTRICITY. 219
in the space beyond the surface a2, we shall have obtained the
complete solution of this particular case.
The resultant force at any point of either sheet is
1
d*i da dsl
or *1 = £=5 < (20)
ttj — a2 JD2 JJ^
If PI be the perpendicular from the centre on the tangent plane
at any point, and Pl the product of the semi-axes of the surface,
then PlDt D^P^
Hence we find /^-^ cpl
or the force at any point of the surface is proportional to the per
pendicular from the centre on the tangent plane.
The surface-density <r may be found from the equation
47T0- = JRl. (22)
The total quantity of electricity on a segment cut off by a plane
whose equation is x — a from one sheet of the hyperboloid is
The quantity on the whole infinite sheet is therefore infinite.
The limiting forms of the surface are : — ow&J'Jv = -A #~
' 'S/Tsi- ~ — ~Z
(l) When a = F(k) the surface is the part of the plane of scz oa '" z
the positive side of the positive branch of the hyperbola whose
equation is #2
45 - ,TZ^ =
(2) When a = 0 the surface is the plane of yz.
(3) When a = — F(k) the surface is the part of the plane of xz
on the negative side of the negative branch of the same hyperbola.
The Hyperboloid of One Sheet.
By making /3 constant we obtain the equation of the hyperboloid
of one sheet. The two surfaces which form the boundaries of the
electric field must therefore belong to two different hyperboloids.
The investigation will in other respects be the same as for the
hyperboloids of two sheets, and when the difference of potentials
is given the density at any point of the surface will be proportional
to the perpendicular from the centre on the tangent plane, and
the whole quantity on the infinite sheet will be infinite.
220 CONFOCAL QUADRIC SURFACES. [150.
Limiting Forms.
(1) When £ = 0 the surface is the part of the plane of xz
between the two branches of the hyperbola whose equation is
written above, (24).
(2) When fi=F(k') the surface is the part of the plane of xy
which is on the outside of the focal ellipse whose equation is
The Ellipsoids.
For any given ellipsoid y is constant. If two ellipsoids, yx and
y2, be maintained at potentials J^ and 7£, then, for any point y in
the space between them, we have
(26)
7i-72
The surface-density at any point is
<7 == —
=5, (27)
=y»P»
where p3 is the perpendicular from the centre on the tangent plane,
and P3 is the product of the semi-axes.
The whole charge of electricity on either surface is given by
and is finite.
When y = F(k) the surface of the ellipsoid is at an infinite
distance in all directions.
If we make P2 = 0 and y2 = F(k\ we find for the quantity of
electricity on an ellipsoid maintained at potential V in an infinitely
extended field, V
-- (29)
The limiting form of the ellipsoids occurs when y = 0, in which
case the surface is the part of the plane of xy within the focal
ellipse, whose equation is written above, (25).
The surface-density on either side of the elliptic plate whose
equation is (25), and whose eccentricity is #, is
" /r^~F~
V "?"^^"
Y
and its charge is Q = c ^rv • (31)
1 (K)
151.] SURFACES OF REVOLUTION. 221
Particular Cases.
151.] If c remains finite, while 6 and therefore k is diminished
till it becomes ultimately zero, the system of surfaces becomes
transformed in the following manner : —
The real axis and one of the imaginary axes of each of the
hyperboloids of two sheets are indefinitely diminished, and the
surface ultimately coincides with two planes intersecting in the
axis of z.
The quantity a becomes identical with 6, and the equation of the
system of meridional planes to which the first system is reduced is
(sin a)2 (cosa)2'
As regards the quantity (3, if we take the definition given in
page 216 (7) we shall be led to an infinite value of the integral at
the lower limit. In order to avoid this we define ft in this
particular case as the value of the integral
:
— A22
If we now put A2 = c sin </>, ft becomes
- — > i.e. loffcott<f>.
sin 9
e$— e~P
Whence cos <£ = -^ ^ >
2
and therefore sin 6 = -5 « •
gP -f g-p
If we call the exponential quantity J (^ -f- e~P) the hyperbolic
cosine of ft, or more concisely the hypocosine of ft, or cosh ft, and if
we call \ (eP—e~P) the hyposine of ft, or sinh /3, and if in the same
way we employ functions of a similar character analogous to the
other simple trigonometrical ratios, then A2 = c sech ft, and the
equation of the system of hyperboloids of one sheet is
(sech/3)2 (tanh/3)2 ~ C*'
The quantity y is reduced to \}r, so that A3 = c cosec y, and the
equation of the system of ellipsoids is
> J • «2 /O £?\
"7 w T 77 v> = C • («OI
(sec y)2 (tan y)2
Ellipsoids of this kind, which are figures of revolution about their
conjugate axes, are called planetary ellipsoids.
222 CONFOCAL QUADRIC SURFACES. [152.
The quantity of electricity on a planetary ellipsoid maintained at
potential V in an infinite field, is
«-'*£-/ _ (37)
where c sec y is the equatorial radius, and c tan y is the polar radius.
If y = 0, the figure is a circular disk of radius c, and
(38)
(39)
152.] Second Case. Let I = c, then 7c = 1 and ¥ = 0,
a = log tan - y whence Aa = c tanha, (40)
and the equation of the hyperboloids of revolution of two sheets
becomes %* f+z*
(tanha)2 (secha)2 ~
The quantity /3 becomes reduced to <£, and each of the hyper
boloids of one sheet is reduced to a pair of planes intersecting in
the axis of x whose equation is
_J^L * = 0. (42)
(sin /3)2 (cos /3)2
This is a system of meridional planes in which /3 is the longitude.
The quantity y as defined in page 216, (7) becomes in this case
infinite at the lower limit. To avoid this let us define it as the
/•oo c(7\
value of the integral / —£ — ^ *
As 3 ~~° r« 7 ,
_ _ ri d\if ,
If we then put A3 = c sec \/r, we find y = J ^T; whence
X3 = c coth y, and the equation of the family of ellipsoids is
*2 _1!±£L. = C2 (43)
(cothy)2 r (cosechy)2 "
These ellipsoids, in which the transverse axis is the axis of revo
lution, are called ovary ellipsoids.
The quantity of electricity on an ovary ellipsoid maintained at
potential Tin an infinite field becomes in this case, by (29),
&> <«>
where c sec \J/0 is the polar radius.
If we denote the polar radius by A and the equatorial by By the
result just found becomes
1 54-] CYLINDERS AND PARABOLOIDS. 223
±1
If the equatorial radius is very small compared to the polar
radius, as in a wire with rounded ends,
AV
^ " log 2A — \og£ '
When both b and c become zero, their ratio remaining finite,
the system of surfaces becomes two systems of confocal cones, and
a system of spherical surfaces of which the radius is inversely
proportional to y.
If the ratio of b to c is zero or unity, the system of surfaces
becomes one system of meridian planes, one system of right cones
having a common axis, and a system of concentric spherical surfaces
of which the radius is inversely proportional to y. This is the
ordinary system of spherical polar coordinates.
Cylindric Surfaces.
153.] When c is infinite the surfaces are cylindric, the generating
lines being parallel to the axes of z. One system of cylinders is
hyperbolic, viz. that into which the hyperboloids of two sheets
degenerate. Since, when c is infinite, Jc is zero, and therefore 6 = a,
it follows that the equation of this system is
x2 ?/2
The other system is elliptic, and since when k = 0, (3 becomes
the equation of this system is
(cosh/3)2 + (sinh/3)2 = **' (48)
These two systems are represented in Fig. X at the end of this
volume.
Confocal Paraboloids.
154.] If in the general equations we transfer the origin of co
ordinates to a point on the axis of x distant t from the centre of
the system, and if for x, A, b, and c we substitute l + tc, t + \, t + b,
and t + c respectively, and then make t increase indefinitely, we
obtain, in the limit, the equation of a system of paraboloids whose
foci are at the points x — b and x = c, viz. the equation is
224 CONFOCAL QUADRIC SURFACES. [154.
If the variable parameter is A for the first system of elliptic
paraboloids, jut for the hyperbolic paraboloids, and v for the second
system of elliptic paraboloids, we have A, b, p, c, v in ascending-
order of magnitude, and
(50)
c-d J
In order to avoid infinite values in the integrals (7) the cor
responding integrals in the paraboloidal system are taken between
different limits.
We write in this case
From these we find
\ = ic-f £) — %(c-b)cosha,\
i-OOBA (51)
(52)
— ^) (cosh y - cos /3- cosh a),
= 2 (c— 5) sinh - sin - cosh - >
22 2
z = 2 (c — ^) cosh | cos ^ sinh |
When 5 = c we have the case of paraboloids of revolution about
the axis of x, and $ = # (e2a—
The surfaces for which ft is constant are planes through the axis,
ft being the angle which such a plane makes with a fixed plane
through the axis.
The surfaces for which a is constant are confocal paraboloids.
When a= — oo the paraboloid is reduced to a straight line terminat
ing at the origin.
I 54-] CYLINDERS AND PARABOLOIDS. 225
We may also find the values of a, /3, y in terms of r, 6, and $,
the spherical polar coordinates referred to the focus as orgin, and
the axis of the parabolas as axis of the sphere,
a = log (fk cos
y = log (r^ sin J0).
We may compare the case in which the potential is equal to a,
with the zonal solid harmonic ri $;. Both satisfy Laplace's equa
tion, and are homogeneous functions of x, y, z, but in the case
derived from the paraboloid there is a discontinuity at the axis, and
i has a value not differing by any finite quantity from zero.
The surface-density on an electrified paraboloid in an infinite
field (including the case of a straight line infinite in one direction)
is inversely as the square root of the distance from the focus, or,
in the case of the line, from the extremity of the line.
VOL. I.
CHAPTER XI.
THEORY OF ELECTRIC IMAGES AND ELECTRIC INVERSION.
155.] WE have already shewn that when a conducting sphere
is under the influence of a known distribution of electricity, the
distribution of electricity on the surface of the sphere can be
determined by the method of spherical harmonics.
For this purpose we require to expand the potential of the in
fluencing- system in a series of solid harmonics of positive degree,
having the centre of the sphere as origin, and we then find a
corresponding series of solid harmonics of negative degree, which
express the potential due to the electrification of the sphere.
By the use of this very powerful method of analysis, Poisson
determined the electrification of a sphere under the influence of
a given electrical system, and he also solved the more difficult
problem to determine the distribution of electricity on two con
ducting spheres in presence of each other. These investigations
have been pursued at great length by Plana and others, who have
confirmed the accuracy of Poisson.
In applying this method to the most elementary case of a sphere
under the influence of a single electrified point, we require to expand
the potential due to the electrified point in a series of solid har
monics, and to determine a second series of solid harmonics which
express the potential, due to the electrification of the sphere, in the
space outside.
It does not appear that any of these mathematicians observed
that this second series expresses the potential due to an imaginary
electrified point, which has no physical existence as an electrified
point, but which may be called an electrical image, because the
action of the surface on external points is the same as that which
would be produced by the imaginary electrified point if the spherical
surface were removed.
156.] ELECTRIC IMAGES. 227
This discovery seems to have been reserved for Sir W. Thomson,
who has developed it into a method of great power for the solution
of electrical problems, and at the same time capable of being pre
sented in an elementary geometrical form.
His original investigations, which are contained in the Cambridge
and Dublin Mathematical Journal, 1848, are expressed in terms of
the ordinary theory of attraction at a distance, and make no use of
the method of potentials and of the general theorems of Chapter IV;
though they were probably discovered by these methods. Instead,
however, of following the method of the author, I shall make free
use of the idea of the potential and of equipotential surfaces, when
ever the investigation can be rendered more intelligible by such
means.
Theory of Electric Images.
156.] Let A and B, Figure 7, represent two points in a uniform
dielectric medium of infinite extent.
Let the charges of A and B be el
and £2 respectively. Let P be any
point in space whose distances from
A and B are r± and r2 respectively.
Then the value of the potential at P
will be Y_e_\ + %m /.M
^ ^ Fig. 7.
The equipotential surfaces due to
this distribution of electricity are represented in Fig. I (at the end
of this volume) when e1 and e2 are of the same sign, and in Fig. II
when they are of opposite signs. We have now to consider that
surface for which V •= 0, which is the only spherical surface in
the system. When e1 and e2 are of the same sign, this surface is
entirely at an infinite distance, but when they are of opposite signs
there is a plane or spherical surface at a finite distance for which
the potential is zero.
The equation of this surface is
e, e0
71+72 = (>- (2)
Its centre is at a point C in AB produced, such that
and the radius of the sphere is
/) 0
A -D el 62
AB^?'
The two points A and B are inverse points with respect to this
228 ELECTRIC IMAGES.
sphere, that is to say, they lie in the same radius, and the radius is
a mean proportional between their distances from the centre.
Since this spherical surface is at potential zero, if we suppose
it constructed of thin metal and connected with the earth, there
will be no alteration of the potential at any point either outside or
inside, but the electrical action everywhere will remain that due to
the two electrified points A and B.
If we now keep the metallic shell in connection with the earth
and remove the point B, the potential within the sphere will become
everywhere zero, but outside it will remain the same as before.
For the surface of the sphere still remains at the same potential,
and no change has been made in the exterior electrification.
Hence, if an electrified point A be placed outside a spherical
conductor which is at potential zero, the electrical action at all
points outside the sphere will be that due to the point A together
with another point B within the sphere, which we may call the .
electrical image of A.
In the same way we may shew that if B is a point placed inside
the spherical shell, the electrical action within the sphere is that
due to B, together with its image A.
157.] Definition of an Electrical Image. An electrical image is
an electrified point or system of points on one side of a surface
which would produce on the other side of that surface the same
electrical action which the actual electrification of that surface
really does produce.
In Optics a point or system of points on one side of a mirror
or lens which if it existed would emit the system of rays which
actually exists on the other side of the mirror or lens, is called a
virtual image.
Electrical images correspond to virtual images in Optics in being
related to the space on the other side of the surface. They do not
correspond to them in actual position, or in the merely approximate
character of optical foci.
There are no real electrical images, that is, imaginary electrified
points which would produce, in the region on the same side of the
electrified surface, an effect equivalent to that of the electrified surface.
For if the potential in any region of space is equal to that due
to a certain electrification in the same region it must be actually
produced by that electrification. In fact, the electrification at any
point may be found from the potential near that point by the
application of Poisson's equation.
1 5 7-] INVERSE POINTS. 229
Let a be the radius of the sphere.
Let/" be the distance of the electrified point A from the centre C.
Let e be the charge of this point.
Then the image of the point is at B, on the same radius of the
sphere at a distance — , and the charge of the image is —e — • ". " —'
J «/
We have shewn that this image
will produce the same effect on the
opposite side of the surface as the
actual electrification of the surface
does. We shall next determine the
surface-density of this electrification
at any point P of the spherical sur
face, and for this purpose we shall
make use of the theorem of Coulomb,
Art. 80, that if R is the resultant force at the surface of a con
ductor, and o- the superficial density,
R = 47T0-,
R being measured away from the surface.
We may consider R as the resultant of two forces, a repulsion
— — acting along AP, and an attraction e —, -^j^ acting along PB.
Resolving these forces in the directions of AC and CP, we find
that the components of the repulsion are
Z>-/* £/T
along AC, and -T-^T along CP.
AP3
Those of the attraction are
a I
x_ s>
BC along AC, and —e— -^j along CP.
/~D t)*3 ^ »»£j *- -J?
S
BP = ~ AP, and B(
the attraction may be written
Now BP = ^ AP, and BC = — , so that the components of
-ef-j^ along AC, and -« £ ~ along CP.
The components of the attraction and the repulsion in the
direction of AC are equal and opposite, and therefore the resultant
force is entirely in the direction of the radius CP. This only
confirms what we have already proved,, that the sphere is an equi-
potential surface, and therefore a surface to which the resultant
force is everywhere perpendicular.
230 ELECTRIC IMAGES. [158.
The resultant force measured along- CP, the normal to the surface
in the direction towards the side on which A is placed, is
(»)
If A is taken inside the sphere f is less than a, and we must
measure R inwards. For this case therefore
*__,-!=£:». (4)
a AP3
In all cases we may write
T? AD. Ad I
R = ~~CP~~AP^'
where AD, Ad are the segments of any line through A cutting the
sphere, and their product is to be taken positive in all cases.
158.] From this it follows, by Coulomb's theorem, Art. 80, that
the surface-density at P is
AD. Ad 1 . .
'
The density of the electricity at any point of the sphere varies
inversely as the cube of its distance from the point A.
The effect of this superficial distribution, together with that of
the point A, is to produce on the same side of the surface as the
point A a potential equivalent to that due to e at A, and its image
— e j at B, and on the other side of the surface the potential is
e/
everywhere zero. Hence the effect of the superficial distribution
by itself is to produce a potential on the side of A equivalent to
that due to the image — e ^ at B, and on the opposite side a
*J
potential equal and opposite to that of e at A.
The whole charge on the surface of the sphere is evidently — e-
since it is equivalent to the image at B.
We have therefore arrived at the following theorems on the
action of a distribution of electricity on a spherical surface, the
surface-density being inversely as the cube of the distance from
ti point A either without or within the sphere.
Let the density be given by the equation
where C is some constant quantity, then by equation (6)
AD. Ad
1 59-] DISTRIBUTION OF ELECTRICITY. 231
The action of this superficial distribution on any point separated
from A by the surface is equal to that of a quantity of electricity
— *> or liraC
AD. Ad
concentrated at A.
Its action on any point on the same side of the surface with A is
equal to that of a quantity of electricity
fAD.Ad
concentrated at B the image of A.
The whole quantity of electricity on the sphere is equal to the
first of these quantities if A is within the sphere, and to the second
if A is without the sphere.
These propositions were established by Sir W. Thomson in his
original geometrical investigations with reference to the distribution
of electricity on spherical conductors, to which the student ought
to refer.
159.] If a system in which the distribution of electricity is
known is placed in the neighbourhood of a conducting sphere of
radius a, which Js maintainedjit ^potential zero by connection with^
-.i^jlji££fe^nen the electrifications due to the several parts of the
system will be superposed.
Let Al} A.^ &c. be the electrified points of the system, fltf2, &c.
their distances from the centre of the sphere, elt e2, &c. their
charges, then the images B^ B2, &c. of these points will be in the
o Q
same radii as the points themselves, and at distances ~ > ~ , &c.
/I /2
from the centre of the sphere, and their charges will be
a a
- f> , f> Xrr>
el f > — 62 f » KC-
•/I /2
The potential on the outside of the sphere due to the superficial
electrification will be the same as that which would be produced by
the system of images B^B.^ &c. This system is therefore called
the electrical image of the system A1} A2t &c.
If the sphere instead of being at potential zero is at potential F,
we must superpose a distribution of electricity on its outer surface
having the uniform surface-density
7
(T = — — •
The effect of this at all points outside the sphere will be equal to
* ~~t,J^ /? I 1 1 * J Lb
T/*- ^Y J ^VCC/tt**-^ ^v t*£-&jG~~" _~ V77~^ &
fot^-t* ^<i^ 4s/ *-^ ij £ **-/ A 4.' &~ -r
232 ELECTRIC IMAGES. [l6o.
that of a quantity Va of electricity placed at its centre, and at
all points inside the sphere the potential will be simply increased
by V.
The whole charge on the sphere due to an external system of
influencing points, Alt A2) &c. is
E=Fa-ei~-e"-&C., (9)
Jl J-2
from which either the charge E or the potential V may be cal
culated when the other is given.
When the electrified system is within the spherical surface the
induced charge on the surface is equal and of opposite sign to the
inducing charge, as we have before proved it to be for every closed
surface, with respect to points within it.
*160.] The energy due to the mutual action between an elec
trified point e, at a distance /from the centre of the sphere greater
than a the radius, and the electrification of the spherical surface
due to the influence of the electrified point and the charge of the
sphere, is
^ Ee 1 e*a*
M=T~ * />(/•-*)' (IO)
where V is the potential, and E the charge of the sphere.
The repulsion between the electrified point and the sphere is
therefore, by Art. 92,
* |The discussion in the text will perhaps be more easily understood if the problem
be regarded as an example of Art. s§fi. Let us then suppose that what is described
as an electrified point is really a small spherical conductor, the radius of which is ?»
and the potential v. We have thus a particular case of the problem of two spheres of
which one solution has already been given in Art. 146, and another will be given in
Art. 173. In the case before us however the radius 6 is so small that we may
consider the electricity of the small conductor to be uniformly distributed over its
surface and all the electric images except the first image of the small conductor to
be disregarded.
We thus have F = - + ,
f ea e
-r -/>-«« + F- .
The energy of the system is therefore, Art. 85,
2a / 2^6 /*(/*_«»)>
By means of the above equations we may also express the energy in terms of the
potentials : to the same order of approximation it is
l6o.] IMAGE OF AN ELECTRIFIED SYSTEM. 233
>• ^ ""
Hence the force between the point and the sphere is always an
attraction in the following cases —
(1) *When the sphere is uninsulated.
(2) When the sphere has no charge.
(3) When the electrified point is very near the surface.
In order that the force may be repulsive, the potential of the
/3
sphere must be positive and greater than e -r— ^ — 2x2 » an^ ^e
\«/ /
charge of the sphere must be of the same sign as e and greater
.
At the point of equilibrium the equilibrium is unstable, the force
being an attraction when the bodies are nearer and a repulsion
when they are farther off.
When the electrified point is within the spherical surface the
force on the electrified point is always away from the centre of
the sphere, and is equal to
The surface-density at the point of the sphere nearest to the
electrified point where it lies outside the sphere is
The surface-density at the point of the sphere farthest from the
electrified point is
When E) the charge of the sphere, lies between
the electrification will be negative next the electrified point and
234
ELECTRIC IMAGES.
[161.
positive on the opposite side. There will be a circular line of division
between the positively and the negatively electrified parts of the
surface, and this line will be a line of equilibrium.
If ,„„___.-, (14)
the equipotential surface which cuts the sphere in the line of equi
librium is a sphere whose centre is the electrified point and whose
radius is v/'2 — a2.
The lines of force and equipotential surfaces belonging to a case
of this kind are given in Figure IV at the end of this volume.
Images in an Infinite Plane Conducting Surface.
161.] If the two electrified points A and B in Art. 156 are
electrified with equal charges of electricity of opposite signs, the
surfaces of zero potential will be the plane, every point of which is
equidistant from A and B.
Hence, if A be an electrified point whose charge is <?, and AD
a perpendicular on the plane, produce AD
to B so that DB = AB, and place at B
a charge equal to — e, then this charge
at B will be the image of A, and will
produce at all points on the same side of
the plane as A, an effect equal to that
of the actual electrification of the plane.
For the potential on the side of A due
to A and B fulfils the conditions that
y277"= 0 everywhere except at A, and that
V — 0 at the plane, and there is only one
form of V which can fulfil these conditions.
To determine the resultant force at the point P of the plane, we
observe that it is compounded of two forces each equal to -j^ ,
one acting along AP and the other along PB. Hence the resultant
of these forces is in a direction parallel to AB and equal to
e AB
Hence JR, the resultant force measured from the surface towards
the space in which A lies, is
R- -*-±™, (15)
•«- Ap* \ I
1 62.] IMAGES IN AN INFINITE PLANE,
and the density at the point P is
eAD
(7 = —
235
(16)
On Electrical Inversion.
162.] The method of electrical images leads directly to a method
of transformation by which we may derive from any electrical
problem of which we know the solution any number of other
problems with their solutions.
We have seen that the image of a point at a distance r from the
centre of a sphere of radius R, is in the same radius and at a distance
r' such that rr = JR2. Hence the image of a system of points, lines,
or surfaces is obtained from the original system by the method
known in pure geometry as the method of inversion, and described
by Chasles, Salmon, and other mathematicians.
If A and £ are two points, A' and B' their images., 0 being the
centre of inversion, and R the radius of the
sphere of inversion,
OA.OA'=R*= OB. OB'.
Hence the triangles OAB, OB' A' are similar,
and AB : A'B' : : OA : OB' ::OA.OB: R\
If a quantity of electricity e be placed at A,
its potential at B will be V =
AB
If e' be placed at A' its potential at B' will be
r = 4-.
~ A'B'
In the theory of electrical images
e:e'::OA:R::R: OA'.
Hence 7 ': V : : E : OB, (17)
or the potential at B due to the electricity at A is to the potential
at the image of B due to the electrical image of A as R is to OB.
Since this ratio depends only on OB and not on OA, the potential
at B due to any system of electrified bodies is to that at B' due
to the image of the system as R is to OB.
If r be the distance of any point A from the centre, and / that
of its image A', and if e be the electrification of A, and J that of A',
also if L, S, K be linear, superficial, and solid elements at A, and
L', S', K' their images at A', and A, a, p, A', </, p' the corresponding
line surface and volume densities of electricity at the two points,
5-
, ' ic
: ft- :
/W'J
236 ELECTRIC IMAGES. [163.
V the potential at A due to the original system, and V the potential
at A' due to the inverse system, then
/ _ L' _ R2 _ r'2 S' _ R± _ /* K' _ RG _ r^ . ^
r~~ L ~'= r* ~ fit* ~S"""^~1^J ~K~~^~r«~~R*':
e' R / X' r R
% e r R A. It r
sm*m»t £.4.*,
7' r R
T = # =7"
f I If in the original system a certain surface is that of a conductor,
' and has therefore a constant potential P, then in the transformed
R
system the image of the surface will have a potential P — . But
by placing at 0, the centre of inversion, a quantity of electricity
equal to — PR, the potential of the transformed surface is reduced
to zero.
Hence, if we know the distribution of electricity on a conductor
when insulated in open space and charged to the potential P, we
can find by inversion the distribution on a conductor whose form is
the image of the first under the influence of an electrified point with
a charge — PR placed at the centre of inversion, the conductor
being in connexion with the earth.
163.] The following geometrical theorems are useful in studying
cases of inversion.
Every sphere becomes, when inverted, another sphere, unless
it passes through the centre of inversion, in which case it becomes
a plane.
If the distances of the centres of the spheres from the centre of
inversion are a and </, and if their radii are a and a', and if we
define the power of a sphere with respect to the centre of in
version to be the product of the segments cut off by the sphere
from a line through the centre of inversion, then the power of the
first sphere is a2 - a2, and that of the second is a"2— a"2. We
have in this case
CL CL & — CL -tt
or the ratio of the distances of the centres of the first and second
spheres is equal to the ratio of their radii, and to the ratio of the
* See Thomson and Tait's Natural Philosophy, § 515,
164.] GEOMETRICAL THEOREMS. 237
power of the sphere of inversion to the power of the first sphere,
or of the power of the second sphere to the power of the sphere
of inversion.
The image of the centre of inversion with regard to one sphere
is the inverse point of the centre of the other sphere.
In the case in which the inverse surfaces are a plane and a
sphere, the perpendicular from the centre of inversion on the plane
is to the radius of inversion as this radius is to the diameter of
the sphere, and the sphere has its centre on this perpendicular and
passes through the centre of inversion.
Every circle is inverted into another circle unless it passes
through the centre of inversion, in which case it becomes a straight
line.
The angle between two surfaces, or two lines at their intersec
tion, is not changed by inversion.
Every circle which passes through a point, and the image of that
point with respect to a sphere, cuts the sphere at right angles.
Hence, any circle which passes through a point and cuts the
sphere at right angles passes through the image of the point.
164.] We may apply the method of inversion to deduce the
distribution of electricity on an uninsulated sphere under the in
fluence of an electrified point from the uniform distribution on
an insulated sphere not influenced by any other body.
If the electrified point be at J, take it for the centre of inversion,
and if A is at a distance f from the centre of the sphere whose
radius is a, the inverted figure will be a sphere whose radius is a'
and whose centre is distant y, where
a' f 7?2
— — f- —
a ~ f ~f*-a*
The centre of either of these spheres corresponds to the inverse
point of the other with respect to A, or if C is the centre and B the
inverse point of the first sphere, C' will be the inverse point, and J5'
the centre of the second.
Now let a quantity / of electricity be communicated to the
second sphere, and let it be uninfluenced by external forces. It-
will become uniformly distributed over the sphere with a surface-
density j
Its action at any point outside the sphere will be the same as
that of a charge er placed at 1? the centre of the sphere.
238 ELECTRIC IMAGES. [165.
At the spherical surface and within it the potential is
F'=7> <22)
a constant quantity.
Now let us invert this system. The centre If becomes in the
inverted system the inverse point B, and the charge / at B/
-n
becomes e' -^ at B, and at any point separated from B by the
J
surface the potential is that due to this charge at B.
The potential at any point P on the spherical surface, or on the
same side as B, is in the inverted system
£A
a' AP'
If we now superpose on this system a charge e at A, where
«=-£*> (23)
the potential on the spherical surface, and at all points on the same
side as B, will be reduced to zero. At all points on the same side
as A the potential will be that due to a charge e at A, and a charge
,*«*
But /*,= -,*.--.*, (24)
as we found before for the charge of the image at B.
To find the density at any point of the first sphere we have
a = cr
iV ^
Substituting for the value of </ in terms of the quantities be
longing to the first sphere, we find the same value as in Art. 158,
/ /><> o\
(26)
On Finite Systems of Successive Images.
165.] If two conducting planes intersect at an angle which is a
submultiple of two right angles, there will be a finite system of
images which will completely determine the electrification.
For let AOB be a section of the two conducting planes per
pendicular to their line of intersection, and let the angle of inter
section AOB = -, let P be an electrified point, and let PO = r,
and POB = 6. Then, if we draw a circle with centre 0 and radius
165.] SYSTEMS OF IMAGES. 239
OP, and find points which are the successive images of P in the
two planes beginning- with OS, we shall find Qi for the image of
P in OB, P2 for the image of Ql in OA, Q3 for that of P2 in OB,
P3 for that of Q3 in OA, and Q2 for that of P3 in OB.
If we had begun with the image of P in AO we should have
found the same points in the reverse order Q2, P3, Q3, P2, Q19
provided AOB is a submultiple of two right angles.
For the alternate images P^
at angular intervals equal to
2 AOB, and the intermediate
images QI9 Q2, Q3 are at inter
vals of the same magnitude.
Hence, if 2 AOB is a submultiple
of 2 IT j there will be a finite
number of images, and none of
these will fall within the angle
AOB. If, however, AOB is not
a submultiple of TT, it will be
impossible to represent the
, , , , -~ , . ,,
actual electrification as the re
sult of a finite series of electrified points.
are ranged round the circle
Fig. 10.
If AOB— -, there will be n negative images Q1} Q2, &c., each
ft
equal and of opposite sign to P, and n—\ positive images P2,
P3, &c., each equal to P, and of the same sign.
The angle between successive images of the same sign is — •
If we consider either of the conducting planes as a plane of sym
metry, we shall find the positive and negative images placed
symmetrically with regard to that plane, so that for every positive
image there is a negative image in the same normal, and at an
equal distance on the opposite side of the plane.
If we now invert this system with respect to any point, the two
planes become two spheres, or a sphere and a plane intersecting
at an angle - , the influencing point P being within this angle.
The successive images lie on the circle which passes through P
and intersects both spheres at right angles.
To find the position of the images we may make use of the
principle that a point and its image are in the same radius of
the sphere, and draw successive chords of the circle beginning at
P and passing through the centres of the two spheres alternately.
240 ELECTRIC IMAGES. [l66.
To find the charge which must be attributed to each image, take
any point in the circle of intersection, then the charge of each
image is proportional to its distance from this point, and its sign
is positive or negative according as it belongs to the first or the
second system.
166.] We have thus found the distribution of the images when
any space bounded by a conductor consisting of two spherical surfaces
meeting at an angle - , and kept at potential zero, is influenced by
ft
an electrified point.
We may by inversion deduce the case of a conductor consisting
of two spherical segments meeting at a re-entering angle - , charged
to potential unity and placed in free space.
For this purpose we invert the system with respect to P. The
circle on which the images formerly lay now becomes a straight
line through the centres of the spheres.
If the figure (ll) represents
a section through the line of
centres AS, and if D, D' are the
points where the circle of in
tersection cuts the plane of the
paper, then, to find the suc
cessive images, draw DA a
radius of the first circle, and
draw DC, D3, &c., making
Fig- 11- angles-, — , &c. with DA.
*=> n n
The points C, £, &c. at which they cut the line of centres will
be the positions of the positive images, and the charge of each
will be represented by its distances from D. The last of these
images will be at the centre of the second circle.
To find the negative images draw DP, DQ, &c., making angles
-, — , &c. with the line of centres. The intersections of these
lines'with the line of centres will give the positions of the negative
images, and the charge of each will be represented by its distance
from D.
The surface-density at any point of either sphere is the sum
of the surface-densities due to the system of images. For instance,
the surface-density at any point S of the sphere whose centre
A, is
167.] TWO INTERSECTING SPHERES. 241
7~) 7?
where ^, B, C, &c. are the positive series of images.
When S is on the circle of intersection the density is zero.
To find the total charge on each of the spherical segments, we
may find the surface-integral of the induction through that segment
due to each of the images.
The total charge on the segment whose centre is A due to the
image at A whose charge is DA is
where 0 is the centre of the circle of intersection.
In the same way the charge on the same segment due to the
image at B is J (DB+ OB), and so on, lines such as OB measured
from 0 to the left being reckoned negative.
Hence the total charge on the segment whose centre is A is
0(7+ &c.),
167.] The method of electrical images may be applied to any
space bounded by plane or spherical surfaces all of which cut one
another in angles which are submultiples of two right angles.
In order that such a system of spherical surfaces may exist, every
solid angle of the figure must be trihedral, and two of its angles
must be right angles, and the third either a right angle or a
submultiple of two right angles.
Hence the cases in which the number of images is finite are —
(1) A single spherical surface or a plane.
(2) Two planes, a sphere and a plane, or two spheres intersecting
at an angle - •
(3) These two surfaces with a third, which may be either plane
or spherical, cutting both orthogonally.
(4) These three surfaces with a fourth cutting the first two
orthogonally and the third at an angle —, . Of these four surfaces
one at least must be spherical.
We have already examined the first and second cases. In the
first case we have a single image. In the second case we have
2n—l images arranged in two series in a circle which passes
through the influencing point and is orthogonal to both surfaces.
VOL. I. R
242
ELECTRIC IMAGES.
[168.
In the third case we have, besides these images, their images with
respect to the third surface, that is, 4^—1 images in all besides the
influencing point.
In the fourth case we first draw through the influencing point
a circle orthogonal to the first two surfaces, and determine on it
the positions and magnitudes of the n negative images and the
n—l positive images. Then through each of these 2n points,
including the influencing point, we draw a circle orthogonal to
the third and fourth surfaces, and determine on it two series of
images, ri in each series. We shall obtain in this way, besides the
influencing point, 2nn'—l positive and 2nn' negative images.
These 4 nn' points are the intersections of n circles with ft' other
circles, and these circles belong to the two systems of lines of
curvature of a cyclide.
If each of these points is charged with the proper quantity of
electricity, the surface whose potential is zero will consist of n + ri
spheres, forming two series of which the successive spheres of the
first set intersect at angles - , and those of the second set at angles
n
—, , while every sphere of the first set is orthogonal to every sphere
n' '
of the second set.
Case of Two Spheres cutting Orthogonally. See Fig. IV at the
end of this volume.
168.] Let A and B, Fig. 12, be the centres of two spheres cutting
each other orthogonally in D and
I/, and let the straight line DJ/ cut
the line of centres in C. Then C
is the image of A with respect to
the sphere B, and also the image
of B with respect to the sphere
whose centre is A. If AD = a,
BD — /3, then AB= \/a2 -f /32, and
Fig. 12. if We place at A, B, C quantities
a/3
of electricity equal to a, {3, and —
respectively, then both
spheres will be equipotential surfaces whose potential is unity.
We may therefore determine from this system the distribution of
electricity in the following cases :
1 68.] TWO SPHERES CUTTING ORTHOGONALLY. 243
(l) On the conductor PDQD' formed of the larger segments of
both spheres. Its potential is 1, and its charge is
a/3
This quantity therefore measures the capacity of such a figure
when free from the inductive action of other bodies.
The density at any point P of the sphere whose centre is A, and
the density at any point Q of the sphere whose centre is B, are
respectively
At the points of intersection, D, D', the density is zero.
If one of the spheres is very much larger than the other, the
density at the vertex of the smaller sphere is ultimately three times
that at the vertex of the larger sphere.
(2) The lens P'DQ'D' formed by the two smaller segments of
the spheres, charged with a quantity of electricity = a^ ,
Va2 + /32
and acted on by points A and J9, charged with quantities a and /3,
is also at potential unity, and the density at any point is expressed
by the same formulae.
(3) The meniscus DPD'Q' formed by the difference of the
segments charged with a quantity a, and acted on by points B
and C, charged respectively with quantities (3 and - , is also
Va2+/32
in equilibrium at potential unity.
(4) The other meniscus QDP'D* under the action of A and C.
"We may also deduce the distribution of electricity on the following
internal surfaces.
The hollow lens P'DQ'D under the influence of the internal
electrified point C at the centre of the circle DD'.
The hollow meniscus under the influence of a point at the centre
of the concave surface.
The hollow formed of the two larger segments of both spheres
under the influence of the three points A, B, C.
But, instead of working out the solutions of these cases, we shall
apply the principle of electrical images to determine the density
of the electricity induced at the point P of the external surface of
the conductor PDQD' by the action of a point at 0 charged with
unit of electricity.
R 2
244 ELECTRIC IMAGES. [l68
Let OA = a, OB = b, OP = r,
^_0=a,
Invert the system with respect to a sphere of radius unity and
centre 0.
The two spheres will remain spheres, cutting each other ortho
gonally, and having their centres in the same radii with A and B.
If we indicate by accented letters the quantities corresponding to
the inverted system,
a
1
_
~>
If, in the inverted system, the potential of the surface is unity,
then the density at the point P' is
If, in the original system, the density at P is <r, then
a- 1
o- /•"
and the potential is -. By placing at 0 a negative charge of
electricity equal to unity, the potential will become zero over the
surface, and the density at P will be
._ _ f j __ _____ • I •
This gives the distribution of electricity on one of the spherical
surfaces due to a charge placed at 0. The distribution on the
other spherical surface may be found by exchanging a and b, a and
/3, and putting q or AQ instead of p.
To find the total charge induced on the conductor by the elec
trified point at 0, let us examine the inverted system.
In the inverted system we have a charge a at A', and ft' at B',
a? 3'
and a negative charge — /( at a point C' in the line dfff,
such that A'C':C'#::a'*:p*.
If OA'= of, OB'= V, OC' = c't we find
/2 _
169.] FOUR SPHERES CUTTING ORTHOGONALLY.
Inverting this system the charges become
245
</ _a {?_ fi_
~tf~a* T==T'
f /->/ t
a' f
a/3
and
Hence the whole charge on the conductor due to a unit of
negative electricity at 0 is
a ^3 a/3
a b J
Distribution of Electricity on Three Spherical Surfaces which
Intersect at Right Angles.
169.] Let the radii of the spheres be a, /3, y, then
+a* AB = V~~
BC =
CA =
Let PQR, Fig. 1 3, be the feet
of the perpendiculars from ABC
on the opposite sides of the tri
angle, and let 0 be the inter
section of perpendiculars.
Then P is the image of B in
the sphere y, and also the image
of C in the sphere (3. Also 0 is
the image of P in the sphere a.
Let charges a, j3, and y be
placed at A, B, and C.
Then the charge to be placed
at Pis
Fig. 13.
A/i + 7
Also ^> =
sidered as the image of P, is
go tbat the ch at 0 con.
^/32y2 + y2a2 + a2/32 /I 1 1
/V -^ + ^2 + y2
In the same way we may find the system of images which are
246 ELECTRIC IMAGES. [170.
electrically equivalent to four spherical surfaces at potential unity
intersecting at right angles.
If the radius of the fourth sphere is 8, and if we make the charge
at the centre of this sphere = 8, then the charge at the intersection
of the line of centres of any two spheres, say a and /3, with their
plane of intersection, is
1
The charge at the intersection of the plane of any three centres
ABC with the perpendicular from D is
and the charge at the intersection of the four perpendiculars is
1
1 I F
¥ + 7 + a2"
System of Four Spheres Intersecting at Eight Angles under the
Action of an Electrified Point.
170.] Let the four spheres be A, B, C, D, and let the electrified
point be 0. Draw four spheres Aly B^ Clt D^ of which any one,
AL, passes through 0 and cuts three of the spheres, in this case B,
C, and D, at right angles. Draw six spheres (ab), (ac), (ad), (be),
(bd), (cd), of which each passes through 0 and through the circle
of intersection of two of the original spheres.
The three spheres B±, Clt D± will intersect in another point besides
0. Let this point be called A', and let B', C', and J/ be the
intersections of C19 D1, Al} of Di} A19 BL, and of A1, B^, C1 re
spectively. Any two of these spheres, A19 B±, will intersect one of
the six (cd) in a point (a'lf). There will be six such points.
Any one of the spheres, Alt will intersect three of the six (ab),
(ac), (ad) in a point a. There will be four such points. Finally,
the six spheres (ab), (ac), (ad), (cd), (db), (be), will intersect in one
point S.
If we now invert the system with respect to a sphere of radius
E and centre 0, the four spheres A, B, C, D will be inverted into
spheres, and the other ten spheres will become planes. Of the
points of intersection the first four A', B', C', V will become the
1 7 1.] TWO SPHERES NOT INTERSECTING. 247
centres of the spheres, and the others will correspond to the other
eleven points in the preceding article. These fifteen points form
the image of 0 in the system of four spheres.
At the point A', which is the image of 0 in the sphere A, we
must place a charge equal to the image of 0, that is, , where a
(t/
is the radius of the sphere A, and a is the distance of its centre
from 0. In the same way we must place the proper charges at
J5', <?', D'.
