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TIIK CilFT OF
I*KOP. Al^BXANDKU ZlWKT
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A TREATISE
ON
ELECTRICITY AND MAGNETISM
MAXWELL
VOL. I.
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jSon^Ott
HENRY FROWDE
Oxford University Press Warkiiousk
Amen Corner, £.C.
113 Fourth Avenue
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A TREATISE
ON
ELECTRICITY AND MAGNETISM
BY
JAMES CLERK MAXWELL, M.A.
LL.D. tDnC , D.C.L., r.E.8S. LONDOH AHD IDINBUROB
HOMOBABT FELLOW OF TRINITT COLLKOB
A!fD LATB PROFESSOR OF EXPERIMEMTAL PHYSICS IN THE UKITKRSITT OF CAMBRIDQX
VOL. I
TRIKD EDITION
Oxford
AT THE CLARENDON PRESS
1892
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' '31- /*??:
Oxfotb
PRINTED AT THE CLARENDON PRESS
DY HORACE HART. PRINTER TO THB UNIVKRSITY
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PREFACE TO THE FIRST EDITION
rilHE fact that certain bodies, after being rubbed, appear
to attract other bodies, was known to the ancients. In
modem times, a great variety of other phenomena have been
observed, and have been found to be related to these pheno-
mena of attraction. They have been classed under the name
of Electric phenomena, amber, {jkeKvpov, having been the sub-
stance in which they were first described.
Other bodies, particularly the loadstone, and pieces of iron
and steel which have been subjected to certain processes, have
also been long known to exhibit phenomena of action at
a distance. These phenomena, with others related to them,
were found to differ from the electric phenomena, and have
been classed under the name of Magnetic phenomena, the
loadstone, iiiyprjs, 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 ElectromagnetisuL
In the following Treatise I propose to describe the 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
4062^7 Digitized by GoOglC
vi PREFACE
having shown how this theory may be applied to the calcula-
tion of phenomena, I shall endeavonr to place in as clear a
light as I can the relations between the mathematical form of
this theory and that of the ftindamental science of Dynamics,
in order that we may be in some degree prepared to determine
the kind of dynamical phenomena 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 ftmdamental 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 phenomena chiefly with
a view to their measurement, describing the methods of
measurement, and defining the standards on which they
depend.
In the application of mathematics to the calculation of elec-
trical 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.
The internal relations of the different branches of the
science which we have to study are more numerous and com-
plex 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 electromagnet-
ism 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.
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TO THE FIRST EDITION. vii
The connexions between the different classes of phenomena
have also been investigated, and the probability of the rigorous
exactness of the experimental laws have been greatly strength-
ened by a more extended knowledge of their relations to each
other.
Finally, some progress has been made in the reduction of
electromagnetism to a dynamical science, by shewing that no
electromagnetic phenomenon is contradictory to the suppo-
sition 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 subjects of enquiry, and ftimish-
ing us with means of investigation.
It is hardly necessaiy to enlarge upon the beneficial 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 ad-
vanced 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 tele-
graphy have also reacted on pure science by giving a com-
mercial value to accurate electrical measurements, and by
affording to electricians the use of apparatus on a scale which
greatly transcends that of any ordinary laboratory. The con-
sequences of this demand for electrical knowledge, and of
these experimental opportunities for acquiring it, have been
already very great, both in stimulating the energies of ad-
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viii PREFACE
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.
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 Transactions of learned societies ;
they do not form a connected 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 useftd
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 considerably
from that of several excellent electrical works, published, most
of them, in Germany, and it may appear that scant justice is
done to the speculations of several eminent electricians and
mathematicians. One reason of this is that before I began
the study of electricity I resolved to read no mathematics 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 phe-
nomena and that of the mathematicians, so that neither he
nor 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
* I take this opportunity of acknowledging my obligations to Sir W.
Thomson and to Profeasor Tait for many valuable suggestions made during
the printing of this work.
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TO THE FIRST EDITION. ix
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 perceived that
his method of conceiving the phenomena was also a mathe-
matical 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 ma-
thematicians.
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 coincided, so that the same phe-
nomena were accounted for, and the same laws of action de-
duced by both methods, but that Faraday's methods resembled
those in which we begin with the whole and arrive at the
parts by analysis, while the ordinary mathematical methods
were founded on the principle of beginning with the parts
and building up the whole by synthesis.
I also found that several of the most fertile methods of
Research discovered by the mathematicians could be expressed
lijiuch better in terms of ideas derived from Faraday than in
tl.eir original form.
\The whole theory, for instance^ of the potential, considered
as '^.quantity which satisfies a certain partial differential equa-
i/iOU, 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
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^
PREFACE
proper place in this treatise, and their appropriate expressions
in terms of conceptions mainly derived fix)m Faraday.
Great progress has been made in electrical science, chiefly
in Germany, by cultivators of the theory of action at a dis-
tance. The valuable electrical measurements of W. Weber are
interpreted by him according to this theory, and the electro-
magnetic speculation which was originated by Gauss, and
EEi carried on by Weber, Riemann, /. 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, 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,
additional 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 fix)m
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
faithftdly 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 electro-
magnetic phenomena, and both of which have attempted to
explain the propagation of light as an electromagnetic phe-
nomenon and have actually calculated its velocity, while at the
same time the fondamental conceptions of what actually takes
place, as well as most of the secondary conceptions of the
quantities concerned, are radically different.
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TO THE FIRST EDITION. xi
I have therefore taken the part of an advocate rather than
that of a judge, and have rather exemplified one 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 supporters, and will be expounded with
a skill worthy of its ingenuity.
I have not attempted an exhaustive account of electrical
phenomena, experiments, and apparatus. The student who
desires to read all that is known on these subjects will find
great assistance from the Traits dCElectriciU of professor A.
de la Rive, and from several German treatises, such as Wiede-
mann's Oalvaniefnius, Biess' BeibungaelektricUdt, Beer's Einlei-
tung in die Elektroatatik, &c.
I have confined myself almost entirely to the mathematical
treatment of the subject, but I would recommend the student,
after he has learned, experimentally if possible, what are the
phenomena to be observed, to read carefully Faraday's Experi-
mental Researches in Electricity. He will there find a strictly
contemporary historical account of some of the greatest elec-
trical 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 orignal memoirs on that subject, for science is always
most completely assimilated when 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 consecutively. 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 accomplish-
ment of one of my principal aims — to communicate to others
the same delight which I have found myself in reading Fara-
day's Researches.
* Lift and Letters ofFaraday^ vol. i. p. 395.
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xii PREFACE TO THE FIRST EDITION.
The description of the phenomena, and the elementary 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 calcu-
lation, 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 dis-
tance, are treated of in the last four chapters of the second
volume.
James Clebk Maxwell.
Feb, 1, 1873.
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PREFACE TO THE SECOND EDITION
TTTHEN I was asked to read the proof-sheets of the second
* * edition of the Electricity and Magnetiani 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 frx)m his premature death. The first nine chapters
were in some cases entirely rewritten, 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 foot-
notes are in square brackets.
There were two parts of the subject in the treatment of
which it was known to me that the Professor contemplated
considerable changes: viz. the mathematical theory of the
conduction of electricity in a network of wires, and the de-
termination of coeflBcients of induction in coils of wire. In
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xiv PREFACE TO THE SECOND EDITION.
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 useftd in calculating coeflBcients 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 expressing this hope as, in reading
some of the proofe, I have had the assistance of various
friends conversant with the work, among whom I may men-
tion particularly my brother Professor Charles Niven, and
Mr. J. J. Thomson, Fellow of Trinity College, Cambridge.
W. D. Niven.
Trikitt Colleob, Cambbidoe,
Oct. 1, 1881.
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PEEFACE TO THE THIRD EDITION
T UNDERTOOK the task of reading the proofs of this
"^ Edition at the request of the Delegates of the Clarendon
Press, by whom I was informed, to my great regret, that Mr.
W. D. Niven found that the pressure of his oflBcial duties
prevented him firom seeing another edition of this work
through the Press.
The readers of Maxwell's writings owe so much to the un-
tiring labour which Mr. Niven has spent upon them, that I am
sure they will regret as keenly as I do myself that anything
should have intervened to prevent this Edition from receiving
the benefit of his care.
It is now nearly twenty years since this book was written,
and during that time the sciences of Electricity and Mag-
netism have advanced with a rapidity almost unparalleled in
their previous history ; this is in no small degree due to the
views introduced into these sciences by this book: many of
its paragraphs have served as the starting-points of important
investigations. When I began to revise this Edition it was
my intention to give in foot-notes some account of the ad-
vances made since the publication of the first edition, not
only because I thought it might be of service to the students
of Electricity, but also because all recent investigations have
tended to confirm in the most remarkable way the views ad-
vanced by Maxwell. I soon found, however, that the progress
made in the science had been so great that it was impossible
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xvi PREFACE TO THE THIRD EDITION.
to cany out this intention without disfiguring the book by a
disproportionate quantity of foot-notes. I therefore decided to
throw these notes into a slightly more consecutive form and
to publish them separately. They are now almost ready for
press, and will I hope appear in a few months. This volume
of notes is the one referred to as the * Supplementary Volume/
A few foot-notes relating to isolated points which could be
dealt with briefly are given. All the matter added to this
Edition is enclosed within { } brackets.
I have endeavoured to add something in explanation of the
argument in those passages in which I have found from my
experience as a teacher that nearly all students find consider-
able difficulties ; to have added an explanation of all passages
in which I have known students find difficulties would have
required more volumes than were at my disposal.
I have attempted to verify the results which Maxwell gives
without proof; I have not in all instances succeeded in
arriving at the result given by him, and in such cases I have
indicated the difference in a foot-note.
I have reprinted from his paper on the Dynamical Theory of
the Electromagnetic Field, Maxwell's method of determining
the self-induction of a coil. The omission of this fix)m previous
editions has caused the method to be frequently attributed to
other authors.
In preparing this edition I have received the greatest pos-
sible assistance from Mr.Charles Chree, Fellow of King's College,
Cambridge. Mr. Chree has read the whole of the proof sheets,
and his suggestions have been invaluable. I have also received
help from Mr. Larmor, Fellow of St John s College, Mr.
Wilberforce, Demonstrator at the Cavendish Laboratory, and
Mr. G. T. Walker, Fellow of Trinity College.
J. J. Thomson.
Cavendish Laboratobt:
Dec, 5, 1891.
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! CONTENTS
PRELIMINARY.
ON THE MEABUBEUBNT OF (jUAKTITIBS.
Alt. Page
1. The expresBion 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
i 1 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 9
11. Meaning of the words Scalar and Vector 10
12. Division of physical vectors into two classes, Forces and Fluxes 11
13. Relation between corresponding vectors of the two classes .. 12
14. Line-integration appropriate to forces, surfiEice-integration to
fluxes 13
15. Longitudinal and rotational vectors 13
16. Line-integrals and potentiab 14
17. Hamilton's expression for the relation between a force and its
potential 16
18. Cyclic regions and geometry of position 17
19. The potential in an acyclic region is single valued 18
20. System of values of the potential in a cyclic region 19
21. Surface-integrals 20
22. Surfaces, tubes, and lines of flow 22
23. Right-handed and left-handed relations in space 26
24. Transformation of a line-integral into a surface-integral .. 27
25. Eflect of Hamilton's operation V on a vector function .. 29
26. Nature of the operation V 31
VOL. I. b
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xviii CONTENTS.
PART I.
ELECTEOSTATIOS.
CHAPTER I.
DESGBIPTIOir OF PHXNOMBKA.
Art. Pi«e
27. Electrification bj friction. Electrification is of two kinds, to
which the names of Vitreous and Resinous, or Positive and
Negative, have been given 32
28. Electrification by induction 33
29. Electrification by conduction. Conductors and insulators 34
50. In electrification by friction the quantity of the positive elec-
trification is equal to that of the negative electrification . . 35
51. To charge a vessel with a quantity of electricity equal and
opposite to that of an excited body 35
32. To discharge a conductor completely into a metallic vessel 36
33. Test of electrification by gold-leaf electroscope 37
54. Electrification, considered as a measurable quantity, may be
called Electricity 37
55. Electricity may be treated as a physical quantity 38
36. Theory of Two fluids 39
37. Theory of One fluid 41
38. Measurement of the force between electrified bodies 43
39. Relation between this force and the quantities of electricity . . 44
40. Variation of the force with the distance 45
41. 42. Definition of the electrostatic unit of electricity. — Its
dimensions 45,46
43. Proof of the law of electric force 46
44. Electric field 47
45. Electromotive force and potential 48
46. Equipotential surfaces. Example of their use in reasoning
about electricity 49
47. Lines of force 51
48. Electric tension 61
49. Electromotive force 61
60. Capacity of a conductor. Electric Accumulators 52
51. Properties of bodies. — Resistance 52
52. Specific Inductive capacity of a dielectric 54
53. * Absorption ' of electricity 56
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I ] > I
l< ■
CONTENTS. xix
Alt Page
54. ImpoBsibilitj of an absolute charge 56
66. Disruptive discharge. — Glow 67
56. Brush 60
57. Spark 60
58. Electrical phenomena of Tourmaline 61
59. Plan of the treatise, and sketch of its results 62
60. Electric polarization and displacement 64
61. The motion of electricity analogous to that of an incompressible
fluid 67
62. Peculiarities of the theory of this treatise 68
CHAPTER II.
ELBXENTAST MATHEMATICAIi THSOBY OF ELECTBICITT.
63. Definition of electricity as a mathematical quantity 71
64. Volume-density, surface-density, and line-density 72
f 65. Definition of the electrostatic unit of electricity 73
f I 66. Law of force between electrified bodies 74
I 67. Besultant force between two bodies 74
I 68. Resultant intensity at a point 75
} 69. Line-integral of electric intensity ; electromotive force .. .. 76
? 70. Electric potential 77
(, 71. Resultant intensity in terms of the potential 78
I 72. The potential of all points of a conductor is the same .. 78
I 73. Potential due to an electrified system 80
jl 74 a. Proofofthe law of the inverse square. Cavendish's experiments 80
i; 74 &. Cavendish's experiments repeated in a modified form .. 81
jj 74 c, cf, «. Theory of the experiments 83-85
75. Surface-integral of electric induction 87
76. Induction through a closed surface due to a single centre of
force 87
77. Poisson's extension of Laplace's equation 89
78 a, h, e. Conditions to be fulfilled at an electrified surface .. 90-92
79. Resultant force on an electrified surface 93
80. The electrification of a conductor is entirely on the surface .. 95
81. A distribution of electricity on lines or points is physically
82. Lines of electric induction 97
83 a. Specific inductive capacity 99
83 6. Apparent distribution of electricity 99
Appendix to Chap, n 101
b a
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CONTENTS.
CHAPTER III.
ON BLECTBICAL WORK AND ENEBOT IN A SYSTEM OP C0NDUCT0K8.
Art. P»««
84. On the Buperposition of electrified systems. Expression for the
energy of a system of conductors 103
85 a. Change of the energy in passing from one state to another .. 104
85 h. Relations between the potentials and the charges 105
86. Theorems of reciprocity 105
87. Theory of a system of conductors. Coejfficients of potential. Ca-
pacity. Coefficients of induction 107
88. Dimensions of the coefficients 110
89 a. Necessary relations among the coefficients of potential . . . . Ill
89 6. Relations derived from physical considerations Ill
89 c. Relations among coefficients of capacity and induction .. .. 112
89 d. Approximation to capacity of one conductor 113
89 e. The coefficients of potential changed by a second conductor .. 114
90 a. Approximate determination of the coefficients of capacity and
induction of two conductors 115
90 &. Similar determination for two condensers .. .. .. .. 115
91. Eelative magnitudes of coefficients of potential 117
92. And of induction 118
93 a. Mechanical force on a conductor expressed in terms of the
charges of the different conductors of the system 118
93 6. Theorem in quadratic functions 119
93 c. Work done by the electric forces during the displacement of a
system when the potentials are maintained constant .. .. 119
94. Comparison of electrified systems 120
CHAPTER IV.
GENERAL THEOREMS.
95 Oy h. Two opposite methods of treating electrical questions 123, 124
96 a. Green's Theorem 126
96 b. When one of the functions is many valued 128
96 c. When the region is multiply connected 129
96 d. When one of the functions becomes infinite in the region .. 130
97 a, 6. Applications of Green's method 131,132
98. Green's Function 133
99 a. Energy of a system expressed as a volume integral .. .. 135
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CONTENTS. xxi
Art. Page
99 b. Proof of unique solution for the potential when its value is
given at every point of a closed surface 136
100 a-e. Thomson's Theorem 138-141
101 (ir-h. Expression for the energy when the dielectric constants
are different in different directions. Extension of Green's
Theorem to a heterogeneous medium 142-147
102 a. Method of finding limiting values of electrical coefficients .. 148
102 6. Approximation to the solution of problems of the distribution
of electricity on conductors at given potentials 150
102 e. Application to the case of a condenser with slightly curved
plates 152
CHAPTER V.
MECHAKIOAIi ACTION BBTWESN TWO ELEOTEIOAL BTSTBM8.
103. Expression for the force at any point of the medium in terms
of the potentials arising from the presence of the two systems 155
104. In terms of the potential arising from both systems .. 156
105. Nature of the stress in the medium which would produce the
same force 157
106. Further determination of the iype of stress 159
107. Modification of the expressions at the surface of a conductor .. 161
108. Discussion of the integral of Art 104 expressing the force
when taken over all space 163
109. Statements of Faraday relative to the longitudinal tension and
lateral pressure of the lines of force 164
110. Objections to stress in a fluid considered 165
111. Statement of the theory of electric polarization 166
CHAPTER VI.
POINTS AND LINES OF BQUIUBBIUM.
112. Conditions for a point of equilibrium 169
113. Number of points of equilibrium 170
114. At a point or line of equilibrium there is a conical point or a
line of self-intersection of the equipotential surface . . . . 172
115. Angles at which an equipotential surface intersects itself .. 172
116. The equilibrium of an electrified body cannot be stable .. .. 174
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xxii CONTENTS.
CHAPTER VIL
FOBMS OF EQT7IP0TEKTIAL SUBFAOES AND LIKES OF FLOW.
Alt. Fa««
117. Practical importance of a knowledge of these forms in simple
cases 177
118. Two electrified points, ratio 4:1. (Fig. I) 178
119. Two electrified points, ratio 4 : - 1 . (Fig. II) 1 79
120. An electrified point in a uniform field of force. (Fig. Ill) .. 180
121. Three electrified points. Two spherical equipotential sur-
faces. (Fig. IV) 181
122. Faraday's use of the conception of lines of force 182
123. Method employed in drawing the diagrams 183
CHAPTER VIII.
SIMPLE CASES OF ELECTRIFICATION.
124. Two parallel planes 186
126. Two concentric spherical surfaces 188
126. Two coaxal cylindric surfaces 190
127. Longitudinal force on a cylinder, the ends of which are sur-
rounded hy cylinders at different potentials 191
CHAPTER IX.
SPHEBICAL HABMONICS.
128. Heine, Todhunter, Ferrers 194
129 a. Singular points r 194
129 6. Definition of an axis 195
129 c. Construction of points of different orders 196
129 <^. Potential of such points. Surface harmonics Fn .. .. 197
130 a. Solid harmonics. jy^ = r*F» 197
130 6. There are 2n+ 1 independent constants in a solid harmonic
of the ntli order 198
131 a. Potential due to a spherical shell 199
1316. Expressed in harmonics 199
131c. Mutual potential of shell and external system 200
132. Value of //F«r^</« 201
133. Trigonometrical expressions for F^ 202
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CONTENTS. xziii
Art Pago
134. Value off/F^ T^ds, when m = n 204
135 a. Special case when F„» is a zonal harmonic 205
1 35 6. Laplace's expansion of a surface harmonic 206
136. Conjugate harmonics 207
137. Standard harmonics of any order 208
138. Zonal harmonics 209
139. Laplace's coefficient or Biaxal harmonic 210
140 a. Tesseral harmonics. Their trigonometrical expansion .. 210
140 6. Notations used hy various authors 213
140 c. Forms of the tesseral and sectorial harmonics 214
141. Surface integral of the square of a tesseral harmonic .. 214
142 a. Determioation of a given tesseral harmonic in the expansion
of a function 215
1 42 6. The same in terms of differential coefficients of the function . . 215
143. Figures of various harmonics 216
144 a. Spherical conductor in a given field of force 217
144 6. Spherical conductor in a field for which Green's function is
known 218
145 a. Distribution of electricity on a nearly spherical conductor .. 220
145 h. When acted on by external electrical force 222
145 c. When enclosed in a nearly spherical and nearly concentric
vessel .. .. 223
146. Equilibrium of electricity on two spherical conductors .. .. 224
CHAPTER X.
COKFOCAL BUBPA0S8 OF THB 8BC0KD DB6RBE.
147. The lines of intersection of two systems and their intercepts
by the third system 232
148. The characteristic equation of Tin terms of ellipsoidal co-
ordinates 233
149. Expression of a, /3, y in terms of elliptic functions 234
150. Particular solutions of electiical distribution on the confocal
surfaces and their limiting forms 235
151. Continuous transformation into a figure of revolution about
the axis of « 238
1 52. Transformation into a figure of revolution about the axis ofx,. 239
153. Transformation into a system of cones and spheres 240
154. Confocal paraboloids 240
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CONTENTS.
CHAPTER XI.
THEOBT OF ELBCTSIO IMAGES.
Art Phge
155. ThomBon's method of electric images 244
156. When two points are oppositely and unequally electrified, the
surface for which the potential is zero is a sphere .. 245
157. Electric images 246
158. Distribution of electricity on the surface of the sphere .. .. 248
159. Image of any given distribution of electricity 249
160. Besultant force between an electrified point and sphere .. .. 250
161. Images in an infinite plane conducting sur&ce 252
162. Electric inversion 253
163. Geometrical theorems about inversion 254
164. Application of the method to the problem of Art. 158 .. .. 255
165. Finite systems of successive images 257
166. Case of two spherical surfaces intersecting at an angle ~ .. 258
n
167. Enumeration of the cases in which the number of images is
finite 259
168. Case of two spheres intersecting orthogonally 261
169. Case of three spheres intersecting orthogonally 263
170. Case of four spheres intersecting orthogonally 265
171. Infinite series of images. Case of two concentric spheres .. 266
172. Any two spheres not intersecting each other 268
173. Calculation of the coefficients of capacity and induction .. .. 270
174. Calculation of the charges of the spheres, and of the force
between them 272
175. Distribution of electricity on two spheres in contact. Proof
sphere 273
176. Thomson's investigation of an electrified spherical bowl .. .. 276
177. Distribution on an ellipsoid, and on a circular disk at
potential V 276
1 78. Induction on an uninsulated disk or bowl by an electrified
point in the continuation of the plane or spherical surface .. 277
179. The rest of the sphere supposed uniformly electrified .. .. 278
180. The bowl maintained at potential V and uninfluenced .. .. 279
181. Induction on the bowl due to a point placed anywhere .. .. 279
Appendix to Chap. XI 281
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CONTENTS. XXV
CHAPTER XII.
CONJUGATE FUNCTIOKS TN TWO DIMENSIONS.
Art. Page
182. Cases in which the quantities are fanctions of x and y
only 284
183. Conjugate functions 285
184. Conjugate functions may be added or subtracted 286
185. Conjugate functions of conjugate functions are themselves
conjugate 287
186. Transformation of Poisson's equation 289
187. Additional theorems on conjugate functions 290
188. Inversion in two dimensions 290
189. Electric images in two dimensions 291
190. Neumann's transformation of this case 292
191. Distribution of electricity near the edge of a conductor formed
by two plane surfaces 294
192. Ellipses and hyperbolas. (Fig. X) 296
193. Transformation of this case. (Fig. XI) 297
194. Application to two cases of the flow of electricity in a con-
ducting sheet 299
195. Application to two cases of electrical induction 299
196. Capacity of a condenser consisting of a circular disk between
two infinite planes 300
197. Case of a series of equidistant planes cut off by a plane at right
angles to them 302
198. Case of a furrowed surface 303
199. Case of a single straight groove 304
200. Modification of the results when the groove is circular .. .. 305
201. Application to Sir W. Thomson s guard-ring 308
202. Case of two parallel plates cut off by a perpendicular plane.
(Fig. XH) 309
203. Case of a grating of parallel wires. (Fig. XIII) 310
204. Case of a single electrified wire transformed into that of the
grating 311
205. The grating used as a shield to protect a body from electrical
influence 312
206. Method of approximation applied to the case of the grating .. 314
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CONTENTS.
CHAPTER XIIL
ELECTROSTATIC INSTBUMEKTS.
Page
The frictional electrical macLine 317
TLe electrophorus of Volta 319
Production of electrification by mechanical work. — Nicholson's
Revolving Donbler 319
Principle of Varley's and Thomson's electrical machines .. 320
Thomson's water-dropping machine 322
Holtz's electrical machine 323
Theory of regenerators applied to electrical machines .. 323
On electrometers and electroscopes. Indicating instruments
and null methods. Difference between registration and mea-
surement 326
Coulomb's Torsion Balance for measuring charges 327
Electrometers for measuring potentials. Snow-Harris's and
Thomson's 330
Principle of the guard-ring. Thomson's Absolute Electro-
meter 331
Heterostatic method 334
Self-acting electrometers. — Thomson s Quadrant Electrometer 336
Measurement of the electric potential of a small body .. .. 339
Measurement of the potential at a point in the air .. ..340
Measurement of the potential of a conductor without touch-
ing it 341
Measurement of the superficial density of electrification. The
proof plane 342
A hemisphere used as a test 343
A circular disk 344
On electric accumulators. The Leyden jar 346
Accumulators of measurable capacity 347
The guard-ring accumulator 349
Comparison of the capacities of accumulators 350
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CONTENTS. xxvu
PAKT 11.
BLECTBOKINBMATIOS.
CHAPTER I.
THE ELEGTBIO CTJBBENT.
Art Page
230. Current produced when conductors are discharged .. 354
231. Transference of electrification 354
232. Description of the Toltaic battery 355
233. Electromotive force 356
234. Production of a steady current 356
235. Properties of the current 357
236. Electrolytic action 357
237. Explanation of terms connected with electrolysis 358
238. Different modes of passage of the current .. .. 359
239. Magnetic action of the current 360
240. The Galvanometer 360
CHAPTER II.
CONDUCTION AND RESISTANCE.
241. Ohm's Law 362
242. Generation of heat by the current. Joule's Law 363
243. Analogy between the conduction of electricity and that of heat 364
244. Differences between the two classes of phenomena 365
245. Faraday's doctrine of the impossibility of an absolute charge .. 365
CHAPTER in.
ELECTBOHOTiyE FORCE BETWEEN BODIES IN CONTACT.
246. Volta's law of the contact force between different metals at the
same temperature 367
247. Effect of electrolytes 368
248. Thomson's voltaic current in which gravity performs the part
of chemical action 368
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CONTENTS.
Page
tier's phenomenon. Deduction of the thermoelectric elec-
romotive force at a junction 368
(beck's discovery of thermoelectric currents 370
gnus's law of a circuit of one metal 371
mming's discovery of thermoelectric inyersions 372
i>mson's deductions from these facts, and discovery of the
Bversible thermal effects of electric currents in copper and
liron 372
Vs law of the electromotive force of a thermoelectric pair .. 374
CHAPTER IV.
ELEGTBOLYSIS.
"aday's law of electrochemical equivalents 375
usius's theory of molecular agitation 377
ctrolytic polarization 377
(t of an electrolyte by polarization 378
Bculties in the theory of electrolysis 378
lecular charges 379
ondary actions observed at the electrodes 381
iservation of energy in electrolysis 383
eisurement of chemical affinity as an electromotive force .. 384
CHAPTER V.
ELBCTBOLYTIC POLABIZATION.
ficulties of applying Ohm's law to electrolytes 387
m's law nevertheless applicable 387
3 effect of polarization distinguished from that of resistance 387
arization due to the presence of the ions at the electrodes.
"he ions not in a free state 388
lation between the electromotive force of polarization and
be state of the ions at the electrodes 389
isipation of the ions and loss of polarization 390
lit of polarization 391
ter's secondary pile compared with the Leyden jar .. .. 391
istant voltaic elements. — DanielFs cell 394
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CONTENTS. xxix
CHAPTER VI.
MATHEMATICAL THEOBT OF THE DISTBIBUTION OF ELECTBIC
CUBBENT8.
Art. Page
273. linear condactors 399
274. Ohm's Law 399
275. Linear conductors in series 399
276. Linear conductors in multiple arc 400
277. Resistance of conductors of uniform section 401
278. Dimensions of the quantities inyolved in Ohm's law .. 402
279. Specific resistance and conductivity in electromagnetic measure 403
280. Linear systems of conductors in general 403
281. Reciprocal property of any two conductors of the system .. 405
282 a, h. Conjugate conductors 406
283. Heat generated in the system 407
284. The heat is a minimum when the current is distributed ac-
cording to Ohm's law 408
Appendix to Chap. VI 409
CHAPTER VII.
CONDUCTION IN THBEE DIMENSIONS.
285. Notation 411
286. Composition and resolution of electric currents 411
287. Determination of the quantity which flows through any
sur£Bkce 412
288. Equation of a surface of flow 413
289. Relation between any three systems of surfaces of flow .. .. 413
290. Tubes of flow 413
291. Expression for the components of the flow in terms of surfaces
offlow 414
292. Simplification of this expression by a pi*oper choice of para-
meters 414
293. Unit tubes of flow used as a complete method of determining
the current 414
294. Current-sheets and current-functions 416
295. Equation of ' continuity ' 415
296. Quantity of electricity which flows through a given surface .. 417
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CONTENTS.
CHAPTER Vm.
RBSISTAKCB AND COKDUCnVITT IK THBKB DIMSNSIOKS.
Art, P»g»
297. Equations of resistance 418
298. Equations of conduction 419
299. Kate of generation of heat 419
300. Conditions of stability 420
301. Equation of continuity in a homogeneous medium 421
302. Solution of the equation 421
303. Theory of the coefficient T. It probably does not exist .. 422
304. Generalized form of Thomson's theorem 423
305. Proof without symbols 425
306. Lord Eayleigh's method applied to a wire of variable section. —
Lower limit of the value of the resistance 426
307. Higher limit 429
308. Lower limit for the correction for the ends of the wire .. .. 431
309. Higher limit 432
CHAPTER IX.
CONDUOnON THBOUOH HBTKSOOBKSOUB MEDIA.
310. Surface-conditions 435
311. Spherical surface 437
312. Spherical shell 438
313. Spherical shell placed in a field of uniform flow 439
314. Medium in which small spheres are uniformly disseminated .. 440
316. Images in a plane surfiftce 441
316. Method of inversion not applicable in three dimensions.. .. 442
317. Case of conduction through a stratum bounded by parallel
planes 443
318. Infinite series of images. Application to magnetic induction 443
319. On stratified conductors. Coefficients of conductivity of a con-
ductor consisting of alternate strata of two different substances 445
320. If neither of the substances has the rotatory property denoted
by I^ the compound conductor is free from it 446
321. If the substances are isotropic the direction of greatest resist-
ance is normal to the strata 446
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CONTENTS. XXXI
Art Pa«e
322. Medimn containing parallelepipeds of another medium .. .. 447
323. The rotatory property cannot he introduced hy means of con-
ducting channels 448
324. Construction of an artificial solid having given coefficients of
longitudinal and transverse conductivity 449
CHAPTER X.
CONDUCTION IN DISLECTBICS.
325. In a strictly homogeneous medium there can he no internal
charge 460
326. Theory of a condenser in which the dielectric is not a perfect
insulator 451
327. No residual charge due to simple conduction 452
328. Theory of a composite accumulator 452
329. Eesidual charge and electrical absorption 454
330. Total discharge 456
331. Comparison with the conduction of heat 458
332. Theory of telegraph cables and comparison of the equations
with those of the conduction of heat 460
333. Opinion of Ohm on this subject 461
334. Mechanical illustration of the properties of a dielectric .. .. 461
CHAPTER XI.
MEASUBBHEMT OF THE ELECTBIC BE8I8TANCE OF C0NDUCT0B8.
335. Advantage of using material standards of resistance in electrical
measurements 465
336. Different standards which have been used and different systems
which have been proposed 466
337. The electromagnetic system of units 466
338. Weber's unit, and the British Association unit or Ohm .. .. 466
339. Professed value of the Ohm 10,000,000 metres per second .. 466
340. Reproduction of standards 467
341. Forms of resistance coils 468
342. Coils of great resistance 469
343. Arrangement of coils in series 470
344. Arrangement in multiple arc 470
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xxxii CONTENTS.
Art. Page
345. On the comparison of resiBtances. (1) Ohm's method .. .. 471
346. (2) Bj the differential galvanometer 472
347. (3) By Wheatstone's Bridge 476
348. Estimation of limits of error in the determination 477
349. Best arrangement of the conductors to be compared .. 478
350. On the use of Wheatstone's Bridge 480
351. Thomsons method for small resistances 482
352. Matthiessen and Hockin's method for small resistances .. .. 485
353. Comparison of great resistances by the electrometer .. 487
354. By accumulation in a condenser 487
355. Direct electrostatic method 488
356. Thomson's method for the resistance of a galvanometer .. .. 489
357. Mance's method of determining the resistance of a battery .. 490
358. Comparison of electromotive forces 493
CHAPTER XII.
ELECTBIC BBSISTANCE OF SUBSTANCES.
359. Metals, electrolytes, and dielectrics 495
360. Resistance of metals 496
361. Resistance of mercury 497
362. Table of resistance of metals 498
363. Resistance of electrolytes 499
364. Experiments of Paalzow 500
365. Experiments of Kohlraosch and Nippoldt 501
366. Resistance of dielectrics 501
367. Gutta-percha 503
368. Glass 504
369. Gases 604
370. Experiments of Wiedemann and RUhlmann 505
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CORRECTIONS AND ADDITIONS. ^
Vol. I. p. 61, last line, /or 28 read 2
ERRATUM. "^
Vol.I,bottomofp.lOO,/or ^' ^ *^^-^' read 51» - ^1^
IfoxMnr/ri SketrieUf and Magnetum, Third Edition.
mental units of Length, lime, ana Mass. Thus the units ot 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
relation 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
definitions.
The formulae at which we arrive must be such that a person
of any nation, by substituting for the different symbols the
VOL. I. B
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r
2 PBELIMINAET. [3.
numerical 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 defined 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 dimensiona
of every unit in terms of the three fundamental units. When a
given unit varies as the nth power of one of these units, it is
said to be of ^ 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 firom 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 diflferent according to the
arbitrary system of units which we adopt*.
The Three Fundamental Units.
8.] (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 mfetre. The mfetre 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
mhtre 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 m^tre. .
* The tiieory of dimeniioiii was fint stated by Fourier, Th^rie de Chaletir, $ 160.
r
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5-] THE THEBB 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 substance such as sodium, which has well-defined lines
in its spectrum. Such a standard would be independent of any
changes in the dimensions 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 [L], II 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 [L],
4.] (2) Time. The standard unit of time in all civilized
countries 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 observations 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 tfme. 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 [T], and the numerical
measure of time t.
5.] (3) Ma88. 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
temperature 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,
B 2
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r
4 PEELIMINAEY. [5,
to be compared directly with the standard, and not deduced fix)m
experiments on water.
In descriptive astronomy the mass of the sun or that of the
earth is sometimes taken as a unit, but in the djrnamical 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^].
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
distance r is by the Newtonian Law -^ • Suppose this attraction
to act for a very small time ^ on a body originally at rest, and to
cause it to describe a space 8, then by the formula of Galileo,
whence m = 2 -^ • Since r and s are both lengths, and ^ is a
time, this equation cannot be true unless the dimensions of nti are
[i3y-2j^ The same can be shewn from any astronomical equa-
* See Prof. J. Losclunidt, * Zur Grogse der Luftmoleciile,' Academy of Vienna^
Oct. 12, 1865 : G. J. Stoney on 'The Internal Motions of Gases/ PhU. Mag., Aug.
1868 ; and Sir W. Thomson on 'The Size of Atoms,* Nature, March 31, 1870.
{ See also Sir W. Thomson on * The Size of Atoms,' Nature, v. 28, pp. 208, 250, 274. }
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6.] DERIVED tJNITS. 5
tion in which the mass of a body appears in some but not in all
of the terms *.
Derived Units.
6.] The unit of Velocity is that velocity in which unit of length
is described in unit of time. Its dimensions are [ijP~^].
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
velocity increases by unity in unit of time. Its dimensions are
The unit of Density is the density of a substance which con*
tains unit of mass in unit of volume. Its dimensions are [ML"^'].
The unit of Momentum is the momentum of unit of mass
moving with unit of velocity. Its dimensions are [MLT-^].
The unit of Force is the force which produces unit of momentum
in unit of time. Its dimensions are [J/iT-*].
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. Li 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 statements involving these quantities
are not complete without a knowledge of the intensity of gravity
in the places where these statements were found to be true.
The abolition, for all scientific pui'poses, of this method of
measuring forces is mainly due to the introduction by Qauss of
* If a oentimetre and a second are taken as nnits, the astronomical unit of mass
wonld be about 1.587 x 10^ grammes, or 15>37 tonnes, according to Baily's repetition
of CavendiRh's experiment. Baily adopts 5*6604 as the mean result of all his experi-
ments for the mean density of the earth, and this, with the values used by Baily for
the dimensions of the eaiih and the intensity of gravity at its surfiue, gives the
above value as the direct result of his experiments.
{Comu's recalculation of Baily's results gives 5-55 as the mean density of the
earth, and therefore 1-50 x 10^ grammes as the aHtronomical unit of mass ; while
Comu's own experiments give 5-50 as the mean density of the earth, and hid x 10^
grammes as the astronomical unit of mass. |
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6 PBELIMINABY. [7.
a general system of making observations of magnetic force in
countries in which the intensity of gravity is diflFerent. 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 [ML^T~^].
The Energy of a system, being its capacity of performing work,
is measured by the work which the system is capable of per-
forming by the expenditure of its whole energy.
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 exp^ss them in terms of any other
unit of the same kind, we have only to remember that eveiy ex-
pression for the quantity consists of two factoi*s, the unit and the
numerical 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 considera-
tion of the continuous existence of a particle of matter in time
and spcu^e. 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 pass-
ing 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 x, and if, while x passes continuously
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8.] OONTINUITT AND DISCONTINUITY. 7
from oDq to x^yU passes continuously from Uq to v^, but when x
passes from x^ to X2,u passes from at/ to U2, Ui being different
from Ui, then u is said to have a discontinuity in its variation
with respect to x for the value x = x^, because it passes abruptly
from Ui to Ui while x passes contiiiuously through x^
If we consider the differential coefficient of u with respect to x
for the value x ^x^aa the limit of the fraction
,
when X2 and Xq are both made to approach x^ without limit, then,
if Xq and X2 are always on opposite sides of a^i, the ultimate value
of the numerator will be u/— it^, and that of the denominator
will be zero. If it is a quantity physically continuous, the dis-
continuity can exist only with respect to particular values of the
variable x. We must in this case admit that it has an infinite
differential coefficient when x^ x^. If u is not physically con-
tinuous, it cannot be differentiated 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 a;^ is a very little less than aJi, and x^ a very little
greater than aj^, then Uq will be very nearly equal to u^ and u^
to Ui\ We may now suppose u to vary in any arbitrary but
continuous manner from u^tov^ between the limits Xq and x.^.
In many physical questions we may begin with a hypothesis of
this kind, and then investigate the result when the values of
Xq and X2 are made to approach that of x^ and ultimately to I'each
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 u is discontinuous.
Discontinuity of a Function of more than One VariaMe.
8.] If we suppose the values of all the variables except a; to be
constant, the discontinuity of the function will occur for particular
values of x, and these will be connected with the values of the
other variables by an equation which we may write
<^ = <^(a;,y,0,&c.) = O.
The discontinuity will occur when <^ = 0. When 4> is positive
the function will have the form F^ {x, y, z, &c.). When <^ is
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8 PRELIMiyAET. [9.
negative it will have the form F^ {x, y, 0, &c.). There need be no
necessary relation between the forms F^ and F2.
To express this discontinuity in a mathematical form, let one
of the variables, say x, be expressed as a function of <^ and the
other variables, and let JF\ and F2 be expressed as functions of
<^, 2/, z, &c. We may now express the general form of the function
by myformula which is sensibly equal to F^jvTh&n^ is positive,
and to JF\ when 0 is negative. Such a formula is the following —
As long as ^ 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 0 is positive, and equal to JF\ when <f) is negative.
Discontinuity of the DeHvatives 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
</) = <^(a;, y, 0...) = 0,
and let F^ and F^ be expressed in terms of ^ and n— 1 other
variables, say {y^ z ...).
Then, when <^ is negative, J^^ is to be taken, and when ^ is
positive i^2 is ^ be taken, and, since F is itself continuous, when
<t> is zero, F^ = F^.
Hence, when <f> is zero, the derivatives -j-^ and -5-^ may be
different, but the derivatives with respect to any of the other
/I Iff gl w
variables, such as -7-^ and -^ y must be the same. The discon-
dy dy
tinuity is therefore confined to the derivative with respect to ^,
all the other derivatives being continuous.
Pei^dic and Multiple Functions,
9.] If u is a function of x such that its value is the same for
Xy aj + a, x-^-na, and all values of x differing by a, u is called a
periodic function of x, and a is called its period.
If aj is considered as a function of it, then, for a given value of
u, there must be an infinite series of values of x differing by
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lO.] VECTOES, OE DIEBOTED QUANTITIES. 9
multiples of a. In this case x is called a multiple function of u,
and a is called its cyclic co^jant.
The differential coefficient -r- has only a finite number of
values corresponding to a given value of tt.
On the Relation of Physical Qaantities 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
geometry by Des Cartes was one of the greatest steps in mathe-
matical 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 lengths 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.
But for many purposes of phjBical 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 primitive 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. wUl 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.
* {For an elementary aoooant of Quaternions, the reader may be referred to Kel-
land and Tait*s 'Introduction to Quatemiona/ Tait's < Elementary Treatise on
Quaternions/ and Hamilton's ' Elements of Quaternions.' [
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/
/
10 PBELIMINAEy. [ll.
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 under-
stood 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, jfrom 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.
There are phy sical^antities ofanother kind which are related
to directions m space^Jbut which are^ m)tjyectors. Stresses and
J fjC^/ strains' in soEd bodies are examples of these, and so axe some of
)Um^^^ *^® properties of bodies considered in the theory of elasticity and
.j^ ./$-/<) in the theory of double refraction. Quantities of this class
' ' require for their definition nine numerical specifications. They
are expressed 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
composition 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 Oeiman capital letter, as 31, S, &c.
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12.] INTENSITIES AND FLUXES. H
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 hydrostatical pressure, the
potential at a point, are examples of scalar functions. The
resultant force at a pointy 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 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 positions
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
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12 PRELIMINAEY. [13.
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 intensities may be considered as producing, or
tending to produce, its corrresponding flux. Thus, electromotive
intensity produces electric currents in conductors, and tends to
produce them in dielectrics. It produces electric induction in
dielectrics, and probably in conductors also. In the same sense,
magnetic intensity produces magnetic induction.
13.] In some cases the flux is simply proportional to the inten*
sity and in the same direction, but in other cases we can only
affirm that the direction and magnitude of the flux are functions
of the direction and magnitude of the intensity.
The case in which the components of the flux are linear
functions of those of the intensity is discussed in the chapter on
the Equations of Conduction, Art. 297. There are in general nine
coefficients which determine the relation between the intensity
and the flux. In certain cases we have reason to believe that six^
of these coeffidenttLfpjmlhree^jairs^of equal jquantities. In such
Zl'I) cases the relation between the line of direction of the intensity
and the normal plane of the flux is of the same kind as that be-
tween a semi-diameter of an ellipsoid and its conjugate diametral
plane. In Quaternion language, the one vector is said to be a
lineal' 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
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15.] LINE-INTEGBALS. 13
intensity. In all cases, however, the product of the intensity
and the flux resolved in its direction, gives a result of scientific
importance^ 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 intensity, we have to take the integral along a
line of the product of an element of the line, and the resolved
part of the intensity along that element. The result of this
operation is called the Line-integral of the intensity. It repre-
sents 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-in^-ftgrftl ^^ ^^^ ^^ry.
It represents 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 m electro-
statics and magnetics as Linesof Induction, and in electrokine-
matics as Lines of Flow.
15.] There is another distinction between difiisrent kinds of
directed quantities, which, though very important from a physical
point of view, is not so necessary to be observed for the sake of
the mathematical methods. This is the distinction between
longitudinalgod rotational properties.
The direction and magnitude of a quantity may depend upon
some action or eflect 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 diflerence in the mathematical
treatment of the two classes, but there may be physical circum-
stances which indicate to which class we must refer a particular
phenomenon. Thus, electrolysis consists of the transfer of cer-
tain substances along a line in one direction, and of certain
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14 PEBLIMINAEY. [l6.
other substances 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
magnetism is a longitudinal phenomenon, while the effect of
magnetism in rotating the plane of polarization of plane polari^d
light distinctly shews that magnetism is a rotational pheno-
menon*.
On Line-iTdegrcUa,
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 Xy y, z he the coordinates of a point P on a line whose
length, measured from a certain point A, is 8. These coordinates
wiU be functions of a single variable 8.
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 i2 cos € is the resolved part of R along the line,
and the integral r*
L = I RcoQcda
Jq
is called the line-integral of R along the line 8.
We may write this expression ^^^7 r
where X, F, Z are the components of R parallel to x^y^z respect-
ively.
THiis quantity is, in general, different for different lines drawn
* (This must not be taken to imply that in any theory in which electric and
magnetic phenomena are suppoBed to be due to the motion of a medium, the electric •
current must necessarily be due to a motion of translation and magnetic force to one
of rotation. There are rotatory effects connected with r current, for example,
a magnetic pole is turned round it, and it is probable that if the medium in which
electrostatic phenomena have their seat has an electric displacement through it
whose components are /, g^ h^ and is moving with the velocity «, r, w, it wiU
be the seat of a magnetic force whose components are 4ir (tr^— vA), 4ir {uh—wf),
iit{iof^ug) respectively: thus, in this case, a motion of translation could produce
a magnetic field. PhU, Mag, July, 1889. }
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1 6.] EELATION BETWEEN FOECB AND POTENTIAL. 15
between A and P. When, however, within a certain region, the
quantity Xdx-^Ydy + Z(h = ^D<f^,
that is, when it is an exact differential within that region, the
value of L becomes
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 con-
tinuous 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
components are X, Y, Z ia said to have a potential % if
x=-0. r=-(f). .=-(f).
When a potential function exists, surfaces for which the
potential is constant are called Equipotential surfaces. The
direction of JR at any point of such a surface coincides with the
normal to the surface, and if 71 be a normal at the point P,
then iJ = — f- •
an
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 Oreen f, who made it the basis of
his treatment of electricity. Green's essay was neglected by
mathematicians till 1846, and before that time most of its im-
portant theorems had been rediscovered by Gauss, Chasles,
Sturm, ^d Thomson {•
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
* M^e. celeste, liv. iii.
f Essay on the Application of Mathematical Analysis to the Theories of Elec-
tricity and Magnetism, Nottinsrham, 1828. Keprinted in CrelWa Journal, and in
Mr. Ferrers* edition of 6reen*s Works.
X Thomson and Ttat, Natural Fhiloeophy, § 483.
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1 6 PRELIMINARY. [ 1 7.
investigations the potential is defined 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
potential 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 fi'om 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 iy jj k are three unit vectors at right angles to each
other, and if X, F, 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 J d ,^.
'^'^^dy'-^dz' ('^
55 = -v4'. (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 4^ increases
fastest, and then to lay off in that direction a vector representing
this rate of increase.
M. Lam^, in his TraiU dee Fonctiona 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 Lam^ uses it, indicates that the quantity referred to has X
dii'ection 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 g the space-variation of the scalar
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1 8.] RELATION BETWEEN FORCE AND POTENTIAL. 17
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
which are those oiXdx + Ydy + Zdz being a complete differential,
are satisfied throughout a certain region of space, and yet the
line-integral from J. 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—\ are
sufficient to join thie p points so as to form a connected system.
Every new line completes a loop or closed path, or, as we shall
caU it, a Cycle. The number of independent cycles in the
diagram is therefore k = i— j9+ 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 Cyclomat^^f nnmbftr.
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 more closed lines, which may in the limiting cases become
double finite lines or points.
* Der Census Rflifmlicher Complete, Gott. Abh., Bd. x. S. 97 (1861). {For »n
elementary acooant of those properties of multiply connected space which are necessary
for physical purposes see Lunb s Treatise on the Motion of FluidSy p. 47« }
VOL. I.
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1 8 PBBLIMINABY. [ 1 9.
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 connections, 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 aU 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 con-
tract inwards, without breaking continuity, till they meet. The
surface will then be reduced to a point in the case of an acyclifi.
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 througliout any acyclic region
Xdx+Ydy-^Zdz^-'I)^
the value of the line-integral from a point A to a point P
taken along any path vxithin 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
corresponding portions of the line-integral being equal and
opposite, the total value is zero.
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20.]
CYCLIC REGIONS
19
Hence if AQP and AQ^P are two paths from -4 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,
therefore 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 -h Ydy + Zdz= -/)*
is everywhere satisfied, the line-integral from AtoP along a
UvjC drawn within the region^ %oill not in general he deter-
Tninate unless the channel of communication between A and
P be specified.
Let N be the cyclomatic number of the region, then iV sections
of the region may be made by surfaces which we may call Dia-
phragms, so as to close up N 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 ^ 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 A A' and PP'^ nearly coincident, but on oppo-
site sides of the diaphragm, the line-integrals along these lines
will be equal*. Suppose each equal to i, then K\ the line-integral
of ^' P', is equal to that of ^'^ + ilP + PP'= -i + ^+i=ir=
that of AP.
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
corresponding 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
{Since X, F, Z, are oontinuoua.}
0 2
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20 PBBLIMINAET. [2 1 .
and p' times in the negative direction, and let p—p^^nfii. Then
the line-integral of the closed curve will be tViK^
Similarly the line-integral of any closed curve will be
niK^+n2K^ + ... + ngKs ;
where Tig represents the excess of the number of positive passages
of the curve through the diaphragm of the cycle S ovei* 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 H- 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, Y, Z to be the com-
ponents of the displacement of a point of a continuous body whose
original coordinates are x, y, z ; the condition then expresses that
these displacements constitute a non-rotatioTial straiwf.
II X, Y, Z represent the components of the velocity of a fluid
at the point x, y, z, then the condition expresses that the motion
of the fluid is irrotational.
If Xj F, Z represent the components of the force at the point
X, 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.
Joy ^
On Surface-Integrals.
21.] Let dS be the element of a surface, and c the angle which
a normal to the surface drawn towards the positive side of the
I surface makes with the direction of the vector quantity iJ, then
/ 1 1 R cos € dS is called the surface-irdegral ofR over the surface St.
* gee Sir W. Thomson * On Vortex Motion,* Tran». R. S. Edin., 1867-8.
t See Thomson and Tait's Natural Fhilosophtf, § 190 (i).
4: { In the following investigations the positive direction of the normal is outwards
from the surface. \
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2 1 .] SURFACE-INTEGRALS. 21
Theorem III. The surface-iTUegral of the flvxc inwards through
a closed eurfdce may he expressed as the volunie-integral of
its convergence taken within the sfwrfa/x. (See Art. 25.)
Let X, F, Z be the components of iJ, and let ?, m, n be the
direction-cosines of the normal to S measured outwards. Then
the surface-integral of R over 8 is ^•^ *^«* ^'^'^'*' * ^
ffRGOS€dS=ffxidS+ffYmdS+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^-'dydZy
and at each exit Zd/S= dydz.
Let a point travelling from a? = — co toa? = +co first enter
the space when a; = «, , then leave it when aj = ojg, and so on ;
and let the values of X at these points be X^, Xj, &c., then
ffxidS= -^ /^{(X.-X^) +(X3-X,) + &c.
+ (^2n-l-X2n)}rft/d0. (2)
If X is a quantity which is continuous, and has no infinite values
between x^ and x^y 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
ffxldS=fff^dxdydz, (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
/S, the total surface-integral of iJ ovei- that surface will be
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22 PEBLIMINABY. [22.
the triple integration being extended over the whole space
within 8.
Let as next suppose that X, F, Z are not continuous within
the closed surface, but that at a certain surface F{x, y,z)=0 the
values of X, Y, Z alter abruptly from Z, F, Z on the negative
side of the surface to X\ F, Z' on the positive side.
If this discontinuity occurs, say, between x^ and x,^^ the value
ofXg-ZiwiUbe ,^
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 iZ over the
closed surface will be expressed by
/yiJco8edS=///(g +^^ + ^)dxdydz+fJ{X'-X)dydz
+j'j'(r-7)dzdx+fJ(Z'-Z)dxdy; (7)
or, if I', m', n' are the direction-cosines of the normal to the sur-
face of discontinuity, and dS' an element of that surface,
^r^^ //i2coscdS=///(g + ^^
+ ff{l\r--X) + m\r^Y) + n'{Z'^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, F, Z are continuous
dX^dY^dZ^^
dx dy dz * ^ ^
and at every surface where they are discontinuous
Vr^m'r + n'Z'=VX + 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 8 the equation
dX dY dZ^^
dx dy dz'^ ^ ^
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22.] SOLENOIDAL DISTRIBUTION. 23
is satisfied. We have as a consequence of this the surface-in tegi'al
over the closed surface equal to zero.
Now let the closed surface S consist of three parts <Si , /Sy,, and
S^. Let Si be a surface of any form bounded by a closed lineii-
Let Sq be formed by drawing lines from every point of Zj always
coinciding with the direction of iJ. If i, 7?i, n are the direction-
cosines of the normal at any point of the surface Sq, we have
jBcos€=Zf+7m + ZTi = 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 8q,
Le* Oi> Oo' Q2 1^ ^^® surface-integrals of the surfaces S^Sq^S^,
and let Q be the surface-integral of the closed surface 8, Then
Q = Qi + Qo + Q2 = 0; (13)
and we know that Qo= 0 ; (14)
therefore Q2 = -Qil (15)
or, in other words, the surface-integral over the surface /Sgis equal
and opposite to that over Si whatever be the form and position
of S2, provided that the intermediate surface Sq is one for which
R is always tangential.
If we suppose L^ a closed curve of small area, Sq 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? ^ ^^0 (16)
dx 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 S bounded by a closed
curve L is equal to the nuniber 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 S depends only on the form of
its boundary Z, and not on the form of the surface within its
boundary.
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24 PRELIMINARY. [22.
On Periphractic Regions.
If, throughout the whole region bounded externally by the
single closed surface S, the solenoidal condition
dX dY dZ ^
dx ay dz
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.
K within one of these enclosed regions, say, that bounded by the
closed surface /S^, the solenoidal condition is not satisfied, let
Q^ = ffRco8€dSi
be the surface-integral for the surface enclosing this region, and
let $2* Qs' ^^' ^ t^® corresponding quantities for the other en-
closed regions 82, S^, &c.
Then, if a closed surface S^ is drawn within the region S, the
value of its surface-integral will be zero only when this surface
fif does not include any of the enclosed regions fif^, iSg, &c. If it
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 i,, ig' ^^* joining the internal surfaces S^, 8^^ &c. to the
external surface 8, 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
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23-] PERIPHBAOTIO REGIONS. 25
number, or the number of internal surfaces. In drawing these
]\nes -w© must remember that any line joining surfaces which are
abready 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
sui*face 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 number of times, it encloses the surface 8i and the
surface-integral is Q^,
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 the latter case we have
where m is the mass, supposed to be at the origin of coordinates.
At any point where r is finite
dX dY^dZ^^
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 47rm.
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 Sjxice.
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*.
* The combined action of the moBcIes of the arm when we turn the upper side of
the right-hand outwards, and at the same time thrust the hand forwards, wiU 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. y tfiAv».,_
Professor W. H. Miller has suggested to me that as the tendrils of the vine are y^S^^ies- j
right-handed screws an4 those of the hop left-handed, the two systems o^ relaOoi^s j^«;u«^>^^^*^.
in space migni be <»lled those of the vine and the hop respectively. L lt% ^
Tlie s^tem of the vine, which we adopt, is that of Linnseus, and of screw-makers /* *
in all civilized countries except Japan. De CandoUe was the first who called the
hop-tendiil right-handed, and m this he is followed by Listing, and by most writers
on the circulitf 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 ordinary
carriages, but they are always called left-handed screws by those who use them.
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26 PRELIMINAEY. [23.
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.
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^ and in Tait's QimtermoTis.
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 a;, y, «, 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 ar 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 ocy may be written either
Ixdy or ^jydx ;
the order of integration being ar, 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 pei-pendicular
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, Zy and negative when the cyclic order is reversed.
* { As in the diagram
l^;
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24.] LINE-INTEGBAL AND SUEFAOE-INTEGRAL. 27
We now proceed to prove a theorem which is useful as estab-
lishing a connection between the surface-integral taken over a
finite surface and a line-integral taken round its boundary.
24.] Theorem IV. A line-dntegral taken round a closed curve
may be expressed in terms of a surface-integral taken ovei*
a surface hounded by the curve.
Let Xy r, Z be the components of a vector quantity 81 whose
line-integral is to be taken round a closed curve s. .
Let S be any continuous finite surface bounded entirely by the
closed curve s, and let f , »?, C be the components of another vector
quantity 93, related to XyY, Z by the equations
dy dz ^ ^ '~ dz dx ^ dx dy ' ^ ^
Then the surface-integral of SB taken over the surface S is equal to
the line-integral of 3[ taken round the curve s. It is manifest that
6 Vf C satisfy of themselves the solenoidal condition.
^ + ^4.^=0.
dx dy dz
Let ly 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 93 may be written
ffm-hmri-hnOdS. (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 )3. If )3 is constant and a varies, the point
(a;, y, z) will describe a curve on the surface, and if a series of values
is given to ^8, a series of such curves will be traced, all lying on
the surface B, 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 oi yz is, by the
ordinary formula,
irfS=(?^-^J)d^da. (3)
^da dp dp da^ ^ ^ ^
The expressions for mdSsskd ndS are obtained from this by
substituting x, y^zia cyclic order.
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28 PRELIMINAEY. [24.
The surface-integral which we have to find is
^{li + mri + nC)d8; (4)
//<'
or, substituting ihe values of ^, i;, ^ in terms of X, Y, Z,
rr, dX dX dY ,dY ,dZ dZ.,„ .,
The part of this which depends on X may be written
rridX rdz dx dz dx\ dX^dxdy ^^ ^y\\^Q^ /r\
Jji'd^ydid0^d^d^)^^^did^''d^da)r^^ ^^^
adding and subtracting -^ — ^ -7-- , this becomes
ff^dx AX dx dXdy dX dz\
jJXd^^dxd^^l^d^^'dzdi)
dx /'dXdx dX dy , dX dz\} ,^ , ..
rpydXdx dXdx\^^j zox
^m-d^Trd-fid^y^^^'^ ^'^
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, a^, and let the last
curve of the series, for which o = aj, coincide with the closed
curve 8.
Let us also suppose that the curves for which fi is constant
form a series of lines drawn from the point at which a^=^ a^
to the closed curve 8, the first, /3o, and the last, )3i, being
identical
Integrating (8) by parts, the first term with respect to o and
the second with respect to /3, the double integrals destroy each
other and the expression becomes
r\x % dp-f'Ux ^) dp-f'\xf) da
+ f'\Xp)da. (9)
Since the point (a, /3j) is identical with the point (a, fi^), the
third and fourth terms destroy each other; and since there is
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/
25.] LINE-INTEGBAL AND SUEPACE INTEGRAL, 29
but one value of x at the point where a =: Oq, the second term is
zero, and the expression is reduced to the first term :
Since the curve a = Oj is identical with the closed curve «, we
may write the expression in the form
x'£ds, (10)
where the integration is to be performed round the curve 8. We
may treat in the same way the parts of the surface-integral
which depend upon Y and Zy so that we get finally,
ffmrar,^nO<l8^f(X^^Y^£^z'£)ds; (U)
where the first integral is extended over the surface S, and the
second round the bounding curve 8*.
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
(HE and IV). The extension of this operator to vector displace-
ments, and most of its further development, are due to Professor
Taitt.
Let o- be a vector function of p, the vector of a variable point
Let us suppose, as usual, that
p = ix -{-jy i-kz,
and (r-iX-hjY+kZ] 0
where X, F, Z are the components of o- in the directions of the
axes.
We have to perform on a- the operation
__ . d! . d J d
" dx dy dz
Performing this operation, and remembering the rules for the
multiplication of % j, kfjwe find that V<r consists of two parts,
one scalar and the otherA^ector.
* This theorem was given by Professor Stokes, Smilh's Prize Examination ^ 1854,
question 8. It is prov^ in Thomson and Tait's Natural Philosophy, $ 190 {j),
t See Proc, R, 8, EtHn., April 28, lfe62. * On Greenes and other allied Theorems,'
Trans, R. 8, Edin., 1869-70, a very valuable paper; and 'On some Quaternion
IntegraU,' Proc, R, 8, Edin,, 1870-71.
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30 PEELIMINAEY. [25,
The scalar part is
8V(r = — ( J- + 77" + 77") ' ®^® Theorem HI,
and the vector part is
Wj/ rf^>' ^dz ^^ ^dx dy'
If the relation between X, Y, Z and i, rj, C is that given by
equation (1) of the last theorem, we may write
Wa = ii+jri + kC See Theorem IV.
It appears therefore that the functions of X, F, Z which occur
in the two theorems are both obtained by the operation V on
the vector whose components are Xy F, Z. The theorems them-
selves may be written
fjfr^^ds^ JJs.aUvds, (in)
and JSadp ^^Jfs.VaUvds', (IV)
where c! 9 is an element of a volume, c^ of a surface, dp of a
curve, and Uv a unit-vector in the direction
V I y of the normal.
^ >^ To understand the meaning of these fiinc-
— ^ . ^ — taons of a vector, let us suppose that o-q is the
value of <r at a point P, and let us examine
/ f \ the value of <r— o-q in the neighbourhood of P.
If we draw a closed surface round P, then,
*^* if the surface-integral of a- over this surface
is directed inwards, SVa- will be positive, and the vector (t—o-q
^ near the point P will be on the whole directed
I i towards P, as in the figure (1).
f ^ I I propose therefore to call the scalar part of V<r
; ^ the fYn'^jfrffpn/^f. of a at the point P.
^' ' To interpret the vector part of V<r, let the direc-
/ tion of the vector whose components are f , rj, ( be
V upwards from the paper and at right angles to it,
^ • \^^ and let us examine the vector 0-—^^ near the point
y P. It will appear as in the figui-e (2), this vector
^ being arranged on the whole tangentially in the
Fig. 3. direction opposite to the hands of a watch.
I propose (with great diffidence) to call the vector part of Vir
the rotation of a at the point P.
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26.] Hamilton's opebatoe v. 31
In Fig. 3 we have an illustration of rotation combined with
convergence.
Let us now consider the meaning of the equation
FV<r = 0.
This implies that Vo-is a scalar, or that the vector o- 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 ^o ^ ^^ value of q at the centre, and q the
mean value of q for all points within the sphere,
SO that the value at the centre exceeds or falls shoii of the mean
value according as V^g is positive or negative.
I propose therefore to call V^g 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 9 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.
i
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PART T.
ELECTROSTATICS.
CHAPTER L
DE8CEIPTI0N OP 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 similai*
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 in Equilibrium/
Cambridge and Dublin Mathematical Journal, March, 1848.
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28.] BLEOTBIFIOATION. 83
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 resinoudy 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.] ExpERraENT n* 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
same in whatever part of the interior space tiie glass Kg. 4.
is suspended t.
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, and leTeral experiments which foUow, are due to Faraday, * On Static
Electrical Inductive Action,' Phil. Mag., 1843, or Exp, Res., yoI. ii. p. 279.
t {TUfl is an iUustration of Art. 100 e. }
VOL. I. D
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34 BLBOTBOSTATIO PHENOMENA. [29.
This electrification of the vessel, which depends on the glass
being within it, and which vanishes when the glass is removed, is
called electrification by Induction.
Similar efiects 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.
ElectriJiccUion by Conduction.
29.] ExPEBTMENT HI. Let the metal vessel be electrified by
induction, as in the last experiment, let a second metallic body
be suspended by white 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-
diLctor of electricity, and the second body is said to be dectrijied
by conduction.
Conductors and Insulators.
ExPEBiMENT 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
Insulators.
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-Qonducting
Digitized by VjOOQ iC
31.] CONDUCTOBS AND INSULATORS, 35
medium. Such a medium, consideFed as transmitting these
electrical effects without conduction, has been called by Faraday
a Dielectric 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 electrification, 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 electrification will disappear from both bodies,
shewing that the electrification of the two bodies was equal and
opposite.
80.] Experiment Y. 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 their electrification.
81.] Experiment YI. Let a seeond insulated metallic vessel,
B, be provided, and let the electrified piece of glass be put into
the first vessel j1, 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 HI. All signs of electrification
will disappear.
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 metal vessel C, it will be found that there is no
D 2
Digitized by
36 ELBCTEOSTATIO PHENOMENA. [32.
electrification outside C, This shews that the electrification of A
is exactly equal and opposite to that of the piece of glass, and
that of £ may be shewn in the same way to be equal and opposite
to that of the piece of resin.
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.
82.] Experiment VII. Let the vessel B, charged with a
quantity 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 outside of C. Now let B be made to touch the inside of
0. No change of the external electrification will be observed.
If J9 is now taken out of C without touching it, and removed to
a sufficient distance, it wil] be found that B is completely dis-
charged, 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 (7, 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 (7, and finally removed without touching (7, the charge of
B is completely transferred 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*
* {The diflScultieB which would have to be overcome to make several of the
preceding experimeDts conclusive are so great as to be almost insurmountable. Their
description however serves to illustrate Uie properties of Electricity in a very
striking way. In Experiment V no method is given by which the electrification of
the outer vessel can^be measured.} Tkc, ^ mJUo U4, #i.C*^*vi. ^ ^if/^%* - m4 i^
^;fc trt/u li^ aV. '
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34-] SUMMATION OP ELECTRIC EFFECTS. 37
33.] Before we proceed to the investigation of the law of
electrical force, let us enumerate the facts we have already
established.
