AN INTRODUCTION
TO THE
MATHEMATICAL
THEOKY OF ATTRACTION.
AN INTRODUCTION
TO THE
MATHEMATICAL
THEORY OF ATTRACTION.
BY
FRANCIS A. TARLETON, Sc.D., LL.D.,
FELLOW OF TRINITY COLLEGE, AND LATE PROFESSOR OF NATURAL PHILOSOPHY IN THE UNIVERSITY OF DUBLIN.
VOL. II.
LONGMANS, GREEN, AND CO.,
39 PATERNOSTER ROW, LONDON, NEW YORK, AND BOMBAY.
1913.
BY PONSONBY & GIBBS.
SRLF
UBfi
Qfi
PREFACE
MY time for some years was so much occupied by administrative duties in Trinity College that I was unable to make any attempt to carry out the intentions stated above fourteen years ago in the Preface to the first volume of this treatise.
I have now to some extent accomplished what I then proposed. I came, however, to the conclusion that a chapter on Conjugate Functions was not suited for such a treatise as the present, and that to a student having a limited amount of time at his disposal some account of Maxwell's Theory of Light would be more interesting and instructive. This theory is not of course part of the Theory of Attraction, but is so intimately connected with the properties of magnetized bodies, electric currents, and dielectrics treated of in the present volume that its introduction does not seem unsuitable.
I should recommend a student reading this book for the first time to omit the whole of Chapter VIII after Article 146.
vi Preface.
Of the more recent developments of the electro- magnetic theory of light I have not attempted to give any account. So -far as I can judge some of these rest on insecure foundations. I imagine, however, that before studying the most recent investigations a preliminary knowledge of Maxwell's theory is required, and I trust, therefore, that my chapter on the subject will not be entirely useless to the student.
I have to thank Mr. S. B. Kelleher, F.T.C.D., for his kindness in reading the proof-sheets of this book, and furnishing me with many valuable corrections.
FRANCIS A. TARLETON.
TRINITY COLLEGE, DUBLIN. April, 1913.
TABLE OF CONTENTS.
CHAPTEE VIII.
SPHERICAL AND ELLIPSOIDAL HARMONICS. SECTION I. — Spherical Surfaces,
Page
Expansion of Potential in Series of Solid Harmonics, .... 1
Laplace's and Legendre's Coefficients, 2
Complete Spherical Harmonics, 3
Application of Spherical Harmonics, 7
Legendre's Coefficients, 15
Spherical Harmonics, 18
Laplace's Coefficients, 20
Reduction of a Function to Spherical Harmonics, 24
Methods of forming Complete Solid Harmonics, 26
Incomplete Harmonics, .34
SECTION II. — Ellipsoids of Revolution.
Solutions of Differential Equation. Prolate Ellipsoid, ... 39
Determination of the Function Qn, 41
Analogues of Tesseral Harmonics, 47
Expansions for Potential, 50
Surface Distribution, 51
Homceoids and Focaloids, 52
Oblate Ellipsoids, 54
Analogues of Tesseral Harmonics, 56
Expression for Potentials, 56
Surface Distribution, 57
Homoeoid and Focaloid, . 58
^ Contents.
SECTION III.— Ellipsoids in general.
Page Ellipsoidal Harmonics, .
Ellipsoidal Harmonics which vanish at infinity,
Ellipsoidal Harmonics expressed as Functions of Cartesian Co-ordinates,
Surface Integral of Product of Harmonics
Identity of Terms in equal Series,
Surface Distribution producing given Potential,
Potential of Homceoid,
Harmonics of the Second Degree in the Coordinates, .... Reduction of Solid Harmonic of the Second Degree,
Potential of Focaloid,
Components of Attraction of Solid Ellipsoid, 77
Potential of Solid Ellipsoid in External Space,
Potential of Solid Ellipsoid in its Interior, 79
CHAPTER IX.
MAGNETIZED BODIES. SECTION I. — Constitution and Action of Magnets.
Magnet of Finite Dimensions 81
Potential of Magnetized Body, 82
Poisson's Equation, 83
Examples of Magnetized Bodies, 83
Expression for Potential as Sum of Force Components, .... 86
Magnetic Force and Induction, 87
Energy due to Magnet, 89
Energy of Magnetic System, 90
Vector Potential of Magnetic Induction, 90
Stokes's Theorem, 91
Determination of Vector Potential, 92
Magnetic Moment and Axis of Magnet. 95
Magnetic Shell, 97
Energy due to Magnetic Shell, .... .100
Contents. ix
SECTION II. — Induced Magnetism.
Magnetic Induction, 102
Magnetism due altogether to Induction , 102
Distribution of Induced Magnetism, . 103
Body Magnetically Anisotropic, 105
Ellipsoid in Field of Uniform Force, 105
SECTION III. — Terrestrial Magnetism.
Earth's Magnetic Potential, 107
Locality of Sources of Earth's Magnetic Force, 108
Earth's Magnetic Poles, 109
CHAPTER X.
ELECTEIC CPEKENTS.
Properties of Electric Currents, 110-
Solenoids, 112
Equivalence of Electric Circuit to Magnetic Shell, . . . .113
Magnetic Potential of Electric Current, 114
Energy due to Electric Current, 115
Force exerted by Current on Magnet-Pole, 116
Energy due to Mutual Action of Currents, 118
Forces between Electric Circuits, 118
Force on Current Element in Magnetic Field, 121
Force exerted by Closed Circuit on Element, 122
CHAPTER XI.
DlELECTEICS.
Influence of Medium, 123
Electric Displacement, 124
Energy due to Electric Displacement, . . . • . . . . 1 25
Conductors and Currents, 125
Distribution of Displacement, . 125
x Contents.
Page
127 Charge on Conductor
Displacement due to Electrified Sphere
Energy due to two small Electrified Spheres 1
Force between Electrified Particles,
Distribution of Electromotive Intensity
Distribution of Electricity on Conductors, .... Conditions at Boundary between Dielectrics, . Attraction on Dielectric,
Crystalline Dielectric,
Differential Equation for Potential '
Distribution of Electricity on Conductors, Energy expressed as Surface Integral, . Energy due to Electrified Particle, System of Charged Conductors,
Force on Electric Particle, 1
Potential due to Spherical Conductor, .
Force due to Spherical Conductor,
Force due to Spherical Particle . . 142
CHAPTER XII. ELECTROMAGNETIC THEORY OF LIGHT.
Energy of Current in Magnetic Field. .144
Energy and Electromotive Force, 145
Maxwell's Theory of Light, 147
Magnetic Induction and Electromotive Intensity, 149
Current Intensity and Magnetic Force, 149
Equations of the Electromagnetic Field, 150
Solution of Equation of Propagation, 151
Direction of Displacement in Isotropic Medium 153
Magnetic Force in Isotropic Medium, 154
Crystalline Medium, 155
Wave-Surface, 157
Wave-Surface for Crystalline Medium, 157
Magnetic Force, 161
Electromotive Intensity, 162
Conditions at a Boundary 162
Contents.
XI
Page
Propagation of Light, . . 165
Keflexion and Refraction, 165
Common Light and Polarized Light, Igg
Intensity of Light, 168
Energy due to Electromagnetic Disturbance, 170
Quantities to be determined in Reflexion and Refraction, . .172
Reflexion and Refraction. Isotropic Media, 172
Reflexion and Refraction. Crystalline Medium 176
Uniradial Directions, 178
Uniaxal Crystals, 178
Uniaxal Crystal. Reflexion and Refraction, 180
Reflexion and Refraction at Interior Surface of Crystal, . . .184
Singularities of the Wave- Surf ace. .... 187
Total Reflexion, ' 191
Absorption of Light, 193
Electrostatic and Electromagnetic Measure, 196
NOTE ON THOMSON AND DIKICHLET'S THEOREM, . . 199
THE
MATHEMATICAL THEOKY OF ATTRACTION.
CHAPTER VIII.
SPHERICAL AND ELLIPSOIDAL HARMONICS.
SECTION I. — Spherical Surfaces.
137. Expansion of Potential in Series of Solid Harmonics.— It was shown in Art. 78 that the potential V at a point P, more distant than any point in the attracting mass from the origin, can be expanded in a series of descending powers of r, where r denotes the distance of P from the origin.
In this case, the series for the potential is of the form
where M denotes the attracting mass, and P,, F2, &c., are functions of 0 and 0, the angular coordinates of P, and of constants depending on the attracting mass, but independent of the position of P.
Since V2 V = 0 for all positions of P outside M, the coefficient of each power of r in V2 V must vanish separately, and therefore
Using for V2 the expression given, equation (17), Art. 48, we obtain
2 Spherical and Ellipsoidal Harmonics.
If Y be a function of 0, and tf> satisfying (1), it is easily seen that Va(r"F) = 0; accordingly, if Ptt = r"Fw, we have yz yn = o? aud ^ is a homogeneous function of x, y, z of the degree n satisfying Laplace's equation. Such a function is called a spherical solid harmonic of the degree n.
It appears from what has been said that if Vn denote a solid harmonic of the degree », then rP"") Vn is also a solid harmonic whose degree is - (» +1).
The function -£ is termed a spherical surface harmonic of
the degree n, and is what has been denoted above by F,,. In the present case, by considering the expression from
TTT
whose expansion -^ was obtained, it is easy to see that Vn
is a rational and integral function of x, y, s. In what follows, Vn will be termed a solid, and Yn a spherical harmonic. 138. Laplace's and Legendre's Coefficients.— If
the attracting muss be concentrated at a point Q whose polar coordinates are /, 0', <p', and whose distance from any point P is r, we have
where X = /I/A' + A/1 - ju* v1 - /"*" cos (tf» ~ *)•
In this case, if P be farther than Q from the origin,
and if P be nearer the origin,
The coefficients L\, L3, &c., in the development of r"1 are
called Laplace's Coefficients. They are obviously spheric* "
harmonics of a particular kind. They may be defined a
the coefficients of the successive poicers of h in the expansion of
(i-2M + A')-i.
These coefficients are plainly symmetrical with respect the angular coordinates of P and Q.
Properties of Complete Spherical Harmonics. 3
If the point Q be on the axis from which 0 is counted, A = //, and Laplace's coefficients become the coefficients of the successive powers of h in the development of
(l-2fih + //2)-4.
In this case, these coefficients are functions of u solely and are called Legendre's Coefficients. They are usually denoted by P1? P,, &c.
It is plain that Pn satisfies the equation
In general, a spherical harmonic of the degree «, whicli is a function of ^ solely, satisfies (2), and is called a zonal harmonic.
139. Properties of Complete Spherical Har- monics.—A spherical harmonic which when expressed as a function of the coordinates is finite and single-mined for all points of space, is said to be complete. If Ym and Yn be complete spherical harmonics of different degrees,
CJ>
This may be proved as follows:— It appears from Art. Jo7 that
satisfy Laplace's equation ; and by (5), Art. 58, if we take as the neld of integration the space outside a sphere S of radius a described round the origin as centre, we have
Eence
tnd therefore, unless m = n, equation (3) must hold good.
B2
4 Spherical and Ellipsoidal Harmonics.
If Fn be a complete spherical harmonic of the degree «, and Ln a Laplacian coefficient of the same degree,
(4)
To prove this, take as the field of integration the space outside a sphere S whose centre is at the origin, and whose radius a is less than r', the distance of the point Q from the origin, and let r denote the distance of any point from Q ; then the function ^ satisfies Laplace's equation,and therefore by (10), Art. 59, we have
but
at all points for which r < /, whence at the surface S we have
also <(S = a*dnd<j>, and
unless m = «». Hence we obtain
- 4, = - - !
from which equation (4) follows by transposition.
If two series of spherical harmonics are equal for all values of /n and $>, each harmonic of one series is equal to the harmonic of the other series whose degree is the same.
Here
Yo + Yl + Tt + &o. - Z« + Zi + Z* * &o.
Properties of Complete Spherical Harmonics. 5
If each side of this equation be multiplied by Ln and integrated, since
rr
by (4) we obtain
4?r 4?r
and as this equation holds good for all values of p and <?/, we get Yn = Zn.
Any function of m and 0 which is finite and single-valued can be expanded in a series of spherical harmonics.
The method of arriving at this result is suggested by what has been already proved. If it be possible to express in the form SFn, we must have
47T/W) = 4jr2r'M = S (2n + 1) f+* P" LnTn d^ d$
J-iJo
f+1 ft*
/(^)S(2n + l)X«^f/0. (5)
.' -iJ o
whence, differentiating and multiplying by 2//, we have 2 (A A - A2)
then by addition to the former equation we get
Accordingly, if the supposed expansion be possible, we must, when h = 1, have
and conversely, if this equation be true, the expansion i possible.
6 Spherical and Ellipsoidal Harmonics.
That equation (ti) is (rue can be shown in the following manner : —
Let Q be a point outside a sphere S whose centre is the origin and whose radius is «, and let r denote the distance of any point on 8 from Q. Then
where r' denotes the distance of Q from the origin ; and if
a h = — , we have
1 -//' _ r' (r/2 - a') (I -2AA + /*')*" r3
As in Art. 42, we have
,«-2?r*,
r
and therefore
CdS = 2iraf 1 1 \ = lira2 m
J r3 r \r' -a r' + a) »•'(»•"-«')'
also dS = tfdfjidQ, and accordingly « 2' 1 - A'
The value of the definite integral above is therefore independent of / ; but h = 1 when r' = a, and in this case each element of the integral in (6) is zero, unless r be infinitely small, in which case /i = //, and <j> = $'. Hence, when h = 1, we have
Application of Spherical Harmonics, 7
140. Application of Spherical Harmonics.— When
the potential is due to mass on one side of a spherical surface 8 and is given at each point of the surface S itself, the potential at any point on the side of S remote from the mass can be represented by a series of solid harmonics. At the surface S this series becomes a series of spherical harmonics representing the known value of the potential at the surface. Hence by Art. 139 each harmonic in this series is determined, and con- sequently so also are the corresponding solid harmonics representing the potential on one side of 8.
If the potential be due to a distribution of mass on the surface S whose density is given, the potential outside S can be represented by the series
r * rn+l and at any point inside by the series
a**
At all points of the surface these two expressions must be equal, whence by Art. 139, Zn = FB. Again, if F and V denote the potentials outside and inside the surface, we have by Art. 46 at the surface
dV <IV
—. --- -=- + 4ir(T = 0,
dr dr
that is, whence and
8 Spherical and Ellipsoidal Harmonics.
141. Potential of Homogeneous Spheroid.— If the
surface of a solid differs but little from a sphere whose centre is at the origin, the radius vector r is given by an equation of the form r = 0(1 + ay), where a denotes the radius of the sphere, y a function of the angular coordinates fi and 0, and a a small constant whose square may be neglected.
