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THE THEORY

OF

ELECTRICITY



CAMBRIDGE UNIVERSITY PRESS
C. F. CLAY, MANAGER



LONDON
FETTER LANE, E.G. 4




EDINBURGH

100 PRINCES STREET



NEW YORK : G. P. PUTNAM'S SONS
BOMBAY, CALCUTTA, MADRAS: MACMILLAN AND CO., LTD.

TORONTO : J. M. DENT AND SONS, LTD.
TOKYO: THE MARUZEN-KABUSHIKI-KAISHA



All rights reserved



THE THEORY

OF

ELECTRICITY



BY
G. H. LIVENS, M.A.

SOMETIME FELLOW OF JESUS COLLEGE, CAMBRIDGE
LECTURER IN MATHEMATICS IN THE UNIVERSITY OF SHEFFIELD



Cambridge :

at the University Press
1918






THIS VOLUME IS IN GBATITUDE DEDICATED TO

THE MASTER AND FELLOWS OF JESUS COLLEGE

CAMBEIDGE





CORRECTIONS AND ADDITIONS.



pp. 39, 158, 661. Until quite recently I have not had an opportunity of examining Einstein's
generalised theory of relativity and was in consequence unable to take up any other
point of view as regards gravit&tional phenomena.

p. 84, 88. The argument of the e^jponential function under the integral sign in the formulae
for if) should read ( - kz) not ( - kr).

p. 338. The values for the electronic charge givi on this page are those first obtained by the
method under discussion: subsequent improvements and corrections have increased
this value by about .Ml ",, to that '_ f iven on p. .'541 and in other parts of the work.

p. 345, 388. In the expression for the potential of a magnetic shell the solid angle is measured
by that area of the unit sphere round the field point which is such that if it were coated
with a double sheet with a similar <][*, osition to that on the given shell, then the direction
of tin- outwap nljus at ajr.\ point would correspond to the positive direction of

magnetisation in this






CONTENTS

MATHEMATICAL INTRODUCTION

STATICS AND KINETICS

CHAP.

I. THE PRODUCTION AND DEFINITION OF THE ELECTROSTATIC FIELD .

II. THE CHARACTERISTIC PROPERTIES OF THE ELECTRIC FIELD

III. THE ELECTRICAL AND MECHANICAL RELATIONS OF A SYSTEM OF CONDUCTORS

IV. THE FARADAY-MAXWELL THEORY OF ELECTROSTATIC ACTION .

V. THE THEORY OF POLARISED MEDIA

VI. MAGNETO -STATICS . .

VII. ELECTRIC CURRENTS IN METALLIC CONDUCTORS

VIII. ELECTRIC CURRENTS IN LIQUID AND GASEOUS CONDUCTORS

DYNAMICS

IX. THE ELECTROMAGNETIC FIELD . . 3$|H

X. SOME SPECIAL ELECTROMAGNETIC FIELDS .

XI. ELECTRODYNAMICS OF LINEAR CURRENTS . ' .

XII. ELECTROMAGNETIC OSCILLATIONS AND WAVKS; .

XIII. REFLEXION, REFRACTION AND CONDUCTION OF ELECTRIC WAVES

XIV. GENERAL ELECTRODYNAMIC THJEORY

XV. THE ELECTRODYNAMICS OF MOVING MEDIA

APPENDIX: EXAMPLES ....
INDEX OF NAMES . ...
GENERAL INDEX . * .




PAGE

1



33
66
126
156
188
235
279
324



343
371
408
459
515
549
614




Ls-

PHY. .,
UBisARY



PREFACE

nnHE following work is offered as a general text book on the mathematical
- aspects of modern electrical theory, and incidentally also as an attempt
to present the complete subject in a consistent form. There seems room
for a comprehensive work of this kind, for in the standard text books on
this subject the treatment, besides being incomplete, is often far from
convincing and at times not free from error.

The treatment is based mainly on the original Faraday-Maxwell theory,
generalised and extended to the case of moving systems by Sir Joseph
Larmor. This form of the theory has been almost entirely abandoned in
recent accounts of the subject, but it remains the only one which appears
to be completely satisfactory from the point of view of mathematical and
physical consistency, and in its generality it is unapproached by any other
form.

Although the present exposition is essentially a mathematical one, much
of the purely analytical mathematics usually associated with the subject
has been omitted. Particular attention has however been given to the
rigorous formulation of underlying physical principles and to their transla-
tion into a mathematical theory. The dynamical aspects of the subject
have been specially emphasised throughout.