The charges for each of the other eleven points may be found from
the expressions in the last article by substituting a', /3', y', 6' for
a, /3, y, 5, and multiplying the result for each point by the distance
of the point from 0, where
« & ft , y «, 8
-?=rf' P^^js±jp' y~- -,:*— 1>> -3*IT»-
[The cases discussed in Arts. 169, 170 may be dealt with as
follows : Taking three coordinate planes at right angles, let us
place at the system of eight points ( + — i + — > ± — ) charges
±e, the minus charges being at the points which have 1 or 3
negative coordinates. Then it is obvious the coordinate planes are
at potential zero. Now let us invert with regard to any point and
we have the case of three spheres cutting orthogonally under the
influence of an electrified point. If we invert with regard to one of
the electrified points, we find the solution for the case of a con
ductor in the form of three spheres of radii a, (3, y cutting ortho
gonally and freely charged.
If to the above system of electrified points we superadd their
images in a sphere with its centre at the origin we see that, in
addition to the three coordinate planes, the surface of the sphere
forms also a part of the surface of zero potential.]
Two Spheres not Intersecting.
171.] When a space is bounded by two spherical surfaces which
do not intersect, the successive images of an influencing point
within this space form two infinite series, all of which lie beyond
the spherical surfaces, and therefore fulfil the condition of the
applicability of the method of electrical images.
Any two non-intersecting spheres may be inverted into two
concentric spheres by assuming as the point of inversion either
of the two common inverse points of the pair of spheres.
248
ELECTRIC IMAGES.
"We shall begin, therefore, with the case of two uninsulated
concentric spherical surfaces, subject to the induction of an elec
trified point placed between them.
Let the radius of the first be b, and that of the second be**, and
let the distance of the influencing point from the centre be r = beu.
Then all the successive images will be on the same radius as the
influencing point.
Let Q0, Fig. 14, be the image of P in the first sphere, P3 that
of $o in the second sphere, Q1 that of Pl in the first sphere, and
so on j then
and OP8.OQs_l
also OQ0 = be~u,
OPl =
Oql =
Hence OPS =
&c.
If the charge of P is denoted by P,
then
Fig. 14.
Next, let Q/ be the image of P in the second sphere, P/ that of
i in the first. &c.,
, OP/= fott-2CT,
Of these images all the P's are positive, and all the §'s negative,
all the P"s and Q's belong to the first sphere, and all the P-'S and
^''s to the second.
The images within the first sphere form a converging series, the
sum of which is
-P
This therefore is the quantity of electricity on the first or interior
sphere. The images outside the second sphere form a diverging
series, but the surface-integral of each with respect to the spherical
surface is zero. The charge of electricity on the exterior spherical
surface is therefore
— 1
~iW-P
172.]
TWO SPHERES NOT INTERSECTING.
249
If we substitute for these expressions their values in terms of
OA, OB, and OP, we find
OA PB
charge on A = —P
^
charge on .B=_P
AP
If we suppose the radii of the spheres to become infinite, the case
becomes that of a point placed between two parallel planes A and B.
In this case these expressions become
charge on A = —P -^ >
A Jj
charge on B = —P - •
Fig. 15.
172.] In order to pass from this case to that of any two spheres
not intersecting each
other, we begin by
finding the two com
mon inverse points 0,
0' through which all
circles pass that are
orthogonal to both
spheres. Then, if we
invert the system with
respect to either of
these points, the spheres
become concentric, as
in the first case.
If we take the point 0 in Fig. 1 5 as centre of inversion, this
point will be situated in Fig. 14 somewhere between the two
spherical surfaces.
Now in Art. 1 7 1 we solved the case where an electrified point is
placed between two concentric conductors at zero potential. By
inversion of that case with regard to the point 0 we shall therefore
deduce the distributions on two spherical conductors at potential
zero, exterior to one another, induced by an electrified point in their
neighbourhood. In Art. 173 it will be shewn how the results thus
obtained may be employed in finding the distributions on two
spherical charged conductors subject to their mutual influence only.
The radius OAPB in Fig. 1 4 on which the successive images lie
becomes in Fig. 1 5 an arc of a circle through 0 and (7, and the
ratio of OfP to OP is equal to Ceu where C is a numerical quantity.
250 ELECTRIC IMAGES.
, O'P . (J A . VB
If we put 0 = log^p, a==lo^o?' P = loS~OB
then (3 — a = w, ^ + a = 0.
All the successive images of P will lie on the arc OAPBO'.
The position of the image of P in A is QQ where
(70
6(Q0) = log-5j = 2a-e.
That of <90 in P is P1 where
Similarly
<>(P,) =
In the same way if the successive images of P in B, A, B, &c.
are Q0', P/, §/, &c.,
e(QQ') = 2p-e,
e(Ps') = e-2S*
To find the charge of any image P8 we observe that in the
inverted figure its charge is
7, /OP.
PA/op-
In the original figure we must multiply this by OPS. Hence the
charge of Ps in the dipolar figure is
/OPS.0'PS
'V OP.C/P'
If we make f = VOP.O'P, and call £ the parameter of the
point P, then we may write
P — AiP
'~ £ -
or the charge of any image is proportional to its parameter.
If we make use of the curvilinear coordinates 6 and </>, such that
where 2/£ is the distance 00', then
£sinh<9 ^ sin <^>
~~
cosh 6 — cos (f> ' ~ cosh ^— cos <$> '
# + y _ co =
(a? + £ coth ^)2 +/ = P cosech2 (93
173-] TWO SPHERES NOT INTERSECTING. 251
cot (b = —7 > coth 0 =
Iky
f= ^k _«.
v cosh 0 — cos e/)
Since the charge of each image is proportional to its parameter,
£, and is to be taken positively or negatively according as it is of
the form P or Q, we find
P v cosh 6 — cos
JL 0 —
A/ cosh (6 -\-2svr) —
P vcosh 6 — cos 0
A/cosh (2 a — 6 — 2 sir) — cos <p
P \/cosh 0 — cos $
A/cosh (0— 2 SOT) — cos ^
PA/cosh0— cos(/>.
Vcosh(2/3— 0 + 2st3-)-
We have now obtained the positions and charges of the two
infinite series of images. We have next to determine the total
charge on the sphere A by finding the sum of all the images within
it which are of the form Q or P'. We may write this
•^-*,S=ao 1
P A/cosh 0 — cose/) 2*-i / i fn \ ?
V cosh (0 — 2 s w) — cos e/>
, _„. ^K *\ S — QO i
— P vcosh 0 — cos d> 2*s=o ~T i / x '
vcosh(2a — 0 — 2 <m) — cose/>
In the same way the total induced charge on B is
1
P A/cosh 0 — cos 0 JLs=i / i //i » \ '
A/ cosh (0+ 25OTJ — cose/)
^ ^ •^.-j .e — rf\
— P A/cosh 0 — cos <
*s~~° A/cosh (2/3 — 0 + 25OT-) — cose/)
173.] We shall apply these results to the determination of the
* In these expressions we must remember that
2cosh0 = ee + e~e, 2sinh0 = ee-e~9,
and the other functions of 9 are derived from these by the same definitions as the
corresponding trigonometrical functions.
The method of applying dipolar coordinates to this case was given by Thomson in
Liouville's Journal for 1847. See Thomson's reprint of Electrical Papers, § 211, 212.
In the text I have made use of the investigation of Prof. Betti, Nuovo Cimento,
vol. xx, for the analytical method, but I have retained the idea of electrical images as
used by Thomson in his original investigation, Phil. Mag., 1853.
252 ELECTRIC IMAGES. [173.
coefficients of capacity and induction of two spheres whose radii are
a and I, and the distance between whose centres is c.
Let the sphere A be at potential unity, and the sphere £ at
potential zero.
Then the successive images of a charge a placed at the centre
of the sphere A will be those of the actual distribution of electricity.
All the images will lie on the axis between the poles and the
centres of the spheres, and it will be observed that of the four
systems of images determined in Art. 1 72, only the first and fourth
exist in this case.
If we put
k k
then sinh a = -- > sinh ft = T •
a o
The values of 6 and $ for the centre of the sphere A are
0 = 2a, 0 = 0.
Hence in the equations we must substitute a or — k -^— r — for P,
sinn a
2 a for 6 and 0 for $, remembering that P itself forms part of the
charge of A. We thus find for the coefficient of capacity of A
for the coefficient of induction of A on B or of B on A
^5=00 1
?«* = *Z*=l^h7^'
We may, in like manner, by supposing B at potential unity and
A at potential zero, determine the value of gbb. We shall find,
with our present notation,
To calculate these quantities in terms of a and b, the radii of the
spheres, and of c the distance between their centres, we observe
that if
we mav write
~ K
- ,
cosh/3 = --—, cosW =
1 74-] TWO ELECTRIFIED SPHERES. 253
and make use of
sinh (a + /3) = sinh a cosh /3 + cosh a sinh /3,
cosh (a -f 0) = cosh a cosh (3 + sinh a sinh (3.
By this process or by the direct calculation of the successive
images as shewn in Sir W. Thomson's paper, we find
*« = a+ A + (c*-b* + a")ll*-b*-ac) +&C"
ad
U = ~ -— -
c c ^-^-
174.] We have then the following equations to determine the
charges Ea and Eb of the two spheres when electrified to potentials
Va and 7£ respectively,
If we put qaa qbb - ^ =D = ,
then the equations to determine the potentials in terms of the
charges are Va = paa Ea +pab Eb,
aa,pab, and pbb are the coefficients of potential.
The total energy of the system is, by Art. 85,
« + 2 E,
The repulsion between the spheres is therefore, by Arts. 92, 93,
where c is the distance between the centres of the spheres.
Of these two expressions for the repulsion, the first, which
expresses it in terms of the potentials of the spheres and the
254 ELECTRIC IMAGES. [174.
variations of the coefficients of capacity and induction, is the most
convenient for calculation.
We have therefore to differentiate the q's with respect to c.
These quantities are expressed as functions of k, a, 0, and &, and
must be differentiated on the supposition that a and b are constant.
From the equations
. , . , sinhasinh/3
k = —a smna = b smh/3 = — c
dk cosh a cosh/3
we find
do sinh txr
da sinh a cosh /3
dc k sinh ur
dj3 __ cosh a sinh (3
dc k sinh t*r
dij? 1
whence we find
dqaa cosh a cosh 8 qaa -^u=oo (sc + b cosh /3) cosh (SVT — <
' k ~
6
dqab cosh a cosh ,8 qa
"^ = sinhtc- T
^65 _ cosh a cosh/8 ql>b -^s=«> (sc— ^ cosh a) cosh (ff + SCT)
"^c~ = sinh -BT 1 ^s=0 c(sinh(j3 + *tsr))2
Sir William Thomsom has calculated the force between two
spheres of equal radius separated by any distance less than the
diameter of one of them. For greater distances it is not necessary
to use more than two or three of the successive images.
The series for the differential coefficients of the #'s with respect
to c are easily obtained by direct differention.
_&
(*-&— acf
dqab _ ab
'^
c*(c2-a2-62 + ab)> (c2 - a2 -b2- ab)2
2al2c 2a2b*c(2c2-2a2-b2) _&c
(c2-a2)2 (c2 -a2 + be)2 (c2 -a2- be)2
1 75-] TWO SPHEEES IN CONTACT. 255
Distribution of Electricity on Two Spheres in Contact.
175.] If we suppose the two spheres at potential unity and not
influenced by any other point, then, if we invert the system with
respect to the point of contact, we shall have two parallel planes,
distant — and — from the point of inversion, and electrified by
the action of a unit of electricity at that point.
There will be a series of positive images, each equal to unity, at
distances s (- + r) from the origin, where * may have any integer
value from — oc to +00.
There will also be a series of negative images each equal to — 1 ,
the distances of which from the origin, reckoned in the direction of
1 A K
0, are - + s ( ~ + T ) •
a ^a b'
When this system is inverted back again into the form of the
two spheres in contact, we have a corresponding series of negative
images, the distances of which from the point of contact are of the
form — -— , where s is positive for the sphere A and negative
for the sphere B. The charge of each image, when the potential
of the spheres is unity, is numerically equal to its distance from the
point of contact, and is always negative.
There will also be a series of positive images whose distances
from the point of contact measured in the direction of the centre
of a, are of the form
WThen s is zero, or a positive integer, the image is in the
sphere A.
When s is a negative integer the image is in the sphere B.
The charge of each image is measured by its distance from the
origin and is always positive.
The total charge of the sphere A is therefore
.„ ^u=oo 1 ab
256 ELECTRIC IMAGES. [I75-
Each of these series is infinite, but if we combine them in the form
the series becomes converging.
In the same way we find for the charge of the sphere J9,
<x> db ab s=-o> 1
The expression for Ea is obviously equal to
-L-i
ab
a + b J0 1—0
in which form the result in this case was given by Poisson.
It may also be shewn (Legendre Traite des Fonctions Mliptiques,
ii, 438) that the above series for Ea is equal to
*\l*
where y = -57712..., and #(#) = — logT(l
The values of * have been tabulated by Gauss (Werket Band iii,
pp. 161-162.)
If we denote for an instant b -r- (a + b) by a?, we find for the
difference of the charges Ea and EbJ
d , . ab
= -7- log sin TT# x
f/a? « +
cot
a + b a + b
When the spheres are equal the charge of each for potential unity
«=« 1
Jj ^Z a y^.c_1 ~ 7~Z -, \ '
= flloge2 = -69314718^.
When the sphere A is very small compared with the sphere B
the charge on A is
^a = j %=r y approximately ;
or
1 77.] SPHEEICAL BOWL. 257
The charge on B is nearly the same as if A were removed, or
Eb = b.
The mean density on each sphere is found by dividing the charge
by the surface. In this way we get
_
245'
" 6
Hence, if a very small sphere is made to touch a very large one,
the mean density on the small sphere is equal to that on the large
n
sphere multiplied by — , or 1.644936.
Application of Electrical Inversion to the case of a Spherical Bowl.
176.] One of the most remarkable illustrations of the power of
Sir W. Thomson's method of Electrical Images is furnished by his
investigation of the distribution of electricity on a portion of a
spherical surface bounded by a small circle. The results of this
investigation, without proof, were communicated to M. Liouville
and published in his Journal in 1847. The complete investigation
is given in the reprint of Thomson's Electrical Papers, Article XV.
I am not aware that a solution of the problem of the distribution
of electricity on a finite portion of any curved surface has been
given by any other mathematician.
As I wish to explain the method rather than to verify the
calculation, I shall not enter at length into either the geometry
or the integration, but refer my readers to Thomson's work.
Distribution of Electricity on an Ellipsoid.
177.] It is shewn by a well-known method"* that the attraction
of a shell bounded by two similar and similarly situated and
concentric ellipsoids is such that there is no resultant attraction
on any point within the shell. If we suppose the thickness of
the shell to diminish indefinitely while its density increases, we
ultimate^ arrive at the conception of a surface- density varying
as the perpendicular from the centre on the tangent plane, and
since the resultant attraction of this superficial distribution on any
* Thomson and Tait's Natural Philosophy, § 520, or Art. 150 of this book.
VOL. I. S
258 ELECTRIC IMAGES. [178.
point within the ellipsoid is zero, electricity, if so distributed on
the surface, will be in equilibrium.
Hence, the surface- density at any point of an ellipsoid undis
turbed by external influence varies as the distance of the tangent
plane from the centre.
Distribution of Electricity on a Disk.
By making two of the axes of the ellipsoid equal, and making
the third vanish, we arrive at the case of a circular disk, and at an
expression for the surface-density at any point P of such a disk
when electrified to the potential V and left undisturbed by external
influence. If o- be the surface-density on one side of the disk,
and if KPL be a chord drawn through the point P, then
7
(T •=•
Application of the Principle of Electric Inversion.
178.] Take any point Q as the centre of inversion, and let R
be the radius of the sphere of inversion. Then the plane of the
disk becomes a spherical surface passing through Q, and the disk
itself becomes a portion of the spherical surface bounded by a circle.
We shall call this portion of the surface the bowl.
If S' is the disk electrified to potential F'and free from external
influence, then its electrical image S will be a spherical segment at
potential zero, and electrified by the influence of a quantity V'R of
electricity placed at Q.
We have therefore by the process of inversion obtained the solu
tion of the problem of the distribution of electricity on a bowl or a
plane disk when under the influence of an electrified point in the
surface of the sphere or plane produced.
Influence of an Electrified Point placed on the unoccupied part of the
Spherical Surface.
The form of the solution, as deduced by the principles already
given and by the geometry of inversion, is as follows :
If C is the central point or pole of the spherical bowl S, and
if a is the distance from C to any point in the edge of the segment,
then, if a quantity q of electricity is placed at a point Q in the
surface of the sphere produced, and if the bowl S is maintained
at potential zero, the density a- at any point P of the bowl will be
1
l8o.] SPHERICAL BOWL. 259
CQ, CP, and QP being the straight lines joining the points, C} Q,
and P.
It is remarkable that this expression is independent of the radius
of the spherical surface of which the bowl is a part. It is therefore
applicable without alteration to the case of a plane disk.
Influence of any Number of Electrified Points.
Now let us consider the sphere as divided into two parts, one of
which, the spherical segment on which we have determined the
electric distribution, we shall call the bowl, and the other the
remainder, or unoccupied part of the sphere on which the in
fluencing point Q is placed.
If any number of influencing points are placed on the remainder
of the sphere, the electricity induced by these on any point of the
bowl may be obtained by the summation of the densities induced
by each separately.
179.] Let the whole of the remaining surface of the sphere be
uniformly electrified, the surface-density being p, then the density
at any point of the bowl may be obtained by ordinary integration
over the surface thus electrified.
We shall thus obtain the solution of the case in which the bowl
is at potential zero, and electrified by the influence of the remaining
portion of the spherical surface rigidly electrified with density p.
Now let the whole system be insulated and placed within a
sphere of diameter /^ and let this sphere be uniformly and rigidly
electrified so that its surface-density is pf.
There will be no resultant force within this sphere, and therefore
the distribution of electricity on the bowl will be unaltered, but
the potential of all points within the sphere will be increased by
a quantity V where y — 2 77 pf.
Hence the potential at every point of the bowl will now be V.
Now let us suppose that this sphere is concentric with the sphere
of which the bowl forms a part, and that its radius exceeds that
of the latter sphere by an infinitely small quantity.
We have now the case of the bowl maintained at potential V and
influenced by the remainder of the sphere rigidly electrified with
superficial density p + p'.
180.] We have now only to suppose p-fp'= 0, and we get the
case of the bowl maintained at potential V and free from external
influence.
260 ELECTRIC IMAGES. [l8l.
If <r is the density on either surface of the bowl at a given point
when the bowl is at potential zero, and is influenced by the rest
of the sphere electrified to density p, then, when the bowl is main
tained at potential V, we must increase the density on the outside
of the bowl by p', the density on the supposed enveloping sphere.
The result of this investigation is that if/ is the diameter of
the sphere, a the chord of the radius of the bowl, and r the chord
of the distance of P from the pole of the bowl, then the surface-
density a on the inside of the bowl is
cr =
and the surface-density on the outside of the bowl at the same
point is y
In the calculation of this result no operation is employed more
abstruse than ordinary integration over part of a spherical surface.
To complete the theory of the electrification of a spherical bowl we
only require the geometry of the inversion of spherical surfaces.
181.] Let it be required to find the surface-density induced at
any point of the bowl by a quantity q of electricity placed at a
point Qy not now in the spherical surface produced.
Invert the bowl with respect to Q, the radius of the sphere of
inversion being R. The bowl 8 will be inverted into its image S'y
and the point P will have P' for its image. We have now to
determine the density </ at P' when the bowl S' is maintained at
potential V, such that q = V'R, and is not influenced by any
external force.
The density o- at the point P of the original bowl is then
QP*
this bowl being at potential zero, and influenced by a quantity q of
electricity placed at Q.
The result of this process is as follows :
Let the figure represent a section through the centre, 0, of the
sphere, the pole, C, of the bowl, and the influencing point Q.
D is a point which corresponds in the inverted figure to the
unoccupied pole of the rim of the bowl, and may be found by the
following construction.
Draw through Q the chords EQE' and FQF, then if we sup-
SPHERICAL BOWL.
261
pose the radius of the sphere of inversion to be a mean propor
tional between the segments into which a chord is divided at Q,
WF' will be the image of EF. Bisect
the arc F'CW in .27, so that F'D'=
ffW, and draw J/QD to meet the
sphere in D. D is the point re
quired. Also through 0, the centre
of the sphere, and Q draw HOQH'
meeting the sphere in If and H' '.
Then if P be any point in the bowl,
the surface-density at P on the side
which is separated from Q by the
completed spherical surface, induced
by a quantity q of electricity at Q, Fig. ig.
will be
'
where a denotes the chord drawn from (?, the pole of the bowl,
to the rim of the bowl.
On the side next to Q the surface-density is
q
27r HH'.PQ;
CHAPTEE XII.
THEORY OF CONJUGATE FUNCTIONS IN TWO DIMENSIONS.
182.] THE number of independent cases in which the problem
of electrical equilibrium has been solved is very small. The method
of spherical harmonics has been employed for spherical conductors,
and the methods of electrical images and of inversion are still more
powerful in the cases to which they can be applied. The case of
surfaces of the second degree is the only one, as far as I know,
in which both the equipotential surfaces and the lines of force are
known when the lines of force are not plane curves.
But there is an important class of problems in the theory of
electrical equilibrium, and in that of the conduction of currents,
in which we have to consider space of two dimensions only.
For instance, if throughout the part of the electric field under
consideration, and for a considerable distance beyond it, the surfaces
of all the conductors are generated by the motion of straight lines
parallel to the axis of z, and if the part of the field where thy*
ceases to be the case is so far from the part considered that the
electrical action of the distant part on the field may be neglected,
then the electricity will be uniformly distributed along each gene
rating line, and if we consider a part of the field bounded by two
planes perpendicular to the axis of z and at distance unity, the
potential and the distributions of electricity will be functions of x
and y only.
If pdxdy denotes the quantity of electricity in an element whose
base is dxdy and height unity, and ads the quantity on an element
of area whose base is the linear element ds and height unity, then
the equation of Poisson may be written
183.] PROBLEMS IN TWO DIMENSIONS. 263
When there is no free electricity, this is reduced to the equation
of Laplace, (py
The general problem of electric equilibrium may be stated as
follows : —
A continuous space of two dimensions, bounded by closed curves
(?1} C2, &c being given, to find the form of a function, F", such that
at these boundaries its value may be Tlt F2, &c. respectively, being
constant for each boundary, and that within this space V may be
everywhere finite, continuous, and single valued, and may satisfy
Laplace's equation.
I am not aware that any perfectly general solution of even this
question has been given, but the method of transformation given in
Art. 190 is applicable to this case, and is much more powerful than
any known method applicable to three dimensions.
The method depends on the properties of conjugate functions of
two variables.
4+fii,J
Definition of Conjugate Functions. .'. ^ - £'{*•+$
183.] Two quantities a and /3 are said to be conjugate functions
of x and y, if a -f \/ — 1 ft is a function of x + \/ — I y. d*t _ 2//* -t V '/
It follows from this definition that "^A ( f • if
*•]*
da. d/3 da d/3 .
— = -p> and — + -^ = 0; (l)
dx dy dy dx
_ _
7 9 H -- 7-9- - ^5 " 7 9' T 7 9 - V.
dx2 dy1 dx*1 dy*
Hence both functions satisfy Laplace's equation. Also
da dj3 da d(3 da
dx dy dy dx dx
«/ 3
da
dx
'<*? . . 7?2 "r (,\
Ty ~E-
If x and y are rectangular coordinates, and if ds± is the intercept
of the curve ((3 = constant) between the curves a and a -f da, and
ds2 the intercept of a between the curves /3 and j3 + d(B, then
d*± d*,^l_
da ~ d$ R
and the curves intersect at right angles.
If we suppose the potential V — F0 + /£a, where k is some con
stant, then V will satisfy Laplace's equation, and the curves (a) will
be equipotential curves. The curves (/3) will be lines of force, and
264 CONJUGATE FUNCTIONS. [184.
the surface-integral of E over unit-length of a cylindrical surface
whose projection on the plane of xy is the curve AB will be Jc(fiB /3 A
where ft A and ftB are the values of ft at the extremities of the curve.
If one series of curves corresponding to values of a in arithmetical
progression be drawn on the plane, and another series corresponding
to a series of values of ft having the same common difference, then
the two series of curves will everywhere intersect at right angles,
and, if the common difference is small enough, the elements into
which the plane is divided will be ultimately little squares, whose
sides, in different parts of the field, are in different directions and of
different magnitudes, being inversely proportional to R.
If two or more of the equipotential lines (a) are closed curves
enclosing a continuous space between them, we may take these for
the surfaces of conductors at potentials (^70 + ^ai)j (^0 + ^2)5 &c-
respectively. The quantity of electricity upon any one of these be-
Jc
tween the lines of force /3X and /32 will be — (/32 — ft).
The number of equipotential lines between two conductors will
therefore indicate their difference of potential, and the number of
lines of force which emerge from a conductor will indicate the
quantity of electricity upon it.
We must next state some of the most important theorems
relating to conjugate functions, and in proving them we may use
either the equations (l), containing the differential coefficients, or
the original definition, which makes use of imaginary symbols.
184.] THEOREM I. If x' and y' are conjugate functions with respect
to x and y> and if x" and y" are also conjugate functions with
respect to x and y, then the functions x' + x" and y' +y" will
~be conjugate functions with respect to x and y.
dx' dy' . dx" dy"
«7 or»H — - ** •
•j — 7 , cl'llU. -. — :: .
ax dy ax dy
.,, n d(x+x} d(y +y")
therefore v 7 — I = ^ /«? J .
dx dy
dx' dy' dx" dy"
Also ^— = ~- , and -^— = ~ :
dy dx dy dx
.-, „ d(x' + ^?//) d(y' + y"\
tneretore — - — ; — = — — - — - :
dy dx
or x+x" andy+y7 are conjugate with respect to x and y.
185.] GEAPHIC METHOD. 265
Graphic 'Representation of a Function which is the Sum of Two
Given Functions.
Let a function (a) of x and y be graphically represented by a
series of curves in the plane of xy, each of these curves corre
sponding to a value of a which belongs to a series of such values
increasing by a common difference, 8.
Let any other function, /3, of x and y be represented in the same
way by a series of curves corresponding to a series of values of /3
having the same common difference as those of a.
Then to represent the function a -f /3 in the same way, we must
draw a series of curves through the intersections of the two former
series, from the intersection of the curves (a) and (/3) to that of the
curves (a + 8) and (/3 — 8), then through the intersection of (a + 2 6)
and (/3 — 28), and so on. At each of these points the function will
have the same value, namely a + /3. The next curve must be drawn
through the points of intersection of (a) and (/3 + 8), of (a + 8) and
(£), of (a + 2 8) and (/3 — 8), and so on. The function belonging to
this curve will be a -f /3 -f 8.
In this way, when the series of curves (a) and the series (/3) are
drawn, the series (a -f/3) may be constructed. These three series of
curves may be drawn on separate pieces of transparent paper, and
when the first and second have been properly superposed, the third
may be drawn.
The combination of conjugate functions by addition in this way
enables us to draw figures of many interesting cases with very
little trouble when we know how to draw the simpler cases of
which they are compounded. We have, however, a far more
powerful method of transformation of solutions, depending on the
following theorem.
185.] THEOREM II. If x" and y" are conjugate functions with
respect to the variables of and y' ', and if xf and y' are conjugate
functions with respect to x and y, then x" and y" will be con
jugate functions with respect to x and y.
dx" dx" dx' dx" dy'
For -j—= -rr^r+ -j-r-r-»
dx dx dx dy dx
dy" dy dy" dx'
& «/ I «y
7/1 7 / 7 '
dy dy dx dy
266 CONJUGATE FUNCTIONS. [185.
daf' dx" daf daf' dy
and -y— = -7-7- -= — h -7-7- -j- »
r/y dx dy dy dy
_ df dyr dy" dx
dy' dx daf dx
~~ dx '
and these are the conditions that %" and /' should be conjugate
functions of x and y.
This may also be shewn from the original definition of conjugate
functions. For x"+*/~^ly" is a function of x' + V— I/, and
a/+ v/^T/ is a function of #+ </^l y. Hence, #"+//— I/'
is a function of x+ \f — \y.
In the same way we may shew that if x and / are conjugate
functions of x and y, then x and y are conjugate functions of x'
and y'.
This theorem may be interpreted graphically as follows : —
Let a?', y' be taken as rectangular coordinates, and let the curves
corresponding to values of x" and of/' taken in regular arithmetical
series be drawn on paper. A double system of curves will thus be
drawn cutting the paper into little squares. Let the paper be also
ruled with horizontal and vertical lines at equal intervals, and let
these lines be marked with the corresponding values of x' and /.
Next, let another piece of paper be taken in which x and y are
made rectangular coordinates and a double system of curves x', y
is drawn, each curve being marked with the corresponding value
of af or /. This system of curvilinear coordinates will correspond,
point for point, to the rectilinear system of coordinates a?', / on the
first piece of paper.
Hence, if we take any number of points on the curve x" on the
first paper, and note the values of x and / at these points, and
mark the corresponding points on the second paper, we shall find
a number of points on the transformed curve x" . If we do the
same for all the curves #", /' on the first paper, we shall obtain on
the second paper a double series of curves x", y" of a different form,
but having the same property of cutting the paper into little
squares.
1 8 6.]
THEOREMS,
267
186.] THEOREM III. If 7 is any function of x and y , and if x'
and yf are conjugate functions of x and y, then
,d*7 d*7^ , rr/d2r
the integration being between the same limits.
For
d7 _d7dx d7dy'
dx ~~ dx dx dy' dx
dx*
' dy d*7dyf
~
dx'dy' dx
and
dx dy'
dx'dy ~dy ~dy
Adding1 the last two equations, and remembering1 the conditions
of conjugate functions (l), we find
,dx'
2 T~> 2
dx
+ T-
dx* dy*
Hence
(f(d*7 d*7
JJ \~d^ + dO*
If F is a potential, then, by Poisson's equation
d2F d*V
J_ + _
and we may write the result
dy ^ dy* \d®
e/ «7
(d*7 d*7^ ,dx_ df dd_ dy\
^dx'* dy'*' Wa? dy dy dx*
F d*V^ ,dx' dy' dx' dy\
* + df^ (di iy - ar 5)
J J p r/^^y = J J p'dafdtf,
or the quantity of electricity in corresponding portions of two sys
tems is the same if the coordinates of one system are conjugate
functions of those of the other.
268 CONJUGATE FUNCTIONS. [187.
Additional Theorems on Conjugate Functions.
187.] THEOEEM IV. If x^ and y15 and also x.2 and y^ are con
jugate functions of x and y, then, if
X=xlx.2-yly^ and Y = as1y2 + aay19
X and Y will be conjugate functions of x and y.
For X+ V^lY =
THEOREM V. If <£ be a solution of the equation
_
~ '
dx* df
TT 2 1
and if
and 0 = — tan-1
dy
R and 0 will be conjugate functions of x and y.
For R and 0 are conjugate functions of -~ and -— , and these
d /^f' are conjugate functions of x and y. *
EXAMPLE I. — Inversion.
188.] As an example of the general method of transformation
let us take the case of inversion in two dimensions.
If 0 is a fixed point in a plane, and OA a fixed direction, and
if r — OP = ae?, and 6 = AOP, and if x, y are the rectangular
coordinates of P with respect to 0,
. +/*£_i_«i2 ft — fnr»-l ?.
(5)
p and 6 are conjugate functions of x and y.
If p '= np and 6'=n0, p' and 6' will be conjugate functions of p
and 6. In the case in which n = — 1 we have
-Ul /=-, and 0'=-0, (6)
r
which is the case of ordinary inversion combined with turning the
figure 1 80° round OA.
A
Inversion in Two Dimensions.
P = log - Vx2+y2, 6 = tan-1 -
a x
In this case if r and / represent the distances of corresponding
points from 0, e and tf the total electrification of a body, 8 and 8'
superficial elements, V and V' solid elements, a- and (/ surface-
1 89.]
ELECTRIC IMAGES IN TWO DIMENSIONS.
269
densities, p and p' volume densities, $ and §' corresponding po
tentials,
^-°L.
S ~ r2
V
a"
^
'"tf
0
= 1.
EXAMPLE II. — Electric Images in Two Dimensions.
189.] Let A be the centre of a circle of radius AQ = b, and let
E be a charge at A, then the potential
at any point P is
AP'
and if the circle is a section of a hollow
conducting cylinder, the surface-density
T7I
at any point Q is J-j -
2 776
Fig. 17.
Invert the system with respect to a point 0, making
AO = ml, and a2 = (m2-l}l2 ;
then we have a charge at A' equal to that at A, where AA'=. -
The density at Q' is
^ b2-AA'
27i b AQ'2
and the potential at any point P' within the circle is
= 2 ^ (log OP'_ log ^'Px- log w). (9)
This is equivalent to a combination of a charge E at A', and a
charge — E at 0, which is the image of ^4', with respect to the
circle. The imaginary charge at 0 is equal and opposite to that
at^f.
If the point P' is defined by its polar coordinates referred to the
centre of the circle, and if we put
p = logr— log£5 and pQ = log A A — log b,
then AP— det>, AA'= be?*, AO = be~^ ; (10)
and the potential at the point (p, B) is
$ = E log (e~2^ — 2 e~^ e? cos 0 4- e2?)
— E log (<?PO — 2 eK e? cos 6 + e2?} + 2^/o0 . (11)
This is the potential at the point (p, 6) due to a charge E, placed
at the point (pQ) 0), with the condition that when p = 0, (/> = 0.
270 CONJUGATE FUNCTIONS.
In this case p and 0 are the conjugate functions in equations (5) :
p is the logarithm of the ratio of the radius vector of a point to
the radius of the circle, and 6 is an angle.
The centre is the only singular point in this system of coordinates,
/i) f)
-j- ds round a closed curve is zero or 2 TT,
according as the closed curve excludes or includes the centre.
EXAMPLE III. — Neumann's Transformation of this Case*.
190.] Now let a and ft be any conjugate functions of x and y,
such that the curves (a) are equipotential curves, and the curves
(/3) are lines of force due to a system consisting of a charge of half
a unit at the origin, and an electrified system disposed in any
manner at a certain distance from the origin.
Let us suppose that the curve for which the potential is a0 is
a closed curve, such that no part of the electrified system except the
half-unit at the origin lies within this curve.
Then all the curves (a) between this curve and the origin will be
closed curves surrounding the origin, and all the curves (/3) will
meet in the origin, and will cut the curves (a) orthogonally.
The coordinates of any point within the curve (a0) will be deter
mined by the values of a and /3 at that point, and if the point
travels round one of the curves (a) in the positive direction, the
value of /3 will increase by 2 TT for each complete circuit.
If we now suppose the curve (a0) to be the section of the inner
surface of a hollow cylinder of any form maintained at potential
zero under the influence of a charge of linear density E on a line of
which the origin is the projection, then we may leave the external
electrified system out of consideration, and we have for the potential
at any point (a) within the curve
0 = 2^(a-a0), (12)
and for the quantity of electricity on any part of the curve a0
between the points corresponding to ^ and /32,
f$. (13)
If in this way, or in any other, we have determined the dis
tribution of potential for the case of a given curve of section when
the charge is placed at a given point taken as origin, we may pass
to the case in which the charge is placed at any other point by an
application of the general method of transformation.
* See Crelle's Journal, 1861.
NEUMANN'S TRANSFORMATION. 271
Let the values of a and /3 for the point at which the charge is
placed be 04 and ft, then substituting in equation (ll) a — a0 for p,
and /3— ft for 0, we find for the potential at any point whose co
ordinates are a and /3,
— 2 ea+«l-2«0 COS (/3 — ft) -f- <?2(« + al-2«o))
2<?a-aicos(/3— /31) + e2(a-ai))-2^(a1-a0). (14)
This expression for the potential becomes zero when a = a0, and is
finite and continuous within the curve a0 except at the point (a1? ft),
at which point the second term becomes infinite, and in its immediate
neighbourhood is ultimately equal to — 2 E log /, where / is the
distance from that point.
We have therefore obtained the means of deducing the solution
of Green's problem for a charge at any point within a closed curve
when the solution for a charge at any other point is known.
The charge induced upon an element of the curve a0 between the
points /3 and (3 -\-dfi by a charge E placed at the point (al5 ft) is3
with the notation of Art. 183,
JL^ 7
"477^ 25
where ds1 is measured inwards and a is to be put equal to a0 after
differentiation.
This becomes, by (4) of Art. 183,
E 1— £2(ai-a0)
" 2^ 1 - 2 <?(«i-«o) cos (/3-ft) + e^i-oo) € *'
From this expression we may find the potential at any point
(aij ft) within the closed curve, when the value of the potential at
every point of the closed curve is given as a function of ft and
there is no electrification within the closed curve.
For, by Art. 86, the part of the potential at (al5 ft), due to the
maintenance of the portion d(3 of the closed curve at the potential
V is n Vt where n is the charge induced on d/3 by unit of electri
fication at (an ft). Hence, if F is the potential at a point on the
closed curve defined as a function of ft and $ the potential at
the point (al5 ft) within the closed curve, there being no electri-
fication within the curve,
2it
o -
272 CONJUGATE FUNCTIONS. [19 1.
EXAMPLE IV. — Distribution of Electricity near an Edge of a
Conductor formed by Two Plane Faces.
191.] In the case of an infinite plane face of a conductor charged
with electricity to the surface-density <70, we find for the potential
at a distance y from the plane
where C is the value of the potential of the conductor itself.
Assume a straight line in the plane as a polar axis, and transform
into polar coordinates, and we find for the potential
7 = C—^-n(rQae^ sin0,
and for the quantity of electricity on a parallelogram of breadth
unity, and length ae? measured from the axis
E = <TQaeP.
Now let us make p = np' and 6 = nb', then, since // and 0' are
conjugate to p and 6, the equations
V = C— 4 TT <r0 aen?' sin n 0'
and E=<rQae*i'
express a possible distribution of electricity and of potential.
If we write r for ae?', r will be the distance from the axis ; we
may also put 0 instead of 0' for the angle. We shall have
V — C—
Twill be equal to C whenever n6 = TT or a multiple of TT.
Let the edge be a salient angle of the conductor, the inclination
of the faces being a, then the angle of the dielectric is 2 TT- a, so
that when 0 = 27r— a the point is in the other face of the con
ductor. We must therefore make
Then F= £-
The surface-density o- at any distance r from the edge is
dE TT
1 9 2.] ELLIPSES AND HYPERBOLAS. 273
When the angle is a salient one a is less than 77, and the surface-
density varies according to some inverse power of the distance
from the edge, so that at the edge itself the density becomes
infinite,, although the whole charge reckoned from the edge to any
finite distance from it is always finite.
Thus, when a = 0 the edge is infinitely sharp, like the edge of a
mathematical plane. In this case the density varies inversely as
the square root of the distance from the edge.
When a = - the edge is like that of an equilateral prism, and
the density varies inversely as the f power of the distance.
When a = - the edge is a right angle, and the density is in
versely as the cube root of the distance.
When a = — the edge is like that of a regular hexagonal prism,
O
and the density is inversely as the fourth root of the distance.
When a = 77 the edge is obliterated, and the density is constant.
When a = | TT the edge is like that in the inside of the hexagonal
prism, and the density is directly as the square root of the distance
from the edge.
When a = -| TT the edge is a re-entrant right angle, and the
density is directly as the distance from the edge.
When a = |77 the edge is a re-entrant angle of 60^, and the
density is directly as the square of the distance from the edge.
In reality, in all cases in which the density becomes infinite at
any point, there is a discharge of electricity into the dielectric at
that point, as is explained in Art. 55.
EXAMPLE V. — Ellipses and Hyperbolas. Fig. X.
192.] We have seen that if
^ = e$ cos \//-, y^ = e$ sin \js, (1)
x and y will be conjugate functions of $ and \j/.
Also, if #2 = er* cos \l/, y^ = —e~$ sin \ff, (2)
#2 and ^ will be conjugate functions. Hence, if
2 a? = ^ + #2 = (d?*-t-<r*)cosi/r, 2y = ^+^2 = (e*— <?-*) sim/r, (3)
x and y will also be conjugate functions of § and \f/.
In this case the points for which $ is constant lie in the ellipse
whose axes are e$ -\ e~$ and e$ — e~^.
VOL. I. T
274 CONJUGATE FUNCTIONS.
The points for which ^ is constant lie in the hyperbola whose
axes are 2 cos \|/~ and 2 sin \^.
On the axis of x, between %=. — 1 and #= + 1 ,
<j> = 0, \f/ = cos-1^. (4)
On the axis of x, beyond these limits on either side, we have
x> 1, V = °» 4> = log(#W^2-l), (5)
#<_!, \l/ = IT, (f) = log(>A2 — 1— #).
Hence, if $ is the potential function, and \j/ the function of flow,
we have the case of electricity flowing from the positive to the
negative side of the axis of OB through the space between the points
— 1 and -f-1, the parts of the axis beyond these limits being
impervious to electricity.
Since, in this case, the axis of y is a line of flow, we may suppose
it also impervious to electricity.
We may also consider the ellipses to be sections of the equi-
potential surfaces due to an indefinitely long flat conductor of
breadth 2, charged with half a unit of electricity per unit of length.
If we make ^ the potential function, and $ the function of flow,
the case becomes that of an infinite plane from which a strip of
breadth 2 has been cut away and the plane on one side charged to
potential TT while the other remains at zero.
These cases may be considered as particular cases of the quadric
surfaces treated of in Chapter X. The forms of the curves are
given in Fig. X.
EXAMPLE VI.— Fig. XI.
193.] Let us next consider x' and / as functions of a? and y, where
sitan-1, (6)
af and y will be also conjugate functions of $ and \fs.
The curves resulting from the transformation of Fig. X with
respect to these new coordinates are given in Fig. XI.
If x' and / are rectangular coordinates, then the properties of the
axis of x in the first figure will belong to a series of lines parallel
to of in the second figure for which / = 6?/7r, where n is any
integer.
The positive values of of on these lines will correspond to values
of x greater than unity, for which, as we have already seen,
-l). (7)
1 9 3.] PARTICULAR CASE OF CONJUGATE FUNCTIONS. 275
The negative values of af on the same lines will correspond to
values of % less than unity, for which, as we have seen,
f!