The potential F at any external point is the sum of the potential due to the sphere and of that due to the shell whose thickness at any point is aay. Hence if p denote the density of the spheroid, p, $' the coordinates of a point on the surface of the sphere, and r the distance of this point from the point r, /u, $ in external space, we have
4 TT pa3 f, 3— + °j
but by Art. 139, y = %YH, and therefore by Art. 138, and by (3) and (4), we get
(7)
For the potential at a point inside the sphere, by Art. 42, we obtain, in like manner,
142. Potential of Heterogeneous Spheroid.— If
a spheroid be composed of homogeneous layers comprised between surfaces given by equations of the form
where Yn is a spherical harmonic which varies with the surface, and a is a variable parameter, we have, for the
Potential of Heterogeneous Spheroid. 9
potential BV of a single layer at a point outside, the equation
pa'da
and at a point inside,
S V = 4irpa da + 4anp
Hence for the potential V of a heterogeneous spheroid at a point outside it, if «t denote the parameter of the external surface, we obtain the equation
V
For the potential of a heterogeneous shell comprised between surfaces whose parameters are ^ and «2, at an internal point, we get
pa da + 4a7r 2 ^~ »*. (
^
By combining the expressions given by (9) and (10), we find for the potential of a heterogeneous spheroid, at an internal point lying on a surface whose parameter is a, the equation
10 Spherical and Ellipsoidal Harmonics.
143. Homogeneous Mass of Revolving Fluid. —
If a homogeneous mass of fluid revolving with a uniform angular velocity be in a state of relative equilibrium under its own attraction, its external surface, if it be nearly spherical, must be an ellipsoid of revolution. This may be proved as follows: —
By Ex. 5, Art. 24, at the free surface of a liquid in relative equilibrium, if V denote the attraction potential, which in this case is a force function, and if the axis of rotation be taken as the axis of s, we have
0; whence, as in Art. 81, we get
IT* ~"'(l -y) - constant. (12)
The last term on the left-hand side of this equation must be small, as otherwise the surface of the liquid could not be approximately spherical. In this term, therefore, we may put r = a, and substituting for V from (7), we get
|irp*(l - «2Fn) + 4airp«* 2 ~i + ~ (1 - /*') = constant.
(13)
In order to make use of this equation, we must express
s2 /u2 by means of spherical harmonics. Since /u7 = -j, it is
plain that the solid harmonic corresponding to the spherical harmonic of highest degree in tf must be 22 + Ar2, where k is an undetermined constant. To determine k, we have
V2{s2 + k (a* + if -i- s2)j = 0.
Hence k = - J, and tf - \ + J is the required expression for i2.
Figure of the Earth. 11
By Art. 139, the sum of the spherical harmonics of each degree above zero in (13) must vanish separately. Hence Yn = 0 if n > 2, and
8 wV ..
•— airptrYz = — (J -/r).
10 xJ
Putting -4 — = 5-, we get n F2 = f ^ (£ - ^u2). Hence the equation of the free surface is of the form
which represents an ellipsoid of revolution nearly spherical whose ellipticity is £ q. See Art. 81.
144. Figure of the Earth.— On the hypotheses that the Earth is composed of homogeneous layers bounded by similar surfaces nearly spherical, and that it is covered with liquid in relative equilibrium, it is easy to show that the external surface of the liquid must be an oblate ellipsoid of revolution whose axis is the axis of rotation.
The attraction potential V of the Earth is given by (11)- At the surface of the liquid, (12) must hold good. Hence, by substitution, we obtain
+ G-]u2) = constant. (14)
Since the surfaces of equal density are similar, Yn does not vary with a, and as «i is the greatest possible value for a if n be not less than 2, we have
(n + 3) tf»* (2n + !)«,«
Hence the multiplier of Yn in (14) cannot be zero, and therefore if n > 2, we have YH = 0.
12 Spherical and Ellipsoidal Harmonics.
If n = 2, we obtain 4;r
By Art. 78, when the centre of inertia is the origin, the coefficient of -j in V is zero. Hence, in the expression for
the potential of a spheroid given by (9), if the surfaces of equal density be similar, and if the centre of inertia be the origin, we must have Fi - 0 ; and in the present case the form of the external surface is determined by the equation r = «t (1 + a Fa), where Y2 is given by (15). The external surface is therefore an oblate ellipsoid of revolution having the axis of rotation as its axis.
It seems improbable that the hypothesis made above with respect to the form of the surfaces of equal density should be correct. In order that it should be true, it is necessary that these surfaces should have been formed under similar con- ditions; but, unless the Earth were of uniform density, this could not have been the case, since the equatorial centrifugal force due to rotation varies as the distance from the centre, whilst the attraction of the sphere having this distance as radius varies in a different manner unless the density be uniform.
A more probable hypothesis is, that the surfaces of equal density are represented by equations of the form
r = a (I + a%),
where h is a parameter varying with a, but constant for each surface, and y a function of /* and <f>, which is the same for all the surfaces.
145. t'lairatit M Theorem. — Whatever be the internal constitution of the Earth, if it be covered with liquid in relative equilibrium whose external surface is an ellipsoid of revolution nearly spherical, the ellipticity, e, the ratio of the centrifugal force at the equator to gravity, q, and the diffe- rence between polar and equatorial gravity divided by the latter, 7, fulfil the relation j + e = f q.
Clairaut's Theorem. 13
This equation was proved in Art. 81 on a particular hypothesis as to the internal constitution of the Earth. Any hypothesis of this kind is, however, unnecessary, as was first pointed out by Sir Gr. Stokes.
At the external surface of the liquid, the Earth's poten- tial F"must satisfy (12); but as this surface is nearly spherical and the term in (12) due to rotation is small, the variable terms in F must be small. Hence, if M denote the mass of the Earth, we may assume
where a is a small constant. Again, by Art. 81, the form of the external surface is represented by the equation
Substituting in (12), we get
— (1 + e O8 - £)) + a 2 — + IT «2 (J - /*') = constant. a a. &
Hence Yn = 0, unless w = 2. If w = 2, we have
r
v2 where
Accordingly,
and
dV M _ «T2 — = — - + da — — • dr rz IA
14 Spherical and Ellipsoidal Harmonics.
Hence if Q denote the acceleration due to gravity at any point on the Earth's surface, in the same manner as m Art. 81, we find
thati8' *-
Hence
7 and therefore
146. Tangential Component of Attraction.— If P
denote the component of the Earth's attraction perpendicular to the radius at any point on its surface, by (16), we have
where A denotes the latitude of the place.
If we compare (16) with (2), Art. 78, we get
Ma-^t - q) Gu* - i) = -57 - (A 4 J5 + C).
Hence C - ^ = ^(2e- y). (19)
o
The equations proved above were arrived at before in Art. 81 by means of a special hypothesis with respect to the internal constitution of the Earth. The facility with which these results have been obtained in the present and preceding Articles without any such hypothesis illustrates the power of the Laplacian method.
Legendre's Coefficients. 15
147. tegendre's Coefficients.— The definition of these coefficients given in Art. 138 enables us to see that they are rational and integral functions of fi. A general expression for these coefficients cannot be readily obtained by the usual methods of expansion. If we put
^_(h_ ~~dx*
we can, by integration, get rid of the negative index ; and thus we obtain
(l-2yx + y2)* z = - — + constant.
y
If we take - for the constant, we get
(>/z - I)2 = 1 - 2yx + y* • whence we obtain _ y_ , _
We have now an expression for z suitable for the applica- tion of Lagrange's theorem (Williamson, Differential Calculus, Art. 125) by which we obtain
whence
Hence, if (1 - Zph + A2)~i = 1 + 2PWA",
««* f.-^g1 ,,.2,.,$(M'-ir. (2D
The development of Pn in powers of n is most easily effected by means of the differential equation satisfied by zonal harmonics.
16 Spherical and Ellipsoidal Harmonics.
If Sn denote a zonal harmonic of the degree n, we may
assume
Sn = atfjL* + fl,.!//"1 + &o.,
where s, &c., must be positive in order that Sn should be finite at every point of space, and Sn must satisfy the equation
Hence we get
+ H(n+l){atfif + &o.} =0. Here the coefficient of p* is
{n(n + l) - *(* + !)!«,, and that of p** is
&(s - l)ff, + (n(n + 1) - (.$• - 2)(.s - 1)} «^2 ;
accordingly, as each coefficient must vanish separately, we obtain
From the first of these we get s = «, or « = - (n 4- 1) : and as the negative value for s is here inadmissible, we have
-w(n-l) -(«-2)(n-3)
-"' ""- 2(2^1)- fl« «-- 4(2-8) ^'
and in general
2.4...2?.(2n-l)(2w-3)...(2n-2g + l)
(23)
Legendre's Coefficients. 17
It is plain that the terms in (22) resulting from
tf,_iyus-1 + <V3/i*"3 + &c.,
must vanish independently of those arising from the series already considered, and that we get for the first term the equation
(s - n - 1) (s + n) rt,_i = 0.
Hence fl«-i, as-3, &c., must each be zero, and we obtain
„ ,
3)" ' &C| (24)
Hence, zonal harmonics of the same order can differ only in the constant factor, and we may write
Sn = aPn, (25)
where a is an undetermined constant.
It is easy to see that the coefficient of nn in (21) is
2n(2n - 1) (2n - 2) ...(» + 1) 2" . f n . 1 . 3 . 5 . . . (2n - 1) _ _ _ or *-~ _
2n n 2" . w . w '
and therefore
2(2w-l)'
&c.
(26)
It follows from the definition of PM that when /x = 1 the value of Pn is unity.
c
18 Spherical and Ellipsoidal Harmonics.
148. Spherical Harmonics. — Since the expansion of
H^^'^r
contains only rational and integral functions of x, yt and z, the coefficients Llt L2, &c., must be rational and integral functions of sin 0 and cos 0, in which each power of sin 0 and cos0 is multiplied by the same power of Hence, as
_4rr_
2n + 1
the spherical harmonic Yn must be a rational and integral function of sin 0 and cos 0 of the nth degree in which each power of sin 0 and cos 0 is multiplied by the same power of •v/(l - j*2). If each power of sin 0 and cos 0 be expanded in a series of sines and cosines of multiples of 0, we see that finally Yn is reducible to the form
2 (AsMt cos «0 + B,NS sin
where As and Bt are undetermined constants, and M, and N, functions of /u.
If we put — = D, and /** - I = w, equation (1) becomes DuDY +!^_ r -
Since the coefficient of the sine or cosine of each multiple of 0 must vanish separately in (27), we have
DuDM,- - + »(» + 1)] Jf,-0, (28)
[U )
Again, since cos s<f> and sin s0 can result only from (cos0)s, (sin0)*, (cos0)4+2, (sin0)m, &c.,
X
Mt must contain t«2 as a factor ; and the other factor must be
Spherical Harmonics. 19
a rational and integral function of ^ Accordingly, we may Ms = tfiv,
where v denotes a rational and integral function of u
From (28), we have then
v + su* v + (2s + 2) u-fjiDv 4 w2 D*v
--. 1
- s2?r t; - n (n + 1) u* v = 0. (29)
Since M = M* - 1, equation (29) is divisible by J, and we get
n&v + (s + 1) Du Dv + ^±11 t>D»rt - n (n + 1) t> = 0. (30)
If we assume * = /)•«., equation (30) becomes
Ds» (uDw) - n(n + 1)D>W = 0. (31)
Since v is a rational and integral function of «, it is plain that, with the exception of a constant factor, it is completely determined by (30) Hence any rational and integral function pt p winch satisfies (30) or (31) must represent v. Equation (31) is satisfied if w satisfy
DuDw - n(n + \)w = 0;
but this equation is the same as (22).
Hence we may put w = Pw, and we have
It is plain that the equations by which XT, is determined are tne same as those for Ms. Accordingly, these two functions can differ only by a constant factor, and we obtain
cos 6-0 + Ba sin ty) i
C2
20 Spherical and Ellipsoidal Hat-monies.
The part of Yn depending upon s<j>, that is, (As cos s0 + Bs sin s0) i?J?Pn,
is termed a tesseral harmonic of degree n and order «, and we may write
Yn = 2 Tns (A. cos s0 + B, sin 80). (33)
If we substitute for Pn its value given by (26), since A, and Bs are undetermined constants, we have
-&c.. (34)
149. L,aplace's Coefficients. — Laplace's coefficients are, as we have seen in Art. 138, a particular kind of spherical harmonic; and as they are functions of <f> and <j>' through being functions of cos (<ft - ^'), and are symmetrical in fi and ju', we must have
Ln = 2fl, cos a (0 - ^ ') « VlD*P,, Z)''^, (36)
where cr, is a definite function of w and s, which may be determined in the following manner: —
By (4) we have
f+lf2* I I
^A, COSS0 + ^Ssms0) dndd> = ^(As cos«d>' -r B. sin s<t>'}n'riD>isP'n.
S.D -4- I v • r /
Laplace's Coefficients. 21
It is plain that the only part of the multiplier of cos s<f>' in the left-hand member of this equation which does not vanish after integration is
[+* f
Hence, if s be not zero, we get
and if s = 0, we have
"I!! *••"'" s£
The first term in Ln is aQPnP'nj and when /A' = 1, all the other terms vanish, and P"« = l. Hence, in this case, Ln = a0Pn ; but Ln becomes Pn when yt = 1, and therefore r/0 = 1. The remaining coefficients can now be found by means of (36) and (37).
Let
then
also by (2), we have
>>(» whence
A! + n(»+ 1)A0 = f1 {PnD(uDPn} + nDPnDP
since z< vanishes at each limit of the integral.
22 Spherical and Ellipsoidal Harmonics.
It is now easy to see that an equation similar to that obtained above holds good for any two successive integrals of the series. In fact, by (2), we have
&"(uDPn} - n(n + l)D°Pn = 0 ;
whence, remembering that D*u = 2, D3u = 0, by Leibnitz's theorem, we have
**Pn + 8(8 +
and therefore
(n -«)(» + * + !) usD>Pn = D(ttsn D°*Pn] . (38) Hence we have A,+i + (n-s) (» + «+!) A,
= J ' D(n^D°Pn l)^Pn] dp = 0. (39)
Accordingly, A,+1 - - (n -«)(« + « + 1) A, ; (40)
and therefore, by (36), if s be not zero, we have
and, by (37), we get
Hence, as a0 = 1, we obtain - 2
and
(44)
Complete Harmonics. 23
150. Complete Harmonics. — The definition of solid and spherical harmonics in general lias been given in Art. 137 ; but the properties of spherical harmonics proved in Art. 139 have been obtained on the hypothesis that these functions are finite and single- valued for every point of space, and in that Art. m and n denote integers.
If YI denote a function of /u and 0 satisfying the equation
* (45)
where * denotes any real numerical quantity, corresponding to Yi, there are two solid harmonics, viz. J*X\ and r~^Y{.
As i is real, one of the quantities i and -(?' + !) is negative.
Hence, selecting the two solid harmonics of negative degree which correspond to YI and Yj, we see that when Y{ and Yj are finite and single- valued, equation (3) holds good, unless i=J, or « = -(/+!). Again, if Yt be finite and single-valued, by a process similar to that employed in proving (4), Art. 139, we have
rr
J-iJo
5-) , (46)
if t be positive, the coefficient of LH being n - i if i be
fr'\i
negative, and that of Y'i being 47r( — I •
Accordingly, by (3), Y't = 0, unless «'=«, or t = -(w+l). In either case YI is a rational and integral function of /u,
v/1 - fj." cos 0, and ^/l - fS sin $
of the degree n.
Hence we conclude that the degree of a complete spherical harmonic must be a positive integer, and that the correspond- ing solid harmonic of positive degree must be a rational and integral function of x, y, and 2.