In the development of the general plan and of the details of the book
I have derived great assistance from my notes of lectures delivered by
Sir Joseph Larmor at Cambridge during the academical year 1909-10,
afterwards supplemented from his various published works, more particularly
the papers, ' On a dynamical theory of the electric and luminif erous medium '
[Phil, Trans. 1894-1897] and the book, Aether and Matter [Cambridge, 1900].
I am extremely grateful to Professor Larmor for his kind permission to
make free use of these notes.



405544



vi Preface

For assistance in the preparation of the work I am indebted also to the
various other standard works on this subject. In addition to the essential
Treatise on Electricity and Magnetism of Maxwell I -may mention especially :
Recent Researches on Electricity and Magnetism,, Oxford, 1893, and Element*
of the Mathematical Theory of Electricity and Magnetism, 3rd ed., Cambridge,
1904, by Sir J. J. Thomson; The Theory of Electricity and Magnetism, Cam-
bridge, 1907, by J. H. Jeans; Modern Electrical Theory, 1st ed., Cambridge,
1907, by N. R. Campbell; Electrical and Optical Wave Motion, Cambridge,
1915, by H. Bateman; The Electron Theory of Matter, Cambridge, 1915, by
0. W. Richardson ; Das elektromagnetische Feld, Leipzig, 190Q, by E. Cohn ;
Die Theorie der Elektrizitdt, 2nd ed., Leipzig, 1907, by M. Abraham and
A. Foppl; The Theory of Electrons, Leipzig, 1910, by*H. A. Lorentz; and
finally the various appropriate articles in the Encyclopaedia Britannica and
Die Encyklopadie der mathematischen Wissenschaften, Bd. v.

My friend Mr H. Spencer Jones, Chief Assistant at the Royal Observatory,
Greenwich, has laid me under the deepest obligation by his generous help
in the production of this book. At a time of great pressure in his own work
he kindly offered to read all the proofs and his criticisms and suggestions
thereon have been of the greatest service to me.

I have lastly to offer my thanks to the officials of the University Press
for their kindness and courtesy in all matters concerning the printing.

G. H. L.

SHEFFIELD,

July llth, 1917.



MATHEMATICAL INTRODUCTION

1. Scalars and Vectors. We shall find it most convenient and certainly
conducive to greater mathematical precision to adopt the system of notation
and analysis which is now associated with the word ' vector-analysis *.'
Owing however to the fact that there is as yet no suitable work of reference
in this subject accessible to the average English student it was deemed
advisable to outline in this introduction the main propositions of this analysis
so far as it is needed for our future work. We may also take the opportunity
of developing in full certain complex mathematical formulae, the derivation
of which in the text would hinder the progress of the reader through the train
of physical reasoning to be there exposed.

In physics we distinguish between two kinds of quantities which we now
usually designate as scalars and vectors. A scalar quantity is completely
determined by the number expressing the ratio of its magnitude to that of
a definitely chosen unit. A vector, on the other hand, has in addition to a
numerical value, also a definite direction, so that vectors are distinguished
from one another not only by their numerical values but also by their direction ;
the numerical value is described as the magnitude of the vector.

Following the usual methods we shall always find it convenient to represent
graphically a vector by a straight line, whose length on a definite scale is equal
to the magnitude of the vector and whose direction and sense (considered as
drawn from one end, the origin, to the other) are the same as of the vector.

Throughout this book we shall always use thick letters of the Clarendon f
type, e.g. A, P, c, n, etc. to represent vector quantities, scalar quantities being
expressed by letters of the ordinary Latin or Greek types. The magnitude
of a vector may occasionally be denoted by the corresponding Latin letter,
e.g. A, P, c, n, etc., or in the usual notation of function theory by enclosing
the vector thus A ; v . We shall always represent a line or curve by s,

* First systematically expounded by Hamilton, Elements of Quaternions. See also Kelland
and Tait, Introduction to Quaternions; Tait, Elementary Treatise on Quaternions; E. B. Wilson.
Vector Analysis founded on the lectures of J. W. Gibbs. The treatment given in the text is on the
lines followed by W. von Ignatowsky, Die Vektoranalysis (Leipzig, 1910). A very illuminating
critical exposition is given by Burali-Forti and Marcolongo, Elements de calcul veclorielle (Paris,
1910).

t This is the notation employed by Heaviside, Electromagnetic Theory (London, 1891-93)
and subsequently by Gibbs, Lorentz and others.