$ = 0, \jf = cos-1*? = cos"1^. (8)
The properties of the axis of y in the first figure will belong to a
series of lines in the second figure parallel to #', for which
y=^(»'+i). (9)
The value of \/r along these lines is \f/ = TT (n + J) for all points
both positive and negative, and
. / *: /!*L \
$ = log(y+ Vy*+ 1) = log \eb + V eb + i) . (10)
[The curves for which <£ and v/f are constant may be traced
directly from the equations
As the figure repeats itself for intervals of -n b in the values of y'
it will be sufficient to trace the lines for one such interval.
Now there will be two cases, according as $ or \jf changes sign
with y'. Let us suppose that 0 so changes sign. Then any curve
for which \jr is constant will be symmetrical about the axis of a/,
cutting that axis orthogonally at some point on its negative side.
If we begin with this point for which </> = 0, and gradually in
crease <£, the curve will bend round from being at first orthogonal
to being, for large values of <j>3 at length parallel to the axis of sf.
The positive side of the axis of x is one of the system, viz. ^r is
there zero, and when/= + \ TT b, \jf = J TT. The lines for which \(,
has constant values ranging from 0 to \i: form therefore a system
of curves embracing the positive side of the axes of x' .
The curves for which <£ has constant values cut the system ^
orthogonally, the values of </> ranging from +00 to — co . For
any one of the curves $ drawn above the axis of so the value of </> is
positive, along the negative side of the axis of x' the value is zero,
and for any curve below the axis of #' the value is negative.
We have seen that the system \jr is symmetrical about the axis
of a?; let PQR be any curve cutting that system orthogonally and
terminating in P and R in the lines /= + \-nl, the point Q being
in the axis of x. Then the curve PQR is symmetrical about the axis
of a?', but if c be the value of 0 along PQ, the value of 0 along QR
will be —c. This discontinuity in the value of <£ will be accounted
T 2
276 CONJUGATE FUNCTIONS. [l94-
for by an electrical distribution in the case which will be discussed
in Art. 195.
If we next suppose that \f/ and not $ changes sign with /, the
values of <£ will range from 0 to oo . When <j> = 0 we have the
negative side of the axis of #', and when $ = oo we have a line
at an infinite distance perpendicular to the axis of af. Along any
line PQR between these two the value of (/> is constant throughout
its entire length and positive.
Any value \js now experiences an abrupt change at the point
where the curve along which it is constant crosses the negative
side of the axis of of, the sign of i/r changing there. The sig
nificance of this discontinuity will appear in Art. 197.
The lines we have shewn how to trace are drawn in Fig. XI
if we limit ourselves to two-thirds of that diagram, cutting oif the
uppermost third.]
194.] If we consider $ as the potential function, and ^ as the
function of flow, we may consider the case to be that of an in
definitely long strip of metal of breadth -nb with a non-conducting
division extending from the origin indefinitely in the positive
direction, and thus dividing the positive part of the strip into two
separate channels. We may suppose this division to be a narrow
slit in the sheet of metal.
If a current of electricity is made to flow along one of these
divisions and back again along the other, the entrance and exit of
the current being at an indefinite distance on the positive side of
the origin, the distribution of potential and of current will be given
by the functions <£ and ^ respectively.
If, on the other hand, we make ^ the potential, and <£ the
function of flow, then the case will be that of a current in the
general direction of/, flowing through a sheet in which a number
of non-conducting divisions are placed parallel to x, extending from
the axis of/ to an indefinite distance in the negative direction.
195.] We may also apply the results to two important cases in
statical electricity.
(1) Let a conductor in the form of a plane sheet, bounded by a
straight edge but otherwise unlimited, be placed in the plane of xz
on the positive side of the origin, and let two infinite conducting
planes be placed parallel to it and at distances \itb on either side.
Then, if ^ is the potential function, its value is 0 for the middle
conductor and \ -n for the two planes.
Let us consider the quantity of electricity on a part of the middle
IQ6.] EDGE OF AN ELECTRIFIED PLATE. 277
conductor, extending to a distance 1 in the direction of z, and from
the origin to #'= a.
The electricity on the part of this strip extending from #/ to #2'
Hence from the origin to x' •=• a the amount is
(ii)
•± 7T
If a is large compared with I, this becomes
_ fl+£loge2 ,12j
Hence the quantity of electricity on the plane bounded by the
straight edge is greater than it would have been if the electricity
had been uniformly distributed over it with the same density that
it has at a distance from the boundary, and it is equal to the
quantity of electricity having the same uniform surface- density,
but extending to a breadth equal to I loge 2 beyond the actual
boundary of the plate.
This imaginary uniform distribution is indicated by the dotted
straight lines in Fig. XI. The vertical lines represent lines of
force, and the horizontal lines equipotential surfaces, on the hypo
thesis that the density is uniform over both planes, produced to
infinity in all directions.
196.] Electrical condensers are sometimes formed of a plate
placed midway between two parallel plates extending considerably
beyond the intermediate one on all sides. If the radius of curvature
of the boundary of the intermediate plate is great compared with
the distance between the plates, we may treat the boundary as
approximately a straight line, and calculate the capacity of the
condenser by supposing the intermediate plate to have its area
extended by a strip of uniform breadth round its boundary, and
assuming the surface-density on the extended plate the same as
it is in the parts not near the boundary.
Thus, if S be the actual area of the plate, L its circumference
and B the distance between the large plates, we have
(13)
7T
278 CONJUGATE FUNCTIONS. [196.
and the breadth of the additional strip is
, ' (14)
so that the extended area is
7T
(15)
The capacity of the middle plate is
Correction for the Thickness of the Plate.
Since the middle plate is generally of a thickness which cannot
be neglected in comparison with the distance between the plates,
we may obtain a better representation of the facts of the case by
supposing the section of the intermediate plate to correspond with
the curve \fr == \//.
The plate will be of nearly uniform thickness, /3 = 26\j/t at a
distance from the boundary, but will be rounded near the edge.
The position of the actual edge of the plate is found by putting
/= 0, whence a/= £ \0ge Cosi/r'. (17)
The value of <£ at this edge is 0, and at a point for which #' = a
it is a + b logc 2
Hence, altogether, the quantity of electricity on the plate is the
same as if a strip of breadth
— (log, 2 + loge cos 2-=) ,
7T ^ ^ -*-)
* ~^ i X ^"/"'\ / 1 o \
had been added to the plate, the density being assumed to be every
where the same as it is at a distance from the boundary.
Density near the Edge.
The surface-density at any point of the plate is
x'
_
4 77 dx
v-i
** \
- &c.A (19)
4:710
1 9 7.] DENSITY NEAR THE EDGE. 279
The quantity within brackets rapidly approaches unity as of
increases, so that at a distance from the boundary equal to n times
the breadth of the strip a, the actual density is greater than the
normal density by about 2n+1 of the normal density.
In like manner we may calculate the density on the infinite planes
V £& +i
When x'= 0, the density is 2~* of the normal density.
At n times the breadth of the strip on the positive side, the
density is less than the normal density by about —~
At n times the breadth of the strip on the negative side, the
density is about — of the normal density.
These results indicate the degree of accuracy to be expected in
applying this method to plates of limited extent, or in which
irregularities may exist not very far from the boundary. The same
distribution would exist in the case of an infinite series of similar
plates at equal distances, the potentials of these plates being
alternately 4- V and — V. In this case we must take the distance
between the plates equal to B.
197.] (2) The second case we shall consider is that of an infinite
series of planes parallel to aoz at distances B = TT£, and all cut off by
the plane of yz, so that they extend only on the negative side of this
plane. If we make <$> the potential function, we may regard these
planes as conductors at potential zero.
Let us consider the curves for which <f> is constant.
When y' = n-nb, that is, in the prolongation of each of the planes,
we have x' = a log J (** + *-*) (21)
when y' •=. (n + ^Jbir, that is, in the intermediate positions
x'= Hogi(^ — erf). (22)
Hence, when $ is large, the curve for which $ is constant is
an undulating line whose mean distance from the axis of y is
approximately a - I (0-loge 2), (23)
and the amplitude of the undulations on either side of this line is
280 CONJUGATE FUNCTIONS. [198.
When (j> is large this becomes be~2$, so that the curve approaches
to the form of a straight line parallel to the axis of/ at a distance
a from that axis on the positive side.
If we suppose a plane for which x'— a, kept at a constant
potential while the system of parallel planes is kept at a different
potential, then, since b$ — a + t>\oge2, the surface-density of the
electricity induced on the plane is equal to that which would have
been induced on it by a plane parallel to itself at a potential equal
to that of the series of planes, but at a distance greater than that
of the edges of the planes by b loge 2.
If B is the distance between two of the planes of the series,
IB = TT b, so that the additional distance is
. = **&*. (25)
198.] Let us next consider the space included between two of
the equipotential surfaces, one of which consists of a series of parallel
waves, while the other corresponds to a large value of </>, and may
be considered as approximately plane.
If D is the depth of these undulations from the crest to the trough
of each wave, then we find for the corresponding value of <£,
D
0=ilog4±1- (26)
F-l
The value of #' at the crest of the wave is
6 log i(^ + <?-*). (27)
* Hence, if A is the distance from the crests of the waves to the
opposite plane, the capacity of the system composed of the plane
surface and the undulated surface is the same as that of two planes
at a distance A -f a', where
« = loge— —„• (28)
* Let 3> be the potential of the plane, <f> of the undulating surface. The quantity
of electricity on the plane per unit area is 1 -=- 4 IT 6. Hence the capacity
= 1 -r 4 IT (A + a'), suppose.
Then ^4 +a'= 6 ($-0).
But
(26).
2OO.] A GROOVED SURFACE. 281
199.] If a single groove of this form be made in a conductor
having the rest of its surface plane, and if the other conductor is
a plane surface at a distance A, the capacity of the one conductor
with respect to the other will be diminished. The amount of this
diminution will be less than the -th part of the diminution due
n
to n such grooves side by side, for in the latter case the average
electrical force between the conductors will be less than in the
former case, so that the induction on the surface of each groove will
be diminished on account of the neighbouring grooves.
If L is the length, B the breadth, and D the depth of the groove,
the capacity of a portion of the opposite plane whose area is 8 will be
S-LB LB S LB a'
If A is large compared with B or a, the correction becomes by (28)
L B\ 2
il0*-- -> (30)
l+e B
and for a slit of infinite depth, putting D = oo, the correction is
To find the surface-density on the series of parallel plates we
must find a = - -- ~ when d> = 0. We find
4ir dx
- -- (32)
The average density on the plane plate at distance A from the
edges of the series of plates is <r = — -, • Hence, at a distance from
4776
the edge of one of the plates equal to na the surface-density is
— of this average density.
200.] Let us next attempt to deduce from these results the
distribution of electricity in the figure formed by rotating the
plane of the figure about the axis^ = — E. In this case, Poisson's
equation will assume the form
dV . .
"- (33)
Let us assume V—$> the function given in Art. 193, and de-
282 CONJUGATE FUNCTIONS. [200.
termine the value of p from this equation. We know that the first
two terms disappear, and therefore
* (34)
If we suppose that, in addition to the surface-density already
investigated, there is a distribution of electricity in space according
to the law just stated, the distribution of potential will be repre
sented by the curves in Fig. XI.
Now from this figure it is manifest that -^ is generally very
small except near the boundaries of the plates, so that the new
distribution may be approximately represented by what actually
exists, namely a certain superficial distribution near the edges of
the plates.
If therefore we integrate / / p dxf dy between the limits if = 0 and
y'=-b} and from x—— oo to x = +oc, we shall find the whole
£l
additional charge on one side of the plates due to the curvature.
deb d\lr .
Since -7-7 = -- =-, » we have
dy dx
[X j , [™ 1
I pdx — I
J .a, -'-co 47r
-- n~ --T-,
-'-co 47r R+y die
Integrating with respect to y't we find
2 p dxdy = - — - —» — log — —^r- (36)
Jo J-n 88^ £&
This is half the total quantity of electricity which we must
suppose distributed in space near the edge of one of the cylindric
plates per unit of circumference. Since it is only close to the edge
of the plate that the density is sensible, we may suppose it all
condensed on the surface of the plate without altering sensibly its
action on the opposed plane surface, and in calculating the attraction
between that surface and the cylindric surface we may suppose this
electricity to belong to the cylindric surface.
200.] CIRCULAR GROOVES. 283
If there had been no curvature the superficial charge on the
positive surface of the plate per unit of length would have been
Hence, if we add to it the whole of the above distribution, this
TO
charge must be multiplied by the factor (l + \ — ) to get the total
charge on the positive side.
*In the case of a disk of radius R placed midway between two
infinite parallel plates at a distance B, we find for the capacity
ofthedisk k2 (38)
* [In Art. 200, in estimating the total space distribution we might perhaps more
correctly take for it the integral ffpln (R + «/') dx'dy', which gives, per unit circum-
1 7?
ference of the edge of radius E, - — - , thus leading to the same correction as in the
text. 6Z U
The case of the disk may be treated in like manner as follows :
Let the figure of Art. 195 revolve round a line perpendicular to the plates and at a
distance + R from the edge of the middle one. That edge will therefore envelope a
circle, which will be the edge of the disk. As in Art. 200, we begin with Poisson's
equation, which in this case will be
dW d*V I dV
We now assume that F = ^, the potential function of Art. 195. We must therefore
suppose electricity to exist in the region between the plates whose volume density /> is
47r R-x' dx
The total amount is B_
p.27r(R-x')dx'dy'.
Now if R is large in comparison with the distance between the plates this result
will be seen, on an examination of the potential lines in Fig. XI, to be sensibly the
same as B
Jo
-^ dx'dy'; that is, —
'0 J-*dx
The total surface distribution if we include both sides of the disk is
„•=(>
If, therefore, the volume distribution between the plates be supposed to be concen
trated on the disk the expression for the capacity, the difference of the potentials
of the plates and disk being ^, becomes
R
T>
result differing from that in the text by — nearly,]
284 CONJUGATE FUNCTIONS. [2OI.
Theory of Thomsons Guard-ring.
201.] In some of Sir W. Thomson's electrometers, a large plane
surface is kept at one potential, and at a distance a from this surface
is placed a plane disk of radius R surrounded by a large plane plate
called a Guard-ring with a circular aperture of radius R' concentric
with the disk. This disk and plate are kept at potential zero.
The interval between the disk and the guard-plate may be
regarded as a circular groove of infinite depth, and of breadth
R' — R, which we denote by B.
The charge on the disk due to unit potential of the large disk,
supposing the density uniform, would be — - •
4 a.
The charge on one side of a straight groove of breadth B and
length L —^^R, and of infinite depth, may be estimated by the
number of lines of force emanating from the large disk and falling
upon the side of the groove. Referring to Art. 197 and footnote
we see that the charge will therefore be
\LBx-—,
RB
i.e. J — 7 — 7 j
A + a
since in this case 4> = 1, </> = 0, and therefore I = A + a.
But since the groove is not straight, but has a radius of curvature
R, this must be multiplied by the factor (l + J — ) -
The whole charge on the disk is therefore
R2 RB , B^
(40)
SA 8A A+a
The value of a cannot be greater than
^Mli = 0.225 nearly.
7T
If B is small compared with either A or R this expression will
give a sufficiently good approximation to the charge on the disk
due to unity of difference of potential. The ratio of A to R
may have any value, but the radii of the large disk and of the
guard-ring must exceed R by several multiples of A.
202.] A CASE OF TWO PLANES. 285
EXAMPLE VII.— Fig. XII.
•91 ^
202.] Helmholtz, in his memoir on discontinuous fluid motion *,
has pointed out the application of several formulae in which the
coordinates are expressed as functions of the potential and its
conjugate function.
One of these may be applied to the case of an electrified plate
of finite size placed parallel to an infinite plane surface connected
with the earth.
Since x±—A$ and y^ — A \jf,
and also #2 = AeP cos \jt and y% = A e$ sin \ff,
are conjugate functions of 0 and \ff, the functions formed by adding
x± to #2 and y^ to y2 will be also conjugate. Hence, if
x =
y = A v/r + A e$ sin \/r.
then OB and y will be conjugate with respect to 0 and \lr} and 0 and
\lt will be conjugate with respect to x and y.
Now let x and y be rectangular coordinates, and let kty be the
potential, then /£0 will be conjugate to &\l/, k being any constant.
Let us put \j/ = TT, then y = ATT, x = A (0 — e&).
If 0 varies from — so to 0, and then from 0 to +00, SB varies
from -co to — A and from — A to — oo. Hence the equipotential
surface, for which \j/ = TT, is a plane parallel to x at a distance
b = irA from the origin, and extending from -co to x = — A.
Let us consider a portion of this plane, extending from
x — —(A + a) to x = —^4 and from z = 0 to z — c,
let us suppose its distance from the plane of xz to be y = 6 = A it,
and its potential to be F= kty = fcir.
The charge of electricity on the portion of the plane considered
is found by ascertaining the values of <£ at its extremities.
We have therefore to determine (/> from the equation
cj) will have a negative value fa and a positive value fa ; at the edge
of the plane, where x = —A, 0 = 0.
Hence the charge on the one side is — ckfa-^^n^ and that
on the other side is c/cfa-+- 47r.
* K'onigl. A~kad. der Wissenschaften, zu Berlin, April 23, 1868.
286 CONJUGATE FUNCTIONS. [203.
Both these charges are positive and their sum is
If we suppose that a is large compared with A,
-4-l+dre,
A
If we neglect the exponential terms in fa we shall find that the
charge on the negative surface exceeds that which it would have
if the superficial density had been uniform and equal to that at a
distance from the boundary, by a quantity equal to the charge on a
strip of breadth A = - with the uniform superficial density.
The total capacity of the part of the plane considered is
The total charge is CV, and the attraction towards the infinite
plane, whose equation is y = 0 and potential \j/ = 0, is
A
— 2 A^5 j
_ 7T Tt & •£*-
The equipotential lines and lines of force are given in Fig. XII.
EXAMPLE VIII. Theory of a Grating of Parallel Wires. Fig. XIII.
203.] In many electrical instruments a wire grating is used to
prevent certain parts of the apparatus from being electrified by
induction. We know that if a conductor be entirely surrounded
by a metallic vessel at the same potential with itself, no electricity
can be induced on the surface of the conductor by any electrified
body outside the vessel. The conductor, however, when completely
surrounded by metal, cannot be seen, and therefore, in certain cases,
an aperture is left which is covered with a grating of fine wire.
Let us investigate the effect of this grating in diminishing the
effect of electrical induction. We shall suppose the grating to
consist of a series of parallel wires in one plane and at equal
intervals, the diameter of the wires being small compared with the
204.] INDUCTION THROUGH A GRATING. 287
distance between them, while the nearest portions of the electrified
bodies on the one side and of the protected conductor on the other
are at distances from the plane of the screen, which are considerable
compared with the distance between consecutive wires.
204.] The potential at a distance / from the axis of a straight
wire of infinite length charged with a quantity of electricity A per
unit of length is F = — 2 A log / -{- (7. ( 1 )
We may express this in terms of polar coordinates referred to an
axis whose distance from the wire is unity, in which case we must
make /2 = 1 - 2 r cos 9 + r2, (2)
and if we suppose that the axis of reference is also charged with
the linear density X', we find
F=-
If we now make
then, by the theory of conjugate functions,
/ ^ 27T# —- '\ 2—
F= —\ log \l-2e a cos- - + e a ) — 2 A/ log 0 « +C, (5)
where x and y are rectangular coordinates, will be the value of the
potential due to an infinite series of fine wires parallel to z in the
plane of xz, and passing through points in the axis of x for which
# is a multiple of a.
Each of these wires is charged with a linear density A.
The term involving A' indicates an electrification, producing a
constant force - in the direction of y.
a
The forms of the equipotential surfaces and lines of force when
A'= 0 are given in Fig. XIII. The equipotential surfaces near the
wires are nearly cylinders, so that we may consider the solution
approximately true, even when the wires are cylinders of a diameter
which is finite but small compared with the distance between them.
The equipotential surfaces at a distance from the wires become
more and more nearly planes parallel to that of the grating.
If in the equation we make y = 6lt a quantity large compared
with a. we find approximately,
Vl = _ (A + A') + C nearly. (6)
If we next make y = — &2 , where #2 is a positive quantity large
compared with a, we find approximately,
288 CONJUGATE FUNCTIONS. [205.
p2 = ±Zp A' +tf nearly. (7)
If c is the radius of the wires of the grating, c being small
compared with a, we may find the potential of the grating itself
by supposing that the surface of the wire coincides with the equi-
potential surface which cuts the plane of osz at a distance c from the
axis of z. To find the potential of the grating we therefore put
x = c, and y — 0, whence
7= -2 A log 2 sin ~+C- (8)
205.] We have now obtained expressions representing the elec
trical state of a system consisting of a grating of wires whose
diameter is small compared with the distance between them, and
two plane conducting surfaces, one on each side of the grating,
and at distances which are great compared with the distance
between the wires.
The surface-density ^ on the first plane is got from the equa
tion (6) d7l 4
That on the second plane <ra from the equation (7)
= ^'. (10)
2
If we now write
a . v\ ,. + \
= -2i*«( T)'
and eliminate A and A/ from the equations (6), (7), (8), (9), (10),
we find
+«b+)=-r1+ri(l+)-r. (13)
When the wires are infinitely thin, a becomes infinite, and the
terms in which it is the denominator disappear, so that the case
is reduced to that of two parallel planes without a grating in
terposed.
If the grating is in metallic communication with one of the
planes, saythe first, 7= 7lt and the right-hand side of the equation
for a-j becomes Fx- 72. Hence the density ^ induced on the first
plane when the grating is interposed is to that which would have
been induced on it if the grating were removed, the second plane
being maintained at the same potential, as 1 to 1 +
206.] METHOD OF APFEOXIMAT10N. 289
We should have found the same value for the effect of the grating
in diminishing the electrical influence of the first surface on the
second, if we had supposed the grating connected with the second
surface. This is evident since b^ and b2 enter into the expression
in the same way. It is also a direct result of the theorem of
Art. 88.
The induction of the one electrified plane on the other through
the grating is the same as if the grating were removed, and the
distance between the planes increased from bl + b.2 to
If the two planes are kept at potential zero, and the grating
electrified to a given potential, the quantity of electricity on the
grating will be to that which would be induced on a plane of equal
area placed in the same position as
M2 : M2 + a(5i + ^)-
This investigation is approximate only when 61 and d.2 are large
compared with a, and when a is large compared with c. The
quantity a is a line which may be of any magnitude. It becomes
infinite when c is indefinitely diminished.
If we suppose c = \ a there will be no apertures between the
wires of the grating, and therefore there will be no induction
through it. We ought therefore to have for this case a = 0. The
formula (11), however, gives in this case
a=-^loge2, =-0.110,
which is evidently erroneous, as the induction can never be altered
in sign by means of the grating. It is easy, however, to proceed
to a higher degree of approximation in the case of a grating of
cylindrical wires. I shall merely indicate the steps of this process.
Method of Approximation.
206.] Since the wires are cylindrical, and since the distribution
of electricity on each is symmetrical with respect to the diameter
parallel to y, the proper expansion of the potential is of the form
7= <?0logr + 2<Vcos^ (14)
where r is the distance from the axis of one of the wires, and 0 the
angle between r and y, and, since the wire is a conductor, when
r is made equal to the radius V must be constant, and therefore
the coefficient of each of the multiple cosines of 6 must vanish.
VOL. i. u
290 CONJUGATE FUNCTIONS. [206.
For the sake of conciseness let us assume new coordinates £, 77, &c.
such that
a£ = 27T#, a-Y] = 2iry, ap = 27rr, a/3 = 2iib, &c., (15)
and let Fft = log (^+ e-(^)-2 cos £). (16)
Then if we make
F=^+4^+4/^ + &c. (17)
by giving proper values to the coefficients A we may express any
potential which is a function of 17 and cos f, and does not become
infinite except when rj + (3 = 0 and cos f = 1.
When /3 = 0 the expansion of F in terms of p and 0 is
,F0 = 2 logp + yV p2 cos 2 0— TTYo P4 cos 40 + &C. (18)
For finite values of (3 the expansion of F is
^ = /3 + 2log(l-er0)+^^pcos0-
In the case of the grating with two conducting planes whose
equations are q = /3j and 77 = — /32, that of the plane of the grating
being 77 = 0, there will be two infinite series of images of the
grating. The first series will consist of the grating itself together
with an infinite series of images on both sides, equal and similarly
electrified. The axes of these imaginary cylinders lie in planes
whose equations are of the form
77= ± 2»(/31 + |82), (20)
n being an integer.
The second series will consist of an infinite series of images for
which the coefficients A0, A.2, A^, &c. are equal and opposite to the
same quantities in the grating itself, while Al9 A3, &c. are equal
and of the same sign. The axes of these images are in planes whose
equations are of the form
77 = 2/32 + 2aw(/31 + j92), (21)
m being an integer.
The potential due to any finite series of such images will depend
on whether the number of images is odd or even. Hence the
potential due to an infinite series is indeterminate, but if we add to
it the function £r]-\-C} the conditions of the problem will be suffi
cient to determine the electrical distribution.
We may first determine Y^ and Y2t the potentials of the two
conducting planes, in terms of the coefficients A0, A1, &c., and of
B and C. We must then determine ^ and crz, the surface-density
at any point of these planes. The mean values of o-j and o-2 are
given by the equations
206.] METHOD OP APPROXIMATION. 291
(22)
"We must then expand the potentials due to the grating itself
and to all the images in terms of p and cosines of multiples of 6,
adding to the result jjp cos e+Cm
The terms independent of 6 then give V the potential of the
grating, and the coefficient of the cosine of each multiple of 0
equated to zero gives an equation between the indeterminate co
efficients.
In this way as many equations may be found as are sufficient
to eliminate all these coefficients and to leave two equations to
determine o-x and <r2 in terms of J\, 7£, and V.
These equations will be of the form
-y). (23)
The quantity of electricity induced on one of the planes protected
by the grating, the other plane being at a given difference of
potential, will be the same as if the plates had been at a distance
^-— — — — instead of #,+&,.
a + y
The values of a and y are approximately as follows,
(24)
U 2
CHAPTER XIII.
ELECTROSTATIC INSTRUMENTS.
On Elecfoostatic Instruments.
THE instruments which we have to consider at present may be
divided into the following* classes :
fl) Electrical machines for the production and augmentation of
electrification.
(2) Multipliers, for increasing electrification in a known ratio.
(3) Electrometers, for the measurement of electric potentials and
charges.
(4) Accumulators, for holding large electrical charges.
Electrical Machines.
207.] In the common electrical machine a plate or cylinder of
glass is made to revolve so as to rub against a surface of leather,
on which is spread an amalgam of zinc and mercury. The surface
of the glass becomes electrified positively and that of the rubber
negatively. As the electrified surface of the glass moves away
from the negative electrification of the rubber it acquires a high
positive potential. It then comes opposite to a set of sharp metal
points in connexion with the conductor of the machine. The posi
tive electrification of the glass induces a negative electrification
of the points, which is the more intense the sharper the points
and the nearer they are to the glass.
When the machine works properly there is a discharge through
the air between the glass and the points, the glass loses part of
its positive charge, which is transferred to the points and so to
the insulated prime conductor of the machine, and to any other
body with which it is in electric communication.
The portion of the glass which is advancing towards the rubber
has thus a smaller positive charge than that which is leaving it
at the same time, so that the rubber, and the conductors in com
munication with it, become negatively electrified.
208.] ELECTROPHORUS. 293
The highly positive surface of the glass where it leaves the
rubber is more attracted by the negative charge of the rubber than
the partially discharged surface which is advancing towards the
rubber. The electrical forces therefore act as a resistance to the force
employed in turning the machine. The work done in turning the
machine is therefore greater than that spent in overcoming ordinary
friction and other resistances, and the excess is employed in pro
ducing a state of electrification whose energy is equivalent to this
excess.
The work done in overcoming friction is at once converted into
heat in the bodies rubbed together. The electrical energy may
be also converted either into mechanical energy or into heat.
If the machine does not store up mechanical energy, all the
energy will be converted into heat, and the only difference between
the heat due to friction and that due to electrical action is that the
former is generated at the rubbing surfaces while the latter may be
generated in conductors at a distance *.
We have seen that the electrical charge on the surface of the
glass is attracted by the rubber. If this attraction were sufficiently
intense there would be a discharge between the glass and the
rubber, instead of between the glass and the collecting points. To
prevent this, flaps of silk are attached to the rubber. These become
negatively electrified and adhere to the glass, and so diminish the
potential near the rubber.
The potential therefore increases more gradually as the glass
moves away from the rubber, and therefore at any one point there
is less attraction of the charge on the glass towards the rubber, and
consequently less danger of direct discharge to the rubber.
In some electrical machines the moving part is of ebonite instead
of glass, and the rubbers of wool or fur. The rubber is then elec
trified positively and the prime conductor negatively.
The Electrophorus of Yolta.
208.] The electrophorus consists of a plate of resin or of ebonite
backed with metal, and a plate of metal of the same size. An
insulating handle can be screwed to the back of either of these
plates. The ebonite plate has a metal pin which connects the metal
* It is probable that in many cases where dynamical energy is converted into heat
by friction, part of the energy may be first transformed into electrical energy and
then converted into heat as the electrical energy is spent in maintaining currents of
short circuit close to the rubbing surfaces. See Sir W. Thomson, ' On the Electro-
dynamic Qualities of Metals.' Phil. Trans., 1856, p. 650.
294 ELECTROSTATIC INSTRUMENTS. [209.
plate with the metal back of the ebonite plate when the two plates
are in contact.
The ebonite plate is electrified negatively by rubbing it with
wool or cat's skin. The metal plate is then brought near the
ebonite by means of the insulating handle. No direct discharge
passes between the ebonite and the metal plate, but the potential
of the metal plate is rendered negative by induction, so that when
it comes within a certain distance of the metal pin a spark passes,
and if the metal plate be now carried to a distance it is found
to have a positive charge which may be communicated to a con
ductor. The metal at the back of the ebonite plate is found to
have a negative charge equal and opposite to the charge of the metal
plate.
In using the instrument to charge a condenser or accumulator
one of the plates is laid on a conductor in communication with
the earth, and the other is first laid on it, then removed and applied
to the electrode of the condenser, then laid on the fixed plate and
the process repeated. If the ebonite plate is fixed the condenser
will be charged positively. If the metal plate is fixed the condenser
will be charged negatively.
The work done by the hand in separating the plates is always
greater than the work done by the electrical attraction during the
approach of the plates, so that the operation of charging the con
denser involves the expenditure of work. Part of this work is
accounted for by the energy of the charged condenser, part is spent
in producing the noise and heat of the sparks, and the rest in
overcoming other resistances to the motion.
On Machines producing Electrification by Mechanical Work.
209.] 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 Philosophical 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/
210.] It was by means of the revolving doubler that Volta
succeeded in developing from the electrification of the pile an
2IO.] THE REVOLVING DOUBLER. 295
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
ducting1 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
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 A and
C, and of two receivers B and D, with two carriers F and G.
Suppose the inductor A to be positively electrified so that its
potential is A, and that the carrier j^is within it and is at potential
F. Then, if Q is the coefficient of induction (taken positive) between
A and Ft the quantity of electricity on the carrier will be Q (F—A).
If the carrier, while within the inductor, is put in connexion with
the earth, then F = 0, and the charge on the carrier will be —QA,
a negative quantity. Let the carrier be carried round till it is
within the receiver B, and let it then come in contact with a spring
so as to be in electrical connexion with B. It will then, as was
shewn in Art. 32, become completely discharged, and will com
municate its whole negative charge to the receiver B.
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
ceivers.
* Specification of Patent, Jan. 27, I860, No. 206.
296 ELECTROSTATIC INSTRUMENTS. [2IO.
But by putting the inductor A in communication with the re
ceiver D, and the inductor C with the receiver B, the potentials
of the inducto -s 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 A and D be U, and that of B
and C, 7, then, since the potential of the carrier is zero when
it is within A, 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 7to7-^U.
If the other carrier has at the same time carried a charge — Q V
from C to D, it will change the potential of A and D from U to
C7_ .3L Y, if Q' is the coefficient of induction between the carrier
A.
and (7, and A the capacity of A and D. If, therefore, Un and 7n
be the potentials of the two inductors after n half revolutions, and
Un+l and 7n+1 after n+1 half revolutions,
F - V JT
' n + I — ' n~ ~£ U n-
Q Q'
If we write p2 = -~ and <f = -j- > we find
X) 4
Hence
Un = U0 ((I -pqY + (1 4 H)") +| FO ((1 -^)n-
+ r0 ((i -^)" + (
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 A and B are in the ratio of p to —q.
On the other hand, the quantity pU—qV continually increases,
so that, however little pUm&y exceed or fall short of qF at first,
the difference will be increased in a geometrical ratio in each
211.] THE RECIPROCAL ELECTROPHORUS. 297
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 Replenishes
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
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.
211.] 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
* Proc. E. S., June 20, 1867.
298
ELECTROSTATIC INSTRUMENTS.
[212.
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.
212.] Several other electrical machines have been constructed
in which the principle of electric induction is employed. Of these
the most remarkable is that of Holtz, in which the carrier is a glass
plate varnished with gum-lac and the inductors are pieces of
pasteboard. Sparks are prevented from passing between the parts
of the apparatus by means of two glass plates, one on each side
of the revolving carrier plate. This machine is found to be very
effective, and not to be much affected by the state of the atmo
sphere. The principle is the same as in the revolving doubler and
the instruments developed out of the same idea, but as the carrier
is an insulating plate and the inductors are imperfect conductors,
the complete explanation of the action is more difficult than in
the case where the carriers are good conductors of known form
and are charged and discharged at definite points.
213.] 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 con
verted 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.
In the figure let A, B, C, A', Bf, C' represent hollow fixed
conductors, so arranged that the carrier P passes in succession
within each of them. Of these A, A' and J5, Bf nearly surround the
Fig. 18.
2I3-] MACHINE WITHOUT SPAKKS. 299
carrier when it is at the middle point of its passage, but C, C' do not
cover it so much.
We shall suppose A, B, C to be connected with a Leyden jar
of great capacity at potential F", and A', B', C' to be connected with
another jar at potential — V .
P is one of the carriers moving in a circle from A to (?', &c.,
and touching in its course certain springs, of which a and a' are
connected with A and A' respectively, and e, e are connected with
the earth.
Let us suppose that when the carrier P is in the middle of A
the coefficient of induction between P and A is — A. The capacity
of P in this position is greater than A, since it is not completely
surrounded by the receiver A. Let it be A -\-a.
Then if the potential of P is V, and that of A, 7, the charge
Now let P be in contact with the spring a when in the middle
of the receiver A, then the potential of P is V> the same as that
of A, and its charge is therefore aV.
If P now leaves the spring a it carries with it the charge aV.
As P leaves A its potential diminishes, and it diminishes still more
when it comes within the influence of £', which is negatively
electrified.
If when P comes within G' 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 P is
(C'+c')U+C'7'=.aTr.
If C'F'=ar,
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 / 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
to earth without a spark, answers to the contrivance called a
regenerator in heat-engines. We shall therefore call it a He-
generator.
Now let P move on, still in contact with the earth-spring /, till
it 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 —BV.
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
300 ELECTROSTATIC INSTRUMENTS. [214.
negative receiver A' its potential will be increasingly negative. At
the middle of A\ if it retained its charge, its potential would be
A'7
A' + a
and if £7 is greater than a'V its numerical value will be greater
than that of 7'. Hence there is some point before P reaches the
middle of A' where its potential is — 7' '. 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 A', still in contact with the spring, and therefore
at the same potential with A'. During this motion it communicates
a negative charge to A'. At the middle of A' it leaves the spring
and carries away a charge —a' 7' 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 J5', during which motion it acquires a positive charge I? 7'
which it finally communicates to the positive receiver A, and the
cycle of operations is repeated.
During this cycle the positive receiver has lost a charge #Fand
gained a charge B'7'. Hence the total gain of positive electricity
is BV'-aV.
Similarly the total gain of negative electricity is B7—a'7/.
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, a 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
C'7'=a7, and C7=a'V.
Since a and a are small the regenerators do not require to be
either large or very close to the carriers.
On Electrometers and Electroscopes.
214.] An electrometer is an instrument by means of which
electrical charges or electrical potentials may be measured. In
struments by means of which the existence of electric charges or
of differences of potential may be indicated, but which are not
capable of affording numerical measures, are called Electroscopes.
An electroscope if sufficiently sensitive may be used in electrical
measurements, provided we can make the measurement depend on
2 1 5.] COULOMB'S TORSION BALANCE. 301
the absence of electrification. For instance, if we have two charged
bodies A and B we may use the method described in Chapter I to
determine which body has the greater charge. Let the body A
be carried by an insulating support into the interior of an insulated
closed vessel C. Let C be connected to earth and again insulated.
There will then be no external electrification on C. Now let A
be removed, and B introduced into the interior of C, and the elec
trification of C tested by an electroscope. If the charge of B is
equal to that of A there will be no electrification, but if it is greater
or less there will be electrification of the same kind as that of B, or
the opposite kind.
Methods of this kind, in which the thing to be observed is the
non-existence of some phenomenon, are called null or zero methods.
They require only an instrument capable of detecting the existence
of the phenomenon.
In another class of instruments for the registration of phe
nomena the instruments may be depended upon to give always the
same indication for the same value of the quantity to be registered,
but the readings of the scale of the instrument are not proportional
to the values of the quantity, and the relation between these
readings and the corresponding value is unknown, except that the
one is some continuous function of the other. Several electrometers
depending on the mutual repulsion of parts of the instrument
which are similarly electrified are of this class. The use of such
instruments is to register phenomena, not to measure them. Instead
of the true values of the quantity to be measured, a series of
numbers is obtained, which may be used afterwards to determine
these values when the scale of the instrument has been properly
investigated and tabulated.
In a still higher class of instruments the scale readings are
proportional to the quantity to be measured, so that all that is
required for the complete measurement of the quantity is a know
ledge of the coefficient by which the scale readings must be
multiplied to obtain the true value of the quantity.
Instruments so constructed that they contain within themselves
the means of independently determining the true values of quan
tities are called Absolute Instruments.
CoulomVs Torsion Balance.
215.] A great number of the experiments by which Coulomb
302 ELECTROSTATIC INSTRUMENTS. [215.
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. See Art. 38.
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 a sin \ 0, where a is the radius
of the torsion-arm, and if F is the force between the spheres the
moment of this force about the axis of torsion is Fa cos J 9.
Let both spheres be completely discharged, and let the torsion-
arm now be in equilibrium at an angle (p with the radius through
the fixed sphere.
Then the angle through which the electrical force twisted the
torsion-arm must have been #—</>, and if M is the moment of
the torsional elasticity of the fibre, we shall have the equation
Hence, if we can ascertain M, we can determine F. the actual
force between the spheres at the distance 2 a sin \6.
To find M, the moment of torsion, let /be the moment of inertia
of the torsion-arm, and T the time of a double vibration of the arm
under the action of the torsional elasticity, then
In all electrometers it is of the greatest importance to know
what force we are measuring. The force acting on the suspended
2I5-] INFLUENCE OF THE CASE. 303
sphere is clue 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 #,
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 E1 and E, and that the angle between their
positions is 6 ; that the fixed sphere is at a distance a^ from the
centre, and that r 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
r*
The image of E^ due to the spherical surface of the case is" a point
in the same radius at a distance — with a charge — Ev — , and the
moment of the attraction between E and this image about the axis
of suspension is
a — sin 0
a2 — 2 — cos 6 + — « ' 2
^ sin 0
If 7j, the radius of the spherical case, is large compared with a
304 ELECTROSTATIC INSTRUMENTS. [216.
and #t ) 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
sin fl JL _
JJL _ ^ = M(6-
Electrometers for the Measurement of Potentials.
216.] 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 po
tentials 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 construction, 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
between them. Hence, by making the areas of the disks large
* Phil. Trans. 1834.
t See an excellent report on Electrometers by Sir W. Thomson. Report of the
British Association, Dundee, 1867.
217.]
PRINCIPLE OF THE GUARD-RING.
305
and the distance between them small, a small difference of potential
may give rise to a measurable force of attraction.
The mathematical theory of the distribution of electricity over
two disks thus arranged is given at Art. 202, but since it is im
possible to make the case of the apparatus so large that we may
suppose the disks insulated in an infinite space, the indications of
the instrument in this form are not easily interpreted numerically.
217.] The addition of the guard-ring to the attracted disk 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 determ
ining 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
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
COUNTERPOISE
Fig. 19.
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
VOL. i. x
306 ELECTROSTATIC INSTRUMENTS. [217.
impossible, for it is part of the inner surface of a closed hollow
conductor all at the same potential.
Thomson's Absolute Electrometer therefore consists essentially
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 piano 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 is
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
2 1 7.] THOMSON'S ABSOLUTE ELECTROMETER. 307
balance and suspended disk, sufficient apertures being left to see
the fiducial marks.
The guard-ring, case, and suspended disk are all in metallic
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 W be the numerical value of the weight, and g the force of
gravity, 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 *9
-i-ir ' -A- -n-
* Let us denote the radius of the suspended disk by E, and that of the aperture
of the guard-ring by E', 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 Z>, and
the difference of potentials between these disks is V, then, by the investigation in
Art. 201, the quantity of electricity on the suspended disk will be
(
I SD 8D D + a
where a = B — ^— , or a = 0.220635 (E' - 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 + z = D', then it appears from the investigation in
Art. 225 that there will be an additional charge of electricity near the edge of
the disk on account of its height z above the general surface of the guard-ring.
The whole charge in this case is therefore, approximately,
^ ' \ 8D 8D
and in the expression for the attraction we must substitute for A, the area of the
disk, the corrected quantity
A =i« & + X*-(K*-&) -- + 8 (B
where E = 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 (K-R).
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.