This last result is usually expressed by saying that every complete solid harmonic is a rational and integral function of x, y, and 2, or can be made so by multiplying by a suitable power of r.
24 Spherical and Ellipsoidal Harmonics.
151. Reduction of a Function to Spherical Harmonics. — It was shown in Art. 139 that a finite and single-valued function of p and 0 can always be expressed by a series of complete spherical harmonics. If this series be finite, so that
= F0 + Pi . . . + FB, we have
rnf= Vn + fVn-i + &o. + r { Fw-i + r'F«_3 + &c.j .
Hence rw/ = fH + rf^ where /„ and /n_i denote rational, integral, homogeneous functions of the coordinates ; and it appears that if a function of p and 0 can be expressed by a finite series of spherical harmonics, the corresponding function of the coordinates must consist of a rational, integral, homogeneous function, together with another such function multiplied by r. Accordingly, the problem to express a given function of /u and <]> in a finite series of spherical harmonics, when soluble, is reduced to that of expressing /„, a rational, integral, homogeneous function of the coordinates in a series of the form Fn + r* Fn_a + &c.
This is effected most easily by means of Laplace's operator.
In fact, by Leibnitz's theorem,
~J~ + T —j + -r — r^ ) + 'lpV2F,n, (47) w dy dij dz c1- '
but
and W=p(p + !)»*-*; also, V2Fm = 0, and ffdVm ^ dVm t ^rfF,,
Accordingly,
rm = {p(p+l) + 2pm}>*-*rm. (48)
Reduction of a Function to Spherical Harmonics. 25
From (48), we get
V% = a2 Fn_2 + as* F«-4 + &c., V4/B = 64 FM.4 + b^ Fn_6 + &c., &e., &c.,
where a2, ait &c., &4, &c., are known numerical coefficients. If this process be repeated sufficiently often, we find ultimately
according as n = 2q, or n = 2q + 1, the coefficients A and k being known numbers. In fact,
li - \ji + 1, and k = — — - \ji.
By the equations previously obtained, we can then determine the other solid harmonics. As a simple example, let
/= Ax* + Bif + Cz~ + 2F//Z then /= F2 + r2F0, and V2/= AF0, where A = 3.2. 1 = 6 ; but V*f=2(A + B+C)', hence F0 = H^ + B + G\ F2 = A& + Bif + Cz* + IFyz + 2Gzx +
and
/ = i { 2^ - B - C}x* + (2B - C - A}f + (2(7 - A -
+ 2Fyz + 2Gzx + 2Hxy + %(A + B + C] r\ Again, let / - x3 + y3 + tfy + ifx,
then ./ = F3 + i* F,, and V2/ = k Fi, where k = | . 3 . 2 = 10 ;
accordingly, 10 Ft = 8(.r + y), hence Fi = f(«! + y), and
F3 = i (*3 + ?/3 + *y + ifx) - i z\x + y).
The method originally given by Laplace for reducing to a series of spherical harmonics a function of p and ^ corre- sponding to a rational and integral function of the coordinates, differs somewhat from that iven above.
26 Spherical and Ellipsoidal Hat-monies.
A rational and integral function of the coordinates corresponds to a rational and integral function of
If the various powers of cos <j> and sin 0 be developed in sines and cosines of multiples of 0, the series multiplying
L (1 - jU2)2 COSS0
will contain all the powers of p not exceeding n - 9, where n is the degree of the given function of the coordinates.
If we collect together the terms containing the highest power of n in each series, we obtain an expression of the form
the function Tn may then be determined by taking its 2» + 1 arbitrary constants, so that the terms of the above form may be equal to those in the expression given above. If we subtract Yn thus determined from /, we get a function, f - Tn of the degree n - 1 in
/*, \/l- n* cos^>, and \/l - /u2 sin 0.
The harmonic Tn-\ can then be determined in a way similar to that employed in finding Yn, and so on.
When the original function of the coordinates is trans- formed into a function of r, /u, and 0, the various powers of r are in / regarded as constants.
It is plain that the total number of terms or of independent constants in / is 1 + 3 + 5 . . . + 2n + 1, that is, (n + I)2.
This is also the number of arbitrary constants in the series
152. Methods of forming Complete Solid Har- monics. — A complete solid harmonic of positive degree is, as we have seen, Art. 150, a rational and integral function of the coordinates. A solid harmonic of the degree «, since
it is homogeneous, contains, therefore, ' - terms.
Methods of Forming Complete Sotid Harmonics. 27
The coefficients of these terms are, however, not all inde- pendent ; for, if Vn denote the harmonic, V2 Vn must vanish
for all values of the coordinates, and therefore - —
equations must be satisfied by the coefficients of Vn.
Accordingly, F» contains 2n + 1 independent arbitrary constants.
Since Wl-0, we Have '' <Y <\V 1 _ 0.
where «, y, /<; denote any integers.
Hence v> ( *Y (£Y ( ')* I _ 0,
Vfo/ vw vfe/ >*
and therefore
(d_ V /rf V / d_ \* 1
w WJ w >~
is a solid harmonic of the degree -(i+j + k + l). If i +J+ k = n, the number of different combinations of the type
lYf-YY-Y
U'/y U;
which can be formed is - - -±-. - 1 . but aii the different
1 functions which result by the use of these operators on -
are not independent. In fact,
(P c^ _tP_\ fd_ y fd_V fd\k' 1 _ 0 dz* + dif + d?) \djcj \d~y) \d») r '
where *' +/ + // = n - 2. Tliere are ^ ~9 ^M equations of
this form which must be satisfied identically by functions of the form
\<tej \dy) V
28 Spherical and Ellipsoidal Harmonics.
where i+J + k = n. Consequently, of these latter functions there are only 2n + 1 independent. Hence, every complete solid harmonic Vn of the degree n is given by the equation
Vn = '•2nt
where » +J + k = n, and where there are 2n + 1 independent functions, and consequently 2» + 1 independent arbitrary constants.
Another method of forming complete solid harmonics depends on the consideration that, if ai, /3i, 71 be the direction cosines of any line,
(L + 3 *_ + *\\
satisfies Laplace's equations, and more generally that this equation is satisfied by
d\
rf , <* rf\ 1
3~ + P« 3- + 7,, 3- -
dx dy ' dzj r It follows from this that the function
satisfies Laplace's equation ; and as it is a rational, integral, homogeneous function of the nth degree, containing 2n + 1 independent arbitrary const ants, every complete solid harmonic of the nth degree can be expressed in this form.
It is not, however, obvious that a set of real values of the coefficients 01, /3i, 71, &c., corresponding to any given com- plete solid harmonic always exists, and that in general there is only one such set.
Methods of forming Complete Solid Harmonics. 29
This proposition, which is necessary to complete Maxwell's method of representing solid harmonics, was proved by Sylvester (Phil. Mag., October, 1876), in the following manner : —
It has been shown above, that by the solution of linear equations for determining the coefficients, we can reduce any complete solid harmonic to the form given by (49).
We have now to show that any rational homogeneous function of the nth degree of the symbols of differentiation
operating on - can be reduced to the product of n real linear
r
factors of the form
d d
Since the symbols of differentiation obey the same laws as quantities, and since
dx* dy~
the theorem just stated is equivalent to asserting that any ternary quantic (x, y, z)n, whose variables are subject to the condition <r2 + yl + £ = 0, can be reduced to the product of n real linear factors, and that this reduction can be effected in only one way.
The equations (x, y, z)H = 0, ar2 -f if + zz = 0 may be regarded as representing plane curves having 2» points of intersection. If these points be joined in pairs, we obtain n straight lines, the coordinates of whose points of determi- nation are obtained by solving for x:y:z the simultaneous equations (x, y, z)H = 0, and a2 + yz + zz = 0.
If - be real, the corresponding value of - given by the latter equation must be imaginary, and so also, therefore, that of - • Hence of the three ratios, x : y : z, two at least are imaginary.
30 Spherical and Ellipsoidal Harmonics.
The equation of the straight line joining the points of, i/, z', and x" ', y", s" is
x(y'z" - zy") + !/ (sV - x'z") + z (x'y" - i/'x") = 0. If we suppose ^— and X— to be each imaginary, and select for
" A/' 2 Z
^7 and — the conjugate imaginaries, each term in the equa- tion of the straight line contains </- 1 as a factor, and the line is therefore real. f/
If the equation of the degree 2n for determining '- have
2m imaginary roots, there are 2m imaginary values of either - or - corresponding, and therefore m real straight lines.
9
Corresponding to the 2 (n - m) real values of -, there must be 2(n-w) imaginary values of - and -, and therefore
(n - m) additional real straight lines. Hence in all there are n real straight lines passing through the points of intersection of (x, y, z)n = 0 and a?2 + y* + sa = 0.
There are no more. For if we seek the values of - which
2
satisfy the equation of a real straight line, and the equation #2 + y* + s2 = 0, these values must be real, or else conjugate
imaginaries ; and in the former case, the values of '- must be conjugate imaginaries, and also those of - • Hence, to obtain a real straight line, each imaginary value of one of the ratios -, &c., satisfying (JT, y, z)n = 0 and x* + yz + s2 = 0, must be
combined with its conjugate ; consequently there are only n such lines.
Let L = 0 denote the equation of n straight lines passing through the 2n points of intersection of (.r, y, z)n = 0, and #8 + y2 + s2 = 0; then, whenever (x,y,z)" and #2 + y* + sa both vanish, so must L, and therefore
L = X (x, y, z)n + Y (x* + y* + s2) . (50)
Methods of forming Complete Solid Harmonics. 31
From the degree of the various functions in this equation we see that X is constant, and Y of the degree n - 2. Since, in general, a ternary quantic of the nth degree contains
- -^ - — constants, and the equation of n straight lines
A
contains In + 1 constants ; and since
(»« + !)(* + 2) ,(»*-!)»»
9 - 9 - '
it is plain that the 9 constants of F can be so deter-
mined that the right-hand side of (50) shall represent n straight lines. It has been proved above that for one of these determinations the n straight lines are real. If ai# + j3i# -f yiS = 0, &c., represent these real lines, then
(ai# + fry + y,s) . . . (anx + finy + yns)
= A 0, y, z)n + Y(o- +if + 22).
Applying the theorem which has been proved for the quantities x, y, z to the symbols of differentiation, by (49), we get
f d\i / <i\f fd\*' ftf ,r- ip\)i
+ tz^) WJ (T*) + * + »
+ ft* + „
rfa? ay
where Fn denotes any solid harmonic of degree », ^4, ^4^, and ii>T constants, and 01, j3i, 71, &c., the direction cosines of straight lines, and where
i +j + k = n, i' +f + k' = n - 2.
32 Spherical and Ellipsoidal Harmonics.
If lines be drawn from the origin, each in one direction, having a,, j3i, 71, &o., as their direction cosines, these lines meet a sphere, having the origin for centre, in « points which are called the poles of the corresponding spherical harmonic.
The mode now described of forming spherical harmonics was given by Clerk Maxwell in his treatise on Electricity and Magnetism.
Maxwell's method of representing spherical and solid harmonics admits of an interesting physical interpretation.
If hj. denote a line whose direction cosines are aM /3i, 7, drawn through the point x, y, a, and h\ the parallel line through the origin,
d d d \ 1 d 1 d 1
and r._^> ._. (52)
dhi dhv. dhn r
Again, ™ expresses the potential of a mass m at the
origin, and ... d 1
- mdh\ —-. —
dh\ r
expresses the potential produced by superimposing on this mass another negative mass of equal magnitude, situated at a point at a distance from the origin infinitely small in the direction h\. If this system be displaced through the distance dh\, reversed, and superimposed on the former, the potential becomes
„, „, d d 1 tn ah \ (in a -77^ —r - , and so on. dn\ dh 2 r
The repetition of this process n times leads to the potential Un, where
..-
an i an 2 an n r
= M— A. A
dh* d/i, ' ' ' dh~n provided A = mdhi dht . . . dhn = M.
Method* of forming Complete Solid Harmonics. 33
If A be a finite constant, m must be an infinitely great quantity of the nth order.
As an easy example, illustrating the foregoing theory, we may consider the question to express in Maxwell's form a solid harmonic of the second degree containing only the squares of the variables.
Here, by Art. 151, the solid harmonic
F2 = ay? + fof -(a + b) z\
<P <f c? Again, a. _ + _ = --, we have
~
d i
+ 3/i//2 - (X + //) >'2 = (2A -
and therefore Vt = ;•* ( X — - + u ~
dx- d i r
Hence, comparing with the former expression for F,, we have 2X - ju = «, 2/i - X = i ; whence
To reduce ( A —i + u -r-n } - to Maxwell's form, we must V <lxz ^ dtp] r
consider the relative values of X and //.
If X and ju have different algebraic signs, and n = - f.i , then
F2 = If X and fi have the same sign and X be the greater,
-dl
34 Spherical and Ellipsoidal Harmonics.
The most general solid harmonic of the second degree is reducible to the form considered above by a transformation of the axes of coordinates.
In general, for the second degree, the reduction of a solid harmonic to Maxwell's form is mathematically the same problem as the determination of the planes of circular section of a quadric surface.
153. Incomplete Harmonics. — We have seen, Art. 150, that if Ti be a complete spherical harmonic whose degree is real, i must be a positive integer, and Y{ a rational and integral function of /u, v/(l - /**) cos 0, and </ (1 - /tt2) sin $.
If i be a negative integer, -(*'+ 1) is zero or a positive integer.
If i be real but not an integer, it is easy to obtain expres- sions for Yi which satisfy the differential equation (45) ; but these expressions become infinite at certain points on a sphere surrounding the origin, or alter in value after having passed continuously through a complete circuit surrounding the axis of z. In the latter case, accordingly, they are not single- valued.
If we assume pt = a0 + a^ + a4W4 + &o.,
qt = fli/i + a3n3 + a+p? + &c., and substitute;^ in the equation
l(l-rt* + f(f + l),-0, (53)
in order that each power of /i should vanish, we find that
(i - n)(i + n + 1) """•• (»+l)(. + 2j '- (M)
Equation (54) is fulfilled also by two successive coefficients in the series denoted by qit provided q, satisfies (53). Hence, if we assume
. '
2.3 ~ 2.:*. 4. 6
16-&C.V
(55)
Incomplete Harmonics. 35
each of the series^ and qi satisfies the differential equation for a spherical harmonic of the degree i, whatever be the value of i.
If * be an integer, one of these series terminates : the other contains an infinite number of terms.
If i be not an integer, both series contain an infinite number of terms.
The sum of each series is finite so long as /u < 1 ; but if fji = 1, either series, if it contains an infinite number of terms, becomes infinite. In fact, (54) may be written
and, accordingly, as n increases without limit, all the terms become of the same algebraical sign, and the value of —
tends to become unity. Hence (Williamson, Differential Calculus, Art. 73), if /u < 1, the series is convergent.
In the case of the more general spherical harmonic F,, whatever be the value of i, we may assume
Yi = S (A, cos s$ + Bg sin
then, as in Art. 148, equation (30), we find that pi, must satisfy the equation
(/£»- l)D*p + 2n(s+l)Dp - (i-s)(i + s + l)p = 0. (56) This equation is satisfied by the series aQ + azfj.z + din* + &c., and by the series «•]/* + «3/i3 + «8ju5 + &c., provided that in each series
LiW-r« / -I \ / o\ '*#•
(n + !)(» + 2) Hence, if we assume
q* = «i
D2
36 Spherical and Ellipsoidal Hai-niotiicn.
Yi = S (A. cos 8$ + 2?, sin 80) (1 - /i')'0>« 4 #,), (59)
where i and « have any values whatever, Fj will be a spherical harmonic of the degree i.