L. 1



2 Mathematical Introduction [1, 2

a surface by / and a volume by v, differential elements of these quantities
being respectively denoted by ds, df and dv; it is to be noticed that the
element ds of a curve may also be required with its direction, it will then
be denoted in the vector sense by ds. In the case of a surface we shall often
have to consider the normal to it ; this is denoted by n. It is always to be
drawn towards a definite side and we shall agree to draw it towards the
outside if we are dealing with a closed surface.

The geometrical space with which we deal will generally be referred to
a system of rectangular coordinate axes with the right-handed screw con-
ventions more usual in such expositions. The coordinates (x, y, z) of any
point of space are then the distances measured parallel to the perpendicular
axes from the respective coordinate planes. In certain special problems
however we shall find it convenient to adopt other types of coordinates. We
shall in fact use both the spherical polar and cylindrical polar coordinate
systems. In the former system the coordinates of a point are its distance
r from a fixed pole and the declination and azimuth angles 6 and </> of this
distance referred to fixed polar axis and azimuth plane through the pole.
In the cylindrical or columnar system the coordinates of a point are its
distances (z, r) from a fixed plane and from a fixed axis perpendicular to the
plane and the azimuth angle 6 of the latter distance.

2. Elementary vector operations, (i) Addition and subtraction of
vectors : suppose we are given two vectors A and B ; at the end of the line
representing A draw a line repre-
senting B. We then say that the
sum of the two vectors A and B is
the vector C which is represented by
the line joining the origin of A to the
end of B, and in this sense. We
express this by the equation

A + B = C.

Fig 1
The difference (A B) between two

vectors A and B is defined as the vector D which is such that

D + B = A.
This involves the conclusion that the sum of the vectors B and ( B) is zero

and also that

(A - B) = A + (- B)

since each of these is equal to the vector

(A - B) + B + (- B).

We have thus sufficient rules for the addition and subtraction of vectors ; we
see at once that the ordinary commutative and associative laws of algebra
are true for vector quantities as well as for scalar.




2, 3] Mathematical Introduction 3

It is also quite clear that we may regard a vector as the sum of any
number of other vectors. In particular we can represent a vector A as the
sum of three vectors whose directions are not all parallel to one plane ; such
a method of resolution is of importance because the knowledge of three such
vectors is sufficient to completely determine the vector A : the magnitudes
of three such vectors are called the components of the vector A along the
three directions respectively. If we take the three directions mutually
perpendicular to one another and respectively parallel to the axes of a rect-
angular coordinate system, then the three components are denoted by A x ,
A y , A. respectively.

All this will be quite familiar to the student who is acquainted with the
elementary ideas of statics and kinetics and it is not necessary to dwell at
any greater length on the significance of the definitions thus involved.

We now introduce the fundamental conception of the unit vector. The
unit vector in any direction is that vector whose magnitude is unity, so that
if we denote it by u, any other vector A is equivalent to the vector An ;

f A = Au.

If we use x, y, z for the unit vectors along the directions of the coordinate
axes, we get from the rule for the addition of vectors

A = A x x + A v y + A z z;

and similarly for any other vector B

B = B w x + B^y + B 2 z,

so that

A + B = (A, + B,) x + (A + BJ y + (A, + B z ) z

and generally the component of the geometric sum of any number of vectors
in any definite direction is equal to the algebraic sum of the components of
the separate vectors in this direction.

3. (ii) Multiplication of vectors; scalar products. From considerations of
the rules of the previous paragraph and of the fundamental concepts under-
lying the idea of a vector it is easy to deduce the general rule for the
multiplication of a vector A by a scalar quantity a, the resulting vector B
defined by

B = aA

being such that its direction is the same as that of A, but its magnitude
a times as large. Thus also

ah . A = a . &A = b . aA
and also

a (A + B) = aA + aB,

(a + 6) A = aA + 6A,
so that the ordinary rules of algebra are still applicable.

12



4 Mathematical Introduction [4, 5

4. The scalar product. We must next define the scalar product of two
vectors A and B. This is defined as the scalar quantity whose value is equal
to the product of the magnitudes of the two given vectors multiplied by the
cosine of the angle included between their directions. We usually denote
this quantity by (A, B) so that

(A, B) - A . B cos AB,

where AB denotes the angle between the positive directions of the vectors
A and B. We deduce then that

(A, B) - (B, A),
(A + B, C) = (A, C) + (B, C),
(A, A) - A 2 = (A 2 ).

A

If A, B are at right angles, then cos AB = and thus

(A, B) - 0.