X 2
308 ELECTROSTATIC INSTRUMENTS. [2 1 8.
If the suspended disk is circular, of radius E, and if the radius of
the aperture of the guard-ring is R', then
A = *& + &, and V=
218.] 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 D
is small, Sir W. Thomson prefers to make all his measurements
depend on differences of the electromotive force V. Thus, if V and
V are two potentials, and D and I/ the corresponding distances,
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 V.
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 F+ 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
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
force varies inversely as the square of the distance, so that any
2 1 9-] GAUGE ELECTROMETER. 309
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 effect of small irregularities of form in the surfaces 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 eminences 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. See Arts. 197, 198.
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
called 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 Idiostatic 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.
219.] The electrometers already described are not self-acting,
but require for each observation an adjustment of a micrometer
310 ELECTROSTATIC INSTRUMENTS.
screw, or some other movement which must be made by the
observer. They are therefore not fitted to act as self- registering
instruments, which must of themselves move into the proper
position. This condition is fulfilled by Thomson's Quadrant
Electrometer.
The electrical principle on which this instrument is founded may
be thus explained : —
A and B are two fixed conductors which may be at the same
or at different potentials. C is a moveable conductor at a high
potential, which is so placed that part of it is opposite to the
surface of A and part opposite to that of Bt and that the proportions
of these parts are altered as C moves.
For this purpose it is most convenient to make C moveable about
an axis, and make the opposed surfaces of A, of B, and of C portions
of surfaces of revolution about the same axis.
In this way the distance between the surface of C and the
opposed surfaces of A or of B remains always the same, and the
motion of C in the positive direction simply increases the area
opposed to B and diminishes the area opposed to A.
If the potentials of A and B are equal there will be no force
urging C from A to B, but if the potential of C differs from that
of B more than from that of A, then C will tend to move so as
to increase the area of its surface opposed to B.
By a suitable arrangement of the apparatus this force may be
made nearly constant for different positions of C within certain
limits, so that if C is suspended by a torsion fibre, its deflexions
will be nearly proportional to the difference of potentials between
A and B multiplied by the difference of the potential of C from
the mean of those of A and B.
C is maintained at a high potential by means of a condenser
provided with a replenisher and tested by a gauge electrometer,
and A and B are connected with the two conductors the difference
of whose potentials is to be measured. The higher the potential
of C the more sensitive is the instrument. This electrification of
6", being independent of the electrification to be measured, places
this electrometer in the heterostatic class.
We may apply to this electrometer the general theory of systems
of conductors given in Arts. 93, 127.
Let A, .8, C denote the potentials of the three conductors re
spectively. Let a, b, c be their respective capacities,^ the coefficient
of induction between B and C, q that between C and A, and r that
2I9-]
QUADRANT ELECTROMETER.
311
between A and B. All these coefficients will in general vary with
the position of <?, and if C is so arranged that the extremities of A
and B are not near those of Cas long as the motion of Cis confined
within certain limits, we may ascertain the form of these coefficients.
If 6 represents the deflexion of C from A towards B, then the part
of the surface of A opposed to C will diminish as B increases.
Hence if A is kept at potential 1 while B and Cave kept at potential
0, the charge on A will be a = #0— a0, where a0 and a are
constants, and a is the capacity of A.
If A and B are symmetrical, the capacity of B is 6 — &Q + a0.
The capacity of C is not altered by the motion, for the only
effect of the motion is to bring a different part of C opposite to the
interval between A and B. Hence c = <?0 .
The quantity of electricity induced on C when B is raised to
potential unity is p = />0— a0.
The coefficient of induction between A and C is q — qQ + a0.
The coefficient of induction between A and B is not altered by
the motion of (7, but remains r = r0 .
Hence the electrical energy of the system is
and if 0 is the moment of the force tending to increase 6,
dW
0 = — , A, J5, C being supposed constant,
dQ
da
db
dc
dp „ do
dr
or 0 = a(A-B) (C-
In the present form of Thomson's Quadrant Electrometer the
conductors A and B are in the form of
a cylindrical box completely divided
into four quadrants, separately insu
lated, but joined by wires so that two
opposite quadrants are connected with
A and the two others with B.
The conductor C is suspended so as
to be capable of turning about a
vertical axis, and may consist of two
opposite flat quadrantal arcs supported
by their radii at their extremities.
In the position of equilibrium these quadrants should be partly
Fig. 20.
312 ELECTROSTATIC INSTRUMENTS. [220.
within A and partly within B} and the supporting radii should
be near the middle of the quadrants of the hollow base, so that
the divisions of the box and the extremities and supports of C
may be as far from each other as possible.
The conductor C is kept permanently at a high potential by
being connected with the inner coating of the Leyden jar which
forms the case of the instrument. B and A are connected, the first
with the earth, and the other with the body whose potential is to be
measured.
If the potential of this body is zero, and if the instrument be
in adjustment, there ought to be no force tending to make C move,
but if the potential of A is of the same sign as that of C, then
C will tend to move from A to B with a nearly uniform force, and
the suspension apparatus will be twisted till an equal force is
called into play and produces equilibrium. Within certain limits
the deflexions of C will be proportional to the product
By increasing the potential of C the sensibility of the instrument
may be increased, and for small values of \ (A -f B) the deflexions
will be nearly proportional to (A—B) C.
On the Measurement of Electric Potential.
220.] 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. If at 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 K' that of this part
221.] MEASUREMENT OF POTENTIAL. 313
of the electrometer, and if 7, V denote the potentials of these
bodies before making contact, then their common potential after
making contact will be
K+K'
Hence the original potential of the conductor was
If the conductor is not large compared with the electrometer,
K' will be comparable with Kt and unless we can ascertain the
values of K and TL' 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 will
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.
To Measure the Potential at any Point in the Air.
221.] First 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 V be the potential at the given point, and a the
radius of the sphere, the charge on the sphere will be — Va = Q,
and if V be the potential of the sphere as measured by an elec
trometer when placed in a room whose walls are connected with
the earth, then Q — ya)
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
314 ELECTROSTATIC INSTRUMENTS. [222.
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
zero.
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 electrified 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.
222.] 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 zinc, are
in metallic contact, then the potential of the air in the hollow
surrounded by zinc 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.
Another convenient method is to fasten a slow match to the
223-] THEOKY OF THE PROOF PLANE. 315
electrode. The potential is very soon made equal to that of the
air at the burning end of the match. Even a fine metallic point
is sufficient to create a discharge by means of the particles of the
air when the difference of potentials is considerable, but if we
wish to reduce this difference to zero, we must use one of the
methods stated above.
If we only wish to ascertain the sign of the difference of the
potentials at two places, and not its numerical value, we may cause
drops or filings to be discharged at one of the places from a nozzle
connected with the other place, and catch the drops or filings
in an insulated vessel. Each drop as it falls is charged with a
certain amount of electricity, and it is completely discharged into
the vessel. The charge of the vessel therefore is continually ac
cumulating, and after a sufficient number of drops have fallen, the
charge of the vessel may be tested by the roughest methods. The
sign of the charge is positive if the potential of the nozzle is positive
relatively to that of the surrounding air.
MEASUREMENT OF SURFACE-DENSITY OF ELECTRIFICATION.
Theory of the Proof Plane.
223.] In testing the results of the mathematical theory of the
distribution of electricity on the surface of conductors, it is necessary
to be able to measure the surface-density at different points of
the conductor. For this purpose Coulomb employed a small disk
of gilt paper fastened to an insulating stem of gum-lac. He ap
plied this disk to various points of the conductor by placing it
so as to coincide as nearly as possible with the surface of the
conductor. He then removed it by means of the insulating stem,
and measured the charge of the disk by means of his electrometer.
Since the surface of the disk, when applied to the conductor,
nearly coincided with that of the conductor, he concluded that
the surface-density on the outer surface of the disk was nearly
equal to that on the surface of the conductor at that place, and that
the charge on the disk when removed was nearly equal to that
on an area of the surface of the conductor equal to that of one side
of the disk. This disk, when employed in this way, is called
Coulomb's Proof Plane.
As objections have been raised to Coulomb's use of the proof
plane, I shall make some remarks on the theory of the experiment.
316 ELECTROSTATIC INSTRUMENTS. [224..
This experiment consists in bringing a small conducting body
into contact with the surface of the conductor at the point where
the density is to be measured, and then removing the body and
determining its charge.
We have first to shew that the charge on the small body when
in contact with the conductor is proportional to the surface-
density which existed at the point of contact before the small body
was placed there.
We shall suppose that all the dimensions of the small body, and
especially its dimension in the direction of the normal at the point
of contact, are small compared with either of the radii of curvature
of the conductor at the point of contact. Hence the variation of
the resultant force due to the conductor supposed rigidly electrified
within the space occupied by the small body may be neglected,
and we may treat the surface of the conductor near the small body
as a plane surface.
Now the charge which the small body will take by contact with
a plane surface will be proportional to the resultant force normal
to the surface, that is, to the surface-density. We shall ascertain
the amount of the charge for particular forms of the body.
We have next to shew that when the small body is removed no
spark will pass between it and the conductor, so that it will carry
its charge with it. This is evident, because when the bodies are
in contact their potentials are the same, and therefore the density
on the parts nearest to the point of contact is extremely small.
When the small body is removed to a very short distance from
the conductor, which we shall suppose to be electrified positively,
then the electrification at the point nearest to the small body is
no longer zero but positive, but, since the charge of the small body
is positive, the positive electrification close to the small body will
be less than at other neighbouring points of the surface. Now
the passage of a spark depends in general on the magnitude of the
resultant force, and this on the surface-density. Hence, since we
suppose that the conductor is not so highly electrified as to be
discharging electricity from the other parts of its surface, it will
not discharge a spark to the small body from a part of its surface
which we have shewn to have a smaller surface -density.
224.] We shall now consider various forms of the small body.
Suppose it to be a small hemisphere applied to the conductor so
as to touch it at the centre of its flat side.
Let the conductor be a large sphere, and let us modify the form
225.] THE PROOF PLANE. 317
of the hemisphere so that its surface is a little more than a hemi
sphere, and meets the surface of the sphere afc right angles. Then
we have a case of which we have already obtained the exact solution.
See Art. 167.
If A and B be the centres of the two spheres cutting each other
at right angles, DD' a diameter of the circle of intersection, and C
the centre of that circle, then if Fis the potential of a conductor
whose outer surface coincides with that of the two spheres, the
quantity of electricity on the exposed surface of the sphere A is
and that on the exposed surface of the sphere B is
\r(AD+BD+BC-CJ)
the total charge being the sum of these, or
If a and £ are the radii of the spheres, then, when a is large
compared with /3, the charge on B is to that on A in the ratio of
Now let a- be the uniform surface-density on A when B is re
moved, then the charge on A is
4 TT a2 o-,
and therefore the charge on B is
377/32cr(l-f i- + &c.),
v o a '
or, when fi is very small compared with a, the charge on the
hemisphere B is equal to three times that due to a surface-density o-
extending over an area equal to that of the circular base of the
hemisphere.
It appears from Art. 175 that if a small sphere is made to touch
an electrified body, and is then removed to a distance from it, the
mean surface-density on the sphere is to the surface-density of the
body at the point of contact as ?r2 is to 6, or as 1.645 to 1.
225.] The most convenient form for the proof plane is that of
a circular disk. We shall therefore shew how the charge on a
circular disk laid on an electrified surface is to be measured.
For this purpose we shall construct a value of the potential
function so that one of the equipotential surfaces resembles a circular
flattened protuberance whose general form is somewhat like that of
a disk lying on a plane.
318 ELECTROSTATIC INSTRUMENTS. [225.
Let <r be the surface-density of a plane, which we shall suppose
to be that of xy.
The potential due to this electrification will be
Now let two disks of radius a be rigidly electrified with surface-
densities — a and + (/. Let the first of these be placed on the plane
of xy with its centre at the origin, and the second parallel to it at
the very small distance c.
Then it may be shewn, as we shall see in the theory of mag
netism, that the potential of the two disks at any point is a></c,
where a> is the solid angle subtended by the edge of either disk at
the point. Hence the potential of the whole system will be
F= — 4 77 0-2-1- (/CO).
The forms of the equipotential surfaces and lines of induction
are given on the left-hand side of Fig. XX, at the end of Vol. II.
Let us trace the form of the surface for which V= 0. This
surface is indicated by the dotted line.
Putting the distance of any point from the axis of z = r, then,
when r is much less than a, and z is small, we find
o> = 27T— 277 - + &c.
a
Hence, for values of r considerably less than a> the equation of
the zero equipotential surface is
zc
0 = — 4 TT vz +2 77 o-'tf — 2 71 </ — +&c. ;
a
</c
or
Hence this equipotential surface near the axis is nearly flat.
Outside the disk, where r is greater than a, o> is zero when z is
zero, so that the plane of xy is part of the equipotential surface.
To find where these two parts of the surface meet, let us find at
dV
what point of this plane -=- = 0.
az
When r is very nearly equal to #, the solid angle o> becomes
approximately a lune of the sphere of unit radius whose angle is
tan-1 {z -*- (r-a)}, that is, w is 2 tan"1 {z -s- (r — a)}, so that
dV _ 2(/c
dz r—a
Hence, when
dV a'c ZQ ,
— = 0, rn — a-\ = a + — , nearly.
dz 2 77 (T 77
*2 2 6.] ACCUMULATORS. 319
The equipotential surface F=0 is therefore composed of a disk-
like figure of radius r0, and nearly uniform thickness #0, and of the
part of the infinite plane of xy which lies beyond this figure.
The surface-integral over the whole disk gives the charge of
electricity on it. It may be found, as in the theory of a circular
current in Part IV, Art. 704, to be
Q = 47ra(r'c {log
?*0 — a,
The charge on an equal area of the plane surface is ir(rr02, hence
the charge on the disk exceeds that on an equal area of the plane
in the ratio of z , STTT .
1 + 8 - log -- to unity,
where z is the thickness and r the radius of the disk, z being sup
posed small compared with r.
On Electric Accumulators and tJie Measurement of Capacity.
226.] An Accumulator or Condenser is an apparatus consisting of
two conducting surfaces separated by an insulating dielectric medium.
A Leyden jar is an accumulator in which an inside coating of
tinfoil is separated from the outside coating by the glass of which
the jar is made. The original Leyden phial was a glass vessel
containing water which was separated by the glass from the hand
which held it.
The outer surface of any insulated conductor may be considered
as one of the surfaces of an accumulator, the other being the earth
or the walls of the room in which it is placed, and the intervening
air being the dielectric medium.
The capacity of an accumulator is measured by the quantity of
electricity with which the inner surface must be charged to make
the difference between the potentials of the surfaces unity.
Since every electrical potential is the sum of a number of parts
found by dividing each electrical element by its distance from a
point, the ratio of a quantity of electricity to a potential must
have the dimensions of a line. Hence electrostatic capacity is a
linear quantity, or we may measure it in feet or metres without
ambiguity.
In electrical researches accumulators are used for two principal
purposes, for receiving and retaining large quantities of electricity
in as small a compass as possible, and for measuring definite quan
tities of electricity by means of the potential to which they raise
the accumulator.
320 ELECTROSTATIC INSTRUMENTS. [227-
For the retention of electrical charges nothing has been devised
more perfect than the Leyden jar. The principal part of the loss
arises from the electricity creeping along the damp uncoated surface
of the glass from the one coating to the other. This may be checked
in a great degree by artificially drying the air within the jar, and
by varnishing the surface of the glass where it is exposed to the
atmosphere. In Sir W. Thomson's electroscopes there is a very
small percentage of loss from day to day, and I believe that none
of this loss can be traced to direct conduction either through air
or through glass when the glass is good, but that it arises chiefly
from superficial conduction along the various insulating stems and
glass surfaces of the instrument.
In fact, the same electrician has communicated a charge to
sulphuric acid in a large bulb with a long neck, and has then her
metically sealed the neck by fusing it, so that the charge was com
pletely surrounded by glass, and after some years the charge was
found still to be retained.
It is only, however, when cold, that glass insulates in this
way, for the charge escapes at once if the glass is heated to
a temperature below 100°C.
When it is desired to obtain great capacity in small compass,
accumulators in which the dielectric is sheet caoutchouc, mica, or
paper impregnated with paraffin are convenient.
227.] For accumulators of the second class, intended for the
measurement of quantities of electricity, all solid dielectrics must be
employed with great caution on account of the property which they
possess called Electric Absorption.
The only safe dielectric for such accumulators is air, which has
this inconvenience, that if any dust or dirt gets into the narrow
space between the opposed surfaces, which ought to be occupied only
by air, it not only alters the thickness of the stratum of air, but
may establish a connexion between the opposed surfaces, in which
case the accumulator will not hold a charge.
To determine in absolute measure, that is to say in feet or metres,
the capacity of an accumulator, we must either first ascertain its
form and size, and then solve the problem of the distribution of
electricity on its opposed surfaces, or we must compare its capacity
with that of another accumulator, for which this problem has been
solved.
As the problem is a very difficult one, it is best to begin with an
accumulator constructed of a form for which the solution is known.
228.] MEASUREMENT OF CAPACITY. 321
Thus the capacity of an insulated sphere in an unlimited space is
known to be measured by the radius of the sphere.
A sphere suspended in a room was actually used by MM. Kohl-
rausch and Weber, as an absolute standard with which they com
pared the capacity of other accumulators.
The capacity, however, of a sphere of moderate size is so small
when compared with the capacities of the accumulators in common
use that the sphere is not a convenient standard measure.
Its capacity might be greatly increased by surrounding the-
sphere with a hollow concentric spherical surface of somewhat
greater radius. The capacity of the inner surface is then a fourth
proportional to the thickness of the stratum of air and the radii of
the two surfaces.
Sir W. Thomson has employed this arrangement as a standard of
capacity, but the difficulties of working the surfaces truly spherical,
of making them truly concentric, and of measuring their distance
and their radii with sufficient accuracy, are considerable.
We are therefore led to prefer for an absolute measure of capacity
a form in which the opposed surfaces are parallel planes.
The accuracy of the surface of the planes can be easily tested,
and their distance can be measured by a micrometer screw, and
may be made capable of continuous variation, which is a most
important property of a measuring instrument.
The only difficulty remaining arises from the fact that the planes
must necessarily be bounded, and that the distribution of electricity
near the boundaries of the planes has not been rigidly calculated.
It is true that if we make them equal circular disks, whose radius
is large compared with the distance between them, we may treat
the edges of the disks as if they were straight lines, and calculate
the distribution of electricity by the method due to Helmholtz, and
described in Art. 202. But it will be noticed that in this case
part of the electricity is distributed on the back of each disk, and
that in the calculation it has been supposed that there are no
conductors in the neighbourhood, which is not and cannot be the
case in a small instrument.
228.] We therefore prefer the following arrangement, due to
Sir W. Thomson, which we may call the Guard-ring arrangement,
by means of which the quantity of electricity on an insulated disk
may be exactly determined in terms of its potential.
VOL. I.
322
ELECTROSTATIC INSTRUMENTS.
[228.
The Guard-ring Accumulator.
Bb is a cylindrical vessel of conducting material of which the
outer surface of the upper face is accurately plane. This upper
surface consists of two parts,
a disk A, and a broad ring
SB surrounding the disk,
separated from it by a very
small interval all round, just
sufficient to prevent sparks
passing. The upper surface
LJ B J
A
B
ft aC/
G
G
n
i i &
of the disk is accurately in
y 21 the same plane with that of
the guard-ring. The disk is
supported by pillars of insulating material GG. C is a metal disk,
the under surface of which is accurately plane and parallel to BB.
The disk C is considerably larger than A. Its distance from A
is adjusted and measured by means of a micrometer screw, which
is not given in the figure.
This accumulator is used as a measuring instrument as follows : —
Suppose C to be at potential zero, and the disk A and vessel Bb
both at potential V. Then there will be no electrification on the
back of the disk because the vessel is nearly closed and is all at the
same potential. There will be very little electrification on the
edges of the disk because BB is at the same potential with the
disk. On the face of the disk the electrification will be nearly
uniform, and therefore the whole charge on the disk will be almost
exactly represented by its area multiplied by the surface-density on
a plane, as given in Art. 124.
In fact, we learn from the investigation in Art. 201 that the
charge on the disk is
(
- a
( 8^4 SA
where R is the radius of the disk, R' that of the hole in the guard-
ring, A the distance between A and C, and a a quantity which
cannot exceed (Rf—R) ^e •
If the interval between the disk and the guard-ring is small
compared with the distance between A and C, the second term will
be very small, and the charge on the disk will be nearly
22Q.] COMPARISON OP CAPACITIES. 323
Now let the vessel Bb be put in connexion with the earth. The
charge on the disk A will no longer be uniformly distributed, but it
will remain the same in quantity, and if we now discharge A we
shall obtain a quantity of electricity, the Value of which we know
in terms of 7t the original difference of potentials and the measur
able quantities R, Rf and A.
On the Comparison of the Capacity of Accumulators.
229.] The form of accumulator which is best fitted to have its
capacity determined in absolute measure from the form and dimen
sions of its parts is not generally the most suitable for electrical
experiments. It is desirable that the measures of capacity in actual
use should be accumulators having only two conducting surfaces, one
of which is as nearly as possible surrounded by the other. The
guard-ring accumulator, on the other hand, has three independent
conducting portions which must be charged and discharged in a
certain order. Hence it is desirable to be able to compare the
capacities of two accumulators by an electrical process, so as to test
accumulators which may afterwards serve as secondary standards.
I shall first shew how to test the equality of the capacity of two
guard-ring accumulators.
Let A be the disk, B the guard-ring with the rest of the con
ducting vessel attached to it, and C the large disk of one of these
accumulators, and let A', _5', and C' be the corresponding parts of
the other.
If either of these accumulators is of the more simple kind, having
only two conductors, we have only to suppress B or Bf9 and to
suppose A to be the inner and C the outer conducting surface, (?,
in this case being understood to surround A.
Let the following connexions be made.
Let B be kept always connected with C', and B' with C, that is,
let each guard-ring be connected with the large disk of the other
condenser.
(1) Let A be connected with B and C' and with /, the electrode
of a Leyden jar, and let A' be connected with B' and C and with
the earth.
(2) Let A, B, and Cf be insulated from /.
(3) Let A be insulated from B and C', and A' from & and C.
(4) Let B and (7 be connected with B' and C and with the
earth.
(5) Let A be connected with A'.
Y 2
324 ELECTROSTATIC INSTRUMENTS. [229.
(6) Let A and A' be connected with an electroscope E.
We may express these connexions as follows : —
(1) o = C'=i' = ^/ | A = £=C'=J.
(2) 0 = C=3'=A' | A = £=C'\J.
(3) Q = C=B'\A' | A\B=C'.
(4) 0 = C=£' \A' | A\ J5=0'=0.
(5) 0 = <?=JS'|^' = A\3=C'=0.
(6) 0 = <?=£' | A'=E = A \3=C'=0.
Here the sign of equality expresses electrical connexion, and the
vertical stroke expresses insulation.
In (l) the two accumulators are charged oppositely, so that A is
positive and A' negative, the charges on A and A being uniformly
distributed on the upper surface opposed to the large disk of each
accumulator.
In (2) the jar is removed, and in (3) the charges on A and A are
insulated.
In (4) the guard-rings are connected with the large disks, so that
the charges on A and A', though unaltered in magnitude, are now
distributed over their whole surface.
In (5) A is connected with A'. If the charges are equal and of
opposite signs, the electrification will be entirely destroyed, and
in (6) this is tested by means of the electroscope E.
The electroscope E will indicate positive or negative electrification
according as A or A' has the greater capacity.
By means of a key of proper construction, the whole of these
operations can be performed in due succession in a very small
fraction of a second, and the capacities adjusted till no electri
fication can be detected by the electroscope, and in this way the
capacity of an accumulator may be adjusted to be equal to that of
any other, or to the sum of the capacities of several accumulators,
so that a system of accumulators may be formed, each of which has
its capacity determined in absolute measure, i.e. in feet or in metres,
while at the same time it is of the construction most suitable for
electrical experiments.
This method of comparison will probably be found useful in
determining the specific capacity for electrostatic induction of
different dielectrics in the form of plates or disks. If a disk of
the dielectric is interposed between A and C, the disk being con
siderably larger than A, then the capacity of the accumulator will
229.] SPECIFIC INDUCTIVE CAPACITY. 325
be altered and made equal to that of the same accumulator when A
and C are nearer tog-ether. If the accumulator with the dielectric
plate, and with A and C at distance #, is of the same capacity as
the same accumulator without the dielectric, and with A and C at
distance x ', then, if a is the thickness of the plate, and K its specific
dielectric inductive capacity referred to air as a standard,
— x
The combination of three cylinders, described in Art. 127, has
been employed by "Sir W. Thomson as an accumulator whose capa
city may be increased or diminished by measurable quantities.
The experiments of MM. Gibson and Barclay with this ap
paratus are described in the Proceedings of the Royal Society, Feb. 2,
1871, and Phil. Trans., 1871, p. 573. They found the specific in
ductive capacity of paraffin to be 1.975, that of air being unity.
F.ABT II.
ELECTRO KINEMATICS.
CHAPTEK I.
THE ELECTEIC CUEEENT.
230.] WE have seen, in Art. 45, that when a conductor is in
electrical equilibrium the potential at every point of the conductor
must be the same.
If two conductors A and B are charged with electricity so that
the potential of A is higher than that of B, then, if they are put
in communication by means of a metallic wire C touching both of
them, part of the charge of A will be transferred to B, and the
potentials of A and B will become in a very short time equalized.
231.] During this process certain phenomena are observed in
the wire C, which are called the phenomena of the electric conflict
or current.
The first of these phenomena is the transference of positive
electrification from A to B and of negative electrification from B
to A. This transference may be also effected in a slower manner
by bringing a small insulated body into contact with A and B
alternately. By this process, which we may call electrical con
vection, successive small portions of the electrification of each body
are transferred to the other. In either case a certain quantity of
electricity, or of the state of electrification, passes from one place
to another along a certain path in the space between the bodies.
Whatever therefore may be our opinion of the nature of elec
tricity, we must admit that the process which we have described
constitutes a current of electricity. This current may be described
232.] THE VOLTAIC BATTERY. 327
as a current of positive electricity from A to B, or a current of
negative electricity from B to A, or as a combination of these two
currents.
According to Fechner's and Weber's theory it is a combination
of a current of positive electricity with an exactly equal current
of negative electricity in the opposite direction through the same
substance. It is necessary to remember this exceedingly artificial
hypothesis regarding the constitution of the current in order to
understand the statement of some of Weber's most valuable ex
perimental results.
If, as in Art. 36, we suppose P units of positive electricity
transferred from A to B, and N units of negative electricity trans
ferred from B to A in unit of time, then, according to Weber's
theory, P = N, and P or N is to be taken as the numerical measure
of the current.
We, on the contrary, make no assumption as to the relation
between P and N, but attend only to the result of the current,
namely, the transference of P + N of positive electrification from A
to B, and we shall consider P -f- N the true measure of the current.
The current, therefore, which Weber would call 1 we shall call 2.
On Steady Currents.
232.] 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 potentials 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 DanielPs Constant
Battery :—
A solution of sulphate of zinc is placed in a cell of porous earth
enware, 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 zinc, and a piece of copper is dipped into the sulphate
of copper. Wires are soldered to the zinc and to the copper above
328 THE ELECTRIC CURRENT. [233.
the surface of the liquid. This combination is called a cell or
element of Daniell's battery. See Art. 272.
233.] 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 zinc is put in contact with H, another insulated conductor
of the same metal as A, then it may be shewn by means of a delicate
electrometer that the potential of A exceeds that of B by a certain
quantity. This difference of potentials is called the Electromotive
Force of the Daniell's Cell.
If A and B are now disconnected from the cell and put in
communication by means of a wire, a transient current passes
through the wire from A to B, and the potentials of A and B
become equal. A and B 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 B. 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 zinc 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 zinc 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
construction, containing different chemical substances and different
metals, provided they are such that chemical action does not go
on when no current passes.
234.] Let us now consider a voltaic battery with its ends insulated
from each other. The copper end will be positively or vitreously
electrified, and the zinc end will be negatively or resinously
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.
236.J ELECTROLYSIS. 329
Properties of the Current.
235.] The current forms a closed circuit in the direction from
copper to zinc through the wires, and from zinc to copper through
the solutions.
If the circuit be broken by cutting any of the wires which
connect the copper of one cell with the zinc of the next in order, the
current will be stopped, and the potential of the end of the wire
in connexion with the copper will be found to exceed that of the
end of the wire in connexion with the zinc by a constant quantity,
namely, the total electromotive force of the circuit.
Electrolytic Action of the Current.
236.] As long as the circuit is broken no chemical action goes
on in the cells, but as soon as the circuit is completed, zinc is
dissolved from the zinc in each of the Darnell's cells, and copper is
deposited on the copper.
The quantity of sulphate of zinc increases, and the quantity of
sulphate of copper diminishes unless more is constantly supplied.
The quantity of zinc dissolved and also that of copper deposited is
the same in each of the Daniell's cells throughout the circuit, what
ever the size of the plates of the cell, and if any of the cells be of a
different construction, the amount of chemical action in it bears
a constant proportion to the action in the Daniell's cell. For
instance, if one of the cells consists of two platinum plates dipped
into sulphuric acid diluted with water, oxygen will be given off
at the surface of the plate where the current enters the liquid,
namely, the plate in metallic connexion with the copper of Daniell's
cell, and hydrogen at the surface of the plate where the current
leaves the liquid, namely, the plate connected with the zinc of
Daniell's cell.
The volume of the hydrogen is exactly twice the volume of the
oxygen given off in the same time, and the weight of the oxygen is
exactly eight times the weight of the hydrogen.
In every cell of the circuit the weight of each substance dissolved,
deposited, or decomposed is equal to a certain quantity called the
electrochemical equivalent of that substance, multiplied by the
strength of the current and by the time during which it has
been flowing.
For the experiments which established this principle, see the
seventh and eighth series of Faraday's Experimental Researches;
330 THE ELECTRIC CURRENT. [237.
and for an investigation of the apparent exceptions to the rule, see
Miller's Chemical Physics and Wiedemann's Galvanismus.
237.] Substances which are decomposed in this way are called
Electrolytes. The process is called Electrolysis. The places where
the current enters and leaves the electrolyte are called Electrodes.
Of these the electrode by which the current enters is called the
Anode, and that by which it leaves the electrolyte is called the
Cathode. The components into which the electrolyte is resolved
are called Ions : that which appears at the anode is called the
Anion, and that which appears at the cathode is called the Cation.
Of these terms, which were, I believe, invented by Faraday with
the help of Dr. Whewell, the first three, namely, electrode, elec
trolysis, and electrolyte have been generally adopted, and the mode
of conduction of the current in which this kind of decomposition
and transfer of the components takes place is called Electrolytic
Conduction.
If a homogeneous electrolyte is placed in a tube of variable
section, and if the electrodes are placed at the ends of this tube,
it is found that when the current passes, the anion appears at
the anode and the cation at the cathode, the quantities of these
ions being electrochemically equivalent, and such as to be together
equivalent to a certain quantity of the electrolyte. In the other
parts of the tube, whether the section be large or small, uniform
or varying, the composition of the electrolyte remains unaltered.
Hence the amount of electrolysis which takes place across every
section of the tube is the same. Where the section is small the
action must therefore be more intense than where the section is
large, but the total amount of each ion which crosses any complete
section of the electrolyte in a given time is the same for all sections.
The strength of the current may therefore be measured by the
amount of electrolysis in a given time. An instrument by which
the quantity of the electrolytic products can be readily measured
is called a Toltameter.
The strength of the current, as thus measured, is the same
at every part of the circuit, and the total quantity of the elec
trolytic products in the voltameter after any given time is pro
portional to the amount of electricity which passes any section in
the same time.
238.] If we introduce a voltameter at one part of the circuit
of a voltaic battery, and break the circuit at another part, we may
suppose the measurement of the current to be conducted thus.
239'] MAGNETIC ACTION. 331
Let the ends of the broken circuit be A and B, and let A be the
anode and B the cathode. Let an insulated ball be made to touch
A and B alternately, it will carry from A to B a certain measurable
quantity of electricity at each journey. This quantity may be
measured by an electrometer, or it may be calculated by mul
tiplying the electromotive force of the circuit by the electrostatic
capacity of the ball. Electricity is thus carried from A to B on the
insulated ball by a process which may be called Convection. At
the same time electrolysis goes on in the voltameter and in the
cells of the battery, and the amount of electrolysis in each cell may
be compared with the amount of electricity carried across by the
insulated ball. The quantity of a substance which is electrolysed
by one unit of electricity is called an Electrochemical equivalent
of that substance.
This experiment would be an extremely tedious and troublesome
one if conducted in this way with a ball of ordinary magnitude
and a manageable battery, for an enormous number of journeys
would have to be made before an appreciable quantity of the electro
lyte was decomposed. The experiment must therefore be considered
as a mere illustration, the actual measurements of electrochemical
equivalents being conducted in a different way. But the experi
ment may be considered, as an illustration of the process of elec
trolysis itself, for if we regard electrolytic conduction as a species
of convection in which an electrochemical equivalent of the anion
travels with negative electricity in the direction of the anode, while
an equivalent of the cation travels with positive electricity in
the direction of the cathode, the whole amount of transfer of elec
tricity being one unit, we shall have an idea of the process of
electrolysis, which, so far as I know, is not inconsistent with known
facts, though, on account of our ignorance of the nature of electricity
and of chemical compounds, it may be a very imperfect repre
sentation of what really takes place.
Magnetic Action of the Current.
239.] 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. See Art. 475.
If a man were to place his body in the line of the current so
that the current from copper through the wire to zinc 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
332 THE ELECTRIC CURRENT. [240.
to point to the north would, when the current flows, tend to point
towards the man's right hand.
The nature and laws of this electromagnetic action will be dis
cussed when we come to the fourth part of this treatise. What
we are concerned with at present is the fact 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 circuit 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.
240.] 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 mea
suring the strength of electric currents. We shall therefore assume
the possibility of constructing such an instrument in studying the
laws of these currents, reserving the discussion of the principles of
the instrument for our fourth part. When therefore we say that
an electric current is of a certain strength we suppose that the
measurement is effected by the galvanometer*
CHAPTEE IL
CONDUCTION AND EESISTANCE.
241.] IF by means of an electrometer we determine the electric
potential at different points of a circuit in which a constant electric
current is maintained, we shall find that in any portion of the
circuit consisting- of a single metal of uniform temperature through-
outj the potential at any point exceeds that at any other point
farther on in the direction of the current by a quantity depending
on the strength of the current and on the nature and dimensions
of the intervening portion of the circuit. The difference of the
potentials at the extremities of this portion of the circuit is called
the External electromotive force acting on it. If the portion of
the circuit under consideration is not homogeneous, but contains
transitions from one substance to another, from metals to elec
trolytes, or from hotter to colder parts, there may be, besides the
external electromotive force, Internal electromotive forces which
must be taken into account.
The relations between Electromotive Force,, Current, and Resist
ance were first investigated by Dr. G. S. Ohm, in a work published
in 1827, entitled Die Galvanische Kette Mathematisch Bearbeitet,
translated in Taylor's Scientific Memoirs. The result of these in
vestigations in the case of homogeneous conductors is commonly
called ' Ohm's Law.'
Ohm's Law.
The electromotive force acting between the extremities of any part
of a circuit is the product of the strength of the current and the
resistance of that part of the circuit,
Here a new term is introduced, the Resistance of a conductor,
which is defined to be the ratio of the electromotive force to
the strength of the current which it produces. The introduction
334: CONDUCTION AND RESISTANCE. [242.
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 conductor is altered.
In the first place, then, the resistance of a conductor is inde
pendent 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.
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. In the sixth chapter we
shall trace its applications and consequences.
Generation of Heat ~by the Current.
242.] We have seen that when an electromotive force causes
a current to flow through a conductor, electricity is transferred
from a place of higher to a place of lower potential. If the transfer
had been made by convection, that is, by carrying successive charges
on a ball from the one place to the other, work would have been
done by the electrical forces on the ball, and this might have
been turned to account. It is actually turned to account in a
partial manner in those dry pile circuits where the electrodes have
the form of bells, and the carrier ball is made to swing like a
pendulum between the two bells and strike them alternately. In
this way the electrical action is made to keep up the swinging
of the pendulum and to propagate the sound of the bells to a
distance. In the case of the conducting wire we have the same
transfer of electricity from a place of high to a place of low potential
without any external work being done. The principle of the Con
servation of Energy therefore leads us to look for internal work in
the conductor. In an electrolyte this internal work consists partly
of the separation of its components. In other conductors it is
entirely converted into heat.
The energy converted into heat is in this case the product of
the electromotive force into the quantity of electricity which passes.
But the electromotive force is the product of the current into the
244-] COMPAEISON WITH PHENOMENA OF HEAT. 335
resistance, and the quantity of electricity is the product of the
current into the time. Hence the quantity of heat multiplied by
the mechanical equivalent of unit of heat is equal to the square of
the strength of the current multiplied into the resistance and into
the time.
The heat developed by electric currents in overcoming the re
sistance of conductors has been determined by Dr. Joule, who first
established that the heat produced in a given time is proportional
to the square of the current, and afterwards by careful absolute
measurements of all the quantities concerned, verified the equation
JN= C2Rt,
where / is Joule's dynamical equivalent of heat, H the number of
units of heat, C the strength of the current, R the resistance of the
conductor, and t the time during which the current flows. These
relations between electromotive force, work, and heat, were first fully
explained by Sir W. Thomson in a paper on the application of the
principle of mechanical effect to the measurement of electromotive
forces*.
243.] The analogy between the theory of the conduction of elec
tricity and that of the conduction of heat is at first sight almost
complete. If we take two systems geometrically similar, and such
that the conductivity for heat at any part of the first is proportional
to the conductivity for electricity at the corresponding part of the
second, and if we also make the temperature at any part of the
first proportional to the electric potential at the corresponding point
of the second, then the flow of heat across any area of the first
will be proportional to the flow of electricity across the corre
sponding area of the second.
Thus, in the illustration we have given, in which flow of elec
tricity corresponds to flow of heat, and electric potential to tem
perature, electricity tends to flow from places of high to places
of low potential, exactly as heat tends to flow from places of high
to places of low temperature.
244.] The theory of potential and that of temperature may
therefore be made to illustrate one another ; there is, however, one
remarkable difference between the phenomena of electricity and
those of heat.
Suspend a conducting body within a closed conducting vessel by
a silk thread, and charge the vessel with electricity. The potential
* Phil Mag., Dec. 1851.
336 CONDUCTION AND RESISTANCE. [245.
of the vessel and of all within it will be instantly raised, but
however long and however powerfully the vessel be electrified, and
whether the body within be allowed to come in contact with the
vessel or not, no signs of electrification will appear within the
vessel, nor will the body within shew any electrical effect when
taken out.
But if the vessel is raised to a high temperature, the body
within will rise to the same temperature, but only after a con
siderable time, and if it is then taken out it will be found hot,
and will remain so till it has continued to emit heat for some time.
The difference between the phenomena consists in the fact that
bodies are capable of absorbing and emitting heat, whereas they
have no corresponding property with respect to electricity. A body
cannot be made hot without a certain amount of heat being
supplied to it, depending on the mass and specific heat of the body,
but the electric potential of a body may be raised to any extent
in the way already described without communicating any electricity
to the body.
245.] Again, suppose a body first heated and then placed inside
the closed vessel. The outside of the vessel will be at first at the
temperature of surrounding bodies, but it will soon get hot, and
will remain hot till the heat of the interior body has escaped.
It is impossible to perform a corresponding electrical experiment.
It is impossible so to electrify a body, and so to place it in a
hollow vessel, that the outside of the vessel shall at first shew no
signs of electrification but shall afterwards become electrified. It
was for some phenomenon of this kind that Faraday sought in
vain under the name of an absolute charge of electricity.
Heat may be hidden in the interior of a body so as to have no
external action, but it is impossible to isolate a quantity of elec
tricity so as to prevent it from being constantly in inductive
relation with an equal quantity of electricity of the opposite kind.
There is nothing therefore among electric phenomena which
corresponds to the capacity of a body for heat. This follows at
once from the doctrine which is asserted in this treatise, that
electricity obeys the same condition of continuity as an incom
pressible fluid. It is therefore impossible to give a bodily charge
of electricity to any substance by forcing an additional quantity of
electricity into it. See Arts. 61, 111, 329, 334.
CHAPTER III.
ELECTROMOTIVE FORCE BETWEEN BODIES IN CONTACT.
The Potentials of Different Substances in Contact.
246.] IF we define the potential of a hollow conducting vessel
as the potential of the air inside the vessel, we may ascertain this
potential by means of an electrometer as described in Part I,
Art. 222.
If we now take two hollow vessels of different metals, say copper
and zinc, and put them in metallic contact with each other, and
then test the potential of the air inside each vessel, the potential
of the air inside the zinc vessel will be positive as compared with
that inside the copper vessel. The difference of potentials depends
on the nature of the surface of the insides of the vessels, being
greatest when the zinc is bright and when the copper is coated
with oxide.
It appears from this that when two different metals are in
contact there is in general an electromotive force acting from the
one to the other, so as to make the potential of the one exceed
that of the other by a certain quantity. This is Volta's theory of
Contact Electricity.
If we take a certain metal, say copper, as the standard, then
if the potential of iron in contact with copper at the zero potential
is /, and that of zinc in contact with copper at zero is Z, then
the potential of zinc in contact with iron at zero will be Z— /.
It appears from this result, which is true of any three metals,
that the differences of potential of any two metals at the same
temperature in contact is equal to the difference of their potentials
when in contact with a third metal, so that if a circuit be formed
of any number of metals at the same temperature there will be
electrical equilibrium as soon as they have acquired their proper
potentials, and there will be no current kept up in the circuit.
VOL. I. Z
338 CONTACT FORCE. [247,
247.] If, however, the circuit consist of two metals and an elec
trolyte, the electrolyte, according- to Volta's theory, tends to reduce
the potentials of the metals in contact with it to equality, so that
the electromotive force at the metallic junction is no longer balanced,
and a continuous current is kept up. The energy of this current
is supplied by the chemical action which takes place between the
electrolyte and the metals.