If / - s be a positive integer or i + 8 a negative integer, one of the series pis and q^ terminates, and the other contains an infinite number of terms.
In any other case, botli series contain an infinite number of terms.
When the number of terms is infinite,
(l-M')'jfc and (1-rfqt.
are each finite if /A < 1 ; but if fi = 1, each of these expressions becomes infinite.
In order to prove this, we observe that (57) may be written
n« 5 - -.
»' + 6n + 2
When n becomes very great, the ratio of «*« to «„ tends towards l + !ii^.
n
Again, if we put (1 -/u2)'* = 1 + 62u2 + biui + &c., we find that
*
As n becomes very great, the ratio of AW+J to bn tends towards
Hence, as ^ approaches 1, the functions pu and ^ tend to become quantities of the same order as (l-/ul)-»; and therefore, if s be positive,
and (l are finite so long as ^ < 1 ; but if ^ = 1, they become infinite.
Incomplete Harmonics. 37
The same thing is true if s be negative. In this case, (l-^)7 becomes infinite when M = 1, and the products
(* - ^)>» and (1 - ^yqis become infinite as before.
When i = s, we have pia = «0 ; and when * = s + 1, the series qis = alf*.
It appears from what has been said that if we assume
(- Ft = (A cos /0 + B sin fy) (1 - ^)*,
where / is positive, Y> is always finite; but if i be not an integer, Yt is not single-valued, for when 0 increases by Zir the functions cos fy and sin i<p do not return to their original values.
If
and i = ns, where » is any integer, and s = -, the function T
satisfies Laplace's equation, vanishes at infinity, and is zero at the planes for which 0 = 0 and 0 = «. At the surface of a sphere of radius a we have
Thus on this sphere Fis a function of ^u and 0, which vanishes at each of two great circles, and is finite and single- valued tor the intercepted portion of the surface. By bringing in a sufficient number of terms, and properly determining the arbitrary constants, it may be possible to make this function equal at least approximately, to an assigned function having the characteristics above.
/? iS£me cases> a function satisfying Laplace's equation, fulfilling certain boundary conditions, can be found by
38 Spherical and Ellipsoidal Harmonics.
means of spherical harmonics of imaginary degrees. We have seen already that whatever be t and *,
S (A, cos s<t> + Bt sin $9) (1 - /u2) '(>,•, -f ?,,)
is a possible form of a spherical harmonic of the degree *. If i be imaginary in order that ptt and j,, should be real, it is necessary only that 1(1+ i) should be real. If we put
f (*' + 1) =/, we get i = - | ± ^/(f+ i) ; whence,
if i be imaginary, / must be negative and greater than £ in absolute magnitude. Accordingly, putting / = - kt we obtain
then, since Ti depends only on the value of *'(»'+ I), we have Yi = Ti', and both these functions are real. If we now assume
tlie function V satisfies Laplace's equation, and we have
Y- 9V
'
If we assume v/- 1 V = r* Yt - r*' Fi- , we get in like manner
2 T7*- V' = ~/f,B™ x' wnere X = */(k ~ i) log r. In order that
V or F' should vanish at a sphere of radius a, we have only to assume
Ellipsoids of Revolution. 39
In the first case, we have
and, in the last,
* = d— Vlog,
Incomplete splierical harmonics are here briefly described in order to give the student an idea of their nature and of the kind of conditions which they can be made to satisfy. They are useful in some departments of mathematical physics.
SECTION II. — Ellipsoids of Revolution.
154. Solutions of Differential Equation.— When
the surfaces with which we have to do are not approximately spherical, the expansions for the potential which have been investigated are of little use. In the case of ellipsoids of revolution, equations (35) and (41), Art. 98, enable us, by an extension of the theory of spherical harmonics, to arrive at suitable forms for the potential.
Equation (41), Art. 98, if we write $ instead of x> becomes by transposition
(i)
If such a form be assigned to F as to make each member of this equation equal to the same quantity, the equation is satisfied ; but, by Art. 148, if Pn satisfy equation (22), Art. 147, then
cos «0 + B sin *0) satisfies equation (27), Art. 148, if substituted for YH.
40 Spherical and Ellipsoidal Harmonics.
Hence
' B sin
must satisfy (1), aud if V satisfy Laplace's equation through- out the region inside a prolate ellipsoid of revolution, we may put
V = STATUS) (A. ooss^ + B, sin «0). (2)
The value of F given by (2) becomes infinite along witli £ at points at an infinite distance from the centre of the ellipsoid. Accordingly, (2) does not give a suitable form for F in the space outside the ellipsoid.
It appears, however, from Art. 147 that there are two solutions of equation (22) of that Article. One of these is Pn; the other, which may be denoted by Qn, contains only negative powers of p. Accordingly, when £ becomes infinite, Qn(£) becomes zero.
Hence, if we put
(3)
and V= S2Jn, (4), we see that this form of F satisfies Laplace's equation throughout the space outside the ellipsoid and is zero at infinity.
If we denote by Unt the function corresponding to Tnl in equation (33), Art. 148, we have
and
tW (A, cos*0 + li. sin «0). (6)
At the surface of the ellipsoid, where £ is constant, becomes a spherical harmonic Fw.
Determination of the Function Qn. 41
155. Determination of tbe Function Qn. — The
differential equation (<J2), Art. 147, being of the second order, has two particular integrals; one of these is Pny the other Qn. Putting £ instead of /u, by Art. 147, we have
(7) By Art. 147, if ff, be the coefficient of £* in this series,
Hence, as - « increases, the ratio a,_2 : r^ tends to become unity, and if £ > 1, the series is convergent; but if £ = J, it is divergent. Hence, in the space inside the ellipsoid of revolution, 2),, is not a suitable form for the potential.
156. Determination of Qn as au Integral. — If we
write y for Sn and x for /*, equation (22), Art. 147, becomes
cfy 2x di, H(H + I- _
* + * y"°'
which is of the form
If we put y = vi/i, equation (10) becomes
If ^ be a solution of (10), we get
d-\
(12)
42 Spherical and Ellipsoidal Harmonics.
whence, by integration,
and if v = 0 when x = ar0, we have v-C,
y?
In the present case,
X\ = -^ — r, and 6' also, y, - P«. Accordingly,
By putting x = -, expanding the expression under the
integral sign in ascending powers of s, and integrating, it is easy to see that when x = oo, or z = 0, we have v = 0. Hence
If we choose - 1 for the value of dt we make Qn perfectly definite, and we obtain
157. Expression of Qn by means of a Finite
Series. — In order to express Qn as a finite series, it is necessary first to prove some relations which exist between successive coefficients of Legendre and the functions obtained from them by differentiation.
Expression of Qn. 43
.
Du = 2*, D*u = 2, .D»« = 0 ;
If we put #2 - 1 = «, — = Z), we have dx
and we get
Substituting for DMlun by a formula similar to that just obtained, we get
= (n + 1) (Du DMl un + (»+!) V-u D" «") = 2(w + l)#Dn+1 wn 4- 2(w + l)2Z>we<n.
D»+2w'<+i = 2(n but by (21), Art. 147, we have
and tlierefore, dividing by 2n+1 1^(« + 1), we get
DPM = tfDP^ + nxPn^ + (n + l)Pn, and subtracting DPn.i, we have
DPn,, - DPn_, = uDPn_i + nxPH-i + (n + l)Pn. (16)
The right-hand side of (16) can be expressed in terms of Pn, for we have
Dnttn = D"u nn~l = wD"^-1 + 2nxD"-1 un-1 + 2"(^~1) D"-2^-1 ;
also,
Z^wn = E*-lDun = ZP-hi un~lDu = 2nxDn-1tin-1 + 2n(n -l)!*"-*^-1.
Comparing the two expressions for Z>"wn, we find
uB" un~l = n (n - I ) D*"1 un~l . (17)
Equation (17) shows that IPu" satisfies (22), Art. 147, a result which has been already proved in Art. 147. If we
Spherical and Ellipsoidal Harmonics.
now substitute uD»nn-1 for «(n- OD"-^"-1 in the first of the expressions for Z^M" given above, we get
D»un = '2uDnu^1 + 2nzDw-lwn-1, (18)
and dividing by 2n |_n , we have
JV-it*ZHV, + «rfV», (19)
n
whence uDPn.\ + nxPn.\ - wP».
Substituting in (16), we obtain
(20) From (20), we get immediately
DPn - (2n - l)Pn-i + (2n - 5)Pn_3 . . . + (2» - 4s + 3) Pn.,,+1 + &o.
(21)
We can now express Qn as a finite series by treating the equation
in a manner somewhat different from that previously employed. If we put y = vy\-w, and substitute in (22), we get
Expression of Qn. 45
If we next suppose y\ to be a solution of (22), and determine v in such a manner as to satisfy
^d-vg-O, (23)
«(»+l)|«.2(*.-l)^'. (24)
From (23), by integration, we obtain
(l-#2)^ = constant. ' dx
If we choose 1 as the value of this constant, we get, by integration,
and (24) becomes
£„_„£.„<„*!>,._„£. (26)
If we assume
w = 4,Pn.i -f- AzPn-z . . . f Ati-iP^M + &c.,
and make yl = Pn, by (21) of the present Article, and (2), Art. 138, we get
2 (2n - 4s + 3)
= Aw [n(n + 1) - (n - Vs + !)(« - 2* + 2)}
* AM {«' + w - [w2 - (4s - 3) w + 2 (* - 1) (2s - 1)] }
whence ^ 2. - 4. + 3
(2s - !)(« - s + 1)
(28)
46 Spherical and Ellipsoidal Harmonics.
Thus we obtain
i . ^'IA i kir'^J -
where w is giveu by (^8).
We have now arrived at three expressions for Qn of which one is perfectly definite, and the other two contain constants which can be determined by comparison with (15).
From (29) and (15) we get
whence, by division and differentiation, we get
C that is,
P -«•
1 ' * dx dx
but when x = 1 we have Pn = 1, and therefore C = 1. In order to find ^ in (7) we put x = - , and identify the
coefficients of the lowest powers of 2 in the expressions for Qn(#) given by (7) and (15). By Art. 147, we have
- Art1 {1 + ascending series in z\ , where ^ = — 5---(2n-1) .
w therefore,
,« (1 + &c.) s2»( I - s1)-1
Analogue* of 'federal Harmonics. 47
Here the lowest power of a after integration is plainly sn+1, its that is,
and its coefficient is ' accordinS1y>
K =
1.3.5 . . . (til T 1)
For Q(£) we have, then, three expressions given by the equations
Li
(33) 158. — Analogues of Tessera! Harmonics. — We saw
in Art. 148 that the multiplier of M^COSS^ in the spherical harmonic Yn must satisfy equation (30) of that Article. This equation has two particular integrals, DsPn and -D*QM; but, by means of (30), Art. 148, the latter can be expressed in terms of the former. In fact, (30), Art. 148, may be written
0- (34)
If we compare this with (10), Art. 156, we see that
and consequently
48 Spherical and Ellipsoidal Harmonics.
Accordingly, if yt be a particular integral of (34), the other particular integral i/z is given by the equation
and therefore, adopting the notation of Art. 154, if we put Tnt = u^y^ we have
(35)
We may regard Ung as defined by (5), Art. 154, and consider Tni as given by the equation
It is now easy to show that in (35) the value of C is
For
and if we put
1.8.5... (&,-!) 1
— in_8 — ~ = A7, and - = 2,
we get Tnt = Nx"(l + Z\ where Z denotes a series i
Analogues of Tesseral Harmonics. 49
ascending powers of s, then
dz
flr
.(-^V
(l + Z) (l -
but I n
Qn = - != - (ar(«*i) + &c.) V
and
(36)
n + s
- 2"+l + &c.
. „ ,
1.3.0 . . .
Hence, equating the coefficients of the lowest powers of z in the two expressions for Unt, we get
C \ n + s
--'
(2n+l)N ' 1.3.5 . . . (2n +1)'
whence
and as s0 = 0, and therefore XQ = co , we have
It is to be observed that, in order to avoid the introduc- tion of imaginary quantities, Tnt (x) is regarded as having a somewhat different signification according as x < 1, or x > 1.
50 Spherical and Ellipsoidal Harmonics.
In fact, in Art. 148,
and, in the present Article,
)', (38)
but TM (0 = (V - I)' & Pn (£). (39)
It is obvious that, whichever signification be attributed to Ttu(x), it satisfies the same linear differential equation.
159. Expansions for External and for Internal Potential. — We can now write down the series express- ing the potential, inside and outside a prolate ellipsoid of revolution, due to a distribution of mass on its surface.
Let V denote the potential inside, and V that outside, the ellipsoid whose semi-axis major is #£0 ; then we may put
, cos sty + Bnt sin z\ and (40)
(41)
At the surface, where ^ = £0, these two expressions become the "same series of spherical harmonics which can be made equal to any assigned function of £ and d> which is finite and single-valued.
Surface Distribution corresponding to Potential. 51
160. Surface Distribution corresponding to Potential. — If the internal and external potentials, V and F"', due to a surface distribution whose density is a, be given by the equations
<r can be found from the equation at the surface of the ellipsoid £0.
If dsi be an element, drawn outwards, of the normal to the surface, equation (12), Art. 46, becomes
dV dV
3 --- j — = 4?r(T J dsl dsi
but by Ex. 2, Art. 75, ttsi = - '• — , and therefore we have
dTT dt, _i
-+ ^'-
; wlience
= ^r0(^-i)^o ,-
(42)
When the density of the surface distribution is given, the expressions for the potential inside and outside the
ellipsoid may be determined by expanding -, expressed as
a function of !; and ^> in a series of spherical harmonics, and determining each of the functions T(£) and the corre- sponding constants by means of (42). The potentials V and V are then given by (40) and (41).
E2
52 Spherical and Ellipsoidal Harmonics.
161. Potentials of Homoeoid and Focaloid.— If
the surface distribution be homceoidal, the density varies as p, and the multiplier of p on the right-hand side of (42) must be constant. Hence, V = constant,
(43)
•mr
When £ becomes very great, V' tends to become — ,
M *'
that is j-y, where M denotes the total mass of the homceoid.
K(+
Hence C ' = , and
For a focaloidal distribution the density varies inversely as jo, and - varies as — , that is, as ?0 - £*. Accordingly, - is of the form AP0 + BP2 (£), and
It is easily seen that
and that
1 r 1
£ + 1
Hence, by integration, we have
£•
2
V> ' lo +
(45)
Potentials of Homceoid and Focaloid. 53
By a method similar to that employed in the case of the homoaoid, we find that AQ = — , where M is the mass of
the focaloid.