From this we deduce the important properties of the fundamental unit
vectors x, y, z, viz. (xy) _ (yz) = (o) = Q

and (x 2 ) = (y 2 ) = (z 2 ) = I.

Thus (A, B) = (A^x + A,y + A z z, B x x + E y y + B 2 z)
= A X B X + A y By + A 2 B.j,

expressing the vector products of two vectors in terms of their components.

5. (iii) The vector product. Suppose we are given a plane surface element
df : call one side of it the positive side and the other side the negative. On
the positive side draw a unit vector n x normal to the surface

through its mean centre. This surface element will be bounded
by a curve s and we choose the sense in this curve so that the
positive direction of its description corresponds to the direction
of the vector iij in the same way as translation to rotation in
a left-handed screw (Fig. 2). We can now specify this surface
element as a vector df whose magnitude is df and direction that

ofn *' thus tt=M/.

Such a vector is called an axial vector as it has associated with

Fig. 2

it a definite sense round its direction as axis.

Since the sum of any number of vectors is still a vector we may have
associated with any unclosed surface / a vector C defined by

C = I df,



-I,



the integral being extended over the whole of the surface/. In the particular
case in which the surface / is the parallelogram whose sides are the two



Mathematical Introduction



vectors A and B, this third vector C is called the vector Drodnct of the two
vectors A and B. The implied direction about the axis of C is ?iven by the
order in which the vectors A and B are taken and in the standard case
exhibited in the figure and in which A is taken first

C - nAB sin AB,

the angle AB being that described in rotation round the axis C from the
plane (CA) to the plane (CB) in the positive direction. This is usually

expressed in the form

C - [A, B] .



f




We see at once that

and

And thus also



Fig. 3

[A, B]= -[B, A], [A, A] =

[A + B, D] = [A, D] + [B, D].

[x, y] - z, [y, z] = x, [z, x] = y,

[x, x] - [y, y] = [z, z] - 0.



E y y + B 2 z]
- A X E Z ) -f z



Therefore also

[A, B] - [A x x + A^y + A z z,

= x (A,B S - A z E y ) + y
or in determinant form

[A, B] = x, y, z

Z J Ay , Aj.

Z , "By , Bj,

The components of the vector product are the minors of x, y, z and are
denoted by [A, B] x , etc.



6 Mathematical Introduction [6, 7

6. (iv) Products of three vectors. There are several 'important results
involving three vectors which it would be as well to set out in detail here,
for future reference.

(a) The product of a vector and the scalar product of two other vectors ;
A (B, C).

Since (B, C) is a scalar quantity the product is a vector parallel to A
whose magnitude is equal to ABC cos BC.

(b) The scalar product of a vector and the vectorial product of two
other vectors; (A, [B, C]).

This is equal to
A x [B, CL + A, [B, C], + A z [B, C] z = A, (B,C, - B.C.) + .'.. + ...

A A A

**a;> *i/5 **z >

j ft ri ri

so that (A, [B, C]) - (B, [C, A]) - (C, [A, B])

and each is equal to the volume of the pa allelepiped whose three edges are
A, B, C.

(c) The vector product of a vector and the vector product of two other
vectors ; [A, [B, C]] . Call this the vector D, then its cc-component is

D a = A, (B.C, - B,C) - A 2 (B.C. - B.C,)

= B K (A, C) - G x (A, B).
Thus D = B (C, A) - C (A, B).

7. Differentiation of scalars and vectors. In the general application
of the present analysis we shall have continually to deal with what are
called scalar and vector fields. That is instead of single isolated scalars and
vectors we have to discuss infinite groups of these quantities inasmuch as
each point of a finite or infinite space has associated with it certain scalar
and vector quantities the magnitudes, and in the latter case also the directions
of which in general vary continuously not only from point to point in the
field but also in time. In such cases the rates of variation of the magnitudes
of the scalar quantities and both the magnitudes and directions of the vector
quantities are matters of fundamental importance. We must therefore
establish rules for the differentiation of scalar and vector quantities with
respect to their scalar (time) or vector (space position) arguments.

The differentiation of a scalar quantity with respect to a scalar argument
follows the usual rules of the calculus and need not now be discussed.