By placing any electrified system inside an insulated hollow
conducting 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 com-
munication of electricity 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
electroscope.
We may suppose the electroscope to consist of a strip of gold
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 hoUow vessel wiUi a gold leaf electroscope.
This method was used by Faraday in his very admirable
demonstration of the laws of electrical phenomena'^.
84.] 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 improved, the loss of electrification becomes less. We may
therefore assert that the electrification of a body placed in a
perfectly insulating 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
* < On Static Electrical Indactive Action,' PhiL Mag., 1843, or Exp. Bet., vol. ii.
p. 279.
Digitized by VjOOQ iC
38 ELBCTBOSTATIO PHENOMENA, [$$.
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.
ni- When electrification is produced by friction, or by any
oth^ known method, equal quantities of positive and negative
electrification 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 electrifi-
cation of the parts of the system may be, the electrification of
the whole, as indicated by the gold leaf electroscope, is in-
variably 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 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 ifco speak of any electrified
body as * 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
increase 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
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36.] ELECTRICITY AS A QUANTITT. 39
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 understood, 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 quantities 'Electricity' and 'Potential/
when multiplied together, produce the quantity * Energy.' It is
impossible, therefore, that electricity and eneigy should be
quantities of the same category, for electricity is only one of the
fectors of energy, the other factor being ' Potential.'*
^e^gyt which is the product of these factors, may also be
considered 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 num-
ber 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 electrifi-
cation which, being combined, annul each other, and since
we cannot conceive of two substances annulling each other, a
distinction has been drawn between FV^a TClAg^.riAif.y and
Combined Electricity.
* { It ia ihown afterwrnrds that ' Potential ' ia not of zero dimensiona. }
Digitized by VjOOQ iC ^
40 BLECTBOSTATIO PHENOMENA, [36.
Theory of Two Fluids,
In what 18 called the Theory of Two Fluids, all bodies, in
their unelectrified state, are supposed to be charged with equal
quantities 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 process of electrification, according to this theory,
consists in taking a certain quantity P of positive electricity
from the body A and communicating it to J?, or in taking
a quantity N of negative electricity from B and communicating
it to il, or in some combination 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 + JV is called
the Free electricity, the i*est 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, weight, and elasticity, are not attributed to them by
those who have used the theory for merely mathematical pur-
poses ; 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 in-
telligible 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 veri-
fication of the mathematical 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 efiect of any one of three diflFerent processes which would
lead to the same result. We have already supposed it produced
Digitized by
Google
37-] THEORY OF ONE FLUID. 41
by the transfer of P units of positive electricity from A U) B^
together with the transfer of N units of negative electricity from
JS to il. But if P-f JV units of positive electricity had been
transferred from il to -B, or if P + i\r units of negative electricity
had been transferred from Bio 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 phe-
nomena 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 in^
capable of variation. The first of these alternatives presents no
difficulty to the mere mathematician, who attributes no pro-
perties 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 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 electrifica-
tion exactly equal quantities of the two fluids are transferred in
opposite directions, so that the total quantity of the two fluids
in any body taken together 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 pre-
dicting non-existent phenomena.
Theory of One Fluid.
87.] 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.
Digitized by VjOOQ IC
42 BLECTEOSTATIC PHENOMENA. [37.
perties and name of Ordinary Matter, while the other retams
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 Under-
charged, 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 when so charged,
this quantity of electric fluid must be communicated to it, so
that the whole quantity required to saturate it must be greater
tban 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 supposed
to account for the attraction of gravitation.
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 suflicient 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
* {The apparent man of a body ia inereaeed by a charge of electricity whether
Titxeous or resinoiu (lee Phil. Mag. 1881, v. xi. p. 229).}
Digitized by VjOOQ iC
38.] THEORY OF ONE FLUID* 43
electricity repd 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.
It is manifest however that the heavenly bodies, according to
this theory, if they consisted of matter uncombined with elec-
tricity, 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 con-
tain the normal charge of electricity, and the only action between
them is the residual force lately mentioned. The artificial manner,
however, in which this residual force is introduced is a much
more valid objection to the theory.
In the present treatise I propose, at difierent stages of the in-
vestigation, to test the difierent 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 connected and mathematical foim, 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 Foixe 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,
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 Harris, and is that
Digitized by VjOOQ iC
44 ELECTROSTATIC PHENOMENA, [39.
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 a&
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
determination of the force of gravitation between small bodies,
and was used by Cavendish for this purpose. Coulomb, working
independently 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 researches where small forces have to be
measured. See Art. 215.
89.] 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 I
either body, and we shall suppose that both bodies are so |
suspended in air as to be at a considerable distance from other 1
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 I
electricity, they will repel each other with a force proportional
to the product of e and e'. If either e or e' 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 tf are negative the force is
again repulsive.
We may suppose the first body, A, charged with in units of
positive and n units of negative electricity, which may be con-
ceived separately placed within the body, as in Experiment V. ^
Let the second body, jB, be charged with m' units of positive
and n' units of negative electricity.
Digitized by VjOOQ iC
41.] MEASURBMBNT OF BLBCTEIO POECES. 45
Then each of the m positive units in A will repel each of the
w/ positive units in B with a certain force, say/, making a total
effect equal to mm^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 mn^f.
Similarly the n negative units in A will attract the m' positive
units in B with a force nvi'f, and will repel the Ttf negative units
in B with a force nny.
The total repulsion will therefore be (mm^-^nvf)f; and the
total attraction will be (mn*-^rn/n)f.
The resultant repulsion will be
{mmf + Titt' — mnf — nm^) f or (m — n) (m' — n')/.
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 ee'f, 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 /? "^ and the general expression for the repulsion
between e units and e^ units at distance r will be
fee'r'\
Definition of the Electrostatic Unit of Electricity.
41.] We have hitherto useda 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
principle, and in order that this unit may belong to a general
system we define it so that / may be unity, or in other words —
The electrostatic unit of electricity is that quantity of positive
Digitized by VjOOQ iC
46 ELBCTEOSTATIC PHENOMENA. [43,
electricity which, wJien placed at unit of distance from an equal
quardity^ repeU 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 p - ^g' ^-2 . ^^^
The repulsion between two small bodies cliarged respectively
with e and e' units of electricity is num^rncally 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, e' the numerical values of particular quantities ; if [X] is
the unit of length, and r the numeiical value of the distance ; and
if [jP] is the unit of force, and F the numerical value of the force,
then the equation becomes
whence [Q]=[-^J^*]
This unit is called the Electrostatic Unit of electricity. Other
units may be employed for practical purposes, and in other de-
partments of electrical science, but in the equations of electro-
statics quantities of electricity are understood to be estimated in
electrostatic 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 unceitain, 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
* {in this definition and in the enunciation of the law of electrical action the
medium surrounding the electrified bodies is supposed to be air. See Art. 94.}
Digitized by VjOOQ iC
44-] I^^W OP BLBCTBIO FORCE. 47
capable of carrying charges sufficient to produce measui*able
forces. The action of each body will then produce an effect on
the distribution of elecUicity 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 calcu*
lated 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 gi*eatly simplified by
Sir W. Thomson in his Theory of Electrical Images. See Ai-te.
J 72-175.
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 chai'ge 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 measur-
ing electrical effects, have been greatly improved since the time
of Coulomb, particularly by Sir W. Thomson ; but the perfect
accuracy 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 bya mathematical consideration of the
phenomenon desciibed as Experiment YII, namely, that an elec-
trified conductor By if made to touch the inside of a hollow closed
conductor C and then withdrawn without touching C, is per- '
fectly discharged, 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
mathematical theory given at Arts. 74c, 74d!, 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 elec-
tric phenomena. It may be occupied by air or other bodies, or
it may be a so-called vacuum, from which we have withdrawn
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48 J5LE0TEOSTAT1C PHENOMENA. [45.
every subBtance wbich 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.
But if the body is very small, and its charge also very small,
the electrification of the other bodies will not be sensibly dis-
turbed, and we may consider the position of the body as deter-
mined by its centre of mass. The force acting on the body will
then be propoiiional 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 6, or F=R€
where R depends on th^ 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 @.
Total Electromotive Force and Potential.
45.] If the small body carrying the small charge e be moved
from one given point, A^ to another jB, 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 AB. If the path
forms a complete circuit, and if the total electi*omotive force round
the circuit does not vanish, the electricity cannot be in equi-
librium but a current will be produced. Hence in Electrostatics
the total electromotive force round any closed circuit must be
zero, so that if A and B are two points on the circuit, the total
electromotive force from ^ to £ 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 total electro-
motive force from Aix)B isihfi..fiaiaiBJ!or all paths from A to B.
If jB is taken as a point of reference for all other points, then the
total electromotive force from il to £ is called the Potential of A.
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46.] ELBCTBIO POTENTIAL. 49
It depends only on the position o{ A. In mathematical investi-
gations, B is generaUy 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,
potential, and aHbody charged negatively tends to move in the
opposite direction.
In a conductor the electrification is free to move relatively to
the conductor. K 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 thereforejsannot be
in electrical equilibrium unless every point in it has the same
potential. This^tential is called the Potential of the Conductor.
EquipotentiaZ Surfaces,
46.] K 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 surfjB.ce to
another, because the potential is the same at every point. An
equipotential surface is therefore a surface of equilibrium or a
level surface.
The resultant force at any point of the surface is in the direc-
tion 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 V to the surface V is F— F'.
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 neces-
sarily an equipotential surface. If the electrification of the con-
ductor 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 seri^ of surfaces of lower
potential.
But if (owing to the action of external electrified bodies) some
VOL. I. B
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50 ELBCTEOSTATIC PHENOMENA. [46.
TegionB of the conductor are charged positively and others ne-
gatively, the complete equipotential surface wHl 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
parts and negatively in others, there must be some other electri-
fied 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 particle will continually diminish till
the particle reaches either a negatively charged surface at a poten-
tial 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 smface, 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 charges 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 negatively charged surface at a lower potential. But the whole
interior surface has the same potential. Hence it can have no
charge t.
* {See Arts. 80, 114.}
t {To make the proof rigid it is necessary to state that by Art 80 the force cannot
vanish where the surface is charged, and that by Art 112 the potential cannot have a
maximum or minimum value at a point where Uiere is no electrification.}
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49-1 ELECTRIC TENSION, 51
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 indicating both the direction and the intensity at every
point.
Electric Tension.
48.] Since the surface of a conductor is an equipotential surface,
the resultant intensity is normal to the surface, and it will be
shewn in Art 80 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 intensity and
the density, that is, proportional to the square of the resultant
intensity.
This force, which acts outwards as a tension on every part of the
conductor, will be called el6ctric 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
E %
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52 ELECTROSTATIC PHENOMENA. [5 1.
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
magnets, inequalities of temperature, &c.
Capacity of a Conductor,
50.] If one conductor is insulated while all the surrounding
conductors are kept at the zero potential by being put in commu-
nication with the earth, and if the conductor, when charged with
a quantity E of electricity, has a potential V, the ratio of ^ to F
is called the Canacitv of the condi^ctor. If the conductor is
completely 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 wiU 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. Accu-
mulators 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 supei*ficial density.
PBOPERTTES OF BODIES IN RELATION TO STATICAL ELECTRICITY.
Resistance to the Passage of Electricity through a Body.
51.] When a charge of electricity b 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
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51.] BLBOTBIC EBSISTANCB. 53
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. 362, 364, and 367, which will be explained in treating of
Electric Currents.
All the metals are good conductors, though the resistance of lead
is 12 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.
Many liquids conduct electricity by electrolysis. This mode of
conduction will be considered in Part 11. For the present, we
may regard all liquids containing water and all damp bodies as
conductors, far inferior to the metals but incapable of insulating
a charge of electricity for a sufficient time to be observed. The
resistance 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 dectricity
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 perceptible 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 gase§ takes place, in general, by disruptive
discharge, and does not begin till the electromotive intensity has
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54 ELECTROSTATIC PHENOMENA. [52,
reached a certain vi^ue. The value of the electromotive intensity
which can exist in a dielectric without a discharge taking place
is called the Eectrio Strength of the dielectric. The electric
strength of air diminishes as the pressure is reduced from the
atmospheric 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 intensity
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
therefore an insulator of very great electric strength.
The electric strength of hydrogen is much less than that of air
at the same pressure.
Certain kinds of glass when cold are marvellously perfect in^
sulators, and Sir W. Thomson has preserved charges of electricity
for years in bulbs hermetically sealed. The same glass, however,
becomes a conductor at a temperature below that of boiling water.
Outta-percha, caoutchouc, vulcanite, paraffin, and resins are
good insulators, the resistance of gutta-percha at 75"" F. being
about 6x10^* times that of copper.
Ice, crystals, and solidified electrolytes, are also insulatora.
Certain liquids, such as naphtha, turpentine, and some oils, are
insulators, but inferior to the best solid insulators.
DIELECTBICS.
Specific Inductive Capacity.
52.] All bodies whose insulating power is such that when they
are placed between two conductors at different potentials the
electromotive force acting on them does not immediately dis-
tribute 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 t that he had, before 1773, measured the capacity of
plates of glass, resin, bees- wax, and shellac, and had determined
* {The pressure at which the electrio strength is » Tnimmntn depends on the
shape and size of the vessel in which the gas is contained.}
t {See Eltdrical Sesearehts of the BonourMe Henry ^^Spdi^K],,,^^
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J
53-] ELECTRIC ABSORPTION. 55
the ratios in which their capacities 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 subsiuEuice.
By a more delicate method of measurement Boltzmann succeeded
in observing the variation of the inductive capacities of gases at
different pressures.
This property of dielectrics, which Faraday called Specific In*
ductivft Hitpii^jty^ is also called the Dielectric Constant of the
substance. 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
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 Residual Discharge.
* {Colin and Arons ( Wiedemann's AnnaUn, ▼. 38, p. 13) have investigAied the
speoifio indactive capacities of tome non-ixunlating fluids such as water and alcohol :
they find that these are Tery large ; thns, that of distilled water is about 76 and that of
ethyl alcohol about 26 times that of air.}
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56 ELECTEOSTATIC PHENOMENA. [54.
The instantaneouB discharge appears always to be proportional
to the difference of potentials at the instant of discharge, and
tiie ratio of these quantities is the true capacity of the accumu-
lator ; but if the contact of the discharger is prolonged so as to
include some of the residual dischai^e, the apparent capacity of
the accumulator, calculated firom such a discharge, will be too
great.
The accumulator if chained 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 electrification 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 absorption of electricity, by supposing the dielectric
in some degree heterogeneous.
That the phenomena called Electric Absorption are 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 substance is charged and insulated, the vessel be instan-
taneously 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 t.
In fact it appears from the result of every experiment which
has been tried that in whatever way electrical actions may take
♦ I For a detailed account of the phenomena of Electric absorption, see Wiedemann**
Slektricitat, v. 2, p. 88.}
f £xp. Res., vol. i. series xi. ^ ii. < On the Absolute Charge of Matter,' and $ 1244.
Digitized by VjOOQ iC
55-] DISEUPnVE DISCHABGE. 57
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
BO 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 ordinary 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 mattei* can by any change of state evolve or
render latent one kind of electiicity or the other. He therefore
regarded induction as 'the essential function both in the first
development and the consequent phenomena of electricity.' 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.
Diaruptive Discharge.*
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
permanent rupture of the dielectric.
The electromotive intensity when this takes place is a measure
of what we may call the electric strength of the dielectric.
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
in every case, if the electromotive force be made great enough,
the dielectric gives way and its insulatiog 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.
* See Faraday, Exp, Ees,, vol. i., series xii. and xiii.
•jSo many inyestigations have been made on the passage of electricity through
gases since the first edition of this book was published that the mere enumeration of
them would stretch beyond the limits of a foot-note. A summary of the results
obtained by these researches will be given in the Supplementary Volume.}
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55 ELECTEOSTATIC PHENOMENA. [55.
The Electric Olow.
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
electromotive intensity, 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 intensity in the neighbourhood 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 intensity 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 as 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 intensity which it would have
experienced 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 poi*tion 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
the newly charged particles move off, and so on as long as the body
remains electi*ified.
** { Or dost ? It it doubtful whether air free frt>m dust and aqueous vapour can be
electrified except at yeiy high temperatures; see Supplementary Volume.}
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55-] ELECTEIC GLOW. 59
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
prevent 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
atmosphere, 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
Convective Dischaxge.
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 sun-ounding 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 Hittory of Eleetrieittf^ pp. 117 and 591 ; and Cavenduh'a < Elec-
trical Besearchee,' PhU, Tram,, 1771, % 4, or Art. 125 of EUatrioal Researches of the
Honourable Htnry Cavendish,
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60 ELECTEOSTATIC PHENOMENA. [57.
This and many other differences between positive and negative
electrification must be studied by those who desire to discover
something 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 dischargesj 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,
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.58.] ELBCTBIO PHENOMENA OP TOUBMALINB. 61
80 that if the electricity, for example, is passing along a tube
containing a very small quantity of gas, a number of luminous
disks will be seen arranged transversely at nearly equal intei'vals
along 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 wiU arrange themselves in
closer order. In a tube described by Mr. Qassiot* 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, tiiey must be considered as outside the
domain of the mathematical theory of electricity.
Electric Phenomena of Touvvialin£'f.
58.] Certain crystals of tourmaline, and of other minerals,
possess what may be called Electric Polaiity. Suppose a crystal
of tourmaline to be at a uniform temperature, and apparently
free from electrification on its surface. Let its temperature be
now raised, the crystal remaining insulated. One end will be
found positively and the other end negatively electrified. Let
the surface be deprived of this apparent electrification by means
of a flame or otherwise, then if the crystal be made still hotter,
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
crystallographic 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
hemihedral 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 neutitJize,
* InidUetMal Observer, March 1866.
f { For a fiiUer aooount of this property and the electrification of cryttala by radiant
light and heat, see Wiedemann*$ EUktriciUit, y. 2l, p. 316. \
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62 ELECTEOSTATIC PHENOMENA, [59.
for all external points, the effect of the internal polarity. The
crystal then has no external electrical action, nor any tendency
to change its mode of electrification. But if it be heated or cooled
the interior polarization of each particle of the crystal is altered,
and can no longer be balanced by the superficial electrification,
80 that there is a resultant external action.
Plan of this Treatise.
69.] In the following treatise I propose first to explain the
ordinary theory of electrical action, which considers it as de-
pending only on the electrified bodies and on their relative
position, without 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, tiie theory of the poten-
tial, and the equations of Laplace and Foisson. 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 electrified bodies.
We shall then investigate certain general theorems by which
Green, Qauss, 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 be-
tween 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
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59.] I'LAN OP THIS TREATISE, 63
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.
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 (f^.V^)
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
enei'gy of the medium per unit of volume is half the product of
the electromotive intensity and the electric polarization multi-
plied 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
betweeen 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 pressurejnall
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
electromotive intensity multiplied by the specific inductive
capacity of the medium.
* Exp, Bet., series xi. 1297.
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64 ELECTEOSTATIC PHENOMENA. [6o.
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 procedure to assume the actual existence of this
state of stress, and to follow the assumption into its consequences.
Finding the phrase dectric tenaian used in several vague senses,
I have attempted to confine it to what I conceive to have been
in the minds 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
x(i*?*J^^ foot before a spark passes.
IvetHAH. 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 character 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 magnitude of these forces being equal, and each pro-
portional 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.
* {This Btatement requires modification : the diBtribntion of stress referred to is
only one among many sach distributions whioh will all produce the required effect}
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60.] STRESS IN DIBLECTBICS. 65
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 i^l^fttrift 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 {continue to} flow through
the medium, but the electricity is displaced within the medium
in the direction 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 or diminishes in
the same ratio.
The amount of the displacement is measured by 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 intensity in
producing 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 displacement cannot exceed a certain value without causing
disruptive discharge, they cannot be continued indefinitely in
the same direction, 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
temperature, and does not require an external electromotive force
to produce it. If the interior of a body were in a state of
permanent electric 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
* {If we MBume the views enanouted in the preceding paragraph. }
VOL. I. F
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66 ELEOTROSTATIC PHENOMENA. [60.
external superficial charge could not be detected by any of tiie
ordinary testa, 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 electrification would then be no longer capable
of neutralizing the external effect of the internal polarization,
and an apparent electrification would be observed, as in the case
of tourmaline.
K a charge e is uniformly distributed over the surfSeu^e of a
sphere, the resultant intensity at any point of the medium sur-
rounding the sphere is proportional to the charge e divided
by the square of the distance from the centre of the sphere.
This resultant intensity, according to our theory, is accompanied
by a displacement of electricity in a direction outwards from the
sphere.
If we now draw a concentric spherical surface of radius r, the
whole displacement, E^ through this surface will be proportional
to the resultant intensity multiplied by the area of the spherical
surface. But the resultant intensity is directly as the charge e
and inversely as the square of the radius, while the area of the
surface is directly as the square of the radius.
Hence the whole displacement, E^ is proportional to the charge
6, and is independent of the radius.
To determine the ratio between the charge 6, 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-^hE. If T^
and J^ denote the potentials at the inner and the outer of these
surfaces respectively, the electromotive force by which the
additional displacement is produced is Tf — T^, so that the work
spent in augmenting the displacement is (^- ^2)^^-
If we now make the inner surface coincide with that of the
electrified sphere, and make the radius of the outer infinite, T^
becomes F, the potential of the sphere, and T^ becomes zero, so
that the whole work done in the surrounding medium ia VhE.
But by the ordinary theory, the work done in augmenting the
charge is Vbe, and if this is spent, as we suppose, in augmenting
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6 1.] THEORY PK0P08ED. 67
the displacement, IE =^ le, and since E and e yanish 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
accumulator formed of two conducting plates A and B, separated
by a stratum of a dielectric C. Let TT be a conducting wire
joining A and £, and let us suppose that by the action of an
electromotive force a quantity Q of positive electricity is trans-
ferred along the wire from Bio 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 dis">
placement, as measured by the quantity of electricity forced
across an imaginary section of the 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 B towards j1, so as to cross every section of
the wire, the same quantity of electricity crosses every section
of the dielectric from A towards B by reason of the electric dis^
placement.
The displacements of electricity during the discharge of the
accumulator will be the reverse of these. In the wire the dis-
charge will be Q from AioB, and in the dielectric the displace-
ment 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 recognized, 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
electricity are like those of an incompi*essible fluid, so that the
total quantity within an imaginary fixed closed surface remains
F 2,
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68 ELBCTBOSTATIO PHENOMENA, [62.
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 quan-
tity of electricity within that space. But we must remember
that the ordinary theory takes no account of the electric dis-
placement in the substance 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 con-
ductor 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-
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 phe-
nomena.
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
direction as the intensity, and is numerically equal to the intensity
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62.] THEOET PROPOSED. 69
multiplied by j-K^ where K is the specific inductive capacity of
the dielectric.
That the energy per unit of volume of the dielectric arising
from the electric polarization is half the product of the electro-
motive 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
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
understand 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 elec-
tricity, 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 displaced 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
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70 ELECTEOSTATIO PHENOMENA.
at every instant as much must flow out of any given closed
surface as flows into it.
It follows from this that evegr 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 dis-
tance 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 de-
duced 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 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 intensity 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.
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CHAPTER IL
BLEMENTAEY MATHEMATICAL THEORY OP STATICAL
ELECTRICITY.
Definition of Electricity as a Mathematical Quantity,
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 — c 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, +e 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 un-
equal 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 concep-
tion 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.
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72 ELBCTEOStATICS. [64,
ON ELECTRIC DENSITY.
Distribviion in Three JDimensions.
64.] Definition. The electric volume-density at a given point
in space is the limiting ratio of the quantity of electricity within
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
positive 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 con-
tained within the sphere, when its radius is diminished without
limit.
We shall denote the surface-density by the symbol <r.
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 p^, or that value of p
which would result from the paiticles having the closest contact
of which they are capable. It is manifest that on this theory
Po^= ^'
When (T 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
electric 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.
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65.] TJNIT OF ELEOTBIOITY. 73
Didribution 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.
K A denotes the line-density, then the whole quantity of elec-
tricity on a curve is e= I XdSy where ds is the element of the
curve. Similarly, if o- is the surface-density, the whole quantity
of electricity on the surface is
=/p*
where dSia the element of surface.
If p is the volume-density at any point of space, then the
whole electricity with a certain volume is
e = 1 1 1 pdxdydz,
where dxdydz is the element of volume. The limits of in-
tegration in each case are those of the curve, the surface, or the
portion of space considered.
It is manifest that e, X, 0- and p are quantities differing in kind,
each being one dimension in space lower than the preceding, so
that if Z be a line, the quantities e, ZX, Pa^ and Pp will be all of
the same kind, and if [L] be the unit of length, and [A], [a], [p]
the units of the different kinds of density, [e], [Xa], [X^<r], and
[L^p] 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 AB, be charged with equal
quantities of positive electricity and placed at A and B respect-
ively, 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 *.
K the charge of the body at B were a unit of negative
* {In this definition the dielectric separating the chaiged bodies is supposed to be
air.}
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74 ELECTEOSTATICS. [67.
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 n^ative, 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 ef units At B ib eef, the distance
AB being unity. See Art. 39.
Law of Force between Charged Bodies.
66.] Coulomb shewed by experiment that the force between
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 e^ and placed at a distance r is
eff
^'
We shall prove in Arts. 74 c, 74 cZ, 74 c that this law is the only
one consistent 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 measurements of Coulomb.
Resultant Foixe 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
pp {x— x')dxdydzdx' d}f d^
where x, y, z are the coordinates of a point in the first body at
msh
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68.] EESULTANT INTENSITY AT A POINT. 75
which the electrical density is p, and x\ 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.
Resultant 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.
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 iJe, and let the
direction-cosines of the force be i, m, ti, then we may call R the
resultant electric intensity at the point (a;, y, z).
If X, Y^ Z denote the components of JB, then
X = Rl, Y=:Rm, Z=Rn.
In speaking of the resultant electric 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 jR and
the negative electricity in the opposite direction. Hence the
quantity jR is also called the Electromotive Intensity at the
point {x, y, z).
When we wish to express the fact that the resultant intensity
is a vector, we shall denote it by the German letter (S. If the
body isadielectrio, then, aooordimy to the theory adnpf/^H m
this treatise, the electricity is displaced within i^-, «ft ^V^<*^' ^-^^
* The Electric and Magnetic Intensities correspond, in electncity and magnetism,
to the intensity of g^vity, commonly denoted by g, in the theory of heavy bodies.
M
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76 ELECTROSTATICS. [69.
quantity of electricity^ich is fqrcedjp^the direction of^aOTOSS
unit of area fixed perpendicular to (£ is
' where 2) is the displacement, @ the resultant intensity, and K the
Rjigfiifip. ^pHn<»tive 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
maintained as long as @ acts on the medium.
Lirie-IntegraZ of Electric Intensity y or Electromotive Force
along an Arc of a Curve,
69.] The Electromotive force along a given arc AP of a curve
is numerically measured by the work which would be done by
the electric intensity on a unit of positive electricity carried along
the curve from -4, the beginning, to P, the end of the arc.
If 8 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 €
with the tangent drawn in the positive direction, then the work
done on unit of electricity in moving along the element of the
curve d!« will be jj cos c da,
nd the total electromotive force E will be
E = I Rcos^dsy
he integration being extended from the beginning to the end
f the arc.
K we make use of the components of the intensity, the ex-
iression becomes
Jo ^ ds da da^
If X, y, and Z are such that Xdx + Ydy + Zdz is the complete
lifferential of — F, a function of a;, y, «, then
E=r{Xdx-\-Ydy-\-Zdz)=z-f''drz= Va-VpI
vhere the integration is performed in any way from the point A
0 the point P, whether along the given curve or along any other
ine between A and P.
Digitized by VjOOQ iC
70.] POTENTIAL FUNCTIONS. 77
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 V is 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
intensity is a function of the distances from any number of
points. For if r^ be the distance of one of the points from the
point (ic, 2/, z), and if iij be the repulsion, then
^' = ie,$:-s
Y — 7? X — .^i p
with similar expressions for 1^ and Z^, so that
X^dx + Xdy + Zidz = Ri dr^ ;
and since iZ^ is a function of r^ only, R^ dr^ is an exact differ-
ential of some function of r, , say — TJ.
Similarly for any other force R.^ , acting from a centre at dis-
*^<^ ^2» X^dx+ Y^dy^Z^dz = R^dr^ = -cZTJ.
But X = Xj + Xg + &c., and Y and Z are compounded in the same
way, therefore
Xdx+ Ydy^-Zdz = -dTJ'-cJ!TJ-&c. = -dV.
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
Digitized by VjOOQ iC
78 ELBCTBOSTATICS, [72.
the word Potential. Clausins 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.^homson.
Definition of Potential. The Potential at a Point is the work
which would be done on a unit of positive electricity by the
electric forces if it were placed at that point without disturbing
the electric distribution, and carried from that point to an in^
finite 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
if we put ds for the arc AB we shall have for the intensity re-
solved in the direction of ds,
i£cos€ = — J-;
ds
whence, by assuming db parallel to each of the axes in succession,,
we get
dx dy ' dz*
^dx\ dy
'^t!'
We shall denote the intensity itself, whose magnitude, or
tensor, is R and whose components are X, F, Z, by the German
letter (S, as in Art. 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 intensity acting within the
Digitized by VjOOQ iC
72.] POTENTIAL. 79
conductor. Hence R = 0 throughout the whole space occupied
by the conductor. From this it follows that
dV ^ dV ^ dV ^
and therefore for every point of the conductor
r=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 he J^ and T^
respectively, then the electromotive force along a wire joining
A and B will be 15 — ^
in the direction AB, that is, positive electricity will tend to pass
from the conductor of higher potential to the other.
Potential, in electrical science, has the same rftUtinn to Elec-
Jtricity that Pressure, in Hydrostatics, has to Fluid, or that Tem^
perature, in Thermodynamics^ has to Heat. Electricity, Fluids,
and Heat aU tend to pass from one place to another, if the
Potential, 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 analogies 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 i9
either a substance like water, or a state of agitation like heat.
* {If there is any discontinuity in the potential as we pass from the dielectric to
the condactor it is necessary to state whether the electrified point is brought inside
the conductor or merely to Uie surface. }
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80 ELECTROSTATICS. . [74 a.
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
x\ 2/', / from it, then
J poo po
•6 , e
r* r
Let there be any number of electrified points whose coordinates
*^^ (*i> 2/i» ^i)> (^2> ^2* ^2)9 &®' ^^^ their charges Cj, 62, fee, and
let their distances from the point {x\ y', /) be rj, r2, &c., then
the potential of the system at {x\ y', /) will be
r=s(«-).
Let the electric density at any point (a?, y, z) within an elec-
trified body be p, then the potential due to the body is
V^fff^dxdydz;
where ^ = {(aj-a:')' + (y-2/')' + (^~^)'}*>
the integration being extended throughout the body.
On the Proof of the Law of the Inverse Square,
74 a.] 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 ex-
periments 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, 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
Digitized by VjOOQ iC
74^-] PEOOP OP THE LAW OP POBOE, 81
together, formed an insulated spherical shell concentric with the
globe.
The globe could then be made to communicate with the hemi-
spheres 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
electrometer.
No indication of any charge of the globe could be detected by
the pith ball electrometer, which at that time (1773) was con-
sidered 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 ^V ^^ ^^ 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 ^V there would have
been a charge on the globe equal to yV o{ that of the whole
apparatus, and therefore capable of being detected by the
electrometer.
746.] 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 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
VOL. I. G
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82 BLECTEOSTATIOS. [74 6.
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 shoH wire which made the connection 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 de-
scribed in Art. 219. The case of the electrometer and one of the
electrodes were always connected to earth, and the testing
electrode was connected 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 {the testing electrode remaining in contact with
the globe} then shewed a positive deflection, D.
The negative charge of the brass ball was about yV ^f ^^^ 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. Hence when the ball was put to earth the potential
of the shell, as indicated by the electrometer, was about i\^ of
its original potential.
Digitized by VjOOQ iC
74 c] PROOF OP THE LAW OP FOECB. 83
But if the repulsion had been as r*"^, the potential of the globe
would have been — 0-1478 q of that of the shell by equation (22),
p. 85.
Hence if ±c? be the greatest deflection of the electrometer
which could escape observation, and D the deflection observed in
the second part of the experiment, {since -1478 qV/-^\^ Fmust be
less than d/D^\ q cannot exceed
1 d
±725*
Now even in a rough experiment D was more than 300 d, so
that q cannot exceed -
. 1 o<i tcnro ofy
^21600' '
Theory of the Ea^periment.
74 c] To find the potential at any point due to a uniform
spherical shelly the repulsion between two units of matter being
any given function of the distance.
Let <f> (r) be the repulsion between two units at distance r, and
let/(r) be such that
^(^fir))=.rf,l>{r)dr. (1)
Let the radius of the shell be a, and its surface density a; then,
if a denotes the whole charge of the shell,
o = 47ra*<r. (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
r* = a2 + 62-2a6cos^. (3)
The mass of the element of the shell is
<ra^am6d<f>d$9
and the potential due to this element at the given point is
f'(r)
Ga?B\nd^-^^ded<b\
T
a 2
Digitized by VjOOQ IC
84 ELECTROSTATICS, [74 c.
and this has to be integrated with respect to <^ from ^ = 0 to
^ = 2 ir, which gives
2ir<ra^Bme^^de, (6)
which has to be integrated from $ := 0 to 0 sx v.
Differentiating (3) we find
rdrzsahsmOdO. (7)
Substituting the value of (20 in (6) we obtain
27ta^f{r)dr, (8)
the integral of which is
F=2,r<r|{/(r0-/(r,)}, (^)
where r^ is the greatest value of r, which is always a + 6, and r^
is the least value of r, which is 6— a when the given point is
outside the shell and a—b when it is within the shell.
If we write a for the whole chai^ of the shell, and V for its
potential at the given point, then for a point outside the shell
y=^{f{b + a)-f{b-a)}. ^ (10)
For a point on the shell itself
and for a point inside the shell
^=^if(<' + b)-f(<'-b)i- ^ (12)
We have next to determine the potentials of two concentric
spherical shells, the radii of the outer and inner shells being a
and 6, and their charges a and /9.
Calling the potential of the outer shell A, and that of fche
inner B, we have by what precedes
^ = 2^«^(2a) + ^{/(a + 6)-/(a-ft)}, (13)
^ = ^/(26) + 2^{/(«+6)-/(a-6)}. (14)
In the first part of the experiment the shells communicate by
the short wire and are both raised to the same potential, say V.
« {Strictly /(2 a) -/(O), bnt the condurions arrived at in Art. 74 (^ are not altered
if we write /(2a) -/(O) for/(2a) and/(2&)-/(0) for/(26) aU through,}
Digitized by VjOOQ iC
74^0 PEOOF OF THE LAW OF FOEOE. 86
By putting A = B = V, and solving the equations (13) and
(14) for fi, we find for the charge of the inner shell
fl_„Tr. bf(2a)-a[f(a + b)-f(a-b\^
In the experiment of Cavendish^ the hemispheres forming the
outer shell were removed to. a distance which we may suppose
infinite, and discharged. The potential of the inner shell (or
globe) would then become
B, = ^J{2h). (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 il = 0. In this case we find for the potential
of the inner globe in terms of V
74 rf.] Let us now assume, with Cavendish, that the law of
force is some inverse power of the distance, not differing much
from the inverse square, and let us put /
<^(r) = r«-«; ~J^y (18)
then /(r) = j^r«+i* (19)
K we suppose ^ to be small, we may expand this by the ex-
ponential theorem in the form
/(^) = 1372^1 l + ?logr+ ^(3 logr)* + &c.J; (20)
and if we neglect terms involving q\ equations (16) and (17) be-
come
from which we may determine q in terms of the results of the
experiment. CtvA^, A4. H, J>^ V)-
746.] 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 f.
* {Strictly /(r) -/(O) = j— ^♦•'** if g" be lew than unity.}
+ Mee, CeL, L 2.
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S6 BLBCTEOSTATICS. [74^-
If we suppose that p in equation (15) is always zero, we may
apply the method of Laplace to determine the form of /(r). We
have by (15),
6/(2a)-a/(a + 6) + a/(a-&) = 0.
Differentiating twice with respect to 6, and dividing by a, we
If this equation is generally true
/"(r) = Cq, a constant.
Hence, /'(r) = ^or + Ci;
and by (1) J%{r)dr ^^-^ = C,+ ^.
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 fimction of the distance, it is the only one consistent
with the fact that similar surfaces can be electrified so as to
have similar electrical properties, {so that the lines of force are
similai"}.
For if the force were any function of the distance except a
power of the distance, the ratio of the forces 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 hypo-
thesis as to the constitution of the electric fiuid, it is impossible for
the distribution of electricity to be accurately similar in two con-
ductors 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 f, but that as soon as the particles
come very close together, their repulsion begins to increase at a
much greater rate with any further diminution of their distance.
* [Electrical Beteareha of the Bon, E. Cavendish, pp. 27, 28.}
t {Idem,Note2, p. 370.)
Digitized by VjOOQ iC
76.] ELBCTEIO INDUCTION. 87
Surfdce-IrUegral of Electric iTiduction, and Electric
Displacement through a surface.
75.] Let R be the resultant intensity at any point of the
surface, and c the angle which R makes with the normal drawn
towards the positive side of the surface, then R cos € 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, i
:^KRcoB€d8.
Since we do not at present consider any dielectric except air,
jsr= 1.
We may, however, avoid introducing at this stage the theory
of electric displacement, by calling Roos^dS the Induction
through the element dS. This quantity is well known in
mathematical physics, but the name of induction is borrowed
firom Faraday. The surface-integral of induction is
//■
RooatdS,
and it appears by Art. 21, that ii Xy Y, Z are the components
of i2, and if these quantities are continuous within a region
bounded by a closed surface 8, the induction reckoned from
within outwards is ** r i
the integration being extended through the whole space within
the surface.
Induction through a Closed Surface due to a dngle
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
intensity at that point is ii = er"^ 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
Digitized by VjOOQ iC
88 ELECTEOSTATICS. [76.
from the surface, cos c will be positive, and where it enters, cos €
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(o from the surface of
the sphere, and small elements dSi^ dS^, &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 f,
ciS= +r^sec€da);
and, since R = ev^ we shall have
RcosfdS = ±ed(o;
the positive sign being taken when r issues from the surface, and
the negative when it enters it.
K the point 0 is without the closed surface, the positive values
are equal in number to the negative ones, so that for any
direction of r, 2 ii cos c c?5 = 0,
and therefore / / -R cos € d S = 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 c cZo),
and then has an equal number of entrances and issues, so that in
this case ^Rcos€dS=^ edio.
Extending the integration over the whole closed surface, we
shall include the whole of the spherical surface, the area of which
is 4ir, so that r r p^
I lii cos €d/S = e I jdoo = 4ire.
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 -ne when 0 is within
the surface.
{Since iiTair the displacement is equal to the induction divided
by 47r, the displacement through a closed surface, reckoned out*
wards, is equal to the electricity within the surface.
CoroUaiTf. 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 » is the solid angle subtended by the closed curve
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77-] EQUATIONS OF LAPLACE AND POISSON. 89
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 the EquatioTis of LapUice and Paisson.
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
6|, 62, &c. within the surface, and e^^ e^^ &c. without the surface,
we shall have /. /.
/ Rco8€d8== 4ire;
where e denotes the algebraical sum of the quantities of elec-
tricity at all the centres of force within the closed surface, that
is, the total electricity within the surface, resinous electricity
being reckoned negative.
If the electricity is so distributed within the surface that the
density is nowhere infinite, we shall have by Art 64,
and by Art. 75,
K we take as the closed surface that of the element of volume
dxdydz, we shall have, by equating these expressions,
dX dY dZ^^^
dx dy dz '^^
and if a potential V exists, we find by Art. 71,
d^F dW d^r ^
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
d^V dJ'V cP7 , ^™
and we may express Poisson's equation in words by saying that
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^/
90 BLEC3TE0STATICS. [78 a.
^jA the electric density multiplied by 4ir is the concentration of the
,£oten.tiaJ. Where there is no electrification, the potential has no
concentration, and this is the interpretation of Laplace's equation.
By Art. 72, F is constant within a conductor. . Hence within
a conductor the volume-density is zero, an(^ the whole charge
must be on the surface.
If we suppose that in the superficial and ^near distributions
of electricity the volume- density p remains finite, and that the
electricity 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 inter-
preted in accordance with the actual circumstances.
Variation of the Potential at a Charged Surfax^e.
78 a.] The potential function, F, must be physically continuous
in the sense defined in Art. 7, except at the bounding surface of
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 F with respect to x, y, or z may be
discontinuous, and, by Art. 8, the points at which this discon-
tinuity occurs must lie on a surface, the equation of which may
be expressed in the form
^^^Y'\ .1 (^ = 0 (a:, y, «) = 0. (1)
*^>^<^^^ IThis surface separates the region in which (f> is negative firom the
region in which 0 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 <f> = 0, and which may
be said to belong to both regions,
r,+c=v„ (2)
where C is the constant excess of potential, if any, in the sub-
stance on the positive side of the surface.
Let 2, m, n be the direction-cosines of the normal v^ drawn
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^'^
78 6.] POTENTIAL NEAR A CHABGED SUEFACE. 91
from a given point of the surface into the positive region. Those
of the normal vi drawn from the same point into the negative
region will be — Z, — m, and —71.
The rates of variation of V along the normals are
^ = -Z^-m^-n^, (3)
dvi dx dy dz ' ^ '
dV, jdT^ dV^ dX ,.v
dv^ dx dy dz ^ ^
Let any line be drawn on the surface, and let its length, measured
from a fixed point in it, be 8, then at every point of the sur£Eu;e,
and therefore at every point of this line, TJ— If = (7. DiflFeren-
tiating this equation with respect to 8, we get
fdJ^_d^.<^ fd^ d^^dy_^fd^ jlll^^-n. (^\
^dx dx^ds^^dy dy^ds^^dz dz^da'^ ' ^^
and since the normal is perpendicular to this line
,dx dy dz ^ ,^.
\A/0 UrO XJVO
From (3), (4), (6), (6) we find
^_^=i(^+^). . (7)
dx dx ^di^i 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.
78 6.] 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.
• |Sinoe (5) mnd (6) are trnefor an infinite number of values of ^- : ^ : ^ » we have
15^1? rf^_d^ d^^d^
dx dx dy dy dz Az ,dV^ ^^\^^l^* ^\ ^«/^ ^»\
and therefore by equations (8) and (4) each of these ratios — ^ + ~ I •
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92 BLBCTBOSTATICS. [78 C.
The surface integral,
I jRcosedS^
extended over this surface, is equal toi-ne, where e is the quantity
of electricity within the closed surface.
Proceeding as in Art. 21, we find
+JJ{l(X,-X,)+m{Y,-y;) + niZ,-Z,)}dS, (10)
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),
4.. =fffA.pd.dyd.-ff{§^ + g) dS. (11)
But by the definition of the volume-density, p, and the surface-
density, <r, /•/•/• rr
4 7rc= 4l1[ pdxdydz-^iiT <rd8. (12)
Hence, comparing the last terms of these two equations,
dV dK ^ ^ ,,^,
a^i dv^ ^ '
This equation is called the chajaoteiiflti(L^uation of Tat an
electrified surface of which the surface-density is o-.
78 c] If F is a function of a?, y, z which, throughout a given
continuous region of space, satisfies Laplace's equation
dW dW d^_^
dx' ■*" dy^ ■*" rf^ ^ '
and if throughout a finite portion of this region Fis constant and
equal to (7, then F must be constant and equal to C throughout
the whole region in which Laplace's equation is satisfied*^.
If F is not equal to C throughout the whole region, let 8 be
the surface which bounds the finite portion within which F= C.
At the surface S, F= G,
Let r be a normal drawn outwards from the surface 8. Since
8 is the boundary of the continuous region for which F= (7, the
value of Fas we travel from the surface along the normal begins
* {It would perhaps be clearer to say that the potential is eqnal to C at any point
which can be reached from the region of constant potential without passing through
electricity.}
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79-] FOECE ACTING ON A CHARGED SURFACE. 93
dV
to differ from C, Hence -5— just outside the surface may be
positive 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
and TT- = 0.
dv
Hence, at every point of the surfioce except certain boundary
1^^> dV dV, ^ .
is a finite quantity, positive or negative^ and therefore the surface
S has a contiauous distribution of electricity over all parts of it
except certain boundary lines which separate positively from
negatively 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 F = C, includes the whole of the continuous
region within which Laplace*s equation is satisfied.
Force Acting on a Charged Surface.
79.] The general expressions for the components of the force
acting on a charged body parallel to the three axes are of the form
A = fffpXdxdydz, (14)
with similar expressions for B and C, the components parallel to
^and^.
But at a charged surface p is infinite, and X may be discon-
tinuous, so 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 discontiauous, but that
it changes continuously from X^ to X^ while x changes from x-^
to x^, IS the result of our calculation gives a definite limiting
value for the force when x^-^x^ is diminished without limit, we
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94 ELEOTEOSTATICS. [79.
may consider it correct when a^ = ^> ^Jid the charged surface
has no thickness.
Substituting for p its value as found in Art 77,
Integrating this expression with respect to x from a; = o^ to
a; = 0^2 it becomes
This is the value of A for a stratum parallel to yz of which the
thickness is ajg— ^i*
Since Zand Z are continuous, -^ + -^ is finite, and since X
is also finite, ^
L
xi ^dy dz^
where C is the greatest value of (y- + ■-r')X between x=^ x^
and a? = aj^, ^
Hence when ajg— aJi is diminished without limit this term must
ultimately vanish, leaving
^={fi„{^^-Xi')dydz, (17)
where X^ is the value of X on the negative and X^ on the positive
side of the surface.
dV dV
But by Art. 786, Zg-Z^ = ^ -^«= 4 7r<r, (18)
so that we may write
A =JJ\ {X^^X^adydz. (19)
Here dydz is the element of the surface, a is the surface-density,
and i (Zj + Xj) is the arithmetical mean of the electromotive in-
tensities 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
electromotive intensities on the two sides of the surface.
Since the other two components of the electromotive intensity
are not discontinuous, there can be no ambiguity in estimating
the corresponding components of the force acting on the surface.
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8o.] CHAEGED SUKFAOB OP A CONDUCTOR. 95
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 expressions for the components of the force on the element
of jurface^ ^ ^ i(Zi + Z,)<rd5,x '
B = \{Y,+Y^)ad8A (20)
Charged Surface of a Conductor.
80.] We have ahready shewn (Art. 72) that throughout the
substance of a conductor in electric equilibrium X = F = Z = 0,
and therefore V is constant.
„ dX dY dZ ^
and therefore p must be zero throughout the substance of the
conductor, or there can be no electricity in the interior of the
conductor.
Hence a superficial distribution of electricity is the only
possible distribution in a conductor in equilibriunL
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
direction of the normal and equal to 4 tto-, acting outwards from
the conductor.
This relation between the surface-density and the resultant in-
tensity close to the surface of a conductor is known as Conlomb'a-
LaWj Coulomb having ascertained by experiment that the elec-
tromotive intensity near a given point of the surface of a con-
ductor is normal to the surface and proportional to the surface-
density at the given point. The numerical relation
JZ = 4gg
was established by Poisson.
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,)
iRadS ^ 2i:<T^d8 = :^R^dS.
OTT
This force acts along the normal outwards from the conductor,
whether the charge of the surface is positive or negative.
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96 ELECTBOSTATICS. [8 1.
Its value in dynes per square centimetre is
8 If
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 da 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 = C(r, and the resultant electric intensity close to the
surface will be ^^
4ir<r = 4ir— •
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 {since a finite charge would make the potential
infinite} 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 n^ligible 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 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.
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82.] LINES OF POBCB, 97
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,
by § 80, 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 ftorraflpondlnff
pointaj)n 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
//■
jRcos€(2i8f for the two ends must be equal in magnitude
but opposite in sign.
If these surfaces are the surfaces of conductors
€ = 0 and jR = — 4iro-,
and ffR cos c d8 becomes - 4 ir fja- dS, or the charge of the sur-
face multiplied by 4 tt t-
Hence ^e 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.
Several important results may be deduced from the properties
of lines of force.
* From awX^K, a tube. Faraday uses (8271) the tenn < Sphondyloid * in the same
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f {R here is drawn outwards from the tvhe,}
VOL. I. H
98 ELECTROSTATICS. [82.
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 0I
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
conducting vessel, provided no charged body be placed inside it.
If a conductor within a closed hollow conducting vessel is
placed in communication with the vessel, its potential becomes
the same as that of the vessel, and its surface becomes con-
tinuous with the inner surface of the vessel. The conductor is
therefore free from charge.
If we suppose any charged surface divided into elementary
portions such that the charge of each element is unity, and ii
solenoids having these elements for their bases are drawn through
the field of force, then the surface-integral for any other surface
will be represented 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.
Wo 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 ElectricTnduction.
In the ordinary cases the lines of induction indicate the direc-
tion 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,
however, in which it is important to remember that these lines
indicate primarily the induction, and that the intensity is
directly indicated by the equipotential surfaces, being normal
to these surfaces and inversely propoitional to the distances
of consecutive surfaces.
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836.] SPECinO INDUCTIVE CAPACITY. 99
On Specific Inductive Capacity.
83 a.] In the preceding investigation of surfaoe-integralB we
have adopted the ordinary conception of direct action at a dis-
tance, and have not taken into consideration any effects de-
pending 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 intensity
into the coefficient of specific inductive capacity of that medium.
If we denote this coefficient by K, then in every part of the in-
vestigation of surface-integrals we must multiply X, Y^ and Z
by K, so that the equation of Poisson will become
dx' dx dydy dz dz ^ • w
At the surface of separation of two media whose inductive
capacities are iT^ and K2, and in which the potentials are T^ and
T^, the characteristic equation may be written
where r^, rg, 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.
83 b.] If we begin with the actual distribution of the potential
and deduce from it the volume density p' and the surface density
a on the hypothesis that K is everywhere equal to unity, we
* {See note ftt the end of thii ohapter.}
H 2
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100 ELBOTEOSTATICS. [83 b.
may call p^ the apparent volume density and a 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 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 dis-
tinguish 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,
Comparing this with the equation (1) above, we find
, - „,, dKdV dKdV dKdV „ ,,,
4,(p-irp) + -^^+^^ + -^^ = 0. (4)
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 everywhere equal to unity.
The apparent surface charge, </, is that deduced from the
electrical forces in the neighbourhood of the surface, using the
ordinary characteristic equation
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.
Hence ^13- + -K^- -=-? = 0,
* dv^ * dv^ '
The surface-density a 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
jluring the action of the inducing force the apparent electrifica-
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836.] SPECIFIC INDUCTIVE CAPACITY. 101
tion 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 o-'^.
APPENDIX TO CHAPTER II.
^^^ l^^)
The equations
are the expressions of the condition that the displacement across any
closed surface is 47r times the quantity of electricity inside it. The first
equation follows at once if we apply this principle to a parallelepiped
whose faces are at right angles to the co-ordinate axes, and the second if
we apply it to a cylinder enclosing a portion of the charged surface.
If we anticipate the results of the next chapter, we can deduce these
equations directly from Faraday's definition of specific inductive capacity.
Let us take the case of a condenser consisting of two infinite parallel
plates. Let F, , F, be the potentials of the plates respectively, d the
distance between them, and £ the charge on an area A of one of the
plates, then, if K is the specific inductive capacity of the dielectric
separating them,
V — V
ind
Q, the energy of the system, is by Art. 84 equal to
or if F is.the electromotiTe intennty at any point between tlie plates
Q=^KAdF*.
If we r^fard the energy as resident in the dielectric there will be
Q/Ad units of energy per unit of volume, so that the energy per unit
volume equals EF*/Stt. This result will be true when the field is not
• See Fandfty*! 'Remarks on Static Induction,* Proeeeding$ of the Soyal In-
siiimiion, Feb. 12, 1S58.
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102 APPENDIX TO OHAPTBR II.
uniform, so that if Q denotes the energy in any electric field
^ = ± fffKF^dxdydz
Let us suppose that the potential at any point of the field is increased
by a small quantity h 7 when 5 T is an arbitrary function of a;, y, «, then
hQ, the variation in the energy, is given by the equation
this, by Green's Theorem,
where dv^ and dv^ denote elements of the normal to the surfieu^ drawn from
the first to the second and from the second to the first medium respectively.
But by (Arts. 85, 86)
hQ = S(e« V) = ffahYdS^ fffph Ydxdydz,
and since 5 F is arbitrary we must have
dv^ * dv^
which are the equations in the text.
In Faraday's experiment the flame may be regarded as a conductor in
connexion with the earth, the effect of the dielectric may be represented
by an apparent electrification over its surface, this apparent electrifica-
tion acting on the conducting flame will attract the electricity of the
opposite sign, which will spread over the surface of the dielectric while
it will drive the electricity of the same sign through the flame to
earth. Thus over the surface of the dielectric there will be a real elec-
trification masking the effect of the apparent one ; when the inducing force
is removed the apparent electrification will disappear but the real electri-
fication will remain and will no longer be masked by the apparent
electrification.
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CHAPTER in.
ON ELECTRICAL WOEK AND ENEEGY IN A SYSTEM
OP C0NDUCT0B8.
84] On the Work 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
definition of potential (Art. 70), Vbe.
The effect of this operation is to increase the charge of the
given part of the system by 6e, 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
W = l(Jrbe); (1)
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 F 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 F+ 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 ti to 1.
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104 STSTBM OP CONDUCTORS. [85 a.
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 tiF, and we may represent
the process of charging by supposing n to increase continuously
from 0 to 1. .
While n increases from nU} n-^-hn^ any portion of the system
whose final charge is e, and whose final potential is F, receives
an increment of charge «d7i, its potential being n F, so that the
work done on it during this operation is eVnhn.
Hence the whole work done in charging the system is
2(6F) ^ndn = 12(6F), (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
Tr=J2(«F) (3)
the electric energy of the system, expressed in terms of the charges
of the different parts of the system and their potentials.
85 a.] Let us next suppose that the system passes from the
state {€, F)to the state (c', V) 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 poi*tion of the system
is c + 7i(c'— 6), its potential will be F+n(F— F), and the work
done in altering the charge of this portion will be
/'
(e'-e) [F+n{r-7)]dw = J («'_<,) (7'+ 7);
so that if we denote by TP the energy of the system in the state
{e\V) Tr'-Tr=4S(€'-e)(F+F). (4)
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86.] , BECIPBOCAL RELATIONS. 105
But Tr=42(<?7),
and W' = i2{^r).
. Substituting these values in equation (4), we find
2(6r)=2(e'7). y (6)
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 states of the system, we may deduce
a number of useful results.
856.] 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,
F'«Tr=42(e' + e)(7'-F). (6)
If the increments are infinitesimal, we may write (4) and (6)
dW^:^{Vbe) = 2{ebV)i (7)
and if we denote by "R^ and Tfp the expressions for W in terms
of the chaiges and the potentials of the system respectively, and
by A^, 6^, and T^ a particular conductor of the system, its charge,
and its potential, then
86.] If in any fixed system of conductors, any one of them,
which we may denote by A^, is without charge, both in the initial
and final state, then for that conductor e^ = 0, and e/ = 0, so
that the terms depending on A^ vanish from both members of
equation (5).
If another conductor, say il« , is at potential zero in both states
of the system, then T^ = 0 and T^^ = 0, so that the terms de-
pending on A^ vanish from both members of equation (5).
If, therefore, all the conductors except two, A^, and A^, are
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106 SYSTEM OP CONDUCTORS. [86.
either insulated and without charge, or else connected to the
earth, equation (5) is reduced to the form
eX + eX--e;V,-^e:j:. (10)
If in the initial state
«^ = 1 and 6, = 0,
and in the final state
e/ = 0 and e/ = 1,
equation (10) becomes TJ^'= Tf; (11)
or if a unit charge communicated to A^ raises A^ when insulated
to a potential F, then a unit charge communicated to A^ will
raise A^ when insulated to the same potential F, 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 poten-
tial 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 solutions 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"^, 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-^.
Let us next suppose that in the initial state
T^= 1 and Tf = 0,
and in the final state
T^= 0 and T^'= 1,
equation (10) becomes e, = e/ ; (1 2)
or if, when ^r is raised to unit potential, a charge e is induced
on A, put to earth, then if il, is raised to unit potential, an equal
charge e will be induced on ^r put to earth.
Let us suppose in the third place, that in the initial state
T? = 1 and c, = 0,
and that in the final state
Tj:'=0 and e/= 1,
equation (10) becomes in this case
g/+T:=0. (13),
Hence if when A, is without charge, the operation of charging
A^ to potential unity raises A^ to potential F, then if ^,. is kept
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87.] COEPPIOIBNTS OP POTENTIAL AND OF INDUCTION. 107
at potential zero, a unit charge communicated to A^ will induce
on A^ a negative charge, the numerical value of which is V.
In all these cases we may suppose some of the other con-
ductors 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 A, of the system.
Let ri^ be the charge of A^ under these circumstances. Then
if we suppose A, without charge, and the other bodies raised each
to a different potential, the potential of A, will be
jr=.^l(n,Vr). (14)
Thus if we have ascertained the surface density at any given
point of a hollow conducting vessel at zero potential 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.
Theory of a system of condvjctors.
87.] Let ^1, ^2> . • — 4, be n conductors of any form ; let e^^e^^
.,.f« be their charges ; and Tf , TJ, ... TJ[ 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
^.'=\Pu^i'\'Pi^i^i'^\P22^^'^Pi^e^'^P2^e^^\P^z^-^^^^ (15)
The suffix e indicates that W is to be expressed as a function
Digitized by VjOOQ iC
108
SYSTEM OP CONDUCTORS.
[87.
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
zero to the given potential, and since this work is spent in
increasing TT, we have only to differentiate WJ with respect to the
charge of the given conductor to obtain its potential y We thus
obtain /
+Al«r
^B^PU^
+ ;>«•««
(IG)
a system of n linear equations which express the n potentials in
terms of the n charges.
The coefficients p^, fee., are called coefficients of potential. Each
has two suffixes, the first corresponding to that of the charge,
and the second to that of the potential
The coefficient p^^, 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 p^^, in which the two suffixes are different,
denotes the potential of A, when A^, receives a charge unity, the
charge of each of the other conductors, except A^^ being zero.
We have already proved in Art. 86 that p^^ = jt?,^, but we may
prove it more briefly by considering that
dV, d dW: ddK dK
^" "■ del "de.de, ^ de, de ~ de. ""^•^
(17)
The number of different coefficients with two different suffixes
is therefore i ti (n— 1), being one for each pair of conductors.
By solving the equations (16) for e^, e^, &c., we obtain
equations giving the charges in terms of the potentials
«i = ?iili'— +?iflir— +?i»^> ^
n
^n= ?nlK- +?«X- +?»^«» J
(18)
•^ 3^ 6rHM*V ^»^ ^ ^ T^M-^ ^Z'
//e.
Digitized by VjOOQ iC
87.] COEFFICIENTS OF POTENTIAL AND OF INDUCTION. 109
We have in this case also g^, = q^, for
_cUr _ d dWy _ d dWy _ de, _ , V
By substituting the values of the charges in the equation for
the electric energy
W=\[e,V,+ ...+e,%...+e,V,\ (20)
•we obtain an expression for the energy in terms of the potentials
+ gi3K^+?23^K+i?33l?' + &C, (21)
A coefficient in which the two suffixes are the same is called
the Electric Capacitv 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 g^,, denotes the charge of A^ when A^ is raised to
potential unity, the potential of all the conductors except A,
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 con-
ductor when no other conductor or electrified body is within a
finite distance of the conductor referred to.
It is sometimes convenienfc^ when we are dealing with capacities
and coefficients of induction only, to write them in the form [-4.P],
this symbol being understood to denote the charge on A when
Digitized by VjOOQ iC
110 SYSTEM OP CONDUCTORS. [88.
P is raised to unit potential {the other conductors being all at
zero potential}.
In like manner [(J. + 5) . {P-{-Q)\ would denote the charge on
A-\'B when P and Q are both raised to potential I ; and it is
manifest that since
[(J+£).(P+Q)] = [^.P] + [^.(2] + [5.P] + [B.Q]
= [(P+Q).(^ + 5)].
the compound symbols may be combined by addition and multi-
plication as if they were symbols of quantity.
The symbol \A . A] denotes the charge on A when the potential
of ^ is 1, that is to say, the capacity of A.
In like manner [{A + ^) . (.4 + 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
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 J. is Tj and that of
B is T^, the sum T^+ 1^ has no physical meaning bearing on the
phenomena, though TJ^— TJ represents the electromotive force
from A U) B.
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 :
[A.B]^\[liA^B).{A-\-B)\^k[A.A^^\[B.B\
Dvmensione 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
represented as the reciprocal of a line.
Digitized by VjOOQ iC
896.]
PROPEBTIBS OP THE COBFPIOIBNTS.
Ill
On certain conditions which the coejfficients must satisfy,
89 a.] In the first place, since the electric energy of a system
is an essentially positive quantity, its expression as a quadratic
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 numbgr, and
may be written
Pn > 0, \
PiuPu
P-m P22
>0,
Pll"'Pln
> 0.
(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
sjrmmetrically from the coefficients belonging to any combin-
ation of the n conductors, and the number of these combinations
is 2««1.
Only n, however, of the conditions so found can be inde-
pendent.
The coefficients of capacity and induction are subject to con-
ditions of the same form.
89 6.] The coejfficients of potential are all positive, bid none
of the coefficients ^„ is greater than p^ or ^„.
For let a charge unity be communicated to A^ the other con-
ductors being uncharged. A system of equipotential surfaces
will be formed. Of these one will be the surface of A^ and its
potential will be p^r If A, is placed in a hollow excavated in
J ^ 80 as to be completely enclosed by it, then the potential of
Ag will also be ^^.
If, however. A, is outside of A^ its potential p^ will lie between
p„ and zero.
* See Williamsun's DifferenHal Caleulue, 8rd edition, p. 407.
Digitized by VjOOQ iC
112 SYSTEM OP CONDUCTORS. [89 c.
For consider the lines of force issuing from the charged con-
ductor A^. 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 potential. 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 p^, that of the charged
body A^ , the lowest must be that of space at an infinite distance,
which is zero, and all the other potentials such as p^, must lie
between p^r and zero.
If A^ completely surrounds A^^ then p„ =^Pri-
89 c.] None of the coefficients of induction are positive^ and the
sum of all those hdon^ng to a single conductor is n^t
numerically greater than the coeffixdent of capacity of that
conductor^ which is always positive.
For let A^ be maintained at potential unity while all the other
conductors are kept at potential zero, then the charge on ii^
is qrr, and that on any other conductor A^iB q^.
The number of lines of force which issue from Ar is qrv 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 potential zero.
No line of force can issue from any of the other conductors
such as Ag, because no part of the field has a lower potential
than Ag. If il, is completely cut off from A^ by the closed surface
of one of the conductors, then q^, is zero. If -4, is not thus cut
off, qrt is a negative quantity.
If one of the conductors A^ completely surrounds -A,., then all
the lines of force from A^ fall on A^ and the conductors within
it, and the sum of the coefficients of induction of these con-
ductors with respect to -4,. wiU be equal to qrr with its sign
changed. But if ilr is not completely surrounded by a conductor
Digitized by VjOOQ iC
Sgd.] PROPEBTIES OP THE COEPPICIENTS. 113
the arithmetical sum of the coefficients of induction q„, &c. will
be less than q„.
We have deduced these two theorems independently by means
of electrical considerations. We may leave it to the mathe-
matical student to determine whether one is a mathematical
consequence of the other.
89 d.] When there is only ^ne conductor in the field its
coefficient of potential qnjteelf^Jbbe.rftfiipi'^^^l ^f ^fa» ftfl.pfi/»if.y.
The centre of mass of the electricity when there are no ex-
ternal forces is called the electric centre of the conductor. If
the conductor 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 #. /»« #» a^.
;(• + ?) •^JC-»?)*i
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 the electric centres, by more than ; a and b being
the greatest distances of any part of the surfaces of the bodies
from their respective electric centres.
#
* {For let p be the density of the electricity at any point, then if we take the line
joining the electric centre to P as the axis of z, the potential at P is
fff'-^ -Iff' I ^ i^'^^^^ * - i -*"•
where e is the distance of P from the electric centre. The first term equals e/c, the
second Tanishes since the origin is the electric centre, and the greatest yalae of the
*^^ {-^^ 1.4aK^. Digitized by GoOglc
114 SYSTEM OP CONDUCTORS. [89^-
896.] 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 {having the same specific inductive capacity as air}
free from charge in any part, then when one of the conductors,
ill, receives a charge Cj, the distribution of the electricity on the
conductors of the system will not be disturbed by £, as £ is still
without charge in any part, and the electric energy of the system
will be simply i ^^K = J e^^p,^
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
i e^^Pii must diminish.
But ei remains constant, therefore Pn must diminish.
Also if B increases by another body b being placed in contact
with it, ^ji will be further diminished.
For let us first suppose that there is no electric communication
between B and 6; the introduction of the new body b will
diminish p^^. Now let a communication be opened between B
and 6. 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 p^.
third 18 when the electricity is concentrated at the pointi for which the third term
inside the bracket has its greatest value, which is a*/^, thos the greatest value of the
third term is ea^/e^; the least value of this term is when the electricity is concen-
trated at the points for which the third term inside the backet has its greatest nega-
tive value which is ^\a^/c^ ; thus the least value of the third term is ^\t€?/<?.