It follows from (42) that the coefficients of P0 and (£2 - £)
in 47r&2£0(£02 - 1) - are A0 and -^^ ; but - varies as £02- £2,
P £o ~ $ P
that is, as £02 - J - (%* - |), and therefore we must have At = - A. Hence
(46)
By Art. 83, the potential of a focaloid in external space is the same as that of the solid ellipsoid of equal mass of which it is the boundary. Hence (46) may be verified by comparing it with (2), Art. 78. This verification is readily effected by taking a point on the axis of revolution. Here
£ = 1, and r = k% ; then putting -z = s, from (46) we get
M( n 3/1 li M
(47)
Again in (2), Art. 78, for points on the axis of revolution, 7 = A ; and, since C = B, we have
A + B + C - 37 _ Mk*
~2~ ="
Hence, putting kZ, for r, we get
r=^ + 5#T3'
which agrees with (47).
54 Spherical and Ellipsoidal Harmonics.
162. Oblate Ellipsoid of Revolution.— When we
have to do with oblate ellipsoids of revolution, Laplace's equation takes the form given by (35), Art. 98. If we put £ = - Z'v/^Tl in this equation, and write 0 instead of x» we get an equation in £', £, and $, which is the same as (1). Hence, in the case of the oblate ellipsoid, we may put for V and V the expressions given by (2) and (4), or by (40) and (41), provided we put £' instead of £. In order to determine completely^ these expressions for Fand V', we have then to put £v/~ 1 f°r £'» and accordingly we have to find what P(£')> Q(O, r(O» and *7"(O become when £v/^l is substituted for £'.
163. Determination of jo (£) and y ($). If we put -l instead of ^ in the expression for Pn (£), given by (26), Art. 147, we get
and we may put
(48) Also, putting </- 1 = {, we have
-P. 00 = tnpn(?). (49)
In like manner, from (31) and (32), we have
(50)
Oblate Ellipsoid of Revolution. 55
It is plain that the right-hand members of (50) and (51) can differ only by a factor which is some power of t from the expressions for Q(/£), given by (31) and (32); and, as the
term involving the lowest power of j is the same in the two forms of qn(Z), they are consistent. From (31), we see that
Qn(£) = cWqn(S). (52)
In order to find a third expression for gn(Z), we must consider what log ^ — - becomes when £ is changed into t£.
If we put j = z = tan 6, we have
. i£ + 1 . 1 - iz . cos 0 - t sin 0 log -J — - = log - - = log - — — - = - 2i9. 0 £ - 1 ° 1 + iz ° cos 0 + i sm 0
Hence,
and
iP»(i?) log
also,
PwC
Accordingly, by (33),
but £„(£) = tn+1QM(t^), and i2n+4 = (- 1)M ; and therefore
- &c (53)
an- - p^ (?) + } P(IJ(?) - &c. .
56 Spherical and Ellipsoidal Harmonics.
164. Analogues of Tesseral Harmonics. — When we put t% for £ in Tnl(K), we get
and we may write
tns(Z) = (? + Also, we may write
^-(P + lfD-fcK); (55)
whence VM(&) = r^i^f?). (56)
Another form of un»(Z) is obtained from (37) from which, by means of (56J, we have
whence
_
(57)
165. Expression for Potentials.— If Fand V denote the potentials inside and outside an oblate ellipsoid of revolu- tion, due to a distribution of mass on its surface, we may write
(58)
(59)
Surface Distribution corresponding to Potential 57
It is plain from what precedes that Fand V satisfy each Laplace's equation, that Fis finite when £ = 0, and V is zero when £ = oo , and that V and V are identical at the surface of the ellipsoid. Hence they satisfy all the conditions required.
166. Surface Distribution corresponding to Potential. — Here we may proceed as in Art. 160. If the internal and external potentials V and V be given by the equations
(60)
and dsi be an element of the normal to the ellipsoid,
\d\ **-—'•
but in this case, by Art. 98, we have X2 = A-3(£2 + 1), and therefore
P
Accordingly, as in Art. 160, we get
and
Ap
-
(61)
"\yhen the density of the surface distribution is given, the potentials may be determined in a manner similar to that described in Art. 160.
58 Spherical and Ellipsoidal Harmonics.
167. Potentials of Homoeoid and Focaloid. —
These may be obtained from (60) and (61) iu a manner similar to that employed in Art. 161 ; but if the expressions given by (44) and (46) are already known, we can get from them the corresponding expressions for an oblate ellipsoid.
Putting i£ for £ in (44), we get
Hence we may conclude that P"', the potential of an oblate homceoid of revolution, is given by the equation
F/=fta'rl£' (62)
It is easy to see that this expression for V must be correct, since it satisfies Laplace's equation, vanishes at oo ,
is constant at the surface, and tends towards the value — ;
at points very distant from the centre when r tends to become equal to k£.
To get the potential of a focaloid in external space, we may put t£ for £ in (46), and we get
since i = t4, and t2 = - 1, this may be written
Hence we may put
(63)
Potentials of Homwoid and Focaloid. 59
This expression for V satisfies Laplace's equation, vanishes at infinity, and at points very distant from the centre tends
towards the value — ; but to prove that it satisfies all the con- ditions of the question, we must show that the corresponding
distribution of mass varies as - •
P
If the external potential Vf be given by (63), the internal potential F", due to the surface-distribution producing V, is given by the equation
(64)
since this expression for V satisfies Laplace's equation, remains finite inside the ellipsoid, and is equal to V at the surface. We have, then,
_dV _ dT
(65) Differentiating and reducing, it is easy to see that
Hence, '
60 Spherical and Ellipsoidal Harmonics.
and at the surface we have
dV dV M £a + g2
p~
where (7 denotes a constant.
C1 1
Accordingly a = . ^ ., -, and therefore the distribution 47T& C0 ;?
of mass producing the potential is fooaloidal.
SECTION III. — Ellipsoids in General.
168. Ellipsoidal Harmonics.— When the surface, at which the potential or mass-distribution is given, is an ellipsoid not of revolution, the preceding methods are inap- plicable. The most general method of determining solutions of Laplace's equation which can be made use of in questions of this kind depends on the employment of functions called ellipsoidal harmonics.
We have seen, Art. 92, that if X, /«, v be the primary semi-axes of the three confocal quadrics passing through a point, Laplace's equation may be written in the form
o*-*S? + <v-oy+e»-*>J£-* a)
where a, j3, y are given by (17), Art. 92. If a, b, c denote the semi-axes of an ellipsoid of the confocal system, we may change the variables by assuming
If we put X = (a2 + £)* (ft2 + £)* (<* + £)*, by (17), Art, 92, we have
d_=d\d% d__<& d_ da da d\ dt, k d£'
Ellipsoidal Harmonics. 61
If we assume
D = («' + n)* (&2 + ,)* (c2 + i,)* 3 = («* + 0* (&* + £)* ('* + 0*, in like manner we get
rf 2§)y/ri rf rf_=23^
rf/3 A rfi,' dy /r ^'
and Laplace's equation becomes
(4)
Following the analogy suggested by the methods em- ployed in the case of ellipsoids of revolution, we may suppose V to be the product of E, a function of £, and of H a function of r\ and H' a function of £. If these functions be such that
and H'-KH-^J?', (5)
where m and y are disposable constants ; we may put V = CEHH', where C is an arbitrary constant, and we have
rr- A f(i-Q(«
where A denotes the other factor of V2 F", since the expression inside the bracket vanishes identically.
We have now to find forms of the functions E, H, and H which will satisfy (5).
If we suppose E to be a rational and integral function of £, we may put
E - |» + l$«-i + ....+ p..
62 Spherical and Ellipsoidal Harmonics.
Operating on £n, we get
g) + (fr+ £)(* + 0 + (c* + £)(«'
+ « (n - |)(W + 6V + cV)
gn
Hence, in order that
we must have the series of « + 2 equations
n (n + J) = m, (n - 1) (n - £) />i + n2 (a2 + i2 + c2) =/ i- »np,,
w (« - i) (a'J2 + 6V + cV) + (M - I)2 (a2 + 62 +
+ (« - 2) (w - 1);;, - jpl + mp9, &c.,
The first of these determines m, the second pi as a linear function of/. By substituting this value of pi in the third, p-t is determined as a quadratic function of /, and so on. Thus, finally, an equation of the degree n + I is obtained to determine/. Each root of this equation corresponds to u set of values of pl} ;>2, &c., which furnishes a function of the required form for E. There are thus n + 1 functions of the degree n in £ which are of the required form. It is plain that £ is of the second degree in the coordinates x, y, 2. Hence the forms of V corresponding to those found for E must be of even degree in each of the coordinates x, y, z. To determine forms of odd degree in these coordinates, we are guided by the formulae for expressing the Cartesian in terms of the elliptic coordinates of a point.
Ellipsoidal Harmonics. In fact (Salmon's Geometry of Three Dimensions],
.r2
(6)
If we consider only the factor containing £, we see that x corresponds to
v/V + £, y to v62 + £, and % to and we are thus led to consider whether
and
where fij is a rational and integral function of £, are possible forms of E.
Operating on v/«2 + £ fw, we get
' (7)
[nV + (n + ^)2 (62 + c2)]|n + [» (M - |) a2 N (» + i) 6V] C"-1 + n(w - l)«2iV5n-8}
Hence, we see that if E = -y/V + ^ Sn, where Kn is a rational and integral function of £ of the degree n, we have
and therefore, that by properly determining m,y, and the coefficients in !En, we can make
64 Spherical and Ellipsoidal Harmonics.
It is easy to see that the final equation for determining j is of the degree n + 1.
If we operate on the functions
we find that
~Yj{(«'+
+ [(«
+ n (n + J) («2 + &2K] I"'1 + w (n -
(8) and that
(9)
x (n +
Hence we conclude that
and that
and, accordingly, that */{(a*+ ^)(62+ |)!^n and 3Efi» are possible forms of E.
If J£ be of the degree v in ^, and v be an integer, the forms we have found for E are
We have found also that there are v + 1 different functions of the first type, and v of each of the others ; so that there are 4v + 1 in all.
Ellipsoidal Harmonics which vanish at Infinity. 65
If v = n + |, where n is an integer, the forms found for E are
Also, there are n + 1 functions of each of the first three types, and n of the fourth, so that there are 4n + 3 ; that is, 4v + 1, in all.
Hence, in any case, there are 4v + 1 determinable func- tions of £ of the degree v, any one of which may be taken for E in order to satisfy (5).
It is plain that if H be the same function of rj, and Hf of £, as E is of £, the product CEHH', where C is an arbitrary constant, will then satisfy Laplace's equation.
169. Ellipsoidal Harmonics which vanish at Infinity. — The functions considered in the preceding Article do not vanish at infinity, and are therefore unfit to represent the potential of a finite mass in space outside itself. The form of the differential equation for E enables us, however, to obtain another function which will fulfil this condition.
In fact, if
we have
y =
but, as was shown in Art. 156, by assuming y = 3/1 w, if y^ be a solution of the equation
d?y dy ,.
— *- + Zi -/- + X2y = 0,
dxz dx
then
is also a solution.
66 Spherical and Ellipsoidal Harmonics.
In the present case, fXidat** log £ ; and, therefore, if E be a solution of (10), so also is
CE
If now we take for E one of the forms found in the last Article, by writing E as the product of £" and a series of descending powers of £, it is plain that
vanishes when f is infinite.
Hence we see that, if V denote the potential inside an ellipsoid, whose semi-axes are a, b, c, of a distribution of mass on its surface, and V the potential of the same distribution in external space, and if V = CEHH', then
V = C'EHH
where
C"
,r *
Jl *tf"
r # c
J.CT a
170. Ellipsoidal Harmonics expressed as func- tions of Cartesian Coordinates. — If Bn be a rational and integral function of £ whose factors are £ - alt £ - a2, £ - an, and E = £„, then
GEEK- («'+ ai)(&'+ ai) (c»+ ai)f
Z -1
*' -1
+ as c* -I- a,
(11)
Functions of Cartesian Coordinates. 67
For the expression
vanishes when u = £, or u = rj, or u = £, where £, ij, £ are the elliptic coordinates of the point whose Cartesian coordinates are x, y, z.
Hence, whatever be u, we have
-JT, (5 -«)(,-«)({:-«),
and therefore,
-JT.CS-aOdi- «,)(£- a,),
where JTi and ^2 denote quantities independent of the coordinates.
Hence, if we denote the right-hand member of the equation (11) by &, we have
x (£-a,)(£-a2)... (£-«„)
By using the expressions for x, //, and s given by (6) we find, in like manner, that when
~ It* -*»(*# I' (12)
= V/K«2 + £)^2+E)}#n |
>, (13) (14)
(a2 - bt
We have seen that there are 4v + 1 ellipsoidal har- monics of degree v in £, that is, of degree 2v in Cartesian coordinates.
F2
68 Spherical and Ellipsoidal Harmonics.
A rational and integral function of x, y, 2, of the degree i, contains
(i + 1) (i + 2) (*' f 3)
i - ' v /v - - constants ;
D
but if it satisfies Laplace's equation, these constants must satisfy
(t-l)i(*+l)
-*-^~ -'equations,
and therefore such a function contains only (« + I)2 indepen- dent constants. Now, if we take all the different ellipsoidal harmonics from the degree 0 up to the degree i in x, y, 2, or \i in £, we have 1 + 3 + 5 . . . . + 2i + 1 in all ; the sum of this series is (t + I)2.
Hence, as each harmonic may be multiplied by an arbitrary constant, we can express any rational and integral function of z, y, 2 of the degree *, which satisfies Laplace's equation, by a series of ellipsoidal harmonics, whose degrees in x, y, 2 range from i to 0.
At the surface of an ellipsoid of the confocal system any rational and integral function of x, y, z can be expressed as a series of ellipsoidal harmonics.
For, if a, b, c be the semi-axes of the ellipsoid, at its surface
and therefore, by multiplication, the degree of any function of x, yt 2 can be increased by 2 without altering its value. Hence a rational and integral function of the degree i can be reduced to two homogeneous functions of the degrees i
and i - 1. Of these, the first contains (J.+ 1K*'+2) inde_
2
pendent constants, and the second * (* * ^ . Hence the two
together contain (t + I)2 independent constants, and can therefore be expressed as a series of ellipsoidal harmonics.
Surface Integral of Product of Harmonics. 69
171. Surface Integral of Product of Harmonics. —
If K: = E.H.H',, and F2 = E,H,H\, and 8 and S' denote two confocal ellipsoids of the system, whose normals drawn into the space between them are v and v', by Green's theorem, we have
also dv = £- , where p is the central perpendicular on the tangent plane to S, and dv = - -?, and therefore
that is,
At corresponding points on the surfaces S and S' the coordinates TJ and £ are the same, and therefore, so also are the values of Hly H\, Hz, H\. Also, by Ex. 7, Art. 90, the volume elements pd8 and p'dS' are proportional to the products of the semi-axes of S and $', that is, to X and £'.
7 Tjl
Hence, as E and -&- are constant over the surface S, we have at,
If we equate to zero the first factor of the left-hand side of this equation, we get
70 Spherical and Ellipsoidal Harmonics.
Since one surface may be taken as fixed and the other as variable, this equation is equivalent to
£#i2 -= ( -jr ) = constant = C,
whence E* = CEl j^-2 + C'E,.
Accordingly, either Ez and E^ differ only by a constant factor, or E* is the external harmonic corresponding to the internal JSi. In either case Hi is the same as H2, and H\ as -Z7'2. If we reject the alternatives considered above, we must have
and therefore,
\V\VzpdS = 0.