248.] The electric effect may, however, be produced without
chemical action if by any other means we can produce an equali
zation of the potentials of two metals in contact. Thus, in an
experiment due to Sir W. Thomson *, a copper funnel is placed in
contact with a vertical zinc cylinder, so that when copper filings
are allowed to pass through the funnel, they separate from each
other and from the funnel near the middle of the zinc cylinder,
and then fall into an insulated receiver placed below. The receiver
is then found to be charged negatively, and the charge increases
as the filings continue to pour into it. At the same time the zinc
cylinder with the copper funnel in it becomes charged more and
more positively.
. If now the zinc cylinder were connected with the receiver by a
wire, there would be a positive current in the wire from the cylinder
to the receiver. The stream of copper filings, each filing charged
negatively by induction, constitutes a negative current from the
funnel to the receiver, or, in other words, a positive current from
the receiver to the copper funnel. The positive current, therefore,
passes through the air (by the filings) from zinc to copper, and
through the metallic junction from copper to zinc, just as in the
ordinary voltaic arrangement, but in this case the force which keeps
up the current is not chemical action but gravity, which causes the
filings to fall, in spite of the electrical attraction between the
positively charged funnel and the negatively charged filings.
249.] A remarkable confirmation of the theory of contact elec
tricity is supplied by the discovery of Peltier, that, when a current
of electricity crosses the junction of two metals, the junction is
heated when the current is in one direction, and cooled when it
is in the other direction. It must be remembered that a current
in its passage through a metal always produces heat, because it
meets with resistance, so that the cooling effect on the whole
conductor must always be less than the heating effect. We must
therefore distinguish between the generation of heat in each metal,
* North British Keview, 1864, p. 353; and Proc. R. S., June 20, 1867.
249-] PELTIER'S PHENOMENON". 339
due to ordinary resistance, and the generation or absorption of heat
at the junction of two metals. We shall call the first the frictional
generation of heat by the current, and, as we have seen, it is
proportional to the square of the current, and is the same whether
the current be in the positive or the negative direction. The second
we may call the Peltier effect, which changes its sign with that
of the current.
The total heat generated in a portion of a compound conductor
consisting of two metals may be expressed by
H=~C*t-UCt,
where // is the quantity of heat, / the mechanical equivalent of
unit of heat, R the resistance of the conductor, C the current, and
t the time ; n being the coefficient of the Peltier effect, that is,
the heat absorbed at the junction by unit of current in unit of
time.
Now the heat generated is mechanically equivalent to the work
done against electrical forces in the conductor, that is, it is equal
to the product of the current into the electromotive force producing
it. Hence, if E is the external electromotive force which causes
the current to flow through the conductor,
JN= CEt = RC2 1 - J n Ct,
whence E = RC—JU.
It appears from this equation that the external electromotive
force required to drive the current through the compound conductor
is less than that due to its resistance alone by the electromotive
force JYl. Hence JU represents the electromotive contact force
at the junction acting in the positive direction.
This application, due to Sir W. Thomson *, of the dynamical
theory of heat to the determination of a local electromotive force
is of great scientific importance, since the ordinary method of
connecting two points of the compound conductor with the elec
trodes of a galvanometer or electroscope by wires would be useless,
owing to the contact forces at the junctions of the wires with
the materials of the compound conductor. In the thermal method,
on the other hand, we know that the only source of energy is the
current of electricity, and that no work is done by the current
in a certain portion of the circuit except in heating that portion
of the conductor. If, therefore, we can measure the amount of the
* Proc. K. S. Edin., Dec. 15, 1851 ; and Trans. E. S. Edin., 1854.
Z 2
340 CONTACT FOKCE. [250.
current and the amount of heat produced or absorbed, we can
determine the electromotive force required to urge the current
through that portion of the conductor, and this measurement is
entirely independent of the effect of contact forces in other parts of
the circuit.
The electromotive force at the junction of two metals, as de
termined by this method, does not account for Volta's electromotive
force as described in Art. 246. The latter is in general far greater
than that of this Article, and is sometimes of opposite sign. Hence
the assumption that the potential of a metal is to be measured by
that of the air in contact with it must be erroneous, and the greater
part of Volta's electromotive force must be sought for, not at the
junction of the two metals, but at one or both of the surfaces which
separate the metals from the air or other medium which forms the
third element of the circuit.
250.] The discovery by Seebeck of thermoelectric currents in
circuits of different metals with their junctions at different tem
peratures, shews that these contact forces do not always balance
each other in a complete circuit. It is manifest, however, that
in a complete circuit of different metals at uniform temperature the
contact forces must balance each other. For if this were not the
case there would be a current formed in the circuit, and this current
might be employed to work a machine or to generate heat in the
circuit, that is, to do work, while at the same time there is no
expenditure of energy, as the circuit is all at the same temperature,
and no chemical or other change takes place. Hence, if the Peltier
effect at the junction of two metals a and b be represented by Ha6
when the current flows from a to $, then for a circuit of two metals
at the same temperature we must have
na& + nba = o,
and for a circuit of three metals a, 6, c, we must have
n6o+ nca+nab = o.
It follows from this equation that the three Peltier effects are not
independent, but that one of them can be deduced from the other
two. For instance, if we suppose c to be a standard metal, and
if we write Pa = /flac and Pb = JUbc , then
Jnat = Pa-Pb.
The quantity Pa is a function of the temperature, and depends on
the nature of the metal a.
251.] It has also been shewn by Magnus that if a circuit is
25I-] THERMOELECTRIC PHENOMENA. 341
formed of a single metal no current will be formed in it, however
the section of the conductor and the temperature may vary in
different parts.
Since in this case there is conduction of heat and consequent
dissipation of energy, we cannot, as in the former case, consider this
result as self-evident. The electromotive force, for instance, between
two portions of a circuit might have depended on whether the
current was passing from a thick portion of the conductor to a thin
one, or the reverse, as well as on its passing rapidly or slowly from a
hot portion to a cold one, or the reverse, and this would have made
a current possible in an unequally heated circuit of one metal.
Hence, by the same reasoning as in the case of Peltier's phe
nomenon, we find that if the passage of a current through a
conductor of one metal produces any thermal effect which is re
versed when the current is reversed, this can only take place when
the current flows from places of high to places of low temperature,
or the reverse, and if the heat generated in a conductor of one
metal in flowing from a place where the temperature is a? to a
place where it is y, is H, then
and the electromotive force tending to maintain the current will
be Sxy.
If os, y, z be the temperatures at three points of a homogeneous
circuit, we must have
syz+szx+szv = o,
according to the result of Magnus. Hence, if we suppose z to be
the zero temperature, and if we put
QX=SXZ and Qv = Syz)
^ find Sxv=Q*-Qy,
where Qx is a function of the temperature #, the form of the
function depending on the nature of the metal.
If we now consider a circuit of two metals a and b in which
the temperature is x where the current passes from a to #, and
y where it passes from I to a, the electromotive force will be
where Pax signifies the value of P for the metal a at the tempera
ture x or
Since in unequally heated circuits of different metals there are in
342 CONTACT FORCE. [252-
general thermoelectric currents, it follows that P and Q are in
general different for the same metal and same temperature.
252.] The existence of the quantity Q was first demonstrated by
Sir W. Thomson, in the memoir we have referred to, as a deduction
from the phenomenon of thermoelectric inversion discovered by
Gumming "*, who found that the order of certain metals in the ther
moelectric scale is different at high and at low temperatures, so that
for a certain temperature two metals may be neutral to each other.
Thus, in a circuit of copper and iron if one junction be kept at the
ordinary temperature while the temperature of the other is raised,
a current sets from copper to iron through the hot junction, and
the electromotive force continues to increase till the hot junction
has reached a temperature T, which, according to Thomson, is
about 284°C. When the temperature of the hot junction is raised
still further the electromotive force is reduced, and at last, if the
temperature be raised high enough, the current is reversed. The
reversal of the current may be obtained more easily by raising the
temperature of the colder junction. If the temperature of both
junctions is above T the current sets from iron to copper through
the hotter junction, that is, in the reverse direction to that ob
served when both junctions are below T.
Hence, if one of the junctions is at the neutral temperature T
and the other is either hotter or colder, the current will set from
copper to iron through the junction at the neutral temperature.
253.] From this fact Thomson reasoned as follows : —
Suppose the other junction at a temperature lower than T.
The current may be made to work an engine or to generate heat in
a wire, and this expenditure of energy must be kept up by the
transformation of heat into electric energy, that is to say, heat
must disappear somewhere in the circuit. Now at the tempera
ture T iron and copper are neutral to each other, so that no
reversible thermal effect is produced at the hot junction, and at
the cold junction there is, by Peltier's principle, an evolution of
heat by the current. Hence the only place where the heat can dis
appear is in the copper or iron portions of the circuit, so that either
a current in iron from hot to cold must cool the iron, or a current
in copper from cold to hot must cool the copper, or both these
effects may take place. By an elaborate series of ingenious experi
ments Thomson succeeded in detecting the reversible thermal action
of the current in passing between parts of different temperatures,
* Cambridge Transactions, 1823.
254-] EXPERIMENTS OF TAIT. 343
and he found that the current produced opposite effects in copper
and in iron*.
When a stream of a material fluid passes along1 a tube from
a hot part to a cold part it heats the tube, and when it passes
from cold to hot it cools the tube, and these effects depend on
the specific capacity for heat of the fluid. If we supposed elec
tricity, whether positive or negative,, to be a material fluid, we
might measure its specific heat by the thermal effect on an un
equally heated conductor. Now Thomson's experiments shew that
positive electricity in copper and negative electricity in iron carry
heat with them from hot to cold. Hence, if we supposed either
positive or negative electricity to be a fluid, capable of being
heated and cooled, and of communicating heat to other bodies, we
should find the supposition contradicted by iron for positive elec
tricity and by copper for negative electricity, so that we should
have to abandon both hypotheses.
This scientific prediction of the reversible effect of an electric
current upon an unequally heated conductor of one metal is another
instructive example of the application of the theory of Conservation
of Energy to indicate new directions of scientific research. Thomson
has also applied the Second Law of Thermodynamics to indicate
relations between the quantities which we have denoted by P
and Q, and has investigated the possible thermoelectric properties
of bodies whose structure is different in different directions. He
has also investigated experimentally the conditions under which
these properties are developed by pressure, magnetization, &c.
254.] Professor Taitf has recently investigated the electro
motive force of thermoelectric circuits of different metals, having
their junctions at different temperatures. He finds that the elec
tromotive force of a circuit may be expressed very accurately by
the formula
where ^ is the absolute temperature of the hot junction, t2 that
of the cold junction, and tQ the temperature at which the two metals
are neutral to each other. The factor a is a coefficient depending
on the nature of the two metals composing the circuit. This law
has been verified through considerable ranges of temperature by
Professor Tait and his students, and he hopes to make the thermo
electric circuit available as a thermometric instrument in his
* ' On the Electrodynamic Qualities of Metals.' Phil. Trans., 1856,
t Proc. R. S. Edin., Session 1870-71, p. 308, also Dec. 18, 1871.
344 CONTACT FORCE. [254.
experiments on the conduction of heat, and in other cases in which
the mercurial thermometer is not convenient or has not a sufficient
range.
According to Tait's theory, the quantity which Thomson calls
the specific heat of electricity is proportional to the absolute tem
perature in each pure metal, though its magnitude and even its
sign vary in different metals. From this he has deduced by ther-
modynamic principles the following results. Let 7cat, kbt, Jcct
be the specific heats of electricity in three metals a, b, c, and let
Tbc, Tca, Tab be the temperatures at which pairs of these metals are
neutral to each other, then the equations
(kb-kc}Tbc+(kc-ka) Tca+(ka-kb)Tab = 0,
express the relation of the neutral temperatures, the value of the
Peltier effect, and the electromotive force of a thermoelectric circuit.
CHAPTER IV.
ELECTROLYSIS.
Electrolytic Conduction.
255.] I HAVE already stated that when an electric current in
any part of its circuit passes through certain compound substances
called Electrolytes, the passage of the current is accompanied by
a certain chemical process called Electrolysis, in which the substance
is resolved, into two components called Ions, of which one, called
the Anion, or the electronegative component, appears at the Anode,
or place where the current enters the electrolyte, and the other,
called the Cation, appears at the Cathode, or the place where the
current leaves the electrolyte.
The complete investigation of Electrolysis belongs quite as much
to Chemistry as to Electricity. We shall consider it from an
electrical point of view, without discussing its application to the
theory of the constitution of chemical compounds.
Of all electrical phenomena electrolysis appears the most likely
to furnish us with a real insight into the true nature of the electric
current, because we find currents of ordinary matter and currents
of electricity forming essential parts of the same phenomenon.
It is probably for this very reason that, in the present imperfectly
formed state of our ideas about electricity, the theories of electro
lysis are so unsatisfactory.
The fundamental law of electrolysis, which was established by
Faraday, and confirmed by the experiments of Beetz, Hittorf, and
others down to the present time, is as follows : —
The number of electrochemical equivalents of an electrolyte which
are decomposed by the passage of an electric current during a given
time is equal to the number of units of electricity which are trans
ferred .by the current in the same time.
The electrochemical equivalent of a substance is that quantity
346 ELECTROLYSIS.' [255.
of the substance which is electrolysed by a unit current passing
through the substance for a unit of time, or, in other words, by the
passage of a unit of electricity. When the unit of electricity is
defined in absolute measure the absolute value of the electro
chemical equivalent of each substance can be determined in grains
or in grammes.
The electrochemical equivalents of different substances are pro
portional to their ordinary chemical equivalents. The ordinary
chemical equivalents, however, are the mere numerical ratios in
which the substances combine, whereas the electrochemical equi
valents are quantities of matter of a determinate magnitude, de
pending on the definition of the unit of electricity.
Every electrolyte consists of two components, which, during the
electrolysis, appear where the current enters and leaves the elec
trolyte, and nowhere else. Hence, if we conceive a surface described
within the substance of the electrolyte, the amount of electrolysis
which takes place through this surface, as measured by the elec
trochemical equivalents of the components transferred across it
in opposite directions, will be proportional to the total electric
current through the surface.
The actual transfer of the ions through the substance of the
electrolyte in opposite directions is therefore part of the phenomenon
of the conduction of an electric current through an electrolyte. At
every point of the electrolyte through which an electric current
is passing there are also two opposite material currents of the anion
and the cation, which have the same lines of flow with the electric
current, and are proportional to it in magnitude.
It is therefore extremely natural to suppose that the currents of
the ions are convection currents of electricity, and, in particular,
that every molecule of the cation is charged with a certain fixed
quantity of positive electricity, which is the same for the molecules
of all cations, and that every molecule of the anion is charged with
an equal quantity of negative electricity.
The opposite motion of the ions through the electrolyte would
then be a complete physical representation of the electric current.
We may compare this motion of the ions with the motion of gases
and liquids through each other during the process of diffusion,
there being this difference between the two processes, that, in
diffusion, the different substances are only mixed together and the
mixture is not homogeneous, whereas in electrolysis they are chemi
cally combined and the electrolyte is homogeneous. In diffusion
257-] THEORY OF CLAUSIUS. 347
the determining1 cause of the motion of a substance in a given
direction is a diminution of the quantity of that substance per
unit of volume in that direction, whereas in electrolysis the motion
of each ion is due to the electromotive force acting on the charged
molecules.
256.] Clausius*, who has bestowed much study on the theory
of the molecular agitation of bodies, supposes that the molecules
of all bodies are in a state of constant agitation, but that in solid
bodies each molecule never passes beyond a certain distance from
its original position, whereas in fluids a molecule, after moving
a certain distance from its original position, is just as likely to
move still farther from it as to move back again. Hence the
molecules of a fluid apparently at rest are continually changing
their positions, and passing irregularly from one part of the fluid
to another. In a compound fluid he supposes that not only the
compound molecules travel about in this way, but that, in the
collisions which occur between the compound molecules, the mole
cules of which they are composed are often separated and change
partners, so that the same individual atom is at one time associated
with one atom of the opposite kind, and at another time with another.
This process Clausius supposes to go on in the liquid at all times, but
when an electromotive force acts on the liquid the motions of the
molecules, which before were indifferently in all directions, are now
influenced by the electromotive force, so that the positively charged
molecules have a greater tendency towards the cathode than towards
the anode, and the negatively charged molecules have a greater
tendency' to move in the opposite direction. Hence the molecules
of the cation will daring their intervals of freedom struggle towards
the cathode, but will continually be checked in their course by
pairing for a time with molecules of the anion, which are also
struggling through the crowd, but in the opposite direction.
257.] This theory of Clausius enables us to understand how it is,
that whereas the actual decomposition of an electrolyte requires an
electromotive force of finite magnitude, the conduction of the
current in the electrolyte obeys the law of Ohm, so that every
electromotive force within the electrolyte, even the feeblest, produces
a current of proportionate magnitude.
According to the theory of Clausius, the decomposition and
recomposition of the electrolyte is continually going on even when
there is no current, and the very feeblest electromotive force is
* Fogg. Ann. bd. ci. s. 338 (1857).
348 ELECTROLYSIS. [258.
sufficient to give this process a certain degree of direction, and so
to produce the currents of the ions and the electric current, which
is part of the same phenomenon. Within the electrolyte, however,
the ions are never set free in finite quantity, and it is this liberation
of the ions which requires a finite electromotive force. At the
electrodes the ions accumulate, for the successive portions of the
ions, as they arrive at the electrodes, instead of finding- molecules of
the opposite ion ready to combine with them, are forced into com
pany with molecules of their own kind, with which they cannot
combine. The electromotive force required to produce this effect
is of finite magnitude, and forms an opposing electromotive force
which produces a reversed current when other electromotive forces
are removed. When this reversed electromotive force, owing to
the accumulation of the ions at the electrode, is observed, the
electrodes are said to be Polarized.
258.] One of the best methods of determining whether a body
is or is not an electrolyte is to place it between platinum electrodes
and to pass a current through it for some time, and then, dis
engaging the electrodes from the voltaic battery, and connecting
them with a galvanometer, to observe whether a reverse current,
due to polarization of the electrodes, passes through the galvano
meter. Such a current, being due to accumulation of different
substances on the two electrodes, is a proof that the substance has
been elect rolytically decomposed by the original current from the
battery. This method can often be applied where it is difficult,
by direct chemical methods, to detect the presence of the products
of decomposition at the electrodes. See Art. 271.
259.] So far as we have gone the theory of electrolysis appears
very satisfactory. It explains the electric current, the nature of
which we do not understand, by means of the currents of the
material components of the electrolyte, the motion of which,
though not visible to the eye, is easily demonstrated. It gives a
clear explanation, as Faraday has shewn, why an electrolyte which
conducts in the liquid state is a non-conductor when solidified, for
unless the molecules can pass from one part to another no elec
trolytic conduction, can take place, so that the substance must
be in a liquid state, either by fusion or by solution, in order to be
a conductor.
But if we go on, and assume that the molecules of the ions
within the electrolyte are actually charged with certain definite
quantities of electricity, positive and negative, so that the elec-
260.] MOLECULAR CHARGE. 849
trolytic current is simply a current of convection, we find that this
tempting hypothesis leads us into very difficult ground.
In the first place, we must assume that in every electrolyte each
molecule of the cation, as it is liberated at the cathode, commu
nicates to the cathode a charge of positive electricity, the amount
of which is the same for every molecule, not only of that cation
but of all other cations. In the same way each molecule of the
auion when liberated, communicates to the anode a charge of
negative electricity, the numerical magnitude of which is the same
as that of the positive charge due to a molecule of a cation, but
with sign reversed.
If, instead of a single molecule, we consider an assemblage of
molecules, constituting an electrochemical equivalent of the ion,
then the total charge of all the molecules is, as we have seen, one
unit of electricity, positive or negative.
260.] We do not as yet know how many molecules there are
in an electrochemical equivalent of any substance, but the molecular
theory of chemistry, which is corroborated by many physical con
siderations, supposes that the number of molecules in an elec
trochemical equivalent is the same for all substances. We may
therefore, in molecular speculations, assume that the number of
molecules in an electrochemical equivalent is JV, a number unknown
at present, but which we may hereafter find means to determine *.
Each molecule, therefore, on being liberated from the state of
combination, parts with a charge whose magnitude is — , and is
positive for the cation and negative for the anion. This definite
quantity of electricity we shall call the molecular charge. If it
were known it would be the most natural unit of electricity.
Hitherto we have only increased the precision of our ideas by
exercising our imagination in tracing the electrification of molecules
and the discharge of that electrification.
The liberation of the ions and the passage of positive electricity
from the anode and into the cathode are simultaneous facts. The
ions, when liberated, are not charged with electricity, hence, when
they are in combination, they have the molecular charges as above
described.
The electrification of a molecule, however, though easily spoken
of, is not so easily conceived.
We know that if two metals are brought into contact at any
* See note to Art. 5.
350 ELECTROLYSIS. [260.
point, the rest of their surfaces will be electrified, and if the metals
are in the form of two plates separated by a narrow interval of air,
the charge on each plate may become of considerable magnitude.
Something like this may be supposed to occur when the two
components of an electrolyte are in combination. Each pair of
molecules may be supposed to touch at one point, and to have the
rest of their surface charged with electricity due to the electro
motive force of contact.
But to explain the phenomenon, we ought to shew why the
charge thus produced on each molecule is of a fixed amount, and
why, when a molecule of chlorine is combined with a molecule of
zinc, the molecular charges are the same as when a molecule of
chlorine is combined with a molecule of copper, although the elec
tromotive force between chlorine and zinc is much greater than
that between chlorine and copper. If the charging of the molecules
is the effect of the electromotive force of contact, why should
electromotive forces of different intensities produce exactly equal
charges ?
Suppose, however, that we leap over this difficulty by simply
asserting the fact of the constant value of the molecular charge,
and that we call this constant molecular charge, for convenience in
description, one molecule of electricity.
This phrase, gross as it is, and out of harmony with the rest of
this treatise, will enable us at least to state clearly what is known
about electrolysis, and to appreciate the outstanding difficulties.
Every electrolyte must be considered as a binary compound of
its anion and its cation. The anion or the cation or both may be
compound bodies, so that a molecule of the anion or the cation
may be formed by a number of molecules of simple bodies. A
molecule of the anion and a molecule of the cation combined to
gether form one molecule of the electrolyte.
In order to act as an anion in an electrolyte, the molecule which
so acts must be charged with what we have called one molecule
of negative electricity, and in order to act as a cation the molecule
must be charged with one molecule of positive electricity.
These charges are connected with the molecules only when they
are combined as anion and cation in the electrolyte.
When the molecules are electrolysed, they part with their charges
to the electrodes, and appear as unelectrified bodies when set free
from combination.
If the same molecule is capable of acting as a cation in one
2 6 1.] SECONDARY PRODUCTS OF ELECTROLYSIS. 351
electrolyte and as an anion in another, and also of entering into
compound bodies which are not electrolytes, then we must suppose
that it receives a positive charge of electricity when it acts as a
cation, a negative charge when it acts as an anion, and that it
is without charge when it is not in an electrolyte.
Iodine, for instance, acts as an anion in the iodides of the metals
and in hydriodic acid, but is said to act as a cation in the bromide
of iodine.
This theory of molecular charges may serve as a method by
which we may remember a good many facts about electrolysis.
It is extremely improbable that when we come to understand the
true nature of electrolysis we shall retain in any form the theory of
molecular charges, for then we shall have obtained a secure basis
on which to form a true theory of electric currents, and so become
independent of these provisional theories.
261.] One of the most important steps in our knowledge of
electrolysis has been the recognition of the secondary chemical
processes which arise from the evolution of the ions at the elec
trodes.
In many cases the substances which are found at the electrodes
are not the actual ions of the electrolysis, but the products of the
action of these ions on the electrolyte.
Thus, when a solution of sulphate of soda is electrolysed by a
current which also passes through dilute sulphuric acid, equal
quantities of oxygen are given off at the anodes, and equal quan
tities of hydrogen at the cathodes, both in the sulphate of soda
and in the dilute acid.
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 can be examined separately,
it is found that at the anode of the sulphate of soda there is an
equivalent 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 seem 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 admission that the same
current which passing through dilute sulphuric acid electrolyses
one equivalent of water, when it passes through solution of sulphate
352 ELECTROLYSIS. [261.
of soda electrolyses one equivalent of the salt as well as one equi
valent of the water, and this would be contrary to the law of
electrochemical equivalents.
But if we suppose that the components of sulphate of soda are
not SO3 and NaO but SO4 and Na, — not sulphuric acid and soda
but sulphion and sodium — then the sulphion travels to the anode
and is set free, but being unable to exist in a free state it breaks
up into sulphuric acid and oxygen, one equivalent of each. At
the same time the sodium is set free at the cathode, and there
decomposes the water of the solution, forming one equivalent of
soda and two of hydrogen.
In the dilute sulphuric acid the gases collected at the electrodes
are the constituents of water, namely one volume of oxygen and
two volumes of hydrogen. There is also an increase of sulphuric
acid at the anode, but its amount is not equal to an equivalent.
It is doubtful whether pure water is an electrolyte or not. The
greater the purity of the water, the greater the resistance to elec
trolytic conduction. The minutest traces of foreign matter are
sufficient to produce a great diminution of the electrical resistance
of water. The electric resistance of water as determined by different
observers has values so different that we cannot consider it as a
determined quantity. The purer the water the greater its resistance,
and if we could obtain really pure water it is doubtful whether it
would conduct at all.
As long as water was considered an electrolyte, and was, indeed,
taken as the type of electrolytes, there was a strong reason for
maintaining that it is a binary compound, and that two volumes
of hydrogen are chemically equivalent to one volume of oxygen.
If, however, we admit that water is not an electrolyte, we are free
to suppose that equal volumes of oxygen and of hydrogen are
chemically equivalent.
The dynamical theory of gases leads us to suppose that in perfect
gases equal volumes always contain an equal number of molecules,
and that the principal part of the specific heat, that, namely, which
depends on the motion of agitation of the molecules among each
other, is the same for equal numbers of molecules of all gases.
Hence we are led to prefer a chemical system in which equal
volumes of oxygen and of hydrogen are regarded as equivalent,
and in which water is regarded as a compound of two equivalents
of hydrogen and one of oxygen, and therefore probably not capable
of direct electrolysis.
262.] DYNAMICAL THEORY. 353
While electrolysis fully establishes the close relationship between
electrical phenomena and those of chemical combination, the fact
that every chemical compound is not an electrolyte shews that
chemical combination is a process of a higher order of complexity
than any purely electrical phenomenon. Thus the combinations of
the metals with each other, though they are good conductors, and
their components stand at different points of the scale of electri
fication by contact, are not, even when in a fluid state, decomposed
by the current. Most of the combinations of the substances which
act as anions are not conductors, and therefore are not electrolytes.
Besides these we have many compounds, containing the same com
ponents as electrolytes, but not in equivalent proportions, and these
are also non-conductors, and therefore not electrolytes.
On the Conservation of Energy in Electrolysis.
262.] Consider any voltaic circuit consisting partly of a battery,
partly of a wire, and partly of an electrolytic cell.
During the passage of unit of electricity through any section of
the circuit, one electrochemical equivalent of each of the substances
in the cells, whether voltaic or electrolytic, is electrolysed.
The amount of mechanical energy equivalent to any given
chemical process can be ascertained by converting the whole energy
due to the process into heat, and then expressing the heat in
dynamical measure by multiplying the number of thermal units by
Joule's mechanical equivalent of heat.
Where this direct method is not applicable, if we can estimate
the heat given out by the substances taken first in the state before
the process and then in the state after the process during their
reduction to a final state, which is the same in both cases, then the
thermal equivalent of the process is the difference of the two quan
tities of heat.
In the case in which the chemical action maintains a voltaic
circuit, Joule found that the heat developed in the voltaic cells is
less than that due to the chemical process within the cell, and that
the remainder of the heat is developed in the connecting wire, or,
when there is an electromagnetic engine in the circuit, part of the
heat may be accounted for by the mechanical work of the engine.
For instance, if the electrodes of the voltaic cell are first con
nected by a short thick wire, and afterwards by a long thin wire,
the heat developed in the cell for each grain of zinc dissolved is
greater in the first case than in the second, but the heat developed
VOL. i. A a
354 ELECTROLYSIS. [263,
in the wire is greater in the second case than in the first. The
sum of the heat developed in the cell and in the wire for each grain
of zinc dissolved is the same in both cases. This has been estab
lished by Joule by direct experiment.
The ratio of the heat generated in the cell to that generated
in the wire is that of the resistance of the cell to that of the wire,
so that if the wire were made of sufficient resistance nearly the
whole of the heat would be generated in the wire, and if it were
made of sufficient conducting power nearly the whole of the heat
would be generated in the cell.
Let the wire be made so as to have great resistance, then the
heat generated in it is equal in dynamical measure to the product
of the quantity of electricity which is transmitted, multiplied by
the electromotive force under which it i& made to pass through
the wire.
263.] Now during the time in which an electrochemical equi
valent of the substance in the cell undergoes the chemical process
which gives rise to the current, one unit of electricity passes
through the wire. Hence, the heat developed by the passage of
one unit of electricity is in this case measured by the electromotive
force. But this heat is that which one electrochemical equivalent
of the substance generates, whether in the cell or in the wire, while
undergoing the given chemical process.
Hence the following important theorem, first proved by Thomson
(Phil. Mag., Dec. 1851)':—
' 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.'
The thermal equivalents of many chemical actions have been
determined by Andrews, Hess, Favre and Silbermann, &c., and from
these their mechanical equivalents can be deduced by multiplication
by the mechanical equivalent of heat.
This theorem not only enables us to calculate from purely thermal
data the electromotive forces of different voltaic arrangements, and
the electromotive forces required to effect electrolysis in different
cases, but affords the means of actually measuring chemical affinity.
It has long been known that chemical affinity, or the tendency
which exists towards the going on of a certain chemical change,
is stronger in some cases than in others, but no proper measure
of this tendency could be made till it was shewn that this tendency
in certain cases is exactly equivalent to a certain electromotive
263.] CALCULATION OF ELECTROMOTIVE FORCE. 355
force, and can therefore be measured according to the very same
principles used in the measurement of electromotive forces.
Chemical affinity being therefore, in certain cases, reduced to
the form of a measurable quantity, the whole theory of chemical
processes, of the rate at which they go on, of the displacement of
one substance by another, &c., becomes much more intelligible than
when chemical affinity was regarded as a quality sui generis, and
irreducible to numerical measurement.
When the volume of the products of electrolysis is greater than
that of the electrolyte, work is done during the electrolysis in
overcoming the pressure. If the volume of an electrochemical
equivalent of the electrolyte is increased by a volume v when
electrolysed under a pressure p, then the work done during the
passage of a unit of electricity in overcoming pressure is vp, and
the electromotive force required for electrolysis must include a
part equal to VJQ, which is spent in performing this mechanical
work.
If the products of electrolysis are gases which, like oxygen and
hydrogen, are much rarer than the electrolyte, and fulfil Boyle's
law very exactly, vp will be very nearly constant for the same
temperature, and the electromotive force required for electrolysis
will not depend in any sensible degree on the pressure. Hence it
has been found impossible to check the electrolytic decomposition
of dilute sulphuric acid by confining the decomposed gases in a
small space.
When the products of electrolysis are liquid or solid the quantity
vp will increase as the pressure increases, so that if v is positive
an increase of pressure will increase the electromotive force required
for electrolysis.
In the same way, any other kind of work done during electro
lysis will have an effect on the value of the electromotive force,
as, for instance, if a vertical current passes between two zinc
electrodes in a solution of sulphate of zinc a greater electromotive
force will be required when the current in the solution flows
upwards than when it flows downwards, for, in the first case, it
carries zinc from the lower to the upper electrode, and in the
second from the upper to the lower. The electromotive force
required for this purpose is less than the millionth part of that
of a Daniell's cell per foot.
A a 2
CHAPTER V.
ELECTROLYTIC POLARIZATION.
264.] WHEN an electric current is passed through an electrolyte
bounded by metal electrodes, the accumulation of the ions at the
electrodes produces the phenomenon called Polarization, which con
sists in an electromotive force acting in the opposite direction to the
current, and producing an apparent increase of the resistance.
When a continuous current is employed, the resistance appears
to increase rapidly from the commencement of the current, and
at last reaches a value nearly constant. If the form of the vessel
in which the electrolyte is contained is changed; the resistance is
altered in the same way as a similar change of form of a metallic
conductor would alter its resistance, but an additional apparent
resistance, depending on the nature of the electrodes, has always
to be added to the true resistance of the electrolyte.
265.] These phenomena have led some to suppose that there is
a finite electromotive force required for a current to pass through
an electrolyte. It has been shewn, however, by the researches of
Lenz, Neumann, Beetz, Wiedemann*, Paalzowf, and recently by
those of MM. F. Kohlrausch and W. A. NippoldtJ, that the con
duction in the electrolyte itself obeys Ohm's Law with the same
precision as in metallic conductors, and that the apparent resistance
at the bounding surface of the electrolyte and the electrodes is
entirely due to polarization.
266.] The phenomenon called polarization manifests itself in
the case of a continuous current by a diminution in the current,
indicating a force opposed to the current. Resistance is also per
ceived as a force opposed to the current, but we can distinguish
* Galvanismus, bd. i. f Berlin Monatshericht, July, 1868.
J Pogg. Ann. bd. cxxxviii. s. 286 (October, 1869).
267.] DISTINGUISHED FROM RESISTANCE. 357
between the two phenomena by instantaneously removing or re
versing the electromotive force.
The resisting force is always opposite in direction to the current,
and the external electromotive force required to overcome it is
proportional to the strength of the current, and changes its direc
tion when the direction of the current is changed. If the external
electromotive force becomes zero the current simply stops.
The electromotive force due to polarization, on the other hand,
is in a fixed direction, opposed to the current which produced it.
If the electromotive force which produced the current is removed,
the polarization produces a current in the opposite direction.
The difference between the two phenomena may be compared
with the difference between forcing a current of water through
a long capillary tube, and forcing water through a tube of moderate
length up into a cistern. In the first case if we remove the pressure
which produces the flow the current will simply stop. In the
second case, if we remove the pressure the water will begin to flow
down again from the cistern.
To make the mechanical illustration more complete, we have only
to suppose that the cistern is of moderate depth, so that when a
certain amount of water is raised into it, it begins to overflow.
This will represent the fact that the total electromotive force due
to polarization has a maximum limit.
267.] The cause of polarization appears to be the existence at
the electrodes of the products of the electrolytic decomposition of
the fluid between them. The surfaces of the electrodes are thus
rendered electrically different, and an electromotive force between
them is called into action, the direction of which is opposite to that
of the current which caused the polarization.
The ions, which by their presence at the electrodes produce the
phenomena of polarization, are not in a perfectly free state, but
are in a condition in which they adhere to the surface of the
electrodes with considerable force.
The electromotive force due to polarization depends upon the
density with which the electrode is covered with the ion, but it
is not proportional to this density, for the electromotive force does
not increase so rapidly as this density.
This deposit of the ion is constantly tending to become free,
and either to diffuse into the liquid, to escape as a gas, or to be
precipitated as a solid.
The rate of this dissipation of the polarization is exceedingly
358 ELECTROLYTIC POLARIZATION. [268.
small for slight degrees of polarization, and exceedingly rapid near
the limiting value of polarization.
268.] We have seen, Art. 262, that the electromotive force acting
in any electrolytic process is numerically equal to the mechanical
equivalent of the result of that process on one electrochemical
equivalent of the substance. If the process involves a diminution
of the intrinsic energy of the substances which take part in it,
as in the voltaic cell, then the electromotive force is in the direction
of the current. If the process involves an increase of the intrinsic
energy of the substances, as in the case of the electrolytic cell,
the electromotive force is in the direction opposite to that of the
current, and this electromotive force is called polarization.
In the case of a steady current in which electrolysis goes on
continuously, and the ions are separated in a free state at the
electrodes, we have only by a suitable process to measure the
intrinsic energy of the separated ions, and compare it with that
of the electrolyte in order to calculate the electromotive force
required for the electrolysis. This will give the maximum polari
zation.
But during the first instants of the process of electrolysis the
ions when deposited at the electrodes are not in a free state, and
their intrinsic energy is less than their energy in a free state,
though greater than their energy when combined in the electrolyte.
In fact, the ion in contact with the electrode is in a state which
when the deposit is very thin may be compared with that of
chemical combination with the electrode, but as the deposit in
creases in density, the succeeding portions are no longer so in
timately combined with the electrode, but simply adhere to it, and
at last the deposit, if gaseous, escapes in bubbles, if liquid, diffuses
through the electrolyte, and if solid, forms a precipitate.
In studying polarization we have therefore to consider
(1) The superficial density of the deposit, which we may call
<T. This quantity a- represents the number of electrochemical
equivalents of the ion deposited on unit of area. Since each
electrochemical equivalent deposited corresponds to one unit of
electricity transmitted by the current, we may consider a as re
presenting either a surface-density of matter or a surface-density of
electricity.
(2) The electromotive force of polarization, which we may call p.
This quantity p is the difference between the electric potentials
of the two electrodes when the current through the electrolyte
270.] DISSIPATION OF THE DEPOSIT. 359
is so -feeble that the proper resistance of the electrolyte makes no
sensible difference between these potentials.
The electromotive force p at any instant is numerically equal
to the mechanical equivalent of the electrolytic process going on at
that instant which corresponds to one electrochemical equivalent of
the electrolyte. This electrolytic process, it must be remembered,
consists in the deposit of the ions on the electrodes, and the state
in which they are deposited depends on the actual state of the
surface of the electrodes, which may be modified by previous
deposits.
Hence the electromotive force at any instant depends on 'the
previous history of the electrode. It is, speaking very roughly,
a function of o-, the density of the deposit, such that p = 0 when
o- = 0, but j» approaches a limiting value much sooner than a- does.
The statement, however, that p is a function of a cannot be
considered accurate. It would be more correct to say that p is
a function of the chemical state of the superficial layer of the
deposit, and that this state depends on the density of the deposit
according to some law involving the time.
269.] (3) The third thing we must take into account is the
dissipation of the polarization. The polarization when left to itself
diminishes at a rate depending partly on the intensity of the
polarization or the density of the deposit, and partly on the nature
of the surrounding medium, and the chemical, mechanical, or thermal
action to which the surface of the electrode is exposed.
If we determine a time T such that at the rate at which
the deposit is dissipated, the whole deposit would be removed in
the time T, we may call T the modulus of the time of dissipation.
When the density of the deposit is very small, T is very large,
and may be reckoned by days or months. When the density of
the deposit approaches its limiting value T diminishes very rapidly,
and is probably a minute fraction of a second. In fact, the rate
of dissipation increases so rapidly that when the strength of the
current is maintained constant, the separated gas, instead of con
tributing to increase the density of the deposit, escapes in bubbles
as fast as it is formed.
270.] There is therefore a great difference between the state of
polarization of the electrodes of an electrolytic cell when the polari
zation is feeble, and when it is at its maximum value. For instance,
if a number of electrolytic cells of dilute sulphuric acid with
platinum electrodes are arranged in series, and if a small electro-
360 ELECTROLYTIC POLARIZATION".
motive force, such as that of one Darnell's cell, be made to act
on the circuit, the electromotive force will produce a current of
exceedingly short duration, for after a very short time the elec
tromotive force arising from the polarization of the cell will balance
that of the Daniell's cell.
The dissipation will be very small in the case of so feeble a state
of polarization, and it will take place by a very slow absorption
of the gases and diffusion through the liquid. The rate of this
dissipation is indicated by the exceedingly feeble current which
still continues to flow without any visible separation of gases.
If we neglect this dissipation for the short time during which
the state of polarization is set up, and if we call Q the total
quantity of electricity which is transmitted by the current during
this time, then if A is the area of one of the electrodes, and <r
the density of the deposit, supposed uniform,
Q=A<r.
If we now disconnect the electrodes of the electrolytic apparatus
from the DauielPs cell, and connect them with a galvanometer
capable of measuring the whole discharge through it, a quantity
of electricity nearly equal to Q will be discharged as the polari
zation disappears.
271.] Hence we may compare the action of this apparatus, which
is a form of Hitter's Secondary Pile, with that of a Leyden jar.
Both the secondary pile and the Leyden jar are capable of being
charged with a certain amount of electricity, and of being after
wards discharged. During the discharge a quantity of electricity
nearly equal to the charge passes in the opposite direction. The
difference between the charge and the discharge arises partly from
dissipation, a process which in the case of small charges is very
slow, but which, when the charge exceeds a certain limit, becomes
exceedingly rapid. Another part of the difference between the charge
and the discharge arises from the fact that after the electrodes
have been connected for a time sufficient to produce an apparently
complete discharge, so that the current has completely disappeared,
if we separate the electrodes for a time, and afterwards connect
them, we obtain a second discharge in the same direction as the
original discharge. This is called the residual discharge, and is a
phenomenon of the Leyden jar as well as of the secondary pile.
The secondary pile may therefore be compared in several respects
to a Leyden jar. There are, however, certain important differences.
The charge of a Leyden jar is very exactly proportional to the
271.] COMPARISON WITH LEYDEN" JAR. 361
electromotive force of the charge, that is, to the difference of
potentials of the two surfaces, and the charge corresponding- to unit
of electromotive force is called the capacity of the jar, a constant
quantity. The corresponding quantity, which may be called the
capacity of the secondary pile, increases when the electromotive
force increases.
The capacity of the jar depends on the area of the opposed
•surfaces, on the distance between them, and on the nature of the
substance between them, but not on the nature of the metallic
surfaces themselves. The capacity of the secondary pile depends
on the area of the surfaces of the electrodes, but not on the distance
between them, and it depends on the nature of the surface of the
electrodes, as well as on that of the fluid between them. The
maximum difference of the potentials of the electrodes in each
element of a secondary pile is very small compared with the maxi
mum difference of the potentials of those of a charged Leyden jar,
so that in order to obtain much electromotive force a pile of many
elements must be used.