The result at the end of Art. 89 d may be deduced as foUows. Suppose the chnrge
is on the first conductor, then the potential due to the electricity on this conductor
by the above is less than ^ ^s
where R is the distance of the point from the electric centre of the first conductor ;
in the second term if we are only proceeding as far as c*, we may put i? » c for any
point on the second conductor. The first term represents the potential to which the
second conductor is raised by a char^ e at the electric centre of the first, but by
Art. 86, this is the same as the potential at the electric centre of the first due to a
charge e on the second conductor, but we have just seen that this must be less than
thus the potential of the second conductor due to a charge e on the first must be less
than f « ...
e c (a* + ft')
This however is not in general a very dose f4>proximation to the mutual potential
of two conductors. I
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90 6.] APPEOXIMATE VALUES OP THE COEFFICIENTS. 115
Hence the diminution of p^ by the body B is greater than
that which would be produced by any conductor the surface of
which can be inscribed in B, and less than that produced by any
conductor the surface of which can be described about B.
We shall shew in Chapter XI, that a sphere of diameter 6 at a
distance r, great compared with 6, diminishes the value of p^
by a quantity which is approximately i -^* •
Hence if the body B is of any other %ure, and if 6 is its
greatest diameter, the diminution of the value of ^^ must be less
b^
than 4 -7 •
Hence if the greatest diameter of jS is so small compared with
its distance from A^ that we may neglect quantities of the order
b^
I -J , we may consider the reciprocal of the capacity of A^ when
alone in the field as a sufficient approximation to p^.
90a.] Let us therefore suppose that the capacity of Ai when
alone in the field is iT^, and that of A^, K^^ and let the mean
distance between A^ and A2 be r, where r is very great compared
with the greatest dimensions of Ai and A^, then we may write
_ 1 _ 1 _ 1
1^11 — ~J^~ * Pl2 — ~ ' P22 — ^ »
Hence ?„ = K^ (1 -ZiiT^r-^j-S
Of these coefficients q^ and ^22 ^^^ ^^^ capacities of A^ 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.
* {S6eaqaatioii(43),Art.H6.}
I 2
Digitized by VjOOQ iC
116 SYSTEM OF CONDUOTOBS. [QO 6.
Let L be the capacity of A^ N that of B^ and M the coefficient
of mutual induction. (We must remember that M is essentially
negative, so that the numerical values of Z+if and M-^^N are
less than L and N.)
Let us suppose tiiat a and h 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 ah when
alone be 2, n, m. Let us calculate the effect of one of the
condensers on the coefficients of the other.
Let D=:LN-AP, and d^ln-^m^;
then the coefficients of potential for each condenser by itself are
Paa= -D-W, p^= dr^n,
Pab = -D^^M, p^ = -d-^m,
Pbb= J>~^L, pf^= drH.
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 2i-\ so that
PAa = PAh =PBa = 1>B& = -B~*-
The equations of potential are therefore
]i= D-WeA-Jy-^MeB-^^R^'e^ + R-^,
1^ = - D-Wca + ly^Les + R-'ea + U-"'«fc ,
K = R'^eA + R^^B + drH Ca — d^hne,, ,
Tj; = R-^eA+R-^eB—d-^mea-^d'^le,,.
Solving these equations for the charges, we find
a - r^ r , (L^M)^{l+2m + n)
qAA-^ - ^-^ jR2^{L + 2M+J>/){U2m + ny
\r u^ (Z-hJf)(Jf+iy)(Z-f2m + n)
g^ =ir = Jf + ^2— (xT2if.f^(f-f2m.fti) '
*^« " J?^-(Z + 2Jf+iv)(i + 2m + n)'
_ JZ(Z-fil/)(m + n)
where i', if, iV' are what Z, M, N become when the second con-
denser is brought into the field.
Digitized by VjOOQ iC
91.] APPROXIMATE VALUES OP THE COBFHCIENTS. 117
If only one conductor, a, is brought into the field, m = n = 0,and
If there are only the two simple conductor, A and a,
Jf=J\r=m = 7i = 0,
, r , LH RLl
and qAA = ^+ ^^ — j^^ qAa
expressions which agree with those found in Art. 90 a.
The quantity L + 2if +i\r is the total charge of the condenser
when its electrodes are at potential 1. It cannot exceed half
the greatest diameter of the condenser''^.
Z + if 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 consider-
able distance from other conductors.
91.] When a round conductor, J.3, of small size compared with
the distances between the conductors, is brought into the field,
the coefficient of potential of A^ on A^ will be increased when A^
is inside and diminished when A^ is outside of a sphere whose
diameter is the straight line A-^ A^.
For if A^ receives a unit positive charge there will be a distri-
bution of electricity on J-g, +6 being on the side furthest fh)m A^^
and — e on the side nearest A^. The potential at A^ due to this
distribution on A^ will be positive or negative as +« or —6 is
nearest to A^y and if the form of ilg is not very elongated this
will depend on whether the angle A^ A^ A^ is obtuse or acute,
and therefore on whether A^ is inside or outside the sphere
described on A^A^dA diameter.
* {For we may proye, as in Art. 89 0, that the capadty of a condenser aU of whose
parts are at the same potential is less than that of the sphere drcomacribing it, and
the capacity of the sphere is equal to its radins.}
Digitized by VjOOQ iC
fls
118 SYSTEM OF CONDUCTOES. [93 «•
If A^ 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
drawn through the points A^, ^3, A^ it may increase the
potential of A^^ even when it is entirely outside the sphere, and
that if it is placed 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 A^ 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 J.^ and diminish
the negative charge of each of the other conductors.
93 a.] Work done by the electric forces during the diapldcevient
of a system of insulated charged conductors.
Since the conductors are insulated, their charges remain
constant during the displacement. Let their potentials be T^,
IJ,... T^^ before and T^\ T^'j-'-KI' after the displacement. The
electric energy is W = \l(eV)
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
Tr-Tr = i2[6(F-r)].
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 <f} be the variable whose variation corresponds to
the kind of displacement, and let <P be the corresponding force,
reckoned positive when the electric force tends to increase 0,
then 4>d!<^=— dl^,
^ Digitized by VjOOQIC
93 C-] MECHANICAL FORCES. 110
where W^ denotes ihe expression for the electric energy as a
quadratic function of the charges.
936.] To prove that ^ + ^ = o.
We have three different expressions for the energy of the system,
0) F=i2(eF).
a definite function of the n charges and n potentials,
(2) K^i^lie^e.p^),
where r and 8 may be the same or different, and both ra 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 thesa
(3) FK=i2S(^]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 con-
figuration of which (j) is one.
Since W=W;=Wfr,
W^+Wy-2W=0.
Now let the n charges, the n potentials, and ^ vary in any
consistent manner, and we must have
x[(3-.H..[(f-..)..j].(f^f>* = o.
Now the n charges^ the n potentials, and ^ are not all inde-
pendent of each other, for in fact only ii+l of them can bo
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
dWr_
and that lastly, ^W, dWy _ i^j
Work done by the electric forces during the diaplacemerd of a
system, whose potentials are maintained constant
dW
93 c.] It follows from the last equation that the force * = -tt- >
and if the system is displaced under the conditiontHat all the
Digitized by VjOOQ IC
120 SYSTEM OF CO!a)UOTOES. [94.
potentials remain constant, the work done by the electric forces is
f<Pd<l> = fdWr = Fp - Wfr;
or the work done by the electric forces in this case is equal to the
iiKyrement of the electric enei^.
Here, then, we have an increase of energy together with a
quantity 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 tiie battery is twice the
work done by the system of conductors during the displacement.
On the comparison of similar dectrified 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 Z to Z^ 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 X to i'. 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 e\ then the potentials V and V at coiresponding points
B and R, due to this electrification, will be
F=^.andF = ^.
But AB is to A'ff as Z to Z', so that we must have
ei^iiLViL'r.
But if the inductive capacity of the dielectric is different in the
two systems, being Kin the first and fin 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 F to F*, and if the quantities
of electricity on corresponding parts are as e and e\ we shall have
e:e'::LVK:rrir.
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 composed
of dielectric media of which the specific inductive capacities at
corresponding points are in the proportion o{ K to K\ and in the
Digitized by VjOOQ iC
94-] SIMILAR SYSTEMS. 121
third place so electrified that the potentials of corresponding
points are as F to y.
From this it appears that if g be any coefficient of capacity or
induction in the first system, and €[ the corresponding one in the
second, q.((..LKxL'K's
and if jt> and^' denote corresponding coefficients of potential in
the two systems, \ \
If one of the bodies be displaced in the first system, and the
corresponding body in the second system receives 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 -FZ to FL\
But the total electric 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
electric energies in the two systems respectively,
and the differences of energy after similar displacements in the
two systems will be in the same proportion. Hence, since FL
is proportiqnal to the electrical work done during the displace-
™e^^ FL'.rL'iieVier.
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 sytem is
F:F:: V^KiV^K^
The first of these proportions shews that in similar systems the
force is proportional to the sqjift^^ ^^ th^ electromotive f<?rffft »^"^
to the inductive capacity pf the dielectric, but is injepfiuHftnt^
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 fhft gn^tity nf ftlftct^city
on each body is given, the forceBM;e^proportional to the squares
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122 SYSTEM OP CONDUCTOBS.
of the charges and inversely to thejguares 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 by
air and charged with the same quantities of electricity*.
* {It follows from the preceding inveetigmtioii that the force between two electri-
fied bodiea ■nrronnded by a medimn whose ■pedfiomdnctive capacity ia kin ee'/Kt^,
where e and ^ are the charges on the bodies and r is^he distance between {hemT}
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CHAPTEE IV.
QENEBAL THEOBEMS.
95 a.] In the second chapter we have calculated the potential
function and investigated some of its properties on the hypo-
thesis that there is a direct action at a distance between electri-
fied 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 in every
case to a determinate result, and to establish certain general
theorems deduced from Poisson's partial differential equation,
<PV (PV dW ,
The mathematical ideas expressed by this equation are of a
different kind from those expressed by the definite integral
/•+« f+« /•+• «
F=/ / / ^dx'dj/dz'.
•/— 00 •/— GOV— • ^
In the differential equation we express that the sum of the
second derivatives of Fin the neighbourhood of any point is
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124 OENEEAL THEOREMS. [95^-
related to 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
ihe point (x\ ^, 2^), at which p exists, from the point {xy 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 ex-
pression 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
equivalence 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.
95 6.] 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 vary 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 spMfi» 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 TrrnfAtinpajl diatribution.
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 convei^-
ence [Art. 25] is zero at every point. Such a distribution may
be called a Solenoidal distribution. The velocity of an incom-
pressible fluid is distributed in a solenoidal manner.
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95&.] IBEOTATIONAL AND SOLENOIDAL DISTBIBUTIONS. 125
When the central forces which, as we have said, give rise to
an irrotational distribution of the resultant force, vary according
to the inverse square of the distance, then, if these centres are
outside the field, the distribution within the field will be sole-
noidal 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 de-
pending on the distance, or of surface pressures, on a frictionless
fluid originally at rest, the distribution of velocity is irrotational
as weU as solenoidaL
When we have to specify a distribution which is at once irrota-
tional aud solenoidal, we shall call it a Laplaci«-n ^jfltr^"^'^'"" »
Laplace having pointed out some of the most important pro-
perties 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 irrotationaUy in all cases of electric equi-
librium. It is shewn that if the surface-potentials be given, then
of all irrotational disfadbutions, that which is also solenoidal has
the least^nergy ; whence it also follows that there can be
only one Laplacian distribution consistent with the surface
potentials.
In the second group of theorems^ including Thomson's Theorem,
the energy is expressed in terms of thq ftlftpi.rir* diap^ftnement, a
vector of which the distribution is solenoidal. It is shewn that
if the surface-charges are givftn, thftn of all solenoidal distribu-
tions that has least energy wbiohLJaL also irrotational^ whence it
also follows that there can be only one Laplagiftji d^«^"b^]t^ftp
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 rect-*^"^
angular axes, we make use in each case of the result of Theorem^.. -^ '
III, Art. 21''^, where the relation between a volume-integral and
the corresponding surface-integral is fully worked out. All that
* This theorem seems to have been first given by Ostrofpntdsky in a paper read in
1828, but pablished in 1831 in the M6m, de VAcad, deSt. Piiershourff, T. I. p. 39. It
may be regarded, however, as a form of the equation of continuity.
Digitized by VjOOQ iC
126 GENEBAL THEOREMS. [96 a.
we have to do, therefore, is to substitute for Z, Y, 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
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 electrostaticB
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 4^ as a special symbol for the electric
potential.
Green's Theorerti.
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 1; be a
normal drawn from the surface 8 into the fields and let 2, m, n be
the direction cosines of this normal, then
li*A.^^^ n—=z— (I)
dx dy dz " dv ^ '
will be the rate of variation of the function ^ in passing along
d^
the normal v. Let it be understood that the value of -7- is to be
taken at the surface itself, where 1; = 0.
Let us also write, as in Arts. 26 and 77,
d^^ d^^ d^^ „„, ,^,
and when there are two functions, 4^ and 4>, let us write
d^d^ d^d^ d'i^dfp _^ ..
dx dx dy dy dz dz " ' ' ^ *
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96 a.] green's theorem. 127
The reader who is not acquainted with the method of Quater-
nions may, if it pleases him, regard the expressions V'^' and
^.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 reason, however, why we use as our abbreviations these ex-
pressions 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 ^ gives the space-variation of that function, and the
expression — /S.V*V<I> is the scalar part of the product of two
space- variations, or the product of either space- variation into the
resolved part of the other in its own direction. The expression
y- is usually written in Quaternions S. UvV'ff, 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
{non-isotropic} media.
Statement of Greens Theorem.
Let 4^ and * be two functions of x, y, 0, which, with their first
derivatives, are fijiite and continuous within the acyclic region 9,
bounded by the closed surface 0, then
where the double integi'als 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,
X = *^. F = *^, Z=<ifp, (5)
dx dy dz ^ ^
^ /»ci4> d^ d^\
then lccos€=— ^'(^j- +m-i-+w-T- )
^ dx dy dz'
= -*^.by(i); (6)
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128 GENERAL THEOREMS. [966.
d^d^ d^d^ d^d<t>
dx dx dy dy dz dz
= -.4'V24> _ s. v^'V*, by (2) and (3). (7)
But by Theorem III
or by (6) and (7)
If''T.'^-IIh'*<^' =///^.V*V*d. (8)
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).
966.] We have next to shew that Green's Theorem is true
when one of the functions, say 4^, is a many-valued one, provided
that its first derivatives are single- valued, and do not become
infinite within the acyclic region ;.
Since V^ and V4> are single-valued, the second member of
equation (4) is single- valued ; but since * is many- valued, any
one element of the first member, as ^V^4>, is many- valued. If,
however, we select one of the many values of 4^, as %, at the
point A within the region 9, then the value of 4^ at any other
point, P, will be definite. For, since the selected value of * is
continuous within the region, the value of 4^ at P must be that
which is arrived at by continuous variation along any path from
J. to P, beginning with the value % 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 y, the closed curve must be entirely
without the region ; and since the region is acycUc, two paths
within the region cannot embrace anything outside the region.
* < / \-j~ dx + —dy + — dzj is the same for all reoonoileable paths, and
since the region ia acyclic all paths are reconoileable. f
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y^^V -■ar
96 c.] green's theorem. 129
Hence, if % is given as the value of ^ at the point A, the
value at P is definite.
If any other value of % say %-^nK, had been chosen as the
value at A, then the value at P would have been * + 7iic. 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
UK
m^-UH^'^'
and this, by Theorem m, Art. 21, is zero.
96 c] If the region 9 is doubly or multiply connected, we may
reduce it to an acyclic region by closing each of its circuits with
a diaphragm, {we can then apply the theorem to the region
bounded by the surface of 9 and the positive and negative sides
of the diaphragm}.
Let 8] be one of these diaphragms, and k^ the corresponding
cyclic constant, that is to say, the increment of 4^ in going once
round the circuit in the positive direction. Since the region 9 lies
on both sides of the diaphragm 8^, every element of 8^ will occur
twice in the surface integral.
K we suppose the normal v^ drawn towards the positive side
of c28|, and i^/ drawn towards the negative side,
and ^1// = 4^1 + (1:1)
so that the element of the surface-integral arising from ds^ will be,
since dv^\A the element of the inward normal for the positive
surface, ^^ ^ ,d^ . d^ .
Hence if the region 9 is multiply connected, the first term of
equation (4) must be written
where c2y is an element of the inward normal to the bounding
surface and where the first surface-integral is to be taken over
the bounding surface, and the others over the dififerent 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-
VOL. I. K
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130 GENERAL THE0BBK8. [96 d,
connected regions was first shewn to be necessary by Helmboltz'^,
and was first applied to the theorem by Thomson f.
96 c2.] Let us now suppose, with Green, that one of the
functions^ sav ^^ does not satisfy the condition that it and its
^rst derivatiyes 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 <1> is
4>o + g/rt, where 4>o is a finite and continuous quantity, and r is
the distance from P. This will be the case if <l> is the potential
of a quantity of electricity e concentrated 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, <1> presents no singularity,
we 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
Li forming the volume-integrals we have to subtract firom the
volume-integral arising from the whole region that arising from
the small sphere.
Now / / / ^V^'ifdxdydz for the sphere cannnot be numericaUy
or (VH)^{2ir«a«+4»a«*y}, ^»— y^^'^ ' ' ^
where the suffix, ^, 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 a^, and may be
neglected when a diminishes and ultimately vanishes.
The other volume^integral
jjj^V^^dxdydz
we shall suppose taken through the region between the small
sphere and the surface /S, so that the region of integration does
not include the point at which ^ becomes infinite.
* ' Ueber Integrale der bydrodynamiBchen Gleichnngen welche den Wirbelbewe-
gungen entspreehen/ CrelU, 1858. T»iisl»ied by Prof. Tait. PhU. Mag., 1867 (I).
t 'On Vortex Motion/ Tram, R. S. Edin. zxv. parti, p. 241 (1867.)
X ^6 mark / separates the numerator from the denominator of a fraction.
97^0 obsen's theorem. 131
The surface-integral / /<!> -?- cb^ for the sphere cannot be nu-
merically greater than % 1 1 j- d^*
Now by Theorem HI, Art. 21,
since dv ia here measured outwards from the sphere, and this
cannot be numerically greater than (V^4')^|lra^ and <t>, at the
e C C d^
surface is approximately - , so that / / 4> -7- da cannot be numeri-
cally greater than i ^^2^ (V**)^,
and is therefore of the order a^, and may be neglected when a
vanishes.
But the surface-integral for the sphere on the other side of
the equation, namely, rr ^^
does not vanish, for / / -j-dff = — 4ire ;
dv being measured outwards from the sphere, and if 4^^ be the
value of 4^ at the point P,
//■
^^^'^ ="*'**<>•
Equation (4) therefore becomes in this case
97 a.] We may illustrate this case of Green's Theorem by em-
ploying it as Oreen does to determine the surface-densitv 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 V^'if = 0, and
outside V'4^= 0 where ^ and ^^ denote the potentials inside and
outside the surface.
Oreen begins with the direct process, that is to say, the distri-
* {In this equation dv ii drawn to the inside of the sniface and yyy^V'^ dxdydz
is not taken through the space oooupied by a smaU s|^ere whose centre is the point at
which ^ becomes infinite.}
K 7,
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132 GENEEAL THEOEEMS. [97 &^
bution of the surface density, o-, being given, the potentials at an
internal point P and an external point P' are found by integrat-
ing the expressions
*-=//>• ^^=11?^'^ («)
where r and r' are measured from the points P and P' respect-
ively.
Now let 4> = 1/r, then appljdng Green's Theorem to the space
within the surface, and remembering that V^^ — 0 and V^^^ = 0
throughout the limits of integration we find
where *p is the value of * at P.
Again, if we apply the theorem to the space between the
surface a 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 \/a and may be neglected, and we have
Now at the surface, * = 4^, and since the normals v and / are
drawn in opposite directions,
/ + /'=«•
dv dv
Hence on adding equations (10) and (11), the left-hand mem-
bers destroy each other, and we have
97 6.] Green also proves that if the value of the potential 9
at every point of a closed surface fiJ)r^^_^TVM\ ^r>^if.rft.ri)yj_the
potential at any point inside_or^if^iHft fhft ain^ft^-ft may be
determined, provided V^ = 0 inside or outside the surface.
For this purpose he supposes the function <l> to be such that
near the point P its value is sensibly 1/r, while at the surface
8 its value is zero, and at every point within the surface
V2<1) = 0.
* (in equatioDB 10 and 11 dv^ \a drawn to the inside of the surface and ir to the
outside. }
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98.] green's function. 133
That 6ach a functioii must exist, Green proves from the
physical consideration that if 8 is a conducting surface connected
to the earth, and if a unit of electricity is placed at the point P,
the potential within a must satisfy the above conditions. For
since 8 is connected to the earth the potential must be zero at
every point of 8, and since the potential arises from the electricity
at P and the electricity induced on 8, V^4> = (> at every point
within the surface.
Applying Qreen's Theorem to this case, we find
where, in the surface-integral, ^ is the given value of the
potential at the element of surface ds; and since, if ap is the
density of the electricity induced on 8 by unit of electricity at P,
/7(f>
4^<^P + ^,= 0, (14)
we may write equation (13)
^J'prz- /Y^crffe* (16)
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 o- 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.
Green's Function.
98.] Let a closed surface a be maintained at potential zero.
Let P and Q be two points on the positive side of the surface a
(we may suppose either the inside or the outside positive), and
let a small body chained 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
* {Thk 11 the Mune u equatioii (14), p. 107.}
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134 GENERAL THEOREMS. [98.
other is due to the action of the electricity induced on 8 by P.
The latter part of the potential is called Green's Function, and is
denoted by Op^.
This quantity is a function of the positions of the two points
P and Q, the form of the function depending on the surface 8.
It has been calculated for the case in which 8 is a sphere, and for
a very few other cases. It denotes tbft pot^ntJAl n.^ Q dua in the
AlAnfriAif.y ihHiiaaH qji p by nnij of electricity at P,
The actual potential at any point Q due to the electricity at P
and to the electricity induced on 8 is l/r^^ + 0,^, where r^ denotes
the distance between P and Q.
At the surface 6, and at all points on the negative side of 8, the
potential is zero, therefore
<^^^~' 0)
'pa
where the suffix ^ indicates that a point A on the surface 8 is
taken instead of Q.
Let (Tp^ denote the surface-density induced by P at a point A^
of the surface 8, then, since 0^^ is the potentiaJ at Q due to the
Rupei'ficial distribution,
where dfa^ is an element of the surface 8 at A\ and the integra-
tion is to be extended over the whole surface 8.
But if unit of electricity had been placed at Q, we should have
had by equation (1),
where <r^ is the density at A of the electricity induced by Q, cfe
is an element of surface, and r^ is the distance between A and
-4.'. Substituting this value of l/r^^ in the expression for (7^,
we find r r r Ttr «■
Since this expression is not altered by changing , into ^ and
,into„wefindthat q^^q^^.^ (6j
a result which we have already shewn to be necessary in Art. 86,
=-//,
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99^-] UNIQUE MINIMUM OP W^. 135
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 a, 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 distribution 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
8i, 82> &<^> and whose potentials are %, %, &c., with surfeu^e-
densities o-^, o-^, &c., then
V«* = 4irp, (17)
^ = -4.<r, (18)
since dvia drawn outwards from the conductor, and
■^<fe^=-4,re„ (19)
//:
where ei is the charge of the surface 8^ .
Dividing (16) by — 8ir, we find
^(*i6, + *j62 + &c.) + iyyy*pda;d!ycfe
=i^///[(S)*-(f)^(S)>*<^ (-)
* llioiiiBon And Taii*8 Natural Philo*<tphjf, $ 526.
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136 GENERAL THEOREMS. [996-
The first term is the electrie energy of the system arising
from the surface-distributions, and the second is that arising
from the distribution 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 05, y, z.
As we shall often have occasion to employ this volume-integral,
we shall denote it by the abbreviation T^, so that
If the only charges are those on the surfaces of the conductoi-s,
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 TT.
996.] Let ^ be a function of a;, t/, Zy subject to the condition
that its value at the closed surface 8 is ^, a known quantity for
every point of the surface. The value of 9 at points not on the
surface 8 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 *
which satisfies the surface condition and also satisfies Laplace's
Equation y2^^ ^ q ^23)
at every point within the surface, then Hf, the value of W
corresponding to ^1, is less than that corresponding to any func-
tion 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
^J' = *i + 4'2; (24)
then ^2 is ^ function which is zero at eveiy point of the
surface.
* {The ezpresBion on the right-lumd side of (20) does not represent the eneigy where
the conductor! »re surronnded by any dielectric other than air.}
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99 6.] UNIQUE MINIMUM OP W^ 137
The value of W for ^ will be evidently
W = W,+ W,+-JJJ{^^^^+^-^' + ^^^)dxdydz. (25)
By Green's Theorem the last term may be written
The volume-integral vanishes because V^*i = 0 within the
surface, and the surface-integral vanishes because at the surface
*2 = 0. Hence equation (26) is reduced to the form
W=Jf[-\-W^. (27)
Now the elements of the integral J^ being sums of three
squares, are incapable of negative values, so that the integral
itself can only be positive or zero. Hence if TV^ is not zero it
must be positive, and therefore W greater than Hf . But if "BJ
is zero, every one of its elements must be zero, and therefore
'^«=0 ^'=0 ^=0
dx * dy * dz
at every point within the surface, and ^2 ix^^st be a constant
within the surface. But at the surface ^j = ^> therefore % = 0
at every point within the surface, and * = ♦,, so that if W is
not greater than T?f , ♦ must be identical with ^^ at every point
within the surface.
It follows from this that % is the only function of a;, y, z
which becomes equal to 4' 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 %,
then T^ must be less than any other value of W. But we have
already proved that Tfjis less than any other value, and therefore
than TIJ. 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, 8, and any number of
interior surfaces, 8], ^2' ^*) ^^^ when the conditions are that the
value of ♦ shall be zero at 8, *i at «i, *2 ** ®2> *^<^ ^^ ^^> where
%y %f &c. are constant for each surface, as in a system of
conductors, the potentials of which are given.
Of aU values of 4^ satisfying these conditions, that gives the
minimum value of TIJ for which V^^' = 0 at every point in the
field.
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138 GBNEBAL THBOBBMS. [1006.
Thomson's Theorem.
Le^n/md,
100 a.] Let 4^ be any function of a?, y^ z which is finite and
continuous within the closed surface 8, and which at certain
closed surfaces, 8^, s^, ...,8„, &c.,has the values *i, *2> ••• > *j»> &<^-
constant for each surface.
Let n^ V, w be functions of a?, y, «, which we may consider as
the components of a vector @ subject to the solenoidal condition
a „ir du dv dw ^ ..
-^•^•^ = 5^ + ^ + ^ = "' ('«)
and let us put in Theorem III ^ fi^i^CT/^
X = ^u, 7=:*t;, Z^^w; (29)
we find as the result of these substitutions
JJ<yp{lpU+m,v + n,w)ds,-\-JJJ<y (^ + ^ + ^)dxdydz
the surface-integrals being extended over the different surfaces
and the volume-integrals being taken throughout the whole
field, and where 2p, m,, n, are the direction cosines of the normal
to Sp drawn from the surface into the field. Now the first
volume-integral vanishes in virtue of the solenoidal condition
for u, v^ Wf and the surface-integrals vanish in the following
cases : —
(1) When at every point of the surface 4' = 0.
(2) When at every point of the surface lui-mv-k-TW) = 0.
(3) When the surfeu^ is entirely made up of parts which
satisfy either (1) or (2).
(4) When ^ is constant over each of the closed surfaces, and
2p
I j {lv,+mV'j'nw)ds = 0.
Hence in these four cases the volume-integral
d^ d* d*x
1006.] Now consider a field bounded by the external closed
surface a, and the internal closed surfaces Si, s^f &c.
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looc] Thomson's thboeem. 139
Let 4^ be a function of x^ y^ z, which within the field is finite
and continuous and satisfies Laplace's Equation
V^^ = 0, (32)
and has the constant, but not given, values %, %, &c. at the
surfaces 8^, 82, &c. respectively, and is zero at the external
surface a.
The charge of any of the conducting surfaces, as 8^, is given
by the surface integral
the normal Vi being drawn from the surface 81 into the electric
field.
100 c] Now let /, jr, A be functions of », y, z, which we may
consider as the components of a vector 2), subject only to the
conditions that at every point of the field they must satisfy the
selenoidal equation
and that at any one of the internal closed surfaces, as s^, the
surfiEtce-integral
{lif-^rnig+n^h)d8 = (5,, (36)
//<
where ^j, m^, % are the direction cosines of the liormal v^ drawn
outwards from the surface 8^ into the electric field, and ei is the
same quantity as in equation (33), being, in fact, the electric
charge of the conductor whose surface is e^.
We have to consider the value of the volume-integral
^ = ^^fff(P'^9'-^h^)dxdy<h, (36)
extended throughout the whole of the field within 8 and without
8}, 82, &c., and to compare it with
the limits of integration being the same.
Let us write
4irda? ^ ^TT dy ^ir dz ^ ^
and W^ =i 2Tr ffnu^+v^'^w^)dxdydz; (39)
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140 GENERAL THEOREMS. [lOOC.
then since
, , , 1 r d* d* d*"!
IT. = IT, + W,-Jff(uf^ + .g +«,g)cirrf2/<fe. (40)
Now in the first place^ u, v, w satisfy the solenoidal condition
at every point of the field, for by equations (38)
dw dv dw_^ dg dh l^ , .
dx^ dy^ dz' dx^dy^ dz ^it^ ' ^^
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
/ / (^1 ^ + ^h V + % tc;) cfei
but by (35) the first term of the second member is 6|,and by (33)
the second term is — 6^, so that
I i{liU + 7n^v-\-niW)dSi = 0. (43)
Hence^ since ^^ is constant, the fourth condition of Art. 100 a
is satisfied, and the last term of equation (40) is zero, so that the
equation is reduced to the form
T»i=T»i + T^. (44)
Now since the element of the integral "% is the sum of three
squares, u^ + v^ + w^, 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 Wd must have a positive value, and T^ must
therefore be greater than f^. But the values u=iV = w = 0 B,t
every point satisfy the conditions.
Hence, if at every point
- Id* Id*, Id* ..
•^=-4^di' S^=-4^d^' ^ = -4^d^' (^^^
then W^ = W*, (46)
and the value of W^ corresponding to these values of /, gr, h, is
less than the value corresponding to any values of /, g, h,
differing from these.
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lOOe.] INTENSITY AND DISPLACEMENT. 141
Hence the problem of determining the displacement and
potential, 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 stilted V^ ^» ^'/
by Sir W. Thomson *. We shall afterwards show of what gene- /' ^^
ralization it is capable.
100 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 dh , .
where p is a finite quantity, whose value is given at every point
in the field, and which may be positive or negative, continuous
or discontinuous, its volume-integral within a finite region
being, however, finite.
We may also suppose that at certain surfaces in the field
Z/+wiflr + 7iA + r/ + mY + ^'A' = <^, (48)
where 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 /, g, h and /', g^y h' re-
spectively, and (T is a quantity given at all points of the surface,
the surface-integral of which^ over a finite surface, is finite.
100 6.] We may also alter the condition at the bounding sur-
faces by supposing that at every point of these surfaces
If-^-mg-^-nhssa, (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,
and the surface condition
d^ d^ ^ ^ .^,.
:t- +:7-> + 47r<r= 0. (51)
dv dv ^ '
* Cambridge and DMin Mathematical Journal, February, 1848.
Digitized by VjOOQ IC
142 GENERAL THEOREMS. [lOI a.
For if, as before, -
.Id* \ d'^ , 1 d*
'' iTrdx ^ A-n dy in dz
then u, V, w will satisfy the general solenoidal condition
dAi dv dw _
cte d!y dz'^ ^
and the surface condition
Zu + mt; + ntc; + ZV+mV + n'tc/'= 0,
and at the bounding surface
lu + mv-^nw = 0,
whence we find as before that
and that T»i=T^ + T^.
Hence as before it is shewn that W^ is b, unique minimum
when T^ = 0, which implies that u^+v^ + v^ is everywhere
zero, and therefore
._ l_d*^ l^d^ ,_ 1 d^
•'"" ivdx^ """ ivdy ' "" 47rrf2f*
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
electromotive intensity just outside it, as expressed in the law
of Coulomb iJ = 4 w.
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 diflbrent in different media. One of these is the electro-
c/*j9*^T'(^ rngtixfiuintensity, the other is the ftl^^trig dJRplacement. The
electromotive intensity is connected by equations of invariable
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lOI c] INTENSITY AND DISPLACEMENT. 143
form with the potential, and the electric displacement is con-
nected by equations of mvariable 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 experiments on dielectrics.
101 J.] The electromotive intensity is a vector defined in
Art. 68, as the mechanical force on a small quantity e of elec- *
tricity divided by e. We shall denote its components by the
letters P, Q, JB, and the vector itself by @.
In electrostatics, the line integral of (S is always independent
of the path of integration, or in other words (S is the space-
variation of a potential Hence
P=-— , 0=- — , ie=- — ,
dx ' dy^ dz *
or more briefly, in the language of Quaternions
101 c] The electric displacement in any direction is defined
in Art. 60, as the quantity of electricity carried through a small
area A^ the plane of whidi is normal to that direction, divided
by A, We shall denote the rectangular components of the
electric displacement by the letters /, gr, A, and the vector itseK
by 3).
The volume-densitv at any point is determined by the equation
df^dg^dh^
dx dy dz'
or in the language of Quaternions
p = -iS.va).
The surface-density at any point of a chai^d surface is deter-
mined by the equation
where/, g, h ai*e 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 i, m, n, and/', jf', A' and l\ m\ vf
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
a=-[fif. I7i;2)+i8f. I7i;'2)'],
Digitized by VjOOQ iC
144 GENERAL THEOREMS, [lOI e,
where Uv^ TJv' 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/', gr', K and 2)' are zero, the equation
is reduced to the form
<r = lf-\'7ag + nh\
= -S. 171/3).
The whole charge of the conductor is therefore
e = i j{lf-\'mg-\'nh)d8\
101 d,] The electiic energy of the system is, as was shewn in
Art. 84, half the sum of the products of the charges into their
respective potentials. Calling this energy "W,
+ ^ff'i^{lf+mg-\'nh)d8;
where the volume-integral is to be taken throughout the electric
field, and the surface-integral over the surfaces of the con^
ductors.
Writing in Theorem III, Art. 21,
X=z^f, Y=^g, Z = ^h,
we find, if 2, m, 71 are the direction cosines of the normal facing
the surface into the field.
Substituting this value for the surface-integral in W we find
or W ='^ffJ(fP + gQ + hR)dxdydz.
101 6.] We now come to the relation between S) and @.
The unit of electricity is usually defined with reference to
Digitized by VjOOQ iC
lOI e.] PBOPBBTIBS OP A DIBLECTEIO. 145
experiments conducted in air. We now know from the ex-
periments 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 coiTCcted 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 measure-
ments of extreme accuracy.
In the standard medium ^c/. /• • 7 f/
or 4ir/=P, 4irgr = Q, 4irA = jR.
In an isotropic medium whose dielectric constant is K
4irS) = ire,
There are some media, however, of which glass has been the
most carefully investigated, in which the relation between 2) and
@ is more complicated, and involves the time variation of one
or both of these quantities, so that the relation must be of the
form
F{% (S, 1), e, S), e, &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
y JH R l,inear and vector function of (S.
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 4nf = K„P+K,,Q^K,.R,
Aitg = K,^P+K,, Q + K^,R,
47iA = if„P + ir.,Q + ir„jB;
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
VOL. I. L
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146 GENERAL THEOREMS. [lOI g,
nine independent coeflBcients. When the coefficients which have
the same pair of suffixes are equal, the function is said to be
self-coixiugate.
If we express (5 in terms of 2) we shall have
(g = 47r(^-i(S)),
or P = 4 IT (*,, /+ k^, g + k^h\
Q = 47r(ft,y f^Kyg'tK^h),
jB = 4 TT (*,, /+ k^^ g + A;„ A).
101/.] The work done by the electromotive intensity whose
components are P, Q, iJ, in producing a displacement whose com-
ponents are df, dg^ and dh, in unit of volume of the medium, is
dW^Pdf-^-Qdg^-Rdh.
Since a dielectric {in a steady state} under electric displace-
ment is a conservative system, W must be a function of/, jr, A,
and since/, gr, h may vary independently, we have
T. dW r, dW T. dW
Hence dP^ d^W ^ d^W ^dQ
dg " dgdf "" dfdg " df
But -T- = iirk^gy the coefficient of gr in the expression for P,
and -j^ =: Aitk^y the coefficient of/ in the expression for Q.
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 <^~^ is a self-conjugate function.
Hence it follows that 4> also is self-conjugate, and K^ = K^^,
101 gr.] The expression for the energy may therefore be written
in either of the forms
or •^2K^RP+2K^^Q]dxdydz,
Wi = 27r fff[k„P + k,yg^+k„ h^ + 2k,, gh
^^^ '\-2k„hf'k-2k^yfg]dxdydz,
where the suffix denotes the vector in terms of which TT is to be
expressed. When there is no suffix, the energy is understood to
be expressed in terms of both vectors.
Digitized by VjOOQ iC
lOI A.] EXTENSION OP GBEEN's THEOREM. 147
We have thus, in all, six different expressions for the energy
of the electric field. Three of these involve the charges and
potentials of the surfaces of conductors, and are given in Art. 87.
The other three are volume-integrals taken throughout th$
electric field, and involve the components of electromotive in-
tensity 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 h.] To extend Green's Theorem to the case of a hetero-
geneous anisotropic {non-isotropic} medium, we have only to
write in Theorem HI, Art. 21,
and we obtain, if Z, m, 7i are the direction cosines of the outward
normal to the surface (remembering that the order of the suffixes
of the coefficients is indifferent),
+ (K„l + Z,.m + K„n) -^] ds
rcTi r<^ /IT <?* V d<p „ d*\
+ ^ (^"^ +^" d^ + ^'" dr)
d /^ d<P rr d^ rr d^ \'\ , , ,
dy
L z
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148 GENERAL THEOREMS. [l02 a.
=///[
^"dx dx"^ ^" dy dy ^^"dzdi
^ "^dy dz^ dz dy>^^''\d^d^-^did^)
Using quaternion notation, the result may be written more
briefly,
Limits between which the electric capacity of a coTiductor
must lie,
102 a.] The capacity of a conductor or system of conductors
has been abready defined as the charge of that conductor or system
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. 306.
Let Si denote the surface of the conductor, or system of con-
ductors, whose capacity is to be determined, and Sq the surface of
all other conductors. Let the potential of s^ be 4^1, and that of
«oi ^0* ^^^ ^^® charge of a^ be e^. That of Sq will be — e^.
Digitized by VjOOQ iC
I02a.] LIMITING VALUES OP CAPACITY. 149
Then if g is the capacity of s^ ,
?=^. (1)
and if TT is the energy of the system with its actual distribution
of electricity F = I Cj (*i - *o), (2)
To find an upper limit of the value of the capacity : assume
any value of ^ which is equal to 1 at 8^ and equal to zero at 8^,
and calculate the value of the volume-integral
extended over the whole field.
Then as we have proved (Art. 99 6) that W cannot be greater
than T^, the capacity, q, cannot be greater than 2 T^.
To find a lower limit of the value of the capacity : assume
any system of values of/, jr, h, which satisfies the equation
df ^dg^dh_^ (5)
dx dy dz ' ^ '
and let it make ffih f-^'^ig-^'^h.^ ^^i = ^i • (^)
Calculate the value of the volume-integral
W^ = 27tfff{f+g^+h^)d^dydz, (7)
extended over the whole field ; then as we have proved (Art. 1 00 c)
that W cannot be greater than Tfi, the capacity, q, cannot be less
than g> 2
.\- (»)
The simplest method of obtaining a system of values of/, g, A,
which will satisfy the solenoidal condition, is to assume a distribu-
tion of electricity on the surface of 8^, and another on 8^, the sum
of the charges being zero, then to calculate the potential, 4^, due
to this distribution, and the electric energy of the system thus
arranged.
If we then make
^ " iirdx' ^ " ^itdy' ^-ndz'
these values of/, g, h will satisfy the solenoidal condition.
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150 GENERAL THEOREMS, [l02 6.
But in this case we can determine T(^ without going through
the process of finding the volume-integraL For since this solu-
tion makes V^^^ = 0 at all points in the field, we can obtain T^
in the form of the surface-integrals,
where the first integral is extended over the surface s^ and the
second over the surface 8^.
If the surface ^q is at an infinite distance from 8^ the potential
at 8q is zero and the second term vanishes.
102 6.] 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 8i be the surface of a conductor or system of conductors
maintained at potential 1 , and let 8q 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
8i to «o.
Along each of these lines, assume ^ so that it is equal to 1 at
8^, and equal io 0 nt 8^. Then if P is a point on one of these
lines {8| and 8q the points where the line cuts the surfaces} we may
Ps
take ^i = — ' as a first approximation.
We shall thus obtain a first approximation to ^ which satisfied
the condition of being equal to unity at 8, and equal to zero at Sq.
The value of Fi calculated from % would be greater than W.
Let us next assume as a second approximation to the lines of
force
The vector whose components are/, g,hiB normal to the surfaces
for which ^^ is constant. Let us determine j9 so as to make
/, g, h satisfy the solenoidal condition. We thus get
P^dx" ^ dy' ^ dz'f^dxdx ^ dy dy ^ Tz~d^ - ^- ^"^
If we draw a line from «i to s^ whose direction is always normal
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I02 6.] CALCULATION OF CAPACITY, 151
to the surfaces for which ^^ is constant, and if we denote the
length of this line measured from 8^ by 8, then
P^— — ^i pC^_ _^i pC?g_ d%
ds'^ dx ^ d8'^ dy ds" dz *
where R is the resultant intensity = - — -^-, so that
dpd% dp d% dp d% _ _ j^dp
dx dx dy dy dz dz ^ ds'
(12)
=*'^.'
(13)
and equation (11) becomes
PV^* = B^^^,
(14)
r*i V**,
whence p=Ceitp.j —^ d%,
(16)
the integral being a line integral taken along the line 8.
Let us next assume that along the line 8,
d% .dx dy , dz
= -.^. (.6)
then % = cf*' (exp.p-^ d%) d%, (17)
the integration being always understood to be performed along
the line 8.
The constant C is now to be determined from the condition
that ^2 = 1 &^ ^1 when also % = 1, so that
This gives a second approximation to ♦, and the process may
be repeated.
The results obtained from calculating "WJ^, "W^,, TJi„ &c., give
capacities alternately above and below the true capacity and
continually approximating thereto.
The process as indicated above involves the calculation of the
form of the line 8 and integration along this line, operations
which are in general too difficult for practical purposes.
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152 GENERAL THEOREMS. [l02 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
whichaxe 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(«.y) = a (19)
for the surface whose potential is zero, and
«2 = /2(«»y)=6 (20)
for the surface whose potential is unity, a and 6 being given
functions of x and y^ of which h is always greater than a. The
first derivatives of a and 6 with respect to x and y are small
quantities 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 0, in which case
/=o, sr = o, g = o. (21)
Hence A is constant along each individual line of induction,
^^^ *= -4irrAd«=-4irA(«-a). (22)
When 2; = 6, ^ = 1, hence
4ir(6— a) ^ '
and * = ^, (24)
which ^ves a first approximation to the potential, and indicates
a series of equipotential surfaces' the intervals between which,
measured parallel to z, are equal.
To obtain a second approximation to the lines of induction,
let us assume that they are everywhere normal to the equi-
potential surfaces as given by equation (24).
This is equivalent to the conditions
^""^^^d^^ ^""^^^d^* ^'^'^ = ^d^' (2«)
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I02C.] POTENTIAL BETWEEN TWO NEARLY FLAT SURFACES. 153
where X is to be determined so that at every point of the field
and also so that the line-integral
taken along any line of induction from the surface a to the
surface 6, shall be equal to —1.
Let us assume
X = l+il + 5(«-a) + C(«-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 s(denoidal condition then gives
5=-V«a, C = -.|^y"'^\ (29)
where v. = _(^+^J. (30)
If instead of taking the line-integral along the new line of
induction, we take it along the old line of induction, parallel to
Zf the second condition gives
1 = l+^ + i5(6-a) + i(7(6-a)».^
Hence il = i^(6-a)V2(2a+6), (31)
and
k=l-^l{b-^a)V^{2a + b)^{z^a)V^a^i^P^V^b^a). (32)
We thus find for the second approximation to the components
of displacement,
.4^/= ^ [^ + rf(fc~a)g-.al ^ \
•^ b — atdx dx 6— aj'
"■ '^^^b-aldy "*" dy 6— aJ
0 — a
(33)
and for the second approximation to the potential,
-i^*(^-«)(i^- <^^>
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154 GENERAL THEOREMS.
If (T^ and 0-5 are the surface-densities and ^^ and ^5 the poten-
tials of the surfaces a and h respectively,
* {This myeetiffation is not very rigorous, and the expressions for the surface density
do not agree witL the results obtained by rigorous methods for the cases of two
spheres, two cylinders, a sphere and plane, or a cylinder and plane placed close
together. We can obtain an expression for the surface density as foUows. Iiet us
assume that the axis of s is an axis of symmetry, then the axis wiU cut all the eqni-
potential surfaces at right angles, and if F is the potential, Ri Mf the principal radii
of currature of an equipotential surface where it is cut by the axis of z, the solenoidal
condition along the axis of z may easily be shown to be
^ yAi Vb i^re the potentials of the two surfaces respectively, t the distance between
the sur&oes along the axis of z,
or if Ra^, Ba^ denote the principal radii of curvature of the first surfaces, substituting
for -3-7 from the differential equation, we get
dz
VB-Vji
'OA'-*'{k'^)\-
when ffA is ^^^ surface density where the axis of z cut the first surface, hence
similarly ^^^ i^ ~^J~^ r'^'^^iw "*" S~}\ '^l^P^^^^^J*
and these expressions agree in the cases before mentioned with those obtained by
rigorous methods. |>
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CHAPTEK V.
MECHANICAL ACTION BETWEEN TWO ELECTRICAL SYSTEMS.
103.] Let E^^ and E.^ be two electrical systems the mutual
action between which we propose to investigate. Let the dis-
tribution of electricity in E^ be defined by the volume-density,
Px, of the element whose coordinates are x^, j/p %• Let pj ^ ^^^
volume-density of the element of E^, whose coordinates are
*^i y29 ^2*
Then the o^K^omponent of the force acting on the element of ^j
on account of the repulsion of the element of E^ will be
where r^ = (x,''X^y'\-{yi^y^f'\-(zi'^z^,
and if il denotes the a;-component of the whole force acting on E^
on account of the presence of E^ ^j
^ ^j J J J J j^'^^'^f^^f^^^^^yi^i^^ (^)
where the integration with respect to a?i, 2/i> ^i is extended
throughout the region occupied by E^, and the integration with
respect to ajg, 2/2> ^2 ^ extended throughout the region occupied
by^2-
Since, however, p^ is zero except in the system E^^ and P2 is zero
except in the system E.^^ 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 +00.
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> ^^ potential at the point ajj, y^, z^y
arising from the presence of the system E^^ by the equation
156 MECHANICAL ACTION. [1O4.
4^2 will vanish at an infinite distance, and will everywhere satisfy
the equation ^2ip^ ^ 4^^^^ (3)
We may now express A in the form of a triple integral
A = -jjj^^^P,db:,dy,dz,. (4)
Here the potential ^^ ^ supposed to have a definite value at
every point of the field, and in terms of this, together with the
distribution, p,, 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 ^2-
Now let ^'j be the potential arising from the first system,
expressed as a function of Xy y^ z, and defined by the equation
%^fff%^idy,dz,, (5)
4^1 will vanish at an infinite distance, and will everywhere satisfy
the equation V^ vp^ = 4 i:p^. (6)
We may now eliminate p^ 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 in-
different what limits are prescribed, provided they include the
whole of the system E^^ . In what follows we shall suppose the
systems E^ and ^^ to be such that a certain closed surface a
contains within it the whole of E^ but no part of E^.
Let us also write
then within 8, ^^ — q^ ^ — p^^
and without 8, ^ ^ q, p = pg. (9)
Now iln = - jyj^Pi^i^i^^i i^^)
represents the resultant force, in the direction x, 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.
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105.] MECHANICAL ACTION. 157
We may therefore write
where ^ is the potential arising from both systems, the integration
being now limited to the space within the closed surface 8, which
includes the whole of the system E^ but none of E^.
105.] K the action of E^ on E^ is eflFected, not by direct action
at a distance, but by means of a distribution of stress in a medium
extending continuously from E^io E^, it is manifest that if we
know the stress at every point of any closed surface 8 which
completely separates E^ from E^j we shall be able to determine
completely the mechanical action of E^ on Ej^. For if the force
on ^1 is not completely accounted for by the stress through 8,
there must be direct action between something outside of 8 and
something inside of 8.
Hence if it is possible to account for the action of E^ on E^ 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 a which completely separates
E2 from E^ .
Let us therefore endeavour to express
in the form of a surface integral.
By Theorem III, Art. 21, we may do so if we can determine X,
Y and Z, so that
d;P.d^ d^,d^._dX dY dZ
dx ^da? ■*" dy"^ ^ dz') '^ dx ^ dy ^ dz' ^ ^ ^
Taking the terms separately,
d^^^ _ d (d^d^\ _ d^ d^^
ft/n ii'ii^ tint V ti^r. rJ^i '
~ dy^dx dy' 2dx^dy^
o- ., , d*6P^ d ,d^d*s 1 d .d*-.^
Similarly -^_-;j^ = ^{;n;^) - ^d^i:^) '
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158
HEOHANIOAL ACTION.
[I05-
If, therefore, we write
,d*s' rd^y} /(?+N« „
'dy
,d*s* .d*x« .d*x« „
\
^dy-
,d*>,2 .d*x« .d+.« „
^dy>
I
(14)
then
=///(
t'*^*'i--)^«%^
(15)
the integration being extended throughout the space within 8.
Transforming the volume-integral by Theorem III, Art. 21,
A =JJ{lPxx + "ff^Ppx + np„) dsy
(16)
where (28 is an element of any closed surface including the whole
of E^ but none of ^2) <^^ ^> ^> ^ ^^ ^^^ direction cosines of the
normal drawn from ds outwards.
For the components of the force on E^ in the directions of y
and 0, we obtain in the same way
(17)
(18)
If the action of the system E^ on E^ does in reality take place
by direct action at a distance, without the intei-vention of any
medium, we must consider the quantities p^ &c. as mere abbre-
viated forms for certain symbolical expressions, and as having
no physical significance.
But if we suppose that the mutual action between E^ 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 re-
sultant force arising from the action, on the outside of the surface
8, of the stress whose six components are p^ &c., we must
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I06.] STRESS IN A MEDIUM. 159
consider p^ &c. as the components of a stress actually existing
in the medium. " '
106.] To obtain a clearer view of the nature of this stress let
us alter the form of part of the surface a 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 ^1 or include any part of E^,)
Let 1/ be a normal to ds drawn outwards.
Let jB = — J- be the intensity of the electromotive intensity
in the direction of r, then
^* P7 d^ ^ d<if ^
Hence the six components of the stress are
ft
p„=^B? (n'-l»-m'), p^ = i- mm.
oTT 4 IT
a = ^i>«r + ^/>,« + ^i>«r= ^I^%
If a, 6, c are the components of the force on da per unit of area,
Stt"
c= —RH.
Sit
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 ^- R\
o 77
Let us next suppose that the element ds is at right angles
to the equipotential surfaces which cut it, in which case
Now Sn{lp^^mp^ + np„) = Z[(^) - (^) - (^) J
^ d^d^ ^ d^d'i^ ,^.
+ 2m;T-T- +271:7- -r-- (20)
ax ay ax dz ^ '
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160 MEOHANIOAL ACTION. [l06.
Multiplying (19) by 2-t- and subtracting from (20), we find
67^{lp„ + mp^+np„) = -l[(^) + {^) + (^) J
= -IR^. (21)
Hence the components of the tension per unit of area of da are
Bv
Hence if the element ds 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 such an axis is a pressure whose
numerical magnitude is also p.
The stress as thus defined is not of the most geneitkl 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, accordingly it is
completely determined by the three components of the electro-
motive intensity
d^ d'if d*
dx ' dy dz
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lO;.] COMPONENTS OF STBBSS. 161
The three relations between the six components of stress axe
P'm = {P,, +P„) (Pm+Ppp), [ (23)
107.] Let ns now examine whether the results we have obtained
will require modification when a finite quantity of electricity is
collected on a finite surface so that the volume-density becomes
infimte at the surfskce.
In this case, as we have shewn in Arts. 78a, 786, the com-
ponents of the electromotive intensity are discontinuous at the
surface. Hence the components of stress will also be discon-
tinuous at the surface.
Let 2, m, ti be the direction cosines of the normal to ds. Let
P, Q, 12 be the components of the electromotive intensity on the
side on which the normal is drawn, and P', Q', JJ' their values
on the other side.
Then by Arts. 78a and 786, if <r is the surface-density
P-P'=4iraZ, \
Q-Q'siiircrm, > (24)
Let a be the a;-component of the resultant force acting on
the sur£Eu^ per unit of area, arising from the stress on both sides,
then
a = ^(i>«-/«)+m(p^-/J + n(2>^-i>'J,
= ^l {(P«-P^)««?-<2^).(lP-ii^)}
= l«{(P-PO(P+P')-(Q-<n(Q+QO-(^-^(^+^}
oV
= \l<r { l{P+I^)-m{Q+Q^-n(R+B')}
+ lm<r{l{Q+Qf)+m{P+I^}+lnir{l{R+Sr) + n{P+P')],
= l«r(P+PO- (25)
VOL. I. M
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162 MECHANICAL ACTION. [107.
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 intensities on the two sides of the
surface.
This is the same result as we obtained in Art. 79 by a process
essentially similar.
Hence the hjrpothesis 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 2^0- 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 P' 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 Pq 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 a;-component for A will be
P^P^ + 2TTal,
and for A' P' = J?- 2Tr(rl.
Hence ^ = i(P+P').
Now the resultant mechanical force on the element of surface
must arise entirely from the action of electricity at a finite distance,
* This method is due to Laplace. See Poisfon, ' Surla Diitribution de rtlectricit^
&c.* M^ de VlruiUut, 1811, p. 30.
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I08.] FORCE ON A CHABGED SURFACE. 163
since the action of the element on itself must have a resultant zero.
Hence the o^component of this force per unit of area must be
108.] If we define the potential (as in equation (2)) in terms
of a distribution of electricity supposed to be given, then it follows
from the fact that the action and reaction between any pair of
electric particles are equal and opposite, that the a;-component of (/, ^f'h)
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 4^ as a function of x^ y, z which satisfies the
equation y2xp _ q
at every point outside the closed surface 8, and is zero at an infinite
distance, the fact, that the volume-integral extended throughout
any space including s is zero, would seem to require proof.
One method of proof is founded on the theorem (Art. 1 00 c), that
if V^ is given at every point, and 4^ = 0 at an infinite distance,
then the value of ^ at every point is determinate and equal to
4/ = -^ fff^ V^* dx dy dz, (27)
where r is the distance between the element dx dy dz at which the
concentration of ♦ is given = V^* and the point x\ y', / at which
4^ is to be found.
This reduces the theorem to what we deduced from the first
definition of 4^.
But when we consider ♦ as the primary function of a?, y, z, from
which the others are derived, it is more appropriate to reduce (26)
to the form of a surface-integral,
A = ff{lp„'\'mp^ + npJ}dS, (28)
and if we suppose the surface S to be everywhere at a great
distance a from the surface «, which includes every point where
V^ differs from zero, then we know that 4' cannot be numerically
greater than e/a, where 4 ve is the volume-integral of V*4', and that
R cannot be greater that— d4'/rfa or c/a^, and that the quantities
PrgiPxgiPxM call none of them be greater than p, i.e. IP/Stt or
eys ira*. Hence the surface-integral taken over a sphere whose
M 2
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164 MECHANICAL ACTION, [109.
radius is very great and equal to a cannot exceed ey2 a^, and
when a is increased without limit, the surface-integral must be-
come ultimately zero.
Eut 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 ^8^, provided S encloses every
point at which V^* differs from zero. Hence, since the integral
is zero when a is infinite, it must also be zero when the limits of
integration are defined by any surface which includes every
point at which V^* differs from zero.
109.] The distribution of stress considered in this chapter is
precisely that to which Faraday was led in his investigation
of induction through dielectrics. He sums up in the following
words : —
*(1297) The direct inductive force, which maybe 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 attractive force which exists amongst the
particles of the dielectric in the direction of the induction is
accompanied by a repulsive or a diverging force in the transverse
direction.
'(1298) Induction jippears to consist ip. a^ertain polarized
state of the particles, into which they are thrown by the elec-
trified 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 contiwaed 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
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no.] FAEADAy's THBOEr. 165
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 prApArfi^^n^.! 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. Sut
in the deduction of this principle from a consideration of the
mobility 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
experiences 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 incon-
sistent with the equilibrium of a fluid dielectric.
The quantity TT, which was investigated in Chapter IV,
Art. 99 a, 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
* {The subject of the strew in the medium will be farther considered in the Sap-
plementary Volume, it nmy however be noticed here that the problem of finding ft
system of stresses which will produce the same forces as those existing in the electric
field is one which has an infimte number of solutions. That adopted by Maxwell is
one that coold not in general be produced by strains in an elastic solid. }
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166 MECHANICAL ACTION. [ill.
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. Elechnc 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
Leyden jar, of which the inner coating is charged positively and
the outer coating negatively, the direction of the displacement
of positive electricity in the substance of the glass is from within
outwards.
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 if/4ir,
where K is the specific inductive capacity of the dielectric.
n. Surface charge of the particles of the dielectric. Conceive
any portion of the dielectric, large or small, to be separated (in
imagination) 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 surfa.ce 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
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III.] BLBOTBIO POLAEIZATION. 167
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. The 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 surfebce multiplied by 4 v.
For what we have called the induction through the surface is
simply the electric displacement multiplied by 4 tt, and the total
displacement outwards is necessarily equal to the total charge
within the surface.
The theory also accounts for the impossibility of communi-
cating 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
EleC^T'^ fnlRrizRtinii-
A dielectric medium, when thus polarized, is the seat of
electric energy, and the energy in unit of volume of the
medium is numerically equal to the electric tension on unit of
area, both quantities being equal to half the product of the
displacement and the resultant electromotive intensity, or
where p is the electric tension, !D the displacement, (S the electro-
motive intensity, and K the specific inductive capacity.
If the medium is not a perfect insulator, the state of con-
straint, 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 con-
straint is converted 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 polarization sinks to half its original value. In copper,
a similar change is effected in less than the biUionth of a second.
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168 HEOHAKIGAL AOTION.
We have supposed the medium after being polarized to be
simply left to itself. In the phenomenon caUed the electric
current the constant passage of electricity through the medium
tends to restore the state of polarization as fast as the con-
ductivity 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 maintaining 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.
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CHAPTER VL
ON POINTS AND LINES OF EQUILIBBITJM.
112.] If at any point of the electric field the resultant force is
zero, the point is called a Point of equilibrium.
If eyeiy 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
— =0 — =0 — =0
dx ^ dy ' dz
At such a point, therefore, the value of F 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 the po-
tential must be stationary, and not a maximum or a minimum.
In fact, a condition for a maximum or minimum is that
^, ^, and —
da^ ' dy^ ' d«*
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 V, then,
at all other points in the immediate neighbourhood of P, the
value of F is less than at P. Hence P will be surrounded by a
series of closed equipotential surfaces, each outside the one before
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170 POINTS AND LINES OP EQUILIBRIUM. [113.
it, and at all points of any one of these surfaces the electrical
force will be 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 n. Now, in this case the force is every-
where outwards, so that the surface-integral is necessarily posi-
tive, and therefore there is a positive charge within the surface,
and, since we may take the surface as near to P as we please,
there is a positive charge at the point P.
In the same way we may prove that if F 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 Pis
a point of stable equilibrium for some displacements, and of
unstable equilibrium for other displacements._
TIST] 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
TJ be the lowest, and T^ the highest potential existing in the
electric field. If we make (7 = TJ, 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 wiU
expand, and new negative regions will be formed round nega-
tively charged bodies. For every negative region thus formed
the surrounding positive region acquii*es one degree of peri-
phraxy.
As the different negative regions expand, two or more of them
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113.] THEIB NUMBEK. 171
may meet in a point or a line. I£ 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 nth degree.
When C becomes equal to T^ 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 degi'ees of the positive points and lines of equilibrium will be
^—1, and that of the negative ones n—l. The surface which
surrounds 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 con-
Digitized by VjOOQ iC
172 POINTS AND LINES OP BQXnUBBIUM. [115.
ductorSj the number arising in this way cannot exceed
(71— 1) (ti— 2), where n is the number of bodies*.
114.] The potential close to any point P may be expanded in
the series v= TJ+JTi + Sa + fcc;
where H^^ H^, &c. are homogeneous functions of a?, y, z, whose
dimensions are 1, 2, &c. respectively.
Since the first derivatives of V vanish at a point of equi-
librium, -ETj = 0, if P be a point of equilibrium.
Let ff^ 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 H^.
Now J7» = 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.
K 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 n sheets of the same equipotential surface intersect,
they must intersect at angles each equal to ir/n.
For let the tangent to the line of intersection be taken as the
axis of Zj then cPV/ds^ = 0. Also let the axis of a; be a tangent
to one of the sheets, then cPV/dx^ = 0. It follows firom this, by
Laplace's equation, that cPV/dy^ = 0, or the axis of y is a tangent
to the other sheet.
This investigation assumes that ff^ ^^ fii^te. If H^ vanishes,
let the tangent to the line of intersection be taken as the axis
* {I have Bot been able to find any place where this result ii proved.)
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115.] THBIB PBOPBBTIBS. 173
of z^ and let a; = r cos 0, and ^ = r sin 0, then, since
the solution of which equation in ascending powers of r is
F= Vq + ilircos(^ + ai) + A^r^eo^{2 d + Og) + &c. + il^r*cos(n^+ o^).
At a point of equilibrium ii^ is zero. If the first term that does
not vanish is that in r^ then
F— % = -4»r* cos {nO-\- a^) + terms in higher powers of r.
This equation shews that n sheets of the equipotential surface
F = T^ intersect at angles each equal to it/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
siurfiEtce. During the next stage of the process, the equipotential
surface round the second body which has the same potential as
the first body wiU cut the surface of the second body at right
angles in a dosed curve, which is a line of equilibrium. This
* 'Suninary of the ProperUea of oerUin SteMm JAo&b,* TkU. Mag,, Oct. 1864.
See elBO, Thomion And T*iV8 Natural Pkilatopkj, f 780 ; and Bankine and Siokes,
in the Proc. S, 8., 1867, p. 468 ; alM W. R Smith, Proe, B. 8. Edin, 1869-70, p. 79.
{Thif inyeetigation ii not tatis&otoiy ae tPV/dg^ only Taniihei along the aads of g.
Bankine's original proof ia rigid. Hm may be written as
«,«""" + tt»f !«■'""*+ • •••I
where «. , «a+i>«- *i^ homogeneoni innctionB of a;, y of doffreee n, « + 1 respeotively, the
axis of « is a singular line of degree n. Since Jim satisfies V JETm s 0, we must have
or v« — ilr* cos (fi0 + a) ; but «. » 0 is the equation of the tangent planes from the
axis of s to the oone Hm » 0, that is of the n sheets of the equipotential surface, hence
these cut at angle v/n.}
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174 POINTS AND LraES OF EQUILIBRIUM. [l l6.
closed curve, after sweeping over the entire surface of the con-
ductor, 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.
Eamsharva 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 JS, 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 £, and let e be the
electricity on a small portion of the moveable body A surround-
ing this point. Then the potential energy of A with respect to
JB will be if=2(ye),
where the summation is to be extended to every charged portion
of A
Let a, 6, c be the coordinates of any charged part of A with
respect to axes fixed in A, and parallel to those of x, y, z. Let
the absolute coordinates of the origin of these axes be f, y;, C
Let us suppose for the present that the body A is constrained
to move parallel to itself, then the absolute coordinates of the
point a, 6, c will be
«=f+a, y = »i + 6, z=C+c.
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 a, 6, c and f, i}, C, and the sum of these
terms is a function of the quantities a, 6, c, which are constant
for each point of the body, and of f, »;, C, 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 6m (PM_
rif« ■•■ dri^ '^ dC^
Now let a small displacement be given to Ay so that
rff = IdVy drj = mcZr, dC= ndri
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m
1 1 6.] EQUILIBEIUM ALWAYS UNSTABLE. 175
and let dMhe the increment of the potential o{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 R = dM/dr tending to diminish r and
to restore j1 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 r, 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 V W = 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 oth^ 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,
aild its equilibrium is therefore necessarily unstable.
The body therefore is imstable even when constrained to move
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 moveable 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 d 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 parallel
to itself 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.
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176 POINTS AND LINES OF BQUILIBBITTM.
Hence the total increment of the potential when the electricity
is free to move will be
and the force tending to bring A back towards its original
position will be dM ^
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.
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CHAPTER VIL
FORMS OF THE EQUIPOTENTIAL SUBFACES 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
dW dW d^F_
da^ ■*" dy' ■*" rf«^ "" '
V being a function of Xy y, and 0, which is always finite and con-
tinuous, which yanishes 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 func-^
tion 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 solu-
tion has been Qonstructed 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 constructed
by the inverse process.
This historical knowledge of results can be turned to account
in two ways. If we are required to devise an instrument for
making electrical measurements with the greatest accuracy, we
may select those forms for the electrified surfaces which corre-
VOL, I. N
Digitized by VjOOQ iC
178 BQUIPOTBNTIAL 8UEFACBS. [ll8,
spond to cases of which we know the accurate solution. If, on
the other hand, we are required to estimate what will be the
electrification of bodies whose forms are given, we may begin
with some case in which one of the equipotential surfaces takes
a form somewhat resembling the given form, and then by a
tentative method we may modify the problem 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 calculation of the
electrification. It appears, therefore, that what we want is a
knowledge of the forms of equipotential surfaces and lines of
induction in as many different cases as we can collect together
and remember. In certain classes of cases, such as those relating
to spheres, 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 surfaces, 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
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119] USE OF DIAGBAMS. 179
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 distribu-
tion of electricity on one of the oval figures, larger at one end
than the other, which surround both centres. Such a body, if
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 equipotendal surfekce, 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 ssero 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 equi-
librium, and somewhat 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 nega-
tive. In this case one of the equipotential surfaces, that, namely,
corresponding to potential zero, is a sphere. It is marked in the
diagram by the dotted circle Q. The importance of this spherical
surface will be seen when we come to the theory of Electrical
Images.