The rules for the ! differentiation of a vector function with respect to a
scalar argument are easily established. Suppose that a vector A regarded



7, 8] Mathematical Introduction 7

as a vector function of the scalar quantity t undergoes a small variation and
changes to the slightly different vector A' corresponding to the value t + 8t
of the argument t ; the vector

SA = A' - A

is the differential variation of the vector A with respect to the argument t ;
8A of course refers to a variation not only of the magnitude of A but also of

dA
its direction. The differential quotient -j~ of the vector A with respect to

the scalar quantity t is now defined as the vector represented by the limiting

value of the quotient

SA

8t '

when 8t is made indefinitely small. The implied existence of a limit to this
quotient is involved in the definition. *

Our rules for vector operation are identical with those of ordinary algebraic
analysis so that it is evident that we may employ the elementary rules of
the differential calculus for the differentiation of vectors and functions of
vectors. Thus if 6 is any scalar function of the variable t

, 6A 7 dA . db

d "IT* - = b j7 + A j7 >
at dt at

dA dA dt

and also -T- = -,.- j- .

da dt da

We can similarly deduce that



and also

d FA BI

^ [A - B]

and many other results are directly deducible.

8. We have next to discuss the differentiation of scalar and vector
quantities with respect to other vectors: the most frequent operation of
this nature and the only one with which we shall now deal involves the
differentiation of a scalar or vector quantity along a line element ds, the
result depending essentially on the direction of this element.

The rate of increase of any scalar quantity </> along the line element ds
is as usual



x d(f> dy d(f> dz
ds dx ds dy ds 'dz ds '

and this may be regarded as the component along ds of the vector

dd> dd> d<b

A = x ~r + y a + z -f
ox oy oz



8 Mathematical Introduction [8, 9

The components of the vector A in the three principal directions are

(A,, A,, A z )
so that



A = Y/ +

V

and its direction is parallel to the direction

dcf> d(f> d(f>

dx ' dy ' dz'
We call this vector A the gradient of the scalar and express the relation in

the form

A = grad </>,

or A = V</>,

where V is the Hamiltonian vector operator

a a a

V = x^ + y^ + z^-.
ox dy oz

As we shall see later the great importance of this latter form of operator lies
in the fact that V may be treated by the ordinary rules of this analysis just
as if it were a vector with components

v -1 v -1 v-1

x dx' * ty' dz'

The gradient of a vector quantity B may also be expressed in the same
way but the result is rather different and is better approached in the indirect
way through the important analytical theorem of the next section.

9. Green's Lemma. Let A x be any function of the three rectangular
coordinates (x, y, z) which is continuous inside any finite space v. The region
v is bounded by the closed surface / which has at each point of it a definite
normal n (or if we wish to imply direction we use a vector n). Now consider

the integral

fdA x , fl[dA x , , ,
-~^ dv= -s- 2 dxdydz

J dx JJJ ox

taken throughout the whole space v. If we assume that a line parallel to
the x-axis cuts the surface /in two points only, then integration with respect
to x gives

dydz [ yrr - dydz (A x " - A^},

J x' OX

where A x ", A x ' are the values of the function A x at the points (x r , y, z), (x", y, z)
on the surface/. The surface element dydz in the Oyz plane is the common
projection of the surface elements df, df" which a small prism with its axis
parallel to the x-axis cuts on the surface. Since the normal n' forms an
obtuse angle with Ox and n" an acute angle, we have
dydz = - n lx 'df - + n lx "df"



9] Mathematical Introduction 9

n/ and ii/' denoting respectively the unit vectors along the normals n' and
n". Thus

dydz I -~ dx = + A x 'n lx 'df + A x "n lx "df".

, If the surface / is more complicated so that a line parallel to the se-axis
cuts it in more than two points x', x", x'", ..., these points will be alternately
points of entry and exit from the space v. The number is even and the values
of n la /, n lC ", n la /", ... are alternately negative and positive and thus

dydz f X d ^ x dx = dydz (- A x ' + A x " - A x '" + ...)

^-< I 7 f



flffi


.-1


-v



Fig. 4

If now we take the sum over all the elements dydz every element df of the
surface occurs once in the total sum and thus we get

O^jl -i / i /



the second integral being taken over the whole of the closed surface /.
Similar formulae can be obtained by changing x into y and z respectively :
combining them together we have Green's Lemma*
- dA v dA.
+ dy + 3?

* 'An Essay on the Application of Mathematical Analysis to the Theory of Electricity and
Magnetism' (Nottingham, 1828) reprinted in Mathematical Papers (Paris, 1903).



10 Mathematical Introduction [9-12

expressing a surface integral taken over a closed surface as a volume
integral taken through the volume enclosed. A y , A z are any other functions
of (x, y, 2).

10. Let us firstly suppose that

A X = Ay = A, = <f>

we then see that our theorem proves that

f V<f>dv = I n l( f>df,



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