On the other hand, the superficial density of the charge in the
secondary pile is immensely greater than the utmost superficial
density of the charge which can be accumulated on the surfaces
of a Leyden jar, insomuch that Mr. C. F. Varley *, in describing
the construction of a condenser of great capacity, recommends a
series of gold or platinum plates immersed in dilute acid as prefer
able in point of cheapness to induction plates of tinfoil separated
by insulating material.
The form in which the energy of a Leyden jar is stored up
is the state of constraint of the dielectric between the conducting
surfaces, a state which I have already described under the name
of electric polarization, pointing out those phenomena attending
this state which are at present known, and indicating the im
perfect state of our knowledge of what really takes place. See
Arts. 62, 111.
The form in which the energy of the secondary pile is stored
up is the chemical condition of the material stratum at the surface
of the electrodes, consisting of the ions of the electrolyte and the
substance of the electrodes in a relation varying from chemical
combination to superficial condensation, mechanical adherence, or
simple juxtaposition.
The seat of this energy is close to the surfaces of the electrodes,
* Specification of C. F. Varley, ' Electric Telegraphs, &c.,' Jan. 1860.
362 ELECTKOLYTIC POLARIZATION.
and not throughout the substance of the electrolyte, and the form
in which it exists may be called electrolytic polarization.
After studying* the secondary pile in connexion with the Leyden
jar, the student should again compare the voltaic battery with
some form of the electrical machine, such as that described in
Art. 211.
Mr. Varley has lately * found that the capacity of one square
inch is from 175 to 542 microfarads and upwards for platinum
plates in dilute sulphuric acid, and that the capacity increases with
the electromotive force, being about 175 for 0.02 of a Daniell's
cell, and 542 for 1.6 Daniell's cells.
But the comparison between the Leyden jar and the secondary
pile may be carried still farther, as in the following1 experiment,
due to Buff f. It is only when the glass of the jar is cold that
it is capable of retaining a charge. At a temperature below 100CC
the glass becomes a conductor. If a test-tube containing mercury
is placed in a vessel of mercuiy, and if a pair of electrodes are
connected, one with the inner and the other with the outer portion
of mercury, the arrangement -constitutes a Leyden jar which will
hold a charge at ordinary temperatures. If the electrodes are con
nected with those of a voltaic battery, no current will pass as long
as the glass is cold, but if the apparatus is gradually heated a
current will begin to pass, and will increase rapidly in intensity as
the temperature rises, though the glass remains apparently as hard
as ever.
This current is manifestly electrolytic, for if the electrodes are
disconnected from the battery, and connected with a galvanometer,
a considerable reverse current passes, due to polarization of the
surfaces of the glass.
If, while the battery is in action the apparatus is cooled, the
current is stopped by the cold glass as before, but the polarization
of the surfaces remains. The mercury may be removed, the surfaces
may be washed with nitric acid and with water, and fresh mercuiy
introduced. If the apparatus is then heated, the current of polar
ization appears as soon as the glass is sufficiently warm to conduct it.
We may therefore regard glass at 100°C, though apparently a
solid body, as an electrolyte, and there is considerable reason
to believe that in most instances in which a dielectric has a
slight degree of conductivity the conduction is electrolytic. The
* Proc. E. S. Jan. 12, 1871.
t Annalen der Chemie und Pharmacie, bd. xc. 257 (1854).
272.] CONSTANT VOLTAIC ELEMENTS. 363
existence of polarization may be regarded as conclusive evidence of
electrolysis, and if the conductivity of a substance increases as the
temperature rises, we have good grounds for suspecting that it is
electrolytic.
On Constant Voltaic Elements.
272.] When a series of experiments is made with a voltaic
battery in which polarization occurs, the polarization diminishes
during the time 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 necessary 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 zinc
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 sulphuric 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 circuit. 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
means, such as stirring the liquid, or rubbing the surface of the
negative plate. In Smee's battery the negative plates are vertical,
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
364: ELECTROLYTIC POLARIZATION. [272.
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 coating 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 method is by using as the liquid an electrolyte, the
cation of which is a metal highly negative to zinc.
In DanielFs battery a copper plate is immersed in a saturated
solution of sulphate of copper. When the current flows through
the solution from the zinc 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 S O4 travelling towards the zinc.
When these conditions are not fulfilled hydrogen is evolved at
the cathode, but immediately acts on the solution, throwing down
copper, and uniting with SO4 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
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 zinc is immersed should be
free from sulphate of copper. If any of this salt makes its way
to the surface of the zinc it is reduced, and copper is deposited
272.]
THOMSON'S FORM OF DANIELL'S CELL.
365
on the zinc. The zinc, copper, and fluid then form a little circuit
in which rapid electrolytic action goes on, and the zinc is eaten
away by an action which contributes nothing1 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 zinc, 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 increase 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 is that diffusion is always going on, whereas 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 zinc and spoils the battery.
To retard this result indefinitely, Sir W. Thomson * has constructed
Daniell's battery in the following form.
ELECTRODES
SIPHON—
Fig. 22.
In each cell the copper plate is placed horizontally at the bottom
* Proc.R. S.t Jau. 19,1871.
366 ELECTROLYTIC POLARIZATION. [272.
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
than that of sulphate of zinc alone, so that it cannot get to the
zinc 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 zinc and copper,
and the other in a vessel outside the cell, so that the liquid is
very slowly drawn off near the middle of its depth. To supply
its place, water, or a weak solution of sulphate of zinc, 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 zinc 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 SO4 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 zinc 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.
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 VI.
LINEAR ELECTRIC CURRENTS.
On Systems of Linear Conductors.
273.] 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 E.
But if there be more possible electrodes than two, the conductor
may have more than one independent current through it, and these
may not be conjugate to each other. See Art. 2.82.
Ohm's Law.
274.] Let E be the electromotive force in a linear conductor
from the electrode At to the electrode A2. (See. Art. 69.) 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 A: A2 in unit of time, and let R be the resistance of
the conductor, then the expression of Ohm's Law is
E = CR. (1)
Linear Conductors arranged in Series.
275.] Let A19 A.± be the electrodes of the first conductor and let
the second conductor be placed with one of its electrodes in contact
368 LINEAR ELECTRIC CURRENTS. [276.
with A2, so that the second conductor has for its electrodes A2, A%.
The electrodes of the third conductor may be denoted by A3
and A4.
Let the electromotive forces along these conductors be denoted
by E12, EH , UM) and so on for the other conductors.
Let the resistances of the conductors be
Then, since the conductors are arranged in series so that the same
current C flows through each, we have by Ohm's Law,
-#12 = CRY1, ^23 = £^23 » #34 = ^34' (2)
If E is the resultant electromotive force, and R the resultant
resistance of the system, we must have by Ohm's Law,
E = CE. (3)
NOW ^=^ + 43 + ^34, (4)
the sum of the separate electromotive forces,
= C (^12 + ^23 + ^34) hy equations (2).
Comparing this result with (3), we find
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, B a point between
them, a, c, and I the potentials of these points respectively. Let
El be the resistance of the part from A to B, R2 that of the part
from B to C, and R that of the whole from A to C, then, since
a—b = R1C, t—c = R.2C, and a-c = EC,
the potential at B is
3 = ^±^f, (6)
/L
which determines the potential at B when the potentials at A and
C are given.
Resistance of a Multiple Conductor.
276.] Let a number of conductors ABZ, ACZ, ADZ be arranged
side by side with their extremities in contact with the same two
points A and Z. They are then said to be arranged in multiple
arc.
Let the resistances of these conductors be R^ R2J JR2 respect-
2 77-] SPECIFIC EESISTANCE AND CONDUCTIVITY. 369
ively, and the currents C^ C2, C3, and let the resistance of the
multiple conductor be R, and the total current C. Then, since the
potentials at A and Z are the same for all the conductors, they have
the same difference, which we may call E. We then have
E = Cl R! = C2R2 = CB R3 = CR,
but C^Q+Cz + Cv
whence i=i + i+i3- (?)
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.
Ciirrent 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
R1 the resistance of that branch,
4=C|, (8)
where C is the total current, and R is the resistance of the multiple
conductor as previously determined.
Longitudinal Resistance of Conductors of Uniform Section.
277.] 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 I, and whose section is unity. This is equivalent to I
cubes arranged in series. The resistance of the conductor is there
fore I p.
Finally, consider a conductor of length I and uniform section s.
This is equivalent to s conductors similar to the last arranged in
multiple arc. The resistance of this conductor is therefore
7? l?
M = — •
S
When we know the resistance of a uniform wire we can determine
VOL. i. B b
370 LINEAR ELECTRIC CURRENTS. [2?8.
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 r is this resistance, I the length, and m the mass of a wire, then
pf
£1 = •
m
On the Dimensions of the Quantities involved in these Equations.
278.] The resistance of a conductor is the ratio of the electro
motive force acting on it to the current produced. The conduct
ivity of the conductor is the reciprocal of this quantity, or in
other words, the ratio of the current to the electromotive force
producing it.
Now we know that in the electrostatic system of measurement
the ratio of a quantity of electricity to the potential of the con
ductor on which it is spread is the capacity of the conductor, and
is measured by a line. If the conductor is a sphere placed in an
unlimited field, this line is the radius of the sphere. The ratio
of a quantity of electricity to an electromotive force is therefore a
line, but the ratio of a quantity of electricity to a current is the
time during which the current flows to transmit that quantity.
Hence the ratio of a current to an electromotive force is that of a
line to a time, or in other words, it is a velocity.
The fact that the conductivity of a conductor is expressed in the
electrostatic system of measurement by a velocity may be verified
by supposing a sphere of radius r charged to potential F, and then
connected with the earth by the given conductor. Let the sphere
contract, so that as the electricity escapes through the conductor
the potential of the sphere is always kept equal to V. Then the
charge on the sphere is rV at any instant, and the current is
-=7 (rV), but, since V is constant, the current is -=7- F, and the
dt dt
electromotive force through the conductor is V.
The conductivity of the conductor is the ratio of the current to
dr
the electromotive force, or — , that is, the velocity with which the
dt
radius of the sphere must diminish in order to maintain the potential
28o.] SYSTEM OF LINEAR CONDUCTORS. 371
constant when the charge is allowed to pass to earth through the
conductor.
In the electrostatic system, therefore, the conductivity of a con
ductor is a velocity, and of the dimensions [I/T'1].
The resistance of the conductor is therefore of the dimensions
The specific resistance per unit of volume is of the dimension of
[T], and the specific conductivity per unit of volume is of the
dimension of [27-1].
The numerical magnitude of these coefficients depends only on
the unit of time, which is the same in different countries.
The specific resistance per unit of weight is of the dimensions
279.] We shall afterwards find that in the electromagnetic
system of measurement the resistance of a conductor is expressed
by a velocity, so that in this system the dimensions of the resist
ance of a conductor are [Z77"1].
The conductivity of the conductor is of course the reciprocal of
this.
The specific resistance per unit of volume in this system is of the
dimensions [Z2!7-1], and the specific resistance per unit of weight
is of the dimensions [L~1T~1M].
On Linear Systems of Conductors in general.
280.] The most general case of a linear system is that of n
points, Alt A2,...An, connected together in pairs by \n(nr-l}
linear conductors. Let the conductivity (or reciprocal of the re
sistance) of that conductor which connects any pair of points, say
Ap and Aqt be called Kpq9 and let the current from Ap to Aq be Cpq.
Let Pp and Pq be the electric potentials at the points Ap and Aq
respectively, and let the internal electromotive force, if there be
any, along the conductor from Ap to Aq be Epq.
The current from Ap to Aq is, by Ohm's Law,
cf, = Kft(Pf-P,+Efq}. (i)
Among these quantities we have the following sets of relations :
The conductivity of a conductor is the same in either direction,
or Kpq = Kqp. (2)
The electromotive force and the current are directed quantities,
sothat Epq = -Zqp, and CM = -CW. (3)
Let P1} P2, ...Pn be the potentials at A19 A2, ... An respectively,
and let Qlt Q2)... Qn be the quantities of electricity which enter
B b 2
372 LINEAR ELECTRIC CURRENTS. [280.
the system in unit of time at each of these points respectively.
These are necessarily subject to the condition of ' continuity '
Qi + Q*... + Qn=0, (4)
since electricity can neither be indefinitely accumulated nor pro
duced within the system.
The condition of { continuity ' at any point Ap is
QP = Cpl+Cp2 + &c. + Cpn. (5)
Substituting the values of the currents in terms of equation
(l), this becomes
Qf = (Zrl + K» + &c. + £,„) Pr - (KA Pl + KA P2 + &c. + *„£)
+ (KtlEtl + &X.+KtxEf,). (6)
The symbol Kpp does not occur in this equation. Let us therefore
give it the value
Kn=-(K,l + K# + te. + Kr.)i (7)
that is, let Kpp be a quantity equal and opposite to the sum of
all the conductivities of the conductors which meet in Ap. We
may then write the condition of continuity for the point Ap,
TT TF D ( Q.\
-£pntipn—(£p. (8)
By substituting 1 , 2, &c. n for j) in this equation we shall obtain
n equations of the same kind from which to determine the n
potentials Plt P2, &c., Pn.
Since, however, if we add the system of equations (8) the result
is identically zero by (3), (4) and (7), there will be only n—l in
dependent equations. These will be sufficient to determine the
differences of the potentials of the points, but not to determine the
absolute potential of any. This, however, is not required to calcu
late the currents in the system.
If we denote by D the determinant
TT JT T?
A1U ^-12» -^iCn-l)'
•^21 > -^22* ^2(n-l)> (9)
and by Bpq, the minor of Kpq, we find for the value of Ip—Pn,
rtSpt-QJDM+i«s. (10)
In the same way the excess of the potential of any other point^
say Aq, over that of An may be determined. We may then de
termine the current between Ap and Aq from equation (l), and so
solve the problem completely.
282 a.] SYSTEM OF LINEAR CONDUCTORS. 373
281.] We shall now demonstrate a reciprocal property of any
two conductors of the system, answering to the reciprocal property
we have already demonstrated for statical electricity in Art. 88.
The coefficient of Qq in the expression for Pp is -~. That of Qp
in the expression for Pq is -— - •
Now Dpq differs from Dqp only by the substitution of the symbols
such as Kqp for Kpq. But, by equation (2), these two symbols are
equal, since the conductivity of a conductor is the same both ways.
Hence £pq=--Dqp. (11)
It follows from this that the part of the potential at Ap arising-
from the introduction of a unit current at Aq is equal to the part of
the potential at Aq arising from the introduction of a unit current
at Ap.
We may deduce from this a proposition of a more practical form.
Let A, .B, C, D be any four points of the system, and let the
effect of a current Q, made to enter the system at A and leave it
at B, be to make the potential at C exceed that at D by P. Then,
if an equal current Q be made to enter the system at C and leave
it at D, the potential at A will exceed that at B by the same
quantity P.
If an electromotive force E be introduced, acting in the conductor
from A to B, and if this causes a current C from X to 7, then the
same electromotive force E introduced into the conductor from X to
T will cause an equal current C from AtoJB.
The electromotive force E may be that of a voltaic battery intro
duced between the points named, care being taken that the resist
ance of the conductor is the same before and after the introduction
of the battery.
282 #.] If an electromotive force Epq act along the conductor
Ap Aq, the current produced along another conductor of the system
Ar As is easily found to be
There will be no current if
But, by (11), the same equation holds if, when the electromotive
force acts along ArAt9 there is no current in ApAq. On account
of this reciprocal relation the two conductors referred to are said to
be conjugate.
The theory of conjugate conductors has been investigated by
LINEAR ELECTRIC CURRENTS. [282 I.
Kirchhoff, who 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
sum of the products of the current in each conductor multiplied by
the resistance of that conductor.
We obtain this result by adding equations of the form (1) for the
complete circuit, when the potentials necessarily disappear.
*282 £.] If the conducting wires form a simple network and if
we suppose that a current circulates round each mesh, then the
actual current in the wire which forms a thread of each of two
neighbouring meshes will be the difference between the two
currents circulating in the two meshes, the currents being reckoned
positive when they circulate in a direction opposite to the motion
of the hands of a watch. It is easy to establish in this case the
following proposition :— Let x be the current, E the electromotive
force, and R the total resistance in any mesh ; let also y, z, ... be
currents circulating in neighbouring meshes which have threads
in common with that in which x circulates, the resistances of those
parts being s, t, . . . ; then
Rx—sy—tz—&c. = E.
To illustrate the use of this rule we will take the arrangement
known as Wheatstone's Bridge, adopting the figure and notation of
Art. 347. We have then the three following equations repre
senting the application of the rule in the case of the three circuits
OUC, OCA, OAB in which the currents x, y, z respectively circulate,
Viz. (0+/3+y)a -yy -$Z=Et
—y # + (#+y + a)j/ — az = o,
—(3 x —ay + (c + a + p)z= 0.
From these equations we may now determine the value of x— y
the galvanometer current in the branch OA, but the reader is
referred to Art. 347 et seq. where this and other questions connected
with Wheatstone's Bridge are discussed.
Heat Generated in the System.
283.] The mechanical equivalent of the quantity of heat generated
* [Extracted from notes of Professor Maxwell's lectures by Mr. J. A. Fleming, B.A.,
St. John's College.]
284.] GENERATION OF HEAT. 375
in a conductor whose resistance is R by a current C in unit of time
is, by Art. 242, Jff = EC\ (13)
We have therefore to determine the sum of such quantities as
RC2 for all the conductors of the system.
For the conductor from Ap to Aq the conductivity is Kpq, and the
resistance Em, where Kpq . Rpq = 1. (14)
The current in this conductor is, according to Ohm's Law,
Cpq = Kpq(Pp-Pq). (15)
We shall suppose, however, that the value of the current is not
that given by Ohm's Law, but Xpq, where
XM=CM+Ypq. (16)
To determine the heat generated in the system we have to find
the sum of all the quantities of the form
7? Y2
j.ipq A. pq ,
or JH=^{RpqC\q + 2RMCpqYM + RpqY\<1}. (17)
Giving Cpq its value, and remembering the relation between KiHi
and Rpq, this becomes
2 [(P,-P.) (Cpq + 2Yfq) + Rpq Y\q}. (18)
Now since both C and X must satisfy the condition of continuity
at Ap, we have Qp = Cpl + CP2 + &c. + Cpn, (19)
Qp = Xpl+XP2 + te>. + Xpn9 (20)
therefore 0 = Ypl + YP2 + &c. + Tpn . (21)
Adding together therefore all the terms of (18), we find
z(jzMz%.)=sp,g,+ssM7V (22)
Now since 72 is always positive and Y2 is essentially positive, the
last term of this equation must be essentially positive. Hence the
first term is a minimum when Yis zero in every conductor, that is,
when the current in every conductor is that given by Ohm's Law.
Hence the following theorem :
284.] In any system of conductors in which there are no internal
electromotive forces the heat generated by currents distributed in
accordance with Ohm's Law is less than if the currents had been
distributed in any other manner consistent with the actual con
ditions of supply and outflow of the current.
The heat actually generated when Ohm's Law is fulfilled is
mechanically equivalent to I,PpQq) that is, to the sum of the
products of the quantities of electricity supplied at the different
external electrodes, each multiplied by the potential at which it is
supplied.
CHAPTEE VII.
CONDUCTION IN THREE DIMENSIONS.
Notation of Electric Currents.
285.] AT any point let an element of area dS be taken normal
to the axis of x, and let Q units of electricity pass across this area
from the negative to the positive side in unit of time, then, if
^becomes ultimately equal to u when dS is indefinitely diminished,
u is said to be the Component of the electric current in the direction
of x at the given point.
In the same way we may determine v and w, the components of
the current in the directions of y and z respectively.
286.] To determine the component of the current in any other
direction OR through the given point 0, let I, m, n be the direction-
cosines of OR ; then if we cut off from the axes of x, y. z portions
equal to r r , r
y > — > and -
I m n
respectively at A, B and C, the triangle ABC
will be normal to OR.
The area of this triangle ABC will be
^=$jf_,
Imn
and by diminishing r this area may be diminished
without limit.
The quantity of electricity which leaves the tetrahedron ABCO
by the triangle ABC must be equal to that which enters it through
the three triangles OBC, OCA, and OAB.
The area of the triangle OBC is \ — , and the component of
2'8;.] COMPONENT AND RESULTANT CURRENTS. 377
the current normal to its plane is u, so that the quantity which
enters through this triangle is \ r2 — •
The quantities which enter through the triangles OCA and OAB
respectively are „ w
\r*-v and Jr^.
If y is the component of the velocity in the direction OR, then
the quantity which leaves the tetrahedron through ABC is
i/2JL
Imn
Since this is equal to the quantity which enters through the three
other triangles,
i .
nl Im
, . , . , 2 Imn
multiplying by — ^— , we get
y = lu + mv + nw. (1)
If we put «2 + v2 + w* = F2,
and make £', m' ', »' such that
u = IT, v = mT, and «; = »T ;
then y = T (IV + mm' + w^x). (2)
Hence, if we define the resultant current as a vector whose
magnitude is F, and whose direction-cosines are Vy m', n't and if
y denotes the current resolved in a direction making an angle 9
with that of the resultant current, then
y = T cos 6 ; (3)
shewing that the law of resolution of currents is the same as that
of velocities, forces, and all other vectors.
287.] To determine the condition that a given surface may be a
surface of flow, let
F(B,y,*) = \ (4)
be the equation of a family of surfaces any one of which is given by
making A constant ; then, if we make
dk
dx
d\
dy
d\
1
)
N*
the direction-cosines of the normal, reckoned in the direction in
which A increases, are
7d\ d\ ^TdX
l=N-=-t m = N-^-j n = N-r-- (6)
dx dy dz ^ '
378
CONDUCTION IN THREE DIMENSIONS.
[288.
d\}
dz]
Hence, if y is the component of the current normal to the surface,
^f ^A dX dX
\ dx dy
If y = 0 there will be no current through the surface, and the
surface may be called a Surface of Flow, because the lines of motion
are in the surface.
288.] The equation of a surface of flow is therefore
u/\ ci/X CvX . .
dx dy dz ~~ ^ '
If this equation is true for all values of A, all the surfaces of the
family will be surfaces of flow.
289.] Let there be another family of surfaces, whose parameter
is A', then, if these are also surfaces of flow, we shall have
d\' dX' dX'
u-^j- + v—r- + w--r- — 0. (9)
dx dv dz v '
If there is a third family of surfaces of flow, whose parameter
is A", then ,, ,, ,,
— -
das dy dz
If we eliminate between these three equations, n, v, and w dis
appear together, and we find
Vbl\.
dX'
dx
dX"
It A.
dy
d\'
U/\
dz
dX'
dy
dX"
flu
dz
dX"
fa
f?,nr>.
= 0;
or
(11)
(12)
A"=4>(A,A');
that is, A" is some function of A and A'.
290.] Now consider the four surfaces whose parameters are A,
A + 8A, A7, and A' + 8 A'. These four surfaces enclose a quadrilateral
tube, which we may call the tube 6A.5A'. Since this tube is
bounded by surfaces across which there is no flow, we may call
it a Tube of Flow. If we take any two sections across the tube,
the quantity which enters the tube at one section must be equal
to the quantity which leaves it at the other, and since this quantity
is therefore the same for every section of the tube, let us call it
£5 A . 8 A' where L is a function of A and A', the parameters which
determine the particular tube.
293-] TUBES OP PLOW. 379
291.] If bS denotes the section of a tube of flow by a plane
normal to x, we have by the theory of the change of the inde
pendent variables,
SA . BA'= bS(— — - T-)> (13)
v ay dz dz dy '
and by the definition of the components of the current
Hence
(15)
dz dz
cr .1 ! T ,d\ d\r dX d\\
Similarly v = L ( -=- , ) ,
\dz dx dx dz '
j. /cl/X dX cl>X clX \
dy dx '
292.] It is always possible when one of the functions A or A' is
known, to determine the other so that L may be equal to unity.
For instance, let us take the plane of yz, and draw upon it a series
of equidistant lines .parallel to yt to represent the sections of the
family Ax by this plane. In other words, let the function Ax be
determined by the condition that when cc = 0 A'= z. If we then
make L = 1, and therefore (when x — 0)
— judy
then in the plane (oc = 0) the amount of electricity which passes
through any portion will be
\\udydz =JJd^ ^A'. (16)
Having determined the nature of the sections of the surfaces of
flow by the plane of yz, the form of the surfaces elsewhere is
determined by the conditions (8) and (9). The two functions A
and A' thus determined are sufficient to determine the current at
every point by equations (15), unity being substituted for L.
On Lines of Mow.
293.] Let a series of values of A and of A' be chosen, the suc
cessive differences in each series being unity. The two series of
surfaces defined by these values will divide space into a system
of quadrilateral tubes through each of which there will be a unit
current. By assuming the unit sufficiently small, the details of
the current may be expressed by these tubes with any desired
amount of minuteness, Then if any surface be drawn cutting the
380 CONDUCTION IN THREE DIMENSIONS. [294.
system of tubes, the quantity of the current which passes through
this surface will be expressed by the number of tubes which cut it,
since each tube carries unity of current.
The actual intersections of the surfaces may be called Lines of
Flow. When the unit is taken sufficiently small, the number of
lines of flow which cut a surface is approximately equal to the
number of tubes of flow which cut it, so that we may consider
the lines of flow as expressing not only the direction of the current
but its strength, since each line of flow through a given section
corresponds to a unit current.
On Current-Sheets and Current-Functions.
294.] A stratum of a conductor contained between two con
secutive surfaces of flow of one system, say that of A", is called
a Current- Sheet. The tubes of flow within this sheet are deter
mined by the function A. If \A and \P denote the values of A at
the points A and P respectively, then the current from right to
left across any line drawn on the sheet from A to P is XP— A^.
If AP be an element, ds, of a curve drawn on the sheet, the current
which crosses this element from right to left is
d\ 7
— ds.
ds
This function A, from which the distribution of the current in
the sheet can be completely determined, is called the Current-
Function.
Any thin sheet of metal or conducting matter bounded on both
sides by air or some other non-conducting medium may be treated
as a current-sheet, in which the distribution of the current may
be expressed by means of a current-function. See Art. 647.
Equation of ' Continuity?
295.] If we differentiate the three equations (15) with respect to
#, y, z respectively, remembering that L is a function of A and A",
wefind *+*+^ = 0. (17)
dx dy dz
The corresponding equation in Hydrodynamics is called the
Equation of < Continuity/ The continuity which it expresses is
the continuity of existence, that is, the fact that a material sub
stance cannot leave one part of space and arrive at another, without
going through the space between. It cannot simply vanish in the
2Q5-] EQUATION OF CONTINUITY. 381
one place and appear in the other, but it must travel along* a con
tinuous path, so that if a closed surface be drawn, including- the
one place and excluding1 the other, a material substance in passing*
from the one place to the other must go through the closed surface.
The most general form of the equation in hydrodynamics is
d(pu) d(pv) d(Pw) dp _
dx dy ~dT^dt~ ( }
where p signifies the ratio of the quantity of the substance to the
volume it occupies, that volume being in this case the differential
element of volume, and (pu), (pv), and (pw) signify the ratio of the
quantity of the substance which crosses an element of area in unit
of time to that area, these areas being normal to the axes of x3 y^ and
z respectively. Thus understood, the equation is applicable to any
material substance, solid or fluid, whether the motion be continuous
or discontinuous, provided the existence of the parts of that sub
stance is continuous. If anything, though not a substance, is
subject to the condition of continuous existence in time and space,
the equation will express this condition. In other parts of Physical
Science, as, for instance, in the theory of electric and magnetic
quantities, equations of a similar form occur. We shall call such
equations { equations of continuity ' to indicate their form, though
we may not attribute to these quantities the properties of matter,
or even continuous existence in time and space.
The equation (17), which we have arrived at in the case of
electric currents, is identical with (18) if we make p = 1, that is,
if we suppose the substance homogeneous and incompressible. The
equation, in the case of fluids, may also be established by either
of the modes of proof given in treatises on Hydrodynamics. In
one of these we trace the course and the deformation of a certain
element of the fluid as it moves along. In the other, we fix our
attention on an element of space, and take account of all that
enters or leaves it. The former of these methods cannot be applied
to electric currents, as we do not know the velocity with which the
electricity passes through the body, or even whether it moves in
the positive or the negative direction of the current. All that we
know is the algebraical value of the quantity which crosses unit
of area in unit of time, a quantity corresponding to (pu] in the
equation (18). We have no means of ascertaining the value of
either of the factors p or u, and therefore we cannot follow a par
ticular portion of electricity in its course through the body. The
other method of investigation, in which we consider what passes
382 CONDUCTION IN THEEE DIMENSIONS. [296.
through the walls of an element of volume, is applicable to electric
currents, and is perhaps preferable in point of form to that which
we have given, but as it may be found in any treatise on Hydro
dynamics we need not repeat it here.
Quantity of Electricity which passes through a given Surface.
296.] Let F be the resultant current at any point of the surface.
Let dS be an element of the surface, and let e be the angle between
T and the normal to the surface, then the total current through
the surface will be r r
I jTcosedS,
the integration being extended over the surface.
As in Art. 2 1 , we may transform this integral into the form
in the case of any closed surface, the limits of the triple integration
being those included by the surface. This is the expression for
the total efflux from the closed surface. Since in all cases of steady
currents this must be zero whatever the limits of the integration,
the quantity under the integral sign must vanish, and we obtain
in this way the equation of continuity (17).
CHAPTER VIII.
RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS.
On the most General Relations between Current and Electro
motive Force.
297.] LET the components of the current at any point be u, v, w.
Let the components of the electromotive force be X, J, Z.
The electromotive force at any point is the resultant force on
a unit of positive electricity placed at that point. It may arise
(1) from electrostatic action, in which case if 7 is the potential,
X = ~~^' Ys=~'fy' Z="~Tz> W
or (2) from electromagnetic induction, the laws of which we shall
afterwards examine; or (3) from thermoelectric or electrochemical
action at the point itself, tending to produce a current in a given
direction.
We shall in general suppose that X, Y} Z represent the com
ponents of the actual electromotive force at the point, whatever
be the origin of the force, but we shall occasionally examine the
result of supposing it entirely due to variation of potential.
By Ohm's Law the current is proportional to the electromotive
force. Hence X, J", Z must be linear functions of ^, v, w. We
may therefore assume as the equations of Resistance,
-»-* -^.
(2)
We may call the coefficients E the coefficients of longitudinal
resistance in the directions of the axes of coordinates.
The coefficients P and Q may be called the coefficients of trans
verse resistance. They indicate the electromotive force in one
direction required to produce a current in a different direction.
384 EESISTANCE AND CONDUCTIVITY. [298.
If we were at liberty to assume that a solid body may be treated
as a system of linear conductors, then, from the reciprocal property
(Art. 281) of any two conductors of a linear system, we might shew
that the electromotive force along z required to produce a unit
current parallel to y must be equal to the electromotive force along
y required to produce a unit current parallel to z. This would
shew that P± = Qi» an(i similarly we should find P2 = Q.2, and
P3 = Q3. When these conditions are satisfied the system of co
efficients is said to be Symmetrical. When they are not satisfied it
is called a Skew system.
We have great reason to believe that in every actual case the
system is symmetrical, but we shall examine some of the con
sequences of admitting the possibility of a skew system.
298.] The quantities ut v, w may be expressed as linear functions
of X, Y9 Z by a system of equations, which we may call Equations
of Conductivity,
, j
, V
')
v = &X+r2Y+plZ, V (3)
we may call the coefficients r the coefficients of Longitudinal con
ductivity, and p and q those of Transverse conductivity.
The coefficients of resistance are inverse to those of conductivity.
This relation may be defined as follows :
Let [PQjB] be the determinant of the coefficients of resistance,
and [pqr\ that of the coefficients of conductivity, then
P1Q1R1-P2Q2R2-P3QA (4)
[PQE] [pgr] = 1, (6)
[PQR] Pl = (P2 P3- qi R& [pgr] Pl = 0^3-?i *i), (7)
&c. &c.
The other equations may be formed by altering the symbols
P, Q, R,p, q, r, and the suffixes 1, 2, 3 in cyclical order.
Rate of Generation of Heat.
299.] To find the work done by the current in unit of time
in overcoming resistance, and so generating heat, we multiply the
components of the current by the corresponding components of the
electromotive force. We thus obtain the following expressions for
Wt the quantity of work expended in unit of time :
3OO.] COEFFICIENTS OF CONDUCTIVITY. 385
= Xu+Yv + Zw; (8)
', (9)
. (10)
By a proper choice of axes, either of the two latter equations may
be deprived of the terms involving the products of u, v, w or of
X, Y, Z. The system of axes, however, which reduces W to the form
is not in general the same as that which reduces it to the form
It is only when the coefficients P13 P2, P3 are equal respectively
to Qu Q%) Qs that the two systems of axes coincide.
If with Thomson * we write
^ = ff-2';)
and p = s + t, <i — s — t\\
then we have
ttlltt
and [PQS-]fl =
(13)
If therefore we cause -S^ 52, ^ to disappear, *j will not also dis
appear unless the coefficients T are zero.
Condition of
300.] Since the equilibrium of electricity is stable, the work
spent in maintaining the current must always be positive. The
conditions that W may be positive are that the three coefficients
jftl5 R2J R3, and the three expressions
(14)
lBlSt-(Pt+<^f,)
must all be positive.
There are similar conditions for the coefficients of conductivity.
* Trans. R. S. Edin., 1853-4, p. 165.
VOL. I. C C
386 RESISTANCE AND CONDUCTIVITY. [301.
Equation of Continuity in a Homogeneous Medium.
301.] If we express the components of the electromotive force
as the derivatives of the potential V, the equation of continuity
du dv dw
becomes in a homogeneous medium
i^2-2T!- 3T-T <i-7— r-2j-r 3j—r
*• dx? L dy* 6 dz* dydz * dzdx dxdy
If the medium is not homogeneous there will be terms arising
from the variation of the coefficients of conductivity in passing
from one point to another.
This equation corresponds to Laplace's equation in an isotropic
medium.
302.] If we put
[rs] = rlr2r3+2s1s2sB-r1s12-r2s22-r.ds32, (17)
and [AS] = A1A2A3+2B1B2B3-A1B12-A2B22-A3B<2, (18)
where |/f]-^i — rzrs — si2) }
[rs\B1=: ^a-r-i*!, |- (19)
and so on, the system A, B will be inverse to the system r, s, and
if we make
Al x2 + A2y* + A3z2 + 2£lyz+2£2zz+2 B, xy = [AS] P2, (20)
we shall find that
7T p
is a solution of the equation.
In the case in which the coefficients T are zero, the coefficients A
and B become identical with R and 8. When T exists this is not
the case.
In the case therefore of electricity flowing out from a centre in an
infinite, homogeneous, but not isotropic, medium, the equipotential
surfaces are ellipsoids, for each of which p is constant. The axes of
these ellipsoids are in the directions of the principal axes of con
ductivity, and these do not coincide with the principal axes of
resistance unless the system is symmetrical.
By a transformation of this equation we may take for the axes
of a?, ?/, z the principal axes of conductivity. The coefficients of the
forms s and B will then be reduced to zero, and each coefficient
303.] SKEW SYSTEM. 387
of the form A will be the reciprocal of the corresponding coefficient
of the form r. The expression for p will be
«,2 f/2 «2 A2
^-+^ + - = ^ -- (22)
^1 f2 T3 ?lr2r3
303.] The theory of the complete system of equations of resist
ance and of conductivity is that of linear functions of three vari
ables, and it is exemplified in the theory of Strains *, and in other
parts of physics. The most appropriate method of treating it is
that by which Hamilton and Tait treat a linear and vector function
of a vector. We shall not, however, expressly introduce Quaternion
notation.
The coefficients 2\, T2, Tz may be regarded as the rectangular
components of a vector T, the absolute magnitude and direction
of which are fixed in the body, and independent of the direction of
the axes of reference. The same is true of t^ t^ £3, which are the
components of another vector t.
The vectors T and t do not in general coincide in direction.
Let us now take the axis of z so as to coincide with the vector
T, and transform the equations of resistance accordingly. They
will then have the form
(23)
Z — S2 u -f S-L v + R3 w.
It appears from these equations that we may consider the elec
tromotive force as the resultant of two forces, one of them depending
only on the coefficients R and S, and the other depending on T alone.
The part depending on E and 8 is related to the current in the
same way that the perpendicular on the tangent plane of an
ellipsoid is related to the radius vector. The other part, depending
on 1] is equal to the product of T into the resolved part of the
current perpendicular to the axis of T, and its direction is per
pendicular to T and to the current,, being always in the direction in
which the resolved part of the current would lie if turned 90° in
the positive direction round T.
If we consider the current and T as vectors, the part of the
electromotive force due to T is the vector part of the product,
Tx current.
The coefficient T may be called the Rotatory coefficient. We
have reason to believe that it does not exist in any known sub-
* See Thomson and Tait's Natural Philo^y, § 154.
C C 2
388 RESISTANCE AND CONDUCTIVITY. [304.
stance. It should be found, if anywhere, in magnets, which have
a polarization in one direction, probably due to a rotational phe
nomenon in the substance.
304.] Assuming then that there is no rotatory coefficient, we
shall shew how Thomson's Theorem given in Art. 100 may be
extended to prove that the heat generated by the currents in the
system in a given time is a unique minimum.
To simplify the algebraical work let the axes of coordinates be
chosen so as to reduce expression (9), and therefore also in this case
expression (10), to three terms; and let us consider the general
characteristic equation (16) which thus reduces to
Also, let a, b, c be three functions of x, y, z satisfying the condition
da db dc . .
-7- + -7- + -7-= 0; (25)
dso d dz
and let
—
3 dz
Finally, let the triple-integral
(27)
be extended over spaces bounded as in the enunciation of Art. 100 ;
such viz. that Fis constant over certain portions or else the normal
component of the vector #, #, c is given, the latter condition being
accompanied by the further restriction that the integral of this
component over the whole bounding surface must be zero : then W
will be a minimum when
u = 0, v = 0, w = 0.
For we have in this case
7-^=1, r2R2=l, 7-3^3 =1;
and therefore, by (26),
305.] EXTENSION OF THOMSON^ THEOREM. 389
du dv dw
But since -?- + — + — = 0, (29)
dx dy dz
the third term vanishes by virtue of the conditions at the limits.
The first term of (28) is therefore the unique minimum value of W.
305.] As this proposition is of great importance in the theory of
electricity, it may be useful to present the following proof of the
most general case in a form free from analytical operations.
Let us consider the propagation of electricity through a conductor
of any form, homogeneous or heterogeneous.
Then we know that
(1) If we draw a line along the path and in the direction of
the electric current, the line must pass from places of high potential
to places of low potential.
(2) If the potential at every point of the system be altered in
a given uniform ratio,, the currents will be altered in the same ratio,
according to Ohm's Law.
(3) If a certain distribution of potential gives rise to a certain
distribution of currents, and a second distribution of potential gives
rise to a second distribution of currents, then a third distribution in
which the potential is the sum or difference of those in the first
and second will give rise to a third distribution of currents, such
that the total current passing through a given finite surface in the
third case is the sum or difference of the currents passing through
it in the first and second cases. For, by Ohm's Law, the additional
current due to an alteration of potentials is independent of the
original current due to the original distribution of potentials.
(4) If the potential is constant over the whole of a closed surface,
and if there are no electrodes or intrinsic electromotive forces
within it, then there will be no currents within the closed surface,
and the potential at any point within it will be equal to that at the
surface.
If there are currents within the closed surface they must either
be closed curves, or they must begin and end either within the
closed surface or at the surface itself.
But since the current must pass from places of high to places of
low potential, it cannot flow in a closed curve.
Since there are no electrodes within the surface the current
cannot begin or end within the closed surface, and since the
potential at all points of the surface is the same, there can be
no current along lines passing from one point of the surface to
another.
390 EESISTANCE AND CONDUCTIVITY. [306.
Hence there are no currents within the surface, and therefore
there can be no difference of potential, as such a difference would
produce currents, and therefore the potential within the closed
surface is everywhere the same as at the surface.
(5) If there is no electric current through any part of a closed
surface, and no electrodes or intrinsic electromotive forces within
the surface, there will be no currents within the surface, and the
potential will be uniform.
We have seen that the currents cannot form closed curves, or
begin or terminate within the surface, and since by the hypothesis
they do not pass through the surface, there can be no currents, and
therefore the potential is constant.
(6) If the potential is uniform over part of a closed surface, and
if there is no current through the remainder of the surface, the
potential within the surface will be uniform for the same reasons.
(7) If over part of the surface of a body the potential of every
point is known, and if over the rest of the surface of the body the
current passing through the surface at each point is known, then
only one distribution of potentials at points within the body can
exist.
For if there were two different values of the potential at any
point within the body, let these be 7l in the first case and F2 in
the second case, and let us imagine a third case in which the
potential of every point of the body is the excess of potential in the
first case over that in the second. Then on that part of the surface
for which the potential is known the potential in the third case will
be zero, and on that part of the surface through which the currents
are known the currents in the third case will be zero, so that by
(6) the potential everywhere within the surface will be zero, or
there is no excess of 7l over 7Z) or the reverse. Hence there is
only one possible distribution of potentials. This proposition is
true whether the solid be bounded by one closed surface or by
several.
On the Approximate Calculation of the Resistance of a Conductor
of a given Form.
306.] The conductor here considered has its surface divided into
three portions. Over one of these portions the potential is main
tained at a constant value. Over a second portion the potential has
a constant value different from the first. The whole of th<f remainder
of the surface is impervious to electricity. We may suppose the
306.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 391
conditions of the first and second portions to be fulfilled by applying
to the conductor two electrodes of perfectly conducting1 material,
and that of the remainder of the surface by coating it with per
fectly non-conducting material.
Under these circumstances the current in every part of the
conductor is simply proportional to the difference between the
potentials of the electrodes. Calling this difference the electro
motive force, the total current from the one electrode to the other
is the product of the electromotive force by the conductivity of the
conductor as a whole, and the resistance of the conductor is the
reciprocal of the conductivity.