We may see from this diagram that if two round bodies are
charged with opposite kinds of electricity they will attract each
other as much as two points having the same charges but placed
somewhat nearer together than the middle points of the round
bodies.
* {This can be seen by comparixig the distance! between the equipotential sarCftoes
in TBiious parts of the field. }
N 2
Digitized by VjOOQ iC
180 BQUIPOTBNTIAL SUBPAOES [l20.
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
equUibrium.
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 ttiis surface we have others having a rounded
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.
If we consider points on the axis on the further side of the
point jB, 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
perpendicular to the axis, if is a point of minimum force rela-*
tively to neighbouring points on that surface.
120.] Figure HI 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
conductor, and another a long way below A as the surface of
another conductor at a difierent potential, the system of lines
* (Maxwell does not give the strength of the field. M. Comodowever has calca>
lated the strength of the uniform field &om the diagram of the lines of force and finds
that its eleotromotiYe intensity before the introduction of the charged bod/, was
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121.] AND LINES OP INDUCTION. 181
and surfaces between the two conductors will indicate tlie distri-
bution 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 lY 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 j5 12 units of nega-
tive 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 = 26.
In this case, the surface for which the potential is zero consists
of two spheres whose centres are A and C and whose radii are 1 5
and 20. These spheres intersect in the circle which cuts the plane
of the paper at right angles in D and 2)', so that B is the centre of
this circle and its radius is 12. This circle is an example of a
line of equilibrium, for the resultant force vanishes at every
point of this line.
If we suppose the sphere whose centre is j1 to be a conductor
with a charge of 3 units of positive electricity, 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
below the small circle DIX will be negatively charged by the
influence of C. All the rest of the sphere will be positively
charged, and the small circle DI/ 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 influenced by 15 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 JDiX, charged with 23 units of positive elec-
tricity.
We shall return to the consideration of this diagram as an
illustration of Thomson's Theory of Electrical Images. See
Art. 168.
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182 BQUIPOTBNTIAL 8UEFACB8 [l22.
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 contem-
plated at once, is called Stress, and may be described as a trans-
ference 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 line of Force,
but it is more correct to call it a line of Induction.
Now the quantity of electricity in a body is measured, accord-
ing 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 eartii, or on the
heavenly bodies, and wherever they terminate there is a quantity
of electricity exactly equal and opposite to that on the part of
the body from which they proceeded. By examining the dia-
grams tiiis 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 continuous 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.
Digitized by VjOOQ iC
Digitized by VjOOQ IC
n.erk MajcA/vpJJ.'s ElecU^ioJfy, VolJ .
To fajceP.163.
riG.6.
Lines of Force'
Etjuipotenticd Surfaces.
Method ofdra^ving
Lines of Force andi Equip otertticLL Surfaces.
UrdMersWyPra^s, Oxford.. ^.^^^^^ ^^ GoOglc
123.] ^^ I'INBS OF INDUCTION. 183
128.] These diagrams are constructed in thefoUowing 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=i e/r;
hence, if we make r = e/V, we shall find r, the radius of the sphere
for which the potential is F. 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 numben^. 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 sur&cee belonging to it, and if we now
wish to find the form of the equipotential surfaces due to both
centres together, we must remember that if ]{ be the potential due
to one centre, and T^ that due to the other, the potential due to
both will be T[4- 1^= V. Hence, since at every intersection of
the equipotential surfaces belonging to the two series we know
both TJ" and T^, we also know the value of V. If therefore we
draw a surface which passes through all those intersections for
which the value of F is the same, this surface will coincide with
a true equipotential surface at all these intersections, and if the
original systems of surfaces 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
pointj we must draw them so that they mark out on the equi-
potential 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
Digitized by VjOOQ IC
184 EQUIPOTENTIAX SUEFACBS [123.
through the centre of force. Any straight line radiating from
the centre and making an angle 6 with the axis will then trace
out a cone, and the surface-integral of the induction through that
part of any surface which is cut off by this cone on the side next
the positive direction of the axis is 2 ir 6 (1 —cos 0).
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
J e (1 — cosd) =s *, say ;
and 0 = cos^^ (l — 2 -)•
If we now give to <l> a series of values 1, 2, 3... e, we shall find
a corresponding series of values of 6^ and if 6 be an integer, the
number of corresponding lines of force, including the ^is, 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 suiface 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 chaiges are 10 and —10 respect-
ively.
If there are two centres of force on the axis of the figure we
may draw the lines of force for each axis corresponding to values
of <t>i and <t>2, and then, by drawing lines through the consecutive
intersections of these lines for which the value of <t>i + ^^ is the
same, we may find the lines of force due to both centres, and in
the same way we may combine any two systems of lines of force
which are symmetrically situated about the same axis. The
continuous curves on the left-hand side of Fig. 6 represent the
lines of force due to the two chained 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*
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123.] ^ND LINES OP INDUCTION. 185
In the case of any such system of finite dimensions the line of
force whose index number ^ <t> has an asymptote which passes
through the electric centre (Art. 89 d) of the system, and is in-
clined to the axis at an angle whose cosine is 1 — 2 4>/e, where e
is the total electrification of the system, provided <l> 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 con-
jugate functions*.
* See » paper ' On the Flow of ElectHdtj in Conducting Sorfaoes/ by Prof. W. B.
Snutli, Proe, £.8. Edin., 1869-70, p. 79.
ajfjL^/CC fi^r^-fjM, M./K4^ ^ ^f^ tL^ ^ tfi4^ s^ ^i^
JJ^ TJiuJc jls^w -//^w^^ X^^^^^^ f^.r,M.
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CHAPTER Vm.
SIMPLE CASES OF ELECTEIPICATION.
Two Parallel Planes.
124.] We shall consider, in the first place, two parallel plane
conducting surfaces of infinite extent, at a distance c firom 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 jB, 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
cPV ^
the integral of which is
r^C, + C,z;
and since when 0 = 0, F= A, and when 0 = c, F= jB,
For all points between the planes, the resultant intensity is
normal to the planes, and its magnitude is
C
In the substance of the conductors themselves, Ji = 0. Hence
the distribution of electricity on the first plane has a surface-
density (T, where
47r(r= jK = •
C
On the other surface, where the potential is jB, the surface-
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SIMPLE GASES. PABALLEL PLANES. 187
density t/ mil be equal and opposite to (r, and
c
Let us next consider a portion of the first surface whose area
is j9, taken so that no part of 8 is near the boundary of the
surface.
The quantity of electricity on this surface is c^ = flfa-, and, by
Art. 79, the force acting on every unit of electricity is iiJ, so
that the whole force acting on the area fif, and attractrug it
towards the other plane, is
Sir Stt c*
Here the attraction is expressed in terms of the area S^ the
difference of potentials of the two surfaces (ji— j5), and the dis-
tance between them o. The attraction, expressed in terms of the
charge e^^ on the area £•, is ^ _ 2^ ,
The electric energy due to the distribution of electricity on
the area £i, and that on the coiTCsponding area S on the surface
B defined by projecting 8 on the surface j5 by a system of lines
of force, which in this case are normals to the plane, is
\2
= *4,
_ B{A-Bf
'4
Sir
2ir
S.^^'
= Jb,
The first of these expressions is the general expression of elec-
tric energy (Art. 84).
The second gives the energy in terms of the area, the distance,
and difference of potentials.
The third gives it in terms of the resultant force iZ, and the
volume 8c included between the areas 8 and 8f^ and shews that
the energy in unit of volume is p where Btt j9 = J2^.
The attraction between the planes is ^8^ or in other words,
there is an electrical tension (or negative pressure) equal to jp on
every unit of area.
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188 SIMPLE CASES. [125.
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.
To express the charge in terms of the difference of potentials,
we have 1 ^ / ^ »v /a »\
The coefficient 9 represents the charge due to a difference of
potentials equal to unity. This coefficient is called the Capacity
of the surface &y due to its position relatively to the opposite
surface.
Let us now suppose that the medium between the two surfaces
is no longer air but some other dielectric substance whose specific
inductive capacity is Ky then the charge due to a given difference
of potentials vrill be K times as great as when the dielectric ia
air, or ^^ r a »\
^ 4 ire ^ '
The total energy will be
The force between the surfaces will be
-^ Sir c*
2ir
Hence the force between two surfaces kept at given potentials
varies directly as K^ the specific inductive capacity of the dielec-
tric, but the force between two surfaces charged with given
quantities of electricity varies inversely as K.
Two ConcerUric Spherical Surfaces.
125.] Let two concentric spherical surfaces of radii a and 6, of
which b is the greater, be maintained at potentials A and B
respectively, then it is manifest that the potential F is a function
of r the distance from the centre. In this case, Laplace's equa-
tion becomes cPV 2 dV ^ ^
dr^ r dr ^ '
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125-] CONOENTEIO SPHEEIOAL SUEFACES. 189
The solution of this is
and the conditions that V=A when r = a, and V=^B when r = 6,
give for the space between the spherical surfaces,
^ Aa-Bb A-B .
a— 6 a^ — 6^
dV _ A-B
If (Tj, (Tj are the sorfiace-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 «i and Bg are the whole charges of electricity on these
surfaces,
ei = 4woVi = ^=i— pr = -«2-
jL
The capacity of the enclosed sphere is therefore r — •
K the outer surface of the shell is also spherical and of radius c,
then, if there are no other conductors in the neighbourhood, the
charge on the outer surface is
^3 = Be,
Hence the whole charge on the inner sphere is
and that on the outer shell
^2 + ^3 = 5^(^-^) + ^^-
If we put 6 = 00, we have the case of a sphere in an infinite
space. The electric capacity of such a sphere is a, or it is
numerically equal to its radiu&
The electric tension on the inner sphere per unit of area is
The resultant of this tension over a hemisphere is ira^ = ^
normal to the base of the hemisphere, and if this is balanced by
a surface tension exerted across the circular boundary of the
hemisphere, the tension on unit of length being T, we have
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190 SIMPLE OASES. [126.
Hence ^ = -^^1 4 = ^»
le^ra (6— a)*
If a spherical soap bubble is electrified to a potential J, then,
if its radius is a, the charge will be Aa^ and the surface-density
will be 1 ^
~ 47ra *
The resultant intensity just outside the surface will be 47r(r,
and inside the bubble it is zero, so that by Art. 79 the electric
force on unit of area of the surface will be 2 irir*, acting outwarda
Hence the electrification will diminish the pressure of the air
within the bubble by 27r<r*, or
^_^
But it maybe shewn that ilT^ 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 2TQ/a. If
the electric force is just sufficient to keep the bubble in equi-
librium 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 an inner surface of a hollow
cylinder, having the same axis vrith the first, be 6. Let their
potentials be A and B respectively. Then, since the potential V
is in this case a function only of r, the distance from the axis,
Laplace's equation becomes
dr^ r dr '
whence F= C^ + C^ log r.
Since F= A when r^^ay and F=5 when r = 6,
h T
A\og- + J?l0flr -
y ^r ^_a
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127.]
COAXAL CTLINDBBS.
191
If a-i, ^2 are the sorface-densitiea on the inner and outer
surfifuses,
A-B . B-A
4ir<ri=-
4ir<r2 =
alog-
Mog-
If Ci and €2 are the charges on the portions of the two cylinders
between two sections transverse to the axis at a distance I from
eachother^ A-^B
Ci = 2ira2<ri = i — r- I = —e^*
The capacity of a length I of the interior cylinder is therefore
logs
If the space betwen the cylinders is occupied by a dielectric of
specific inductive capacity K instead of air, then the capacity of
a length I of the inner cylinder is
b '
The energy of the electrical distribution on the part of the
infinite cylinder which we have considered is
Kg. 5.
127.] Let there be two hollow cylindric conductors A and 5,
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 coordinates.
Let a cylinder C of length 22 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 hollow cylinder on the positive side be
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192 SIMPLE CASES. [127.
Af that of the one on the negative side B^ and that of the internal
one (7, and let us put o for the capacity per unit of length of C
with respect to A, and p 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
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=ia(^ + «)(<?-^)^ + ii3(i-aj)((7-5)2 + quantities
independent of x ;
and the resultant force parallel to the axis of the cylinders since the
energy is expressed in terms of the potentials will by Art. 93 6 be
If the cylinders A and B are of equal section, a = ^, and
X ^ a{B-'A){C^i{A+B)).
It appears, therefore, that there ia a constant force acting on
the inner cylinder tending to draw it into that one of the outer
cylinders from which its potential differs most.
If (7 be numerically large and A-hB 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 (7, 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
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12 7.] COAXAL OYLINDEBS. 193
A is zero, and that of B and (7 is F, then the quantity of elec-
tricity on il wiU be ^3 = (gi3 + a(Z+a;)) F;
where q^^ is a quantity depending on the distribution of electricity
on the ends of the cylinder but not upon x^ so that by moving C
to the right till x becomes 0? + ^ the capacity of the cylinder C
becomes increased by the definite quantity af, where
1
a =
a and b being the radii of the opposed cylindric surfaces.
VOL. I.
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CHAPTEE IX.
SPHERICAL HARMONICS.
128.] The maihematical theory of spherical harmonics has
been made the subject of several special treatises. The Handbuch
der Kugelfvmctionen 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 Beitrdge
zur Theorie der Kugdfunctionen (Leipzig, Teubner, 1878). The
treatment of the subject in Thomson and Tait*s Natural Philo-
Bophy is considerably improved in the second edition (1879), and
Mr. Todhunter's Elementary Treatise on Lapla/^es FunctionSy
Lamii's Functions, and BesaeTs Functions, together with Mr.
Ferrers' Elementary Treatise on Spherical Harmonics and
sufy'ects connected with them, have rendered it unnecessary to
devote much space in a book on electricity to the purely mathe-
matical 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, il^, of electiicity is uniformly spread over
the surface of a sphere the coordinates of whose centre are
(a, 6, c), the potential at any point {x, y, z) outside the sphere is,
by Art. 125, a
^ F=^, (1)
where r* = (aj-a)« + (y-6)2 + (0-c)«. (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 A^ is placed on the surface
of an infinitely small sphere, which is sensibly the same as a
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SPHBEIOAL HARMONICS. 195
mathematical point. We have already (Arts. 55, 81) shewn that
there is a limit to the siirfiEkce-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 (1) 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 Aq
condensed at the mathematical point (a, b, c), and we may call
the point a singular point of order zero.
There are other kinds of singular points, the properties of
which we shall presently 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
correspond to them.
1296.] 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 /rom the centre in the direction
of the axis meets the surface. This point is called the Pole of
the a^. An axis has therefore one pole only, not two.
If /A is the cosine of the angle between tiie axis k and any
vector r, and if p^ ^^^ (3)
p is the resolved part of r in the direction of the axis h.
Different axes are distinguished by different suffixes, and the
cosine of the angle between two axes is denoted by X^^, where
m and n are the suffixes specifying the axis.
Differentiation with respect to an axis, A, whose direction
cosines are Z, Jf, N^ is denoted by
From these definitions it is evident that
dK ^
If we now suppose that the potential at the point (aj, y, z) due
to a singular point of any order placed at the origin is
02
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196 SPHEBICAL HARMONICS. [129C.
then if such a point be placed at the extremity of the axis A,
the potential at (x^ y, z) will be
Af[{x^Lh\ (y-Mk), {z^Nh)l
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
r=^Af[{x^Lh\ {y-Mh\ {z^Nh)]^Afix,y,zl
= -^Ah^f{x, y, z) + terms containing A*.
If we now diminish A 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
F=-il'^/(a^y.4 (8)
lif{x, y, z) satisfies Laplace's equation, then, since this equation
is linear, K, which is the difference of two functions, each of
whidi separately satisfies the equation, must itself satisfy it.
129 c.] Now the potential due to a singular point of order zero,
Vo = ^,l' (9)
satisfies Laplace's equation, therefore every function formed from
this by differentiation with respect to any number of axes in
succession must also satisfy that equation.
A point of the first oi'der may be formed by taking two points
of order zero, having equal and opposite charges —A^ and ^0,
and placing the first at the origin and the second at the extremity
of the axis h^ . The value of h^ is then diminished and that of
4o increased indefinitely, but so that the product Aq h^ is always
equal to Ai- The ultimate result of this process, when the two
points coincide, is a point of the first order whose moment is A^
and whose axis is hi. A point of the first order is therefore a
double point Its potential is
mtiai 18 y
(10)
By placing a point of the first order at the origin, whose
moment is — ili, and another at the extremity of the axis h^
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I30a.] SOLID HAEMONIO OP POSITIVE DEGREE. 197
whose moment is il^, and then diminishing h^ and increasing il^,
«otl«* A,h,^iA„ (11)
we obtain a point of the second order, whose potential is y
^ A,i'Ji!tipb.: ^ (12)
We may call a point of the second order a quadruple point
because it is constructed by making four points of order zero
approach each other. It has two axes hi and A^ and a moment
A 2. 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 n*^ order. It will be
the product of three factors, a constant, a certain combination of
cosines, and r~(»+i). It is convenient, for reasons which will
appear as we go on, to make the numerical value of the constant
sudi that when all the axes coincide with the vector, the co-
efficient of the moment is r~<*+^). We therefore divide by n
when we differentiate with respect to h^.
In this way we obtain a definite numerical value for a par-
ticular potential, to which we restrict the name of The Solid,
Harmonic of degree — fa -hi), namely
y. /i\» ^ d d d I ,»
^•"^"^^ 1.2.3...ndh^'dh^'''dh,'r' ^ ^
If this quantity is multiplied by a constant it is still the
potential due to a certain point of the n^ order.
129 d.] The result of the operation (13) is of the form
lJ=i;r-<*+i), (14)
where ^ is a function of the n cosines fAi.../A« of the angles
between r and the n axes, and of the in(n^ 1) cosines Aj2, &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 }^ as a quantity
varying from point to point on that surface, being a function of the
I n (n + 1) distances between the n poles of the axes and the pole
of tie vector. We therefore call I^ The Surfa^^ TTi^prmnift nf
order n.
180a,] We have next to shew that to every surface-harmonic
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198 SPHEEIOAL HAEMONICS4 [1306.
of order n there corresponds not only a solid harmonic of degree
— (ti + 1) but another of degree n, or that
5;, = I3;r-= TJr2"+i • (16)
satisfies Laplace's equation.
For ^= (2n+l)r2-ia?Tj:+r2«+i^,
'^" = (27i+l)[(2n-l)ar^ + r2]r2«-3]r+2(27i+l)r2— iflj^
^^ d^
Hence
Now, since 1^ is a homogeneous function of x, y, and z, of
negative degree «+ 1,
dV d^ dV , ,.rr /,,x
The first two terms therefore of the right-hand member of
equation (16) destroy each other, and, since J^ satisfies Laplace's
equation, the third term is zero, so that H^ 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 (x, y, z) is a function
of X, y, and z which satisfies Laplace's equation, then there exists
another function, a„^a^x a^y a^Zy.
which also satisfies Laplace's equation. See Art. 162.
1806.] The surface harmonic IJ[ contains 2n arbitrary vari-
ables, 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 T^ and H^ also contain 2n arbitrary
variables. Each of these quantities, however, when multiplied
by a constant, will satisfy Laplace's equation.
To prove that AH^ is the most general rational homogeneous
function of degree n which can satisfy Laplace's equation, we
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I3I&.] SOLID HABMONIO OF POSITIVE DBGEEE. 199
observe that f, the general rational homogeneous function of
degree ti, contains J (n+ 1) (71 + 2) terms. But V^K is a homo-
geneous function of degree 71— 2, and therefore contains ^ 11 (71— 1)
terms, and the condition V^K = 0 requires that each of these
must vanish. There are therefore 1 71(71— 1) equations between
the coefficients of the i(7i-f l)(n+2) terms of the function K,
leaving 271 + 1 independent constants in the most general form
of the homogeneous function of degree n which satisfies Laplace's
equation. But J?^, when multiplied by an arbitrary constant,
satisfies the required conditions, and has 2 7i + 1 arbitrary con-
stants. It is therefore of the most general form.
131a.] 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 J^ = IJjr~^"+^) satisfies the condition of vanishing
at infinity, but becomes infinite at the origin.
The function H^ = I^r* is finite and continuous at finite dis-
tances from the origin, but does not vanish at an infinite distance.
But if we make a*^r~<*+^) the potential at all points outside
a sphere whose centre is the origin, and whose radius is a, and
a-<*+*)IJir" the potential at all points vrithin the sphere, and if
on the sphere itself we suppose electricity spread with a surface
density cr such that
4w<ra« = (271+1)^, (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 every-
where finite and are continuous exeept at the charged surface,
where they satisfy *the equation
av dv
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 c it is the only distribution which
can satisfy them.
181 6.] The potential due to a sphere of radius a whose surface-
density is given by the equation
4wa2(r=(27i+l)Ii, (20)
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200 SPHBEIOAL HARMONICS. [^3^0.
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 4^ is the potential
due to this system, and let us find the value of 2{^e) for the
singular point. This is the part of the electric eneigy depending
on the action of the external system on the singular point.
If ilo is the charge of a singular point of order zero, then the
potential energy in question is
Ttj; = A)*. (21)
K 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 hi, then the potential energy will be
and when A^ increases and hi diminishes indefinitely, but so that
AqKi = Au the value of the potential energy for a point of the
first order will be
Similarly for a point of order n the potential energy will be
W^^—^A^^r'^ » (23)
181c.] If we suppose the charge of the external system to
be made up of parts, any one of which is denoted by dE^ and
that of the singular point of order ^ to be made up of parts
any one of which is de, then
* = 2(id^). (24)
But if T^ is the potential due to the singular point,
K=S(^de). (25)
^nd the potential energy due to the action of j& on 6 is
Tlj; = 2 (♦(fe) = 22 i^-dEde) = '2,(y^dE), (26)
the last expression being the potential energy due to the action
of e on E.
* We shftll find it oonvenient, in wli»t follows, to denote the prodnot of the posi-
tive integral nnmben 1.2.8... n by fil
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132.] SINGULAE POINT EQUIVALENT TO A CHAEGED SHELL. 201
Similarly, if ada is the charge on an element ds of the shell,
since the potential dud to the shell at the external system E
is IJ, we have
TIJ = 2 {V^dE) = 22 (^dBirds) = 2 (*<rcfo). (27)
The last term contains a summation to be extended over the
surface of the sphere. Equating it to the first expression for W^y
we have
ff<ifad8 = 2(*de)
1 . d*SP , .
^^r^dh,..,dK' ^ ^
If we remember that ivaa* = (2n + l)']^, and that A, = a", this
becomes .
This equation reduces the operation of taking the surface
integral of ^IJ^ds over every element of the surface of a sphere of
radius a, to that of differentiating 4^ with respect to the n axes
of the harmonic and taking the value of the differential coeffi-
cient at the centre of the sphere, provided that ^ satisfies
Laplace's equation at all points within the sphere, and I^ is a
surface harmonic of order n.
132.] Let us now suppose that 4^ is a solid harmonic of positive
degree m of the form _ _-.tt -. /„/v\
^ ♦ = a"^]^r*. (30)
At the spherical surface, r^s a, and >P = ]^, so that equation
(29) becomes in this case
J:^7,ds = -,il_^a— « ^^ , (31)
* iil(2ti+l) dhi...dh^ ^ '
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?, j/, and z of degree m— ti, the value of
which at the centre of the sphere is zero. If ti is equal to m the
result of the differentiation is a constant, the value of which we
shall determine in Art. 134. If the differentiation is carried
further, the result is zero. Hence the surface-integral I j T^T^da
vanishes whenever m and n are different.
The steps by which we have arrived at this result are all of
//
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202 SPHERICAL HASMONICS. [133.
them purely mathematical, for though we have made use of terms
having a physical meaoing, 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 to follow a mathematical calculation, will
understand it all the better if each of the steps of the calculation
admits of a physical interpretation.
183.] We shall now determine the form of the surface har-
monic I^ as a function of the position of a point P on the sphere
with respect to the n poles of the harmonic.
We have
and so on.
Every term of 3J[ therefore consists of products of cosines,
those of the form /x, 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 8 cosines with double suffixes,
there must hen^28 cosines with single suffixes.
Let the sum of all products of cosines in which a of them have
double suffixes be written in the abbreviated form
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 /a's only or among the A's only, we write it as a suffix to the
fjL or the X. Thus the equation
2 (m"~^*X') = 2 (mI"^^') + 2 (fx"-2.x'J (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.
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1 33-] TBIGONOMBTBIOAL EXPRESSION. 203
Let us now assume that for a particular value of n
+ ^,..2(m-2-X') + &C., (34)
where the A'q are numerical coefficients. We may write the
series in the abbreviated form
7, = 8[A,^Ml^'-^'^')l (35)
where 8 indicates a summation in which all values of 8, including
zero, not greater than in, are to be taken.
To obtain the corresponding solid harmonic of negative degree
{n-\- 1) and order 71, we multiply by r~^"+^), and obtain
jr = fif[ii, .r2'-2-i2(^«-2«V)], (36)
putting TfA = J?, as in equation (3).
If we differentiate J^ with respect to a new axis h^ we obtain
—(71+1) TJ+i, and therefore
(ii+ 1) TJ+i = 8[A^,. (271+ l-28)r2-2-8 2 (pl'^'^h')
-^,..r2-2-i2(2)-2*-iA;;,"*'')]. (37)
If we wish to obtain the terms containing s cosines with
double suffixes, we must diminish 8 by unity in the last term,
and we find
(n+1) T;;^! = /8f[r2-2»-s|^^^(2ti«28+l)2(^;;,-*'"''\')
-il....,2(p-«-^iA'J}]. (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 X'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 71+ 1 for 1^ in the expression
for TJ and multiplying by 71+ 1, we obtain the following equa-
^"""^^ (7l+l)ii.+i.. = (271-28+ l)il,.. = -^...-1. (39)
If we put a = 0, we obtain
(7l+l)il,^,., = (27l+l)ii,.o; (40)
and therefore, since J.^ q = 1,
_ 27i! ,
^••<>"2"(7il)«* -^ ^
and from this we obtain the general value of the coefficient
-^-..-l ^^2«-* 71 1 (71-8) I' ^"^
and finally the trigonometrical expression for the surface har-
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204 SPHERICAL HARMONICS. [134.
monic, as f9^^^fi\i
This expression gives the value of ihe 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 har-
monic will have its sign reversed. For any cosine involving 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 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 surftu^e
integral l Y^li^de 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 T^r^ and
to differentiate it with respect to each of the n axes of 1^.
Any term of I^r* of the form r'"/A"'~^*X* may be written
^'l^m *'^1»' I^iflferentiating this n times in succession with
respect to the n axes of 1^, we find that in differentiating r**
* {We mAjr dedaoe from this that
C2»-l)(2n-»)
^ £• (^f,.a:»»- V*' + «<^««'^"'«' + r<?6-a V«'"*
(2n-l)(2n-3)(2»-6)'
where n « j? + g -1- r uid E*^z* + j^ + «', and mC. denotes the number of peimutations
of m things n at a time divided by 2^(^)t-}
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1 35 a] ff^m^nds. 205
with respect to 8 of these axes we introduce a of the ^.'s, and
the numerical factor
28(28— 2).. .2, or 2*8!.
In continuing the differentiation with respect to the next 8 axes,
the p.'s become converted into A^k'S) ^^^ ^o numerical factor is
introduced, and in differentiating with respect to the remaining
n*-28 axes, the p^% become converted into A«m'S} so that the
result is 2'8!a;;^x;;^a;;;;*'.
We have therefore, by equation (31),
and by equation (43),
K^=4(-.y,J?:7(lt),M''-i':-o]- («)
Hence, performing the differentiations and remembering that
m = n, we find
185 a.] The expression (46) for the surface-integral of the pro-
duct of two surface-harmonics assumes a remarkable form if we
suppose all the axes of one of the harmonics, ^, to coincide with
each other, so that 1^ becomes what we shall afterwards define
as the zonal harmonic of order m, denoted by the symbolic.
In this case all the cosines of the form A^^ ^^7 ^^ written //.,
where fi^ denotes the cosine of the angle between the common
axis of i^ and one of the axes of }^. The cosines of the form
Aw«» will all become equal to unity, so that for SAJ[^ we must
put the number of combinations of 8 symbols, each of which is
distinguished by two su^xes out of ti, no suffix being repeated.
« I We can see thU if we conader how many permutatioiu of the suffixes of one
term in the expreenon SX^ we can form. The luffizes conaiit of t groups of two
naniben each, by altering the order of the groups we can form s ! arrangements, and
by interchanging the order of the numbers inside the groups we can form from any
one of these arrangements 2' other arrangements, so Uuht from each of the groups of
suffixes we can get 2' s ! arrangements ; thus, if ^ be the number of tenns in the series
•^Kmk*^^' 1 arrangements of the n numbers taken 2s at a time maybe made, but the
whole number of arrangements thus made is evidently the number of permutatiouH of
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206 SPHBRIOAIi HARMONICS. [l35&-
The number of permutations of the remaining n—28 indices of
the axes of i^ is (n~ 28) 1 Hence
sec") = («-2«)l /-"-*•• (")
Equation (46) therefore becomes, when all the axes of 1^
coincide with each other,
A 2
= 2^^ ^(~)> l>y equation (43), (50)
where ^(^) denotes the value of 1^ at the pole of j^.
We may obtain the same result by the following shorter
process : —
Let a system of rectangular coordinates be taken so that the
axis of z coincides with the axis of ij,, and let ^r" be expanded
as a homogeneous function of x, y, z of degree n.
At the pole of i2», ic = y = 0 and 0 = r, so that if Cz^ is the
^ term not involving x or y, C is the value of 1^ at the pole of ^.
Equation (31) becomes in this case
4wa^ 1 dr
If
* " 271+1 nld2^^^ *
As m is equal to 71, the result of differentiating Csf^ is 71 1 (7, and
is zero for the other terms. Hence
//
• " 271+1 '
C being the value of Xk ^t the pole of i^.
135 6.] This result is a very important one in the theory of
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 da the element of
surface at a point Q of the spherical surface, then if we multiply
Fda by ij, 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
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136.] CONJUGATE HAEMONIOS, 207
ds of the surface a zonal harmonic is constructed having its pole
at Q and having a coefficient Fda.
We shall thus have a system of zonal harmonics superposed
on each other with their poles at every point of the sphere where
jPhas a value. Since each of these is a multiple of a surface
harmonic of order 71, their sum is a multiple of a surface har-
monic (not necessarily zonal) of order n.
The surface integral / / FP^da considered as a function of the
point P is therefore a multiple of a surface harmonic H^ ; so that
is also that particular surface harmonic of the n^^ order which
belongs to the series of harmonics which expresses Fy provided
F can be so expressed.
For if F can be expressed in the form
J"=^]^ + iliI[+&c. + ^,i;; + &c.,
then if we multiply by P^da and take the surface integral over
the whole sphere, all terms involving products of harmonics of
different orders will vanish, leaving
//
271+1 * •
Hence the only possible expansion of jPin spherical harmonics is
F==Z'^^[JjFP^d8+kQ. + {2n+l)JjFP^d8'^&<^^ (51)
CoryugcUe 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 surfiEU^
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 n^ order, then, in
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208 SPHBEICAL HAEM0NIC8. [137.
order i^t a second harmonio 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 (2n+ 1)^ harmonic will have
2n conditions to satisfy by means of its 2 n variables, and will
therefore be completely determined.
Any multiple Al^ of a surfeice harmonic of the n*^ order can
be expressed as the sum of multiples of any set of 2 71 + 1 con-
jugate harmonics of the same order, for the coefficients of the
271+1 conjugate harmonics are a set of disposable quantities
equal in number to the 2 n variables of ^ and the coefficient A.
In order to find the coefficient of any one of the conjugate
harmonics, say Y^ suppose that
az = a^y;;+&o.+a^yi +&c.
Multiply hj Yl ds and find the surface integral over the sphere.
All the terms involving products of harmonics conjugate to each
other will vanish, leaving
a/JkIT da = A,ff(r:yds, (52)
an equation which determines A^*
Hence if we suppose a set of 27i+l conjugate harmonics
given^ any other harmonic of the n^ 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.
187.] We have seen that if a complete system of 2 71+ 1 har-
monics of the n^ 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 71 + 1 harmonics there are 2 7i(2 71 + 1)
variables connected by 7i(27i+l) equations, 71 (2 71 + 1) 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 thaty of them coincide at the pole of the
axis of oj, k at the pole of y, and I (= n^j^k) at the pole of z.
The 71+1 distributions for which 2=0 and the n distributions
for which 1=1 being given, all the others may be expressed in
terms of these.
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138.] ZONAL HAEMONICS. 209
The system which has been actually adopted by all mathe-
maticians (including Thomson and Tait) is that in which ti— <r
of the poles are made to coincide at a point which we may call
the Positive Pole of the sphere, and the remaining o- 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 Mj, /ut,, ... /i„_o.are each of them equal to cos d, which
we shall denote by fi. If we also write if for sin d,/ut„_<,.+i, ... ij^
are of the form v cos (0— iS), where p is the azimuth of one of the
poles on the equator.
Also the viJue of k^^ is unity if p and q are both less than
71—0-, zero when one is greater and the other less than this
number, and cossv/a- when both are greater, 8 being an integral
number less than <r.
188.] When all the poles coincide at the pole of the sphere,
<r = 0, and the harmonic is called a Zonal harmonic. As the
zonal harmonic is of great importance we shall reserve for it the
symbol I^.
We may obtain its value either from the trigonometrical
expression (43) or more directly by differentiation, thus
p _ 1.3.S...(2^-l) r n{n-l)
"~ 1.2.3...W r 2.(2»-l)'^
'n{n-l){n-2){n-3) ._4_fe- 1
+ 2.'i.{2n-l){2n-3f '^j
= ^[(-lY {2n-2p)\ 1 . .
L^ ^2*pl{n-p)lin-2p)l'* J' ^ *''
where we must give to p every integral value from zero to the
greatest integer which does not exceed i n.
It is sometimes convenient to express i, as a homogeneous
function of cos 0 and sin 6, or, as we write i^em, ix and p,
=^[(-'^»6..;(L»rti'---""]- ("'
It is shewn in the mathematical treatises on this subject that
VOL. I. p
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210 SFHEBIOAL HABMOXICS. [140a.
iJ(/A) is the coefficient of A* in the expansion of (1 — 2/iA + A*)""*
1 d^
{and that it is also equal to — — • j-iiCM^^O'l-
The surface integral of the square of the zonal harmonic, or
Hence £^{PMfd^ = 2liTT " ^"^
139.] If we consider a zonal harmonic simply as a function
of M) and without any explicit reference to a spherical surface, it
may be called a Legendre's Coefficient.
K we consider the zonal harmonic as existing on a spherical
surface the points of which are defined by the coordinates 0 and <^,
and if we suppose the pole of the zonal harmonic to be at the point
(^, <^'), then the value of the zonal harmonic at the point (d, <^)
is a function of the four angles d', <l>\ 0, <f>, and because it is a
function of fx, the cosine of the arc joining the points (d, <^) and
((^, <^'), it will be unchanged in value if 0 and e\ and also <^ and <l>\
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 iv, y, z which satisfies Laplace's
equation 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. Li 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 2^1+1
variables, is the more restricted surface harmonic multiplied by
an arbitrary constant. The more general surface harmonic, when
expressed in terms of 0 and 0, is called a Laplace's Function.
140 a.] To obtain the other harmonics of the symmetrical sys-
tem, we have to differentiate with respect to <r axes in the plane
of aey inclined to each other at angles equal to ir/<r. This may
be most conveniently done by means of the system of imaginary
coordinates given in Thomson and Tait's Naturol Philosophy^
vol. I, p. 148 (or p. 185 of 2nd edition).
If we write i = x + iy, ri = x—iy,
where i denotes a/— 1, the operation of differentiating with respect
to the (T axes if one of these axes makes an angle a with x may
be written when o- is odd in the form
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I40a.] TESSEBAL HABlfOinOS. 211
This equals
cos <ra
{d' d'\. Ad' d') ,^„,
If 0- is even we may prove that the operation of differentiating
may be written
-we may express the operation of differentiating with respect to the
a axes in terms oi D8,D c. These are, of course, real operations,
and may be expressed without the ose of imaginary symbols, thus :
We shall ako write
-J-— 2)8 = 2)8, and ^^r=zDc=iDc; (62)
80 that Ds and 2)c denote the operations of differentiating with
n n
respect to n axes, ti— <r of which coincide with the axis of z,
while the remaining o- make equal angles with each other in the
plane o£ xy, Ds being used when the axis of y coincides with
« la)
one of the axes, and Do when the axis of y bisects the angle
n
between two of the axes.
The two tesseral surface harmonics of order n and type o- may
now be written /^x * (<^) ^
r8=(-l)"-Lr"+i2)8i, (63)
7c=(-l)«f.r-«2)ci. (64)
n Til nV
Writing /ut = cosd, j; = sin^, p^ = a? + y^^ r*=rf?;+«*,
so that 2f = fir, p ^ vr, a? = p cos 0, y = p sin ^,
we have 2)^I^i = (-i).WLi(,.-f.)-2_ , (es)
p a
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and
212 SFHEBIOAL HABMONICS. [140 a.
in which we may write
IC'y'-ff) = P'sincrc^, lii' + r) = p'COBa<l>. (67)
We have now only to differentiate with respect to z, which we
may do so as to obtain the result either in terms of r and z, or as
a homogeneous function of z and p divided by a power of r,
^n^^ ^ _ (2n)l2^a\ 1
[ — '''7(t:r)"'''--"'-^H'" ""
°' dzf^<'i^'*i~^ ■' (2«r)l r«"+i
[^._("-;)(»-p')^-V.^^]. (6,)
If we write
» L 2(211—1) '^
^ 2.4(2«-l)(2«-3) '^ ^J *- '
50 = „,L— _ (n-<r)(n-^-l) ,^
«!r'=%'':jur"■^^ (">
so that these two functions differ only by a constant factor.
We may now write the expressions for the two tesseral har-
monics of order n and type a in terms either of 0 or ^^
^U -Mi_0(')28m.« = |?±fii5r28ina^. (73)
1^0= (^^)^ 0<->2cos<rd>=^r-^^)^.5<->2coS(r(^t. (74)
* {Equation (68) may eaaily be proved by notioing that the left-hand nde is («— o') !
r 1 1 » If ^hz + h\ *
times the coefficient of A*"^ in | ^ ^ /^ ^ ^Nt J 'OT;i^+ril+ — ^3 — J ' "
we write this a8^j^^|(l + f»-.) +F"pl ' and pick out the coefficient of *—',
we get equation (69)}.
t {This value must be halved when c^^J]
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I40&.] TESSEBAL HABMONICS. 213
We must remember that when <r = 0, sin a <^ = 0 and cos o-c^ = 1 .
For every value of <r from 1 to n inclusive there is a pair of
(0) (0)
harmonics, but when <r = 0, Fe = 0 and Fc = ^, the zonal har-
monic. The whole number of harmonics of order n is therefore
2n+ 1, as it ought to be.
1406.] The numerical value of F adopted in this treatise is
that which we find by differentiating r~^ with respect to the n
axes and dividing hj n\ It is the product of four factors, the
sine or cosine of <r<^, u^, a function of /a (or of /i and v), and a
numerical coefficient
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 i^ into a series
of descending powers of /Lt> the first term being fx*"*, it is the
function which we, following Thomson and Tait, denote by 0.
The function which Heine {HaTidbuch der KiLgd/unctumeny
§ 47) denotes by j^'*^ and calls eine zugeordnete Function erster
Art, or, as Todhunter translates it, an ' Associated Function of
the First Blind,' is related to 0|^^ by the equation
0^'> = (-l)^^<">. (76)
The series of descending powers of /i, beginning with /i*~^, is
expressed by Heine by the symbol 5p^"\ and by Todhunter by the
symbol «r (a, n).
This series may also be expressed in two other forms,
_ 2-{n-a)\n\ d- . .
- (2nj\ d^^""' ^ ^
The last of these, in which the series is obtained by differentiating
the zonal harmonic with respect to ii, seems to have suggested the
symbol T^^^ adopted by Ferrers, who defines it thus
2<-)-^*lp-_ii!^)i_0<'>. (77)
When the same quantity is expressed as a homogeneous
function of /ut and y, and divided by the coefficient of m*""' ^^'j ^^
is what we have already denoted by ^^.
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214 SPHERIOAL HABKONIOS. [14I.
140c.] 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.
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 n^^ degree, all whose roots lie
between — 1 and + 1, and therefore correspond to n parallels of
latitude on the sphere.
The zones included betwete 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 spaca
The other harmonics of the symmetrical system odsur in pairs,
one involving the cosine and the other the sine of o-<^. They
therefore become zero at <r meridian circles on the sphere and
also at 71— 0- parallels of latitude, so that the spherical surface is
divided into 20- (71— o-— 1) quadrilaterals or tesserae, together with
4 o- triangles at the poles. They are therefore useful in investiga-
tions 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 2 n 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 F^'^
into a solid haimonic of positive degree by multiplying it by r".
We differentiate this solid harmonic with respect to the n axes of
the harmonic itself, and then make a: = j/ = 2; = 0, aiid we
A 2
multiply the result by —7-7- -^ •
^•^ -^711(271+1)
These operations are indicated in our notation by
Writing the solid harmonic in the form of a homogeneous
function of z and f and 1;, viz..
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142 &.] SITBPAOB INTBGBALS, 215
we find that on performing the differentiations with respect
to 0, all the terms of the series except the first disappear, and
the factor (Ti—a) I is introduced.
Continuing the differentiations with respect to f and i; we
get rid also of these variables and introduce the factor — 2i 0*1, so
that the final result is
JJ^ n^ 271+1 2^^n\n\ ^ '
We shall denote the second member of this equation by the
abbreviated symbol [?i, o-].
This expression is correct for all values of <r from 1 to ti inclu-
sive, but there is no harmonic in sin o-^ corresponding to o- = 0.
In the same way we can shew that
J J K^V "^ - 271+ 1 2^''7ll7ll ^^^f
for aU values of a from 1 to ti inclusive.
When o- = 0, the harmonic becomes the zonal harmonic, and
a result which may be obtained directly from equation (50) by
putting I^ = i^ and remembering that the value of the zonal
harmonic at its pole is unity.
142 a.] 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 -4^ be the
coefficient of Y^J^ in the expansion of this function in surface
harmonics of the symmetrical system, then
ffFy:'ds = A^'Jf(Yt')'d8 = 4'>[n, «r] . (83)
where [71, a] is the abbreviation for the value of the surface in-
tegral given in equation (80).
1426.] Let ^ be any function which satisfies Laplace's equa-
tion, 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.
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216 SPHERICAL HABM0NI08. [l43-
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.
But a more convenient method, and one which does not involve
integration, is by differentiation with respect to the axes of the
harmonics of the symmetrical system.
For instance^ let us suppose that in the expansion of % there is
(<r) (<r)
a term of the form Ac Yc r\
n n
K we perform on * and on its expansion the operation
and put X, y, z equal to zero after differentiating, all the terms
(<r)
of the expansion vanish except that containing Ac,
Expressing the operator on 4^ in terms of differentiations with
respect to the real axes, we obtain the equation
dz'-' Ida^ 1 . 2 diC-* dy^ ^ J
= i?(!i±4fc^% (84)
from which we can determine the coefficient of any harmonic
of the series in terms of the differential coefficients of ^ with
respect to x^yyZ 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
equipotential 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 at 120** in the plane of the
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I44<*-] DIAGRAMS OF SPHERICAL HARMONICS. 217
paper, and this difference is the harmonic of the second type in
which <r = 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 at
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. Yin represents the difference of two zonal harmonics of
the fourth order whose axes are at right angles. The result is a
tesseral harmonic for which n = 4, o- = 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 con-
nected part of the spherical surface which lies outside the ovals,
the harmonic 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 mag-
netism if it were distributed according to the most simple tjrpe.
144 a.] 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 il„r*I^ be one of these, then since within the conducting
sphere the potential is uniform, there must be a term — il^r*]^
arising from the distribution of electricity on the surface of the
sphere, and therefore in the expansion of 4 inr there must be a
term ^ita^ = {2n+ l)a*~i A^Y^.
' Digitized by VjOOQ iC
218 SPHBEIOAL HABMONICS. [^44 &•
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 4ir<ro = a~*c.
The potential of the sphere is
** a
144 &.] Let us next suppose that the sphere is placed in the
neighbourhood of conductors connected with the earth, and that
Green's Function, (?, tss been determined in terms of a;, y, z and
^') j/} ^i the coordinates of any two points in the region in which
the sphere is placed.
K the surface density on the sphere is expressed in a series
of spherical harmonics, then the electrical phenomena outside the
sphere, arising 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 difierent orders corresponding to the harmonics
which express the surface density.
Let Green's function be denoted by Gpj/, where p indicates the
point whose coordinates ai-e x, y, z, and p' the point whose co-
ordinates are x\ y^, z\
If a charge Aq is placed at the point p\ then, considering
aj', y', / as constants, G^^f becomes a function of x,y,z\ and the
potential arising from the electricity induced on surrounding
bodies by il^ is * = Af^G^^/. (1)
If, instead of placing the charge Aq at the point p\ it were
distributed uniformly over a sphere of radius a having its centre
at p\ 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
^Tta^a^z AQ-\-SAiYi-^&c. + {2n+l)AJ'^+.... (2)
The potential arising from any term of this distribution, say
47raV^ = (27i+l)^X i^)
will be — ;^ A^y^ for points inside the sphere, and -^^ A^T^ for
points outside the sphere.
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144 &•] gbbbn's PUNOTioisr. 219
Now the latter expression, by equations (13), (14), Arts. 129 c
and 1 29 d is equal to . v^ . a* d* 1
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 Ai . . . A^ and whose moment is
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.
The potential, therefore, at the point jt>, or (aj, y, «), due to the
induced electrification of surrounding bodies, is
where the accent over the c2's indicates that the differentiations
are to be performed with respect to x\ y", z\ These coordinates are
afterwards to be made equal to those of the centre of the sphere.
It is convenient to suppose Y^ broken up into its 27i+ 1 con-
stituents of the symmetrical system. Let ^^ Y^"^ be one of
these, then d'* . , i^s.
It is unnecessary here to supply the affix a or c, which indicates
whether sin o-^ or cos <r^ occurs in the harmonic.
We may now write the complete expression for *, the potential
arising from induced electrification,
♦ = ^G'+22[(-l)M^"^"i)^')(?] . (6)
But within the sphere the potential is constant, or
♦ + 1^0+22 [^^|;;.>1^.>]= constant. (7)
Now perform on this expression the operation D^^ , where the
difierentiations are to be with respect to a;, y^ 0, and the values
of n^ and o-} are independent of those of n and a. All the terms
of (7) will disappear except that in Y^^^ and we find
g(7h + <ri)!(^~<ri)l 1 .<o
2^1 Til I a«»+i \
= ^2)<^>(?+22[(-l)«< Jj2)i^^>^^^^^ (8)
We thus obtain a set of equations, the first member of each of
Digitized by VjOOQ iC
220 SPHERICAL HARMONICS. [l45 «.
which contains one of the coefficients which we wish to deter-
mine. The first term of the second member contains A^, 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
**» 2(7ii + cri)!('ni-(ri)I ^ \ ^ ^
If the shortest distance from the centre of the sphere to the
nearest of the surrounding conductors is denoted by 6,
If, therefore, b is large compared with a, the radius of the
sphere, the coefficients of the other spherical harmonics are very
small compared with -4^,. The ratio of a term after the first on
the right-hand side of equation (8) to the first term will there-
fore be of an order of magnitude similar to (r)
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(I+i7, (1)
where ^ 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 ^ be expanded in the form of a series of surface harmonics
^=/o+/iir+/2i$+&c-+/x (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 /^
will disappear.
The second term, that in /j, depends on the distance of the
centre of mass of the conductor, supposed of uniform density,
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I45«-] NBAELY 8PHBEICAL CONDUOTOBS. 221
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
Aq , and that no external electrical force acts on it. The potential
outside the conductor must therefore be of the form
r=A, I +^,^'i +&c.+^X^+.... (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 jP, and
n^lecting the square and higher powers of jP, we have
+A,±.j:il-(n+l)]l)+.... (i)
Since the coefficients il^, &c. are evidently small compared
with Aq^ we may begin by neglecting products of these co-
efficients 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
« = A^' (6)
A,Ti^=A,af,X=0, (6)
AJ:' = Aoa'f,7,. (7)
It follows from these equations that the Vb must be of the
same type as the Fs, and therefore identical with them, and
that -4.1 = 0 and A^ = A^a^ f^.
To determine the density at any point of the surface, we have
the equation ^ dV dV • x i /«x
^ 4ircr = —-7- = — -r~^8 ^> approximately ; (8)
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 0 and (^ to be
small, we may put cos e = 1, so that
4^<r=-^ = ^„l+&c. + («+l)^.^^,+ .... (9)
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222 SPHBBIOAL HABH0NIC8. [l45&*
Expanding the powers of r in terms of a and F, and neglecting
products of F into -4.^, we find
4,r(r=iloi(l-2^ + 8w5. + («+l)^.^,^. (10)
Expanding jPin spherical harmonics and giving A^ its value
as already found, we obtain
4^<r=^i[l+AIJ+2/3^+&0. + (n-l)/.^. (11)
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 ti, the ratio of the difference of the surface
densities at any two points to their sum will be 7i~l times
the ratio of the difference of the radii at the same two points to
their sum.
145 &.] If the nearly spherical conductor (1) is acted on by
external electric forces, let the potential, Uy 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
?7' = 5o+^i^ir'+^8^5J'+&c. + 5^r*i;;'+..., (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
F= A,\-B,^X'-&c.-B,''^T:-.... (13)
Let the actual potential arising from the surface charge be
F+ TT, where
Tr = C,iF/' + &c. + (7«^F."+...; (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 F is small.
The condition to be fulfilled is that, when r = a (1 +^,
Cr+ F+ F = constant = Aq--¥B^,
the potential of the conductor.
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H5^-] NEARLY SPHEBIOAL VESSELS. 223
Expanding the powers of r in terms of a and F, and retaining
the first power of ^ when it is multiplied by A or B, but neglect-
ing it when it is multiplied by the small quantities (7, we find
+ C,ir + &c. + ^«^^"+...= 0. (15)
To determine the coefficients C,we must perform the multipli-
cation indicated in the first line, 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 con-
ductor.
The product of two surface spherical harmonics of orders n
and m, is a rational function of degree n + m in. a/r, y/r, and z/r,
and can therefore be expanded in a series of spherical harmonics
of orders not exceeding m + ti. If, therefore, F can be expanded
in spherical 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 harmonics of orders
not exceeding m + n.
This surface density can then be found from the potential by
the approximate equation
47r(r+^([7"+r+Tr) = 0. (16)
145 c.] A nearly spherical conductor enclosed in a nearly
spherical and nearly concentric conducting vessel.
Let the equation of the surface of the conductor be
r = a(l+^, (17)
where ^=/i Jr+&c.+/^'> y^^^^^. (18)
Let the equation of the inner surface of the vessel be
r = 6(1 + 0), (19)
where » = fl^i 3?+ &c. +g^;> y^<'>, (20)
the/*8 and gr's being small compared with unity, and F; being
the surface harmonic of order n and type <r.
Let the potential of the conductor be a, and that of the
vessel )9. Let the potential at any point between the conductor
and the vessel be expanded in spherical harmonics, thus
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224 SPHEBIOAL EABHONIOS. [146.
then we have to determine the constants of the forms h and k so
that when r=a(l +i^, 4' = a, and when r = 6(1 + G), 4^ = /3.
It is manifest, from our former investigation, that all the JCs
and A;'s except h^ and k^ will be small quantities, the products of
which into F may be neglected. We may, therefore, write
« = *o+*o^(l-fO+&c.+ (AWa- + A;W_l^)y<')+..., (22)
We have therefore , , 1 ,„ .
« = ^+^o^' (24)
fi = K + K\, (25)
*o^yi'> = A::^a"+Ar^. (26)
whence we find for k^, the charge of the inner conductor,
*o = («-i8)5??^„. (28)
and for the coefficients of the harmonics of order n
h^:^=klIlL—lil^,
^6an+l_^2n+l
(29)
where we must remember that the coefficients f^\ ^^\ h^^\ k^^^ are
those belonging to the same type as well as order.
The surface density on the inner conductor is given by the
equation
_/;^{(^ + 2)^'"^^ +(^- l)fe'^^^ } -9n\^n+ 1) a^^'b- ^ ,3^.
where il„— j!>aii+i_^2n+i ' ^ ^
146.] As an example of the application of zonal harmonics,
let us investigate the equilibrium of electricity on two spherical
conductors.
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146.] TWO SPHBfilCAL OONDUCTOBS, 226
Let a and h be the radii of the spheres, and c the distance
between their centres. We shall also, for the sake of brevity,
write a = cx^ and & = c^^ 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 8 the distance of the same point from that of the
second sphere.
Let the surface density, o-i, of the first sphere be given by the
equation
4ir(ria*=:il + iliiJ+3ilj-^4-&c. + (2m+l)il^ii, (1)
so that A is the total charge of the sphere, and A^, &c. are the
coefficients of the zonal harmonics P^, &o.
The potential due to this distribution of charge may be repre-
sented by
U'=\[A^A,P:-^A,^^,^^^A^P,g\ (2)
for points inside the sphere^ and by
D'=i[il + ^i,^+A^^V&c. + .l.P.^] (3)
for points outside.
Similarly, if the surface density on the second sphere 'is given
by the equation
^Tta^lfl = 5+ 5i^+&c. + (27i+ 1)5^^, (4)
the potential inside and outside this sphere due to this charge
may be represented by equations of the form
F'=i[5 + £i^| + &c. + B.P.i;j, (5)
F = l[£+5,^^ + &c.+B.P.^;]. (6)
where the several harmonics are related to the second sphere.
The charges of the spheres 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 tT' + F = a. (7)
VOL. I. Q
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226
SPHEBIGAL HABMONICS.
[146.
Similarly, if the potential of the second sphere is /3, for points
vrithin that sphere, jj^ V'^, ff. (8)
For points outside both spheres the potential is % where
U+r=*. (9)
On the axis, between the centres of the spheres,
r-^a^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
(H)
A
1
dV
da ~
0.
A
- +
dii'~
0.
•
A
• ■
ml
• *
d'V
*«^.m+l
where, after differentiation, a is to be made equal to c.
If we perform the differentiations, and write a/c = x and
h/c = y, these equations become
0 = ilj + J5a;3 + 35i»3y + 6^2^? V + &c. + i (71 + 1) (ti + 2) 5^aj V»
0 = il^ + 5a;-»-i + (m+l)5ia;"-*-iy + 4(^^+0(^ + 2)52aj"'+V
ml 71 1 ' '
By the correspondiog operations for the second sphere we find,
0 = Bi + Ay^+3AiXy'^+6A^as^^ + gm. + i(vi+l){m+2)A^x''y\
0 = B, + ily"+H (to + 1) Aa^"** + K« + 0(« + 2)-4ja!*y"+^ + &c.
(m+^ ,
ml 71 1 ^ /
(12)
(13)
To determine the potentiab, a and /3, of the two spheres we
have the equations (7) and (8), which we may now write
ca=zAl + B + B,y + B^y^ + kc. + B,y-, (14)
cp=zB--{-A + A^x + A^ix? + &c,'i-A^x'^. (15)
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146,] TWO SPHBEIOAL CONDUOTOES. 227
If, therefore, we confine our attention to the coefficients A^ to
A^ and B^^ to £„, we have m + n equations from which to deter-
mine these quantities in terms of A and B, 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 o{ B^.^.B^ from equa-
tions (13), we find
ili = -5ar* + iliB*y8[2.1 + 3.1y2 + 4.1y* + 5.l2/» + 6.1y» + ...]
+ il2ajV[2.3 + 3.62/« + 4.lOy* + ...]
-hA^a^y^[2A'{^3.10y^-\'...]
+ A^afly^[2.5 + ...] (16)
+
A^=i'-Bx^ + Aa^y^[3.l'i-6Ay^+l0.ly^+15.ly^ + ...]
+ ilia^y3|-3.2 + 6.3j/^+10.4y* + ...]
+ A^afy^[3.3 + e.Gy^+...]
+ i!3^V[3.4 + ...] (17)
+
il3 = -5aj* + ila:*t/^[4.l + 10.1y2+20.1y* + ...]
+ ilia:«i/[4.2 + 10.3j/2 + ...]
+ A^a:f^y^[4.3 + ...] (18)
+
A^ = -Bafi + A afy^[5.l + l5Ay^+...]
^A,afiy^[5.2 + ...] (19)
+
By substituting in the second members of these equations the
approximate values of Ai &c., and repeating the process for
further approximations, we may carry the approximation to the
coefficient to any extent in ascending powers and products of x
Budy. If we write ^^= ;>*^-?«^,
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2^8 SPHEBIOAL HABMONIGS. [146.
we find
^,j = (T^t/s [ 2 + 3^2 + 4y* + 5y^+ ef-\- ly^"" + %y^^ + 9y'* + &c.]
+ aJ*2/'[ 8+ 302/2+ 75y*+1542/» + 280jf» + &c.]
+ ajV[l®+ 90y2 + 288y* + 7352/« + &c.]
+ a?» y« [32 + 200t/« + 7802/* + &c.]
+ a;"2/*[50 + 3752/*+&c.]
+ aj8 2^ [32 + 1 922/2 -h&c]
(20)
gi = rc2
+ a:«2/'[4+ 9^+ 162/*+ 252/«+ 362/*+ 492/^<^ + 642/^2 + 8cc.]
+ ajV[ 6 + 1^2^+ 402/*+ 762/«+l262/» + 196yi^ + &c.]
+ iC^2/^[ 8 + 302/2+ 802/*+176y» + 3362/® + &C.]
+ aj"y8[l0 + 452/2 +1402/* + 3502/* + 8cc.]
+ a;i3y8[l2 + 63y2 + 2242/*+&c.]
+ «iV[l^ + 842/^ + &c.]
+ a;^^2/3[-i6^&o.-j
+ aj8 2/«[ 16+ 722/2+ 2092/* + 4882/« + 8m5.]
+ a^V[ 60+ 342y2+i2222/*+&c.]
+ a;^^ [ 150 + 1 0502/2 + &C.]
+ aj^V[308 + &c.]
+ iB"y«[ 64 + &0.] (21)
+
It will be more convenient in subsequent operations to write
these coefficients in terms of a, 6, 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
^i = 2a263c-« + 3a26«c-'' + 4a26^c-» + (5a269 + 8a«6«)c-"
+ (6a26n + 30a»68 + l%a?}fi)c'^^
+ {7a?h^^ + 75a«6«> + ^Qa?l^ + Z2a?¥)c''^^
+ (8a26w + 154a«6i2+ 288a'^6io + ^2a^i^ + 200a»68 + 50ai*6<^) c--^^
Digitized by VjOOQ IC
146.] TWO 8PHBEI0AL CONDUOTOES. 229
+ (9a*&" + 280a«6"+ 735a'^6^2 + 192a86" + 7B0a^b^^
+ 144a^«6» + 376a"68 + 72a"6«)c-i>> + ... . (22)
+ (lOaii6«+30a«6»+16a86« + 40a''6^ + 26a*6»)c-i*
+ (1 2a"63 + 45a^i6« + 60ai<^6« + BOa^V'
-|-72a«6« + 76a'6« + 36a»6")c ^«
+ (14a"68 + 63a"6«+160a^26«+140aii6'^ + 342a*o68
+ 175a»6» + 209 a* 610 + 126a'^6ii + 49a«6i>)c-"
+ (16ai''6» + 84ai«6» + 308a"6« + 224ai«6^+1060ai26»
+ 414aii6» + 1222ai«6" + 336a»6" + 488a86iHl96a^6"
+ 64a«6")c-«>+.... (23)
p^ = 3a»68c-« + 6a56«(j-8 + lOa^i^c-^o ^ (i2a«6« + ISan^)^^^
+ (27a»6« + 54a«6« + 21a3in)c-i*
+ (48aio6«+162a86« + 168a«6io+28a86i8)c-i«
+ (76ai*6« + 360aio6« + 48a»6» + eOGa^ft*®
+ 372a«6i* + 36a36i6)c-"^.... (24)
q.^ = a8c-»+ 6a»63c'« + (9a«63+ i8a«6«)c-"
+ (12aio6« + 36a86« + 40a«6^)c-i3
+ (l5ai2j8^60ai«6« + 24a*6«+100a»6'' + 75a«6»)c-i*
+ (18ai*6» + 90a^6* + 90aii6« + 200aio6^
+ 126a« 6® + 225a«6» + 126a« 6^1)0-1^
+ (21ai«63+126ai*6»+225a"6« + 350ai2j7^ 594^11^8
+ 626a^<^6* + 418a«6io + 441a«6ii + 196a«6i8)c-" + .... (26)
p^ = 4a*6»c-'' + 10a*6«c-» + 20a*Vc''^^ + (I6a^6« + 36a*6»)c-i3
+ (36a»6« + 84a"^68 + 56a*6")c-"
+ (64a"6« + 262a»68 + 282a^6iH84a*6i^)c-" + -... (26)
^3 = a*c-*4-8a''6^c-io + (12a»68 + 30a''6«)c-i2
+ (lea^ift* + 60a^6** + 80a^6'')c-i*
+ (20ai«6^ + 100a"6« + 32ai«6« + 200a»6^+175a^6»)c-"
+(24ai«6Hl50ai86*+120a"6« + 400a"6^ + 192aio6«
+ 626a»6»+336a''6")c-i« + .... (27)
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230 6PHEBI0AL HABMONICS. [146.
+ (46aioi«+120a«6» + 126a«6")c-" + .,.. (28)
+ (25a"6»+160a^«6«+40a"6« + 360aio6'^ + 360a*6»)c-" + ... (29)
Pf^ = 6a«63c-» + 21 a«6»c-"+ 56a«6^c-"
+ (24a«6« + 126a«6»)c-i«+.... (30)
q^ = a«c-« + 12a»63c-^2 + (I8a"63 + 63a»6«)c-i*
+ (24ai863^126a"6'^ + 224a«&^)c-i« + .... (31)
p^ = 7a''68c-" + 28a^6»c-^^ + 84a''&''c-" + ... , (32)
g, = a''c-^+14aio63c-i» + (21a«63 + 84aio6«)c-" + .... (33)
2>7 = 8a«6»c-" + S6a«6'^c-i^ + -... (34)
57 = a«c-«+ 16a"6'^c-i* + ... . (35)
p^-9aH^c^^^ ■{-.... (36)
g8 = a»c-5 + .... (37)
The values of the r's and 8*a may be written down by inter-
changing a and b in the q*u and 2>*8 respectively.
If we now calculate the potentials of the two spheres in terms
of these coefficients in the form
a=zlA +w5, (38)
pz= mA'+nB, (39)
then 2, m, 71 are the coefficients of potential (Art. 87), and of these
m = c~"^ +piac~'^ i-p.^a^c~^ + &c., (40)
71 = 6"* — q^ac^^q^a^c^^ico., (4 1 )
or, expanding in terms of a, 6, c,
7/1 = c-i + 2a^¥c-'^ + 3a368(a» + fc«)c-» + a868(4a* + Sa^I^ + 46*)c-ii
+ a»68[6a«+10a*6* + 8a»6»+10a*6* + 56«]c-i8
+ a86»[6a« + 16a«6«+30a«6» + 20a*6*
+ 30a»6«4- 16a*6« + 66«]c-i«
+ a368[7a^o + 21 a»6« + 76a'^63+36a«6*+144a«6»
+ 36a*6« + 76a36' + 21a26« + 76i<^]c-"
+ a86»[8a^« + 28aio62+164a»6» + 66a86* + 446a'6«+102a«6«
+ 446a«6'' + 66a*6« + 154a»6« + 28a*6i<> + 86^«]c-"
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146.] TWO SPflEBIOAL CONDUOTOBS. 231
+ 1668a''6'' + 318a«6» + 1107a«6» + 84a*6^o+280a36^»
+ 36a26i2^96i*]c-2i+.... (42)
-(a»+12a263 + 96«)a«c-i2-(a^ + 25a*68 + 36a26«+166^)a«c-"
— (a» + 44a«63 + 96a*6«+ 16a86« + 80a26^ + 256»)a«c-i«
-(a" + 70a863 + 210a«6« + 84a«fc« + 260a*6^
+ 72a^¥ + ISOa^i® + 366ii)a«c-"
-(a^8+104a^«63^406a86« + 272a*'6« + 680a«6*' + 468a«6»
+ 675a*6* + 209a36io + 252a^b^^ + 496^3)^6^-20
-(a^5 + 147a^2j8^720ai056^ 693^9 J6^ 1548^857^ 1836a^68
+ 1814a«6^ + 1640a* 6^^+ 11 13a*6" + 488a36^2
+ 392a26i8 + 646i5)a«c-2»+.... (43)
The value of I can be obtained from that of n by interchanging
a and 6.
The potential energy of the system is, by Art. 87,
W =^ HA^-^mAB-hinB^, (44)
and the repulsion between the two spheres is, by Art. 93a,
The surface density at any point of either sphere is given by
equations (l) and (4) in terms of the coefHcients A^ and B^.
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CHAPTER X.
CONPOOAL QUADEIO STJEPACBS *.
147.] Let the general equation of a confooal system be
-^+-^ + -^=1 (1)
A2-.a« + A2_ft2 + x*_^» - 'y VU
where A is a variable parameter, which we shall distinguish by a
suffix for the species of quadric, viz. we shall take X^ for the
hyperboloids of two sheets, X^ ^^^ ^® hyperboloids of one sheet,
and A3 for the ellipsoids. The quantities
^> Aj, Oy A29 C, Aj
are in ascending order of magnitude. The quantity a is intro-
duced for the sake of symmetry, but in our results we shall
always suppose a = 0.