Hence we conclude that the surface integral of the pro- duct of two ellipsoidal harmonics and the central perpendicular on the tangent plane, taken over an ellipsoid of the confocal system, is zero, unless the two harmonics have a constant ratio to each other, or be a corresponding pair of harmonics, one internal and the other external.
172. Identity of Terms in equal Series.—//* two
series of internal or of external harmonics be equal to each
other, each harmonic of one series must be identical with a
corresponding harmonic of the other.
To prove this, let the series
Fo + F, + F2 . . . Vn = U0 + ZTi . . . + Un ;
multiply each side of this equation by 7mpdS, and integrate over the surface S, then all the integrals on the left-hand side vanish except J Vm2pdS, and on the right-hand side they all vanish, unless Um = CVm, in which case we have
\UmVmPdS= C\Vn*pdS.
Hence 0=1, and a harmonic of the right-hand series is identical with F.
Density of Surface Distribution. 71
If two series of harmonics be equal throughout the whole of the space inside or outside an ellipsoid, both series must be composed of harmonics of the same kind, either internal or external, since an internal harmonic becomes infinite in external space at an infinite distance from the centre, and the differential coefficient of an external harmonic becomes infinite at the focal ellipse in the plane of xy.
It is easy to show by multiplication and integration over the surface 8 that, if two series of harmonics be equal at the surface of the ellipsoid S whose semi-axes are a, b, c, and one series be composed of internal harmonics, the other of external ; then, if a term Vm of the first series be given by the equation
Vm = EmHmH'm, there must be a term Um in the second such that
Um = where
•r
3EA
173. Density of Surface Distribution producing given Potential. — If V denote the potential inside the ellipsoid a, b, c of a distribution of mass on its surface, and V the potential in external space of the same distribution, and if
we have seen, Art. 169, that
In this case, if a denote the density of the distribution. we have
dV dV
— + - — r + 47r<7 = 0 ; dv dv
but dv' = y dv = ~'
72 Spherical and Ellipsoidal Harmonics.
whence
2™ = dv _ d_r
that is,
If the potential due to the surface distribution whose density is a- be the sum of a number of harmonics F0, Fi, F2, &c., it may be shown in a similar manner that
?-. - 0. + . JJT, + - JTJT. + &o., (16)
^ (A i)0 (/i»)0
where
" " &c.
When o- is assigned, (16) enables us to determine the functions £Tj, H\y &c., and from thence Fi, &c.
174. Potential of Homoeold. — As an example of the mode of procedure described in the preceding article, we may find the potential of a homoaoid. Here u varies as p,
and - is constant, whence P
To determine C0 we have
— = 2 ) — = - ^— ° dv' d% 3E0
Hence, — — - $pdS = fNdS
where M is the total mass of the homcaoid, but fpdS = 4ir«bc,
Ellipsoidal Harmonics of the Second Degree. 73
and therefore 2(70= M\ accordingly, F', the potential of the homceoid in external space, is given by the equation*
This agrees with the result found in Ex. 3, Art. 75.
175. Ellipsoidal Harmonics of the Second Degree in the Coordinates. — The forms of E which correspond to functions of the second degree in the coordinates are
and ((• - a).
There are two functions of the last form, f - a\, and £ - a2. We proceed to determine the values of eti and a2. By (5), Art. 168, we have
= | f + (a2 + b* + c2) ^ + i- («8 whence
n = -|, j - ma = a? + b~ + c2, - ja = | (a2b* + bzcz + c2a2).
Eliminating m and/, \ve obtain
3a2 + 2(a2 + bz + c2)a + a262 + 62c2 + c2^ = 0. (17) Hence,
(a^+c)+v("T } i
74
Spherical and Ellipsoidal Harmonics.
176. Reduction of Solid Harmonic of the Second Degree to Ellipsoidal Harmonics. — The general form of a solid harmonic of the second degree is
rhere
0.
As regards the terms containing products of the coordi- nates, we have only to substitute for each product its value in terms of f, »», and £. Thus, 2f\yz becomes
and the remaining two products are reduced to expressions of a similar kind.
We may next assume
where a± and at have the values given by (18). Jn this manner we get
«!«• + drf + a3s2 = C8 + Ci (a2 + a,)(68 + a,)(c2 + a
«2 4 a,
i ' Cs + a, ) 2 4 o,)
«-
Hence we have
CM&2 -fa,) (c2 +«,)+ (73(62
4- ai)(rt2 + a,) + Ca(c* 4- a,)(a» + o,).
(19)
Reduction of Solid Harmonics. 75
In virtue of equation (18), which determines a: and a2, and of the relation between «i, «2, and a3, the values of <7i and (72, found from (19), satisfy the equation
«3 = Ci (a2 + 01) (b* + ai) + C2(a2 + 03) (b2 + a2). From the last two equations (19), we find
„ (iiaz-chbz + (a\ - fl2)a2 ~ di* - * -
I = 2z ' Z =
(20)
At the surface of the ellipsoid a, b, c, the quadratic function a\x* + (hy2 + a3zz can be reduced to ellipsoidal harmonics, whatever be the values of a\, aZy and a3. For, in this case, we have
and therefore we may substitute for the given function the expression
We can then determine e so as to satisfy the equation
and putting al + — for «i , and «z + n for tf2, proceed in
the same manner as before. In this case, the right-hand side of the first of the equations (19) is not zero but - e, so that
C0 = <?, («2 + ai)(~j* + ai)(6-2 + o,) + C7,(a* + a8)(6s 4 a2)(c2 + 02) - e.
76 Spherical and Ellipsoidal Harmonics.
177. Potential of a Focalold.— We have seen, Art. 83, that, for a focaloid distribution of mass, the surface density a is given by the equation
2K
47HT = ,
P
where
the density of the solid ellipsoid of equal mass being denoted by p. By Art. 173, the potential £7 of the focaloid in internal space is given then by a series of the form
°J0 * J0 Eft Jo E&
provided that
2irabc _ Kobe K ~~ "
It is plain that an expression of the form
A0 + A^ri - o,)(^ - o.) + Ai(*i - a»)(£ - a2)
TS~
can be made identical with — ij£ by suitably determining
the constants A0, A\> and At. To determine these constants is unnecessary for our present purpose ; but if we proceed to do so, we shall find
Components of Attraction of Focaloid. 77
By substituting the equivalent expressions in #, y, and z for the functions
(S-aOOt-o.XS-aO and (g - «,)(, - „,) (£ - «,), the potential Z7 becomes of the form
where a0, &o., are constants.
178. Components of Attraction of Focaloid and Homogeneous Ellipsoid. — From the form of £7 it follows that the components of force in internal space due to a focaloid are -
At the outside of the surface the components of force become px
- 2«i # + 47TCT — , &C.
Hence, if X, T, Z denote the components of force at the outside surface of the focaloid,
(21) By Art. 83, the attraction of a solid homogeneous ellipsoid
in external space is the same as that of a focaloid of equal
mass on its surface.
Hence, if X, Y, Z denote the components of force due
to a solid ellipsoid, by (21), we have at its surface
X-M, Y-hy, Z=b*z,
where ix, b2, and b$ are constants.
Since X satisfies Laplace's equation in external space, and is equal to
at the surface of the ellipsoid, we must have at any point in external space
X = ax\ „—• — TT , where a ^-r-, — =r = bi . Ji *(«' + £)' J0 £(«' + £)
78 Spherical and Ellipsoidal Harmonics.
In like manner,
Z =
At a great distance r from the centre, towards the values
MM , M
-x, -y, and -«,
, and Z tend
where J/ denotes the mass of the ellipsoid.
If we expand in descending powers of £ the functions under the integral sign in the expressions for X, F, and Z, and integrate, we find that X tends towards | «#£"§, and that P tends towards f/3y£~t, and Z towards f7z£-f.
Hence we have a = /3 = 7 = fJ/, and we get
r =
Z-fJf.l -
(22)
179. Potential of Ellipsoid in External Space.—
If V denote the potential of the ellipsoid in external space,
V = - J (Xdx + Tdy + Zdz) + constant. Integrating by parts, we find
f dxx r —~- - r d^ {**
where in the second integral x is to be regarded as a function of £, y, and s ; and y and z are to be looked upon as constant in the integration.
Potential of Ellipsoid in its Interior. 79
Similar results hold good for the functions contained in T and Z. Hence we have
_ No constant is to be added, since the right-hand side of this equation vanishes at infinity. The three integrations involved in the last integral on the right-hand side afe to be performed on three different hypotheses; but, as
we have, finally,
(23)
fUf 'eeflhf this.r.esult a£rees with (11), Art. 87, remembering t that equation *, y, , are not regarded as functions of u.
180. Potential of Ellipsoid in its Interior.— The
-f force due to a focaloid at i\
that is, if we put
they are Ax, By, and Gz.
80 Spherical and Ellipsoidal Harmonics.
Hence, at any point inside the focaloid, if these component^ be denoted by X, Y, and Z, we have
For these expressions satisfy Laplace's equation throughout the interior of the focaloid, and take the proper values at i$ inner surface.
Hence the potential U of the focaloid is given by the 1 equation
U= -
where ^40 denotes an undetermined constant.
By Art. 83, if F denote the potential of the ellipsoid inside itself,
Hence,
V = K+ A, - \(Ax* + By*
At the surface of the ellipsoid, V = V, and therefore, by (23), we have
accordingly,
^ ^r ^ 2r ^ .r_^_
-n-^^-^J.(l?Tp-'J.5r^
(24)
Magnet of Finite Dimensions. 81
CHAPTER IX.
MAGNETIZKD BODIES.
SECTION I. — Constitution and Action of Magnets.
181. Magnet of Finite Dimensions. — A magnetized body is composed of elements each of which is a magnetic particle (Art. 17). When such a body is placed in a uniform field of magnetic force, each particle is acted on by a couple, and the resultant of all these couples tends to bring the body into a position in which a certain line in the body is in the direction of the uniform force. When the body is in the position in which this couple is the greatest possible, its ratio to the force is the magnetic moment of the body.
If a body be composed of a number of infinitely thin parallel bars, magnetized at their extremities so that the pole strength of each bar is proportional to its orthogonal section, it is plain that the magnetic moment of the body is proportional to the sum of the products obtained by multiplying the length of each bar by the area of its orthogonal section — in other words, to the volume of the body. Hence we may assume that the magnetic moment of an element of a magnetized body is proportional to the volume of the element, and we may denote this magnetic moment by the expression Id<&, where d<& denotes the volume of the element, and / the intensity of magnetization. This latter is defined as the ratio of the magnetic moment of the element to its volume. Magnetization is a directed quantity, and its direction is that of the parallel bar magnets which are regarded as composing the element whose magnetic axis is a line in this direction.
82 Magnetized Bodies.
182. Potential of Magnetized Body.— By (28), Art. 54, the potential of one element of the body is
If s, y, 2 denote the coordinates of the element ; f , TJ, £ those of the point at which the potential is required ; A, /u, v the direction-cosines of the magnetic axis of </@, we have
/, dr dr dr\
= -(\-r + u-r + v—}
\ dx ^ dy dz)
The quantities 7X, 7/u, and Iv are termed the components of magnetization, and may be denoted by A, £, and G. If V denote the potential of the magnetized body, we have, then,
. (1)
If I, mt n denote the direction-cosines of the normal to the surface S, which is the boundary of the body, we get, by integration,
ff// m^s \\\(<*A dB <tC\d®
V - \\(IA + mB + nC) -- \-r + -5- + ^r) — • JJ r / / / \ dm dy dz J r
(2)
Hence the potential of a magnetized body is equivalent to that of a volume-distribution, throughout the space occupied by the body, of mass whose density is
dx dy
together with a distribution on the surface bounding the body whose density is
IA + mB + nC.
Poisson's Equation. 83
183. Poisson's Equation. — From the expression for the potential given by (2) we have
(3) dy dz
at any point inside the body.
In space outside the body Laplace's equation obviously holds good.
184. Examples of Magnetized Bodies. — As an
example of a magnetized body, we may take a sphere magnetized in a uniform direction so that the magneti- zation at any point is a function of its distance from the centre. Here, if r' denote the distance from the centre of any point of the sphere whose coordinates are x, y, z' ; V the magnetic potential of the sphere at an external point whose coordinates are x, y, z ; and r the distance between the points xyz and x'y'z', we have, the direction of magnetization being parallel to x,
where X denotes the component of force due to a sphere whose density at any point is /(/) Hence
where « denotes the radius of the sphere, and therefore the magnetic action of the sphere at an external point is the same as that of a small magnet at the centre whose magnetic moment is expressed by
4ir
If the magnetization be of uniform intensity /, the magnetic moment becomes
47T/
G2
84 Magnetized Bodies.
If an ellipsoid be uniformly magnetized in the direction of its longest axis, the potential F, at an external point xyz, is given by the equation
'(!)«„./' [2- jj
ei\rj dx] r
de
where X denotes the component of force of a solid homo- geneous ellipsoid whose density is /. Hence by (12), Art. 87, we have
V = 2irlabcx
du
\i / • a» (4)
where g is the greatest root of the equation
(See Ex. 1, Art. 52.)
If the integral in (4) be denoted by £, and the corre- sponding integrals for the other two axes by §J and 3» it is easy to see that the potential V of an ellipsoid, uniformly magnetized in a direction inclined to the axes at angles whose cosines are A, /u, v, is given by the equation
V = 2;r labc (A.r£ + py® + v*3)- (5)
From (5) we can obtain the components of the magnetic force exerted by the ellipsoid at an external point. By differentiation we have
d» -v "*dp W^TW^;'
but
^ _ 1 rfy
^ "
Examples of Magnetized Bodies. 85
If we denote the semi-axes of the ellipsoid passing through the point xyz and conf ocal with the given ellipsoid by a', b', c', we have
Hence, Up' denote the central perpendicular on the tangent plane to the ellipsoid ab'c' at the point xyz, we have
due
Hence ^
rfa?
In like manner,
rf# rt3 3 c flfo ft
accordingly, by substitution, we obtain
a'b'c' «'2 V «* b" t
If TOJ, w2, t73, and 5r denote the angles which the normal to the ellipsoid a'b'c' at the point xyz makes with the axes, and with the direction of magnetization, we have
p'x _
—ft = cos OTU &c. ; a
and cos S = X cos OTJ 4- n cos t<r2 + v cos w3.
Hence we obtain
dV 4-trIfibc
— = Zirlabc\£ -^rrr- cos S cos -ssl ;
dx ab c
86 Magnetized Bodies.
and, since the components of magnetic force a, /3, y are
f?V expressed by --T» &c-> we have
. , , COS S COS OTi -
abc
cos 3
4irlabc _, o r j. i'
•v = — 777-7- cos.31 cos TOS - Zirlabc 1/3 . ' a o c
(6)
From equations (6) it appears that the force exerted by the magnetized ellipsoid abc at an external point P is the resultant of two forces of which one is in the direction of the normal at P to the ellipsoid «'6Y, and is expressed by
the other is the force due to a homogeneous solid ellipsoid, coinciding with abc, at the point Q in which a line drawn from the centre in the direction of the magnetization meets the surface of the ellipsoid a'b'c', the density of the solid
ellipsoid, supposed attractive, being -^7, where M denotes the distance of Q from the centre.