It is only when a conductor is approximately in the circumstances
above defined that it can be said to have a definite resistance, or
conductivity as a whole. A resistance coil, consisting of a thin
wire terminating in large masses of copper, approximately satisfies
these conditions, for the potential in the massive electrodes is nearly
constant, and any differences of potential in different points of the
same electrode may be neglected in comparison with the difference
of the potentials of the two electrodes.
A very useful method of calculating the resistance of such con
ductors has been given, so far as I know, for the first time, by
Lord Rayleigh, in a paper on the Theory of Resonance *.
It is founded on the following considerations.
If the specific resistance of any portion of the conductor be
changed, that of the remainder being unchanged, the resistance of
the whole conductor will be increased if that of the portion is
increased, and diminished if that of the portion be diminished.
This principle may be regarded as self-evident, but it may easily
be shewn that the value of the expression for the resistance of a
system of conductors between two points selected as electrodes,
increases as the resistance of each member of the system in
creases.
It follows from this that if a surface of any form be described
in the substance of the conductor, and if we further suppose this
surface to be an infinitely thin sheet of a perfectly conducting
substance, the resistance of the conductor as a whole will be
diminished unless the surface is one of the equipotential surfaces
in the natural state of the conductor, in which case no effect will
be produced by making it a perfect conductor, as it is already in
electrical equilibrium.
* Phil. Trans., 1871, p. 77. See Art. 102.
392 RESISTANCE AND CONDUCTIVITY. [306.
If therefore we draw within the conductor a series of surfaces,
the first of which coincides with the first electrode, and the last
with the second, while the intermediate surfaces are bounded by
the non-conducting surface and do not intersect each other, and
if we suppose each of these surfaces to be an infinitely thin sheet
of perfectly conducting- matter, we shall have obtained a system
the resistance of which is certainly not greater than that of the
original conductor, and is equal to it only when the surfaces we
have chosen are the natural equipotential surfaces.
To calculate the resistance of the artificial system is an operation
of much less difficulty than the original problem. For the resist
ance of the whole is the sum of the resistances of all the strata
contained between the consecutive surfaces, and the resistance of
each stratum can be found thus :
Let dS be an element of the surface of the stratum, v the thick
ness of the stratum perpendicular to the element, p the specific
resistance, E the difference of potential of the perfectly conducting
surfaces, and dC the current through dS, then
dC=E±-dS, (1)
and the whole current through the stratum is
the integration being extended over the whole stratum bounded by
the non-conducting surface of the conductor.
Hence the conductivity of the stratum is
ds, (a)
E JJ pv
and the resistance of the stratum is the reciprocal of this quantity.
If the stratum be that bounded by the two surfaces for which
the function F has the values Fand F+dF respectively, then
(IF
and the resistance of the stratum is
-VFdS
P
To find the resistance of the whole artificial conductor, we have
only to integrate with respect to F} and we find
307.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 393
P
The resistance R of the conductor in its natural state is greater
than the value thus obtained, unless all the surfaces we have chosen
are the natural equipotential surfaces. Also, since the true value
of R is the absolute maximum of the values of Rl which can thus
be obtained, a small deviation of the chosen surfaces from the true
equipotential surfaces will produce an error of R which is com
paratively small.
This method of determining a lower limit of the value of the
resistance is evidently perfectly general, and may be applied to
conductors of any form, even when p, the specific resistance, varies
in any manner within the conductor.
The most familiar example is the ordinary method of determining
the resistance of a straight wire of variable section. In this case
the surfaces chosen are planes perpendicular to the axis of the
wire, the strata have parallel faces, and the resistance of a stratum
of section S and thickness ds is
77?
1 =
and that of the whole wire of length s is
^
where S is the transverse section and is a function of s.
This method in the case of wires whose section varies slowly
with the length gives a result very near the truth, but it is really
only a lower limit, for the true resistance is always greater than
this, except in the case where the section is perfectly uniform.
307.] To find the higher limit of the resistance, let us suppose
a surface drawn in the conductor to be rendered impermeable to
electricity. The effect of this must be to increase the resistance of
the conductor unless the surface is one of the natural surfaces of
flow. By means of two systems of surfaces we can form a set
of tubes which will completely regulate the flow, and the effect, if
there is any, of this system of impermeable surfaces must be to
increase the resistance above its natural value.
The resistance of each of the tubes may be calculated by the
method already given for a fine wire, and the resistance of the
whole conductor is the reciprocal of the sum of the reciprocals of
the resistances of all the tubes. The resistance thus found is greater
394 RESISTANCE AND CONDUCTIVITY. [307.
than the natural resistance, except when the tubes follow the
natural lines of flow.
In the case already considered, where the conductor is in the
form of an elongated solid of revolution, let us measure as along- the
axis, and let the radius of the section at any point be b. Let one
set of impermeable surfaces be the planes through the axis for each
of which <j) is constant, and let the other set be surfaces of revolution
for which yz=\l/lz, (9)
where \j/ is a numerical quantity between 0 and 1.
Let us consider a portion of one of the tubes bounded by the
surfaces $ and (£ + f7$, \jf and \}/ -\-d\ff, x and x-\- dx.
The section of the tube taken perpendicular to the axis is
ydyd$ = ±l*dtyd<t>. (10)
If 6 be the angle which the tube makes with the axis
tan0 = ^~ (11)
The true length of the element of the tube is clx sec 0, and its
true section is i ^ d^ ^ cos ^
so that its resistance is
Let A =J jf dx, and B = j £ Q <fo, (13)
the integration being extended over the whole length, x, of the
conductor, then the resistance of the tube d\\r dcf) is
2^
d\^f (,
and its conductivity is
To find the conductivity of the whole conductor, which is the
sum of the conductivities of the separate tubes, we must integrate
this expression between <£ = 0 and $ = 2 TT, and between i/r = 0
and \// = 1 . The result is
i-JurCi + J). (")
which may be less, but cannot be greater, than the true con
ductivity of the conductor.
308.] HIGHER AND LOWER LIMITS. 395
77 -n
When — is always a small quantity — will also be small, and we
CISC -£L
may expand the expression for the conductivity, thus
The first term of this expression, — ^-, is that which we should
J3.
have found by the former method as the superior limit of the con
ductivity. Hence the true conductivity is less than the first term
but greater than the whole series. The superior value of the
resistance is the reciprocal of this, or
If, besides supposing the flow to be guided by the surfaces <£ and
\l/, we had assumed that the flow through each tube is proportional
to d\l/d(p, we should have obtained as the value of the resistance
under this additional constraint
(17)
which is evidently greater than the former value, as it ought to be,
on account of the additional constraint. In Lord Rayleigh's paper
this is the supposition made, and the superior limit of the resistance
there given has the value (17), which is a little greater than that
which we have obtained in (16).
308.] We shall now apply the same method to find the correction
which must be applied to the length of a cylindrical conductor of
radius a when its extremity is placed in metallic contact with a
massive electrode, which we may suppose of a different metal.
For the lower limit of the resistance we shall suppose that an
infinitely thin disk of perfectly conducting matter is placed between
the end of the cylinder and the massive electrode, so as to bring
the end of the cylinder to one and the same potential throughout.
The potential within the cylinder will then be a function of its
length only, and if we suppose the surface of the electrode where
the cylinder meets it to be approximately plane, and all its dimen
sions to be large compared with the diameter of the cylinder, the
distribution of potential will be that due to a conductor in the form
of a disk placed in an infinite medium. See Arts. 151, 177.
If E is the difference of the potential of the disk from that of
the distant parts of the electrode, C the current issuing from the
396 RESISTANCE AND CONDUCTIVITY. [309.
surface of the disk into the electrode, and p the specific resistance
of the electrode ; then if Q is the amount of electricity on the disk,
which we assume distributed as in Art. 151, we have
171
p'C = i.47r§ = 2u— , by Art. 151,
7T
(18)
Hence, if the length of the wire from a given point to the
electrode is L, and its specific resistance p, the resistance from that
point to any point of the electrode not near the junction is
K L "'
E = f^ + Ta'
and this may be written
R=-P-fL + L^, (i9)
-no? \ p 4 J
where the second term within brackets is a quantity which must
be added to the length of the cylinder or wire in calculating its
resistance, and this is certainly too small a correction.
To understand the nature of the outstanding error we may
observe, that whereas we have supposed the flow in the wire up
to the disk to be uniform throughout the section, the flow from
the disk to the electrode is not uniform, but is at any point in
versely proportional to the minimum chord through that point. In
the actual case the flow through the disk will not be uniform,
but it will not vary so much from point to point as in this supposed
case. The potential of the disk in the actual case will not be
uniform, but will diminish from the middle to the edge.
309.] We shall next determine a quantity greater than the true
resistance by constraining the flow through the disk to be uniform
at every point. We may suppose electromotive forces introduced
for this purpose acting perpendicular to the surface of the disk.
The resistance within the wire will be the same as before, but
in the electrode the rate of generation of heat will be the surface-
integral of the product of the flow into the potential. The rate of
C
flow at any point is — -, and the potential is the same as that of
an electrified surface whose surface-density is o-, where
' being the specific resistance.
30Q.] CORRECTION FOR THE ENDS OF THE WIRE. 397
We have therefore to determine the potential energy of the
electrification of the disk with the uniform surface-density o\
* The potential at the edge of a disk of uniform density cr is easily
found to be 4 a a: The work done in adding a strip of breadth
da at the circumference of the disk is 2 n a a- da . 4#o-, and the
whole potential energy of the disk is the integral of this,
or P=~a*o2. (21)
In the case of electrical conduction the rate at which work is
done in the electrode whose resistance is Bf is C2R'. But from the
general equation of conduction the current across the disk per unit
area is of the form 1 d y
p' dv
47T
or -7-0".
P
Hence the rate at which work is done is
4*
7
We have therefore
V«=jP, (22)
whence, by (20) and (21),
7?'- 8p/
" SirV
and the correction to be added to the length of the cylinder is
/ 8
Js^*'
this correction being greater than the true value. The true cor-
/
rection to be added to the length is therefore - an, where n is a
7T 8 P
number lying between - and — , or between 0.785 and 0.849-
4 o TT
fLord Rayleigh, by a second approximation, has reduced the
superior limit of n to 0.8282.
* See a Paper by Professor Cayley, London Math. Soc. Proc. vi. p. 47.
f Phil. Mag., Nov. 1872. Lord Eayleigh subsequently obtained -8242 as the
superior limit. See London Math. Soc. Proc. viii. p. 74.
CHAPTER IX.
CONDUCTION THROUGH HETEROGENEOUS MEDIA.
On the Conditions to le Fulfilled at the Surface of Separation
between Two Conducting Media.
310.] THERE are two conditions which the distribution of currents
must fulfil in general, the condition that the potential must be
continuous, and the condition of ' continuity ' of the electric currents.
At the surface of separation between two media the first of these
conditions requires that the potentials at two points on opposite
sides of the surface, but infinitely near each other, shall be equal.
The potentials are here understood to be measured by an elec
trometer put in connexion with the given point by means of an
electrode of a given metal. If the potentials are measured by the
method described in Arts. 222, 246, where the electrode terminates
in a cavity of the conductor filled with air, then the potentials at
contiguous points of different metals measured in this way will
differ by a quantity depending on the temperature and on the
nature of the two metals.
The other condition at the surface is that the current through
any element of the surface is the same when measured in either
medium.
Thus, if V\ and ^ are the potentials in the two media, then at
any point in the surface of separation
*; = r» (^
and if u19 vl9 w^ and w29 v2, w.2 are the components of currents in the
two media, and I, m, n the direction-cosines of the normal to the
surface of separation,
Uil+vLm -\-w^n = U2l + v2m + w2n. (2)
In the most general case the components u, v, w are linear
3IO.] SURFACE-CONDITIONS. 399
functions of the derivatives of F, the forms of which are given in
the equations
u =
v =23X+r.Y+Jp1Z,
(3)
where X, J, Z are the derivatives of V with respect to as, yt z
respectively.
Let us take the case of the surface which separates a medium
having these coefficients of conduction from an isotropic medium
having a coefficient of conduction equal to r.
Let X', Y', Zf be the values of X, Y} Z in the isotropic medium,
then we have at the surface
r=7', (4)
or Xdx + Ydy + Zdz = X'dx + Y'dy + Z'dz, (5)
when I dx -\- m dy >\- n dz = 0. (6)
This condition gives
X'=X+4Tr<rl, 7'= r+47ro-M, Z'=Z + 4iT(yn, (?)
where cr is the surface-density.
We have also in the isotropic medium
u' = r X', v' = rY', w' = rZ', (8)
and at the boundary the condition of flow is
u'l+tfm+w'n = ul-\-vm + wn, (9)
or r(lX+mY-\- nZ+kvcr)
whence
a(ra—r))Z. (11)
The quantity a- represents the surface- density of the charge
on the surface of separation. In crystallized and organized sub
stances it depends on the direction of the surface as well as on
the force perpendicular to it. In isotropic substances the coeffi
cients p and q are zero, and the coefficients / are all equal, so that
4:7(7 = (A _ i) (IX+mY+nZ), (12)
where ^ is the conductivity of the substance, r that of the external
medium, and I, m, n the direction-cosines of the normal drawn
towards the medium whose conductivity is r,
When both media are isotropic the conditions may be greatly
400 CONDUCTION IN HETEROGENEOUS MEDIA. [SH«
simplified, for if Jc is the specific resistance per unit of volume, then
1 dV I dV l dV
U=—j^-t V=— y-y-J W = — y — - , (13)
k dx K dy k dz v '
and if v is the normal drawn at any point of the surface of separa
tion from the first medium towards the second, the condition of
continuity is I dFl I dV2
~J 7 "~~ ~7 7 * V •*• ^ )
% dv #2 du
If Ol and 62 are the angles which the lines of flow in the first and
second media respectively make with the normal to the surface
of separation, then the tangents to these lines of flow are in the
same plane with the normal and on opposite sides of it, and
#]_ tan Sl = k2 tan 02 . (15)
This may be called the law of refraction of lines of flow.
311.] As an example of the conditions which must be fulfilled
when electricity crosses the surface of separation of two media,
let us suppose the surface spherical and of radius a, the specific
resistance being k^ within and k2 without the surface.
Let the potential, both within and without the surface, be ex
panded in solid harmonics, and let the part which depends on
the surface harmonic S4 be
r1 = (4i'+.B1i-<'«>)*<, (i)
r. = (A, t+st r-««>) st (2)
within and without the sphere respectively.
At the surface of separation where r = a we must have
Fi=F2 and l"i '"i. (3)
^ dr k.2 dr
From these conditions we get the equations
These equations are sufficient, when we know two of the four
quantities Alt A^ Blt B2, to deduce the other two.
Let us suppose Al and B± known, then we find the following
expressions for A2 and B2,
r^v-r./i-,
(2»+l)
i+l
SPHERICAL SHELL. £01
In this way we can find the conditions which each term of the
harmonic expansion of the potential must satisfy for any number of
strata bounded by concentric spherical surfaces.
312.] Let us suppose the radius of the first spherical surface
to be al} and let there be a second spherical surface of radius a2
greater than %, beyond which the specific resistance is kz. If there
are no sources or sinks of electricity within these spheres there
will be no infinite values of Yt and we shall have Bl = 0.
We then find for A3 and JB3, the coefficients for the outer medium,
(6)
The value of the potential in the outer medium depends partly
on the external sources of electricity, which produce currents in
dependently of the existence of the sphere of heterogeneous matter
within, and partly on the disturbance caused by the introduction of
the heterogeneous sphere.
The first part must depend on solid harmonics of positive degrees
only, because it cannot have infinite values within the sphere. The
second part must depend on harmonics of negative degrees, because
it must vanish at an infinite distance from the centre of the sphere.
Hence the potential due to the external electromotive forces must
be expanded in a series of solid harmonics of positive degree. Let
AS be the coefficient of one of these, of the form
Then we can find A19 the corresponding coefficient for the inner
sphere by equation (6), and from this deduce A2, B2, and .Z?3. Of
these £3 represents the effect on the potential in the outer medium
due to the introduction of the heterogeneous spheres.
Let us now suppose £3 = < so that the case is that of a hollow
shell for which k — /£2, separating an inner from an outer portion of
the same medium for which k = klt
If we put
o = ! „
VOL. I. D d
402 CONDUCTION IN HETEROGENEOUS MEDIA.
then A1 = ^(2*+ l)2 CAB,
B.2 = 2i+l
The difference between A3 the undisturbed coefficient, and Al its
value in the hollow within the spherical shell, is
- A, = (k,- ktfi (i + 1) (l - )2'+1) CA,. (8)
4r
Since this quantity is always positive whatever be the values
of k± and £2, it follows that, whether the spherical shell conducts
better or worse than the rest of the medium, the electrical action
in the space occupied by the shell is less than it would otherwise
be. If the shell is a better conductor than the rest of the
medium it tends to equalize the potential all round the inner
sphere. If it is a worse conductor, it tends to prevent the
electrical currents from reaching the inner sphere at all.
The case of a solid sphere may be deduced from this by making
#! = 0, or it may be worked out independently.
313.] The most important term in the harmonic expansion is
that in which i = 1, for which
1
(• (9)
i — Ql- I- C< A A Q I' (9 If _i_ I- \ r 4
Lj — t) A>J K<£ \J*O-n j 2 ~~~ 2 \ 1 ' 2/ 3 '
The case of a solid sphere of resistance /£2 may be deduced from
this by making ^ = 0. We then have
"
__ 21 3 A
jL/'j — . -. tA/n ^iQ •
^i + 2/^2 2
It is easy to shew from the general expressions that the value
of -Z?3 in the case of a hollow sphere having a nucleus of resistance
&15 surrounded by a shell of resistance /£2, is the same as that of
a uniform solid sphere of the radius of the outer surface, and of
resistance K, where
-
314-] MEDIUM CONTAINING SMALL SPHERES. 403
314.] If there are n spheres of radius a^ and resistance $15 placed
in a medium whose resistance is /£2> at such distances from each
other that their effects in disturbing* the course of the current
may be taken as independent of each other, then if these spheres
are all contained within a sphere of radius #2, the potential at a
great distance from the centre of this sphere will be of the form.
V = Ar + n£cos0, (12)
where the value of B is
The ratio of the volume of the n small spheres to that of the
sphere which contains them is
>-$•• . <">
The value of the potential at a great distance from the sphere
may therefore be written
Now if the whole sphere of radius a2 had been made of a material
of specific resistance K, we should have had
That the one expression should be equivalent to the other,
2^ + ^ + X^i-^) *
' *'
This, therefore, is the specific resistance of a compound medium
consisting of a substance of specific resistance k.2, in which are
disseminated small spheres of specific resistance kl} the ratio of the
volume of all the small spheres to that of the whole being p. In
order that the action of these spheres may not produce effects
depending on their interference, their radii must be small compared
with their distances, and therefore^? must be a small fraction.
This result may be obtained in other ways, but that here given
involves only the repetition of the result already obtained for a
single sphere.
When the distance between the spheres is not great compared
^ _ fc
with their radii, and when — r— — f- is considerable, then other
2 Kl -f- #2
terms enter into the result, which we shall not now consider.
In consequence of these terms certain systems of arrangement of
D d 2,
404 CONDUCTION IN HETEROGENEOUS MEDIA. [315.
the spheres cause the resistance of the compound medium to be
different in different directions.
Application of the Principle of Images.
315.] Let us take as an example the case of two media separated
by a plane surface, and let us suppose that there is a source S
of electricity at a distance a from the plane surface in the first
medium, the quantity of electricity flowing from the source in unit
of time being 8.
If the first medium had been infinitely extended the current
at any point P would have been in the direction SP, and the
potential at P would have been — where E = — - and /, = SP.
r-L 477
In the actual case the conditions may be satisfied by taking
a point 7, the image of S in the second medium, such that 7$
is normal to the plane of separation and is bisected by it. Let r2
be the distance of any point from 7, then at the surface of separation
i _ 2 /«x
J~v- -3T
Let the potential F^ at any point in the first medium be that
due to a quantity of electricity E placed at 8t together with an
imaginary quantity E2 at 7, and let the potential F2 at any point
of the second medium be that due to an imaginary quantity El at
8, then if E E2 El
Y — --- j_ -A and F2 = — - i (3)
*i ^ ri
the superficial condition V^ = F2 gives
(4)
and the condition
Aj Av ~~ A2 dv
gives 1-(E-E2)=E1, (6)
The potential in the first medium is therefore the same as would
be produced in air by a charge E placed at 8t and a charge E%
at 7 on the electrostatic theory, and the potential in the second
medium is the same as that which would be produced in air by
a charge El at S.
31 7-] STRATUM WITH PARALLEL SIDES. 405
The current at any point of the first medium is the same as would
have been produced by the source 8 together with a source 2~ 1 S
K-\ -\- /C.)
placed at I if the first medium had been infinite, and the current
at any point of the second medium is the same as would have been
2k S
produced by a source 2 placed at/? if the second medium had
been infinite.
We have thus a complete theory of electrical images in the case
of two media separated by a plane boundary. Whatever be the
nature of the electromotive forces in the first medium, the potential
they produce in the. first medium may be found by combining their
direct effect with the effect of their image.
If we suppose the second medium a perfect conductor, then
£2 = 0, and the image at / is equal and opposite to the source
at 8. This is the case of electric images, as in Thomson's theory
in electrostatics.
If we suppose the second medium a perfect insulator, then
k2 — oo, and the image at / is equal to the source at 8 and of the
same sign. This is the case of images in hydrokinetics when the
fluid is bounded by a rigid plane surface.
316.] The method of inversion, which is of so much use in
electrostatics when the. bounding surface is supposed to be that
of a perfect conductor, is not applicable to the more general case
of the surface separating two conductors of unequal electric resist
ance. The method of inversion in two dimensions is, however,
applicable, as well as the more general method of transformation in
two dimensions given in Art. 190 *.
Conduction through a Plate separating Two Media.
317.] Let us next consider the effect of a plate of thickness AS of
a medium whose resist
ance is k.2, and separating \
two media whose resist
ances are k^ and /£3, in "t J J~
altering the potential due
to a source S in the first
medium.
The potential will be Fis- 24-
* See Kirchhoff, Pogg. Ann. Ixiv. 497, and Ixvii. 344 : Quincke, Pogg. xcvii. 382;
and Smith, Proc. R. S. Edin., 1869-70, p. 79.
406 CONDUCTION IN HETEROGENEOUS MEDIA.
equal to that due to a system of charges placed in air at certain
points along1 the normal to the plate through S.
Make
AI=SA, BI^SB, AJ^I.A, BI^^B, AJ2=I2A, &c. ;
then we have two series of points at distances from each other equal
to twice the thickness of the plate.
318.] The potential in the first medium at any point P is equal to
PS PI '
that at a point P in the second
*
•P8 T PI P'IL ^ P'72
and that at a point Px/ in the third
r + + +&c" (10)
where /, 7', &c. represent the imaginary charges placed at the
points 7, &c., and the accents denote that the potential is to be
taken within the plate.
Then, by the last Article, for the surface through A we have,
(11)
J !
For the surface through B we find
7. 7. o 7.
^_^ ^/>s=_^3
yt3 + ^2 ^2 +
Similarly for the surface through A again,
7. _ 7, 07:
7/i 2r' " —
and for the surface through 7?,
- ' 9 Jf
/ TT_ ^^3 T/ /14\
1> /1-J-
If we make = = and =
we find for the potential in the first medium,
r= Ts -p-J
3I9-] STRATIFIED CONDUCTORS. 40?
For the potential in the third medium we find
If the first medium is the same as the third, then k^ = £3 and
p = p', and the potential on the other side of the plate will be
If the plate is a very much better conductor than the rest of the
medium, p is very nearly equal to 1. If the plate is a nearly perfect
insulator, p is nearly equal to — 1, and if the plate differs little in
conducting power from the rest of the medium, p is a small quantity
positive or negative.
The theory of this case was first stated by Green in his ' Theory
of Magnetic Induction' (Essay, p. 65). His result, however, is
correct only when p is nearly equal to 1 *. The quantity g which
he uses is connected with p by the equations
2p k-^ — k2 3y k-± — k2
If we put p = - , we shall have a solution of the problem of
1 ~\~ 2i 7T K
the magnetic induction excited by a magnetic pole in an infinite
plate whose coefficient of magnetization is K.
On Stratified Conductors.
319.] Let a conductor be composed of alternate strata of thick
ness c and <f of two substances whose coefficients of conductivity
are different. Required the coefficients of resistance and* conduc
tivity of the compound conductor.
Let the plane of the strata be normal to Z. Let every symbol
relating to the strata of the second kind be accented, and let
every symbol relating to the compound conductor be marked with
a bar thus, X. Then
Y= X = X', (c + c')u = cu+c'u',
(c + c)~Z = cZ+ c'Z', w = w = w'.
We must first determine u, u, v> v, Z and Z' in terms of
X, Y and w from the equations of resistance, Art. 297, or those
* See Sir W. Thomson's 'Note on Induced Magnetism in a Plate,' Camb. and
Dub. ]\LatU. Journ., Nov. 1845, or Reprint, art. ix. § 156.
408 CONDUCTION. IN: HETEROGENEOUS MEDIA. [320.
of conductivity, Art. 298. If we put D for the determinant of the
coefficients of resistance, we find
vr3D = RiJ-
Similar equations with the symbols accented give the values
of u'', v' and Z'. Having found u, v and w in terms of X, Fand ^
we may write down the equations of conductivity of the stratified
conductor. If we make k = — and /$'= ~, we find
_
-
_
=
. .
_
. .
320.] If neither of the two substances of which the strata are
formed has the rotatory property of Art. 303, the value of any
P or p will be equal to that of its corresponding Q or q. From
this it follows that in the stratified conductor also
Pi = £L» Pz = £2> Ps = ?s»
or there is no rotatory property developed by stratification, unless
it exists in the materials.
321.] If we now suppose that there is no rotatory property, and
also that the axes of a, y and z are the principal axes, then the
j} and q coefficients vanish, and
If we begin with both substances isotropic, but of different
322.] STRATIFIED CONDUCTORS, 409
conductivities, then the result of stratification- will be to make
the resistance greatest in the direction of a normal to the strata,
and the resistance in all directions in the plane of the strata will
be equal.
322.] Take an isotropic substance of conductivity r, cut it into
exceedingly thin slices of thickness #, and place them alternately
with slices of a substance whose conductivity is s, and thickness k±a.
Let these slices be normal to x. Then cut this compound con
ductor into thicker slices, of thickness b, normal to yt and alternate
these with slices whose conductivity is s and thickness kzb.
Lastly, cut the new conductor into still thicker slices, of thick
ness c, normal to z, and alternate them with slices whose con
ductivity is s and thickness kzc.
The result of the three operations will be to cut the substance
whose conductivity is r into rectangular parallelepipeds whose
dimensions are a, b and c, where ~b is exceedingly small compared
with c9 and a is exceedingly small compared with b, and to embed
these parallelepipeds in the substance whose conductivity is s, so
that they are separated from each other k^a in the direction of xy
kjb in that of y, and k$c in that of z. The conductivities of the
conductor so formed in the directions of #, yy and z are to be found
by three applications in order of the results of Art. 321. We
thereby obtain
{!
_ -f- c-c + / 4- / s
'3 =
r + C1
The accuracy of this investigation depends upon the three dimen
sions of the parallelepipeds being of different orders of magnitude,
so that we may neglect the conditions to be fulfilled at their edges
and angles. If we make klt k2 and k% each unity, then
s 3r+5s
If r = 0, that is, if the medium of which the parallelepipeds
are made is a perfect insulator, then
410 CONDUCTION IN HETEROGENEOUS MEDIA. [323.
If r — oo, that is, if the parallelepipeds are perfect conductors,
fl == ~T~ S, Tn ~~~ — ,9 V 9 ?
1 ^^ 4^ ) '2 ~~ 2 J '3 — *5 « •
In every case, provided ^ = £2 = /£3, it may be shewn that
ri> r2 and 7*3 are in ascending order of magnitude, so that the
greatest conductivity is in the direction of the longest dimensions
of the parallelepipeds, and the greatest resistance in the direction
of their shortest dimensions.
323.] In a rectangular parallelepiped of a conducting solid, let
there be a conducting channel made from one angle to the opposite,
the channel being a wire covered with insulating material, and
let the lateral dimensions of the channel be so small that the
conductivity of the solid is not affected except on account of the
current conveyed along the wire.
Let the dimensions of the parallelepiped in the directions of the
coordinate axes be a, b, c, and let the conductivity of the channel,
extending from the origin to the point (abc\ be abcK.
The electromotive force acting between the extremities of the
channel is aX+bY+cZ,
and if C' be the current along the channel
C'=Kabc(aX+bY+cZ\
The current across the face be of the parallelepiped is bcu, and
this is made up of that due to the conductivity of the solid and
of that due to the conductivity of the channel, or
bcu = bc(
or u = (^ + Ka2) Z+ (p3 + Ka b} Y+ fa + Kca) Z.
In the same way we may find the values of v and w. The
coefficients of conductivity as altered by the effect of the channel
will be
Pi + Kbc,
In these expressions, the additions to the values of plt &c., due
to the effect of the channel, are equal to the additions to the values
of qlt &c. Hence the values of p^ and q1 cannot be rendered
unequal by the introduction of linear channels into every element
of volume of the solid, and therefore the rotatory property of
Art. 303, if it does not exist previously in a solid, cannot be
introduced by such means.
324.] COMPOSITE CONDUCTOR. 411
3.24.] To construct a framework of linear conductors which shall
have any given coefficients of conductivity forming a symmetrical
system.
Let the space be divided into equal small cubes, of which let the
figure represent one. Let the coordinates of the
points 0, L} M, N, and their potentials be as
follows : — x y z Potential
0 000 X+Y+Z
L 0 1 1 X
M 1 0 1 Y
#"110 Z
Let these four points be connected by six conductors,
OL, OM, ON, MN, NL, LM,
of which the conductivities are respectively
A, B, C, P, Q, E.
The electromotive forces along these conductors will be
Y+Z, Z+X, X+Y, Y-Z, Z-X, X-Y,
and the currents
A (Y+Z), B(Z + X\ C(X+Y\ P(Y-Z), Q(Z-X), R(X-Y).
Of these currents, those which convey electricity in the positive
direction of so are those along LM, LN, OM and ON, and the
quantity conveyed is
u = (B+C+q + R)X+(C-R)Y +(B-q)Z.
Similarly
v = (C-R)X +(C+A
w=(B-Q)X +(A-P)Y
whence we find by comparison with the equations of conduction.
Art. 298,
4 A = r:6 + r3—r1+2p1, 4P = r2 + r3-r1-2jp1,
4 Q = r3 + rl — r2—2_p2,
CHAPTER X.
CONDUCTION IN DIELECTRICS.
325.] WE have seen that when electromotive force acts on a
dielectric medium it produces in it a state which we have called
electric polarization, and which we have described as consisting
of electric displacement within the medium in a direction which,
in isotropic media, coincides with that, of the electromotive force,
combined with a superficial charge on every element of volume
into which we may suppose the dielectric divided, which is negative
on the side towards which the force acts, and positive on the side
from which it acts.
When electromotive force acts on a conducting medium it also
produces what is called an electric current.
Now dielectric media, with very few, if any, exceptions, are also
more or less imperfect conductors, and many media which are not
good insulators exhibit phenomena of dielectric induction. Hence
we are led to study the state of a medium in which induction and
conduction are going on at the same time.
For simplicity we shall suppose the medium isotropic at every
point, but not necessarily homogeneous at different points. In this
case, the equation of Poisson becomes,- by Art. 83,
d /rdV d dV d /d
where K is the ' specific inductive capacity.'
The ' equation of continuity' of electric currents becomes
jlfl^U^fi^H Afl^-^=o (2
dx^r dx}~^ dy\r dy> + dz\ dz' dt ~ {
where r is the specific resistance referred to unit of volume.
When K or r is discontinuous, these equations must be trans
formed into those appropriate to surfaces of discontinuity.
326.] THEORY OF A CONDENSER. 413
In a strictly homogeneous medium r and K are both constant, so
that we find
d^V d2T d2V p dp
dx2 dy* dz2 ~ K " dt '
whence p — Ce Kr ; (4)
Kr --
or, if we put T = — , p — Ce T . (5)
This result shews that under the action of any external electric
forces on a homogeneous medium, the interior of which is originally
charged in any manner with electricity, the internal charges will
die away at a rate which does not depend on the external forces,
so that at length there will be no charge of electricity within
the medium, after which no external forces can either produce or
maintain a charge in any internal portion of the medium, pro
vided the relation between electromotive force, electric polarization
and conduction remains the same. When disruptive discharge
occurs these relations cease to be true, and internal charge may
be produced.
On Conduction through a Condenser.
326.] Let 67 be the capacity of a condenser, R its resistance, and
E the electromotive force which acts on it, that is, the difference of
potentials of the surfaces of the metallic electrodes.
Then the quantity of electricity on the side from which the
electromotive force acts will be CE, and the current through the
substance of the condenser in the direction of the electromotive
force will be -=-•
If the electrification is supposed to be produced by an electro
motive force E acting in a circuit of which the condenser forms
part, and if -~ represents the current in that circuit, then
Let a battery of electromotive force EQ and resistance rt be
introduced into this circuit, then
jj f\ 1jl Ijl Tfl rJtfl
CtlcJ JJjn—Jl/ "j .^Ct/jll
dt r^ li dt
Hence, at any time tlt
'wherer'=!fv «
414 CONDUCTION IN DIELECTRICS. [327.
Next, let the circuit r^ be broken for a time t.2,
E( = E2) = El6~^ where T2 = CR. (9)
Finally, let the surfaces of the condenser be connected by means
of a wire whose resistance is r3 for a time t^,
_^_ ntr
E( = St) = E^'i where 73 = %g . (10)
If QB is the total discharge through this wire in the time t3 ,
-*-±-\ -f-2-f -^-^
- *)•*(*-*)' <">
In this way we may find the discharge through a wire which
is made to connect the surfaces of a condenser after being charged
for a time f^, and then insulated for a time t2. If the time of
charging is sufficient, as it generally is, to develope the whole
charge, and if the time of discharge is sufficient for a complete
discharge, the discharge is
_
327.] In a condenser of this kind, first charged in any way, next
discharged through a wire of small resistance, and then insulated,
no new electrification will appear. In most actual condensers,
however, we find that after discharge and insulation a new charge
is gradually developed, of the same kind as the original charge,
but inferior in intensity. This is called the residual charge. To
account for it we must admit that the constitution of the dielectric
medium is different from that which we have just described. We
shall find, however, that a medium formed of a conglomeration of
small pieces of different simple media would possess this property.
Theory of a Composite Dielectric.
328.] We shall suppose, for the sake of simplicity, that the
dielectric consists of a number of plane strata of different materials
and of area unity, and that the electric forces act in the direction
of the normal to the strata.
Let alt a2, &c. be the thicknesses of the different strata.
Xlf X2, &c. the resultant electrical forces within the strata.
> &c. the currents due to conduction through the strata.
' &c- tne electric displacements.
«!, u2, &c. the total currents, due partly to conduction and partly
to variation of displacement.
328.] STKATIFIED DIELECTEIC. 415
rlt r2, &c. tlie specific resistances referred to unit of volume.
Klt E2, &c. the specific inductive capacities.
£T, k2, &G. the reciprocals of the specific inductive capacities.
FJ the electromotive force due to a voltaic battery, placed in
the part of the circuit leading from the last stratum towards the
first, which we shall suppose good conductors.
Q the total quantity of electricity which has passed through this
part of the circuit up to the time t.
RQ the resistance of the battery with its connecting wires.
o-12 the surface-density of electricity on the surface which separates
the first and second strata.
Then in the first stratum we have, by Ohm's Law,
X1 = r1p1. (1)
By the theory of electrical displacement,
^ = 4^^. (2)
By the definition of the total current,
dfi
Ui=Pi + ^> (3)
with similar equations for the other strata, in each of which the
quantities have the suffix belonging to that stratum.
To determine the surface-density on any stratum, we have an
equation of the form ^ _ y2__ f^ u\
and to determine its variation we have
By differentiating (4) with respect to t, and equating the result
to (5), we obtain
f2
=u>™J> (6)
or, by taking account of (3),
«]_ = u2 =. &c. = u. (7)
That is, the total current u is the same in all the strata, and is
equal to the current through the wire and battery.
We have also, in virtue of equations (l) and (2),
1 . 1 dX.
» = ^ + 4^-lT' <8>
from which we may find Xx by the inverse operation on uy
416 CONDUCTION IN DIELECTKICS. [329.
The total electromotive force E is
E=alX1 + a2Xz + &c., (10)
or ^ = ,1l+)-V,2I+)-1 + &c.5 (11)
an equation between U, the external electromotive force, and ut the
external current.
If the ratio of r to Tc is the same in all the strata, the equation
reduces itself to
w> (12)
which is the case we have already examined, and in which, as we
found, no phenomenon of residual charge can take place.
If there are n substances having different ratios of r to k, the
general equation (11), when cleared of inverse operations, will be
a linear differential equation, of the nth order with respect to E
and of the (n— l)th order with respect to u, t being the independent
variable.
From the form of the equation it is evident that the order of
the different strata is indifferent, so that if there are several strata
of the same substance we may suppose them united into one
without altering the phenomena.
329.] Let us now suppose that at first f^ , f^ , &c. are all zero,
and that an electromotive force E is suddenly made to act, and let
us find its instantaneous effect.
Integrating (8) with respect to tt we find
q = fudt - — fx^+ -^r-Zj+const. (13)
J T-i J 4 77 K j
Now, since X1 is always in this case finite, / X1df must be in
sensible when t is insensible, and therefore, since Xl is originally
zero, the instantaneous effect will be
X1 = 4w£1Q. (14)
Hence, by equation (10),
E = 47T (Vl + V2 + &C') Q> (15)
and if C be the electric capacity of the system as measured in this
instantaneous way,
329-] ELECTRIC 'ABSORPTION/ 41?
This is the same result that we should have obtained if we had
neglected the conductivity of the strata.
Let us next suppose that the electromotive force E is continued
uniform for an indefinitely long time, or till a uniform current of
conduction equal to p is established through the system.
We have then X1 = r±j)9 etc., and therefore by (10),
E = fa % + r2 a2 + &c.)j?. ( 1 7)
If R be the total resistance of the system,
E = — = r1fl1 + r202 + &o. (18)
In this state we have by (2),
so that ^-(Jij. __£_),. ' (19)
If we now suddenly connect the extreme strata by means of a
conductor of small resistance, E will be suddenly changed from its
original value EQ to zero, and a quantity Q of electricity will pass
through the conductor.
To determine Q we observe that if !"/ be the new value of Xlt
then by (13), ^'=^ + 4^. (20)
Hence, by (10), putting E = 0,
0 = a1X1 + &c. + 4v(a1&1 + a2&2 + &G.)Q, (21)
or o = fio+-Q. (22)
Hence Q = — CU0 where C is the capacity, as given by equation
(T6). The instantaneous discharge is therefore equal to the in
stantaneous charge.
Let us next suppose the connexion broken immediately after this
discharge. We shall then have u = 0, so that by equation (8),
_47T/ti
Xi = r* n , (23)
where X' is the initial value after the discharge.
Hence, at any time t,
The value of S at any time is therefore
=^o{(^p-4Mi*iC)r^1|+ (^p-^^^r^'+ftc.!, (24)
VOL. I. E 6
418 CONDUCTION IN DIELECTRICS. [330.
and the instantaneous discharge after any time t is EG. This is
called the residual discharge.
If the ratio of r to Jc is the same for all the strata, the val ue of E
will be reduced to zero. If, however, this ratio is not the same, let
the terms be arranged according to the values of this ratio in
descending order of magnitude.
The sum of all the coefficients is evidently zero, so that when
t = 0, E = 0. The coefficients are also in descending order of
magnitude, and so are the exponential terms when t is positive.
Hence, when t is positive, E will be positive, so that the residual
discharge is always of the same sign as the primary discharge.
When t is indefinitely great all the terms disappear unless any
of the strata are perfect insulators, in which case ^ is infinite for
that stratum, and R is infinite for the whole system, and the final
value of E is not zero but
E = EQ(l-^'nalklC). (25)
Hence, when some, but not all, of the strata are perfect insulators,
a residual discharge may be permanently preserved in the system.
330.] We shall next determine the total discharge through a wire
of resistance 11 Q kept permanently in connexion with the extreme
tstrata of the system, supposing the system first charged by means
of a long-continued application of the electromotive force E.
At any instant we have
E = air1p1 + aar2p2 + bc.+S0u = 0, (26)
and also, by (3), u - = pl + ± • (27)
Hence (R + BQ}u = a,r + V22+ &c. (28)
Integrating with respect to t in order to find Q, we get
(R + S<))Q = a, r, (//-/,) + a. r, (// -/2) + &e., (29)
where/j is the initial, and/i' the final value of/].
In this case //= 0, and by (2) and (20) fa = E0 ( *,' — C) •
Hence (R+R0) Q = A_ (f! + + &c.) -E9CR, (30)
where the summation is extended to all quantities of this form
belonging to every pair of strata.
33 !•] RESIDUAL DISCHARGE. 419
It appears from this that Q is always negative, that is to say, in
the opposite direction to that of the current employed in charging
the system.
This investigation shews that a dielectric composed of strata of
different kinds may exhibit the phenomena known as electric
absorption and residual discharge, although none of the substances
of which it is made exhibit these phenomena when alone. An
investigation of the cases in which the materials are arranged
otherwise than in strata would lead to similar results, though
the calculations would be more complicated, so that we may
conclude that the phenomena of electric absorption may be ex
pected in the case of substances composed of parts of different
kinds, even though these individual parts should be microscopically
small.
It by no means follows that every substance which exhibits this
phenomenon is so composed, for it may indicate a new kind of
electric polarization of which a homogeneous substance may be
capable, and this in some cases may perhaps resemble electro
chemical polarization much more than dielectric polarization.
The object of the investigation is merely to point out the true
mathematical character of the so-called electric absorption, and to
shew how fundamentally it differs from the phenomena of heat
which seem at first sight analogous.