If we consider the three surfaces whose parameters are
^i> ^2) ^8> ^^ ^^> ^7 elimination between their equations, that
the value of a^ at their point of intersection satisfies the
equation
x'{b^^a'){c^^a^) = (Ai8-a«) (A2«-a*)(A32-a«). (2)
The values of y^ and s^ may be found by transposing a, 6, c
symmetricaUy.
Differentiating this equation with respect to A^, we find
dA^-A^^-a^^- ^^^
If {foi is the length of the intercept of the curve of intersection
of A2 and A3 cut off between the surfaces A^ and A^ + c^A^, then
d8i
dA^
dx
\dy
^ dXi
dz
^ d\.
* This inyeBtigation is chiefly borrowed from a very interesting work, — Le^om »nr
lei Fonotiom Inverses dee Tratueendantee et les Surfaces Isothermee, Par 6. Lam^
Pant, 1857.
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CONFOCAL QUADEIO SUEFAOBS, 283
The denominator of this fraction is the product of the squares
of the semi-axes of the surface X^.
If we put
D^« = A3»«A,^ D,« = A3«-A^^ and D3« = V-V, (6)
and if we make a = 0, then
^=-=i^ (6)
^^1 -v/fta-Ai^V^^IV
It is easy to see that D^ 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 D3 is parallel to cfeji <^d
D^ to ds^.
If we also substitute for the three parameters A^, A,, A3 their
values in terms of three functions a, fi, y, defined by tiie equations
-5:
=r
V(6*-V)(c«-V)
cdk^
/9=r, '^"' (7)
then cfojs-DjDsda, ^8, = -DgDjti^, d8^ = -LiD^dy. (8)
C 0 c
148.] Now let F be the potential at any point a, /9, y, then the
resultant force in the direction of d^ is
■^~ (fo,~ do<fo, ~ daD^D^' ^'
Since da^^da^, and c^s, are at right angles to each other, the
surface-integral over the element of area da^da^ is
R,da,da,^-^^^-^.-L-^.d^dy
Now consider the element of volume intercepted between the
surfaces a, j3, y, and a+cJo, fi + dfi, y-hdy. 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 p and P+dp^ y
and y+dy.
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234 CONFOCAL QUADEIC SUBPAOES. [l49-
The surface-integral for the corresponding element of the
surface a + da will be
da c da^ 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 d^VD^
-j—o —^dadBdy.
da^ c
Similarly the surface-integrals for the other two pairs of faces
^^^ dWD^^, ,^ , , dWD^^, ,^ ,
^-^—=-dadfidy and -^ -^dadpay.
These six fapes enclose an element whose volume is
2) 22) 22) «
da^ds^ds^ = ^ J — - dadfidy,
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 tt, is
zero, or, dividing hy dad^dy,
d^-^'-^d^^^'-^W '^ ^^ c^ ^ ' ^ ^
which is the form of Poisson's extension of Laplace's equation
referred to ellipsoidal coordinates.
If P = 0 the fourth term vanishes, and the equation is equi-
valent to that of Laplace.
For the general discussion of this equation the reader is
referred to the work of Lam^ already mentioned.
149.] To determine the quantities a, ^, y, we may put them in
the form of ordinary elliptic integrals by introducing the auxiliary
angles 0, </>, and ^, where
Xi = 6sin^, (12)
Xg = Vc^wnV+ft^cos^, (13)
Aj^csec^^. (14)
If we put 6 = faj, and yP + A;'^ = 1, we may call k and J(f the
two complementary moduli of the confocal system, and we find
a=r^'.., (15)
an elliptic integral of the first kind, which we may write ac-
cording to the usual notation F{k 0).
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150.] DISTBIBUTION OP ELEOTBICITT. 235
In the same way we find
^ =.[*-— ^ = F{k')-F(k',<l>), (16)
Jo Vl—A;'^ 008^0
where F{k') is the complete function for modulus k\
Here a is represented as a function of the angle B, which is
accordingly a function of the parameter Aj, j9 as a function of 4*
and thence of A^, and y as a function of yjf and thence of A3.
Sut these angles and parameters may be considered as func-
tions of o, ft y. The properties of such inverse functions, and of
those connected with them, are explained in the treatise of
M. Lam^ on this 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 o, ft y: the periods of A^ and A3 are iF{k), and that
ofA2is2^(ifc').
Particular SclutioTis.
150.] If F is a linear function of a, /3, or y, the equation is
satisfied. Hence we may deduce from the equation the distri-
bution of electricity on any two confocal surfaces of the same
family maintained at given potentials, and the potential ai) any
point between them.
The Hyperholoids 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 a^, Oj be the values of a corresponding to two single sheets,
whether of different hyperholoids or of the same one, and let
T^, T^, be the potentials at which they are maintained. Then, if
we make y^a^V^-a,V^-^a{r^'V,) ,^3)
the conditions will be satisfied at the two surfaces and throughout
the space between them. If we make V constant and equal to V[
in the space beyond the surface c^, and constant and equal to
f^ in the space beyond the surface a^^ we shall have obtained
the complete solution of this particular case.
Digitized by VjOOQ iC
236 C0N70CAL QUADBIO 8UBFAGES. [150.
The resultant force at any point of either sheet is
If Pi be the perpendicular from the eentre on the tangent
plane at any point, and II the product of the semi-axes of the
surface, then p^ DJD^ = P^.
Hence we find p X-^<^P\ ro^\
or the force at any point of the surface is proportional to the
perpendicular ftt)m the centre on the tangent plane.
The surface-density a- may be found from the equation
4ir<r = J2i. (22)
The total quantity of electricity on a segment cut off by a plane
whose equation iQx^d from one sheet of the hyperboloid is
The quantity on the whole infinite sheet is therefore infinite.
The limiting forms of the surface are : —
(1) When a = F{k) the surface is the part of the plane of xz
on the positive side of the positive branch of the hyperbola
whose equation is ^% ^
(2) When a = 0 the sur&ce is the plane of yz.
(3) When a = — jP(A;) 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.
Sy making /3 constant we obtain the equation of the hyper-
boloid of one sheet. The two surfaces which form the boun-
daries 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
suiface will be proportional to the perpendicular from the centre
on the tangent plane, and the whole quantity on the infinite
sheet will be infinite.
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150.] DISTBIBUnON OP BLECTBIOITY. 237
Limiting Forms.
(1) When jS = 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 /3 = F{}(f) the surface is the part of the plane of xy
which is on the outside of the focal ellipse whose equation is
Hie Ellipsoids.
For any given ellipsoid y is constant. If two ellipsoids, y^
and 72) ^ maintained at potentials T^ and 1^, then, for any
point y in the space between them, we have
^^yjIzM±Z(2zl). (26)
The surface-density at any point is
where p^ is the perpendicular from the centre on the tangent
plane, and 1^ is the product of the semi-axes.
The whole charge of electricity on either surface is given by
e. = cKzl=_Q„ (28)
/l — /2
and is finite.
When y = F(k) the surface of the ellipsoid is at an infinite
distance in all directions.
If we make 1J= 0 and y^ = F(k), we find for the quantity of
electricity on an ellipsoid y maintained at potential F in an
infinitely extended field, v
The limiting form of the ellipsoids occurs when y = 0, in which
case the surface is the part of the plane of ooy within the focaJ
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 A;, is
r 1 1
(30)
and its charge is Q = c-p--. . (31)
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238 CONFOCAL QUADEIO SURFACES. [151.
Particular Cases.
151.] If c remains finite, while b 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 ^ _ _J^ ^ 0. (32)
(sino)* (cosa)^ ^ ^
As regards the quantity /3, if we take the definition given in
page 233, (7), we shall be led to an infinite value of the integral at
the lower limit. In order to avoid this we define /8 in this
particular case as the value of the integral
cdX^
A;
If we now put A2 = c sin <^, /3 becomes
nd<t>
J^ sin<^'
i.e. logcot J<^;
6^ — 6"'*
whence cos <^ = ^ _^ > (33)
2
and therefore sin <^ = ^ _^ . (34)
If we call the exponential quantity J (e^+e-^) the hyperbolic
cosine of jS, or more concisely the hypocosine of /3, or cosh /3, and
if we call J (e^— e-^) the hyposine of ^, or sinh ft and if in the
same way we employ functions of a similar character analogous
to the other simple trigonometrical ratios, then Ag = c sech fi, and
the equation of the system of hyperboloids of one sheet is
fl^ + y' _ ^ _ ^2 /o,rX
(sech;3)^ (tanh/S)^ - • ^'"''^
The quantity y is reduced to <^, so that A3 = c secy, and the
equation of the system of ellipsoids is
(secy)^ (tany)* ^ '
Ellipsoids of this kind, which are figures of revolution about
their conjugate axes, are called planetary ellipsoids.
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152.] SURFACES OF REVOLUTION. 239
The quantity of electricity on a planetary ellipsoid maintained
at potential F in an infinite field, is
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
V
2wVc2-r2
(38)
« = ^^- (39)
152.] Second Caee. Let 6 = c, then A; = 1 and k'= 0,
IT 4- 2d
a = log tan — - — , whence Xj = c tanh o, (40)
and the equation of the hyperboloids of revolution of two sheets
becomes cc^ V^-^^ ^ 2 /^ix
The quantity p becomes reduced to <f>, 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
y'
2
(sinfif (cos^)2"^- ^^^^
This is a system of meridional planes in which fi is the longitude.
The quantity y as defined in page 233, (7), becomes in this case
infinite at the lower limit. To avoid this let us define it as the
value of the integral r* cdk^
If we then put A3 = csecV^, we find y=/ ~^"^' whence
A3 = c coth y, and the equation of the family of ellipsoids is
(cothy)^ (cosechy)* ^ ^
These ellipsoids, in which the transverse axis is the axis of
revolution, are called ovary ellipsoids. .
The quantity of electricity on an ovary ellipsoid maintained
at potential Fin an infinite field becomes in this case, by (29),
cV^r-i^^ (44)
where c sec V^q is the polar radius.
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240 COOTOOAL QUADBIO SUBPACES. [154.
If we denote the polar radius hj A and the equatorial by B, the
result just found becomes
, A+'/A^-B^
log 5
(45)
If the equatorial radius is very small compared to the polar
radius, as in a wire with rounded ends,
^ = log2^-log5- ^*®>
When both h 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 radii are in-
versely proportional to y.
If the ratio of 6 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 radii are inversely proportional to y. This
is the ordinary system of spherical polar coordinates.
Cylindric Surfaces.
158.] When c is infinite the surfi&ces are cylindric, the generat-
ing 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, k is zero, and therefore
^ = a, it follows that the equation of this system is
-^ 4- = *"- (47)
sm*o cos^a ^ '
The other system is elliptic, and since when A; = 0, /9 becomes
or Ag = 6cosh)3,
r
the equation of this system is
(coshi3)2"'"(8inh^)2^*'' ^^^^
These two systems are represented in Fig^X at the end of this
volume.
Confocal Paraholoida.
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, X, 6, and c we substitute t + Xft-^X^t-^b,
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154.]
CYLINDERS AND PABAB0L0ID8.
241
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=zb and x = c, viz. the equation is
2/» ^
4(aj-A) + -
= 0.
(49)
If the variable parameter is X for the first system of elliptio
paraboloids, ii for the hyperbolic paraboloids, and v for the second
system of elliptic paraboloids, we have X, 6, ^, c, v in ascending
order of magnitude, and
aj = X + ^+v— c— 6,
c—b
(60)
-X)
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
** "A -/(6-x)(c-;
Jb ^/{|JL^b){c-^l)
^^Jc V{v^b){y^cy
From these we find
X = i(c + 6)-J(c— 6)cosho,
fiz= J(c + 6)-J(c-6)cosi3,
p=: J(c + 6) + J(c-6)coshy;
or = J (c + 6) + J {c—b) (coshy— cos/3— cosho),
y = 2(c— 6)sinh|sin^cosh|>
0=r2(c— 6)co8h|cos|sinh|- j
When 6 = c we have the case of paraboloids of revolution
about the axis of x, and {see foot note}
y = 2a6«+ycos/3, (53)
z= 2ae*+^sin/3.
VOL. I. B
(51)
(62)
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242 CONFOOAL QUADMO SUBFAOES. [154.
The surfaces for which /9 is constant are planes through the
axis, /3 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 o=— 00 the paraboloid is reduced to a straight line
terminating at the origin.
We may also find the values of a, )3, y in terms of r, d, and <^,
the spherical polar coordinates referred to the focus as origin,
and the axis of the paraboloids as the axis of 0,
a = log (r* cos i ^),
/8 = *, (54)
y = log (r* sin \S).
We may compare the case in which the potential is equal to a,
with the zonal solid harmonic r^Q^. Both satisfy Laplace's
equation, and are homogeneous functions of x, y^ 0, but in the
case derived from the paraboloid there is a discontinuity at the
axis {since a is altered by writing d+ 2ir for d}.
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 *.
* {The results of Art. 154 can be deduced as follows. Obanging the yariables from
X, y, z ix) A, /I, V, Laplace's equation becomes
d\ 1 (;i_d)* (c-A»)* ("-ft)* (v-c)* d\ S
or (.-M){6-A}*{c-X}*^j(5-A)»(c-A)*g|
+ 0..X){._6}l(..e}*|.](.-6)*(.-c)*gj-0;
or if da 1
d\ " (5_x)i (c-A)* '
dfi ^ 1
dy ^ 1
rfi'" (k-6)*(v-o)*'
Laplace's equation becomes
So that a linear function of a, jB, 7 satisfies Laplace's equation.
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^54-] CYLINDEES AND PABABOLOIDS. 243
When 6 = c, we may take
a
jo &-^'
7-
725 -&'
A.
= {J1-.-},
v<
.d{l + *Y}.
(M-
-h)
= i(c-5){l-coe^},
(c.
-M)
-K<»-^){l + o«ii8};
X
«6 + 6(tfV_e«),
y'
«4d»«y+*8m»|,
From (61)
hence from (50),
«»-4b»fly+»ooe»|.
If we take the origin at the focus x^h, and write 2pf Ux fit a^^ totlt^f a^^
for l>f ^we get x^^^ -«*■'*
y-2a€*'+y'ooer.
From which eauationa of the form (54) may easily be deduoed.
Since from these equations the force along the radius varies as 1/r, the normal torce,
and therefore the surface-density, will vary as - • - where p is the perpendicular
from the focus on the tangent plane, thus the sur&ce-density varies as 1/|>, and there-
fore inversely as the square root of r. }
R 2
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CHAPTER XL
THEOBY OP BLECTEIO IMAGES AND ELEOTBIC INVBESION,
155.] Wb have already shewn ibat 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-
fluenced 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 harmonics, and to determine a second series of solid
harmonics which express the potential, due to the electiification
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 was removed.
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BLEOTKIO IMAQES. 245
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 presented in an elementary geometrical form.
His original investigations, which are contained in the Canfh-
bridge and DvJblin MathevncUical Jownudy 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 firee use of the idea of the potential and
of equipotential surfaces, whenever the investigation can be
rendered more intelligible by such means.
Theory of Electric Images.
156.] Let A and £, Figure 7, represent two points in a uniform
dielectric medium of infinite extent.
Let the charges of A and B he e^
and 62 respectively. Let F be any
point in space whose distances from
A and B are r^ and Tg respectively.
Then the value of the potential at P
will be v=^+^' (1)
n ^2 Fig. 7.
The equipotential surfaces due to
this distribution of electricity are represented in Fig. I (at the
end of this volume) when e^ and 62 are of the same sign, and in
Fig. n when they are of opposite signs. We have now to
consider that surface for which F= 0, which is the only
spherical surface in the system. When e^ and 62 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 over which the potential is zero.
The equation of this surface is
^ + ^ = 0. (2)
lie centre is at a point C in AB produced, such that
AC.BC:: e^:ei,
and the radius of the sphere is
^1 ""^2
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246 ELECTRIC IMAGES. [157.
The two points A and B are inverse points with respect to this
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 wUl
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 electri*
fication.
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 £ 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
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1 5 7-] INVBESE POINTS, 247
to a certain electrification in the same region it must be actually
produced by that electrification. In fact, the electrification at
any pcont may be found from the potential near that point by
the application of Poisson's equation.
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 By on the same radius of the
sphere at a distance -7 , and the charge of the image is — e ? •
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 Pof the spherical sur-
face, and for this purpose we shall p. I"
make use of the theorem of Coulomb,
Art. 80, that if J2 is the resultant force at the surface of a con-
ductor, and (T the superficial density,
i2= 4ir(r,
R being measured away from the surface.
We may consider R as the resultant of two forces, a repul-
sion jpg acting along AP^ and an attraction e ^ ^^ acting
along PB.
Resolving these forces in the directions of AC and CP, we
find that the components of the repulsion are
^ along A (7, and ^ along CP.
Those of the attraction are
"^JBP^^^ along AC, and -c^ -gp along CP.
CL fit
Now BP = -^ AP, and BC = ^ , so that the components of the
attraction may be written
1 • /2 1
-^f-JP^ along AC, and -« -JF *^^^ ^^*
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248 ELBOTEIO IMAGES. [158.
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 radias CP.
This only confirms what we have already proved, that the
sphere is an equipotential surface, and therefore a surface to
which the resultant force is everywhere perpendicular.
The resultant force measured along CP, the normal to the
surface in the direction towards the side on which A is placed, is
If ^ is taken inside the sphere / is less than a, and we must
measure 12 inwards. For this case therefore
JK = -«^^==^'to- (4)
a AF^ ^ '
In all cases we may write
^ AD.Ad 1 ,^.
^^"""—CT^AF^' <^)
where AB^ 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 ,
^"--%,r.CPlP5- W
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 Ay and its
image ^e ^ at JS, and on the other side of the surface the poten-
tial is 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 — ^ j at J?, and on the
opposite side a potential equal and opposite to that of e at A.
The whole charge on the surface of the sphere is evidently
— 6 ^ since it is equivalent to the image at B.
We have therefore arrived at the following theorems on the
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1 59-] DISTBIBUnON OF ELBCTKICITT. 249
action of a distribution of electricity on a spherical surface, the
surface-density being inversely as the cube of the distance from
a 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)
C = -e^^. (8)
The action of this superficial distribution on any point
separated from A by the surface is equal to that of a quantity
of electricity — e, or 4iraC
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
f.AD.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 ^ is within the sphere, and to the''
second if ^ is without the sphere.
These propositions were established by Sir W. Thomson in his
original geometrical investigations with reference to the distribu-
tion 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 is maintained at potential zero by connection
with the earth, then the electrifications due to the several parts
of the system will be superposed.
Let jlj, ^2) ^* ^ ^^ electrified points of the system, /|,/2,&c
their distances from the centre of the sphere, Cj, c^, &c. their
charges, then the images B^B^, &c. of these points will be in the
same radii as the points themselves, and at distances 7- ' 7- , &c
/l J2
from the centre of the sphere, and their charges will be
a ^ ft.
/l /2
The potential on the outside of the sphere due to the superficial
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250 ELECTRIC IMAGES. [l6o.
electrification will be the same as that which would be produced
by the system of imi^s B^^ B^, &c. This system is therefore
called the electrical image of the system A^^ A^, &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
V
<r = .
47ra
The effect of this at all points outside the sphere will be equal to
that of a quantity Va of electricity placed at its centre, and at
all points inside the sphere the potential will be simply increased
byF.
The whole charge on the sphere due to an external system of
influencing points, il^, il^, &c. is
^=Fa-^5-e,^-&c, (9)
/i /a
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 ihe mutual action between an elec-
trified point 6, at a distance / from the centre of the sphere
* The diBcnmion in the text wiU periutps be more eMily imdentood if the problem
be regarded as an example of Art. 86. Let us then suppose that what is described
as an electrified point is really a small spherical conductor, the radius of which is h
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 & is so small that we may
consider the electricity of the smaU conductor to be uniformly distributed over its
surface and aU the electric images except the first image of the small conductor to
be disregarded. Since the charge E on the sphere is given, we must in addition to
the charge —ea/f at the image have a charge ea/f at the centre of the sphere.
We thus have F=-+~,
« /
The energy of the system is therefore, Art. 85,
E* Ee ^fl a» ^
2^"^ /"*■ 2^6 TiJ^^ay'
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
2 f 2^ /«-a*' J
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l6o.] IMAGE OP AN ELECTBiriBD SYSTEM. 251
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
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,
p- /F ef ^
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
sphere must be positive and greater than e -t-jt' — 2\2"» ^^^ ^®
charge of the sphere must be of the same sign as e and greater
thane-27^^ — 2^'
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
e^af
The surface-density at the point of the sphere nearest to the
electrified point when it lies outside the sphere is
- 4 ^a«r */(/-«)*) ^ ^
The surface-density at the point of the sphere farthest from
the electrified point is
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252
ELEOTBIO IMAGES.
[i6i.
When E^ the charge of the sphere, lies between
the electrification will be negative next the electrified point and
positive on the opposite side. There will be a circular line of
division between tiie 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/^— a^
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.] K the two electrified points A and B in Art 156 are
electrified with equal charges of electricity of opposite signs, the
surface of zero potential will be the
plane, every point of which is equidistant
from A and B.
Hence, if -4. be an electrified point
whose charge is e, and AD a perpen-
dicular on the plane, produce AD to
B so that DB=zAD, and place at JB a
charge equal to —6, then this charge
at B will be the image of -4, and will
produce at all points on the same side
of the plane as -4, an efiect 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 V*F= 0 everywhere except at -4, and
that F= 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, wo
Fig. 8.
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1 62.] IMAGES IN AN INFINITE PLANE. 253
observe that it is compounded of two forces each equal to j-p^ i
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
AP^' AP'
Hence iZ, the resultant force measured from the surface towards
the space in which A lies, is
ii=-^. (.5)
and the density ai the point P is
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 22 is in the same radius and at a
distance r' such that rr^^RK 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.
K A and B 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'.
Hfflice the triangles Oil JB, OVA' are similar,
and AB : A'B^ - OA : OB" : : OA.OB: R\
K a quantity of electricity e be placed at
At its potential at B will be F= -j^ •
If e^ be placed at A', its potential at B' will be
In the theory of electrical images
e:e'::OA:R::R: 0A\
Hence F:F::JB:OJB, (17)
or the potential at B due to the electricity at il is to the
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e
_R
X'
r
~5^
r'
T
~ R^
R'
r
r
iJ
v~
5~
7'
254 ELEOTEIO IMAGES. [163.
potential at the image of £ due to the electrical image of ^ as i2
is to OB.
Since this ratio depends only on OB and not on 0-4, the poten-
tial at B due to any system of electrified bodies is to that at^
due to the image of the system as i2 is to OB,
If r be the distance of any point A from the centre, and / that
of its image A\ and if c be the electrification of Ay and ^ that of
A\ also iS L, Sy K he linear, superficial, and solid elements at Ay
and L\ ^, K^ their images at A\ and X, a, p, V, </, p' the corre-
sponding line-surface and volume-densities of electricity at the
two points, V the potential at A due to the original system, and
y the potential at A' due to the inverse system, then
r ■" i" r« ""Jt^' /Sfr^"^*' K " r^ " R^'
} *(18)
If in the original system a certain surface is that of a con-
ductor, and has therefore a constant potential P, then in the
transformed 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 con-
ductor 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, imder the influence of an
electrified point with a charge — PjB 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
* See Thomson and Twt*B Natural Philosophy, § 515.
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164.] GEOMBTEICAL THEOEEMS. 255
of inyersion are a and a^, 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 a^— a*, and that of the second is a'*— o'*. We
have in this case
a' _a _ R^ _o!^^a^ ..^.
a "■ a " i^3^ " "E^' ^^^^
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
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 ai*e a plane and a
sphere, the perpendicular &om 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 A, take it for the centre of in-
version, and if ii is at a distance / 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/, where
The centre of either of these spheres corresponds to the inverse
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256 ELBCTBIC IMAGES. [164.
point of the other with respect to -4., or if C is the centre and B
the inverse point of the first sphere, (/ will be the inverse point,
and R the centre of the second.
Now let a quantity e' of electricity be communicated to the
second sphere, and let it be uninflaenced by external forces. It
will become uniformly distributed over the sphere with a surface-
density <r'=j^- (21)
Its action at any point outside the sphere will be the same as
that of a charge e' placed at ff the centre of the sphere.
At the spherical surface and within it the potential is
P'=^. (22)
a constant quantity.
Now let us invert this system. The centre R becomes in the
inverted system the inverse point JB, and the charge ^ at ^
becomes e' ^ at JS, and at any point separated from B by the
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 JS, is in the inverted system
a'AP'
If we now superpose on this system a charge e at A, where
e=-j'iJ. (23)
the potential on the spherical surface, and at all points on the
same side as JS, will be reduced to zero. At aU points on the
same side as A the potential will be that due to a charge e at A,
and a charge e^^ at B.
But 6'^= -e-^= -e^r, (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
<' = <''S' (25)
Substituting for the value of a' in terms of the quantities be-
longing to the first sphere, we find the same value as in Art. 158,
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165.] STSTEMS OF IMAGES. 267
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. Then, if we
draw a circle with centre 0 and radius OP, and find points which
are the successive images of P in the two planes beginning with
OB, we shall find Q^ for the image of P in OB, i^ for the image
of Qi in OA, Q3 for that of ij in OB, ij for that of Q3 in OA,
Q2 for that of ^ in OB, and so on.
K we had begun with the image of P in AO we should have
found the same points in the
reverse order Q^, ij, Q3, ^, Q^,
provided AOB is a submultiple
of two right angles.
For the electrified point and
the alternate images 2 2, J^
are ranged round the circle at
angular intervals equal to 2 AOB,
and the intermediate images
Qi) Qa Q3 ^^ At intervals of
the same magnitude. Hence, ** Fiir 10
if 2 AOB is a submultiple of
2ir, 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 result of a finite series of electrified points.
If AOB = -, there will be n negative images Qi, Qj, &c., each
n
equal and of opposite sign to P, and n—1 positive images ig)
ij, &c., each equal to P, and of the same sign.
2ir
The angle between successive images of the same sign is — •
If we consider either of the conducting planes as a plane of
symmetry, we shall find the electrified point and the positive
and negative images placed symmetrically with regard to that
VOL. I. s
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258 ELEOTBIO IMAGES. [l66.
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 inter-
secting at an angle -, the influencing point P, the inverse point
of 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 in a sphere are in the
same radius of the sphere, and draw successive chords of the
circle on which the images lie beginning at P and passing
through the centres of the two spheres alternately.
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 fii*st 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 an electrified point.
We may by inversion deduce the case of a conductor consisting
IT
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 of planes with re-
spect to P and change the signs
of the charges. The circle
on which the images formerly
lay now becomes a straight
p. j^ line thi-ough the centres of
the spheres.
If the figure (11) represents a section through the line of
centres AB^ and if D, 1/ are the points where the circle of
intersection cuts the plane of the paper, then, to find the sue-
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167.] TWO INTEESBCTING SPHERES. 259
cessive images, draw DA a radius of the first circle, and draw
IT 2'7r
DCj DE, &c., making angles -» — > &c. with DA. The
° ° n n
points Ay C, E, &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 distance from D. The last of these images
will be at the centre of the second circle.
To find the negative images draw 2)Q, 2)12, &c., making angles
IT 2ir
— > — , &c with the line of centres. The intersections of these
n n
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 {for if E and Q are inverse points in the
sphere A the angles ADE, AQD are equal}.
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 is ^, is
1 C- ,.^„ .^.DB ,.^« .^.DC
a =
|l + (ili)^-^52)^ + (ilD2-ilC^)g +&C.}:
4TrDA
where A, 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 one 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 £ is i {DB 4 OB), and so on, lines such as OB measured
frx)m 0 to the left being reckoned negative.
Hence the total charge on the segment whose centre is ul is
Ki)il + 2)S+iX7+&c.) + i(Oil + 0£ + OC+&c.)
-i(Z)P+2)Q + &c.)-J(0P+0Q + &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.
s 2
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260 ELEOTEIO IMAGES. 1 1 ^7-
In order that such a system of spherical surfaces may exist,
eyery 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 inter-
secting at an angle — •
(3) These two surfaces with a thirds which may be either plane
or spherical, cutting both orthogonally.
(4) These three surfaces with a fourth, plane or spherical,
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
271—1 images arranged in two series on a circle which passes
through the influencing point and is orthogonal to both surfaces.
In the third case we have, besides these images and the in-
fluencing point, their images with respect to the third surface,
that is, 4n— 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
ti— 1 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, vf in each series. We shall obtain in this way, besides
the influencing point, 2n7i'— 1 positive and 2nW negative
images. These i.nvf points are the intersections of circles
belonging to the two systems of lines of curvature of a cydide.
If each of these points is charged with the proper quantity of
electricity, the surface whose potential is zero will consist of
ii+7i' spheres, forming two series of which the successive spheres
of the first set intersect at angles — , and those of the second set
at angles -^ , while every sphere of the first set is orthogonal to
every sphere of the second set.
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i68.]
TWO SPHEEBS CUTTING OBTHOGONALLY.
261
Case of Two Spheres cutting Orthogonally. See Fig. IV
at the end of this volume.
168.] Let A and JB, Fig. 12, be the centres of two spheres
cutting each other orthogonally
in a circle through D and iX, and
let the straight line Dl/ cut the
line of centres in C. Then C is the
image of A with respect to the
sphere J3, and also the image of
B with respect to the sphere
A. If il2) = a, BD = p, then
and if we place
Fig. 12.
afi
AB^Va'-^fi^
at Ay B,C quantities of electricity equal to a, /9, 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) On the conductor PDQIf formed of the larger segments of
both spheres. Its potential is unity, and its charge is
a + /3-.
^AD + BD^CD.
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 JB,
are respectively
T^O-cA)") -^ i^(-(A))-
On the circle of intersection 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) On the lens FDi^If formed by the two smaller segments of
the spheres, charged with a quantity of electricity = — .~ ^t
and acted on by points A and By charged with quantities a and fi
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262 BLECTEIO IMA.GB8. [l68.
at potential anity, and the density at any point is expressed
by the same fonnula.
(3) On the meniscus DPD'Qf charged with a quantity a, and
acted on by points B and C charged respectively with quantities
p and , , which is also in equilibrium at T)otential
(4) On the other meniscus QDP^I/ charged with a quantity
/3 under the action of A and C.
We may also deduce the distribution of electricity on the
following internal surfaces —
The hollow lens T'BQ^If under the influence of the internal
electrified point G at the centre of the circle DIX.
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^ JB, 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 PLQIY by the action of a point at 0
charged with unit of electricity.
Let OA-a, OB=^b, OP = r, BP = p,
AD^ay BD = fi, AB^^/^?TW.
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^-a^' ^-62_^2' «--^2Z^> ^-"WZT^'
If, in the inverted system, the potential of the surface is
unity, then the density at the point P' is
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169.] FOUB SPHBBBS OUTTINa OBTHOGONALLT. 263
I^ in the original system, the density at P is o-, then
and the potential is -. By placing at 0 a negative charge of
electricity equal to unity, the potential will become zero over
the original surface, and the density at P will be
This gives the distribution of electricity on one of the spherical
segments due to a charge placed at 0. The distribution on the
other spherical segment may be found by exchanging a and 6, a
and /9, and putting q or AQ instead of p.
To find the total charge induced on the conductor by the
electrified point at 0, let us examine the inverted system.
In the inverted system we have charges a' at A\ and p^ at JB',
a ^
and a negative charge , at a point Cf in the line A'B^.
such that A'Cr iCTffii a'« : pTK
UOA'^a\ OB!^h\ OCr^(f, we find
Inverting this system the charges become
J^ 1 a^
and
::? —
Hence the whole charge on the conductor due to a unit of
negative electricity at 0 is
a 6 ya«/3« + 6«a«-a2/3«
IHstribviion of Electricity on Three Spherical Surfaces
which Intersect at Right Angles.
169.] Let the radii of the spheres be a, )3, y, then
Let PQRi Fig. 13, be the feet of the perpendiculars from ABC
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264
ELECTEIC IMAGES.
[169.
on the opposite sides of the triangle, 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, /3, and y be
placed hi A, B, and C.
Then the charge to be placed
at Pis
py 1
Fig. 13.
Vfi' + y"
VB^y^ + y^a^ + a^d^
Also AP = '-^--^—p=== , so that the charge at 0, con-
sidered as the image of P, is
apy
1
^PY + y^aJ' + a^fi'
2«2
Vi
•^2 + y*
In the same way we may find the system of images which are
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 = 5, then the charge at the
intersection of the line of centres of any two spheres, say a and
p, with their plane of intei-section, is
1
V
1
>«
The charge at the intersection of the plane of any three centres
ABC with the perpendicular from the centre D is
1
+ / - >
i3' V
and the charge at the intersection of the four perpendiculars is
1
V:
1
1
«2+-^2+y2 + 52
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I70.] FOUR SPHEEES INTERSECTING AT EIGHT ANGLES. 265
System of Four Spheres Intersecting at Right Angles, at zero
potential, under the Action of an Electrified Unit Point
170.] Let the four spheres be A, B, (7, D, and let the electrified
point be 0. Draw four spheres A^ B^, (7j, D^, of which any
one, Ai, passes through 0 and cuts three of the spheres, in this
case B, C, and 2), at right angles. Draw six spheres (ab), {ac\
(od), (6c), (6c?), (cd), of which each passes through 0 and through
the circle of intersection of two of the original spheres.
The three spheres B^, Cj, Dj will intersect in another point
besides 0. Let this point be called A\ and let ^, (/, and 1/ be
the intersections of C^, Dj, A^, of D^, A^y -Bj, and of Ai, B^, £7,
respectively. Any two of these spheres, A^, Bj, will intersect
one of the six (cd) in a point (a' 6'). There will be six such
points.
Any one of the spheres, Aj^, 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 in addition to 0.
If we now invert the system with respect to a sphere of radius
unity 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\ S, (f. If will become
the centres of the spheres, and the others will correspond to the
other eleven points described above. 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
a is the radius of the sphere A, and a is the distance of its centre
from 0. Li the same way we must place the proper charges at
BT, a, ly.
The charge for any of the other eleven points may be found
from the expressions in the last article by substituting a', ff, /, 6'
for a, p, y, h, and multiplying the result for each point by the
distance of the point from 0, where
""- a^^a^' '^ 6^^/32' ^"" c^^y»' d^-b^'
[The cases discussed in Arts. 169, 170 may be dealt with as
follows : Taking three coordinate planes at right angles, let us
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266 ELECTBIO IMAQES. [l7l.
place at the system of eight points (±— » i: sis i-ir)
^ 2 a 2p Zy'
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 ortho-
gonally 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 conductor in the form of three spheres of radii
a, ^9, y cutting orthogonally and £reely 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.]
Tivo Spheres not Iviersecting.
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, none of which lie
between 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.
We shall begin, therefore, with the case of two uninsulated
concentric spherical surfaces, subject
to the induction of an electrified point
P placed between them.
Let the radius of the first be 6, and
that of the second he^^ and let the
distance of the influencing point from
the centre be r = 6^.
Then all the successive images
will be on the same radius as the
influencing point.
Let Qq, Fig. 14, be the image of P in the flrst sphere, I{ that
of Qo ^ ^^G second sphere, Q^ that of ^ in the first sphere, and
soon; then OP,.OQ,^b\
and OP..OQ.^^ = b^(^,
Fig. 14.
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1 7 1.] TWO SPHERES NOT INTERSECTING. 267
alao OQo = 6«"^,
Hence OP, =66(-^2*t«r),
If the charge of P is denoted by P, that of ij by i?, then
P, = Pe^, Q. = «Pe-<"+'^.
Next, let Qi be the image of P in the second sphere, ^ that
of Q/ in the first, &c., then
0Q{ = 6e««^-, 0^ = 66-««^,
e/ = ^Pc^^-, If = P6-^^.
Of these images all the P's are positive, and all the Q's
negative, all the P^'s and Q s belong to the first sphere, and
all the P*s and Q^'b to the second.
The images within the first sphere form two converging series,
the sum of which is c^"^- 1
— P— OT 7^'
This therefore is the quantity of electricity on the first or
interior sphere. The images outside the second sphere form two
diverging series, but the surface-integral due to each with respect
to the spherical surface is zero. The charge of electricity on the
exterior spherical surface is therefore
If we substitute for these expressions their values in terms of
OA, OB, and OP, we find
, ^ j,OAPB
charge on il ^-^qPAB'
charge on 5 =:-P^-jg-
K 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 il = — P ^ d >
AP
charge onB = — P -^-g-
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268
BLEOTBIO IMAGES.
[172.
Fig. 16.
172.] In order to pass from this case to that of any two spheres
not intersecting each other, we begin by finding the two common
inverse points O, O'
through which all cir-
cles pass that are ortho-
gonal 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.
Ijf we take the point
O in Fig. 15 as centre
of inversion, this point
will be situated in Fig. 14 somewhere between the two spherical
surfaces.
Now in Art. 171 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 O we shall there-
fore deduce the distributions induced on two spherical conductors
at potential zero, exterior to one another, 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 distri-
butions 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. 15 an arc of a circle through O and O^, and
the ratio of O'P to OP is equal to Ce" where C is a numerical
quantity.
If we put ^ = logQp, o = logQ^, ^ = logQQ,
then ^ — o = «r, u + a=^*.
All the successive images of P will lie on the arc O'APBO.
The position of the image of P in A is Q^ where
fl(Q,) = log^«=2«-ft
* {Since O' iuvertB into 0, the common centre of the spberee, we hare by Art. 162
O'P OP O'A OA ., . O'P.OA OP
OP 00 • OA 00
^,80 that
OP.O'A OA
-Hi-"}
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172.] TWO SPHERES NOT INTEESEOTING. 269
That of Qo in B is I? where
Similarly
d(F=!) = ^ + 281^, ^(Q.) = 2a-^-28i]T.
In the same way if the successive images of P in B, A, B, &c.
are Qo', P/, Q/, &c.,
^(Q,^ = 2p^e, e(r^) = ^-2tsr ;
e{r^- ^-28tsr, e{Q/) = 2p^e + 28m.
To find the charge of any image F? we observe that in the
inverted figure (14) its charge is
In the original figure (15) we must multiply this by OF^. Hence
the charge of fj in the dipolar figure as P = P/OP, is
V op.o'P
If we make f = -/OP.O'P, and call ( the parameter of the
point P, then we may write
P= ^P
or the charge of any image is proportional to its parameter.
If we make use of the curvilinear coordinates 6 and (f>, such
x+ ^/^ly + k
where 2 A; is the distance 00\ then
__ fcsinh^ _ A;sin<f>
"~ cosh^— cos</>' ^ "" cosh^— cos</>'
a^ + (y—k cot (f>Y = k^ cosec* <^,
(x-i-k coth 0)^ •\-y^^l(? cosech* ^,
^ 2% 2A:a5
f= , -^^ t.
V cosh ^— cos <^
* {Hence ^ if constant for all pointa on the arc along which the images are
ntnated. }
f In these expressions we must remember that
2ooehtf - e* + e-*, Ssinhtf « e^-e"*,
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270 BLBOTEIO IMAGES. [^73*
Since the charge of each image is proportional to its parameter,
f, and is to be taken positively or negatively according as it is
of the form P or Q, we find
p__ P\/co8h^— cos<^
V^cosh {6+ 2 8cr)— cos <t>
^ Pv^cosh^— cosd)
v^cosh(2a—^— 28X0') — cos <^
p,_ Pycosh^— cos<^
Vcosh (d— 28X0') — cos <l>
Qf P -/cosher— COS <^
Vcosh(2/i — ^ + 28cr) — C08<^
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
P ycoshd— cos<^ 2! I r / 1./^ » \ "^ '
^-^'=^ VCOSh(d— 28«r) — C08<^
P— -/cosh ^— COS (f) 2!ir / V /» /I o \ '
^-^•^0 v^cosh(2a— ^ — 28w)— cos<^
In the same way the total induced charge on B is
P v^cosh ^ — cos 6 2!ir / t, //> o \ "^ *
^^i-i v^cosh(d + 28tsr)-cos0
— p ycosh d- cos <^ 2!ir / 1, /» ^ z» o \ '
^^i^o v^cosh(2/3-£^+28cr)— cos<^
178.] We shall apply these results to the determination of the
coefficients of capacity and induction of two spheres whose radii
are a and 6, and the distance between whose centres is c.
Let the sphere ^ be at potential unity, and the sphere B at
potential zero.
Then the successive images of a charge a placed at the centre
and the other functions of 0 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*8 Journal for 1847. See Thomson's reprint of Electrical Papervy §§ 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., 1858.
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1 73-] TWO ELBCTBIFIBD SPHERES. 271
of the sphere A will be those of the actual distribution of elec-
tricity. All the images will lie on the axis between the poles
and the centres of the spheres, and it wiU be observed that of
the four systems of images determined in Art. 172, only the third
and fourth exist in this case.
If we put
, -/a* + 5* + c*-26V-2c2a2-.2a262
fc= »
2c
k . k
then sinh a = — » sinh i9 = r *
a 0
The values of 0 and (f> for the centre of the sphere A are
d = 2o, <^ = 0.
Hence in the equations we must substitute a or —A; -r-r —
^ smha
for P, 2 a for 0 and 0 for <f>, remembering that P itself forms part of
the charge of A. We thus find for the coefficient of capacity of A
_ , '^#=00 1
for the coefficient of induction of ^ on £ or of j8 on ^
We might, in like manner, by supposing B at potential unity
and A at potential zero, determine the value of g^b* ^^ should
find, with our present notation,
?66= *2!;rriKh03+^)'
To calculate these quantities in terms of a and 6, the radii of
the spheres, and of c the distance between their centres, we
observe that if
Z= v^a* + 6* + c*-26'-^c2-2c2a»-2a^6^
we may write
sinha = — - — , sinh/3=T-r-, sinh ct = ;r~T »
2ac 26c' 2a6
cosh a = — , cosh /3 = ^r-r — , cosh tu = — jr-r — ;
2ca 2cb 2ab
and we may make use of
sinh(a + y3) = sinh a cosh ^ + cosh a sinh ft
cosh(a + j3) = cosh a cosh ^3 + sinh a sinh /3.
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272 ELECTBIC IMAGES. [174.
By ihis process or by the direct calculation of the successive
images as shewn in Sir W. Thomson's paper, we find
a'b ^ „
3-a - « + ^::^ + (c«_6«+ac)(c*-62_ac)'''®®''
ah a«6« a^
*"* c c(c*-a»-6«) c(c*-a»-6* + a6)(c*-a«-6«-a6)~**'"
, 06' . ^ .„
?» - '' +0"^^^^ "^ (c*-a» + 6c)(c«-a«-6c) "^ **'
174.] We have then the following equations to determine
the charges E^ and Ei, of the two spheres when electrified to
potentials Tij[and TJ respectively,
If we put ?.«5'»-?<** = ^ = 5> '
«"^ p.a=quiy, p.k=-q^iy, p,*=q.»iy,
whence ^ ^ ^2 iv.
then the equations to determine the potentials in terms of the
charges are y,^ p^E, +Pa,E,,
^—PahEa-^-PbbEh,
and p^y pahi and pi^ are the coefficients of potential.
The total energy of the system is, by Art. 85,
Q^iiEJ^ + E.r,)
= \{EJ^Paa + 2E,E,p^+E,^p^).
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
variations of the coefficients of capacity and induction, is the
most convenient for calculation.
We have therefore to differentiate the g's with respect to c.
These quantities are expressed as functions of fc, a, ft and w, and
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1 75-] TWO SPHEEES IN CONTACT. 273
must be differentiated on the supposition that a and b are con-
stant From the equations
, • , sinhasinh/S
k = — asmha = 6smhi3 = — c t—t »
smho-
dk coshacosh/9
we find
dc "^ sinhv
da __ sinhacosh^
dc "" iksinhv
dp coshasinh)3
dc A^sinho- '
whence we find
dqaa _, cosh Q cosh jSg^a '^#=00 (»c + 6cosh/3)cosh(8Br— a)
dc""" sinhw T "" -^'=o c(sinh(«i]T-a))2 '
dqab _ cosh Q cosh ff^qft ^«=oo scoshecr
dc " sinhtsr k -^*=i (sinh ecr)^ '
dqj^ _ cosh Q cosh fi g^ ^#=0 (sc -f a cosh a) cosh (ff -f gqr)
dc "" sinhw T "" ^'=® c(sinh()3 + 8tu))'^
Sir William Thomson 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 neces-
sary to use more than two or three of the successive images.
The series for the differential coefficients of the q'a with respect
to c are easily obtained by direct differentiation,
dc "" (c«-62^ (c«-62H-ac)«(c«-6«-ac)« ^''
dq^_ab aH^jSc^^a^^b^)
dc ^ c''^ c«(c2-a2-57
a»6«{(5c»-a«-6»)(c^-a^-6»)~a«&^}
dq^_ 2ab^c 2a^6»c(2c^-2a^~6^)
dc ■" {(^--a^f (c2-a2 + 6c)^(c2^a^-6c)« ^*
Dutr^ibviion of Electricity on Two Spheres in Contact.
175.] If we suppose the two spheres at potential unity and
not influenced by any point, then, if we invert the system with
respect to the point of contact, we shall have two parallel planes,
VOL. I. T
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274 ELBCTEIO IMAGES. [175.
distant -— and ^ from the point of inversion, and electrified by
the action of a positive unit of electricity at that point.
There will be a series of positive images, each equal to unity,
at distances « (- + r) from the origin, where 8 may have any
integral value from — 00 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 a, are - + « f- + t) •
When this system is inverted back again into the form of the
two spheres in contact, we have corresponding to the positive
images a series of negative images, the distances of which from
the point of contact are of the form — - — --, where « is positive
/I 1\
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 aJso be a series of positive images corresponding to
the negative ones for the two planes, whose distances from the
point of contact measured in the direction of the centre of a,
are of the form
1 /I 1\
When 8 is zero, or a positive integer, the image is inside
the sphere A,
When 8 is a negative integer the image is inside the sphere B,
The chaise of each image is numerically equal to its distance
from the origin and is always positive.
The total charge of the sphere A is therefore
p __ '^#=00 1 ab '^fsoD 1
"■"-^'=« 1 .1 k"^+6-^'«i 8*
Each of these series is infinite, but if we combine them in the
^o™ _ ^,=,« an
^a - ^,.1 8(a + 6)|8(a + fe)-a}
the series becomes convergent.
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1 75-] "^^ SPHERES IN CONTACT. 275
In the same way we find for the charge of the sphere B,
„ '^tssto ah ah ^«ss-ao 1
* - -^'=1 8{a + b)-h ~ ^+6 -^'=-1 1
" -^'=1 «(a4-6){«(aH-5)-6} '
The expression for E^ is obviously equal to
a + bJo i-e "^^^
in which form the result in this case was given by Poisson.
It may also be shewn (Legendre, TraiU des Fonctiona EUip-
tiqueSy ii. 438) that the above series for E^ is equal to
«-l^+*(^6)l5Ti
where y = -57712..., and ♦(aj) = ^logr(l +i»).
The values of * have been tabulated by Gauss (Werke, Band iii,
pp. 161-162).
If we denote for an instant 6 -r (a + 6) by x, we find for the
difference of the charges E^ and Ej,,
-^^iogr(.)r(i-.)x^,.
ah d -
irab ^ irb
= rCOt-
aH-6 a + b
When the spheres are equal the charge of each for potential
^^yis -j,=« 1
^a = aZ,-i2«(28-l)
= a(l-i + J^} + &c.)
= alog,2 = •69314718a.
When the sphere A is very small compared with the sphere B^
the charge on il is
K--T 2l«r S approximately,
or ^a = Jj-
T %
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276 ELECTRIC IMAGES. [^77*
The charge on £ is nearly the same as if -4 were removed, or
The mean density on each sphere is found by dividing the
charge by the surface. In this way we get
- ^a - -^
_ ^6 _ 1
""*" 47x62 ^47r6'
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 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 1 8 4 7. 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 dis-
tribution 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 sheU. If we suppose the thickness of
the shell to diminish indefinitely while its density increases, we
ultimately arrive at the conception of a surface-density varying
as the perpendicular from the centre on the tangent plane, and
* Thomson and Tait's Natural FhiU>8ophy, § 520, or Art. 150 of this book.
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178.] SPHBBIOAL BOWL- 277
since the resultant attraction of this superficial distribution on
any 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.
Distribtdion 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 ex-
ternal 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
r
2 1^ Vkp.pl
Application of the Principle of Electric Inversion.
178.] Take any point Q as the centre of inversion, and let jB
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 port**n of the spherical surface bounded by a
circle. We shall call this portion of the surface the boivL
If /S' is the disk electrified to potential V^ and free firom 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
solution of the problem of the distribution of electricity on a bowl
or a plane disk at zero potential when under the influence of an
electrified point in the surface of the sphere or plane produced.
Injltience of an Electrified Point plkuced on the unmcupied
part of tlie Spherical Surface.
The form of the solution, as deduced by the principles ab-eady
given and by the geometry of inversion, is as follows :
If (7 is the central point or pole of the spherical bowl S, and
if a is the distance fix)m 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
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278 BLEOTRIO IMAGES. [179.
at potential zero, the density o- at any point P of the bowl will be
1 q /W^\
CQy CPy 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
influencing 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 re-
maining 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 p\
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 T where y^ lisp'f.
Hence the potential at every point of the bowl will now be F.
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.
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1 8 1.] SPHERICAL BOWL. 279
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+ // = 0, and we get the
case of the bowl maintained at potential Fand free from external
influence.
If 0- 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 electiified to density p, then, when the bowl is main-
tained at potential F, 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
*^ = 2^ 1 Vfc^ -**^"' V fcSl '
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 uninsulated bowl by a quantity q of electricity
placed at a point Q, 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 S will be inverted into its image S',
and the point P will have P' for its image. We have now to
determine the density rr' at P' when the bowl 8' is maintained at
potential F', such that q = V^Ry and is not influenced by any
external force.
The density <t at the point P of the original bowl is
i/R^
this bowl being at potential zero, and influenced by a quantity q
of electricity placed at Q.
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280
ELECTBIO IMAGES.
[l8l
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.
2) 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 suppose the
radius of the sphere of inversion to
be a mean proportional between the
segments into which a chord is
divided at Q, E^F^ will be the image
of EF. Bisect the arc FVE" in 2/,
so that F'ly^iyEr, and draw lyqB
to meet the sphere in D. i) is the
point required. Also through 0, the centre of the sphere, and Q
draw nOQW meeting the sphere in II and H\ Then if P be
any point in the bowl, the suiface-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, will be
\^ j_ QH.QH' iPQ .CD'^aKi tnn-^[P^ (^^'-''\^M
'''' 2Tt'HH\PQHDQ^a'-CP'^ LDQW-cWJr
where a denotes the chord drawn from C, the pole of the bowl,
to the rim of the bowl *.
On the side next to Q the surface-density is
q QH.QH'
Kg. 16.
(r +
2Tr HH\PQ^
* {For farther inTestigatioiis of the electrical distribution on a bowl, see Ferrer*B
Quarterly Journal of Math, 1882 ; Grallop., Quarterly Journal, 1886, p. 229. In this
paper it is shewn that the capacity of the bowl -> -^ where a is the radins of
the sphere of which the bowl forms a part and a the semi-vertical angle of the cone
passing through the edge of the bowl whose apex is the centre of the sphere.
See also KrusemAn ' On tiie Potential of the Electric Field in tiie neighbourhood of a
Spherical Bowl,' Phil Mag, xxiv. 88, 1887. Basset, Proc, LomL Math. 8oe. xvi.
p. 286.}
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APPENDIX TO CHAPTBB XI. 281
APPENDIX TO CHAPTER XL
{ The electrical distribution over two mutually influencing spheres has
occupied the attention of many mathematicians. The first solution, which
was expressed in terms of definite integrals, was given by Foisson in two
most powerful and fascinating papers, Mem, de VlnsiitvJt. 1811, (1) p. 1,
(2) p. 163. In addition to those mentioned in the text the following
authors among others have considered the problem.- Plana, Mem. di
Torino 7, p. 71, 16, p. 67; Cayley, Phil, Mag, (4), 18, pp. 119, 193;
Kirchhoff, CrelU, 59, p. 89, Wied. Ann. 27, p. 673; Mascart, C. R 98,
p. 222, 1884.
The series giving the charges on the spheres have been put in a very
elegant form by Kirchhoff. They can easily be deduced as follows.
Suppose the radii of the spheres whose centres are A^ B are a, 6, their
potentials 27, V respectively, then if the spheres did not influence each
other the electrical effect would be the same as that of two charges a U^
h V placed at the centres of the spheres. When the distance c between the
centres is finite this distribution of electricity would not make the
potentials over the spheres constant ; thus the charge at A would alter
the potential of the sphere B, If we wish to keep this potential unaltered
we must take the image of il in ^ and place a charge there, this charge
however will alter the potential of A, so we must take the image of this
image and so on. Thus we shall get an infinite series of images which it
will be convenient to divide into four sets a, /3, y, S. The first two sets
are due to the charge at the centre of il, a comprises the images
inside A, /3, the images inside the sphere B^ the other two sets, y and d, are
due to the charge at the centre of B; y consists of those inside B, b of
those inside A, Let 7>«, f^ denote the charge and the distance from
A of the nth image of the first set, /;/, // the charge and the distance
from B of the n^ image of the second set, then we have the following
relations between the consecutive images,
t — " « — -'
Eliminating /,' and p,' from these equations we get
«"-«/.- 6'
c-f.
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282 BLBOTBIO IMAGES.
ab
or
but from (1)
and thus
P.n «*
Pn _ «/«-«'
Pn-l «*
P' , P» _c»-J»-a»
+ = ; »
PnH Pn-l a^
or if we put />,= ^, i>,-,= 5— . l>.+>=p— . we get
■'i •••-I ■'•+1
From the symmetry of the equations we see that if we put p/ = — ^ we
shall get the same sequence equation for i^' as for /J^. »
From the sequence equation we see that
where a and 1/a are the roots of the equation
We shall suppose that a is the root which is less than unity. Then
- g*
and the charge on the sphere due to this series of images b
To determine A and B we have the equations
hence 5 = ~ (fLt^fO. — _ |«, say,
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APPENDIX TO C3HAPTBB XI. 283
,_ g"
^» ~ilV+JS"
,_ abU _ 1
^« ^ A'+Bf*
,_ a*VU g
^> -~c(«'-(a'+J»))~ilV+^*
hence » A'/ff--a\
Hence If j^, and E^ are the charges on the sphere, and if
E^=qJJ+qjr;
where w' = ^ = — - •
These are the series given by Foisson and Kirehhoff.
Since -pl-j = - + 4 / -ottS"**
1 1 1 _ r" sinpe ^,
=s - a"—
1-fV 2 2«loga + 2logf
/•*a'8in(2«loga+2logQ<^^
l-^a"» 2 1-a 2»loga+21ogf
„ r° a»8in(2«logg+21ogO< .
^°'' ^2»loga+21ogf-yo l-g«''"»»«'*'
and
V • • /« 1 «i r\. sin(2<logf)-.a8in(2dogf/a)^
2a*sin(2nloga + 2logf)< = — ^ — ^ — .^ . v . « — '5
V 6 -r fife/ 1 — 2acos(2doga) + a"
hence
r* 8in(2<logf)~asin(2<logf/a) K^
Jo (€2»*-l)(l-2acoB(2«loga) + a«)) '
which is Poisson's integral for these expressions.}
* {De Moigaii, Diff. and Int. Cal, p. 672.}
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CHAPTER XIL
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 surfeuses 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 0, and if the part of the
field where this ceases to be the case is so far from the part con-
sidered that the electrical action of the distant part of the field
may be neglected, then the electricity will be uniformly dis-
tributed along each generating line, and if we consider a part
of the field bounded by two planes perpendicular to the axis of ;a;
and at distance unity, the potential and the distributions of
electricity will be functions of x and y only.
H 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
d^r^d^r^^
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PROBLEMS IN TWO DIMENSIONS. 285
When there is no free electricity, this is reduced to the equa-
tion of Laplace, ^a pr ^y
The general problem of electric equilibrium may be stated as
follows : —
A continuous space of two dimensions, bounded by closed
curves C^, Cg, &c. being given, to find the foim of a function, F,
such that at these boundaries its value may be Fj , Fj , &c. re-
spectively, being constant for each boundary, and that within
this space F 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 problem 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 dimen-
sions.
The method depends on the properties of conjugate functions
of two variables.
Dejimtion of Conjugate Functions.
188.] Two quantities a and /3 are said to be conjugate functions
of X and y, if o -h -/— I )3 is a function of « + -/— 1 y.
It follows from this definition that
da ^dp , da <^i9 _ . .
dx" dy^ dy dx " ' ^ ^
d^a d^a ^ d^fi d^fi ^ ,„,
Hence both functions satisfy Laplace's equation. Also
da dp da dp _ da
dx dy dy dx ~ dx
, da
dy
dp\\dp
dx\ dy
= R\ (3)
If X and y are rectangular coordinates, and if da^ is the inter-
cept of the curve (/3 = constant) between the curves (a) and
{a-^da), and ds.^ the intercept of a between the curves (/3) and
(^ + d^),then _^_rfi2.i u^
da^dp^R' ^^
and the curves intersect at right angles.
If we suppose the potential F= VQ + ka, where k is some con-
stant, then F will satisfy Laplace's equation, and the curves (a)
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286 CONJUGATE FUNCTIONS. [184.
will be equipotential curves. The curves (/3) will be lines of
force, and the surface-integral of jR over unit-length of a cylin-
drical surface whose projection on the plane of ajy is the curve
AB will be k^^B—^A)^ where ^a and /8b are the values of )3 at
the extremities of the curve.
K there be drawn on the plane one series of curves corre-
sponding to values of a in arithmetical progression, and another
series corresponding to a series of values of /3 having the same
common diflference, then the two series of curves wiU everywhere
intersect at right angles, and, if the commQU 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 If + A 04, TJ+io^, &c.
respectively. The quantity of electricity upon any one of these
k
between the lines of force {^^ and {p^ will be t— (i32"-)3i)«
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 (1), 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 tx!' and }f' are also conjugaie
functions with respect to x and y, then the functions af+x"
and 'j/'Vj/' will be conjugate functions with respect to x
andy.
daf ^dy' ^daf' ^dy'^
dx " dy^ dx '^ dy *
therefore djarW') ^dj^^/^ ^
dx dy
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185.] GBAPHIO METHOD. 287
Also d^^_d£ ^^ ^^_^.
dy dx' dy dx *
dy dx
or oi Jfx" and y' + y" are conjugate with respect to x and y.
Oraphic Representation of a Function which is the Sum
of Tvx) Oiven Functions.
Let a function (a) of x and y be graphically represented by a
series of curves in the plane of icy^ each of these curves corre-
sponding to a value of a which belongs to a series of such values
increasing by a common difference, b.
Let any other function, ()9), 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 + /9) 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— i), then through the intersection of
(a + 2 5) and ()3— 2d), 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 ()9 + d), of (a + b) and (fi), of {a + 2b) and ()3— 8), and so on.
The function belonging to this curve will he (a + fi + b).
In this way, when the series of curves (a) and the series (fi) are
drawn, the series (a + /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 n. If x'* and jf' are conjugate fun^^ions with
respect to the variaJbles of and 3^, and if x' and jf are con-
jugate functions with respect to x and y, then oi' and y" wHL
he conjugate functions with respect to x and y.
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288 CONJUGATE FUNCTIONS. [185.
dx Ib^ dx 'd/^ dx '
dy' dy dx' dy '
_ dr.
~ dy'
and — '= ^^ + ^'^,
dy dx' dy dy' dy '
^_df_d/_dy^d^
dy' dx dx' dx '
~ dy'
and these are the conditions that x'' and y^' should be conjugate
functions of x and y.
This may also be shewn firom the original definition of conjugate
functions. For af'+ -/_i^' is a function of «'+ -/^ j^', and
x' i- -/— 1^^ is a function of a;+ -/^t/. Hence, a?"+ >/— ly"
is a function of a;+ ^/^ly.
In the same way we may shew that if a?' and ^ are conjugate
functions of x and y, then a? and y are conjugate functions of x'
and y'.
This theorem may be interpreted graphically as follows : —
Let of, y^ be taken as rectangular coordinates, and let the
curves corresponding to values of a?" and of y" 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 corre-
sponding values of x' and y'.
Next, let another piece of paper be taken in which x and y are
made rectangular coordinates and a double system of curves x\ 'if
is drawn, each curve being marked with the corresponding value
of x' or if. This system of curvilinear coordinates will correspond,
point for point, to the rectilinear system of coordinates x\ ^f 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 oi and ^ at these points, and
mark the corresponding points on the second paper, we shall find
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1 86.] THEOBEMS. 289
a number of points on the transformed curve a;". K we do the
same for all the curves x'\ y'' on the first paper, we shall obtain
on the second paper a double series of curves Qi\ 'if' of a different
form, but having the same property of cutting the paper into
little squares.
186.] Theorem in. If V is any function of x' and /, and if
x' and if are conjugaie functions of x and y, then
the integration being between the same limits.
For " =^,^+" ^,
dx ax dx dy dx
dW _ ^ /^\* dW d^ d^ d^r^\'
da? ~ dx'^^dx^ "^ dx'dy'dx dx "•" dy'^^dx ^
dVd^af dVd^
"^ da^ dx" '^d/dx^'
, d^r_dW.dx''' „ d^r dx'dy' dW.dy'.^
df ~ daf'^^dy^ ■•" dx'd-ifdy dy "'' dy'^ ^dyf
dVd^ dV^
"•" dx' dy' ■*■ d^dy' '
Adding the last two equations, and remembering the conditions
of conjugate functions (1), we find
dW d!'V_ dJ'Vf.d^.^ (d^^\ . <PVi(^\\ (d^^X
dx^ "^ dy^ -d^'X'dxf "^ '^dy M d'lf^X'dx^ '^^dy)y
~ ^dx'^ "•" d/v ^dx dy dy dx^'
Hence
rCfd^V (PFx , , rCfdW d^r..d(/dy' dafdj/^, ,
If V is a potential, then, by Poisson's equation
dW d?Y ,
and we may write the result
Jfpclxdy ^jjp'dx'dT/,
VOL. I. V
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290 CONJUGATE FUNCTIONS. [l88.
or the quantity of electricity in corresponding portions of two
systems is the same if the coordinates of one system are conjugate
functions of those of the other.
Additional Theorems on Conjugate Functions.
187.] Theorem IV. If a^ and y^ and also x^ and 2/2 > «^^
conjugate functions of x and y, then^ if
X = x^x.^-y^y^, and Y= Xiy^-^x.^y^,
X and Y uill be conjugate functions of x and y.
For Jr+ v^^lF= (aji+ ^'^y^{x^'^ -/^^a)-
Theorem V. If<l>bea solution of the equation
da?'^ dy'^^'
.d<t>
d<f>
«^^/2i2 = log(£ +£), and 0=-tan-^_,
dy
, dx
d4>
dy
R and 0 will be conjugate functions of x and y.
For R and © are conjugate functions of -^ and ^ > and these
are conjugate functions of a; 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.
K 0 is a fixed point in a plane, and OA a fixed direction, and
if r = OP = ae^, and 0 = AOPy and if a?, y are the rectangular
coordinates of P with respect to 0,
P = logVi^TF. <>=tau-|.j ^^^
x = ae'^cos 0^ y = ae^sin $, )
thus p and 0 are conjugate functions of a; and y.
If p^=inp and 6^=: n$, p^ and ^ will be conjugate functions of
p and 0. In the case in which n = —l we have
r'=-, Bsider^^e, (e)
which is the case of ordinary inversion combined with turning
the figure 180'' from OA.
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1 89.]
ELECTBIO IMAGES IN TWO DIMENSIONS.
291
Invei^&ion in Two Dimensions,
In this case if r and r' represent the distances of coiTesponding
points from 0, e and 6' the total electrification of a body, 8 and 8'
superficial elements, F and V solid elements, a and a surface-
densities, p and p volume densities^ 0 and if/ corresponding po-
tentials,
r" 8
a^
a*
P_
9
a"
) (7)
and since by hypothesis 4/ is got from <f> by expressing
the old variables in terms of the new, ~- = 1.
<P
Example II. — Electt^ic Images in Two Dimensions.
189.] Let A be the centre of a circle of radius ^IQ = 6 at zero
potential, and let ^ be a charge at A^
then the potential at any point P is
0 = 2^1og^; (8)
and if the circle is a section of a
hollow conducting cylinder, the surface-
density at any point Q is — ^— r •
Fig. 17.
Invert the system with respect to a point 0, making
AO=mb, and a^ =: {m^^i)b^;
then the circle inverts into itself and we have a charge at -4'
equal to that at -4, where
AA' =
The density at (^ is
m
E b^^AA
712
2Trb A'Q'^
and the potential at any point P^ within the circle is
<^' = <^= 2 i? (log 6- log ^P),
= 2 ^ (log OP'-log A'P'-logm). (9)
This is equivalent to the potential arising from a combination
of a charge E at A\ and a charge —E at 0, which is the image
of A^ with respect to the circle. The imaginary charge at 0 is
thus equal and opposite to that at A\
V %
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292 CONJUGATE FUNCTIONS. [^90.
If the point P' is defined by its polar coordinates referred to
the centre of the circle, and if we put
p = logr— logft, and pQ^logAA'—logb,
then AP'^bef', AA' = bef^, AO=zbe'P^; (10)
and the potential at the point (p, 0) is
(f) = J?log (e^2po_ 2e-f^ e^ cos ^ + e^)
~^log(e2^— 2e^c^co8^ + c^) + 2JS>o- (1^)
This is the potential at the point (p, $) due to a charge E,
placed at the point (p^, 0), with the condition that when p = 0,
In this case p and 6 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 ^ is an angle.
The centre is the only singular point in this system of coor-
-7- ds round a closed curve is
zero or 2tt, according as the closed curve excludes or includes
the centre.