185. Potential of Magnetized Body expressed as Sum of Force Components. — Adopting the notation of Art. 182, we have, by (1),
V = - — ( —— - —
~~~~""
Hence, if we suppose three bodies geometrically identical with the magnetized body, and having for densities A, B, and (7, the magnetic potential is equal to the sum of the force components exercised by the first body parallel to the axis of x, by the second parallel to the axis of y, and by the third parallel to the axis of z.
Magnetic Force. 87
186. Magnetic Force. — The differential coefficients of the potential with their signs changed are termed the com- ponents of magnetic force. Outside the magnetized body these are the actual components of the force which the body would exert on a north magnetic pole of unit intensity.
Inside the body the actual force due to the body is indeterminate. In order to imagine that such a force should act, we must suppose a small cavity inside the body, and, in the case of a magnetized body, the force depends on the shape of this cavity.
The components of the magnetic force are usually denoted by the letters a, /3, 7.
It is easy to see that the normal component of the magnetic force as defined above is not continuous when we pass from the outside to the inside of the magnetized body. This follows from the consideration that the normal compo- nent of that part of the force due to the surface distribution I A + mB + nC is diminished by 4ir(lA + mB + nC).
187. Magnetic Induction. — We can obtain a vector quantity whose components satisfy the solenoidal condition, and whose normal component at the boundary of the magnet is continuous, by adding to each component of magnetic force the corresponding component of magnetization multiplied by 47r. This vector quantity is termed the magnetic induction, and its components are usually denoted by the letters «, b, c. We have, then,
(8)
Outside the magnet a = a, b = |3, c = y ; and
Inside the magnet
da db dc . fdA dB dC
_++_ = _ V2F+ 4ir(-r- + -r- + d% dri dZ, \dx dy
and, by (3), the right-hand member of this equation is zero ; accordingly, a, b, and c always fulfil the solenoidal condition.
88 Magnetized Bodies.
At the surface bounding the magnet, in passing from a point outside to a point inside, la + m/3 + ny is diminished by 4?r (IA + tnJB + w<7) ; but outside the surface la + mb + nc is the same as la + m/3 + My, and inside the former exceeds the latter by 4* (I A + mB + nC).
Hence, in passing through the surface, the value of la + mb + nc is unaltered.
It is now easy to see that the surface integral
(fa + mb + nc) dS
taken over any closed surface is zero.
If the surface be altogether outside or altogether inside the magnetic body, this follows from taking the volume integral of
da db dc
If the surface S be partly outside and partly inside the magnet, the enclosed volume is divided into two parts by the intercepted portion of the surface of the magnet. Through each of these parts the integration may be effected, and in consequence of the continuity of the normal component of magnetic induction, the two surface integrals which are taken over the portion of the magnet surface are equal in magnitude and opposite in algebraical sign, and therefore the surface integral of induction over the closed surface S is zero.
188. Magnetic Force and Magnetic Induction regarded as Forces. — If we imagine a small cylindrical cavity whose axis is in the direction of magnetization, and a north magnetic pole of unit intensity placed at the middle point of this axis, the actual force acting on this pole is the magnetic force when the cylinder is long and narrow, and the magnetic induction when the cylinder is short and broad.
As the cavity is supposed to be small, the removal of the volume distribution with which it was occupied produces no sensible change in the force acting on the magnet-pole, and this force is therefore due to the volume distribution through- out the magnet, the surface distribution on its boundary, and to the surface distribution on the surface bounding the cavity.
Energy due to Magnet. 89
In the case of a cylinder parallel to the magnetization axis, I A -f mB + nC is zero except at the plane ends, where it is - / at the positive end, and + / at the negative. By (3), Art. (14), the force due to the surface-distribution is, therefore,
<_rfi c \
in the direction of magnetization, where c denotes the semi- axis of the cylinder, and a its radius.
When c is large compared with a, this expression becomes zero ; and when a is large compared with c it becomes 4-Tr/. Hence, in the first case, the components of the total force acting on the magnet-pole are a, /3, y ; and in the second /3 + 4irB, y
189. Energy due to Jlagnet. — When a magnet is placed in an independent field of force, if V denote the potential of the field at any point where there is a south pole of strength 2ft, the energy due to the presence of this pole is -WIV, and that due to the corresponding north pole is
where dh is the axis of the particle whose poles are 9ft and - 2ft. Hence the energy due to the particle is
If A, n, v be the direction-cosines of dh, we have dV dV dV dV
also, sffldh = Id&, and therefore
^
dh \ dx dy
90 Magnetized Bodies.
Consequently, if W denote the energy due to the presence of a magnet in an independent field of force,
where A, £, C denote the components of magnetization of the magnet at any point where the potential of the field is V.
190. Energy of Magnetic System.— When the field of force is due to the magnets which are present, it is plain that if the magnetization be everywhere increased in the same ratio, the potential is likewise increased in this ratio. Hence, by reasoning similar to that employed in Art. 50, we see that, if JFdenote the energy of a magnetic system, and V its potential at any point, we have
If we integrate by parts the expression for W given by (10), we get
where the last two integrals are taken throughout the whole of space.
191. Vector Potential of Magnetic Induction.—
"We have seen, Art. 187, that a, b, c, the components of mag- netic induction, fulfil the solenoidal condition throughout the whole of space, and that the surface integral of induction over any closed surface is zero. From hence it follows that this surface integral has the same value for any two surface sheets having a common boundary.
Stokes' 8 Theorem. 91
Hence the integral of induction taken over a surface- sheet 8 must be expressible as a line integral taken round the curve s which is the boundary of S. We have, therefore, an equation of the form
(la + mb + nc] d8=\(F- + G^L+H- j\ ds ds ds
The directed magnitude of which F, O, H are the components is called the vector potential of magnetic induction.
192. Stokes's Theorem. — If u, v, w denote three func- tions of the coordinates, Stokes's theorem is expressed by the equation
ds ds ds
where S is a surface-sheet, and s the curve which forms its boundary.
To prove this, we observe that the terms in the surface integral which contain w may be written
(((dw m dw\ 77CY (((dw m dtv^
\\(Ty--<r*r8' or IK*"*
In this double integral x is regarded as a function of y and z given by the equation of the surface S.
If -7- w denote the differential coefficient of w taken on dy
this hypothesis, we have
d die div dx dy dy dx dy '
but the differential equation of the surface may be written Idx + m dy + n dz = 0,
j ,, „ dx m
and therefore — = - — •
dy I
92 Magnetized Bodies.
d dw m die Hence -r- M ' = 3 r ~r- >
and ( / ?w -r- } dS = \\ ~T~ w dy <
If the terms containing u and v in the double integral be treated in a similar manner, we obtain the right-hand member of (12). If the axes be drawn in the usual manner so that counter-clockwise rotations round x, y, and z bring // to s, 2 to x, and x to y, equation (12) shows that the direction of integration round s is counter-clockwise as viewed from the positive end of the normal to S.
If the surface-sheet S be contained between two curves, the surface integral is equal to the difference between two line integrals.
193. Determination of Vector Potential.— It follows from Stokes's theorem that as a consequence of (12) we may assume
dH dG
dF dff
= --'
_ dG dF ~ ~
(13)
where f, i», £ denote the current coordinates. If FI, <?„ Hi be three functions of £, »j, £ satisfying these equations, it is plain that they will be satisfied also by
where $ is any function of the coordinates.
We see, then, that equations (13) are not sufficient to determine F, 6, and H, and we may assume
dF dG dH
Determination of Vector Potential.
93
From (13) we have db da d*F d2
whence
= ^(^ + ~ -f^l-V'tfj (15)
Equation (15) is similar in form to that for determining the potential of an attracting mass. Hence apparently we have
H = ( V --r]—>
JJJ \dj- dy J r
the integral being taken through the whole of the magnetized body. This integral is, however, indeterminate, as at the surface A, B, and C are discontinuous, and their differential coefficients in the direction of the normal infinite.
If we integrate by parts inside the boundary of the magnet, we get for H the expression
j(
lB-mA}dS+\(A^-B^V- J J V dy
We may therefore assume
F =
G
dz
(16)
provided these forms satisfy the differential equations (13) and (14).
94 Magnetized Bodies.
It is easy to see that this is the case, for since <fr dr
and A, B, C are not functions of £, »j, £, we have dH 'dQ
5|J\ rfa? ^ rfz/ r J r
(17)
but, by (1), the first term in the right-hand member of (17) is a, and in space outside the magnet the remaining term is zero, and inside the magnet, when £»}£ coincides with ,*y/s, its value is 4irA. Hence we obtain
dH dG
— --- ^ = a + 4irA = a. (18)
rfi! rt£
» J7I
We may write — in the form
d2 (C 7, r/2 f^ M 75J- - rf@ - -77-^ — a&8 ; /^rfnj r ^5J r
. rfG' , ^J3" .
and expressing — and — r in a similar manner, we see Ctrl f/t,
that
dF dG dH
vanishes identically.
Vector Potential of Magnetic Particle. 95
194. Vector Potential of Magnetic Particle. — In
the case of a magnetic particle equations (16) become
where 20? denotes the magnetic moment of the particle, and \, ju, v the direction-cosines of its axis.
If 01} Bz, 03 denote the direction-cosines of r, we have
d_ 1 = g-g = 03
dz r r3 r2'
whence
but ju6/3 - vOz = siri e cos &!,
where Si denotes the angle wliich a perpendicular to r and the magnetic axis makes with the axis of z, and € the angle between r and the magnetic axis. Hence
• SO? sin £ 9tt sin £ $? sin e
.F = - - — cos $•,, Gr = - -2 — cos S-o, // = -- - — cos $3.
Accordingly, the magnitude of the vector potential of a magnetic particle at any point is - - — , and its direction
is perpendicular to the axis of the magnet and the line join- ing its centre to the point.
If rotations from x to ?/, from y to z, and from z to x, be counter-clockwise, the rotation from the magnetic axis to radius vector is counter-clockwise as viewed from the positive end of the vector potential. The vector potential of a magnet of finite size is the resultant of the vector potentials of the magnetic particles of which it is composed.
195. Magnetic Moment and Axis of Magnet. — The potential energy of a magnet plaoed in a uniform field of force is determined from (9) by regarding a, /3, 7, the components of force in the field, as constants ; we have then
(19)
96 Magnetized Bodies.
If we assume J Ad<& = Kl, J B<«5 = Km, J CM® = 7T«, /* + m2 + «« = 1,
then /, m, n are the direction-cosines of a line, and we have W = - KH cos 0, (20)
where H denotes the resultant uniform force, and 6 the angle between its direction and that specified by /, m, n. This latter direction is fixed in the magnet, and the direction of H is fixed in space. Hence the magnet is acted on by a couple
expressed by - -^-, that is, - KH sin 0, which tends to
diminish 0 and make the line I, m, n coincide with the direction of H.
Accordingly, the magnetic moment of the body, Art. 181, is expressed by K, and /, m, n are the direction-cosines of the magnetic axis.
If the potential of a magnet be expanded in a series of harmonics so that at an external point P, we have
where r denotes the distance of P from the origin, the first term ~ vanishes, since the total magnetic mass is zero, and
Y.
in the second term, — , the spherical harmonic Yl is - Kco& 0.
This is easily seen if we consider the expression for the potential energy W due to the presence of a mass at the point P. In this case W is given by the equation
If we now suppose ;• to become infinite, but ^ to be finite
and equal to H, we have W = H F,, but as the energy is that due to the presence of the magnet in a uniform field of force whose intensity is H, we have W = - HK cos 8.
Hence Y, = - K cos 8.
Magnetic Shell. 97
196. Magnetic Shell.— A magnetic shell may be defined as a surface magnetized at each point in the direction of the normal.
In this case, the expression for the magnetic moment of an element of the body is of the form I dv dS, where dS denotes an element of the surface, and dv an element of it s normal. The total magnetic moment of such a body is in general infinitely small ; but if we suppose Idv finite, this moment becomes finite. The quantity Idv is termed the strength of the magnetic shell, and may be defined as the ratio of the magnetic moment of an element of the surface to its area. If we put Idv = J, then J denotes the strength of the magnetic shell.
When the strength of a magnetic shell is the same at all its points, J is constant, and the shell is said to be uniform.
197. Potential of Uniform magnetic Shell.— If r
denote the distance of an external point P from any point Q of the shell, by (28), Art. 54, the potential at P of the
element of the shell at Q is — — cos e, where € denotes the
angle between r and the normal at Q.
But if dQ, denotes the solid angle which dS subtends at P, we have r*dQ, = dS cos e. Hence
and the potential Fof the shell at P is given by the equation V = <7Q, (21)
where Q denotes the solid angle subtended by the shell at P.
This potential differs in character from those with which we have hitherto been concerned, as it is discontinuous at the surface of the shell.
If we regard as positive the side of the shell at which the north poles of the elements are situated, or towards which they point, the potential at the positive side exceeds that at the negative by 4iirJ.
98 Magnetized Bodies.
The solid angle subtended at P by the shell is in general the same as that subtended by its bounding curve, but the two solid angles differ in some important respects.
The solid angle subtended by the curve is continuous except at the curve itself, and in a circuit embracing the curve, by passing through its interior, is cyclic. Each time the circuit is completed the value of the solid angle is increased by 4ir.
These characteristics of the two solid angles we shall now consider.
The solid angle subtended by the shell at P with its sign reversed is the same as Gauss's integral of the normal com- ponent of force emanating from a unit mass at P. The sign is reversed, because in Gauss's integral the positive direction of r is from P towards the surface ; but, in the present case, the positive direction is from the surface towards P.
If P be on the positive side of the shell, the lines from P to the shell which fall inside the cone standing on the bounding curve meet the shell once externally, and possibly an even number of times afterwards. Those which fall out- side this cone meet the shell twice, or some other even number of times : first, externally, and then internally, and therefore contribute nothing to the integral representing the solid angle. Accordingly, the two solid angles are the same when P is on the positive side of the shell, and when P is infinitely near the shell on this side, each may be denoted by d.
When P moves across the surface of the shell from the positive to the negative side, the solid angle subtended by the bounding curve remains unaltered, but that subtended by the shell becomes Qi - 47r. To see the truth of this we have only to suppose the closed surface completed of which the shell is part. Then, by Art. 26, the solid angle which the entire closed surface subtends at P is - 4jr ; and it is plain that Qi denotes the absolute magnitude of that part of this angle which is subtended by the portion of this surface which has been added to the shell. Hence the solid angle subtended by the shell is - (4?r - d).
The solid angle subtended at P by the curve bounding the shell is everywhere continuous unless P be infinitely near the curve. As P moves about, the variations of the two solid
Potential of Uniform Magnetic Shell. 99
angles are the same except wlieii P is passing through the surface of the shell. Hence we may take for the potential of the shell at P the expression JQ, where Q denotes the solid angle subtended at P by the curve bounding the shell, with the proviso that when P passes through the shell from the positive to the negative side, 4Jir must be subtracted from the foregoing expression.