331.] If we take a thick plate of any substance and heat it
on one side, so as to produce a flow of heat through it, and if
we then suddenly cool the heated side to the same temperature
as the other, and leave the plate to itself, the heated side of the
plate will again become hotter than the other by conduction from
within.
Now an electrical phenomenon exactly analogous to this can
be produced, and actually occurs in telegraph cables, but its mathe
matical laws, though exactly agreeing with those of heat, differ
entirely from those of the stratified condenser.
In the case of heat there is true absorption of the heat into
the substance with the result of making it hot. To produce a truly
analogous phenomenon in electricity is impossible, but we may
imitate it in the following way in the form of a lecture-room
experiment.
Let Alt AZ9 &c. be the inner conducting surfaces of a series of
condensers, of which £0, JB±, £2t &c. are the outer surfaces.
Let Al} AZ, &c. be connected in series by connexions of resist-
E e 2
420
CONDUCTION IN DIELECTRICS.
[33I-
ance R, and let a current be passed along this series from left to
right.
Let us first suppose the plates _Z?0, JSlt J?2, each insulated and
free from charge. Then the total quantity of electricity on each of
the plates B must remain zero, and since the electricity on the
plates A is in each case equal and opposite to that of the opposed
Fig. 26.
surface they will not be electrified, and no alteration of the current
will be observed.
But let the plates B be all connected together, or let each be
connected with the earth. Then, since the potential of Al is
positive, while that of the plates B is zero, A1 will be positively
electrified and Bl negatively.
If Pls P2> &c. are the potentials of the plates Als Az, &c., and C
the capacity of each, and if we suppose that a quantity of electricity
equal to Q0 passes through the wire on the left, Ql through the
connexion S19 and so on, then the quantity which exists on the
plate Al is Q0— Q19 and we have
Co-«i=tfi3.
Similarly Qi~Q2= <?2P2>
and so on.
But by Ohm's Law we have
If we suppose the values of C the same for each plate, and those
of R the same for each wire, we shall have a series of equations of
the form
332.] THEORY OF ELECTRIC CABLES. 421
cU
If there are n quantities of electricity to be determined, and if
either the total electromotive force, or some other equivalent con
ditions be given, the differential equation for determining any one
of them will be linear and of the nth order.
By an apparatus arranged in this way, Mr. Varley succeeded in
imitating the electrical action of a cable 12,000 miles long.
When an electromotive force is made to act along the wire on
the left hand, the electricity which flows into the system is at first
principally occupied in charging the different condensers beginning
with Alt and only a very small fraction of the current appears
at the right hand till a considerable time has elapsed. If galvano
meters be placed in circuit at R19 E2) &c. they will be affected
by the current one after another, the interval between the times of
equal indications being greater as we proceed to the right.
332.] In the case of a telegraph cable the conducting wire is
separated from conductors outside by a cylindrical sheath of gutta-
percha, or other insulating material* Each portion of the cable
thus becomes a condenser, the outer surface of which is always at
potential zero. Hence, in a given portion of the cable, the quantity
of free electricity at the surface of the conducting wire is equal
to the product of the potential into the capacity of the portion of
the cable considered as a condenser.
If aly a2 are the outer and inner radii of the insulating sheath,
and if K is its specific dielectric capacity, the capacity of unit of
length of the cable is, by Art. 126,
.-- V o>
2io^5
Let v be the potential at any point of the wire, which we may
consider as the same at every part of the same section.
Let Q be the total quantity of electricity which has passed
through that section since the beginning of the current. Then the
quantity which at the time t exists between sections at x and at
, is , dO \ dO^
and this is, by what we have said, equal to cvtix.
422 CONDUCTION IN DIELECTRICS, [333.
Hence CV = ~^' ^
Again, the electromotive force at any section is — ^-, and by
Ohm's Law, dv 7 dO
-E-*;r ^
where k is the resistance of unit of length of the conductor, and
^ is the strength of the current. Eliminating Q between (2) and
(3), we find *dv _ d*v ,.
'kdt~ dx*' ( '
This is the partial differential equation which must be solved
in order to obtain the potential at any instant at any point of the
cable. It is identical with that which Fourier gives to determine
the temperature at any point of a stratum through which heat
is flowing in a direction normal to the stratum. In the case of
heat c represents the capacity of unit of volume, or what Fourier
denotes by CD, and k represents the reciprocal of the conductivity.
If the sheath is not a perfect insulator, and if ^ is the resist
ance of unit of length of the sheath to conduction through it in a
radial direction, then if ft is the specific resistance of the insulating
material, it is easy to shew that
^
The equation (2) will no longer be true, for the electricity is
expended not only in charging the wire to the extent represented
by cv, but in escaping at a rate represented by y. Hence the rate
of expenditure of electricity will be
whence, by comparison with (3), we get
dv __ d2v k
kdt~ da* k
and this is the equation of conduction of heat in a rod or ring
as given by Fourier *.
333.] If we had supposed that a body when raised to a high
potential becomes electrified throughout its substance as if elec
tricity were compressed into it, we should have arrived at^ equa
tions of this very form. It is remarkable that Ohm himself,
* Theorie de la Chaleur, Art. 105.
334-]
HYDROSTATICAL ILLUSTRATION.
423
~A<-
-A -
-
«-Do-
misled by the analogy between electricity and heat, entertained
an opinion of this kind, and was thus, by means of an erroneous
opinion, led to employ the equations of Fourier to express the
true laws of conduction of electricity through a long wire, long
before the real reason of the appropriateness of these equations had
been suspected.
Mechanical Illustration of the Properties of a Dielectric.
334.] Five tubes of equal sectional area A, B, (7, D and P are
arranged in circuit as in the figure. — ^^^
A, B} C and D are vertical and equal, f P0 * p* \
and P is horizontal.
The lower halves of A, B, C, D
are filled with mercury, their upper
halves and the horizontal tube P are
filled with water.
A tube with a stopcock Q con
nects the lower part of A and B
with that of C and D, 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
indicated by A0, BQ, (?0, D0, that
the piston is at P0, and that the
stopcock Q is shut.
Now let the piston be moved from P0 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 Al and Clt and the mercury
in B and D will sink an equal distance a, or to Bl and J)l .
The difference of pressure on the two sides of the piston will
be represented by 4 a.
This arrangement may serve to represent the state of a dielectric
acted on by an electromotive force 4 a.
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
excess of mercury in the tube A 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.
f
• ' '• •«.
"\
- c-
*
-v
-.-
-°«-
i
Q
Fig. 27.
424 CONDUCTION IN DIELECTKIC& [334-
If the piston is free to move it will move back to P0 and be
in equilibrium there. This represents the complete discharge of
the dielectric.
During the discharge there is a 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 Pl.
The level of AL and DL will remain unchanged, but that of B and
C will become the same, and will coincide with BQ and C0 .
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 \a to 2 a 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 P2, and the discharge
will be apparently only half of the charge.
The level of the mercury in A and B will be \a above its
original level, and the level in the tubes C and D will be \a
below its original level. This is indicated by the levels A^ J52,
C,, D,.
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 I?0 and
C0. 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 left free to move, it will again come to equilibrium at a point P3 ,
half way between P2 and PQ. This corresponds to the residual
charge which is observed when a charged dielectric is first dis
charged and then left to itself. It gradually recovers part of its
charge, and if this is again discharged 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
334-] HYDROSTATICAL ILLUSTRATION. 425
discharges, which are \, J, &c. of the original charge, form a series
whose sum is equal 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 pheno
menon called ' electric absorption.'
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 A 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 P, and electromotive force by
the resultant force on the piston.
CHAPTEE XL
THE MEASUREMENT OF ELECTRIC RESISTANCE.
335.] IN the present state of electrical science, the determination
of the electric resistance of a conductor may be considered as the
cardinal operation in electricity, in the same sense that the deter
mination 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 preserve 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.
336.] The unit -of resistance may be an entirely arbitrary one,
as in the case of Jacobi's Etalon, which was a certain copper
wire of 22.4932 grammes weight, 7.61975 metres length, and 0.667
339-] STANDARDS OF RESISTANCE. 427
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.
337.] Finally, the unit may be defined with reference to the
electrostatic or the electromagnetic system of units. In practice
the electromagnetic 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, as we shall she-w at the proper
place, a resistance is a quantity homogeneous with a velocity, and
may therefore be expressed as a velocity. See Art. 628.
338.] 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.
339.] 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 arc, 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
value of the unit is originally chosen so as to represent as nearly
428
MEASUREMENT OF RESISTANCE.
[340.
as possible a quantity derived from the electromagnetic absolute
system.
340.] 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 likely that
any difficulty will be found in obtaining copies of them if the
original standards should be 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. 28.)
Fig. 28.
341-] RESISTANCE COILS. 429
On the Forms of Resistance Coils.
341.] 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 mercury
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. For 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. 347).
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
430
MEASUREMENT OF RESISTANCE.
[342.
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.
342.] 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 Phillips *. 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.
343.] 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
in a box in series.
Fig. 29.
The electrodes consist of stout brass plates, so arranged on the
outside of the box that by inserting a brass plug or wedge between
* Phil. Mag., July, 1870.
344-]
RESISTANCE BOXES.
431
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
64 + 32 + 8 + 2+1 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 1 + l'
and 2, then that of 1 : + I' + 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.
344.] 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, &c.,
and the plugs are inserted at 2 and 8, the conductivity of the
box is J + 1- = f , and the resistance of the box is therefore J-
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. 276.
Fig. 30.
324
MEASUREMENT OF RESISTANCE.
[345-
On the Comparison of Resistances.
345.1 If E is the electromotive force of a battery, and R 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 I when the battery connexions are closed, and I13 72
when additional resistances rlt r2 are introduced into the circuit,
then, by Ohm's Law,
Eliminating E, the electromotive force of the battery, and R
the resistance of the battery and its connexions, we get Ohm's
formula rt _ (I— /t) /2
==
This method requires a measurement of the ratios of I, 1^ and 72,
and this implies a galvanometer graduated for absolute mea
surements.
If the resistances ^ and r2 are equal, then 7j and I2 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 E 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.
346.] The comparison of resistances can be made with extreme
accuracy by either of two methods, in which the result is in-
dependent of variations of R and E.
346.] COMPARISON OF RESISTANCES. 433
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 n they have no resultant effect on the
galvanometer needle.
Let /i , J2 be the currents through the two coils of the galvan
ometer, then the deflexion of the needle may be written
8 = ml-^—nl^.
Now let the battery current 7 be divided between the coils of
the galvanometer, and let resistances A 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 (3 respect
ively, and let the resistance of the battery and its connexions
between C and D be r} and its electromotive force U.
Then we find, by Ohm's Law, for the difference of potentials
between C and D,
and since
4-J^S, 4 = *^. I=* D
The deflexion of the galvanometer needle is therefore
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 galvano
meter, and the accuracy of the observer.
Suppose that B has been adjusted so that there is no apparent
deflexion.
Now let another conductor A' be substituted for A, and let
A' be adjusted till there is no apparent deflexion. Then evidently
to a first approximation A— A.
To ascertain the degree of accuracy of this estimate, let the
altered quantities in the second observation be accented, then
VOL. i. r f
434 MEASUREMENT OF RESISTANCE. [346.
Hence n (A' -A) = ~ &-|^'.
If b and 5', instead of being both apparently zero, had been only
observed to be equal,, then, unless we also could assert that E = ff,
the right-hand side of the equation might not be zero. In fact,
the method would be a mere modification of that already described.
The merit of the method consists in the fact that the thing
observed is 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 8 and 8' are quantities too small to be
observed, and therefore any change in the value of E will not affect
the accuracy of the result.
The actual degree of accuracy of this method might be ascer
tained by making a number of observations in each of which A'
is separately adjusted, and comparing the result of each observation
with the mean of the whole series.
But by putting A' out of adjustment by a known quantity, as,
for instance, by inserting at A or at B an additional resistance
equal to a hundredth part of A or of J3, 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 A and B, and if the
positions of A and B are exchanged, then the second equation
becomes
* This investigation is taken from Weber's treatise on Galvanometry. Gottingen
Transactions, x. p. 65.
346.] DIFFERENTIAL GALVANOMETER. 435
= ~b'i
D jy
whence (m + ri) (B—A) = -=- S — ^- S'.
If m and n, A and H, a and /3 are approximately equal, then
Here 8 — 6' 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 n will vary as the length of the wire
and a as the square of the length. Hence there will be a minimum
, „
value of » - ^ — = - '- when
If we suppose r, the battery resistance, small compared with A,
this gives a=J^;
or, the resistance of each coil of the galvanometer should le one-third
of the resistance to be measured.
We then find 8 A"-
*-A = o !&(*-*)•
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
mE 3nfl.,> , 1 A
A = —. - = — 7- if r = 0 and a = -A.
4 A 3
B-A _ 2 S-57
~I~ ~3~A~
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 n may change
when the position of the needle is changed, and therefore it is
necessary to determine this ratio by proper methods during each
F f 2
436 MEASUREMENT OF RESISTANCE. [S47«
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 errors 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.
347.] Wheatstone's Bridge consists essentially of six conductors
connecting four points. An electromotive
force E is made to act between two of the
points by means of a voltaic battery in
troduced between B and C. The current
between the other two points 0 and A is
measured by a galvanometer.
Under certain circumstances this current
becomes zero. The conductors BC 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 0 must be equal
to that at A. Now when we know the potentials at B and C we
can determine those at 0 and A by the rule given in Art. 275,
provided there is no current in OA,
/3 + y
whence the condition is 1$ = Cyj
where b, c, /3, 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 OA when this
condition is not fulfilled exactly.
Let A, B, C and 0 be the four points. Let the currents along
£C, CA and AB be x, y and z, and the resistances of these
348-]
WHEATSTONES BRIDGE.
437
conductors a, b and c. Let the currents along OA, OB and OC be
£, 17, £ and the resistances a, (B and y. Let an electromotive force
E act along BC. Required the current f along OJ.
Let the potentials at the points A, B, C and 0 be denoted
by the symbols A, B, C and 0. The equations of conduction are
ax^B-C+E, a£=0-A,
ly = CW, £17 = 0-B,
with the equations of continuity
-z= 0,
z—x = 0,
- = 0.
By considering the system as made up of three circuits 0£C,
OCA and OAJB, in which the currents are #, y, z respectively, and
applying Kirchhoff's rule to each cycle, we eliminate the values
of the potentials 0, A, B, C, and the currents £, 17, £ and obtain the
following equations for a?, y and #,
=0,
-y
—yx
Hence, if we put
—ay
—a
-a
we find
and
X7
_
•777
348.] The value of D may be expressed in the symmetrical form,
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
If the electromotive force E were made to act along OA, the
resistance of OA being still a, and if the galvanometer were placed
438 MEASUREMENT OF RESISTANCE. [349.
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 electro
motive force E acting along OA would be equal to the current
in OA due to the electromotive force E acting in BC.
But if we simply disconnect the battery and the galvanometer,
and without altering their respective resistances connect the battery
to 0 and A and the galvanometer to B and C, then in the value of
D we must exchange the values of B and G. If I/ be the value
of D after this exchange, we find
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 /3, y
with the junction of the two conductors of greatest resistance b, c,
or, in other words, we shall suppose that if the quantities #, c, y, /3
are arranged in order of magnitude, b and c stand together, and
y and /3 stand together. Hence the quantities #— ft and c—y are
of the same sign, so that their product is positive, and therefore
D— B' 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.
349.] We shall suppose that we have to determine the ratio of
the resistances of the conductors AB and AC, and that this is to be
done by finding a point 0 on the conductor HOC, such that when
the points A and 0 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 BOG 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.
349-] WHEATSTONE'S BEIDGE. 439
Instead of the whole conductor being a uniform wire, we may
make the part near 0 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 commenced.
Let the whole resistance of SAC be R.
Let c — mR and b = (1 — m) E.
Let the whole resistance of BOC be 8.
Let /3 = nS and y = (1 -n) 8.
The value of n is read off directly, and that of m is deduced from
it when there is no sensible deviation of the galvanometer.
Let the resistance of the battery and its connexions be J9, and
that of the galvanometer and its connexions G.
We find as before
—lmn) BRS,
and if £ is the current in the galvanometer wire
t ERS . .
£ =-2j- (*-*)•
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 (n — m). This may be done by properly choosing the dimensions
of the galvanometer and the standard resistance wire.
It will be shewn, when we come to Galvanometry, Art. 716,
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 8 is the deviation,
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 5 is a maximum, the part
involving G must be made equal to the rest of the expression.
If we also put m = n, as is the case if we have made a correct
observation, we find the best value of G to be
440
MEASUREMENT OF RESISTANCE.
[350-
This result is easily obtained by considering- the resistance from
A to 0 through" the system, remembering that £C, 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
arrangement of the battery is when
Finally, we shall determine the value of 8 such that a given
change in the value of n may produce the greatest galvanometer
deflexion. By differentiating the expression for f we find
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
and if n = i G= \R.
On the Use of Wheatstone's Bridge.
350.] We have already explained the general theory of Wheat
stone's Bridge, we shall now consider some of its applications.
Fig. 33.
The comparison which can be effected with the greatest exact
ness is that of two equal resistances.
35o.] USE OF WHEATSTONE'S BRIDGE. 441
Let us suppose that ft is a standard resistance coil, and that
we wish to adjust y to be equal in resistance to (3. '
Two other coils, b and c, are prepared which are equal or nearly
equal to each other, and the four coils are placed with their electrodes
in mercury cups so that the current of the battery is divided
between two branches, one consisting of (3 and y and the other
of b and c. The coils b and c 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 ft and y with
a point Q of the wire PR, 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 ft 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 ft 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
resistance equal to that of ft.
If the resistances of the coils b and c, each including part of the
wire PR up to its zero reading, are equal to that of b and c
divisions of the wire respectively, then, if x is the scale reading
of Q in the first case, and y that in the second,
c-\-x _ ft c+y _ y
b—x ~ y ' b— y ~~ ft'
y2
whence - = l +
Since b — y is nearly equal to c -f x, and both are great with
respect to as or y, we may write this
and
When y is adjusted as well as we can, we substitute for I and c
other coils of (say) ten times greater resistance.
The remaining difference between ft and y will now produce
a ten times greater difference in the position of Q than with the
442
MEASUREMENT OF RESISTANCE.
[35i.
original coils I and <?, 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
On the Measurement of Small Resistances.
351.] When a short and thick conductor is introduced into a
circuit its resistance is so small compared with the resistance
occasioned by unavoidable faults in the connexions, such as want
of contact or imperfect soldering, that no correct value of the
resistance can be deduced from experi
ments made in the way described above.
The object of such experiments is
generally to determine the specific re
sistance of the substance, and it is re
sorted to in cases when the substance
cannot be obtained in the form of a
long thin wire, or when the resistance
to transverse as well as to longitudinal
conduction has to be measured.
Sir W. Thomson* has described a method applicable to such
cases, which we may take as an example of a system of nine
conductors.
* Proc. R. S., June 6, 1861.
35i.] THOMSON'S METHOD FOR SMALL RESISTANCES. 443
The most important part of the method consists in measuring
the resistance, not of the whole length of the conductor, but of
the part between two marks on the conductor at some little dis
tance from its ends.
The resistance which we wish to measure is that experienced
by a current whose intensity is uniform in any section of the
conductor, and which flows in a direction parallel to its axis.
Now close to the extremities, when the current is introduced
by means of electrodes, either soldered, amalgamated, or simply
pressed to the ends of the conductor, there is generally a want of
uniformity in the distribution of the current in the conductor.
At a short distance from the extremities the current becomes
Fig. 35.
sensibly uniform. The student may examine for himself the
investigation and the diagrams of Art. 193, where a current is
introduced into a strip of metal with parallel sides through one
of the sides, but soon becomes itself parallel to the sides.
The resistances of the conductors between certain marks 8t 8'
and T, T' are to be compared.
The conductors are placed in series, and with connexions as
perfectly conducting as possible, in a battery circuit of small resist
ance. A wire S7T is made to touch the conductors at S and T,
and S'VT' is another wire touching them at S' and T'.
The galvanometer wire connects the points Fand V of these wires.
The wires 8VT and S'V ' T' are of resistance so great that the
resistance due to imperfect connexion at /S, T, S' or T' may be
neglected in comparison with the resistance of the wire, and Vt V
are taken so that the resistances in the branches of either wire
leading to the two conductors are nearly in the ratio of the resist
ances of the two conductors.
Calling 7/and F the resistances of the conductors SS' and T'T.
A and C those of the branches /STand FT.
444
MEASUREMENT OF RESISTANCE.
[352.
Calling P and R those of the branches 8' V and V'T.
„ Q that of the connecting piece S'T'.
„ IB that of the battery and its connexions.
„ G that of the galvanometer and its connexions.
The symmetry of the system may be understood from the
skeleton diagram. Fig. 34.
The condition that B the battery and G the galvanometer may
be conjugate conductors is, in this case,
— — (*L A Q
C" A+\~C~ A> P+Q + fi ~ °'
Now the resistance of the connector Q is as small as we can
make it. If it were zero this equation would be reduced to
L-iL
C ~ A '
and the ratio of the resistances of the conductors to be compared
would be that of C to A, as in Wheatstone's Bridge in the ordinary
form.
In the present case the value of Q is small compared with P
or with R, so that if we assume the points V, V so that the ratio
of R to C is nearly equal to that of P to A, the last term of the
equation will vanish, and we shall have
FiHiiCiA.
The success of this method depends in some degree on the per
fection of the contact between the wires and the tested conductors
at S, S'9 T' and T. In the following method, employed by Messrs.
Matthiessen and Hockin*, this condition is dispensed with.
Fig. 36.
352.] The conductors to be tested are arranged in the manner
* Laboratory. Matthiessen and Hockin on Alloys.
352.] MATTHIESSEN AND HOCKI^S METHOD. 445
already described, with the connexions as well made as possible,
and it is required to compare the resistance between the marks SS'
on the first conductor with the resistance between the marks T'T<ji\
the second.
Two conducting points or sharp edges are fixed in a piece of
insulating material so that the distance between them can be
accurately measured. This apparatus is laid on the conductor to
be tested, and the points of contact with the conductor are then
at a known distance SS'. Each of these contact pieces is connected
with a mercury cup, into which one electrode of the galvanometer
may be plunged.
The rest of the apparatus is arranged, as in Wheatstone's Bridge,
with resistance coils or boxes A and C, and a wire PR with a
sliding contact piece Q, to which the other electrode of the galva
nometer is connected.
Now let the galvanometer be connected to S and Q, and let
A1 and C^ be so arranged, and the position of Q so determined, that
there is no current in the galvanometer wire.
Then we know that XS A
where XS, PQ, &c. stand for the resistances in these conductors.
From this we get
XS_
XT'
Now let the electrode of the galvanometer be connected to Sf,
and let resistance be transferred from C to A (by carrying resistance
coils from one side to the other) till electric equilibrium of the
galvanometer wire can be obtained by placing Q at some point
of the wire, say Q2. Let the values of C and A be now C2 and A2)
and let A2+C2 + PR = A^ + C^ + PR = R.
Then we have, as before,
XS'
XY' R
Whence
In the same way, placing the apparatus on the second conductor
at TT' and again transferring resistance, we get, when the electrode
isinr, XT> _
XY ' R
446 MEASUREMENT OF RESISTANCE. [353'
and when it is in T,
XT
TUTU
Whence
XY' R
T'T ^4 —
-=-= = —^
— - ^
A-! R
We can now deduce the ratio of the resistances SS' and T'T, for
T'T" A±-A^q,Qt
When great accuracy is not required we may dispense with the
resistance coils A and C, and we then find
88' Q.Q,
2"2»~ 6364'
The readings of the position of Q on a wire of a metre in length
cannot be depended on to less than a tenth of a millimetre, and the
resistance of the wire may vary considerably in different parts
owing to inequality of temperature, friction, &c. Hence, when
great accuracy is required, coils of considerable resistance are intro
duced at A and C, and the ratios of the resistances of these coils
can be determined more accurately than the ratio of the resistances
of the parts into which the wire is divided at Q.
It will be observed that in this method the accuracy of the
determination depends in no degree on the perfection of the con
tacts at S, y or T, T'.
This method may be called the differential method of using
Wheatstone's Bridge, since it depends on the comparison of ob
servations separately made.
An essential condition of accuracy in this method is that the
resistance of the connexions should continue the same during the
course of the four observations required to complete the deter
mination. Hence the series of observations ought always to be
repeated in order to detect any change in the resistances.
On the Comparison of Great Resistances.
353.] When the resistances to be measured are very great, the
comparison of the potentials at different points of the system may
be made by means of a delicate electrometer, such as the Quadrant
Electrometer described in Art. 219.
If the conductors whose resistances are to be measured are placed
in series, and the same current passed through them by means of a
battery of great electromotive force, the difference of the potentials
355-] GREAT RESISTANCES. 447
at the extremities of each conductor will be proportional to the
resistance of that conductor. Hence, by connecting the electrodes
of the electrometer with the extremities, first of one conductor
and then of the other, the ratio of their resistances may be de
termined.
This is the most direct method of determining resistances. It
involves the use of an electrometer whose readings may be depended
on, and we must also have some guarantee that the current remains
constant during the experiment.
Four conductors of great resistance may also be arranged as in
Wheatstone's Bridge, and the bridge itself may consist of the
electrodes of an electrometer instead of those of a galvanometer.
The advantage of this method is that no permanent current is
required to produce the deviation of the electrometer, whereas the
galvanometer cannot be deflected unless a current passes through
the wire.
354.] When the resistance of a conductor is so great that the
current which can be sent through it by any available electromotive
force is too small to be directly measured by a galvanometer, a
condenser may be used in order to accumulate the electricity for
a certain time, and then, by discharging the condenser through a
galvanometer, the quantity accumulated may be estimated. This
is Messrs. Bright and Clark's method of testing the joints of
submarine cables.
355.] But the simplest method of measuring the resistance of
such a conductor is to charge a condenser of great capacity and to
connect its two surfaces with the electrodes of an electrometer
and also with the extremities of the conductor. If E is the dif
ference of potentials as shewn by the electrometer, S the capacity
of the condenser, and Q the charge on either surface, E the resist
ance of the conductor and x the current in it, then, by the theory
of condensers, Q — £23.
By Ohm's Law, E = Ex,
and by the definition of a current,
* — *«.
dt
Hence -Q=ES^f
t
and Q = Q0e~**9
where QQ is the charge at first when 1 = 0.
448
MEASUREMENT OF RESISTANCE.
[356.
Similarly E = EQ e Rs'
where EQ is the original reading of the electrometer, and
same after a time t. From this we find
t
the
which gives R in absolute measure. In this expression a knowledge
of the value of the unit of the electrometer scale is not required.
If Sj the capacity of the condenser, is given in electrostatic
measure as a certain number of metres, then R is also given in
electrostatic measure as the reciprocal of a velocity.
If S is given in electromagnetic measure its dimensions are
y , and R is a velocity.
Since the condenser itself is not a perfect insulator it is necessary
to make two experiments. In the first we determine the resistance
of the condenser itself, RQ, and in the second, that of the condenser
when the conductor is made to connect its surfaces. Let this be R'.
Then the resistance, R, of the conductor is given by the equation
JL. JL l
R R/ RQ
This method has been employed by MM. Siemens.
Thomson's * Method for the Determination of the Resistance of
the Galvanometer.
356.] An arrangement similar to Wheatstone's Bridge has been
Gtllvanometer
Fig. 37.
employed with advantage by Sir W. Thomson in determining the
* Proc. R. 8., Jan. 19, 1871.
357-] MANGE'S METHOD. 449
resistance of the galvanometer when in actual use. It was sug
gested to Sir W. Thomson by Mance's Method. See Art. 357.
Let the battery be placed, as before, between B and C in the
figure of Article 347, but let the galvanometer be placed in CA
instead of in OA. If &j3—cy is zero, then the conductor OA is
conjugate to JBC, and, as there is no current produced in OA by the
battery in BC, the strength of the current in any other conductor
is independent of the resistance in OA. Hence, if the galvano
meter is placed in CA its deflexion will remain the same whether
the resistance of OA is small or great. We therefore observe
whether the deflexion of the galvanometer remains the same when
0 and A are joined by a conductor of small resistance, as when
this connexion is broken, and if, by properly adjusting the re
sistances of the conductors, we obtain this result, we know that
the resistance of the galvanometer is
where c, y, and /3 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
can 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.
357.] 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
* Proc, R. S., Jan. 19, 1871.
VOL. I. G g
450 MEASUREMENT OF RESISTANCE. [357.
the strength of the current through it is changed. 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 Mance's method, which is free from this objection, the battery
is placed in BC and the galvanometer in CA. The connexion
between 0 and B is then alternately made and broken.
Now the deflexion of the galvanometer needle will remain un
altered, however the resistance in OB be changed, provided that
OB and AC are conjugate. This may be regarded as a particular
case of the result proved in Art, 347, or may be seen directly on
the elimination of z and ft from the equations of that article, viz.
we then have
If y is independent of a?, and therefore of ft, we must have
a a = cy. The resistance of the battery is thus obtained in terms
of c, y, a.
When the condition a a = cy is fulfilled, the current through
the galvanometer is then
Ea Ey
-> or
To test the sensibility of the method let us suppose that the
condition cy = a a is nearly, but not accurately, fulfilled, and that
Fig. 38.
y0 is the current through the galvanometer when 0 and B are
connected by a conductor of no sensible resistance, and yl the
current when 0 and B are completely disconnected.
To find these values we must make ft equal to 0 and to oo in the
general formula for y^ and compare the results.
357-] COMPARISON OF ELECTROMOTIVE FORCES. 451
The general value for y is
cy + py + ya + afi ^
where D denotes the same expression as in Art. 348. Making use
of the values of y given above we can then easily shew that the
expressions for y0 and yl are approximately
y , c(cy-gq) y2
and y —
y(y-fa) E
From these values we find
cy—aa
y y (c+a)(a+y)
The resistance, c, of the conductor AB should be equal to a,
that of the battery; a and y should be equal and as small as
possible; and b should be equal to a-fy.
Since a galvanometer is most sensitive when its deflexion is
small, we should bring the needle nearly to zero by means of fixed
magnets before making contact between 0 and B.
In this method of measuring the resistance of the battery, the
current in the galvanometer is not in any way interfered with
during the operation, so that we may ascertain the resistance of
the battery for any given strength of current in the galvanometer
so as to determine how the strength of the current affects the
resistance *.
If y is the current in the galvanometer, the actual current
through the battery is a>0 with the key down and ^ with the
key up, where
"^
y y(ct + <?) > a-J-
the resistance of the battery is
cy
a = — -,
and the electromotive force of the battery is
* [In the Philosophical Magazine for 1857, vol. i. pp. 515-525, Mr. Oliver Lodge
has pointed out as a defect in Mance's method that as the electromotive force of the
battery depends upon the current passing through the battery, the deflexion of the
galvanometer needle cannot be the same in the two cases when the key is down or up,
if the equation a a = cy is true. Mr. Lodge describes a modification of Mance's
method which he has employed with success.]
Gg 2
452
MEASUREMENT OF RESISTANCE.
[353.
The method of Art. 356 for finding the resistance of the galva
nometer differs from this only in making and breaking contact
between 0 and A instead of between 0 and J9, and by exchanging
a and /3 we obtain for this case
On the Comparison of Electromotive Forces.
358.] 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 E of the battery be greater than that
of either of the electromotors to be compared, then, if a sufficient
E
Fig. 39.
resistance, P19 be interposed between the points Alt Bl of the
primary circuit EB1A1E, the electromotive force from Bl to AL
may be made equal to that of the electromotor U1. If the elec
trodes of this electromotor are now connected with the points
Alt Bl no current will flow through the electromotor. By placing
a galvanometer G1 in the circuit of the electromotor Elt and
adjusting the resistance between A1 and Blt till the galvanometer
Gl indicates no current, we obtain the equation
where E^ is the resistance between Al and B^ and C is the strength
of the current in the primary circuit.
In the same way, by taking a second electromotor E2 and placing
its electrodes at A2 and j52, so that no current is indicated by the
galvanometer G2)
358.]
COMPARISON OF ELECTROMOTIVE FORCES.
453
where R2 is the resistance between A2 and B2. If the observations
of the galvanometers G± and G2 are simultaneous, the value of C,
the current in the primary circuit, is the same in both equations,
and we find
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 electrometer,
or electromagnetically by means of an absolute galvanometer.
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 :
Daniell I. Amalgamated Zinc HSO4 + 4 aq.
II. „ HS04 + 12 aq.
III. „ HS04+12aq.
Bunsen I. „ „ „
-*••*•• » » ?>
Grove „ HS04+ 4 aq.
Concentrated
solution of
CuS04
CuS04
CuN06
HNO6
sp. g. 1. 38
HN06
Copper
Copper
Copper
Carbon
Carbon
Volts.
= 1.079
= 0.978
= 1.00
= 1.964
= 1.888
Platinum = 1.956
A Volt is an electromotive force equal to 100,000,000 units of the centimetre-gramme-
second system.
CHAPTEE XII.
ON THE ELECTRIC RESISTANCE OF SUBSTANCES.
359.] THERE 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
called 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
360.] RESISTANCE. 455
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 Electric Resistance of Metals.
360.] 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, Vogt, 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 100CC being to that
at 0CC in the ratio of 1.414 to 1, or of 100 to 70.7. For pure iron
the ratio is 1.645, and for pure thallium 1.458.
The resistance of metals hns 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 1000CC, is
where T is the absolute temperature reckoned from — 273CC, and
a, /3, y are constants. Thus, for
Platinum ...... r = 0.039369 T* + 0.00216407 T-0.241 3,
Copper ......... r = 0.026577 T? + 0.0031443 T-0. 22751,
Iron ............ r = 0.072515 T* :4 0.0038133 T- 1.23971.
* Proc. R. S., April 27, 1871.
456 RESISTANCE.
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, it 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.
361.] The electric resistance of some metals changes when the
metal is annealed ; and until a wire has been tested by being
repeatedly raised to a high temperature without permanently
altering its resistance, it cannot be relied on as a measure of
resistance. Some wires alter in resistance in course of time without
having been exposed to changes of temperature. Hence it is
important to ascertain the specific resistance of mercury, a metal
which being fluid has always the same molecular structure, and
which can be easily purified by distillation and treatment with
nitric acid. Great care has been bestowed in determining the
resistance of this metal by W. and C. F. Siemens, who introduced
it as a standard. Their researches have been supplemented by
those of Matthiessen and Hockin.
The specific resistance of mercury was deduced from the observed
resistance of a tube of length I containing a weight w of mercury,
in the following manner.
No glass tube is of exactly equal bore throughout, but if a small
quantity of mercury is introduced into the tube and occupies a
length A of the tube, the middle point of which is distant x from
one end of the tube, then the area s of the section near this point
Q
will be s — -, where C is some constant.
A
362.]
OF METALS.
457
The weight of mercury which fills the whole tube is
w
— p fsdx = pCl, (-)
where n is the number of points, at equal distances along the
tube, where A has been measured, and p is the mass of unit of
volume.
The resistance of the whole tube is
T~i 1*7 '
R = I - dx = -^
s C
— t
n
where r is the specific resistance per unit of volume.
1 72
Hence wR = rp 2 (A) 2 (-} -= ,
v ' vx/ nz
wR n2
and
r =
gives the specific resistance of unit of volume.
To find the resistance of unit of length and unit of mass we must
multiply this by the density.
It appears from the experiments 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.
362.] In the following table R is the resistance in Ohms of a
column one metre long and one gramme weight at 0°C, and r is
the resistance in centimetres per second of a cube of one centi
metre, according to the experiments of Matthiessen *.
1
Silver
Percentage
increment of
Specific resistance for
gravity R r 1°C at 20°C.
10.50 hard drawn 0.1689 1609 0.377
8.95 hard drawn 0.1469 1642 0.388
19.27 hard drawn 0.4150 2154 0.365
11.391 pressed 2.257 19847 0.387
13.595 liquid 13.071 96146 0.072
15.218 hard or annealed 1.668 10988 0.065
Crystalline form 6 x 1 013 1.00
Gold
Lead
Mercury
Gold 2, Silver 1 . .
Selenium at 1 00°C
Phil May., May, 1865.
458 RESISTANCE. [363.
On the Electric Resistance of Electrolytes.
363.] 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.
364.] 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 R: and R.2, the siphons were next filled with mercury, and
their resistances when filled with mercury were found to be li^
and R.2'.
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 It —R
p = ]pn7p *
To deduce from the values of p the resistance of a centimetre in
length having a section of a square centimetre, we must multiply
them by the value of r for mercury at 0°C. See Art. 361.
* Berlin Monatslericht, July, 1868.
365.] OF ELECTEOLYTES. 459
The results given by Paalzow are as follow : —
Mix hires of Sulphuric Acid and Water.
m Resistance compared
with mercury.
H2SO4 15°C 96950
H2SO4+ 14H20 19CC 14157
H2SO4 + 13H20 22°C 13310
H2SO4+499H2O 22CC 184773
Sulphate of Zinc and Water.
ZnSO4-H 23H20 23CC 194400
ZnSO4 + 24H20 23°C 191000
ZnSO4+105H2O 23CC 354000
Sulphate of Copper and Water.
CuSO4+ 45H20 22°C 202410
CuSO4+105H2O 22CC 339341
Sulphate of Magnesium and Water.
MgSO4 + 34H20 22°C 199180
MgSO4+107H2O 22°C 324600
Hydrochloric Acid and Water.
HC1 + 15H20 23CC 13626
HC1 +500H20 23°C 86679
365.] 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 -fT of that of a Grove's cell, and
by means of a thermoelectric copper-iron pair they reduced the
electromotive force to -^/ou •$ 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.
460
RESISTANCE.
[366.
Resistance of Mixtures of Sulphuric Acid and Water at 22°C in terms
of Mercury at 0°C. MM. Kohlrausch and Nippoldt.
Specific gravity
at 18°5
Percentage
of H2SO,
Resistance
at 22°C
(Hg-1)
Percentage
increment of
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
13133
1.317
1.2631
35.2
13132
1.259
1.3163
41.5
14286
1.410
1.3547
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.
366.] 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 cylin
drical coating of the insulator of great 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
charge 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
366.] OF ELECTROLYTES. 461
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.
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 Bitter's
secondary pile, Art. 271.
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 con
necting 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-
462 RESISTANCE. [367.
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
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.
367.] 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 0°C as at 24CC. Messrs. Bright and Clark have found that the
following formula gives results agreeing with their experiments.
If T is the resistance of gutta-percha at temperature T centigrade,
then the resistance at temperature T+ 1 will be
R = rx 0.8878',
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
application 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.
Red Sea 267x 1012 to .362xl012
Malta-Alexandria 1.23 x 1012
Persian Gulf... 1.80 x 1012
Second Atlantic 3.42 x 1012
Hooper's Persian Gulf Core. ..74. 7 x 1012
Gutta-percha at 24CC 3.53 x 1012
368.] The following table, calculated from the experiments of
* Jenkin's Cantor Lectures.
370.] OF DIELECTRICS. 463
M. Buff, described in Art. 271, shews the resistance of a cubic
metre of glass in Ohms at different temperatures.
Temperature. Kesistance.
200CC 227000
250° 13900
300° 1480
350° 1035
400° 735
369.] Mr. C. F. Varley * has recently investigated the conditions
of the current through rarefied gases, and finds that the electro
motive force E is equal to a constant EQ together with a part
depending on the current according to Ohm's Law, thus
For instance, the electromotive force required to cause the
current to begin in a certain tube was that of 323 DanielPs 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 DanielFs 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 results,
described in Art. 57, 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.
370.] MM. Wiedemann and Riihlmann have recently f investi-
* Proc. R. £, Jan. 12, 1871.
t Bericlite der Konigl. Sachs. Gesellschaft, Oct. 20, 1871.
464 RESISTANCE OF DIELECTEICS.
gated the passage of electricity through gases. The electric current
was produced by Holtz's machine, and the discharge took place
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 it 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. 48) 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 stated in Art. 57, 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.
* Exp. lies., 1501.
Vol.. I
PIG. I
Art H8.
Lines of Force and hqiiipolen licit Surfaces.
A = ZO .
AP
FIG. IT.
Art 119
Lines of Force and Eqn {potential Surfaces.
A = 2G B~-3 P, J>oint <
Q, Spherical surface ofZ<f,rojvoteriticU'.
M, Ii>int of Ma.rimusnsf'orce atony fJie a
The dotted Line is the Low o/Jtorce T = O.I f/tt^f _________
AP = 2 AB
For the D&iegales of th& darcndoir Press.
FIG Hi
Art 120
Lines of Force and .Kfjuipvlc.nlial Surfaces.
A = JO .
For ike Delegates of the Ctcuvrulon, Press.
Fro. iv.
Art . 121
infft of Force and ] \cjni potential Surfaces.
FjG . V .
Art 143
N
of Force <Z7wL £pucpo6en£ux£ Surfaces iri a,
section/ of a< xjytt&rical Su/rtzce m which, the ^uperticiat density
M' a, harmonic of the farst de
For the Deleqates of the- Clcur&ncion' Press.
FIG VI
Art J43
Spherical Harmonic of the tkirvL order
7?, = J O • = J ..
For the Delegates of the Clarendon- Press.
FIG VII
Art 143
Sp/terioal ffarrn<7n,ic <?f the third,
For th-e Delegates of I'ha Clo.rsndor>. Press.
of tie, fourth
4 o - z
ror the Delegates of tlie, Clare-ndon, Press.
PIG IX
Art. 143
>Spfie7~tca/ JJarTnvruc of the foicrfk order
For the Delegates of the. Clarendon, Press.
FIG x.
Art 192
Ellipses and
For the Delegates of Ihe Clarendon Press.
Fro. XI.
Art 193 .
Lvnas of Force rveour the edge of ou Plate
Delegates of lh& Clarendon Press.
Fie. XE
Art. 202
Jjcnes of Rmx between
For the Delegates of lh& CUir&ndan
FIG. m
Art. 203
of Force ?tea-r a ffraliny.
For lh.e Deleaves of the- Clarendon> Przss.
/•z
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