Example in. — Neumanns Transformation of this Case*.
190.] Now let a and fi be any conjugate functions of x and y,
such that the curves (a) are equipotential curves, and the curves
(j3) are lines of force due to a system consisting of a charge of
half a unit per unit length 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 a^ 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) ortho-
gonaUy.
The coordinates of any point within the curve (a^) will be
determined by the values of a and )3 at that point, and if the
point travels round one of the curves (a) in the positive direc-
tion, the value of ^ will increase by 2ir for each complete circuit.
If we now suppose the curve {a^ to be the section of the inner
♦ See CreUe'i Journal, lix. p. 835, 1861, also Sohwarz CreUe, Ixiiv. p. 218, 1872.
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190.] Neumann's transformation. S93
surface of a hollow cylinder of any form maintained at potential
zero under the influence of a charge of linear density J? 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
<^=2^(a-ao), (12)
and for the quantity of electricity on any part of the curve a^
between the points corresponding to fii and ^Sg,
Q = ^E(fi,-fi,). (13)
If in this way, or in any other, we have determined the dis-
tribution of potential for the case of a curve of given 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*
Let the values of a and p for the point at which the charge is
placed be a^ and /Sj, then substituting in equation (11) a — a^
for p, Oi— a^ for p^, since both vanish at the surface a^a^, and
P—Pi for 6, we find for the potential at any point whose coor-
dinates are a and fi,
<^ = J? log (1 - 2 6*+-i~H cos (/3 -)3,) -f c2(*+«i-2-o))
- J?log(l-26«— icos (^-^j)4.e2(«--h))-2J?(ai-ao). (14)
This expression for the potential becomes zero when a = a^^,
and is finite and continuous within the curve Oq except at the
point (oj, /3i), at which point the second term becomes infinite,
and in the immediate neighbourhood of that point this term
is ultimately equal to — 2 J? log r', where r' 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 a^ between
the points fi and fi+dfi by a charge E placed at the point (oj, p^)
is, with the notation of Art. 183,
1 d<l>.
iir dsi ^
where dsi is measured inwards and a is to be put equal to a^
after differentiation.
Digitized by VjOOQ IC
294 CONJUGATE FUNCTIONS. [191.
This becomes, by (4) of Art. 183,
E l-e^(*i-'o> , . .
'•^- ~ 2^1-26K-.)cob(/3-^i)+62C-x-%) '^^' ^^^^
From this expression we may find the potential at any point
(«i> P\i within the closed curve, when the value of the potential
at every point of the closed curve is given as a function of /3,
and there is no electrification within the closed curve.
For, by Art. 86, the part of the potential at (a,, )3i), due to the
maintenance of the portion dfi of the closed curve at the potential
Vv&nV^ where n is the charge induced on d^ by unit of electri-
fication at (a|, /9]). Hence, if V is the potential at a point on
the closed curve defined as a function of /9, and ^ the potential
at the point (a^, fi^ within the closed curve, there being no
electrification within the curve,
. ^ 2. /•»' (l~g«(>»-%))Frf^
^ 2TtJti l-2e(-i— •)cos(/3-/3i) + 6^^'i-V ^ '
Example IV. — Distribution of Electi^icity near an Edge of a
Conductor formed by Two Plane Faces,
191.] In the case of an infinite plane face ^ = 0 of a con-
ductor, extending to infinity in the negative direction of y,
charged with electricity to the surface-density <r^, we find for
the potential at a distance y from the plane
F=C-4,r^oy,
where C is the value of the potential of the conductor itself.
Assume a straight line in the plane as a polar axis, and trans-
form into polar coordinates, and we find for the potential
F = C— 4 IT (roac** sin d,
and for the quantity of electricity on a parallelogram of breadth
unity, and length ae^ measured along the axis
E = a^ae^.
Now let us make p = np' and ^ = 71^, then, since p' and $'
are conjugate to p and ^, the equations
r = C- \Tt<r^a€^' sin 71 ^
and i?=(rQae"'»'
express a possible distribution of potential and of electricity.
Digitized by VjOOQ IC
19 1.] DISTRIBUTION OF ELECTEICITY. 295
If we write r for ae^, r will be the distance from the axis ; we
may also put 0 instead of ^ for the angle. We shall then have
7= C— 4^(70-— rr sin 71^,
"a *
V will be equal to C whenever nfl = ir or |i multiple of ir.
Let the edge be a salient angle of the conductor, the inclination
of the faces being a, then the angle of the dielectric is 27— a, so
that when 0 s= 2 7r— a the point is in the other face of the con-
ductor. We must therefore make
7i(2ir— a) = IT, or 71 =
2 IT— a
Then
F=C7-.47r<roa(-) sm„ >.
^ 2»-«
The Borface-densitj (t at any distance r from the edge is
When the angle is a salient one a is less than ?r, 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
o
the density varies inversely as the |th 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
Digitized by VjOOQ iC
296 CONJUGATE FUNCTIONS. [192.
prism, and the density is inversely as the fourth root of the
distance.
When a = TT the edge is obliterated, and the density is con-
stant.
When a = ^ TT the edge is like that of the outside of the
hexagonal prism, and the density is directly as the square
root of the distance from the edge.
When a^-^-n the edge is a re-entrant right angle, and the
density is directly as the distance from the edge.
When a = I TT 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 Y.— Ellipses and Hyperbolas. Fig. X.
192.] We see that if
x^ = e* cos t/r, y^ = e* sin t/r, (1)
x^ and y^ will be conjugate functions of ^ and t/r.
Also, if a;^ = ^"^ ^^ ^9 2^2 = ~ ^"^ ^^ V^> (2)
X2 and 2/2 will be conjugate functions of ^ and ^. Hence, if
2x = Xi-hX2 = {e^ + e-^)co&\lr, 22/ = t/i + J/g = («*-^"*)suiV^j (3)
X and y will also be conjugate functions of <f> and i/r.
In this case the points for which ^ is constant lie on the ellipse
whose axes are e^ + e~^ and c^— «"♦•
The points for which ^ is constant lie on the hyperbola whose
axes are 2 cos^ and 2 sin^.
On the axis of aj, between a; = — 1 and a; = + 1,
<^ = 0, ^ = COS"* x. (4)
On the axis of x, beyond these limits on either side, we have
x> 1, V^=2n7r, <l> = log(aj+ -/ar*— 1),
aj<— 1, V^ = (27i+l)7r, <^ = log(V'ar^-l-a;). (5)
Hence, if <^ is the potential function, and i/r the function of
flow, we have the case of electricity flowing from the positive
to the negative side of the axis of x through the space between
the points — 1 and + 1, the parts of the axis beyond these limits
being impervious to electricity.
Digitized by VjOOQ iC
1 93-] ELLIPSES AND HYPERBOLAS. 297
Since, in this case, the axis of 2/ 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. {This includes the chai-ge on both sides of the flat
conductor.}
If we make yfr the potential function, and 0 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 potential.
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 j/ as functions of x and y,
where ^
x'=blog^^^^Ty^ y'^fttan-i^' (6)
x^ and y' will be also conjugate functions of the <f> and yjf of
Art. 192.
The curves resulting from the transformation of Fig. X with
respect to these new coordinates are given in Fig. XI.
If a^ and y are rectangular coordinates, then the properties of
the axis of a? in the first figure will belong to a series of lines
parallel to oj' in the second figure for which 3/^= bn\ where n'
is any integer.
The positive values of x' on these lines will correspond to
values of a; greater than unity, for which, as we have akeady
seen,
*^ /'iT'
Vr = 7iir, <^ = log(aj+ V^a^-l) = log(6^ + /\/ e" -1). (7)
The negative values of x' on the same lines will correspond
to values of x less than unity, for which, as we have seen,
<^=0, ^ = cos-*a:= cos"^6^- (8)
The properties of the axis o{ y in the first figure will belong
to a series of lines in the second figure parallel to a:', for which
/= 6^(71' H-i). (9)
Digitized by VjOOQ iC
298 CONJUGATE FUNCTIONS. [193.
The value of i/r along these lines is t/r = ir (n + i) for all points
both positive and negative, and
[The curves for which <f> and t/f are constant may be traced
directly from the equations
a;'= ^6 log 1 (^2^ + 6-2^ + 2 cos 2 t/r),
2/'=6tan-i{^^^tanVr).
As the figure repeats itself for intervals of Trb 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 ^ changes sign
with y\ Let us suppose that ^ so changes sign. Then any
curve for which yfr is constant will be symmetrical about the
axis of x\ cutting that axis orthogonally at some point on its
negative side. If we begin with this point for which <> = 0, and
gradually increase ^, the curve will bend round from being at
first orthogonal to being, for lai^e values of 0, at length parallel
to the axis of (xf. The positive side of the axis of a' is one of the
systemj viz. yjr is there zero, and when y' = ± iirft, t/r = iir.
The lines for which yjr has constant values ranging from 0 to iir
form therefore a system of curves embracing the positive side of
the axis of af>
The curves for which 4> has constant values cut the system >fr
orthogonally, the values of ^ ranging from +00 to — oo. For
any one of the curves <^ drawn above the axis of x^ the value of
<l> is positive, along the negative side of the axis of x' the value
is zero, and for any curve below the axis of of the value is
negative.
We have seen that the system ^ is, symmetrical about the axis
of x' ; let PQR be any curve cutting that system orthogonally
and terminating in P andiZ in the lines /= ±l7r6, the point
Q being in the axis of x\ Then the curve PQR is symmetrical
about the axis of x\ but if c be the value of <f> along PQ,
the value of <l> along QR will be — c. This discontinuity in the
value of ^ will be accounted for by an electrical distribution in
the case which will be discussed in Art. 195.
If we next suppose that ^ and not <^ changes sign with y', the
values of <f> will range from 0 to 00. When ^ = 0 we have the
Digitized by VjOOQ iC
I95-] EDGE OP AN BLECTEIPIED PLATE. 299
negative side of the axis of x\ and when ^ = oo we have a line
at an infinite distance perpendicular to the axis of x\ Along any
line PQ-R between these two cutting the V^ system orthogonally the
value of <> is constant throughout its entire length and positive.
Any value ^ now experiences an abrupt change at the point
where the curve along which it is constant crosses the negative
side of the axis of a?', the sign of >/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 >/r as the
function of flow, we may consider the case to be that of an in-
definitely long strip of metal of breadth ir6 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 iJong the other, the entrance and exit
of the current being at an infinite distance on the positive side
of the origin, the distribution of potential and of current will be
given by the functions ^ and ^ respectively.
K, 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 o^, extending
from the axis of y' to an infinite 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 Qcz on the positive side of the origin, and let two infinite con-
ducting planes be placed parallel to it and at distances iirft on
either side. Then, if i/r is the potential function, its value is 0
for the middle conductor and } ir for the two planes.
Let us consider the quantity of electricity on a part of the
middle conductor, extending to a distance 1 in the direction of z,
and from the origin to 0:^= a.
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300 CONJUGATE rUNCTIONS. [196.
The electricity on the part of this strip extending from x{ to
^i'^ ^(*2-*i)-
Hence from the origin to a:'=a the amount on one side of the
middle plate is
If a is large compared with 6, this becomes
1 -
i?=— log2e*,
^a + &log,2 .
47rfc ^ ^
Hence the quantity of electricity on the plane bounded by
the straight edge is greater than it would have been if the elec-
tricity 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 &log^2
beyond the actual boundary of the plate.
This imaginary uniform distribution is indicated by the dotted
sti'aight lines in Fig. XI. The vertical lines represent lines of
force, and the horizontal lines equipotential surfaces, on the
hypothesis that the density is uniform over both planes, pro-
duced to infinity in all directions.
196.] Electrical condensers are sometimes formed of a plato
placed midway between two parallel plates extending con-
siderably beyond the intermediate one on all sides. If the
radius of curvature of the boundary of the inteimediate 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 inter-
mediate 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 £f be the actual area of the plate, L its circumference,
and B the distance between the large plates, we have
h=\B, (13)
Digitized by VjOOQ iC
J 96.] EDGE OP AN ELECTRIFIED PLATE. 301
and the breadth of the additional strip is
(14)
so that the extended area is
S'=S+^^^BL. (15)
The capacity of one side of the middle plate is
1 fir 1 ffif .1, J ,.
Corrections 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 ^ = yjf^
The plate will be of nearly uniform thickness, p =^ 2b\lf\ 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
y' = 0, whence 3.' ^ j i^g^ eos >/.'. (17)
The value of <t> at this edge is 0, and at a point for which
x'= a (a/b being large) it is approximately
a + 6 log^ 2
b •
Hence, altogether, the quantity of electricity on the plate is
the same as if a strip of breadth
^(l0g.2+l0g.C08j|),
i.e. -- log, (2 cos 2^), (18)
had been added to the plate, the density being assumed to be
everywhere 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
y
J_d^__l_^ ^
4'ndx'^ ^Tsb
/x/a"'-
1
2x' 4x'
1 • 2j' 4Jr' V
= -— T(l+i«~T + ge— T + fecJ. (19)
4iro
Digitized by VjOOQ IC
302 CONJUGATE FUNCTIONS. [197.
The quantity within brackets rapidly approaches unity as x'
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 ^^^^^ of the normal density.
In like manner we may calculate the density on the infinite
planes ^
^ ^' . (20)
4 7r6 / tx*
/sj I
h
+1
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 z^^zi of the
normal density.
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 + Y and — F. 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 01! z at distances B z=nih^ and
all cut off by the plane of y'z, 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 ^ is constant.
When ^= 'n.Trft, that is, in the prolongation of each of the
planes, we have a^ = 6 log i (e* + c"*), (2 1 )
when y'= (^ + i)7r6, that is in tlie intermediate positions
a<=61ogi(e*~c-^). (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 ^ ^ j (<^-log, 2), (23)
Digitized by VjOOQ iC
198.] DENSITY NBAE THE EDGE. 303
and the amplitude of the undulations on either side of this line is
When <() is large this becomes be"^^, so that the curve ap-
proaches to the form of a straight line parallel to the axis of y'
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 6^ = a + 6 log, 2, 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 log, 2.
If B is the distance between two of the planes of the series,
£ = ir 6, so that the additional distance is
a = B^^^. (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 (t>, and may be considered as approximately plane.
If Z) is the depth of these undulations from the crest to the
trough of each wave, then we find for the corresponding value of ^,
D
* = Uog^- (26)
6^-1
The value of z' at the crest of the wave is
61ogi(^ + «"^)- (27)
* Hence, if il is the distance from the crests of the waves to
* Let 4 be the potential of the plane, <p of the nndulating snifaoe. The quantity
of electricity on the plane per unit area ii 1 + 4 v&. Hence the capacity
-l+4»(il + o'), Buppoee.
Then il + a'-»&(«-^).
But ^ + 61ogi(a'^ + «"^-6(*-log2);
.-. o'--6^ + 6(log2 + logt(e* + e"*))
-61og(l + e-2*)
.6Ug_?_,by(26).
1 + «"J
Digitized by VjOOQ IC
304 CONJUGATE FUNCTIONS. [199.
the opposite plane, the capacity of the system composed of the
plane surface and the undulating surface is the same as that of
two planes at a distance A H- a\ where
, B. 2
« = -log. D' (28)
199.] If a single groove of thiis 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 to 71 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
gi-oove will be diminished on account of the neighbouring
grooves.
If i 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 S will be
S-LB LB _ _S__ LB a'
47r^ "^ 471(4 + 0')"" 47r^ 47r4'4+a' ^^
If A is large compared with B or a, the correction becomes
by (28) I B\ 2 .,
1+e B
and for a slit of infinite depth, putting D =00, the correction is
To find the surface-density on the series of parallel plates we
must find cr = rX when 6 = 0. We find
Alt ax
"=4^ /-^' (32j
The average density on the plane plate at distance A from the
of the series of plates is ^ = - — r • Hence at a distance
Digitized by VjOOQ iC
200.] A GROOVED SUEPACE. 305
from 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 {a series of co-axial
cylinders in front of a plane} formed by rotating the plane of
the figure in Art. 197 about the axis i/=— iJ. In this case,
Poisson's equation will assume the form
Let us assume F= <^, the function given in Art. 193, and
determine the value of p from this equation. We know that the
first two terms disappear, and therefore
If we suppose that, in addition to the surface-density already
investigated, there is a distribution of electricity in space ac-
cording to the law just stated, the distribution of potential will
be represented by the curves in Fig. XI.
Now from this figure it is manifest that t^ is generally very
small except near the boundaries of the plates, so that the new
distribution may be approximately represented by a certain
superficial distribution of electricity near the edges of the plates.
K therefore we integrate / ( pdx'df/ between the limits 2/^=0
and y' = - 6, and from a;'=— oo toaj = +oo, we shall find the
St
whole additional charge on one side of the plates due to the
curvature.
Since ^=-g, we have
J-J J.»4-aR + ydx'
y
1 1
8R + y'
VOL. I. X
Digitized by VjOOQ IC
Jt J -a
306 CONJUGATE FUNCTIONS. [2OO.
Integratiiig with respect to y', we find
This is half the total quantity of electricity which we must
suppose distributed in space near the edge of one of the cylinders
per unit of circumference. Since it is only close to the edge of
the plate that the density is sensible, we may suppose the elec-
tricity all condensed on the surface of the plate without altering
sensibly its action on the opposed plane surface, and in calcu-
lating the attraction between that surface and the cylindric
surface we may suppose this electricity to belong to the cylindric
surface.
If there had been no curvature the superficial charge on the
positive surface of the plate per unit of length would have been
-r.
1 dif,,, \ ,, , . 1
Hence, if we add to it the whole of the above distribution, this
charge must be multiplied by the £ekotor (l + i p) to get the total
charge on the positive side^.
t la the case of a disk of radius 22 placed midway between two
* {Sinoe there ii % ohwge on the negatiye side of the pUte equal to that on the
podtiye side, it would seem that the total ohuge on ue cylinders per unit cir-
cnmferenoe is -- (l + j^)f •<> ^^^ *he correction for curvatnre is (l + -^\ and not
(lf^|)a.in the text.}
t [In Art. 200, in estimating the total space distribntion we might perhaps more
correctly take for it the integrally/* 2 sr(iJ+y')c2afcy, which gives, per unit circum-
1 B
ference of the edge of radius JB, — r;? ^, thus leading to the same correction as in the
text. 82 E
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 + JB from the edge of the middle one. That edge will theie^e envelope a
circle, which will be the edge of the disk. As in Art. 200, we begin with Poisson'a
equation, which in this case will be
We now assume that r» ^, the potential function of Art. 195. We must therefore
suppose electricity to exist in the region between the plates whose volume density p is
1 1 rf^
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200.] OIEOULAB GBOOVBS, 307
infinite parallel plates at a distance B, we find for the capacity
of a disk P2 i^^ o
^ + 2i2iili?+lA (38)
The total amount is B
2/ ' f /».2ir(B-*')ifa'dy.
i'l:
Now if B is large in oomparison with the distance between the platee this result
will be seen, on an examination of the potential lines in Fig. XI, to be sensibly the
same as B
*8 r«j^
fX
.jd^d^\ that is, -i» A
Hie total surface distribution if we include both sides of the disk is
-X
72
V«=o
..2"l|:,iog(i.x/^I7?)}^
'-'S'-.jr^iog(i+v^iTr^O<^£.
To evaluate the latter integral put
we get approximately if 22/6 is large
R -15
jr»iog(i+yi::7^)«i£-i^** ' i„g(2-«)(^-J)*
so that the quantity of eleotrioity on the plate
Since the difference of potential of the pUtes - ^ and ^ - »6, the capacity is
a result which is less than that in the text by -28 B nearly.] ^^^C,^ ^ JL 1 « s!r^U Li i)^
J.iC^.f=^,(S^>.<.-.^^5
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308 CONJUGATE FUNCTIONS. [2OI.
Theory of Thorii8on*8 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 ittdius R surrounded by a large
plane plate called a Guai*d-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
K—Ry which we denote by B.
The charge on the disk due to unit potential of the large disk,
R^
supposing the density uniform, would be —j •
The charge on one side of a straight groove of breadth B and
length L = 2TrR, 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
. RB
since in this case <I> = 1, <^ = 0, and therefore 6 = ^ +a'.
But since the groove is not straight, but has a radius of curv-
ature R, this must be multiplied by the factor (l + i ^)* •
^ R^
The whole charge on the disk is therefore
R^ , RB . B.
_R^-\-R^ R'^^R^ a'
The value of a cannot be greater than
^-,= 0.22j8 nearly.
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 ^ to i2
♦ I If we take the correction for curvature to be (l +-■=), see foot-note p. 306, the
charge on the disk will be less than that given in the text by B^/IQ {A + o'). }
Digitized by VjOOQ iC
202.] A CASE OF TWO PLANES. 309
may have any value, but the radii of the large disk and of the
guard-ring must exceed R by several multiples of A.
Example Vn.— Fig. Xn.
202.] Helmholfcz, 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 Xj^=:A<f> and y^^Ay^^
and also aj^ = -4e* cos^ and 3/2 = ile* sini/r,
are conjugate functions of <t> and ^, the functions formed by
adding ar^ to x^ and y^ to 2/2 will be also conjugate. Hence, if
a:= il<^ + JLe*cos^,
y = JLi/r + ile^sin^,
then X and y will be conjugate with respect to <t> and >/r, and ^
and >/r will be conjugate with respect to x and y.
Now let X and y be rectangular coordinates, and let i\/f be the
potential, then A;0 will be conjugate to A;^, A; being any constant.
Let us put yjf = 'JT, then y = AiT,x = A (0— e*).
If (j) varies from — oo to 0, and then from 0 to +00 ,x varies
from —00 to —-4 and from — il to — 00 . Hence the equipotential
surface, for which t|f = tt, is a plane parallel to o^ at a distance
b = vA from the origin, and extending from a; = — cx> toaj = — -4.
Let us consider a portion of this plane, extending from
X = — (-4 +a) to aj = —A and from 0 = 0 to 0 = c,
let us suppose its distance from the plane o{ xztohey = b = Av,
and its potential to be F = A;^ = A;ir.
The charge of electricity on the portion of the. plane considered
is found by ascertaining the values of 4> at its extremities.
We have therefore to determine <f> from the equation
aj = — (il+a) = A{<t>-€i^),
<l> will have a negative value <t>i and a positive value (ft^; at the
edge of the plane, where x = —-4, ^ = 0.
Hence the charge on the one side of the plane is — cAj^j -?- 4^7,
and that on the other side is cA;02 -s- ^ir.
* Monatheriehte der Kd.M. Akad. der WiMentehq/ten, zu Berlin, April 23, 1868,
V' 215.
Digitized by VjOOQ iC
310 CONJUGATE FUNCTIONS. [203.
Both these charges are positive and their sum is
If we suppose that a is large compared with A,
*2 = log|3 + l + log(2+l+&c.)j-
If we neglect the exponential terms in 0^ 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 OF, and the attraction towards the infinite
plane, whose equation is j^ = 0 and potential ^ = 0, is
A
The equipotential lines and lines of force are given in Fig. XII.
Example Vin. Theory of a Oratirig of Parallel Wires. Fig. XIII.
208.] 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 elec-
tricity 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
Digitized by VjOOQ iC
204.] INDUCTION THROUGH A GRATING. 311
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 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==-2Alog/+a (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
mustmake 7^2 «. j^grcos^ + r*, (2)
and if we suppose that the axis of reference is also charged with
the linear density A', we find
F=-Alog(l-2rco8^ + r»)— 2A'logr + C. (3)
If we now make
i,y ^ 2'nx f^.
T^^ a, fl= , (4)
then, by the theory of conjugate functions,
2»y ^ 4wy l»y
F = -Alog(l-2e- cos^^+e«)-2\'log6« +C, C^)
a
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 x is a multiple of a, and to planes perpendicular to the
axis of y.
Each of these wires is chaiged with a linear density A.
The term involving A^ indicates an electrification, producing a
constant force in the direction of y.
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 dis-
tance between them.
Digitized by VjOOQ iC
312 CONJUGATE FUNCTIONS. [205.
The eqnipotential 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 = b^^a, quantity large compared
with a, we find ^proximately,
^ = «il[^(X + \') + Cnearly. (6)
(Xf
If we next make y = — Jg* where b^iBA positive quantity large
compared with a, we find approximately,
IJ=i^U' + Cnearly. (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
eqnipotential surface which cuts the plane of xz at a distance c
from the axis of z. To find the potential of the grating we
therefore put oj = c, and y = 0, whence
F=-2Xlog,2sin~ + C. (8)
205.] We have now obtained expressions representing the
electrical 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 a^ on the first plane is got from the
equation (6) dV, 47r, ,,,
that on the second plane 0*2 from the equation (7)
If we now write ^ ^^
and eliminate c, K and Xf from the equations (6), (7), (8), (9), (10),
we find
A^a,(b, + b,+ hh) = V,(l +^^)-r,-V^-^. (12)
4ifa^(b,+b,+^) = -v,+%(i + ^)-r^. (13)
Digitized by VjOOQ iC
205.] INDUCTION THROUGH A GRATING. 313
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, say the first, F = T[, and the right-hand side of the
equation for o-i becomes TJ"— TJ. Hence the density o-j induced
on the first plane when the grating is interposed is to that
which would be induced on it if the grating were removed,
the second plane being maintained at the same potential, as
0(61+62)
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 b^ 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 bi + b^U>
61 + 62 +
6162
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
61 62 : 61 62 + <* (61 + 62)-
This investigation is approximate only when b^ and 62 ^^
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 wiU be no induction
through it. We ought therefore to have for this case a = 0.
The formula (11), however, gives in this case
a=-^log.2, = -Olla,
which is evidently erroneous, as the induction can never be
Digitized by VjOOQ IC
314 CONJUGATE FUNCTIONS. [206.
altered in sign by means of Uie 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 distri-
bution of electricity on each is symmetrical with respect to the
diameter parallel to y^ the proper expansion of the potential is
of the form F= Cologr + SC^r'cosi^, (14)
where r is the distance from the axis of one of the wires, and $
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 $ must
vanish.
For the sake of conciseness let us assume new coordinates
i, 17, &c. such that
af=2vaj, ai; = 2iry, ap=2'nry a/3 = 2ir6,&c., (15)
and let F^ = log(e^+^ + e-(^+'*>-2 cos f). (16)
Then if we make
F=^„^, + 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 17 +/3 = 0 and cos f = 1.
When i8 = 0 the expansion of ^in terms of p and 6 is*
Fq= 2logp + /ip2cos2^-YTVTrP*cos4^ + &c. (18)
For finite values of /3 the expansion of ^ is
^^==/3 + 2log(l-6-0+^i^pcos^-,-^-^ (19)
In the case of the grating with two conducting planes whose
equations are r\^ P\ and 17 = — ^2> *^at of the plane of the
grating being ?/ = 0, there will be two infinite series of images
* {The eipansion of J* can be got by noticing that log («'"'* + €^- 2 ooif) only
differK by a constant from logr* + logri* + logr,* + ... where r, n, r,... are the distanoee
of P from the wires.
We can apply the same method to expand JJj since this corresponds to moving the
wires through a distance — b parallel to y, the expansion however is not of the nme
form as that given in the text.}
Digitized by VjOOQIC"
206.] METHOD OF APPBOXIMATION. 815
of the gratmg. 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
i?=±27iOi + ft), (20)
n being an integer.
The second series will consist of an infinite series of images for
which the coefficients Aq, A^^ A^^ &c. are equal and opposite to
the same quantities in the grating itself, while A^y ^3, &c. are
equal and of the same sign. The axes of these images are in
planes whose equations are of the form
i?=2i32±2m()8i + i3j), (21)
^i being an integer.
The potential due to any infinite 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 £17 + (7, the conditions of the problem will
be sufficient to determine the electrical distribution.
We may first determine T^ and T^, the potentials of the two
conducting planes^ in terms of the coefficients il^, JL^, &o., and
of B and (7. We must then determine cr^ and o-,, the surface-
densities at any points of these planes. The mean values of <t^
and (Tj are given by the equations
4»a, = ^(^,-5), 4,a-, = i^(^„ + B). (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 0^
adding to the result BpcoBB + C.
The terms independent of B then give V the potential of the
grating, and the coefficient of the cosine of each multiple of $
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 a^^ and o-g in terms of TJ", TJ, and F.
These equations will be of the form
Tf-Frr 4wcri(6i + a-y)+4w(r2(a + y),
15~F=47r(rj(a + y)+4ir<r2(62 + a-y). (23)
The quantity of electricity induced on one of the planes
Digitized by VjOOQ iC
316 CONJUGATE FUNCTIONS.
protected by the grating, the other plane being at a given dif-
ference of potential, will be the same as if the planes had been at
a distance
(a-y)(^4-6.)-h^6,-4ay ^^^ ^^
a + y ^ *
The values of a and y are approximately as follows,
^'"2^( ^^2^"" 3'l6a* + 7r*c*
+ 2e-*'~ (l + e"*'- + e-*'« + &cj + &c j ' (2*)
\l-6 « 1-C «/
* { In the Supplementary Volume another method of employing conjugate function!,
by which the capacity of finite plane sur&ceB etc. can be calculated, will be described } .
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CHAPTEE XIII.
ELECTROSTATIC INSTRUMENTS.
On Electrostatic iTistruments.
The instruments which we have to consider at present may
be divided into the following classes ;
(1) 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 positive electrification of the glass induces a
negative electrification of the points, which is tiie more intense
the sharper the points and the nearer they are to the glass.
When the machine works properly there is a dischai-ge 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, or to any other
body with which it is in electric communication.
The portion of the glass which is advancing towards the
Digitized by VjOOQ iC
318 CONJUGATE FUNCTIONS. [207.
rubber has thus a smaller positive charge than that which is
leaving it at the same time, so that the rubber, and the con-
ductors in communication with it, become negatively electrified.
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 over-
coming ordinary friction and other resistances, and the excess is
employed in producing 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 be-
tween 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 suffi-
ciently intense there would be a discharge between the glass and
the rubber, instead of between the glass and the collecting points.
To prevent this, fiaps 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
nibber.
In some electrical machines the moving part is of ebonite
instead of glass, and the rubbers of wool or fur. The rubber
is then electrified positively and the prime conductor negatively.
* It is probable that in many oaees where dynamical energy is oonyerted into heat
by friction, part of the enerey may be iint transformed into electrical energy and
then converted into heat as Uie electrical energy is spent in maintaining currents of
short circuit dose to the rubbing surfaces. See Sir W. Thomson, ' On the Electro-
dynamic QuaUties of Metals.' PkiL Trans., 1866, p. 649.
Digitized by VjOOQ iC
209-] BLBOTBOPHOEUS. 319
The Electrophorua of Volta.
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 plate with the metal back of the ebonite plate when
the ebonite and metal plates are in contact.
The ebonite plate is electrified negatively by nibbing 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 poten*
tial 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 conductor. 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 tiiat the operation of charging the
condenser 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 electrification may be produced entirely by mechanical work
against the electrical forces is of scientific importance if not of
Digitized by VjOOQ iC
320 CONJUGATE FUNCTIONS. [2IO.
practical value. The first machine of this kind seems to have
been Nicholson's Revolving Doubler, described in the Philo-
sophical Transactions for 1788 as *an Instrument which, by the
turning of a Winch, produces theTwo States of Electricity with-
out 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
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 move-
able 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 conducting body. As the inductors and re-
ceivers cannot completely suiTound 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 theoreti-
cally perfect without a pair of regenerators, 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 i), with two carriers F and O.
Suppose the inductor ^ to be positively electrified so that
its potential is A^ and that the carrier F is within it and is at
potential F. Then, if Q is the coefficient of induction (taken
positive) between A and F, 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
— Qi4, 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
♦ Specification of Patent, Jan. 27. 1860, No. 206.
Digitized by VjOOQ iC
2IO.] THE REVOLVING DOUBLEB. 321
will ihen, as was shewn in Art. 32, become completely dis-
charged, and will communicate 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 receivers.
But by putting the inductor A in communication with the
receiver D, and the inductor O with the receiver -B, the poten-
tials of the inductors will be continually increased, and the
quantity of electricity communicated to the receivers in each
revolution will continually increase.
For instance, let the potential of A and Dhe U, and that of B
and C, F, then, since the potential of the carrier is zero when
it is within -A, being in contact with earth, its charge is
= — Qt7. The carrier enters B with this charge and com-
municates it to £. If the capacity of B and C is B^ their
Q
potential will be changed from Fto F— -^ U,
If the other carrier has at the same time carried a charge
— QF from (7 to D, it will change the potential of A and D from
U to U— ~- F, if Q' is the coeflScient of induction between the
carrier and O, and A the capacity of A and D. If, therefore,
U^ and T^ be the potentials of the two inductors after n half
revolutions, and tT.+i and TJ+x after n+1 half revolutions,
91
A
Q
t^.« =1^.-1-^.
F — F— ~ U
Q .^^ ^ _ 0' ^. find
If we write !>* = ^ and ^=^~-9we
pJ^n^i-^qV.^! = (ptr.+ffTD (1 -pq) = iP^o+ 9V0) (1 --pjr S
P^n^i-qVn^i = (p^n-qK) (1 +i>?) = (pu.^qv,) (1 ^pqr'
VOL. I.
Digitized by VjOOQ IC
822 BLECTBOSTATIO INSTEUMENTS. [2 II.
Hence
2U,= fr,((l-2>3)« + (l+^)-)+|^((l-2>3)«-(l+^)«),
^^ = f - ^o((l-2>3)*-(l +^r)-f Vo ((1-1>3)" + (1 +i>?)*) •
It appears from these equations that the quantity pU-^qV
continually diminishes^ so that whatever be the initial state of
electrification the receivers are ultimately oppositely electrified,
so that the potentials of A and B are in the ratio of g to —p.
On the other hand, the quantity pU^qV continually in-
creases, so that, however little pU may exceed or fall short of
qVai first, the difference will be increased in a geometrical ratio
in each revolution till the electromotive forces become so gi*eat
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 called a Beplenisher.
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 thenno-electric pair, then, by turning the
machine, the difference of potentials may be continually multi-
plied 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 carrier 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 into an insulated receiver out of an uninsulated
* Proc Ji. JS,, June 20, 1867.
Digitized by VjOOQ iC
213-] THE BECIPBOOAL ELEOTEOPHORUS. 823
vessel placed inside but not touching an inductor. The receiver
is thus continuaUy supplied with electricity of opposite sign to
that of the inductor. K the inductor is electrified positively, the
receiver will receive a continually increasing charge of negative
electricity.
The water is made to escape from the receiver by means of a
funnel, the nozzle of which is almost surrounded by the metal of
the receiver. The drops falling from this nozzle are therefore
nearly free from electrification. Another inductor and receiver
of the same construction are arranged so that the inductor of
the one system is in connexion with the receiver of the other.
The rate of increase of charge of the receivers is thus no longer
constant, but increases in a geometrical progression with the
time, the charges of the two receivers being of opposite signs.
This increase goes on till the falling drops are so diverted from
their course by the electrical action that they fall outside of the
receiver or even strike the inductor.
In this instrument the energy of the electrification is drawn
from that of the falling drops.
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 be-
tween 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 atmosphere. 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 dis-
charged 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.
* { The induction mnchinee most frequently used at present are those of Voss and
Wimshnrst. A description of these with diagramn will be found in Nature, vol. xxviii.
p. 12.}
Digitized by VjOOQ iC
324 BLEOTEOSTATIO INSTEUMBNTS. [2 1 3.
Now we have shewn that whenever this occurs there is a loss
of energy, and therefore the whole work employed in turning
the machine is not converted into electrification in an available
form, but part is spent in pro-
ducing the heat and noise of
electric sparks.
I have therefore thought it
desirable to shew how an elec-
trical machine may be con-
structed which is not subject
to this loss of efiiciency. 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.
Li the figure let A, By C, A\ JB'y (T represent hollow fixed
conductors, so arranged that the carrier P passes in succession
within each of them. Of these -4, A' and B, B' nearly surround
the cai-rier when it is at the middle point of its passage, but
0 and C do not cover it so much.
We shall suppose A, B, C to he connected with a Leyden jar
of great capacity at potential F, and A\ jB', C to be connected
with another jar at potential — F.
P is one of the carriers moving in a circle from A to (7, fee,
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 coeflBcient of induction between P and A is —-4, The
capacity of P in this position is greater than A, since it is not
completely surrounded by the receiver A, Let it be -4 +a.
Then if the potential of P is IT, and that of -A, F, the charge
on P will be {A^a) U-^AV.
Now let P be in contact with the spring a when in the middle
of the receiver A, then the potential of P is F, 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
Digitized by VjOOQ iC
213.] MACHINE WITHOUT 8PAEKS. 325
more when it comes within the influence of (T, which is
negatively electrified.
If when P comes within C its coefficient of induction on (7
is — C and its capacity is C + c', then, if IT is the potential of
P, the charge on P is
If ar^av,
then at this point U the potential of P will be reduced to zero.
Let P at this point come in contact with the spring e' which
is connected with the earth. Since the potential of P is equal
to that of the spring there will be no spark at contact.
This conductor (f, by which the carrier is enabled to be con-
nected to earth without a spark, answers to the contrivance^
called a regenerator in heat-engines. We shall therefore call it
a Regenerator.
Now let P move on, still in contact with the earth-spring e\
till it comes into the middle of the inductor J3, the potential of
which is F. If — J5 is the coefficient of induction between
P and B at this point, then^ since U ^0 the charge on P will
be -J5F.
When P moves away from the earth-spring it carries this
charge with it. As it moves out of the positive inductor B
towards the negative receiver A' its potential will be increasingly
negative. At the middle of A\ if it retained its charge, its
potential would be
A'r+BV
A' + a' '
and if BV is greater than a'V its numerical value will be
greater than that of V. Hence there is some point before P
reaches the middle of A' where its potential is — F. At this
point let it come in contact with the negative receiver-spring of.
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' IT
towards the positive regenerator C, where its potential is re-
duced to zero and it touches the earth-spring e. It then slides
along the earth-spring into the negative inductor B'^ during
which motion it acquires a positive charge ffV which it finaUy
Digitized by VjOOQ iC
326 ELECTROSTATIC INSTEUMENTS. [214.
communicates to the positive receiver A, and the cycle of opera-
tions is completed.
During this cycle the positive receiver has lost a charge aV
and gained a charge B'V^ Hence the total gain of positive
electricity is B^V'—aV.
Similarly the total gain of negative electricity is BV—a'V\
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 sun'ound
the carrier when it is within them, a and a' may be made very
small, and then the charges of both the Ley den jars will be
increased in every revolution. .
The conditions to be fulfilled by the regenerators are
(TV'^aV, and Cr=a'V\
Since a and a' are small the regenerators must neither be
large nor very close to the carriers.
On Electrometers and Electroscopes,
214.] An electrometer is an instrument by means of which
electric charges or electric 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 Electro-
scopes.
An electroscope if sufficiently sensitive may be used in elec-
trical measurements, provided we can make the measurement
depend on 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
— nected to earth and again insulated. There will then be no
emal electrification on C. Now let A be removed, and B
x)duced into the interior of C, and the electrification of C
\ed by an electroscope. If the charge of B is equal to that
A there will be no electrification, but if it is greater or lesF
re will be electrification of the same kind as that of J3, or
opposite kind.
Methods of this kind, in which the thing to be observed is the
i-existence of some phenomenon, are called niUl or zero
Digitized by VjOOQ iC
215-] coulomb's torsion balance. 327
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 i^at is
required for the complete measurement of the quantity is a
knowledge 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 them-
selves the means of independently determining the true values
of quantities are called Absolute Instruments.
CovZoTtii's Torsion Balance.
215.] A great number of the experiments by which Coulomb
established the fundamental laws of electricity were made by
measuring 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
uppir end of the wire about its own axis any number of
degrees.
Digitized by VjOOQ iC
328 ELBCTBOSTATIO INSTRUMENTS. [215*
The whole of this apparatus is esclosed 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 0 with the radius through the centre of the fixed
sphere. The distance of the centres is then 2a&m\0, where a
is the radius of the torsion-arm, and if jP is the force between the
spheres the moment of this force about the axis of torsion is
FacosiO,
Let both spheres be completely discharged, and let the torsion-
arm now be -in equilibrium at an angle <f> with the radius through
the fixed sphere.
Then the angle through which the electrical force twisted the
torsion-arm must have been d— <^, and if M is the moment of
the torsional elasticity of the fibre, we shall have the equation
Fa COB iez=zM{e -<!>).
Hence, if we can ascertain if, we can determine F, the actual
force between the spheres at the distance 2 a sin i^.
To find Jf, 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
sphere is due partly to the dii-ect 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
measurements 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
Digitized by VjOOQ iC
215-] INPLUENCB OP THE CASE. 329
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 b,
that the centre of motion of the torsion-arm coincides with the
centre of the sphei'e and that its radius is a ; that the charges on
the two spheres are E^ and E^ and that the angle between their
positions is 0 ; that the fixed sphere is at a distance 04 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
_EEj^
and the moment of this force round a vertical axis through the
centre will be j^j? ^^ «;« /i
The image of E^ due to the spherical surface of the case is a
point in the same radius at a distance from the centie — with
b "^
a charge ^E^ —y and the moment of the attraction between E
and this image about the axis of suspension is
, a — sin (9
PP ± ^
=zEE • cui,sm0
^'{l-2-plcos^+^^^^J
If 6, the radius of the spherical case, is large compared with a
and O], the distances of the spheres £rom the centre, we may
neglect the second and third terms of the factor in the de-
nominator. Equating the moments tending to turn the torsion-
arm, we get
^J:,aa, sin^ Ji - i| = J/ (tf-i^).
Digitized by VjOOQ iC
330 ELBOTEOSTATIO INSTBUMENTS. [2 1 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 instru-
ment, provided the radii of the balls are small compared with
their distances from each other and from the sides or opening of
the case.
Coulomb's form of apparatus, however, is not well adapted for
measurements of this kind, owing to the smallness of the force
between spheres at the proper distances when the difference of
potentials is small. A more convenient form is that of the
Attracted Disk Electrometer. The first electrometers on this
principle were constructed by Sir W. Snow Harris *. They have
since been brought to great perfection, both in theory and con-
struction, by Sir W. Thomson f.
When two disks at different potentials are brought face to
face with a small interval between them there will be a nearly
uniform electrification on the opposite faces and very little elec-
trification 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 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 discs thus arranged is given at Art. 202, but since
♦ PhU, Trans. 1834.
t See an excellent report on Electrometer! by Sir W. Thomson. Report of the
Lritish Aseociation, Dundee, 1867.
Digitized by VjOOQ iC
217.]
PBINOIPLB OP THE GUABD-BING.
331
it is impossible 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 inter-
preted 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
determining 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
tmMtnmmtt
Fig. 19.
electrification near the edge is of no importance, as it occurs
on the guard-ring and not on the suspended part of the disk.
Besides this, by connecting the guard-ring with a metal case
surrounding the back of the attracted disk and all its sus-
pending apparatus, the electrification of the back of the disk
is rendered impossible, for it is part of the inner surface of a
closed hollow conductor all at the same potential.
Thomson's Absolute Electrometer therefore consists essentially
Digitized by VjOOQ iC
332 ELEOTEOSTATIO INSTRUMENTS* [217.
of two parallel plates at diflferent 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 move-
able 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^ . nd indicates that the suspended 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
Digitized by VjOOQ iC
217.]
Thomson's absolute blbcteometeb. 333
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 me^ins 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 TT be the numerical value of the weight, and g the force of
gravity, the force is TTgr, and if -4 is the area of the suspended
disk, D the distance between the disks, and V the difference of
the potential of the disks *,
^^=^^' - ^=^V'
* Let ns denote the radius of the saspended dibk by B, and that of the aperture
of the guard-ring by Sf^ then the breadth of the annular interval between the
disk and the ring will be S « JS'-i^.
If the distance between the suspended disk and the large fixed disk is D, and
the difference of potentials between these disks is F, then, by the investigation in
Art. 201, the qoantity of electricity on the suspended disk will be
^ ^ \ ^D 82) D + oJ'
where a-J?^^, or a - 0-220635 (iJ'-B).
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 betweetsn the fixed disk and
the guard-ring it* not D but iJ-1-2 « D\ then it appears from the investigation in
Art. 225 that there wiU be an additional charge of electricity near the edge of
the diuk on account of its height t above the general surfitce of the gnard-ring.
The whole charge in this case is therefore, approximately,
and in the expresj^ion for the attraction we must sobstitute for A, the area of the
disk, the corrected quantity
^ - 4- j B' + B"- (B"-JP) ^ + 8 (B + B') (B'-P) log. *-^?^ \ .
where R a radius of suspended disk,
B'^ radius of a|>erture in the guard-ring,
I) « distance between fixed and suspended disks,
jy le distance between fixed disk and guard-ring,
a = 0.220685 (JT-/?).
When a is small compared with D we may neglect the second term, and when
D'— 2) is small we may neglect the last term. |For another investigation of this see
Supplementary Volume}.
Digitized by VjOOQ IC
334 ELECTROSTATIC INSTRUMENTS. [2 1 8.
If the suspended disk is circular, of radius R, and if the radius
of the aperture of the guard-ring is R\ then
A = i^{R'+R''), and F= 4D A/^f—i'
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 F' are two potentials, and D and 1/ the
corresponding distances,
r^V= (D.D')^^J^.
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
replenisher, the lower disk of the principal electrometer is main-
tained 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
electrometer 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 electro-
meter 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 1/ 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 a conveniently measurable
distance. When the distance is too small a small chanije of
Digitized by VjOOQ iC
2 1 8.] GUAGE ELECTBOMETEE. 335
absolute distance makes a great change in the force, since the
forces varies inversely as the square of the distance^ so that any
error in the absolute distance introduces a large error in the
result unless the distance is large compared with the limits of
error of the micrometer screw.
The effects of small irregularities of form 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
whatever be the form of a corrugated surface, the eminences of
which just i*each 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 aux-
iliary 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
electrification 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 at-
tracted 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 ad-
justing the force of the electric attraction to a constant value.
In these electrometers 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
Digitized by VjOOQ iC
336 ELECTEOSTATIC IN8TBUMENTS. [219.
readings multiplied by a constant to be determined for each
electrometer gives the potential required.
219.] The electrometers akeady described are not self-acting,
but require for each observation an adjustment of a micrometer
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. (7 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 J3, and that the pro-
portions of these parts are altered as G moves.
For this purpose it is most convenient to make G moveable
about an axis, and make the opposed surfaces of A^ of J3, and
of G portions of surfaces of revolution about the same axis.
In this way the distance between the surface of G and the
opposed surfaces of -4 or of £ remains always the same, and the
motion of (7 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 G from il to 5, but if the potential of G differs from that
of B more than from that of A, then G 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 G within certain
limits, so that if (7 is suspended by a torsion fibre, its deflexions
will be nearly proportional to the difference of potential between
A and B multiplied by the difference of the potential of G from
the mean of those of A and B.
G 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 dif-
ference of whose potentials is to be measured. The higher the
potential of G the more sensitive is the instrument. This elec-
trification of C, being independent of the electrification to be
measured, places this electrometer in the heterostatic class.
Digitized by VjOOQ iC
219.] GAUGE BLBCTEOMETER. 337
We may apply to this electrometer the general theory of
systems of conductors given in Arts. 93, 127.
Let A, By C denote the potentials of the three conductors re-
spectively. Let a, by c be their respective capacities, p the co-
efficient of induction between B and (7, q that between C and Ay
and r that between A and B. All these coefficients will in
general vary with the position of C, and if (7 is so arranged that
the extremities of A and B are not near those of (7 as long as
the motion of C is confined within certain limits, we may
ascertain the form of these coefficients. If $ represents the de-
flexion of C from A towards B, then the part of the surface of A
opposed to C will diminish as 0 increases. Hence if ^ is kept
at potential 1 while B and C are kept at potential 0, the charge
on A will be a = a^—aOy where a^ and a are constants, and a is
the capacity of A.
If A and B are symmetrical, the capacity of £ is 6 = &o + ad.
The capacity of (7 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 = Cq.
The quantity of electricity induced on C when B is raised to
potential unity is p = p^^aO.
The coefficient of induction between A and (7 is g = ^o + ad.
The coefficient of induction between A and B is not altered
by the motion of (7, but remains r^^r^.
Hence the electrical energy of the system is
W= \A^a+\B^b + iC^c + BCp + CAq-\-ABry
and if 0 is the moment of the force tending to increase B,
dW
0 = -TT- yAyByC being supposed constant,
or ® = a{A-B){C-\{A+B)}*.
* {This can also be deduced as follows : If the needle is symmetrically placed
within the quadrants there will be no couple when A^ B. Since dW/d$ vanishes
in this case for all possible values of C, we must have
^ da ^ dh dr
^ n
d?-^-
VOL. I.
Digitized by VjOOQ IC
338
ELEOTEOSTATIO INSTRUMENTS.
[219.
Fig. 20.
In the present form of Thomson's Quadrant Electrometer the
conductors A and B are in the form of a cylindrical box com-
pletely divided into four quadrants,
separately insulated, but joined by
wires so that two opposite quadrants
A and A' are connected together as
are also the two others B and 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 sup-
ported by radii at their extremities.
In the position of equilibrium these
quadrants should be partly within A and partly within J5, 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 il is of the same sign as that of (7,
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
If the quadrants entirely surround the needle the couple wiU not be affected by
increasing all the potentials by the same amount, hence
da db dq
de^de'^ do"
If the quadrants are symmetrical 3^ - — 377 and we get tlie expression in the text.
d9 dO
The student should also consult Dr. G. Hopkinson's Paper on the Quadrant Electro-
meter, Phil Mag. 6th series, xix. p. 291, and Hallwachs Wied. Ann. xxix. p. 11.}
Digitized by VjOOQ iC
220.] MEASUREMENT OP ELECTEIO POTENTIAL. 339
certain limits the deflexions of C will be proportional to the
product (il - J5) {C- i (^ + J?)}.
By increasing the potential of C the sensibility of the instini-
ment may be increased, and for small values of i (-4 + B) the
deflexions will be nearly proportional to (-4 — J5) C.
^JOn the Measuremerd of Electric PotentiaL
220.] In order to determine large differences of potential in
absolute measure we may employ the attracted disk electro-
meter, 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 of the electrometer, and if F, F denote the
potentials of these bodies before making contact, then theii*
common potential after making contact will be
Hence the original potential of the conductor was
If the conductor is not large compared with the electrometer,
K' will be comparable with K, and unless we can ascertain the
values of K and if' the second term of the expression will have
a doubtful value. But if we can make the potential of the
* {Large differences of potential are more conveniently measured by means of
Sir WiUiam Thomson's new Voltmeter.}
Z 2
Digitized by VjOOQ iC
340 ELECTEOSTATIO INSTEUMENTS, [221.
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 approxi-
mately, we may charge the electrode by means of a *replemsher '
or otherwise to this approximate potential, and the next experi-
ment 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
compared 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 F 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 F be the potential of the sphere as measured by an
electrometer when placed in a room whose walls are connected
with the earth, then n — y^^
whence F+ F' = 0,
or the potential of the air at the point where the centre of the
sphere was placed is equal but of opposite sign to the potential
of the sphere after being connected to earth, then insulated, and
brought into a room.
This method has been employed by M. Delmann of Creuznach
in measuring the potential at a certain height above the earth's
surface.
SecoTid Method. We have supposed the sphere placed at the
given point and first coimected 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
electrode of the electrometer to the place where the potential is
to be measured. Let the sphere be first dischaiged 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
Digitized by VjOOQ iC
222.] MEA8UEBMENT OP POTENTUL. 341
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
afiected 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 potential 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 touch-
ing 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 sur-
rounded by the conductor, then the potential at any point of
the air in this hollow wiU 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
electrode. The potential is very soon made equal to that of the
air at the burning end of the match. Even a fine metallic poidt
is sufficient to create a discharge by means of the particles of
the air {or dust?} when the difference of potentials is consider-
able, 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
Digitized by VjOOQ iC
342 ELECTEOSTATIO INSTRUMENTS. [223.
Mngs in an insulated vessel. Each drop as it falls is charged
with a certain amount of electricity, and it is completely dis-
charged into the vessel. The charge of the vessel therefore is
continually accumulating, 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 place connected to the nozzle is positive rela-
tively to that of the other place.
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 applied 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 in-
sulating 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 suface 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. A disk, when employed in this way, is
called a 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 experi-
ment.
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-
Digitized by VjOOQ IC
224.] THE PEOOP PLANE. 343
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
ligidly electrified within the space occupied by the small body
may be neglected, and we may treat the surface of the conductor
near the smaU 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 ex-
tremely small. When the small body is removed to a very short
distance from the conductor, which we shall suppose to be elec-
trified positively, then the electrification at the point nearest to
the small body is Ho 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 surfEice-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 sphei'e, and let us modify the
form of the hemisphere so that its surface is a little more than a
hemisphere, and meets the surface of the sphere at right angles.
Then we have a case of which we have already obtained the
exact solution. See Art. 168.
Digitized by VjOOQ iC
344 ELECTEOSTATIO INSTKUMENTS. [225.
If A and B be the centres of the two spheres cutting each
other at right angles, DIX a diameter of the circle of intersection,
and C the centre of that circle, then if F is 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 il is j V{AD-\^BD + AC^CD^BC),
the exposed surface of the sphere B is
\ V{AD^-BD-\'BC^CD^AC),
urge being the sum of these, or
V{AD^-BD^CDy
3 are the radii of the spheres, then, when a is large
ith )3, the charge on £ is to that on il in the ratio of
<r be the uniform-surface density on A when B is
en the charge on A is
4 Tracer,
e the charge on B is
37ri32(T(l+i^+&c.),
is very small compared with a, the charge on the
B is equal to three times that due to a sui-face-density
over an area equal to that of the circular base of the
s from Art. 175 that if a small sphere is made to
ectrified body, and is then removed to a distance
mean surface-density on the sphere is to the surface-
bhe body at the point of contact as ir^ is to 6, or
[.
most convenient form for the proof plane is that of
isk. We shall therefore shew how the charge on a
: laid on an electrified surface is to be measured,
purpose we shall construct a value of the potential
that one of the equipotential surfaces resembles a
iened protuberance whose general form is somewhat
St disk lying on a plane,
he sui'face-density of a plane, which we shall suppose
Digitized by VjOOQ iC
225-] THE PROOF PLANE. 345
The potential due to this electrification will be
Now let two disks of radius a be rigidly electrified with
surface-densities —a' and +<r'. Let the first of these be placed
on the plane of a^ 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 oxr^c,
where co 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 7r<r0+ (t'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 F= 0. This
surface is indicated by the dotted line.
Putting the distance of any point from the axis of « = r, then,
when r is much less than a, and z is small, we find
0) = 277— 2 71 - + &C.
a
Hence, for values of r considerably less than a, the equation
of the zero equipotential surface is
0 =— 477(7 ;jjj+27ra'c — 2710-' -^ + &C.;
d
2<T + <r'-
a
Hence this equipotential surface near the axis is nearly flat.
Outside the disk, where r is greater than a, a> is zero when
z is zero, so that the plane of ay is part of the equipotential
surface.
To find where these two parts of the surface meet, let us find
dV
at what point of this plane — = 0.
When r is very nearly equal to a, the solid angle a> becomes
approximately a lune of the sphere of unit radius whose angle
is tan-> {2^-i-(r-a)}, that is, o) is 2tan-* {«-!-(r-a)}, so that
, ^dV ^ 2t/c . ,,
wnen 5;= 0-j- = — 47r<r+ , approximately.
Digitized by VjOOQ iC
346 ELBOTBOSTATIO INSTEUMBNTS. [226.
Hence, when
-5— = 0, r^ = a + - — ssraH- — 5 nearly,
d2: ' ® 2iro- IT "^
The equipotential surface F= 0 is therefore composed of a disk-
like figure of radius r^, and nearly uniform thickness Zq, and of
the part of the infinite plane of ocy which lies beyond this figure.
The surface-integral over the whole disk gives the charge of
electricty on it. It may be found, as in the theory of a circular
current in Part IV, Art. 704, to be
Q = 4ira<r'c (log— ^-2} ^--nfrr^.
1q — a
The charge on an equal area of the plane surface is ^crr^^,
hence the charge on the disk exceeds that on an equal area of
the plane very nearly in the ratio of
1 + 8 -nog ^^ to unity,
where Zq is the thickness and r^ the radius of the disk, Zq being
supposed small compared with Tq.
On Electric Accumulators and the Measurement of Capo/dty.
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 con-
sidered 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 difierence 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
Digitized by VjOOQ iC
227.] MEASUREMENT OP CAPACITY. 347
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
quantities of electricity by means of the potential to which they
raise the accumulator.
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 electro-
scopes 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 con-
duction either through air or through glass when the glass is
good, but that it arises chiefiy from superficial conduction along
the various insulating stems and glass surfaces of the instru-
ment.
In fact, the same electrician has communicated a charge to
sulphuric acid in a large bulb with a long neck, and has then
hermetically sealed the neck by fusing it, so that the charge was
completely 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 parafl^ 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
Digitized by VjOOQ iC
348 ELECTEOSTATIO INSTRUMENTS, [227,
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 diflScult one, it is best to begin with
an accumulator constructed of a form for which the solution is
known. 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.
Kohlrausch and Weber, as an absolute standard with which
they compared 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, {it has also been used by Prof. Rowland and Mr.
Rosa in their determinations of the ratio of the electromagnetic
to the electrostatic unit of electricity, Phil, Mag. ser. v. 28,
pp. 304, 315,} 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 con-
siderable.
We are therefore led to prefer for an absolute measure of ca-
pacity 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
Digitized by VjOOQ iC
228.] THE GUAED-EING ACCUMULATOR. 349
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 with a small instrument.
228.] We therefore prefer the following arrangement, due to
Sir W. Thomson, which we may call the Guard-ring arrange-
ment, by means of which the quantity of electricity on an
insulated disk may be exactly determined in terms of its
potential.
The Guard-ring AccumvZator.
Bh 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, kaw>m
a disk Ay and a broad ring
BB surrounding the disk,
separated from it by a very *-
small interval all round, just
sufficient to prevent sparks |J
passing. The upper surface ^
of the disk is accurately in Q
the same plane with that of Kg. 2i.
the guard-ring. The disk is
supported by pillars of insulating material OG. 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 (7 to be at potential zero, and the disk A and vessel
Bh botii at potential V, Then there will be no electrification on
DC
\o^
Digitized by VjOOQ iC
350 ELECTEOSTATIO INSTBUMENTS. [229.
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
I 8^ 8^ il + ar
where R is the radius of the disk, iZ' that of the hole in the
guard-ringj A the distance between A and C, and a a quantity
which cannot exceed (R^^R) -^ •
IT
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
R^jhR^
^ SA '
{This is very nearly the same as the charge on a disk uni-
formly electrified with the surface-density F/4 it A, whose radius
is the arithmetic mean between those of the original disk and
the hole.}
Now let the vessel Bb be put in connexion with the earth.
The charge on the disk A will no longer be uniformly dis-
tributed, 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 F, the original
difierence of potentials and the measurable quantities 12, i2'
and A.
On the Comparison of the Capacity of Accv/mulators.
229.] The form of accumulator which is best fitted to have its
capacity determined in absolute measure from the form and
dimensions 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 sur-
rounded by the other. The guard-ring accumulator, on the
Digitized by VjOOQ iC
229.] COMPAEISON OF CAPACITIES. 351
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 accumu-
lators 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', R^ and (f be the corresponding
parts of tlie other.
If either of these accumulators is of the more simple kind,
having only two conductors, we have only to suppress B or 5^,
and to suppose ^ to be the inner and C the outer conducting
surface, C in this case being understood to surround A.
Let the following connexions be made.
Let B be kept always connected with (7, and ff 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 (7 and with «/, the elec-
trode of a Leyden jar with a positive charge, and let A' be
connected with R and C and with the earth.
(2) Let A^ B, and (7 be insulated from J.
(3) Let A be insulated from B and C, and A' from ff and C,
(4) Let B and (7 be connected with E and C and with the
earth.
(5) Let A be connected with A\
(6) Let A and A' be connected with an electroscope E.
We may express these connexions as follows : —
(1) 0 = (7 = J9'=il' I A^B = C'=J.
(2) 0 = C=B'=il' I A^B:=C'\J.
(3) o = C = J9'| A' \ A\ B = C\
(4) 0 = C = B'|il' I A\B = C' = 0.
(6) 0=^0 = ^1 A' = A\B = C' = 0.
(6) 0 = C=zR\A'^E = A \ B^C'=0.
Here the sign of equality expresses electrical connexion, and
the vertical stroke expresses insulation.
In (1) the two accumulators are charged oppositely, so that A
is positive and A^ negative, the charges on A and A^ being
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352 ELBCTEOSTATIO INSTRUMENTS. [229.
uniformly distributed on the upper surfaoe 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 surfaces.
In (5) J. 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 electri-
fication according as -4. or J.' 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 accumu-
lators, 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
difierent dielectrics in the form of plates or disks. If a disk of
the dielectric is interposed between A and C, the disk being
considerably larger than A, then the capacity of the accumulator
will be altered and made equal to that of the same accumulator
when A and C are nearer together. If the accumulator with the
dielectric plate, and with A and C at distance x, 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, ^
Z= V--
a + x —X
The combination of three cylinders, described in Art. 127,
has been employed by Sir W. Thomson as an accumulator whose
* {Such a key is defloribed in Dr. Hopkinson's paper on the Electrostatic Capacity
of Glass and of Liquids, FhiL Trans,, 1881, Part U, p. 360.}
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229-] . SPECIFIC INDUCTIVE CAPACITY. 353
capacity may be increased or diminished by measurable quan-
tities.
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. Tram., 1871, p. 573. They found the
specific inductive capacity of solid paraffin to be 1.975, that
of air being unity.
VOL. I. A a
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PART 11.
ELECTROKINEMATICS.
CHAPTEB I.
THE ELECTRIC CUBBENT.
230.] We have seen, in Art. 45, that when a conductor is in
electrical equilibrium the potential at every point of the con-
ductor must be the same.
If two conductors A and B are charged with electricity so
that the potential of ^ is higher than that of By 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 trans-
ferred 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 (7, which are called the phenomena of the electric
conflict or cuiTent.
The first of these phenomena is the transference of positive
electrification from ^ 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.
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232.] THE VOLTAIC BATTEEY. 355
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 de-
scribed as a current of positive electricity from A to B^ or b,
current of negative electricity from £ to J., 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
transfeiTed from JS to ^ in unit of time, then, according to
Weber's theory, P^N, and P or iV 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 Ny but attend only to the result of the current,
namely, the transference of P-fiV' units of positive electrification
from A to By and we shall consider P + 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 difierent 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 diflference 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 Daniell's Con-
stant Battery : —
A solution of sulphate of zinc is placed in a cell of porou8
A a 2
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356 THE BLBCTBIO CURRENT. [234.
earthenware, 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 the surfaces of the liquids. This combination
is called a cell or element of Darnell'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 con-
nected with the zinc is put in contact with B, 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 ^ 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 (7, and at the same
time connected with the battery as before, then the cell will
maintain a constant current through (7, 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 elec-
tromotive 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 cx)pper end will be positively
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236.] BLEOTEOLYSIS. 357
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.
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.
Electixlytic 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 Daniell'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 de-
posited, is the same in each of the Daniell's cells throughout the
circuity whatever the size of the plates of the cell, and if any one
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 Danieirs 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
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358 THE ELBCTEIO OUBBENT. [237.
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, multi-
plied 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;
and for an investigation of the apparent exceptions to the rule,
see Miller s Chemical Physics and Wiedemann's Galvanisrnus.
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,
electrolysis, 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.
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238.] ELECTEOLTSIS. 359
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 Voltameter.
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. 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 wiU carry from A ioB s, certain
measurable quantity of electricity at each journey. This quan-
tity may be measured by an electrometer, or it may be calculated
by multiplying the electromotive force of the circuit by the
electrostatic capacity of the ball. Electricity is thus can-ied
from il to £ 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 trouble-
some 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 electrolyte 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 experiment may be considered
as an illustration of the process of electrolysis 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
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360 THE BLBOTBTO CUBBENT. - [239.
electricity 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 representation of what really takes place.
Magnetic Action of the Current.
289.] 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 &om 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 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
discussed 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 intro-
ducing 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 elec-
tric current by its magnetic eflTects 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 considered as a great magnet, the north pole of
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240.] THE ELECTRIC CUREBNT. 361
the earth being similar to that end of the compass needle which
points south.
The galvanometer is the most convenient instrument for
measuring 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 galvano-
meter.
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CHAPTEB II.
CONDUCTION AND RESISTANCE.
241.] If by means of an electrometer we determine the elec-
tric 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
throughout, 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 dif-
ference of the potentials at the extremities of this portion of the
circuit is called the External electromotive force acting on it.
'''* '\q portion of the circuit under consideration is not homo-
50US, but contains transitions from one substance to another,
1 metals to electrolytes, or from hotter to colder parts, there
' be, besides the external electromotive force, Internal elec-
lotive forces which must be taken into account,
he relations between Electromotive Force, Current, and
stance were first investigated by Dr. G. S. Ohm, in a work
lished in 1827, entitled Die Galvanische Kette Matftematisch
rbeitet, translated in Taylor s Scientific Memoirs. The result
lese investigations in the case of homogeneous conductors is
monly called * Ohm's Law.'
Ohm's Law,
he electromotive force acting between the extremities of any
t of a circuit is the product of the strength of the current
- the resistance of that part of the circuit.
[ere a new term is introduced, the Resistance of a conductor,
ch is defined to be the ratio of the electromotive force to
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242.] COMPAKISON WITH PHENOMENA OP HEAT. 363
the strength of the current which it produces. The introduction
of this term would have been of no scientific value unless Ohm
had shewn, as he did experimentally, that it corresponds to a
real physical quantity, that is, that it has a definite value which
is altered only when the nature of the 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 as-
surance of the truth of Ohm's Law is now very high *. In the
sixth chapter we shall trace its applications and consequences.
Oeneration of Heat by the Current
242.] We have seen that when an electromotive force causes
a current to flow through a conductor, electricity 5s 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 con-
ducting 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 Conservation of Energy
♦ I For the verificfttion of Ohm*s Law for metaUic oondactow see Chrystal, B. A.
Beport 1866, p. 36, who shews that the resistance of a wire for infinitely weak currents
does not differ from ite resistance for very strong ones by 10~'' per cent. ; for the veriti-
cation of the hiw for electrolytes see Fitzgerald and Trouton, B. A. Report, 1886. |
Digitized by VjOOQ iC
364 CONDUCTION AND BESISTANCE. [243.
therefore leads us to look for internal work in the conductor.
In an electrolyte this internal work consists partly of the separa-
tion of its components. In other conductors it is entirely con-
verted 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 resistance, and the quantity of electricity is the product
of the current into the time. Hence the quantity of heat multi-
plied 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
resistance of conductors has been determined by Dr. Joule, who
first established that the heat produced in a given time is pro-
portional to the square of the current, and afterwards by careful
absolute measurements of all the quantities concerned, verified
the equation jji ^ (jftjn^
where J is Joule's dynamical equivalent of heat, H the number
of units of heat, C the strength of the current, 22 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 measure-
ment of electromotive forces *.
243.] The analogy between the theory of the conduction of
electricity 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 corre-
sponding 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 cori'esponding area of the second.
Thus, in the illustration we have given, in which flow of elec-
tricity coiTesponds to flow of heat, and electric potential to
temperature, electricity tends to flow from places of high to
» PhU. Mag., Doc. 1851.
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245-] COMPARISON WITH PHENOMENA OF HEAT. 365
places of low potential, exactly as heat tends to flow from places
of high to places of low temperature.
244.] The theory of electric 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 of the vessel and of all within it wiU 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, whei*eas
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 experi-
ment. 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 electiification but shall afterwards become
electrified. It was for some phenomenon of this kind that
Faraday sought in vain under the name of an absolute chai-ge
of electricity.
Heat may be hidden in the interior of a body so as to have no
Digitized by VjOOQ iC
366 CONDUCTION AND BESISTANCE.
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 thei*efore 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.
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CHAPTER III.
ELECTEOMOTIVB FOBOB 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 ascer-
tain this potential by means of an electrometer as described in
Parti, Art. 221.
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 Yolta'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
Zy then the potential of zinc in contact with iron at zero will be
^— /, if the medium surrounding the metals remains the same.
It appears from this result, which is true of any three metals,
that the difference 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 tempera-
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368 CONTACT FORCE. [249.
ture 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.
247.] If, however, the circuit consist of two metals and an
electrolyte, 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
equalization 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 ftlnnel in it becomes chaiged more and more posi-
tively.
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 arrange-
ment, 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
electricity is supplied by the discovery of Peltier, that, when
a current of electricity crosses the junction of two metals, the
* North BritiMh Seview, 1864, p. 853 ; and Proc. B. S,, June 20, 1867.
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249-] pbltibb's phenomenon. 369
junction is heated when the current is in one direction, and
cooled when it is in the other direction. It must be remem-
bered that a current in its pass^e 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^ 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
cuiTent.
The total heat generated in a portion of a compound conductor
.consisting of two metals may be expressed by
where J? is the quantity of heat, J 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,
Jif = CEt = RCH^JUCt,
whence E^RC^JU.
It appears from this equatijon 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 JIT. 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
♦ Proc, R. 8. Sdin., I>eo. 15, 1861 ; and Trans, J?. 8. Edin., 1854.
VOL. I. B b
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370 CONTACT rORCB. [250.
electrodes of a galvanometer or electroscope by wires would be
useless, owing to the contact forces at the junctions of the wires
with the mateiials 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 cuirent and the amount of heat
produced or absorbed, we can determine the electromotive force
requii*ed to urge the current through that portion of the con-
ductor, 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
determined by this method, does not account for Volta's electro-
motive 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 6
be represented by 11^ when the current flows from a to 6, then
for a circuit of two metals at the same temperature we must
have n^+n^, = 0,
and for a circuit of three metals a, 6, c, we must have
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251.] THEBMOBLBCTEIO PHENOMENA. 371
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 standai-d metal,
and if we write ^ = Jfl^e and ^ = Jllt^, then
The quantity ^ 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
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
phenomenon, we find that if the passage of a current through
a conductor of one metal produces any thermal effect which is
reversed when the current is reversed, this can only take place
when the current flows from places of high to places of low tem-
perature, or the reverse, and if the heat generated in a conductor
of one metal in flowing from a place where the temperature is x
to a place where it is y, is H, then
and the electromotive force tending to maintain the current will
he 8,^.
li X, y, 2 he the temperatures at three points of a homo-
geneous circuit, we must have
S,, + S„ + 8,,= 0,
according to the result of Magnus. Hence, if we suppose z iohe
the zero temperature, and if we put
Q,= ASf„ and Q^=/S,.,
* { Le Roaz hat ahewn that this doM not hold when there »re Biioh sudden changes
in the section that the temperature changes by a finite amount in a distance com-
parable with molecular distances. }
B b 2
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372 CONTACT POBCE. [253.
we find 5., = Q.-Q„
where Q^ is a function of the temperature x, 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 6, and
y where it passes from 6 to a, the electromotive force will be
where i^^ signifies the value of P for the metal a at the tempera-
ture X, or
P= i„-Q,..-(i?.,-QJ-(iL-Q^) + iJ,-(25..
Since in unequally heated circuits of different metals there are
in general thennoelectric 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 dis-
covered by Gumming*, who found that the order of certain
metals in the thermoelectric 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 con-
tinues to increase till the hot junction has reached a temperature
r, which, according to Thomson, is about 284*'C. When the
temperature of the hot junction is raised still further the elec-
tromotive 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 observed 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.
258.] From this fact Thomson reasoned as follows : —
Suppose the other junction at a temperature lower than T.
* Cambridge Tran$aci%ons, 1823.
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253-] THEEMOELEOTEIO PHENOMENA. 373
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
temperature 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 disappear 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. {This reasoning assumes that
the thermoelectric junction acts merely as a heat engine, and
that there is no alteration (such as would occur in a battery) in
the energy of the substance forming the junction when electricity
passes across it.} 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, and he found that the current produced opposite
effects in copper and in iron *.
When a stream of a material fluid passes along 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 electricity and by copper for negative electricity,
so that we should have to abandon both h3rpotheses.
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 Thermo-
• < On the EleotrodyiiAmio Qualities of MeUls.* PhiL Trant., Part III, 1856.
Digitized by VjOOQ iC
374 CONTACT PORCB.
dynamics 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 Tait* has recently investigated the electro-
motive force of thermoelectric circuits of different metals, having
their junctions at different temperatures. He finds that the
electromotive force of a circuit may be expressed very ac-
curately by the formula
where t^ is the absolute tempei^ature of the hot junction, t^ that
of the cold junction, and t^ 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 tem-
perature by Professor Tait and his students, and he hopes to
make the thermoelectric circuit available as a thermometric
instrument in his 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
temperature in each pure metal, though its magnitude and even
its sign vary in different metals. From this he has deduced by
thermodynamic principles the following results. Let A;^^, A;^^, h^t
be the specific heats of electricity in three metals a, 6, c, and let
Tfccj ^co> T^ be the temperatures at which pairs of these metals
are neutral to each other, then the equations
E^ = {K^h) (t.^t,) [T„,^\ (t, + y]
express the relation of the neutral temperatures, the value of
the Peltier effect, and the electromotive force of a thermoelectric
circuit
♦ Proe. S. 8, Edin,, Sewion 1870-71. p. 308, also Dec. 18, 1871.
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CHAPTER IV.
ELECTROLYSIS.
Electrolytic CoTiduction.
255.] I HAVE already stated that when an electric current in
any part of its circuit passes through certain compound sub-
stances called Electrolytes, the passage of the current is accom-
panied 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 electi'onegative 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 imper-
fectly formed state of our ideas about electricity, the theories of
electrolysis 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 transferred by the current in the same time.
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376 ELECTEOLYSIS. [255.
The electrochemical equivalent of a substance is that quantity
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
proportional to their ordinary chemical equivalents. The
ordinary chemical equivalents, however, are the mere numerical
ratios in which the substances combine, whereas the electro-
chemical equivalents are quantities of matter of a determinate
magnitude, depending cm 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
electrolyte, 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 electrochemical 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 pheno-
menon 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 parti-
cular, 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,
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257-] THEORY OF CLAUSIUS. 377
that, in diffusion, the different substances are only mixed
together and the mixture is not homogeneous, whereas in
electrolysis they are chemically combined and the electrolyte
is homogeneous. In diffusion the determining 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 do
the compound molecules travel about in this way, but that, in
the collisions which occur between the compound molecules, the
molecules 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 electro-
motive 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 during 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 con-
duction of the current in the electrolyte obeys the law of Ohm,
♦ Ppgg. Ann. d. p. 838 (1867).
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378 BLBCTEOLTSIS. [259,
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 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 accumu-
late, 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 company 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.
268.] One of the best methods of determining whether a body
is or is not an electrolyte is to place it between platinum
electi'odes and to pass a current through it for some time, and
then, disengaging 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 galvanometer. Such a current, being due to ac-
cumulation of different substances on the two electrodes, is a
proof that the substance has been electrolytically 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
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26o.] MOLBCULAB OHAEGE. 379
which conducts in the liquid state is a non-conductor when
solidified, for unless the molecules can pass from one part to
another no electrolytic 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-
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,
communicates 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 anion 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 mole-
cular theory of chemistry, which is corroborated by many
physical considerations, supposes that the number of molecules
in an electrochemical equivalent is the same for all substances.
We may therefore, in molecular speculations, assume that the
number of molecules in an electrochemical equivalent is -^, a
number unknovm 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
* See note to Art. 5.
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380 ELBOTBOLTSIS. [260.
exercising our imagination in tracing the electrification of mole-
cules and the discharge of that electrification.
The liberation of the ions and the passage of positive elec-*
tricity from the anode and into the cathode are simultaneous
facts. The ions, when liberated, are not charged with elec-
tricity, 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
point, the rest of their sui'faces 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 con-
siderable magnitude. Something like this may be supposed to
occur when the two components of an electrolyte are in combi-
nation. 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 electromotive 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 electromotive 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
Digitized by VjOOQ iC
26 1.] SECONDARY PRODUCTS OF ELEOTEOLYSIS. 381
bodies. A molecule of the anion and a molecule of the cation
combined together 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 caUed 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
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 however that when we come to undei*-
stand 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 elec-
trodes are not the actual ions of the electrolysis, but the pro-
ducts 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, both in the
sulphate of soda and in the dilute acid, and equal quantities of
hydrogen at the cathodes.
But if the electrolysis is conducted in suitable vessels, such as
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382 ELECTE0LT8I8. [26 1 .
U-shaped tubes or vessels with a porous diaphi'agm, so that the
substance surrounding each electrode can be examined sepa-
rately, 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 an equivalent 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 elec-
trolysed 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 a solution of sulphate of soda electrolyses one
equivalent of the salt as well as one equivalent 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 NagO but SO4 and Nag, — 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 one of hydrogen.
In the dilute sulphuric acid the gases collected at the elec-
trodes are the constituents of water, namely one volume of
oxygen and two volumes of hydrogen. There is also an in-
crease 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
electrolytic 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 deter-
mined 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 *.
* {See F. Kohlrausch, 'Die Elektriache Leitungif^gkeit des im Vacuum dis-
tillirten Wawers.* Wied. Ann, 24, p. 48. Bleekrode Wied. Ann, 3, p. 161, has
Bhewn tliat pure hydrochlorio acid is a non-conductor.}
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262.] DYNAMICAL THEORY. 383
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 mole-
cules 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 r^ai*ded as a
compound of two equivalents of hydrogen and one of oxygen,
and therefore probably not capable of direct electrolysis.
While electrolysis fully establishes the close relationship be-
tween 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 electrification by contact, are not, even when in a
fluid state, decomposed by the current *. Most of the combina-
tions of the substances which act as anions are not conductors,
and therefore are not electrolytes. Besides these we have many
compounds, containing the same components as electrolytes, but
not in equivalent proportions, and these are also non-conductors,
and therefore not electrolytes.
On the Conservation of Enei*gy 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 elec-
trolysed.
* {See Roberts-AoBien, B. A. Report, 1887.}
Digitized by VjOOQ iC
384 • BLECTEOLTSIS. [263.
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 measui*e 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 quantities 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 circuiti,
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 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 established 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 is 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
Digitized by VjOOQ iC
263.] CALCULATION OP ELBCTEOMOTIVE FOBCE. 386
through the wire. Hence, the heat developed by the pa43sage of
one unit of electricity is in this case measured by the electro-
motive 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 Thom-
son {PhU. 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 sub-
stance *.*
The thermal equivalents of many chemical actions liave been
determined by Andrews, Hess, Favre and Silbermann, Thomsen,
&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 arrange-
ments, 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 force^ and can therefore be measured according to
the very same principles used in the measurement of electro-
motive 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 avd generisy
and irreducible to numerical measurement.
* {This theorem only applies when there are no revernble thermal effects in
the cell, when these exist tlie relation between the electromotive force p and the
mechanical eqaivalent of the chemical action, w, ia expressed by the relation
where $ is the absolute temperature of the cell v. Helmholtz, < Die Thermodynamik
chemischer Yorgange.' Wissenscha/tlieAe Ahhandlungen, ii. p. 958.}
VOL. I. C C
Digitized by VjOOQ iC
386 ELECTROLYSIS.
When the yoluine of the products of electrolysis is greater than
that of the electrolyte, work is done during the electrolysis in
overcoming the pressure. If the volome 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 vp, 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 decom-
position 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.
* {Thii result is inconsistent with the Second Law of Thermodynamics, according
to this Law an increase in the pres-^mre increases the ElectromotiTe force required for
Electrolysis. See J. J. Thomson's 'Applications of Dynamics to Phyiics and Chemis^/
p. 86. V. Helmholtz, ' Weitere Untersucbnnffen die Electrolyse des Wassers betreflFend.*
JFied. Ann. 84, p. 737.}
Digitized by VjOOQ IC
CHAPTER V.
ELECTROLTTIO POLABIZATION.
264.] When an electric current is passed through an electro-
lyte bounded by metal electrodes, the accumulation of the ions
at the electrodes produces the phenomenon called Polarization,
which consists 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*, Paalzow f, and recently by
those of MM. F. Kohlrausch and W. A. Nippoldt^, Fitzgerald
and Trouton §, that the conduction 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
♦ Elektricitat, I. 668, bd. 5. f Berln, MonaUhericht, July, 1868.
X Pogg. Ann. bd. cxxxviii. b. 286 (October, 1869). § B.A. Report, 1887.
C C 2
Digitized by VjOOQ iC
388 ELBCTEOLTTIO POLAEIZATIOX. [267.
perceived as a force opposed to the current, but we can distin-
guish between the two phenomena by instantaneously removing
or reversing the electromotive force.
The resisting force is always opposite in direction to the
current, and the external electromotive force required to over-
come it is proportional to the strength of the current, and
changes its direction 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 ciUTent 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 bore up into a cistei^n. 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 electro-
motive 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.
Digitized by VjOOQ iC
268.] DISTINGUISHED FEOM RESISTANCE. 389
This deposit of the ion is constantly tending to become free,
and either to diffiise 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
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 electro*
motive 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 electro-
motive 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 electi*olysis. This will give the maximum
polarization.
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 electro-
lyte. In fact, the ion in contact mth 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
increases in density, the succeeding portions are no longer so
intimately 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
<r. This quantity o- represents the number of electrochemical
equivalents of the ion deposited on unit of area. Since each
electrochemical equivalent deposited corresponds to one unit of
Digitized by VjOOQ iC
390 BLBCTBOLYTIC POLARIZATION. [269.
electiricity transmitted by the current, we may consider a as
representing 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 diflFerence between the electric
potentials of the two electrodes when the cuiTent through the
electrolyte is so feeble that the proper resistance of the electro-
lyte 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 elec-
trodes, and the state in which they are deposited depends on
the actual state of the surfaces of the electrodes, which may be
modified by previous deposits.
Hence the electromotive force at any instant depends on the
previous history of the electrodes. It is, speaking very roughly,
a function of cr, the density of the deposit, such that jo = 0 when
o- = 0, but p approaches a limiting value much sooner than <t
does. The statement, however, that ^ is a function of <r cannot
be considered accuiate. 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 r, 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
Digitized by VjOOQ iC
271.] COMPARISON WITH LEYDEN JAB. 391
gas, instead of contributing 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
polarization 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 electromotive force, such as that of one Daniell'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 electromotive force arising from the polarization of the
cells wi]l 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 separ-
ation 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 ^ is the area of one of the electrodes,
and 0* the density of the deposit, supposed uniform,
Q= Aa.
If we now disconnect the electrodes of the electrolytic ap-
paratus from the Daniell's 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 polarization disappears.
271.] Hence we may compare the action of this apparatus,
which is a form of Bitter'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
afterwards discharged. During the discharge a quantity of
electricity nearly equal to the charge passes in the opposite
direction. The difference between the chaise 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
Digitized by VjOOQ iC
392 BLBCTROLTTIO POLABIZATION. [27!.
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 dis-
charge. This is called the residual discharge, and is a pheno-
menon 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 pro-
portional to the 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 elec-
trodes in each element of a secondaiy pile is very small com-
pared with the maximum difference of the potentials of those of
a charged Leyden jar, so that in order to obtain much electro-
motive 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 that 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
preferable 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
* Specification of C. F. Varley, < Electric Telegraphs, Ac.,* Jan. 1860.
Digitized by VjOOQ IC
271.] COMPARISON WITH LEYDEN JAB. 393
Burfaces, 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 ad-
herence, or simple juxtaposition.
The seat of this energy is close to the surfaces of the elec-
trodes, 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 following 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
lOO^'C the glass becomes a conductor. If a test-tube containing
mei-cury is placed in a vessel of mercury, and if a pair of elec-
trodes 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 connected 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.
* Proe, B. 8. Jan. 12, 1871. For an aooount of other inveftigationi on thia
sabject, see Wiedeinanns EUktricUdt, bd. ii. pp. 744-771.
t Annalen der Chemie und Pharmaeie, bd. xc 257 (1854).
Digitized by VjOOQ IC
394 ELBOTBOLTTIO POLARIZATION. [^7^-
This current is manifestly electrolytic, for if the electrodes are
disconnected from the battery, and connected with a galvano-
meter, 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 polari-
zation of the surface remains. The mercury may be removed,
the surfaces may be washed with nitric acid and with water, and
fresh mercury introduced. If the apparatus is then heated, the
current of polarization 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
existence of polarization may be regarded as conclusive evidence
of electrolysis, and if the cx>nductivity of a substance increases as
the temperature rises, we have good grounds for suspecting that
the conduction 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 mea-
surements, 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 sulphuiic acid. The principal seat of polari-
zation is at the surface of the negative metal. When the fluid
in which the negative metal is immersed is dilute sulphuric acid.
Digitized by VjOOQ iC
272.] CONSTANT VOLTAIC ELEIIBNTS. 395
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 electromotive 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
means. These are of two kinds. In the batteries of Qrove 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 Danieirs 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 jio hydrogen appears
on the copper plate, but copper is deposited on it. When the
Digitized by VjOOQ iC
396 ELECTROLYTIC POLARIZATION. [272.
solution is saturated, and the current is not too strong, the
copper appears to act as a true cation, the anion SO4 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 S O4 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 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 nothing
to the useful effect of the battery.
To prevent this, the zinc is immersed either in dilute sulphuric
acid or in a solution of sulphate of 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 earthenware, 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 electrolytic conduction is a process the mathematical
Digitized by VjOOQ iC
272.] Thomson's form op danibll's cell. 397
laws of which have the same form as those of diffusion, and
whatever interferes with one must interfere equally with the
other. The only difference 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.
Fig. 22.
In each cell the copper plate is placed horizontally at the
bottom and a saturated solution of sulphate of zinc is poured
over it. The zinc is in the form of a grating and is placed hori-
zontally 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,
80 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
♦ Proe. M. 8., Jan. 19, 1871.
Digitized by VjOOQ iC
398 ELECTBOLTTIC POLARIZATION.
from sulphate of copper, and having a very slow downward
motion in the cell, which still further retards the upwcurd 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 zina 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 sufficiently dilute to be lighter than any other
stratum of the liquid in the cell.
Danieirs battery is by no means the most powerful in common
use. The electromotive force of Grove's cell is 192,000,000, of
Darnell's 107,900,000 and that of Bunsen's 188,000,000.
The resistance of Darnell's cell is in general greater than that
of Grove's or Bunsen's of the same size.
These defects, however, ai-e more than counterbalanced in all
cases where exact measurements are required, by the fact that
Darnell's cell exceeds every other known arrangement in con-
stancy of electromotive force *. It has also the advantage of
continuing in working order for a long time, and of emitting
no gas.
* { When a Btandard Electromotivo force Ib required a Clark's ceU it now mo«i
frequently ased. For the precautions which must be taken in the conBtmction and
uBe of such cells, see Lord Rayleigh*s paper on * The Clark C«ll as a Standard of
Electromotive Force.' Phil. Trans, part ii. 1885.}
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CHAPTER VL
LINEAR ELEOTEIO CUBRENTS.
On Systems of Linear Condv/itors.
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 ex-
pressed 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 Arts. 282 a and 2826.
Ohm's Law.
274.] Let E be the electromotive force in a linear conductor
from the electrode Ai to the electrode A^* (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^ A^ 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 ill, ilg be the electrodes of the first conductor and
let the second conductor be placed with one of its electrodes in
Digitized by VjOOQ IC
400 UNEAE ELECTEIO OUEBBNTS, [276.
contact with A^^ySO that the second conductor has for its elec-
trodes ilg, -4 3. The electrodes of the third conductor may be
denoted hy A^ and A^.
Let the electromotive forces along these conductors be denoted
by j&,2, -Sga* -^34 ♦ ^^^ ^^ ^^ ^^^ *'^® other conductors.
Let the resistances of the conductors be
^12 > ^23 > -^34 > ^^'
Then, since the conductors are arranged in series so that the
same current C flows through each, we have by Ohm's Law,
J?j2 = CRi2j E^ = CiZga, -^34 = C^^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 = CR. (3)
Now ^= ^12 + ^23 + ^:34+ &c., (4)
the sum of the separate electromotive forces,
= C (iJi2 + i?23 + i234 + &c.) by equations (2).
Comparing this result with (3), we find
iZ =iJlj + i223 + iZ34 + &C. (5)
Or, the resistance of a series of conductors is the sum of the
resistances of the conductors taken separately.
Potential at any Point of the Series.
Let A and C be the electrodes of the series, B a point between
them, a, c, and 6 the potentials of these points respectively. Let
R^ be the resistance of the part from A to B, R2 that of ike pai*t
from B to C, and -R that of the whole from il to C, then, since
a — 6 = -RiC, b^c^ R^C^ and a^c^RC,
the potential at B is
6=^?^^^ (6)
which determines the potential at B when the potentials at A
and C are given.
Resistarvce of a Multiple Coiiductor.
276.] Let a number of conductora 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.
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277-] SPECIFIC RESISTANCE AND CONDUCTIVITY. 401
Let the resistances of these conductors he Rj^, R.^, R^ respect-
ively, and the currents €^^,0^,0^, and let the resistance of the
multiple conductor be iZ, 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=zC,R, = C^R^ = C^R^ = CR,
C = Cj + Cg + Cg,
whence ^ = i. + i. + ^^. (r)
Or, the reciprocal of the resistance of a multiple condvxstor ia 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 sv/m of the conductivities
of the component conductors.
Current in any Bran/ch of a Multiple Conductoi\
From the equations of the preceding article, it appears that if
C^ is the current in any branch of the multiple conductor, and
i2j the resistance of that branch,
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 Z, and whose section is unity. This is equi-
valent to I cubes arranged in series. The resistance of the
conductor is therefore Ip.
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
VOL. I. D d
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402 LINBAB ELECTRIC CUEBENTS. [^/S.
When we know the resistance of a uniform wire we can deter-
mine 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 deter-
mined 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 resist-
ance 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 jj _ ^
m
On the DimeTvaions of the Quaidities 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 con-
ductivity of the conductor is the reciprocal of this quantity, or
in other T^ords, 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 V, and then connected with the earth by the given con-
ductor. 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 — -^ (rT), but, since V is constant,
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28o.] SYSTEM OP LINEAR CONDUCTORS. 403
the current is — -^ F, and the electromotive force through the
conductor is F.
The conductivity of the conductor is the ratio of the current
to the electromotive force, or — -^> that is, the velocity with
which the radius of the sphere must diminish in order to main-
tain the potential constant when the charge is allowed to pass
to earth through the conductor.
In the electrostatic system, therefore, the conductivity of a
conductor is a velocity, and so of the dimensions [XT"^].
The resistance of the conductor is therefore of the dimensions
The specific resistance per unit of volume is of the dimension
of [3^, and the specific conductivity per unit of volume is of the
dimension of [7"*].
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
[L'^MT].
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 [ZT"*].
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 [i^T"^], and the specific resistance per unit
of weight is of the dimensions [Lr'^T~'^M\.
On Linear Systems of Conductors in general.
280.] The most general case of a linear system is that of
n points, ilj, ilg,...^!,,, connected together in pairs by \n(n'-l)
linear conductors. Let the conductivity (or reciprocfd of the re-
sistance) of that conductor which connects any pair of points,
say A^ and -4,, be called iT^, and let the current from A^ to A^
be C^ Let Pp and i^ be the electric potentials at the points A^
and A^ respectively, and let the internal electromotive force,
if there be any, along the conductor from -4^ to ^, be E^^.
T> A 2
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404 LINEAR BLEOTEIO OUEEENTS. [280.
The current from Ap to A^ is, by Ohm's Law,
C„ = K^{P,-P,+S„). (1)
Among these quantities we have the following sets of re-
lations :
The conductivity of a conductor is the same in either direc-
tion, or K^ = K^p. (2)
The electromotive force and the current are directed quantities,
80 that E^^-E^p, and C^^-C^p. (3)
Le^ ^i> -'^,... Ji he the potentials at A^, il2,...-4„ respectively,
and let Qi, Qgj-'-Ow ^ *^® quantities of electricity which enter
the system in unit of time at each of these points respectively.
These are necessarily subject to the condition of * continuity*
ei+Q2...+Q« = o, (4)
since electricity can neither be indefinitely accumulated nor pro-
duced within the system.
The condition of * continuity * at any point A^ is
Qp = C^i + C^+&c. + C^. (5)
Substituting the values of the currents in terms of equation
(1), this becomes
e, = (ir^,+ir^+&c.+^^)^-(ir,,i>+ir^p,+&c.+^^^)
+(^^i^^i+&c+jr^j?^). (6)
The symbol K^^ does not occur in this equation. Let us
therefore give it the value
K„^^{K,, + K^ + ko. + K^); (7)
that is, let K^^ be a quantity equal and opposite to the sum of
all the conductivities of the conductors which meet in A p. We
may then write the condition of continuity for the point A^,
Kp,F,-hKp2P2'^SLc. + K„Pp + &c.^Kp,P^
= Kp,Ep,-^kc.-^Kp,Ep,--Qp. (8)
By substituting 1,2, &c. 7i for p in this equation we shaJl
obtain n equations of the same kind from which to determine
the n potentials ij, P^, &c, P^.
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.— 1 independent equations. These will be suflScient to deter-
mine the difierences of the potentials of the points, but not
to determine the absolute potential of any. This, however,
is not required to calculate the currents in the system.
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28 1.] SYSTEM OF LINBAB CONDUCTORS. 405
If we denote by D the determinant
-^11 > ^Ui -^l(n-l)>
-^21 > •^22» -^2(ii-l)>
(9)
and by D^^ the minor of K^^^ we find for the value of ^— ii,
+(Z,i^,i+&c. + Z^^^-Q,)2)^ + &c. (10)
In the same way the excess of the potential of any other point,
say Aqy over that of A^ may be determined. We may then de-
termine the current between A^ imd A^ from equation (1), and
so solve the problem completely.
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. 86.
The coeflScient of Q^ in the expression f or -^ is — ^ • That
D
of Qp in the expression for i; is — ^ •
Now D^ differs from D^^ only by the substitution of the
symbols such as K^p for K^^. But by equation (2), these two
symbols are equal, since the conductivity of a conductor is the
same both ways. Hence J) ^ J) . (11)
It follows from this that the part of the potential at Ap arising
from the introduction of a unit current at A^ is equal to the
part of the potential at A^ arising from the introduction of a
unit current at Ap.
We may deduce from this a proposition of a more practical
form.
Let A, By Cy D he 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 By 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 2), the potential at A will exceed that at B by
the same quantity P.
If an electromotive force E be introduced, acting in the con-
ductor from il to -B, and if this causes a current C from X to Y,
then the same electromotive force E introduced into the con-
ductor from X to Y will cause an equal current C from A to B.
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406 LINEAR BIiBOTBIO CUBEBNTS. [282 6.
The electromotive force E may be that of a voltaic battery
introduced between the points named, care being taken that the
resistance of the conductor is the same before and after the
introduction of the battery.
282 a.] If an electromotive force E^ act along the conductor
ApA^y the current produced along another conductor of the
system Ar A, is easily found to be
There will be no current if
2),,+2).,-A,-A„=0. (12)
But, by (11), the same equation holds if, when the electromotive
force acts along A^ A, , there is no current m ApA^. 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
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 multi-
plied 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 6.] 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 tiiis case the following proposition : — Let x be the current, E
the electromotive force, and R the total resistance in any mesh ;
let also y, 0,... be currents circulating in neighbouring meshes
* [Bxtncted from notee of Professor M»zweU*i leotimt by Mr. J. A. Fleming, BJL.,
St. John*8 College. See also a paper by Mr. Fleming in the Phil. Ma^.^ xx. p. 221,
1885.]
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283.] GENERATION OP HEAT. 407
which have threads in common with that in which x circulates,
the resistances of those parts being 8, ^, ... ; then
ite— «y— te— &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 OBC^ OCA, OAB in which the currents a?, y, z respect-
ively circulate, viz.
{a + i8 + y)aj -yy -Pz = E,
— yaj + (6 + y + a)2/ —0;^= 0,
"Px — ay + (c + o + i8)0= 0.
From these equations we may now determine the value of
z—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 in a conductor whose resistance is ii by a current C
in ui^t of time is, by Art 242,
JH^BC^. (13)
We have therefore to determine the sum of such quantities as
RC^ for all the conductors of the system.
For the conductor from A^ to A^ the conductivity is Kp^,
and the resistance Rp^^ where
K„.B^=l. (14)
The current in this conductor is, according to Ohm's Law,
We shall suppose, however, that the value of the current is
not that given by Ohm's Law, but X^^^ where
^«=C'«+^- (»6)
To determine the heat generated in the system we have to
find the sum of all the quantities of the form
or JB^==2{i2^C»j]|'+2'jB^C^]^ + iZ^y«„}. (17)
Giving Cp^ its value, and remembering the relation between
Kp^ and iZp,, this becomes
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408 LINEAE ELBOTEIO CUBEBNT8,
Now since both C and X must satisfy the condition of
continuity at iip, we have
Q, = Z,i + Z„ + &c. + Z^, (20)
therefore 0 = l^li + IJj +&c. + Y,^. (21)
Adding together therefore all the terms of (1 8), we find
S (B«Z« J = ^P,Q,+-S.R^ Y\. (22)
Now since R is always positive and Y^ is essentially positive,
the last term of this equation must be essentially positive.
Hence the first term is a minimum when Y b zero in eveiy
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 conditions of supply and outflow of the current
The heat actually generated when Ohm's Law is fulfilled is
mechanically equivalent to ^PpQp^ 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.
* {We can prove in a nmilar way that when there are electromotive forces in the
different branches the onrrents adjast themselves so that 'XRC^-~%XEC is a minimum,
where E is the electromotive force in the branch when the current is C. If we express
this quantity, which we shall call P^ in terms of the independent currents flowing round
the circuits, the distribution of cuivent d7, y, s, ... among the conductors may be found
from the equations
Thus in the case of Wheatstone^s Bridge considered in Art. 882,
and the equations in that Art. are identical with
dF ^ dP ^ dF ^
dx ' dy dz
This is often the most convenient way of finding the distribution of current among
the conductors. The reciprocal properties of Art. 281 can be deduced by it with
great ease. [
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APPENDIX TO CHAPTEE VI, 409
APPENDIX TO CHAPTER VI.
The laws of the distribution of currents which are investigated in
Art. 280 may be expressed by the following rules, which are easily
remembered.
Let us take the potential of one of the points, say A^, as the zero
potential, then if a quantity of electricity Q, flows into A^ the potential
of a point Ap is shewn in the text to be
D ^*'
The quantities D and Dp, may be got by the following rules. — D is the
sum of the products of the conductivities taken (n— 1) at a time, omitting
all those terms which contain the products of the conductivities of
branches which form closed circuits. Dp, is the sum of the products of
the conductivities taken (n— 2) at a time, omitting all those terms which
contain the conductivities of the branches ApA^ or i4« ii«, or which
contain products of conductivities of branches which form closed
circuits either by themselves or with the Axdof ApAn or A, A^.
We see from equation (10) that the effect of an electromotive force
H^r acting in the branch A^ A^ iB the same as the effect due to a sink
of strength K^^ E^ at Q and a source of the same strength at R^ so that
the preceding rule will include this case. The result of the application
of this rule can however be stated more simply as follows. If an electro-
motive force Ep^ act along the conductor ApA^, the current produced
along another conductor Ar A^is
where D is got by the rule given above, and A = Ai — A^. Where Aj is got
by selecting from the sum of the products of the conductivities taken
(n— 2) at a time those products which contain the conductivities of both
Ap Ar (or the product of the conductivities of branches making
a closed circuit with Ap A^) and A^ A, (or the product of the con*
ductivities of branches making a closed circuit with A, A^), omitting
from the terms thus selected all those which contain the conductivities
oi ArAff or ApA^, or the product of the conductivities of branches
making closed circuits by themselves or with the help of A^A, or
Ap A^; Af corresponds to A|, the branches Ap J«, A^ A^ being taken
instead of A pA^ and A, A^ respectively.
If a current enters at P and leaves at Q, the ratio of the current to
the difference of potential between Ap and A^ib—,*
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410 LINEAR ELECTRIC CURRENTS.
Where A' is the sum of the products of the conductivities taken n— 2
at a time, omitting all those terms which contain the conductivity of
A^ A^ or the products of the conductivities of branches forming a closed
circuit with it.
In these expressions all the terms which contain the product of the
conductivities of branches forming a closed circuit are omitted.
We may illustrate these rules by applying them to a very important
case, that of 4 points connected by 6 conductors. Let us call the points
1, 2, 3, 4.
Then i>=the sum of the product of the conductiyities taken 3 at a
time, leaving out, however, the 4 products K^ K„ K^, K^^ K^ K^,
Kj^ K^ JT^j , K^ K^ K^2 > *8 these correspond to the four closed circuits
(123), (124), (134), (234).
Thus
Let us suppose that an electromotive force ^ acts along (23), the current
through the branch (14)
A, = if,, K^ (by definition),
Hence if no current passes through (14), -^18-^84—^12^43 = ^> this is the
condition that (23) and (14) may be conjugate.
The current through (13)
_ ^n(^U + -g^g4■^^34)^^-^14^94 1? JT IT
= -^ . EK^^ Agj.
The conductivity of the net work when a current enters at (2) and
leaves at (3)
^D
Kwe have 5 points, the condition that (23) and (14) are conjugate is
-^Jf -^84 (-^16 + -^» + -^85 + -^45) + -^12 ^W ^46 + ^M -^51 ^88
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Google
CHAPTER VIL
CONDUCTION IN THREE DIMENSIONS.
Notation qf Electric Currents.
285.] At any point let an element of area dS be taken normal
to the axis of Xy 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 ti;, 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 Z, 7>i, n be the
direction-cosines of OR; then if we cut off irom the axes of
x, yy z portions equal to
r r J r
7> — > and -
L ^n n
respectively at A, B and C, the triangle ABC will be normal
to OR.
The area of this triangle ABC will be
Lnin
and by diminishing r this area may be di-
minished without limit. Fig.
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.
r^
The area of the triangle OBC is \ — » and the component of
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412 CONDUCTION IN THEEE DIMENSIONS. [287.
the current normal to its plane is u, so that the quantity which
enters through this triangle in unit time is i r*
The quantities which enter through the triangles OCA and
OAB respectively in unit time are
Jr*— => and \r^^ — •
If y is the component of the current in the direction OjB, then
the quantity which leaves the tetrahedron in unit time through
Imn
Since this is equal to the quantity which enters through the
three other triangles,
Imn "" ( vm rd Imj
,.. , . , 2lmn .
multiplying by — ^ — , we get
y = lu-hmv+nw* (1)
If we put u^ + v* +^ = r*,
and make l\ m\ n' such that
u ss i'r, V = mT, and it; = nT ;
then y =r r (ii' + mt\^ + tin'). (2)
Hence, if we define the resultant current as a vector whose
magnitude is F, and whose direction-cosines are l\ *n\\ n\ and if
y denotes the current resolved in a direction making an angle B
with that of the resultant current, then
y = rcos^; (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
be the equation of a family of surfaces any one of which is given
by making A. constant ; then, if we make
-2
dX\ dK^ dk
dx\ dy\ dz
2
the direction-cosines of the normal, reckoned in the direction in
which k increases, are
i = iV^^, m^N^, n^N^^ (6)
dx dy dz ^ '
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290.]
TUBES OP FLOW.
413
Hence, if y is the component of the current normal to the
surface. , ^X^ dk^ dk} ,.
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
flow are in the surface.
288.] The equation of a surface of flow is therefore
d\ d\ dk
dx dy dz
(8)
If this equation is true for all values of X, all the surfaces of the
family will be surfaces of flow,
289.] Let there be another family of surfaces, whose parameter
is k\ then, if these are also surfaces of flow, we shall have
d\' dk' dx" ^
u-j — hv-T- -^w-j- = 0.
dx dy dz
(9)
If there is a third family of surfaces of flow, whose parameter
is X'', then d\" d\'' d\" ^
11
dx
dz
If we eliminate tc, t;, and w between these three equations,
we find
dk
dk
dk
dx '
dy
dz
dk'
dk'
dk'
dx '
dy
dz
dk"
dk"
dk"
dx •
dy
dz
= 0;
(H)
(12)
or X''=0(A, X');
that is, X'^ is some function of X and X'.
290.] Now consider the four surfaces whose parameter are X,
X + 8X, X', and X' + 5X', These four surfaces enclose a quadri-
lateral tube, which we may call the tube hk . h\\ 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 Lh\.h\\ where Z is a function of X and X', the
parameters which determine the particular tube,
Digitized by VjOOQ iC
414 CONDUCTION IN THEEB DIMENSIONS, [293.
291.] If 18 denotes the section of a tube of flow by a plane
normal to re, we have by the theory of the change of the iude-
pendent variables,
8X.«X=«^(^-^-^_). (13)
and by the definition of the components of the current
udS = Lbk.bK\ (14)
„ ^ Akdk' d\d\\
Hence u^ Li~j--j -5 — 3-).
^dy dz dz dy^
o,. ., , r /dkdk dKdK'\
SimUarly „ = i(_____),
^^ J /dkdk dKdk\
"" ^dx dy dy dx^' f
292.] It is always possible when one of the functions k or X'
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 paraUel to y, to represent the
sections of the family A' by this plane. In other words, let the
function X' be determined by the condition that when x = 0
X' = z. If we then make i = 1 , and therefore (when x = 0)
(15)
^judyy
then in the plane (x = 0) the amount of electricity which passes
through any portion will be
ffudydz = ffdkdk\ (16)
The nature of the sections of the surfaces of flow by the plane
of yz being determined, 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 Z.
On Lines of Flow,
298.] 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
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295-] EQUATION OP CONTINUITY. 415
amount of minuteness. Then if any surface be di'awn cutting
the system of tubes, the quantity of the current which passes
through this surface will be expressed by the Tvwmher of tubes
which cut it, since each tube carries a unit 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 also its strength^ since each line of flow through a
given section corresponds to a unit current.
On Current'Sheete and Current-Functions.
294.] A stratum of a conductor contained between two con-
secutive surfaces of flow of one system, say that of X', is called
a Current-Sheet. The tubes of flow within this sheet are deter-
mined by the function A. If A^ and Ap 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 il to P is Ap— A^*.
If AP be an element, da, of a curve drawn on the sheet, the
current which crosses this element from right to left is
d\ .
-^r-da.
as
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 X, y, z respectively, remembering that i is a function of
A and A', we find du dv dw ^
di^dy'-Tz^''' (^^)
* (By the 'current aeroM AP* is me«nt the cmrent through the tube of flow
bounded by the lurfiMet Xa> ^p» ^' Mid X' -f 1.}
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416 CONDUCTION IN THREE DIMENSIONS. [295.
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 one place and appear in the other, but it must
travel along a continuous path, so that if a closed surface be
drawn, including the one place and excluding 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
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 x, y, and z respectively. Thus understood,
the equation is applicable to any material substance, solid or
fluid, whether the motion be continuous or discontinuous, pro-
vided the existence of the parts of that substance 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 Hydrody-
namics. In one of these we trace the course and the deforma-
tion 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
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I
296.] ELECTKICITY THROUGH A GIVEN SURFACE. 417
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 particular portion of
electricity in its course through the body. The other method of
investigation, in which we consider what passes 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 r be the resultant current at any point of the
surface. Let dS be an element of the surface, and let € be the
angle between T and the normal to the surface drawn outwards,
then the total current through the surface will be
//■
r cos €dS,
the integration being extended over the surface.
As in Art. 21, we may transform this integral into the form
in the case of any closed surface, the limits of the triple integra-
tion 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).
VOL. I. E e
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CHAPTER VIII.
RESISTANCE AND CONDUCTIVITY IN THREE DIMENSIONS.
On the most General Relations between Current and
Electromotive Force,
297.] Let the components of the current at any point be u,
Vy w.
Let the components of the electromotive intensity be X, F, Z,
The electromotive intensity 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 F is the potential,
-=-£■ --f' --f' 0)
or (2) from electromagnetic induction, the laws of which we
shall afterwards examine ; or (3) from thermoelectric or electro-
chemical 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 intensity at the point, what-
ever 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 electro-
motive intensity. Hence X, F, Z must be linear functions of u
V, V), We may therefore assume as the equations of Resistance,
X ^M^u-hQ^v-^-Jl 'm;, )
Y = P^u-hR.v-hQ,wA (2)
Z=Q2U-hPjV + R.,iv.)
We may call the coefficients R the coefficients of longitudinal
resistance in the directions of the axes of coordinates.
The coefficients P and Q may be called the coefficients of
transverse resistance. They indicate the electromotive intensity
in one dii'ection required to produce a current in a different
direction.
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GENERATION OF HEAT. 419
If we were at liberty to assume that a solid body may be
treated as a system of linear conductors, then, from the recipro-
cal 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 F^z=,Q^^ and similarly we
should find ij = Qg' ^^^ ^ = Qa- When these conditions are
satisfied the system of coefficients 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 u, v, w may be expressed as linear
fimctions of X, Y, Z hy a, system of equations, which we may
call Equations of Conductivity,
v=:q^X + r^Y+PiZ, j (3)
we may call the coefficients r the coefficients of Longitudinal
conductivity, and p and q those of Transverse conductivity.
The coefficients of resistance are inverse to those of conduc-
tivity. This relation may be defined as follows :
Let [PQR] be the determinant of the coefficients of resistance,
and [pqr] that of the coefficients of conductivity, then
[PQR\ = P,P,P, + Q,Q,Q,-¥R,R,R,^rrQ,R,''m,R,--P,Q,R,, (4)
[PQR] [pqr] = 1, (6)
[PQR]p^ = (PPa-Q A). [pqr] ij = {p.P.-q.r^). (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 Oeneration of Heat.
299.] To find the work done by the cun*ent in unit of time
in overcoming resistance, and so generating heat, we multiply
the components of the current by the corresponding components
♦ {Seenoteto Art. 303.}
£ e 2
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=f:f;} (■■)
EESTSTANCE AND CONDUCTIVITY. [300.
le electromotive intensity. We thus obtain the following ex-
dons for W, the quantity of work expended in unit of time :
i+Yv^Zw; (8)
^ji^-hBy+R^w^-\^{P, + Q,)m) + (P^ + Q^)iini + {Ps + Qs)uv; (9)
^' + r,Y' + r,Z' + {p, + q^)YZ + (p, + q,)ZX-^{p, + q,)XY.{lO)
Y a proper choice of axes, (9) may be deprived of the terms
Iving the products of u, v, w or else (10) of those involving
products of X, F, Z. The system of axes, however, which
ces W to the form
R^u^ + Ry + R^m^
)t in general the same as that which reduces it to the form
r^X^ + r^Y^ + r^Z^.
is only when the coeflScients ij, i^, ij are equal respectively
i» Q2» Qd ^^^^ ^^^ ^wo systems of axes coincide,
with Thomson * we write
P=i 8+ty
we have
R] = R,R^R^'^2S,S.,Ss'-Si^Ri^S^^R.;,-S^^R^ ^
2{S,T,T,-^S,T,T, + 8,T,T,)-^R,Ti^+R,T,^ + R,T,^; i (^^^
[PQR]r, = R,R,^S,'-^T,^ .
[PQR]8, = T,T, + SA~^R,S,, I (13)
[PQR]t,=R,T,-^SJ^ + S,T,. ^
therefore we cause Sj^, S.^, 8^ to disappear, the coeflScients s
not also disappear unless the coefficients T are zero.
Condition of Stability.
X).] Since the equilibrium of electricity is stable, the work
it in maintaining the current must always be positive. The
Litions that W must be positive are that the three coefficients
R,^j Rq, and the three expressions
4R,R,-{P, + Q,yA (14)
t all be positive.
here are similar conditions for the coefficients of conductivity.
* Tram. R, 8, Edin., 1853-4, p. 165.
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302.] EQUATION OF CONTINUITY. 421
Equation of Continuity in a Homogeneous Medium.
301.] If we express the components of the electromotive force
as the derivatiyes of the potential 7, the equation of continuity
du dv dw ^ ,,^.
becomes in a homogeneous medium
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 a non-
isotropic medium.
302.] If we put
[ra] = rir2r3 + 28i82«3-ri8i«-r2a2*-r388«, (17)
and [AB] = AT^A2A^'k'2BiBj^B^^AiBi*^A^B^^'-A^B^\ (18)
where [rs] il^ = rg r3 — 8^^, \
[r8]5i = «283-ri8i, I (19)
and so on, the system A, B will be inverse to the system r, a, and
if we make
A,x^ + A^y^-i-A^z^ + 2B^yZ'^2B^zx + 2B^xy = [AB]p^ (20)
we shall find that n i
^=^- (21)
is a solution of the equation *.
* {Sappose that by the transformation
arsa X+6 T+e Z, )
y^a'X+VT-K/zA (1)
the left-hand side of (16) becomes
(PV d^ dW
For this to be the case, we Ree that
must be identical with
which we shall call U,
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422 RESISTANCE AND CONDUCTIVITY. [303.
T" *he case in which the coefficients T are zero, the coefficients
B become identical with the coefficients jR and ^S of Art.
iVhen T exists this is not the case.
le case therefore of electricity flowing out from a centre
afinite, homogeneous, but not isotropic, medium, the eqoi-
al surfaces ai*e ellipsoids, for each of which p is constant
:es of these ellipsoids are in the directions of the prin-
xes of conductivity, and these do not coincide with the
al axes of resistance unless the system is symmetrical,
transformation of the equation (16) we may take for the
a, 2/, z the principal axes of conductivity. The coefficients
forms 8 and B will then be reduced to zero, and each co-
b of the form A will be the reciprocal of the corresponding
mt of the form r. The expression for p will be
^ + y! + ?! = _^. (22)
1 The theory of the complete system of equations of re-
3 and of conductivity is that of linear functions of three
es, and it is exemplified in the theory of Strains *, and in
arts of physics. The most appropriate method of treating
\i by which Hamilton and Tait treat a linear and vector
n of a vector. We shall not, however, expressly introduce
iiion notation.
coefficients 2^, T^^ T^ may be regarded as the rectangular
lents of a vector T, the absolute magnitude and direction
liminate £, i?) C ^7 ^^ equaiioni
.dU dU dU
*"*^' y"*d^' '^*rfC'
x = a (af + a'j; + a"0 + ft {h^ + h'ti + VO-^ c (<?f + <?'»; + c"0» )
z = a" {a £ + a' 17 + a" f) + ^ (pi + *'»/ + &" f) + ^ i^^ + ^'l + <^' C)» )
nee the system AB is inverse to the system rs^
i7 = ^iar» + ^jy« + ^,«» + 2 J?,y« + ... .
>m equations (1) and (3) we see that
X= of + a'i; + a"f,
(2) r « ,_ =^ satisHes the differential equation, henoe l/Vr
Bfyit.
rhomson and Tait's N&tural Philoiophif, § 1^^*
Digitized by VjOOQ IC
304.] SKEW SYSTEM. 423
of which are fixed in the body, and independent of the direction
of the axes of reference. The same is true o{ t^jt^.t^, 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 ;^ so as to coincide with the vector
r, and transform the equations of resistance accordingly. They
will then have the form
F= S^u-^R^v + S^w + Tu, > (23)
Z= S^u+S^v-^R^w. )
It appears from these equations that we may consider the
electromotive intensity 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 jR and S 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 jT, is equal to the product
of T into the resolved part of the current perpendicular to the
axis of T, and its direction is perpendicular 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 direc-
tion round T.
If we consider the current and T as vectors, the part of the
electromotive intensity due to J^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
substance. It should be found, if anywhere, in magnets, which
have a polarization in one direction, probably due to a rotational
phenomenon in the substance ^.
304.] Assuming then that there is no rotatory coefficient, we
shall shew how Thomsons Theorem given in Arts. lOOa-lOOc
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
* { Mr. HaU*8 discovery of the action of magnetism on a permanent electric current
{Phil. Mag. ix. p. 225 ; x. p. 301, 1880) may be described by saying that a conductor
placed in a magnetic field has a rotatory coefficient. See Hopkinson {Phil. Mag, x.
p. 430, 1880.)}
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EESISTANCB AND CONDUCTIVITY. [304.
on (10), to three terms; and let us consider the
cteristic equation (16) which then reduces to
cPV dW
'■irfa;*''"^*
cPV
, 6, c be three functions of x, y, z satisfying the
da db dc _
dx dy dz~ '
dV
'dx
dV
'dy
dV
« = -^^d^+«'
^ = -'"^^+'''
0'=-^s:^+^-
(25)
(26)
J the triple-integral
W=JJJ{R,a^ + RJ)^ + R,c^)dxdydz (27)
over spaces bounded as in the enunciation of Art.
viz. that over certain portions Fis constajit or else
omponent of the vector a, 6, c is given, the former
ng accompanied by the further restriction that the
ihis component over the whole bounding surface
: then W will be a minimum when
16 = 0, t; = 0, -m; = 0.
i^e in this case
r^Ri = 1, r^Ri = 1, rgjRg = 1 ;
, by (26),
dV
'dx
dV
dV
' dz
^dxdydz
(R^u^ + -Kg^ + ^z'^^) dxdydz
■///< ^^
du dv dw ^
d^^d^'-d^^'' (29)
n vanishes by virtue of the conditions at the limits.
3rm of (28) is therefore the unique minimum value
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305-] EXTENSION OF THOMSON'S THEOREM. 425
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 con-
ductor 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 current 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 form 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
i cannot begin or end within the closed surface, and since the
potential at all points of the surface is the same, there can be
j no current along lines passing from one point of the surface to
f another.
Hence there are no currents within the surface, and therefore
there can be no difference of potential, as such a difference would
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426 EESISTANCE AND CONDUCTIVITY. [306.
produce currenta, and therefore the potential within the closed
surface is everywhere the same as at the surface.
(5) K 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 hypo-
thesis 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 potential 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 TJ" in the first case and TJ 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 Tf over TJ, 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 CalcvZation of the Resistance of a
Conductor of a given Forni,
306.] The conductor here considered has its surface divided
into three portions. Over one of these portions the potential is
maintained at a constant value. Over a second portion the
potential has a constant value different from the first The
whole of the remainder of the surface is impervious to electricity.
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306.] RESISTANCE OF A WIRE OF VARIABLE SECTION. 427
We may suppose the conditions of the first and second portions
to be fulfilled by applying to the conductor two electrodes of
perfectly conducting material, and that of the remainder of the
surface by coating it with perfectly non-conducting material.
Under these circumstances the current in every part, of the
conductor is simply proportional to the dificrence 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 circum-
stances above defined that it can be said to have a definite
resistance or conductivity as a whole. A resistance coil, con-
sisting 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 difi*erences
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
conductors 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 is diminished.
This principle may be regarded as self-evident, but it may
easily be shewn that the value of the expression for the re-
sistance of a system of conductors between two points selected
as electrodes, increases as the resistance of each member of the
system increases.
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
♦ Pha. Trans., 1871, p. 77. See Art. 102 a.
Digitized by VjOOQ iC
428 BESISTANCE AND CONDUCTIVITY. [306.
be produced by making it a perfect conductor, as it is already in
electrical equilibrium.
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 opera-
tion of much less difficulty than the original problem. For the
resistance 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 d 5 be an element of the surface of the stratum, v the
thickness 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-d8, (1)
pv
and the whole current through the stratum is
C=Ejfl^d8, (2)
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, (3)
^' J J 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 J* has the values ^and jF+c^jF respectively, then
(5)
and the resistance of the stratum is
dF
fl
-VFdS
Digitized byCjOOQlC
307.] RESISTANCE OF A WIRE OP VARIABLE SECTION. 429
To find the resistance of the whole artificial conductor, we
have only to integrate with respect to -F, and we find
(6)
"•ill
-VFdS
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 R^
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 comparatively 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 deter-
mining 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
dB,=e^. (7)
«.=/^
and that of the whole wire of length a is
''-§' («)
whei*e S is the transverse section and is a function of 8.
This method in the case of wires whose section vaiies 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 per-
fectly uniform.
807.] To find the higher limit of the resistance, let us suppose
a surface drawn in the conductor to be rendered impermeable to
electricity. The efiect 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
efiect, if there is any, of this system of impermeable surfaces
must be to increase the resistance above its natural value.
Digitized by VjOOQ iC
430 RESISTANCE AND CONDUCTIVITT. [307-
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 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 x along
the axis, and let the radius of the section at any point bo 6.
Let one set of impermeable surfaces be the planes through the
axis for each of which ^ is constant, and let the other set be
surfaces of revolution for which
y^ = ykh\ (9)
where ^ is a numerical quantity between 0 and 1.
Let us consider a portion of one of the tubes bounded by the
surfaces <^ and <^ -f (i</i, >/f and yjr -f dyjr, x and x + dx.
The section of the tube taken perpendicular to the axis is
ydyd(i>^ \b^dyl^d<t>. (1^)
If 6 be the angle which the tube makes with the axis
The true length of the element of the tube is dx seed, and its
true section is j b^d\lrd(t> cosd,
so that its resistance is
L.t A=f^dx,^ B=f^{fJ<l^. (13)
the integration being extended over the whole length, x, of the
conductor, then the resistance of the tube d^d0 is
and its conductivity is
d^d(t>
To find the conductivity of the whole conductor, which is the
sum of the conductivities of the separate tubes, we mtist inte-
gmte this expi^ession between ^ = 0 and 0 = 2 jt, and between
Digitized by VjOOQ iC
308.] HIGHER AND LOWER LIMITS. 431
yjf =z 0 and \/r = 1. The result is
which may be less, but cannot be greater, than the true con-
ductivity of the conductor.
When -J- is always a small quantity -j will also be small, and
we may expand the expression for the conductivity, thus
The first term of this expression, -.-, is that which we should
have found by the former method as the superior limit of the
conductivity. Hence the true conductivity is less than the first
tei-m 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 yjfj we had assumed that the flow through each tube is
proportional to d\lfd<p, we should have obtained as the value of
the resistance under this additional constraint
iJ" = 1(^ + 15)* (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 cor-
rection 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
difierent metal.
For the lower limit of the resistance we shall suppose that an
infinitely thin disk of perfectly conducting matter is placed be-
tween the end of the cylinder and the massive electrode, so as to
bring the end of the cylinder to one and the same potential
♦ Lord Rayleigb, I%eory of Saundt ii. p. 171.
Digitized by VjOOQ iC
432 EESISTANCB AND CONDUCTIVITY. [309.
throughout. The potential within the cylinder iivill then be &
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 dimensions to be large compared with the diameter of
the cylinder, the distribution of potential will be that due to &
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
surface of the disk into the electrode, and p' the specific re-
sistance of the electrode ; then if Q is the amount of electricity
on the disk, which we assume distributed as in Art. 151, we see
that the integral over the disk of the electromotive intensity is
p'C=i.47rQ = 2 7r^, byArt. 151,
TT
2
= AaE. (18)
Hence, if the length of the wire from a given point to the
electrode is i, and its specific resistance /o, the resistance from
that point to any point of the electrode not near the junction is
•na^ 4 a
and this mav be written
— 1 * ^ '-
)nd term within brackets is a quantity which
to the length of the cylinder or wire in calcu-
Eince, and this is certainly too small a correction,
id the nature of the outstanding error we may
hereas we have supposed the flow in the wire up
be uniform throughout the section, the flow from
electrode is not uniform, but is at any point in-
ional (Art. 151) to the minimum chord through
the actual case the flow through the disk will not
t it will not vary so much from point to point
posed case. The potential of the disk in the
11 not be uniform, but will diminish from the
dge.
all next determine a quantity greater than the
Digitized by VjOOQ iC
309.] COBEECTION FOB THE ENDS OP THE WIRE. 433
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 sur-
face-integral of the product of the flow into the potential. The
(J
rate otflow at any point is — «, and the potential is the same as
TtOb
that of an electrified surface whose surface-density is <r, where
2,. = ^;. (20)
P^ being the specific resistance.
We have therefore to determine the potential energy of the
electrification of the disk with the uniform surface-density <r.
* The potential at the edge of a disk of uniform density o-
is easily found to be 4a<r. The work done in adding a strip of
breadth da at the circumference of the disk is 2'naada.Aaa'^
and the whole potential energy of the disk is the integral of this,
or P^^a^a^. (21)
o
In the case of electrical conduction the rate at which work is
done in the electrode whose resistance is ii' is C^Bf. But firom
the general equation of conduction the current across the disk
per unit area is of the form
p' dv
27r
or — J <r.
P
The rate at which work is done is, if F is the potential of the disk,
and de an element of its surface,
-no?]
Yds
2C P
= — 7. — > smce
= i^P (by (20)).
P
= \Jv<rd8,
We have therefore
* See a Paper by Professor Cayley, London Math, Soc. Proc. vi. p. 88.
VOL, I. F f
(22)
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434 EBSISTANCB AND CONDUCTIVITY,
whence, by (20) and (21),
and the correction to be added to the length of the cylinder is
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
IT 8
number lying between -and — - » or between 0-785 and 0-849.
* Lord Rayleigh, by a second approximation, has reduced the
superior limit of n to 0-8282.
* PhU, Mag. Nov. 1872, p. 844. Lord Rayleigh lubBequently obtained *8242 m the
snperior limit. See London Ma^ Soe. Proc, yii. p. 74, lUso Theory of Sound, vol. ii.
Appendix A. p. 291.
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CHAPTEE IX.
CONDUCTION THEOUGH HETBEOGENEOUS MEDIA.
On the Conditions to be Fulfilled at the Surface of Reparation
between Two Conducting Media,
810.] 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 electrometer 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
difierent 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 T^ and K are the potentials in the two media, then at
any point in the surface of separation
and if iti , t;, , t&2 and u.^^ v^, W2 are the components of currents in
the two media, and Z, m, n the direction-cosines of the normal to
the surface of separation
u^l-hVim-^Win = u^l-k-v^m-k-w^n. (2)
In the most general case the components u, v, w are linear
F f a
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436 CONDUCTION IN HETEEOGENEOUS MEDIA. [3IO.
functions of the derivatiyes of F, the forms of which are given
in the equations
v=q,X-^r^Y^p,ZA (3)
t/JrrpgZ + grjF+rjjZ, )
where Z, F, Z are the derivatives of V with respect U^ x, y, 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\ F, Z" be the values of Z, F, Z in the isotropic medium,
then we have at the surface
F= F^ (4)
or Xdx + Ydy + Zdz = X'dx + Tdy + Z'dz, (5)
when ldx-¥mdy-hndz= 0. (6)
This condition leads to
Z'=Z + 47r<rZ, F= F+4ir<rm, Z'=Z+4iro-n, (7)
where a is the surface-density.
We have also in the isotropic medium
ti' = rZ', t;' = rr, it;' = rZ' , (8)
and at the boundary the condition of flow is
u74 v^m + w'n = ttZ + twi + t(m, (9)
or r(iZ + mF4-nZ-f 4iT(r)
= i(r,Z+p3F+(7aZ) + m(g3Z + r2F+2?iZ) + n(p,Z + ?iF+r3Z),(10)
whence
4 iro-r = {/(r, — r) + 771^3 + np^}X + {Z;?3 + m(r2 — r) + ti^, } F
'^ {Iq^-^nipi + n {r^-r)\ Z. (11)
The quantity o- 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. Li isotropic substances the coeffi-
cients p and q are zero, and the coefficients r are all equal,
so that ^
47r<r = (-!i - 1) (IX + mT+nZ), (12)
where r, is the conductivity of the substance, r that of the
external medium, and I, vi, 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
simplified, for if fc is the specific resistance per unit of volume,
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3 1 1 -] SUBPAOB-CONDITIONS. 437
^^^^ U = ---^, V=_l^, -.«i^ (13)
k dx ' "" k dy ' ^ " k dz^ ^ ^
and if r is the normal drawn at any point of the surface of
separation from the first medium towards the second, the con-
dition of continuity is
k^ dv k^ dv ^ '
If ^1 and ^2 ftre 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,
*^^ k^\Ane^ = k^td^nO^, (16)
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 k^ 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 S^ be
^=(^»- + 5,r-('+i))iSr„ (1)
^=(jy + 5gr-<'+i))«„ (2)
within and without the sphere respectively.
At the surface of separation where r = a we must hare
V^ = V^, and J-^S^-i^. (3)
From these conditions we get the equations
These equations are sufficient, when we know two of the four
quantities A^, A.^y B^^ B^^io deduce the other two.
Let us suppose A^ and B^ known, then we find the following
expressions for A^ and B^,
. _{k^{i-¥\)-¥k^i]A^^{k^^k^{i+\)B,a-^^'^^^
*i(2i+l) ' \ (5)
J. _ {K^h)iA^a^'^^^{k^i^-k^{i-\^ 1)} B^ ' ^ ^
Digitized by VjOOQ iC
y (6)
438 CONDUCTION IN HBTBEOGENEOUS MEDIA, [3 1 2.
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 02, and let there be a second spherical surface of radius a,
greater than Oj, beyond which the specific resistance is k^. If
there are no sources or sinks of electricity within these spheres
there will be no infinite values of F, and we shall have Bj^ = 0.
We then find for A^ and jB,, the coefficients for the outer
medium,
^A;,A;2(2t+l)2 = r{«:i(i+l) + M}{M^+0 + M}
The value of the potential in the outer medium depends partly
on the external sources of electricity, which produce currents
independently 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 A^ be the coefficient of one of these, of the form
Then we can find A^, the corresponding coefficient for the
inner sphere by equation (6), and from this deduce A^, B^^
and £3. Of these B^ represents the effect on the potential in
the outer medium due to the introduction of the heterogeneous
sphere.
Let us now suppose A^sA;^, so that the case is that of a hollow
shell for which A; = ib^) separating an inner from an outer portion
of a medium for which k = k^.
Digitized by VjOOQ iC
3X3.]
SPHBEICAL SHELL.
439
If we put
(7 = -
2< + lv
then
(7)
A^ = k^{2i + l)(k^{i + l) + k^i)CA,,
B^ = k^i{2i+l)(k^-k^a,^*-'^CA,,
5, = i{k^-k,)(k,{i + l)+k,i){a^'**^-a,"*-')CA^
The difference between A^ the undisturbed coefficient, and A^
its value in the hollow within the spherical shell, is
A,-A, = {k,-k,)H(i+l)(l - Q) )CA,. (8)
Since this quantity is always of the same sign as A^ whatever
be the values of ki and k^, 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 i*ound
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 aj = 0, or it may be worked out independently.
813.] The most important term in the harmonic expansion is
that in which i = 1, for which
9k,k,+2ik,-k,y(i-0)
A^ = 9kikiGA,, A^ = 3kg{2ki + kt)CA,,
S^= 3k^(k^-k^)ai^CAi, B^ = {k3-ii)(2k^ + k^(a^'-a»)CA^
The case of a solid sphere of resistance k^ may be deduced
from this by making o^ = 0. We then have
w
(10)
^»"ifei + 2A:a
It is easy to shew from the general expressions that the value
of J?3 in the case of a hollow sphere having a nucleus of re*
sistance k^, surrounded by a shell of resistance k^^ is the same as
Digitized by VjOOQ IC
440 CONDUCTION IN HETEEOGBNEOUS MEDIA. [3 1 4.
that of a uniform solid sphere of the radius of the outer surface,
and of resistance iT, where
814.] If there are n spheres of radius a^ and resistance k^,
placed in a medium whose resistance is k^^ 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 a^^
the potential at a great distance r from the centre of this sphere
will be of the form
V= {Ar-k-nB^co^e, (12)
where the value of £ is
The ratio of the volume of the n small spheres to that of the
sphere which contains them is
TlCti
8
The value of the potential at a great distance from the sphere
may therefore be written
Now if the whole sphere of radius a^ had been made of a
material of specific resistance K, we should have had
That the one expression should be equivalent to the other,
K^^h±h±A=htu. (17)
This, therefore, is the specific resistance of a compound medium
consisting of a substance of specific resistance k^, in which are
disseminated small spheres of specific resistance k^^ tiie 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 com-
pared with their distances, and therefore p must be a small
fraction.
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315.] APPLICATION OP THE PRINCIPLE OF IMAGES. 441
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
with their radii, and when -^ — ^ is considerable, then other
terms enter into the result, which we shall not now consider.
In consequence of these terms certain systems of arrangement of
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 electiicity at a distance a from the plane surface in
the first medium, the quantity of electricity flowing firom the
source in unit of time being S.
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 Vi := SP.
In the actual case the conditions may be satisfied by taking
a point /, the image of S in the second medium, such that IS
is normal to the plane of separation and is bisected by it. Let
r^ be the distance of any point from /, then at the surface of
separation ^^ ^ ^^^ ^1)
di^^~di' ^^^
Let the potential V^ at any point in the first medium be that
due to a quantity of electricity E placed at S^ together with an
imaginary quantity E^ at i, and let the potential ^ at any
point of the second medium be that due to an imaginary
quantity E^ at 8, then if
^ = ^+^2 ^d TJ = ^\ (3)
the superficial condition T^= Ogives
E^E^^E,, (4)
and the condition 1 ^^ ^ 1 ^ ,gv
k^dv k^dp ^ '
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442 OOKDUCTION IN HBTBEOOBNEOUS MEDIA, [316.
gives ^{E-E^^Ie„ (6)
whence ^i = J^\^' ^» = |t|^- <'^
The potential in the first medium is therefore the same as
would be produced in air by a charge E placed at fif, and a
charge E^ at / on the electrostatic theory, and the potential in
the second mediam is the same as that which would be produced
in air by a charge E^ at S.
The current at any point of the first medium is the same as
would have been produced by the source S together with a
k ^k
source j^ j^ S placed at / if the first medium had been infinite,
and the current at any point of the second medium is the same
2k S
as would have been produced by a source yj- — ^. placed at S if
the second medium had been infinita
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
k^ = 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
^2 = ^) and the image at I is equal to the source at S 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
resistance. The method of inversion in two dimensions is, how-
* { A ninilar inyestigatton wiU give the electric field doe to a obai]^ of electricity
at 6 placed in a dielectric whose specific inductive capacity is ATj , this dielectric being
separated by a plane face from another dielectric whose specific indactive cfwacity is
JT,. Fi and F, will represent the poteiitiab in this case if the charge '^ KiB and if
Digitized by VjOOQ iC
3 1 8.] STBATUM WITH PARALLEL SIDES, 443
ever, applicable, as well as the more general method of trans-
formation in two dimensions given in Art. 190 *.
Conduction through a Plate separating Two Media.
817.] Let us next consider the effect of a plate of thickness
AB o{ a, medium whose
resistance is k^, and ^^
separating two media
whose resistances are , , i
Aj, and A3, in altering '• * '
the potential due to a
source S in the first
medium. Fig 24.
The potential will be
equal to that due to a system of charges placed in air at certain
points along the normal to the plate through S.
Make
AI=SA, BI^=SB, AJ^^I^A, BI^=J,B, AJ^^I^A^Scc.;
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
that at a point I" in the second
and that at a point P"' in the third
where /, I\ &c. represent the imaginary charges placed at the
points /, &c., and the accents denote that the potential is to be
taken within the plate.
* See Kirehhoff, Pogg. Ann. hdy. 497, and Ixrii. 844; Quincke, Pogg. xcvii. 882;
Smith, Pro€, R. 8. Edin,, 1869-70, p. 79. HolantQler, Einfahrung in die Theorie
der iaogonalen Verwandschaflen, Leipzig, 1882. Guebhard, Journal de Phytique,
t. i. p. 483, 1882. W. G. AduinB. Phil. Mag. iv. 60. p. 548, 1876 ; G. C. Foeter and
O. J. Lodge, PAtZ. Mag, iv. 49, pp. 886, 468 ; 60, p. 476, 1879 and 1880; O. J. Lodge,
Phil Mag. (6), i. 878, 1876.
Digitized by VjOOQ iC
444 OONDtJCTION IN HETEROGENEOUS MEDIA. [3 1 8.
Then, by Article 315, we have from the conditions for the
surface through A,
^=^*' ^=Kh/- ('■)
For the surface through B we find
Similarly for the surface through A again,
and for the surface through B,
If we make p = ^1^^ and p'= ^^,
we find for the potential in the first medium,
+p'(l-p«)(pp')-»^+.... (15)
For the potential in the third medium we find
F=(l+p')(l-p)^{^ + ^^ + &c.+ ^ + ...|*. (16)
* {These ezpreasions may be reduced to definite integnds by the relation
V a' + 1>« Jo
where J^ denotes BessePs fuDction of zero order.
Hence if we take /S as the origin of ooordinatee, and the normal to the plate as
the axis of a;,
i's'fl''
(yf)e-''dt.
where c is the thickness of the plate,
and so on. Substitntbg these values in (16), we see that V equals
Jo l-pp'e-'''
The values of this wheny - 0, » « 2nc when n is an integer can easUy be found.)
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3 1 9-] STBATIPIED CONDUCTORS, 446
If the first medium is the same as the third, then k^^ k^ and
p = p\ and the potential on the other side of the plate will be
F=(l-p^)^{^ + ^^ + &c.-,^+...}. (17)
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^^k^ 3g ^k^^k^
^ 3-p k^-\-2k^' ^ 2+gf k^ + k^'
If we put p = ; — - — , we shall have a solution of the problem
lH-27rjc ^
of 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 sti-ata of
thicknesses c and c^ of two substances whose coefficients of con-
ductivity are different. Required the coefficients of resistance
and conductivity of the compound conductor.
Let the planes of the strata be normal to z. Let every symbol
relating to the sti'ata of the second kind be accented, and let
every symbol relating to the compound conductor be marked
with a bar thus, X. Then
Xz:zX==X\ (c + O-a =s cu+cV,
(c + c')Z"=cZ +c'Z', w-w-vf.
We must first determine u, v!, v, i/, Z and Z" in terms of
X, Y and w from the equations of resistance. Art 297, or those
♦ See Sip W. ThomBon^e *Note on Indaced Magnetiflm in a Plate/ Camib. and
Dub, Math. Joarn., Nov. 1845, or Reprint, art. ix. § 156.
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446 CONDUCTION IN HETEEOGBNEOUS MEDIA. [32 1.
of conductivity, Art. 298. K we put D for the determinant of
the coefficients of resistance, we find
Similar equations with the symbols accented give the values
of u', v' and ^. Having found H, v and W in terms of X, Y and
^ we may write down the equations of conductivity of the
c c'
stratified conductor. If we make A = — and A' = — 7 , we find
_ cp^-¥c%' ^h:{q^--q{){q^-q^^
^^" c + c' (A+A')(c + 0 '
- _ cqz-¥c'q^' Jih'{p,^p{){p^-^p^')
^«" c + c' (A + A')(c + 0 '
^ ^ crx + cV/ hh'(p2-P2^(q2-q2')
1- c + c' (A + AO(c + cO
«"~c+7 (A + AO(c + c')
_ crgH-cV hh'{p,^p^'){q^^q,')
820.] 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 ^ will be equal to that of its corresponding Q or q. From
this it follows that in the stratified conductor also
Pi = 5l> P2 = 52» Ps = Ja*
or there is no rotatory property developed by stratification,
unless it exists in one or both of the separate materials.
321.] If we now suppose that there is no rotatory property,
and also that the axes of re, y and z are the principal axes, then
the p and q coefficients vanish, and
n r.
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322.] STRATIFIED CONDUCTORS. 447
If we begin with both subBtances isotropic, but of different
conductivities r and /, then, since r<.^r^ = j~—= — f-. ,
the result of stratification will be to make the resistance greatest
in the direction of a normal to the strata, and the resistances
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 a, and place them
alternately with slices of a substance whose conductivity is 8,
and thickness k^a.
Let these slices be normal to x. Then cut this compound
conductor into very much thicker slices, of thickness 6, normal
to y, and alternate these with slices whose conductivity is 8 and
thickness k^b.
Lastly, cut the new conductor into still thicker slices, of
thickness c, normal to 0, and alternate them with slices whose
conductivity is a and thickness k^c.
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 c, and a is exceedingly small compared with b, and to
embed these parallelepipeds in the substance whose conductivity
is 8, so that they are separated from each other ^^a in the
direction of x, kjb in that of y, and k^c in that of z. The
conductivities of the conductor so formed in the directions of
X, y, and z are to be found by three applications in order of the
results of Art. 321. We thereby obtain
_ {1 -hA?t(l +k2){l +h)}r+(k^ + k^ + k^k,)8^
(l+A:,)(l+A:3)(V + «) '
_ (1 4 feg -^ ^2^3)^ + (ki-hk^-^k^k^-^ kjk^ + A;^ Argfcg)^
"""^ (1+A:3){V + (1 +*! + *! W ^'
^ {l-^h)(r-h(k,+k,-^k,k,)8) ^
The accuracy of this investigation depends upon the three
dimensions 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 kj^ , k^ and ^^3 each unity, then
_6r+38 _ 3r+58 _ 2r-h6g
n-47:^*' ""^-27+6^^' ^^-r+Za""*
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448 CONDUCTION IN HETEEOGBNEOUS MEDIA, [323.
If r = 0, that Ib, if the medium of which the parallelepipeds
are made is a perfect insulator, then
If r = 00, that is, if the parallelepipeds are perfect conductors,
In every case, provided A^j = fe^ = feg, it may be shewn that
r^, r, and r^ 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.
828.] 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, 6, c, and let the conductivity of
the channel, extending from the origin to the point (ct&e), be
ahcK*
The electromotive force acting between the extremities of the
channel is aX + 6F+cZ,
and if (7 be the current along the channel
Cr= Kabc(aX + 6 F+ cZ).
The current across the face be of the parallelepiped is bcuy 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{riX'¥p^Y+q2Z) + Kahc{aX'^bY'\'cZ),
or u = (ri + Ka^)X + (p^ + Kab)Y'\' (q^ + 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
r, + Ka^ r^ + Kb^ r^ + Kc\
Pi-\-Kbc, p^'\'Kca, p^-tKab,
q^^ + KbCy q^ + Kca, q^-^-Kab.
In these expressions, the additions to the values o{ pi, &c., due
to the effect of the channel, are equal to the additions to the
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X
y
z
0
0
0
0
L
0
1
1
M
1
0
1
N
1
1
0
324.] COMPOSITE CONDUCTOR. 449
values of j^, &c. Hence the values of p^ and q^ 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.] To construct a framework of linear conductors which
shall have any given coefjicients of conductivity forming a
symmetrical system.
Let the space be divided into equal small cubes, of which let
the figure represent one. Let the coordin-
ates of the points 0, i, ilf, iV, and their poten- a/
tials be as foUows : —
Potential ^^
X
Y Fig. 25.
z.
Let these four points be connected by six conductors,
OZ, OM, ON, MN, NL, LM,
of which the conductivities are respectively
A, 5, C, P, Q, R.
The electromotive forces along these conductors will be
F+Z, Z+Z, Z+F, F-Z, Z-Z, Z-F,
and the currents
il(FH-Z), 5(Z+X), C{X^7), PiY-Z), Q(Z-Z), R{X-Y).
Of these cun*ents, those which convey electricity in the positive
direction of x are those along LM, LN, OM and ON, and the
(juantity conveyed is
u = (5 + C+Q + ii)Z+(C— R)F +(5-Q)Z.
Similarly
v^{C-'li)X +(a4-il+i2 + P)FH-(il-P)Z;
w^(B--Q)X +(.1-P)F +(^ + 5+p + Q)^;
whence we find by comparison with the equations of conduction,
Art. 298,
^A = rg-f r3-ri + 2;>i, 4P = r^'\-r^--r^'^2p^,
4B = r3 + ri-r2 + 22>2, 4Q = r^ + Ti^r^-^p^,
4(7 = ri + r2-r3 + 2^3, 4JJ= ri + rg-r8-22)3.
VOL. I, o g
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CHAPTER X.
CONDUCTION IN DIBLEOTEICS.
825.] 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 dielectiic 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,
where K is the ' specific inductive capacity.'
The ' equation of continuity ' of electric currents becomes
dx^rdx^ ^ dy^T dy^ "^ dzW dz^ dt^ ' ^ ^
where r is the specific resistance referred to unit of volume.
When if or r is discontinuous, these equations must be trans-
formed into those appropriate to surfaces of discontinuity.
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326.] THEOEY OP A C0NDEN8EE. 451
In a strictly homogeneous medium r and K are both constant,
so that we find
^ + ^+rf^ = "4.^ = r^, (3)
whence p = Ce ^'^ ; (4)
or, if we put T^-^, p=zCe ^. (5)
This result shews that under the action of any external elec-
tric 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, provided 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 C 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 currrent in that circuit, then
dQ_E dE
Let a battery of electromotive force Eq whose resistance
together with that of the wire connecting the electrodes is >'^
be introduced into this circuit, then
dQ_E,-E_E dE
Tt--7r~R'^ ~di' ^')
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452 CONDUCTION IN DIELECTRICS. [328.
Hence, at any time t^ ,
Next, let the circuit r^ be broken for a time t^, putting r^
infinite, we get from (7),
E{= E^) = E^e'^'^ where T^ = CR. (9)
Finally, let the surfaces of the condenser be connected by
means of a wire whose resistance is r^ for a time ^3, then
putting -2^0=0, r^ = rg in (7), we get
E(= E^ = E,e-T. where T, = ^^ • (10)
If Q3 is the total discharge through this wire in the time ^3,
«3 = ^o(-^^^^-:^/i-^0«^'(i-4. (11)
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 t^. If the time of
charging is sufficient, as it generally is, to develop 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 graduaUy 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.
828.] 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.
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328.] STRATIFIED DIELECTRIC. 453
Let 04, aj, &c. be the thicknesses of the different strata.
Xj^, X^i &c. the resultant electrical forces within the strata.
^19 i>2» &c* the currents due to conduction through the strata.
/uA* &^' ^^® electric displacements.
u,, Ug, &c. the total currents, due partly to conduction and
partly to variation of displacement.
r,, rg, &c. the specific resistances referred to unit of volume.
Ki, K29 &c. the specific inductive capacities.
k^, ^2) ^^' ^^^ reciprocals of the specific inductive capacities.
E 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.
<T^ 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,
X,^r,p,. (!)
By the theory of electrical displacement,
^i = 4irV,. (2)
By the definition of the total current,
^ = 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 ^ =/ -/ , (4)
and to determine its variation we have
By differentiating (4) with respect to ^, and equating the result
to (5), we obtain
i'.+^=i>s+^ = ^»»y. (6)
or, by taking account of (3),
Uj = 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.
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454 CONDUCTION IN DIELECTRICS. [3 29-
We have also, in virtue of equations (l) and (2),
1 -- I dX^ ,.
from which we may find X^ by the inverse operation on u,
The total electromotive force E is
^= OiXi + agXa+fec, (10)
an equation between E, the external electromotive force, and u,
the external current.
K the ratio of r to X; is the same in all the strata, the equation
reduces itself to
^+4^ dT " («l^l + «2^2 + &C.)u, (12)
which is the case we have already examined in Art. 326, and in
which, as we found, no phenomenon of residual charge can take
place.
K there are n substances having different ratios of r to X;, 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 (^yi— l)th order with respect to u, ^ being the in-
dependent 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 fi^f^, &c. are all zero,
and that an electromotive force Eq is suddenly made to act, and
let us find its instantaneous effect.
Integrating (8) with respect to ty we find
Q=fudt = iyXid^ + ^ Xi + const. (13)
Now, since X^ is always in this case finite, I X^^dt must be
insensible when t is insensible, and therefore, since X^ is origin-
ally zero, the instantaneous effect will be
Xj^^^-nk^Q. (14)
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329.] ELBCTBIO 'ABSORPTION.' 455
Hence, by equation (10),
Eq = 4ir{k^a^ + k^a2'{-&c,)Q, (15)
and if C be the electric capacity of the system as measured in
this instantaneous way,
C = -^=— i— (16)
Eq ^Tfijc^a^'^'k^a^ + ioo.) ^ '
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 con-
tinued uniform for an indefinitely long time^ or till a uniform
current of conduction equal to ^ is established through the
system.
We have then X^ = r^p^ etc., and therefore by (10),
^0 = (^i«i + ^2«2 + ^^)P' (17)
If R be the total resistance of the system,
E
R=-^ = ria^ + ria^+kc (18)
In this state we have by (2),
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 dq^rmine Q we observe that if X^ be the new value of Xi ,
then by (13), x/= Xi-^4i:k,Q. (20)
Hence, by (10), putting jE = 0,
0 = a^ ^1 + &c. + 4 ir(ai A^i + a^k^ + Sec.) Q, (21)
or 0 = ^o + ^Q- (22)
Hence Q:=^CEq where C is the capacity, as given by
equation (16). The instantaneous discharge is therefore equtJ
to the instantaneous charge.
Let us next suppose the connexion broken immediately after
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456 CONDUCTION IN DIELECTRICS. [33O.
this discharge. We shall then have u = 0, so that by equation (8),
_4ir*i^
Zi = Xi'« n , (23)
where X/ is the initial value after the discharge.
Hence, at any time t, we have by (23) and (20)
The value of E at any time is therefore
^oli"^ -^^<hKC) e'''^' + (^^-■47ra2fc2C)/"?V&c-}. (24)
and the instantaneous discharge after any time t is EC. This is
called the residual discharge.
If the ratio of r to i; is the same for all the strata, the value
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
^ = 0, J? = 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
dischai-ge 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 r^ is infinite for
that stratum, and R is infinite for the whole system, and the
final value of ^ is not zero but
E = EQ(l^4iraik^C). (25)
Hence, when some, but not all, of the strata are perfect in-
sulators, a residual discharge may be permanently preserved in
the system.
830.] We shall next determine the total discharge through a
wire of resistance Rq kept permanently in connexion with the
extreme strata of the system, supposing the system first charged
by means of a long-continued application of the electromotive
force E.
* {This is perhaps more easily seen if we write (24) as
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330.] EESIDUAL DISCHAEGE. 467
At any instant we have
E = aiViPi + a^r^Pi + &c. + Bo« = 0, (26)
and alBO, by (3), u = p, + ^. (27)
Hence (-B + -Bo) « = "i ^ ^ + a2»-2 ^ + &<^ (28)
Integrating with respect to t in order to find Q, we get
{R+R,)Q = air,(//-A)+a,r,(//-/^ + &c, (29)
where /i is the initial, and// the final value of/j.
In this case//= 0, and by (2) and (20) /i = ^©(j^-r-^)*
Hence (iJ+iJ„)Q = _^(?^ + ML+&c) + ^,CJe. (30)
= -^12[a,a,kAQy^f} (31)
where the summation is extended to all quantities of this foitn
belonging to every pair of strata.
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 diffei*ent kinds may exhibit the phenomena known as electric
absorption and residual discharge, although none of the sub-
stances 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
expected in the case of substances composed of parts of different
kinds, even though these individual parts should be micro-
scopically 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
* {Rowland and Nichols hare shewn that crystals of Iceland Spar which are yery
homogeneous shew no Electric Absorption, Phil, Mag. xi p. 414, 1881. Muraoka
found that while paraffin and xylol shewed no residual charge when separate, a layer
of xylol on a layer of paraffin did. Witd. Ann, 40, 881, 1890. }
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458
CONDUCTION IN DIELECTRICS,
[33^'
be capable, and ibis 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.
831.] 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
mathematical 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 J-i, -^2, &c. be the inner conducting surfaces of a series of
condensers, of which Bq, B^yB^, &c. are the outer surfaces.
^'"^^^^^\
Fig. 26.
Let -4.1,-4.2, &c. be connected in series by connexions of resist-
ances jR, and let a current be passed along this series from left to
right
Let us first suppose the plates 5^, B^^ iJg, each insulated and
free from charge. Then the total quantity of electricity on each
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33I-] THEORY OF ELECTRIC CABLES. 459
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
surface they will not be electrified, and no alteration of the
current will be observed.
But let the plates JL be all connected together, or let each be
connected with the earth. Then, since the potential of J.^ is
positive, while that of the plates B is zero, -Aj will be positively
electrified and 5^ negatively.
K ^, i^, &c. are the potentials of the plates -Aj, J-g, &c., and C
the capacity of each, and if we suppose that a quantity of elec-
tricity equal to Qq passes through the wire on the left, Q^ through
the connexion JJ^, and so on, then the quantity which exists on
the plate A^ is Qq—Qi, and we have
Qo-Qi = CPr.
Similarly Qi-Qi^CP^,
and so on.
But by Ohm's Law we have
We have supposed the values of C the same for each plate,
if we suppose those of R the same for each wire, we shall have
a series of equations of the form
If there are n quantities of electricity to. be determined, and
if either the total electromotive force, or some other equivalent
condition be given, the difierential 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 -Aj, and only a very small fraction of the current
appears at the right hand till a considerable time has elapsed.
If galvanometers be placed in circuit at 22^, R^y &c. they will be
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460 CONDUCTION IN DIBLECTEICS, [332.
affected by the current one after another, the interval between
the timee of equal indications being greater as we proceed to the
right.
882.] 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 Oj , a2 are the outer and inner radii of the insulating sheath,
and if jfiT is its specific dielectric capacity, the capacity of unit of
length of the cable is, by Art. 126,
2log^
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 0;+^^, is
and this is, by what we have said, equal to cvhx.
Hence cv = — ~ • (2)
Again, the electromotive force at any section is — -7- , and by
Ohms Law, ^v . dQ
where k is the resistance of unit of length of the conductor, and
-J is the strength of the current. TClimiTiivf.mg Q between (2)
and (3), we find , dv dh)
di^d^^' (^)
This is the partial differential equation which must be solved
in order to obtain the potential at any instant at any point of
Digitized by VjOOQ iC
334-] HYDROSTATIOAL ILLUSTRATION. 461
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 cai)acity 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 k^ is the resist-
ance of unit of length of the sheath to conduction through it in
a radial direction, then if /^^ is the specific resistance of the
insulating material, it is easy to shew that
*x=^Piio&^;- (5)
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 v/Aj. Hence the
rate of expenditure of electricity will be
whence, by comparison with (3), we get
and this is the equation of conduction of heat in a rod or ring
as given by Fourier*.
883.] 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,
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 lUuetration of the Properties of a Dielectric.
834.] Five tubes of equal sectional area -4, 5, C, D and P are
arranged in circuit as in the figure. A, B, C and D are verti-
cal and equal, and P is horizontal
* Thiorie de la ChaUwr, Art. 105.
Digitized by VjOOQ IC
462
CONDUCTION IN DIELECTRICS.
[334.
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 connects 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 hyA^, B^^G^yD^^
that the piston is at i^, and that
^/^ I I r"%^^ the stopcock Q is shut.
/ I* f i' \ Now let the piston be moved
from i^ to ii, a distance a. Then
since the sections of all the tubes
are equal, the level of the mercury
in A and G will rise a distance a,
or to Ay and C^ , and the mercury
in B and D will sink an equal
distance a, or to B^ and D^.
The difference of pressure on
the two sides of the piston will
be represented by 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.
If the piston is free to move it will move back to ij 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
Digitized by VjOOQ iC
334-] HYDROSTATICAL ILLUSTBATION. 463
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 i^.
The levels of A-^ and D^ will remain unchanged, but those of
B and C will become the same, and will coincide with Bq
and Cq.
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
insulated, but their difference of potential diminishes.
In fact, the difference of pressure on the two sides of the
piston sinks from 4 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 i^, 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 ia
below its original level. This is indicated by the levels A^, B^,
K 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
Bq and Cq, 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 ig, half way between ^ and ij. This corresponds to
the residual charge which is observed when a charged dielectric
is first discharged 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 discharges, which are i, ^, &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 ex-
periment, we should have had a case resembling that of the
Digitized by VjOOQ iC
464 CONDUCTION IN DIELECTBICS.
electrification of a dielectric which is a perfect insulator and yet
exhibits the phenomenon 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 elec-
tricities are represented by two real fluids, and the electric
potential is represented by fluid pressure. Charge and discharge
are represented by the motion of the piston P, and electromotive
force by the resultant force on the piston.
Digitized by VjOOQ IC
CHAPTEE XI.
THE MEASUEEMBNT OF ELEOTBIO BESISTANOE.
885.] In the present state of electrical science, the deter-
mination of the electric resistance of a conductor may be con-
sidered as the cardinal operation in electricity, in the same
sense that the determination of weight is the cardinal operation
in chemistry.
The reason of this is that the determination in absolute
measure of other electrical magnitudes, such as quantities of
electricity, electromotive forces, currents, &c., requires in each
case a complicated series of operations, involving generally
observations of time, measurements of distances, and deter-
minations of moments of inertia, and these operations, or at
least some of them, must be repeated for every new deter-
mination, because it is impossible to preserve a unit of elec-
tricity, or of electromotive force, or of current, in an unchange-
able state, so as to be available for direct comparison.
But when the electric resistance of a properly shaped con-
ductor 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 measure-
ment *. Measures of electrical capacity, which are also of great
* {The CUrk*8 cell as a standard of Eleotromotive Force may now claim to be an
exception to this statement.}
VOL. I. H h
Digitized by VjOOQ iC
466 MBASUBBMBNT OF EBSI8TAN0B. [339.
importance, are still defective, on account of the disturbing in-
fluence of electric absorption.
836.] 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 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, Siemen's unit is defined as the resistance of
a column of mercury of one metre in length, and one square
millimetre in section, at the temperature of 0°C.
887.] Finally, the unit may be defined with reference to the
electrostatic or the electromagnetio 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 shew at the proper
place, a resistance is a quantity of the dimensions of a velocity,
and may therefore be expressed as a velocity. See Art. 628.
888.] 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 laige 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 BAl. unit, but in order to
connect it with the name of the discoverer of the laws of
resistance, it is called the Ohm.
889.] 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 electromagnetio 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 Associa-
Digitized by VjOOQ iC
340.] 8TANDAEDS OP EBSISTANOB. 467
tion's material standards, is not really represented by this
velocity, electricians would not alter their standards, but would
apply a correction''^. 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 ab-
solute value of the unit is originally chosen so as to represent
as nearly as possible a quantity derived fix>m the electromagnetic
absolute system.
840.] 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 distributed
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 re-
constructed 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 j- ^g
by the electromagnetic method by which
it was originally determined. This method, which is con-
siderably more laborious than the determination of a foot from
* {Lord Rayleigh's and Mrs. ffidgwiok's experiments have shewn that the British
Association Unit is only 'SSdT earth quadrants a second, it is thus smaller than was in-
tended b^ nearly 1-8 per cent. The Congress of Electricians at Paris in 1884 adopted
a new nnit of resistance, the ' Legal Ohm/ which is defined as the resistance at 0*^0. of
a oolmnn of mercury 106 centimetres long and 1 square millimetre in croH section.}
H h 2
Digitized by VjOOQ iC
468 MBASTTEBMBNT OF EBSISTANOB. [34 1.
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 platinum
in the form of wires from ^S millimetres 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 soUd 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 sup*
port of the coil. (See Fig. 28.)
On the Forma 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 wires or rods well
amalgamated 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 electrodes.
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.
Digitized by VjOOQ iC
342.] BESISTANOE COILS. 469
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 pai*t 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 still used for adjusting
the resistance where accurate measurement is not required.
Resistance coils are generally made of those metals whose
resistance is greatest and whidi 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 modem accuracy.
842.] 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 con-
nected 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.
♦ PhU. Mag., Jidy, 1870.
Digitized by VjOOQ iC
470 MEASUBBMBNT OP EESISTANOB. [344.
848.] There are variouB arrangementB 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.
The electrodes consist of stout brass plates, so arranged on
the outside of the box that by inserting a brass plug or wedge
between two of them as a shunt, the resistance of the corre-
sponding coil may be put out of the circuit. This arrangement
was introduced by Siemens.
Each interval between the electrodes is marked with the
resistance of the corresponding coil, so that if we wish to make
^n ^9
Pig. 29.
the resistance in the 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 quality of 1 and 1^ then that of
1 + 1^ and 2, then that of 1 + 1^ + 2 and 4, and so on.
The only disadvantage of the arrangement is that it requires
a familiarity with the binary scale of notation, which is not
generally possessed by those accustomed to express every number
in the decimal scale.
844.] A box of resistance coils may be arranged in a different
way for the purpose of measuring conductivities instead of
resi3tances.
Digitized by VjOOQ iC
345.]
COMPABISON OF BESISTAKOES.
471
The coils are placed so that one end of each is connected with
a long thick piece of metal which forms one electrode 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 conduc-
tivities of the coils.
In the figure^ in which the
resistances of the coils are
n n A n (V=Q=
jSI fiH [Y] m [il IT
E
Fig. 80.
1, 2, 4, &c., and the plugs are inserted at 2 and 8, the con-
ductivity of the box is I + 1 s= I, and the resistance of the box is
therefore f 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.
On the Compariaon of Resistances.
345.] If JE^ is the electromotive force of a battery, and R the
resistance of the battery and its connexions, including the gal-
vanometer used in measuring the current, and if the strength of
the current is / when the battery connexions are closed, and
Ii, I 2 when additional resistances r^, r, are introduced into the
circuit, then, by Ohm's Law,
Ez^IR^ lAR-^r,) = /a(iJ + r,).
Eliminating E, the electromotive force of the battery, and R
the resistance of the battery and its connexioas, we get Ohm's
formula r^ _ {I'-I^I^
This method requires a measurement of the ratios of /, Ii and
J^, and this implies a galvanometer graduated for absolute
measurements.
If the resistances r^ and r, are equal, then I^ and 7, 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
Digitized by VjOOQ iC
472
MEASUREMENT OF BESISTANOE.
[346.
method than as a practical method of determining resistance.
The electromotive 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,
The first of these methods depends on the use of the differ-
ential 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 71 they have no resultant effect on
the galvanometer needle.
Let /j, I 2 be the currents through the two coils of the gal-
vanometer, then the deflexion of the needle may be written
h = m/j — Ti/g'
Now let the battery current / 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 resistances of the coils and their connexions be a and /3
respectively, and let the resistance of the battery and its con-
nexions between C and D be r, and its electromotive force E.
Digitized by VjOOQ iC
346.] DIPPBBENTIAL GALVANOMETEB. 473
Then we find, by Ohm's Law, for the difference of potentials
between C and D,
and since /i + /g = /,
j^B-\-p J j.A^-a ^iJ + g + J + ff
where D = (ii + o)(5 + i8) + r(ii + a + jB + )3).
The deflexion of the galvanometer needle is therefore
8 = |{m(5 + i9)-7i(^ + a)},
and if there is no observable deflexion, then we know that the
quantity enclosed in brackets cannot differ from zero by more
than a certain small quantity, depending on the power of the
battery, the suitableness of the arrangement, the delicacy of the
galvanometer, 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 il, and let
A' be adjusted till there is no apparent deflexion. Then evi-
dently to a first approximation ^^= A.
To ascertain the degree of accuracy of this estimate, let the
altered quantities in the second observation be accented, then
m(5+i8)-'M(il + a) = ^8,
m(5+^)«7i(il'+a) = ^8'.
Hence n (^A'—A) = ~ 6 — = 5'.
If 5 and h\ instead of being both apparently zero, had been
only observed to be equal, then, unless we also could assert that
E = R, 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 feict 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.
1 Digitized by VjOOQ iC
474 MEASXmBMENT OF BESISTANGS. [346.
Null methods are of great yalne where they can be employed,
bnt 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 5 and If 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 observa-
tion with the mean of the whole series.
But by putting A' out of adjustment by a known quantity,
as, for instance, by inserting at il or at £ an additional resist-
ance equal to a hundredth part of A or of jS, and then observing
the resulting deviation of the galvanometer needle we can esti-
mate 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.
"I" 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 jy
m{A^fi)^n{B + a)^-^b\
D If
whence (m+^)(5— -4) = —5— =5'.
If 771 and 71, A and By a and /3, E and K are approximately
equal, then
^-^ = 2l^(^ + <»)(^+«+2r)(8-8')-
Here 5—5' 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 mini-
mum value of ^ — ' when
n
* This inyestigation is taken from Weber's treatise on Galvanometiy. GUiingtn
Trantattiom, x. p. 65.
Digitized by VjOOQ IC
347-] wheatstonb's bbidgb. 475
If we suppose r, the battery resistance, n^ligible compared
with ^, this gives a = iii;
or, the resistance of each coil of the galvanometer ehovM be
one-third of the resistance to he raeasv/red.
We then find q a%
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
^ nE ZnE .^ ^ , 1 .
£k = i— — :— = T -T- if r = 0 and a = - 4.
^ B-A 25-8'
Hence — -. — = -— r— •
A 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 ti may change
when the position of the needle is changed, and therefore it is
necessary to determine this ratio by proper methods during each
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 galvano-
meter. 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
847.] Wheatstone's Bridge consists essentially of six con-
ductors connecting four points. An electromotive force E is
made to act between two of the points by means of a voltaic
Digitized by VjOOQ iC
476
MEASUBEMENT OF BESISTANGE.
[347.
battery introduced 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 0-4,
Q_By^Cp
Fig. 82.
A^
Bb-hCc
6+c
whence the condition is g o -. ^^
where 6, 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
BCy CA and AB he x^ y and 0, and the resistances of these
conductors a, b and c. Let the currents along OA, OB and OC
be i, rj, C and the resistances a, fi and 7. Let an electromotive
force E act along BC. Required the current ^ along OA.
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:sB-C+E, a£=0--A,
by = C'-A, pri=:O^B,
cz-A-^B, yC^O-C;
with the equations of continuity
i+y-z = o,
YI + Z^X= 0,
(+x^y = 0.
By considering the system as made up of three circuits OBC,
OCA and 0-45, in which the currents axe x, y^ z respectively,
Digitized by VjOOQ IC
348.] wheatstone's beidgb. 477
and appl3dDg Eirchhoff's rule to each cycle, we eliminate the
values of the potentials 0, Ay B, C, and the currents ^, 17, f, and
obtain the following equations for x^ y and z^
(a+)3 + y)aj-yy -fiz = ^,
—yx '\- {b'^y + a)y'-az =0,
-/3a!
Hence, if we put
» » # «^
-ay +(c + a'\-p)z
= 0.
D =
a+p + y -y -i3
—y b + y + a —a
-j3 -a c + a + )3
)
we find f
= |(6^-cy),
and ^ = ;5 {(6 + y)(c + i3)+a(6+c+j9 + y)}.
848.] The value of D may be expressed in the symmetrical
form,
2) = a6c + 6c()3 + y) + ca(y + a)
+ a6(o + i8) + (a + 6 + c)(ey + ya + oi3)*
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 0 the galvanometer resistance for cu We then find
D = 5(?(6 + c+)3 + y) + 5(5+y)(c + i3)
+ (?(6+c)(i3 + y) + 6c(i3 + y) + i3y(6 + c).
If the electromotive force E were made to act along OA, the
resistance of OA being still a, and if the galvanometer were
placed in BC, the resistance of BC being stUl a, then the value
of D would remain the same, and the current in BC due to the
electromotive 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 1/
be the value of D after this exchange, we find
JD-iX = ((?-£) {(5 + c)(^ + y)-(5 + y)(i3 + c)},
= (5-©) {(6-i3) (c-y)}.
* {D if the ram of the prodnotf of the reiiftanceB taken 8 at a time, leaving out
the prodnot of any three that meet in a point.}
Digitized by VjOOQ IC
478 MEASUBEM£NT OF BESISTAXCE. [349.
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 galyano-
meter connects the junction of the two conductors of least
resistance /9, y with the junction of the two conductors of
greatest resistance 6, c, or, in other words^ we shall suppose that
if the quantities 6, c, y, fi are arranged in order of magnitude,
b and c stand together, and y and /9 stand together. Hence the
quantities 6-/3 and c—y are of the same sign, so that their
product is positive, and therefore 2)— 2/ is of the same sign as
JB-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 ex-
changed.
The rule therefore for obtaining the greatest galvanometer
deflexion 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.
849.] 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 BOC, 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
galvanometer needle occurs when the battery is made to act
between B and C.
The conductor BOC may be supposed to be a wire of uniform
resistance divided into equal parts, so that the ratio of the resist-
ances of BO and OC may be read off at once.
Listead 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 resistances of which are accurately
known.
We shall now use a different notation instead of the sym-
metrical notation with which we commenced.
Let the whole resistance of BAG be i2.
Let 0 = mR and b = (1— m)i2.
Digitized by VjOOQ iC
349-] wheatstone's bbidqe. 479
Let the whole resistance of BOC be S.
Let i3 = TiS and y = (1 -n)/S.
The value of 7i is read off directly, and that of m is deduced
from it when there is no sensible deviation of the galva-
nometer.
Let the resistance of the battery and its connexions be £, and
that of the galvanometer and its connexions 0.
We find as before
+ (m-^n—2mn)BIlS,
and if f is the current in the galvanometer wire
f, ERS f .
Li 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.
Li the present case, if d is the deviation,
where C is some constant, and 0 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 d is a maximum, the part
involving 0 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 0 to be
(? = 7i(l-n)(i2 + iS).
This result is easily obtained by considering the resistance
from .^ to 0 through the system, remembering that BC^ being
conjugate to AO^ has no effect on this resistance.
Li the same way we should find that if the total area of the
Digitized by VjOOQ iC
480
MEASUREMENT OF BESISTA17GE.
[350-
acting surfaces of the battery is given, since in this case E is
proportional to VB^ the most advantageous arrangement of
the battery is when jj^
Finally^ we shall determine the value of S such that a given
change in the value of n may produce the greatest galvanometer
deflexion* By differentiating the expression for ^ with respect
to fi^ we find it is a maximum when
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
S = -R, 5= IB, (? = 27i(l -n)li,
and if n = I, © = \R.
On the Use of Wheatstone^a Bridge.
850.] We have already explained the general theory of Wheat-
stone's Bridge, we shall now consider some of its applications.
Fig. ss.
The comparison which can be effected with the greatest
exactness is that of two equal resistances.
Let us suppose that )3 is a standard resistance coil, and that
we wish to adjust y to be equal in resistance to p.
Two other coils, 5 and c, are prepared which are equal or
nearly equal to each other, and the four coils are placed with
Digitized by VjOOQ iC
350.] USE OF wheatstonb's bridge. 481
their electrodes in mercury cups bo that the current of the
battery is divided between two branches, one consisting of fi
and y and the other of b and c. The coils 6 and c are connected
by a wire Pi2, as uniform in its resistance as possible, and fur*
nished with a scale of equal parts.
The galvanometer wire connects the junction of /3 and y with
a point Q of the wire PR, and the point of contact 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 p and y ai*e 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 /9 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 p.
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 a: is the scale reading
of Q in the first case, and y that in the second,
c-fa? _p c-^-y _ y
b'-x'^ y^ b—y^p^
whence L' - i + (^±£lfc^.
whence ^^ " ^ + (c + ^)(6-2/)
Since b—yia nearly equal to c+z, and both are great with
respect to a; or y, we may write this
and y = ^(l + 2|^).
When y is adjusted as well as we can, we substitute for b and c
other coils of (say) ten times greater resistance.
The remaining difference between fi and y will now produce
a ten times greater difference in the position of Q than with
the original coils b and c, and in this way we can continually
increase the accuracy of the comparison.
The adjustment by means of the wire with sliding contact
VOL. I. I i
Digitized by VjOOQ iC
482 MEASUREMENT OF RESISTANCE. [35 1.
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 Snvall Resistances.
851.] 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 experiments made
in the way described above.
The object of such experiments is
generally to determine the specific
resistance of the substance, and it is
resorted to in cases when the substance
PI 34 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.
* Proe. B. 8., Jane 6, 1861.
Digitized by VjOOQ iC
351.] Thomson's method foe small ebsistances. 483
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
distance 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 paraJlel 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
o
^t=^
^
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 S, S"
and r, y are to be compared.
The conductors are placed in series, and with connexions as
perfectly conducting as possible, in a battery circuit of small
resistance. A wire 8VT is made to touch the conductors
at 8 and T, and SVT is another wire touching them at S
andr.
The galyanometer wire connects the points Fand V of these
wires.
The wires SVT and SfVT 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 wirC; and
lia
Digitized by VjOOQ iC
484
MEASUBEMENT OF RESISTANCE.
[351-
F, 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
resistances of the two conductors.
Call H and F the resistances of the conductors 88" and TT.
A and C those of the branches SV and VT.
P and iJ those of the branches S'V and VT.
Q that of the connecting piece S^T.
B that of the battery and its connexions.
0 that of the galvanometer and its connexions.
The symmetry of the system may be imderstood from the
skeleton diagram. Fig. 34.
Fig. 86.
The condition that B the battery and 0 the galvanometer
may be conjugate conductors is, in this case*,
F H fR P\ Q
C^A'^^C Ah+Q+R^ '
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
F H
C^A'
and the ratio of the resistances of the conductors to be compared
would be that of C to ^, as in Wheatstone's Bridge in the
ordinary form.
In the present case the value of Q is small compared with P
or with ii, so that if we select the points F, V so that the
* {Thii nuty easily be dedaoed by the rale given in the Appendix to Chap. ▼!. }
Digitized by VjOOQ iC
352.] MATTHIESSEN AND HOOKINS's METHOD. 485
ratio of 12 to C is nearly equal to that of P to A, the last term
of the equation will vanish, and we shall have
F:H::C:A.
The success of this method depends in some degree on the
perfection of the contact between the wires and the tested con-
ductors at S, S't T and T. In the following method, employed
by Messrs. Matthiessen and Hockin^, this condition is dispensed
with.
852.] The conductors to be tested are arranged in the manner
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
TT on the second.
Two conducting points or sharp edges ate fixed in a piece of
insulating material so that the distance between them can be
accurately measured. Thia apparatus is laid on the conductor to
be tested, and the points of contact with the conductor are then
at a known distance ^^S^. Each of these contact pieces is con-
nected 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 (7, and a wire Pi2
with a sliding contact piece Q, to which the other electrode of
the galvanometer is connected.
Now let the galvanometer be connected to 8 and Q^ and let
A^ and G^ be so arranged, and the position of Q, (viz. Q^,) so
determined, that there is no current in the galvanometer wire.
Then we know that XS A +PQ
where XS, PQi, &c. stand for the resistances in these conductors.
From this we get
XS_ A, + PQ,
Now let the electrode of the galvanometer be connected to S^^
and let resistance be transferred from C to A (by carrying re-
sistance coils from one side to the other) till electric equilibrium
of the galvanometer wire can be obtained by placing Q at some
* Laboratory. MatUueasen and Hookin on Alloys.
Digitized by VjOOQ iC
486 MEASUEBMBNT OF BESI8TAN0B. [35 2.
point of the wire, say Qj. Let the values of C and A be now
C^ and A^, and let
Then we have, as before
XSr _ A^+PQ^
XY~ R
_ SS' A,-A, + Q,Q^
Whence yT~ 'k *
In the same way, placing the apparatus on the second
conductor at TT and again transferring resistance, we get,
when the electrode is in 2",
Xr_A^ + PQ^
XY~ R '
and when it is in T,
XT _ At+PQ^
XY~ R
Wbence ^^^illi^.
We can now deduce the ratio of the resistances S8' and TT,
TT- A^-A, + Q,Q,'
When great accuracy is not required we may dispense with
the resistance coils A and C, and we then find
rT-Q,Q,'
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 milli-
metre, 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 introduced 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
contacts at S, S' or T, T.
This method may be called the differential method of using
Digitized by VjOOQ iC
354-] GBEAT BB8I8TAN0BS. 487
Wheatstone's Bridge, since it depends on the comparison of
observations 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
determination. Hence the series of observations ought always
to be repeated in order to detect any change in the resistances *.
On the Comparison of Oreat Rmatancea.
358.] When the resistances to be measured are very great,
the comparison of the potentials at diffei*ent 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 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 ex-
tremities, first of one conductor and then of the other, the ratio
of their resistances may be determined.
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 galvano-
meter. 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.
854.] When the resistance of a conductor is so great that the
current which can be sent through it by any available electro-
motive force is too small to be directly measured by a galvano-
meter, 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
* {For another method of comparing small reaistancee, fee Lord Bayleigh, Pro-
ceedings of the Cambridge PhUoeophical Societjf, voL y. p. 50.}
Digitized by VjOOQ iC
488 MEASUBBMBNT OF BE8I8TAN0B. [355.
may be estimated. This is Messrs. Bright and Clark's method
of testing the joints of submarine cables.
855.] 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. I{ E is the
difference of potentials as shewn by the electrometer, S the
capacity of the condenser, and Q the charge on either surface,
J2 the resistance of the conductor and x the current in it, then,
by the theory of condensers,
Q^SE.
By Ohm's Law, E ^ Re,
and by the definition of a current.
'' = '' dt
49.
It
Hence «Q = 2J5§,
at
and Q±z Q^e ^,
where Q^ is the charge at first when ^ = 0,
Similarly E ^ E^e" ^
where Eq is the original reading of the electrometer, and E the
same after a time t From this we find
S{hg,E,^log.E}'
which gives 22 in absolute measure. In this expression a
knowledge of the value of the unit of the electrometer scale is
not required.
If S, the capacity of the condenser, is given in electrostatic
measure as a certain number of metres, then J2 is also given in
electrostatic measure as the reciprocal of a velocity.
If ^ is given in electromagnetic measure its dimensions are
-^, and iZ 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, R^, and in the second,
that of the condenser when the conductor is made to connect its
Digitized by VjOOQ iC
356.]
Thomson's method.
489
surfaces. Let this be R\ Then the resistance, J2, of the
conductor is given by the equation
JL- JL J_
R R Rq
This method has been employed by MM. Siemens.
Thomaon'a* Method for the Determination of the Redstan/^
of a Oalvanometer.
856.] An arrangement similar to Wheatstone's Bridge has
been employed with advantage by Sir W. Thomson in de-
termining the resistance of the galvanometer when in actual
Fig.87.
use. It was suggested to Sir W. Thomson by Mance's Method.
See Art. 367.
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. K i^—oy is zero, then the conductor OA is
conjugate to BC, 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
galvanometer 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
♦ Proe. H. S,, Jan. 19, 1871.
Digitized by VjOOQ iC
490 MEASUREMENT OP RESISTANCE. [357!
of small resistance, as when this connexion is broken, and if, by
properly adjusting the resistances of the conductors, we obtain
this result, we know that the resistance of the galvanometer is
where c, y, and fi 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
contact 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 re-
mains constant, in spite of certain changes which we can repeat
at pleasure, we are sore that the current is quite independent of
these changes.
The determination of the resistance of the coil of a galvano-
meter 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 em-
ployed to measure its own resistance.
Mance'a^ Method of determining the Beaistance of a Battery.
857.] The measurement of the resistance of a battery when in
action is of a much higher order of difficulty, since the resistance
of the battery is found to change considerably for some time
after the strength of the current through it 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 EC 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 ilC are conjugate. This may be regarded as a particular
* Proe, B. 8., Jan. 19, 1871.
Digitized by VjOOQ iC
357-] mange's method. 491
case of the result proved in Art. 347, or may be seen directly on
the elimination of z and j3 from the equations of that article,
viz. we then have
(aa— cy)aj + (cy + ca + c6 + 6 a) y = Ea.
If 2^ is independent of x, and therefore of j3, we must have
aa = cy. The resistance of the battery is thus obtained in terms
of 0, y, a.
When the condition aa = cy is fulfilled, the current y through
the galvanometer is given by
_ Eg _ Ey
^ " €b + a{a + b + cy "" a6 + y(a + 6 + c)*
To test the sensibility of the method let us suppose that
the condition cy = aa is nearly, but not accurately^ fulfilled^
Fig. 88.
and that y^ is the current through the galvanometer when
0 and B are connected by a conductor of no sensible resistance,
and y^ 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.
The general value for y is
D ^'
where D denotes the same expression as in Art. 348. Putting
0 = 0, we get ^ yE
^'^■~ ab + y{a + b-\-c) + c{aa—cy)
cfcy— aa)v^ . ^ ,
= 2^ + y{c + a) 'E "PPro^omately.
Digitized by VjOOQ IC
492 MEASUBBMENT OF EB8ISTAN0B, [357.
putting )3 = 00 , we get
_ E
^^ ^ . ah (aa—cy)b
y {y+^)y
^ b(cy^aa)y^
^ y{y^a) E'
From these values we find
yp— yi _ g cy-^aa
y y{<^'^o){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 + y.
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 cuiTcnt in the galvanometer, the actual current
through the battery is x^ with the key down and x^ with the
key up, where
(b olc n / o \
y yya + cy * "^^ a + y^'
the resistance of the battery is
^-^^
a = — I
a
and the electromotive force of the battery is
^ = y(6 + c+^{6 + y)).
The method of Art. 356 for finding the resistance of the galva-
nometer differs from this only in making and breaking contact
* [In the Pkilatophieal Magatine for 1877, vol. i. pp. 615-525, Mr. Oliver Lodge
haa pointed out m 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 Uie key is down or up,
if the equation aa « C7 is true. Mr. Lodffe describes a modification of Manoe s
method which he has employed with success^
Digitized by VjOOQ iC
358.]
COMPAEISON OP ELEOTBOMOTIVE FOECBS.
493
between 0 and A instead of between 0 and B^ and by exchanging
a and ^, a and 6, we obtain for this case
y y(c+i3)(^ + y)'
Oti ^Ac Comparison of Electromotive Foixea,
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 resistance, iJj, be interposed between the points A^^
B^ of the primary circuit EB^A^E, the electromotive force from
B^ to ill may be made equal to that of the electromotor E^.
If the electrodes of this electromotor are now connected with
the points A^, Bi no current will flow through the electromotor.
By placing a ^vanometer G^ in the circuit of the electro-
motor j^i, and adjusting the resistance between Ai and B^
till the galvanometer Gi indicates no current, we obtain the
equation E^=zR^C,
where iZ^ is the resistance between il, and B^, and C is the
strength of the current in the primary circuit.
In the same way^ by taking a second electromotor E^ and
placing its electrodes at A2 and B^y so that no current is
indicated by the galvanometer G29
E^ ^ It^Cf
Digitized by VjOOQ iC
494 MEASUREMENT OF EB8I8TANCB.
where iZg is the resistance between A^ and B^ If the observa-
tions of the galvanometers Q^ and &2 <^^ simultaneous, the
value of C, the current in the primary circuit, is the same in
both equations, and we find
E^ : E^ I '• Ri I R^ •
In this way the electromotive forces of two electromotors may
be compared. The absolute electromotive force of an electro-
motor 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 modi-
fication of PoggendorfTs method, and is due to Mr. Latimer
Clark, who has deduced the following values of electromotive
forces :
Gonoeotnttdd Vai^^
■olution of ^**'"-
DanteUl. Amalgamated Zinc H, SO4 + 4 aq. GUSO4 Copper b1'079
n. „ HaS04 + 12aq. CtiS04 Copper -0978
III. „ H, SO4 + 12 aq. Cu (NOj), Copper - 100
Bumenl. „ „ „ HNO, Carbon » 1-964
II. „ „ „ Bp. g. 1-88 Carbon b1.888
Grove „ H,S04+ 4 aq. HNO3 Platinum » 1.956
A Volt is an electromotive force equal to 100,000,000 unite qf the centimetre'
gr(tmme*eeeond system.
Digitized by VjOOQ IC
CHAPTEB XIL
ON THE ELEOTRIO RESISTANCE OP 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
sulphui'ets, 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 sub-
stance, 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 elec-
trolytes, because the current is associated with a decomposition
of the substance into two components which appear at the elec-
trodes. As a rule a substance is an electrolyte only when in
the liquid form, though certain colloid substances, such as glass
at 1 00°C, which are apparently solid, are electrolytes t. 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.,
* {Cftrbon is an exception to this statement ; and Feunner has lately foand that
the resistance of an alloy of manganese and copper diminishes as the temperature
• increases.} 0
t { W. Kohlransch has shown that the halVid salts of silver conduct electrolytioally
when solid, Wied. Ann, 17. p. 642, 1882.}
Digitized by VjOOQ iC
496 EESISTANCE. [360.
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 is difficult, on account of the great resistance of these sub-
stances, 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.
860.] 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 select-
ing the materials. When any fault occurs in the line, its position
is at once ascertained by measurements of resistance, and these
measurements, 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, Yogt, and Hockin,
and by MM. Siemens, who have done so much to introduce exact
electrical measurements into practical work.
It appears from the researches of Dr. Matthiessen, that the
effect of temperature on the resistance is nearly the same for a
considerable number of the pure metals, the resistance at 100°C
being to that at OT in the ratio of 1*414 to 1, or 100 to 70-7.
For pure iron the ratio is 1*6197, and for pure thallium 1-458.
The resistance of metals has been observed by Dr. C. W.
Siemens^ through a much wider range of temperature, extending
from the freezing-point to SSO^'C, and in certain cases to lOOO^'C.
He finds that the resistance increases as the temperature rises,
but that the rate of increase diminishes as the temperature rises.
The formula, which he finds to agree very closely both with the
resistances observed at low temperatures by Dr. Matthiessen and
with his own observations through a range of 1000^*0, is
* Proe. E, S., April 27, 1871.
Digitized by VjOOQ iC
36 1.] RESISTANCE OF METALS. 497
where T is the absolute temperature reckoned from — 273''C, and
a, /3, y are constants. Thus, for
Platinum r= 0-039369 r* + 0.00216407 7-0-2413*
Copper r= 0-026677 r* + 0.0031443r-0.22751,
Iron r = 0-072645 T* + 00038133 T- 1-23971.
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
constituents, it varies very little with a slight alteration of the
proportions. 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
temperature.
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
* {Mr. Callendar'8 recent researchefl in the CaTendish Laboratory on the Reditanoe
of Platinum haye ihoivn that these ezprenionfl do not aooord with the facts at high
temperatures. Siemen*8 formula for platinom requires the temperature ooeflBcient of
the resistance to become constant at high temperatures and eaual to -0021 ; while the
experiments seem to indicate a much slower rate of increase it not a decrease at Tery
high temperatures. H. L. Callendar, * On the Pnustical Measurement of Temperature^'
PkU. Trant, 178 A. pp. 161-280.}
VOL. I. K k
Digitized by VjOOQ iC
498 KESISTANCE OF SUBSTANCES. [362.
with nitric acid. Great care has been bestowed in determining
the resistance of this metal by W. and C. F. Siemens, who intro-
duced 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 mass
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 X of the tube, the middle point of which is
distant x from one end of the tube, then the area a of the section
Q
near this point will be s = y» where C is some constant
The mass of mercury which fills the whole tube is
/» 1 I
where n is the number of points, at equal distances along the
tube, where X has been measured, and p is the mass of unit of
volume.
The resistance of the whole tube is
R = f^dx = ^2{\)^y
J 8 C ^ 'n
where r is the specific resistance per unit of volume.
Hence wR = rp2 (X) 2 (-) ;^ »
- wR V?
and r = -.^ r-
"' 2(X)S(1)
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 O^C, is 13-071 B. A. units,
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 B.A. units.
362.] In the following table R is the resistance in B.A. units
of a column one metre long and one gramme weight at 0*^0, and
r is the resistance in centimetres per second of a cube of on^
Digitized by VjOOQ iC
363.] OP ELECTROLYTES. 499
centimetre^ according to the experiments of Matthiessen^ as-
suming the B. A. unit to be •98677 Earth quadrants.
Percentage
increment of
Specific resistance for
gravity. J?. r. 1*»C at 20°C.
Silver 10-50 hard drawn 0-1689 1588 0-377
Copper 8-95 hard drawn 01469 1620 0-388
Gold 19-27 hard drawn 0-4150 2125 0-366
Lead 11-391 pressed 2-257 19584 0-387
Mercuryf. . . . 13-595 liquid 13-071 94874 0-072
Gold 2, Silver 1 . 15-218 hard or annealed 1-668 18326 0-065
Selenium at 1 00*^0 crystalline form 6 x 1 0^ * 1-00
On the Electric Resistance of Electrolytes.
863.] The measurement of the electric resistance of electrolytes
is rendered difficult on account of the polarization of the elec-
trodes, 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
compared with the section of the part of the electrolyte whose
resistance 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.
* Phil. Mag., May, 1865.
\ { More recent experiments have given a different value for the specific resistance
of mercury. The following are recent determinations of the resistance in B. A. units
of a column of mercury one metre long and one square millimetre in cross section
at O^C :-
Lord Rayleigh and Mrs. Sidgwick. Phil Tram. Part 1. 1883 . . 95412,
Glazebrook and Fitzpatrick, Phil. Tram. A. 1888 .... 95352,
Hutchinson and Wilkes, Phil. Mag. (5). 28. 17. 1889 . . .95341. }
K k 9
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500 EESISTANCB OP SUBSTANCES. [064.
864.] In the experiments of Dr. Paalzow^ the electrodes were
in the form of lai*ge 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 iZ^ and R29 the siphons were next filled with mercury, and
their resistances when filled with mercury were found to be
JJ/ and B2>
The ratio of the resistance of the electrolyte to that of a mass
of mercury at O^'C of the same form was then found from the
formula ^ Ri-R^
Ri—R/
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 O^C. See
Art. 361.
The results given by Paalzow are as follow : —
Mixtures of Sulphuric Add amd Water.
Temp. Betbtance compared
with menmrj.
HjjSO^ 15^0 96960
H2SO4+ HHjjO 19^C 14157
Hj,S04+ I3H2O 22'C 13310
HjSO^+499Hs50 22'C 184773
Svlphate of Zinc and Water,
ZnS04+ 33H2O 23*'C 194400
ZnS04+ 24H2O 23°C 191000
ZnS04+ 107 Bfi 23T 354000
Sulphate of Copper and Water.
CUSO4+ 45H2O 22**C 202410
CUSO4+ 105 HjO 22^0 339341
Sulphate of Ma^nesiu/m and Water.
MgS04+ 34H2O 22'C 199180
MgS04+ 107 HgO 22T 324600
* Berlin MonaUberiokt, July, 1868.
Digitized by VjOOQ IC
366.]
HCl
HCl
OF DIELBCTBI08.
Hydrochloric Acid and Water.
Temp.
+ ISHjO 23°C
501
Besistanee oompAred
with mercary.
13626
+ 500H.O 23'*C
86679
365.] MM. F. Kohbausch and W. A. Nippoldt* have de-
termined the resistance of mixtures of sulphuric acid and water.
They used alternating magneto-electric currents, the electro-
motive force of which varied from \ to yV of *'tat of a Grove's
cell, and by means of a thermoelectric copper-iron pair they re-
duced the electromotive force to T^vVrnr of ^^^ 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 rC is given in the following table : —
Resistance of Mixtures of Svlphuric Add and
tei^ms of Mercury at 0*^0. MM. Kohlrausch
Specific gravity
at 18'5.
0-9985
1-00
1-0504
1.0989
11431
1-2045
1.2631
13163
13597
13994
1.4482
15026
Percentage
ofHaSOf.
0-0
0.2
8.3
14.2
202
28.0
352
41-5
46.0
50*4
552
60.3
Kesittanoe
at 22"C
(Hg-1).
746300
465100
34530
18946
14990
13133
13132
14286
15762
17726
20796
25574
Water at 22°C in
and Nippoldt
Percentage
increment of
conductivity
for VC.
0.47
0-47
0-653
0.646
0.799
1-317
1.259
1.410
1-674
1582
1417
1.794
On the Electrical Resistance of Dielectrics,
866.] A great number of determinations of the resistance
of gutta-percha, and other materials used as insulating media,
* Pogg., Ann, czzzviii. pp. 280, 870, 1869.
Digitized by VjOOQ IC
502 KESTSTANCE OP SUBSTANCES. [366.
ID 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 cylindrical coating of the insulator of great area and small
thickness.
It is found that when the electromotive force b^ins 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 electro-
motive 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
continues to decrease for at least half an hour, so that a
determination of the resistance deduced from the current will
give a greater value if a certain time is allowed to elapse than
if taken immediately after applying the battery.
Thus, with Hooper's insulating material the apparent resist-
ance 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 polariza-
tion, 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 Hitter'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
electromotive force acting on the series will produce a current,
as indicated by a galvanometer, which will gradually diminish
till the jars are fully charged.
Digitized by VjOOQ iC
367.] OP DIELECTRICS. 503
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 con-
nexion made between the ends of the series, a reverse current
will be observed, the total quantity of which, in the case of
perfect insulation^ will be the same as that of the direct current.
Similar effects are observed in the case of the secondary pile,
with the difference that the final insulation is not so good,
and that the capacity per unit of surface is immensely greater.
In the case of the cable covered with gutta-percha, &c., it is
found that after applying the battery for half an hour, and then
connecting the wire with the external electrode, a reverse
current takes place, which goes on for some time, and gradually
reduces the system to its original state.
These phenomena are of the same kind with those indicated
by the * residual discharge' of the Leyden jar, except that the
amount of the polarization is much greater in gutta-percha, &c.
than in glass.
This state of polarization seems to be a directed property
of the material, which requires for its production not only
electromotive force, but the passage, by displacement or other-
wise, of a considerable quantity of electricity, and this passage
requires a considerable 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 conduction 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
discoveries 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 O^'C as at 24''C. Messrs. Bright and Clark have
found that the following formula gives results agreeing with
their experiments. If r is the resistance of gutta-percha at
temperature T centigrade, then the resistance at temperature
T+^ will be R-TxO,
Digitized by VjOOQ iC
504 KESISTANCB OP SUBSTANCES. [369.
where C varies between 0-8878 and 0-9 for different specimens of
gutta-percha.
Mr. Hockin has verified the curious fact that it is not until
some hours after the gutta-percha has taken its final 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 lO^* to •362x 10^«
Malta-Alexandria 1-23 xlO^*
Persian Gulf 1-80 xlO'*
Second Atlantic 342 x lO^*
Hooper's Persian Gulf Core 74-7 x 10^«
Gutta-percha at 24*^0 3.63 xlO^^
868.] The following table, calculated from the experiments of
M. Buff, described in Art. 271, shews the resistance of a cubic
metre of glass in Ohms at different temperatures.
Temperatare. Beflistanoe.
200**C 227000
260** 13900
300° 1480
360° 1035
400° 735
869.] Mr. C. F. Varleyt has recently investigated the con-
ditions of the current through rarefied gases, and finds that
the electromotive force E is equal to a constant E^^ together with
a part depending on the current according to Ohm's Law, thus
Er^E^-\-RC.
For instance, the electromotive force required to cause the
current to begin in a certain tube was that of 323 Daniell's
cells, but an electromotive force of 304 cells was just sufficient
to maintain the current. The intensity of the current, as
measured by the galvanometer, was proportional to the number
* Jenkm's Cantor Leetares. f Proe. R, 8,, Jan. 12, 1871.
Digitized by VjOOQ iC
370.] OP DIELECTRICS. 505
of cellB 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 160, 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
ifl equal to that of 304 Daniell's cells, and that up to this
electromotive 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 rai*efied 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 ^ ^^tlrf <2L
results, that the electromotive force required to produce a /-y^**^
spark in air was found to be proportional not to the dis-
tance, but to the distance together with a constant quan-
tity. The electromotive force corresponding to this constant
quantity may be regarded as the intensity of polarization of the
electrodes.
370.] MM. Wiedemann and Biihlmann have recently * investi-
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 i-arefied gas. The discharge was in general dis-
continuous, and the interval of time between successive dis-
charges 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 tUl 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.
* JBeriehie der Konigh 8&chi. GegellseJutft, Leipzig, Oct. 20, 1871.
Digitized by VjOOQ iC
506 EESISTANCE OP SUBSTANCES.
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 phe-
nomenon, and they indicate tiiat this condensation is greatest at
the positive electrode.
♦ Exp. He*., 1601.
END OF VOL. I.
Digitized by VjOOQ IC
PLATES.
Vol, I.
Digitized by VjOOQ IC
Digitized by VjOOQ IC
derk. McutyweHs El&ctrici^ ^ol.I.
FIG . I.
Art .118.
Linens of Fcnrce^ ariEL EcnupoterdiaL Stcrfcuces.
A ' 20. B ' 5 . P, Fcfint of Etjidtibrww. AP ' iAB
Unwersi^ Pre^ss, Oxfordu
Digitized by VjOOQ IC
Digitized by VjOOQ IC
CLerhMajy^^eUJ^ Electrioay. Vol.1.
FIGH
ArtH9
Lines of Force aiidy EquzpotervtiaL Surfaces.
A -20 B "-5 F.PoimofEqujibbrwni.
^> , Spheruuii suHcuce of Zej^o potentUd
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Cleric Maxi^veWs Eltcirictty,Voh.I.
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Lines of FcfTcc aruL EqidpotentiaL srurfcuce^
A '10.
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Clerk- 'Maa^yyelUs KUcti^id^. Vol.1.
FIG.IV^.
Art 121.
Line^ of Farce arid^ EoidpotentixiL Suj^aces .
A-15.
B'lZ.
C'20.
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7 Vr? J -«»»-.-,•*.. P«--^.. n-r^rtr^A.
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OerhMaxi^^dLs Electridfy, Vol.1.
Lines of Force and^ EquipotentiaL Surfaoca irt a cUainetral
section of a sphericaZ Surface in whidi the suparficial densify
is a hoOTnanio of the first degree^ .
UroA^ersay Ffiess, (Xx^ord,
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ClerhldcLOOi/^elL's Electria^, Vol.I.
FIG .VI.
Art .143.
Spherical Harmonic ofthe^ third order.
I J nix e*^su\ Pre ss (iT-Kj ^A
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Art.H3.
Spherical' Hamwruc ofth& third order,
n ' 3
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Vruxersity Press . Ojcford
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Clerh MaocwdL's Electria^ Vol.1.
FlGVm.
Art.143.
SpheJixxJ, Harmonir^ of the fourths ordber
71-4 O'Z
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FIG. X.
Art. 192.
ConfocdL Ellipses cmdHyperbolcLS
Umya^sv^ Press . Oxford
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FIG. XI.
Art.193.
Lirus of Forces ixuoar the edge ofcuPlate .
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C2jerk, MojcavMs Electric^, Vol.1.
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Lines ofForce^ between twoPIates
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Clerk Maxwell's Eleetria^ Vol.1.
FIG XIH
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