If i// be a function of the coordinates of a point, and if
-j- ds taken round a closed circuit be zero for every possible
closed circuit, \L is acyclic, but, if for some circuits f —^ ds
jets
taken round the circuit be not zero, i// is cyclic. If a closed circuit s be such that we can draw a surface S, of which s is the boundary, so that at every point of S the function ^ has differential coefficients u,v, ic which are finite and continuous, then by Stokes' Theorem, Art. 192, the function ^ must be acyclic for the circuit s. Again, if a surface fulfilling the conditions stated above be bounded by two curves, SL and s2,
the value of -^ ds taken round the circuit is the same for
J ds
one of these curves as for the other. It is now easy to see that Q, the solid angle subtended at P by the curve * bounding the shell, is acyclic for every circuit which does not embrace this curve, passing through its interior. For since the differential coefficients of Q are finite and continuous for all positions of P not infinitely near the curve s, this follows immediately from what has been said above.
If we suppose P to start from a point at an infinite distance on the positive side of the shell and to move in a straight line to a point at an infinite distance on the negative side, passing in its course through the interior of the curve s, the solid angle Q passes from 0 to 4?r. For if a unit sphere be described round P as centre, the edges of the cone having its vertex at P and standing on s initially converge to a point. As P approaches to s the cone opens out, and the area swept out on the sphere by the edges of the cone increases. After P passes through the interior of s, this area becomes greater than a hemisphere, and finally when P H 2
100 Mayneti&ed Bodies.
reaches au infinite distance on the negative side of the shell, the edges of the cone again converge to a point on the sphere which is now opposite to that to which they originally con- verged. These edges have then swept out the entire sphere or 4rr.
We may now suppose P to return to its original position along a path on the outside of s, and such that all its points are infinitely distant from s. At all these points the diffe- rential coefficients of Q are zero ; and hence the value of Q, is 4?r, when P returns to its original position. It is now easy to see, from Stokes's theorem, that for any circuit passing through the interior of the curve and embracing it once
— ds must be 4ir. J ds
Hence we conclude that the potential of a magnetic shell is expressed by a cyclic function, but that at the surface of the shell discontinuity occurs in the potential though not in the function. In consequence of this discontinuity the principle of the conservation of energy is maintained.
In fact, if P start from a point 0 on the surface of the shell, at the negative side, and travel round the edge of the shell till it reaches the point Of on the positive side of the shell, opposite and infinitely near to 0, the function il in- creases by 47T, but in passing through the shell from 0' to 0 the potential of the shell is diminished by 4;r. Hence the value of the potential at 0 is unchanged by the motion of P round the complete circuit, but the value of Q is increased by 47T.
198. Energy due to Magnetic Shell.— The energy due to a magnetic shell placed in au independent magnetic field is given by (9). If /, m, n denote the direction-cosines of the normal to the element dSof the shell, and Jits strength, we have
Ad® = JldS, Bd& = JmdS, Cd& = JndS,
and if a', /3', 7' denote the components of magnetic force due to the field (9) becomes
r) dS. (22)
Energy due to T\co Magnetic Shells. 101
199. Energy due to Two Magnetic Shells.— If the
magnetic field be due to a second shell 8' whose components of force are «'/3'/, the energy W given by (22) represents the result of the mutual action of the two shells.
Since the one shell is outside the other, we may in (22) substitute the components of induction for those of force and for the components of induction we may put the expressions given by (13), Art. 193. Thus (22) becomes
p y \ tf.£ </#
(23)
where s is the curve bounding the first shell.
The values of F't G' ', H', the components of the vector potential of the second shell, are given by (16) Art 193 In this case
d
but B'd& = J'm'dS', CV3' = J'n'dS',
and hence F - J> \\nt *(1) - n> * (1 } dym J ( dz \r/ dy'\rj)
In Stokes's theorem (12), Art. 192, if we make «=-, v = 0, w = 0, we get F' =
where a' denotes the curve bounding the second shell. In a similar manner we have
0'-J'\W\d*'> H'-J'\%\d«> W
Substituting in (23) the values obtained for F' G' and H' we get
u/ T r' f (Y^'r dx dy (ft/
YV — — t/ 1/ 111 — ~f~ — — *
--^'JJ^*^, (25) where £ denotes the angle between the curve elements ds andtfs'.
102 Magnetized Bodies.
SECTION I. — Induced Magnetism.
200. magnetic Induction. — When a body is placed in a field of magnetic force, in general its magnetism is altered. The magnetism produced by the force is called induced magnetism. When the magnetizing force is small, the in- duced magnetization is, in general, proportional to and co- directional with the total magnetic force acting at the point, so that if A2 denote a component of induced magnetization, and a the corresponding component of the total magnetic force, At = KU, where K is a coefficient depending on the nature of the body, and is called the coefficient of induced magnetization.
It is easy to see that At and a are quantities of the same order, so that K is a numerical, magnitude, which is positive in the case of paramagnetic bodies, and negative in the case of diamagnetic.
If AI denote the component of that part of the magneti- zation which is independent of induction, we have
A = Al + Ka, B = BI + Kj3, C = Ci + icy. (1)
201. Magnetism due altogether to Induction.—
If there be no magnetism in the body independent of the induction due to the field of force, AI = BI = C\ = 0, and
A = Ka, B = Kfi, C=Ky. (2)
In this case, by (3), Art. 183, we have dA dB dC (da dp dy\ fdA
dB dC\ dy^dnf
whence ^+*? + *?_0. (3)
dx dy dz
When the components of magnetization fulfil this con- dition, the distribution of magnetism is said to be solenoidal, and the potential corresponding may be regarded as due to a surface distribution of mass whose density is
I A + mB + nC.
Components of Induction. 103
202. Components of Induction. — When a body has no magnetism independent of that induced by the acting force, the components of induction are given by the equations
a = (1 + 47rK)a, b = ( L + 4™) j3, o - (1 + 4™)?. (4)
If we put 1 + 47r/c = •&, the quantity ra is called by Maxwell the specific magnetic inductive capacity, and by Thomson the magnetic permeability, and in the case of a body magnetically isotropic, having no permanent magnetism independent of induction, we have, then,
a = Tza, b = tcrjS, c = zsy. (5)
203. Distribution of Induced magnetism.— Let U
denote the total magnetic potential, inside the body in which the distribution of induced magnetism is to be determined, U' the total potential in the external medium ; then, as the distribution of induced magnetism is solenoidal, and there is no other magnetism inside the field in which U and U' are to be determined, we have
V2£7=0, V2Z7'=0,
also U = Uf at the surface bounding the magnetized body, and since, Art. 187, the normal component of induction is continuous,
dU ,dU' fl
W— + W-7^=0, (6)
(tv ctv
where zs and OT' denote the coefficients of permeability of the body and the external medium, and, v and v the normals drawn into them at the separating surface.
If U' be assigned at the surface bounding the field exter- nally, U and Uf can be determined in only one way so as to satisfy the given conditions. Let us suppose that the equations could be satisfied by two pairs of functions Z7i, U'i and U2, U't, and let
104 Magnetized Bodies.
then, if & be the surface bounding the field externally, we have
If we multiply the first of these equations by OT, the second by in', and add, observing that at the surface S' we must have $ = 0, and that 0 = $' at S, we get
w
Since w + w' = 0 at S, and V2* = 0, VV = 0, «i/ r/v
the left-hand member of this equation is zero. The coefficients TO and TO' are always essentially positive, even if K or K' be negative. Hence each member of the right-hand side, and each of the terms under the integral signs, must vanish separately, and therefore ^' = 0, <f> = 0.
204. External Medium not Magnetic.— If the ex-
ternal medium is not capable of being magnetized, we have K =0, TS = 1 ; also
U= F+Q, U'= F+ii';
where V denotes the potential of the forces producing the induction, and Q and Q' the potentials, inside and outside the body, of the induced magnetism. In this case V is supposed to be given, and 12' is zero at infinity.
Iwtropic Ellipsoid in Uniform Field. 105
205. Aiiisotropic medium. — When a magnetic medium is anisotropic or crystalline, the induced magnetism is not, in general, codirectional with the magnetic force ; but the com- ponents of induced magnetization are linear functions of the components of force, so that we have
A = KUa + K12|3 + Kis
B = K^a + K-22/3 + K23y, > (7)
, \ 22 23
C = Ks
By (9), Art. (189), we see that, to increase by ca the force acting in the element rf<2>, the work required is - A$a, and therefore we conclude that
JW__ _iW _4W=C
da d(3 (ty
Hence K->I = K-12, &c., and (7) become
A = Kiid + Kit ft + Kisy, \
B = K120 + K22/3 + KMy, | (9)
C = KKO. + (C23/3 + K33y.
206. Isotropic Ellipsoid in Field of Uniform Force. — If an ellipsoid, free from magnetism and sur- rounded by a non-magnetic medium, be placed in a field of uniform force, the distribution of induced magnetism can readily be determined. In fact, we may suppose the ellipsoid to be uniformly magnetized in a direction to be determined ; and if the conditions of the question can thus be satisfied, we know by Art. 203 that we have reached the correct solution of the question.
Let / denote the intensity of the induced magnetization, and X, ju, v its direction-cosines; then, by (5), Art. 184, the potential V of the induced magnetism is given by the equation
V = I(\Lx + pMy + vNz), (10)
106 Magnetized Bodies.
where, by (17), Art. 22, the constants L, M, N denote
«•+«)! (ft" +*)*(*+«)* '
and the two other integrals obtained by interchanging I and c with a. Hence, if the components of the uniform force due to the field be denoted by Flt Ft, Fs, the total magnetic force a, parallel to the axis of x, is Fi - 7XL, and we have
with two similar equations. Accordingly, we get (l + KL)I\ = Ktf,
(11)
The values of /, A, n, and v obtained from these equations satisfy the conditions of the question.
207. Anisotropic Ellipsoid surrounded by \on- IWagnetic Medium in Uniform Field of Force.—
In this case, if we proceed in a manner similar to that of the last Article, we get
I\ = Kll(Fl - /XL) + Klt(F3 - J/i J/) + Ku(F* - and two similar equations ; whence we have
* (12)
i+K»Ft+K#F»)
Hence 7, X, ju, and v are determined.
Earth's Magnetic Potential] 107
SECTION III. — Terrestrial Magnetism.
208. Earth's Magnetic Potential. — The components of the Earth's magnetic force at any place can be deter- mined by observation. This can be done either by finding the time of oscillation of a magnet, free to move in a horizontal or in a vertical plane, when disturbed from its position of equilibrium, or by arranging a position of equilibrium under the combined action of the Earth and magnets whose strengtli and position are known. The investigation of the Earth's magnetic potential was initiated by Q-auss. In the British Islands some of the earliest observa- tions were carried out by Lloyd in the magnetic observatories of Trinity College, Dublin.
When the Earth's horizontal force has been determined at a sufficient number of places, the question of the existence of an acyclic magnetic potential can be investigated.
If s denote any portion of a closed path on the Earth's surface, H the horizontal component of magnetic force at any point, and 9 the angle which its direction makes with that of *, on the hypothesis that a magnetic potential V
dV
exists, we have H cos 0 = r. Hence, if an acyclic
ds
magnetic potential exists, /.ETcos 6 ds taken round the closed path is zero. By finding a sufficient number of values of H and 0 the numerical value of the integral can be deter- mined approximately. In fact, if Sj and «2 correspond to stations not too far apart, we have
H cos Bds = l (Hl cos 0i + H* cos 02) (sz - Sj)
approximately. It is found in this way that J H cos 9 ds taken round a closed path is always zero.
Hence we conclude that the magnetic action of r the Earth can be represented by an acyclic potential, and con- sequently that electric currents passing from the outer atmosphere to the ground cannot be the cause of any part of this action.
108 Magnetised Bodies.
209. Locality of the Sources of the Earth's Magnetic Force. — If the Earth's magnetic action be due to magnetism, or closed electric currents in its interior, the magnetic potential V at any point P outside its surface can be expanded in descending powers of r, the distance of P from the centre of the Earth. The difference between the numerical values- of V &t any two places can be determined from observations of the horizontal force. If the magnetic action be due to magnets or currents outside, the potential at any point nearer to the centre than the nearest of these sources of action can be expanded in ascending powers of r.
Hence for a point P close to the Earth's surface at its exterior we have
r-stf}—, + *Yii> (i)
where a denotes the radius of the Earth and Ui and spherical harmonics. At the surface of the Earth
and if «» denote a coefficient in U^ and bi the coefficient of the corresponding term iu Y{, the coefficient of this term in V i& «i + bi. By taking a sufficient number of numerical values of V at known places on the Earth's surface we can deter- mine as many of these coefficients as we please so that we may regard a^ 4 l{ as known.
If we now consider the vertical component Z, towards the centre, of the Earth's magnetic force, we have
At the surface (2) becomes
7i -(*'+!) 71). (3)
Earth's Magnetic Poles. 109
Hence, from the observation of a sufficient number of values of Z we can determine
iff,- -(/+!) h,
and consequently #,- and bi are each known.
It is found that «»• is always zero, and accordingly we conclude that the Earth's magnetic action is due altogether to sources inside its surface, and that V, the potential of the Earth's magnetic action, is given by the equation
r -•**£. (4)
210. JEarth's Magnetic Poles. — A magnetic pole is a point at which the horizontal force vanishes. At such a point this force changes sign so that at each side of the pole the same end of the needle points towards the pole.
If there be two poles of the same kind on the Earth's surface in going from one to the other along a magnetic meridian, the horizontal force must change sign and therefore vanish. Hence there must be a third pole between the two former. The end of the needle which pointed towards these poles points away from the intermediate one at each side.
As a matter of fact there are only two magnetic poles on the Earth's surface, and these two are of opposite kinds. The proximity of these poles to the extremities of the Earth's axis of rotation appears to indicate a connexion between the Earth's magnetism and the Earth's rotation. From the properties of electric currents it is easy to see that such currents circulating round the Earth, and approximately parallel to the equator, would account for the magnetic phenomena exhibited.
110 Electric Currents.
CHAPTER X.
KLECTRIC CURRENTS.
211. Introductory. — Not long after the discovery of current electricity it was observed by Oersted that a wire through which an electric current is passing exercises an attractive or repulsive force upon the pole of a magnet - needle. It was found also that wires along which electric currents are passing attract or repel one another.
By a combination of experimental and mathematical investigations Ampere succeeded in arriving at the laws which regulate the attraction of currents on each other and on magnets.
His original investigations must ever be regarded as worthy of the highest admiration, but some of his experi- ments, combined with the theoretical developments of other physicists, enable us to arrive at his results by methods shorter and simpler than those employed by him.
212. Electric Currents. — An electric current may be produced in various ways ; but in all cases the maintenance of an electric current requires an expenditure of energy supplied by an external source.
The source of energy may be chemical, as when two substances unite chemically, or mechanical, such as the action of a steam-engine or water-mill, but in all cases there must be a source of energy outside the current itself on which its continuance depends. The force due to an electric current is not therefore a permanent natural force, and pro- positions depending on the principle of energy cannot be applied to it in the same manner as to gravitation, or to the attraction of static electricity.
The currents whose attraction we are about to consider are those transmitted along a wire of small section.
Electric Currents. Ill
The quantity of electricity which passes through an orthogonal section of the wire in the unit of time is called the strength of the current. The quantity which passes through the unit of area is called its intensity. When a steady current is established, the strength is uniform through- out the wire. The force which causes and keeps up the current is the electric force. When there is a potential corresponding to this force, the force is the rate of diminution
of the potential, or — — , where s denotes an element of
els
length along the wire. As the current is supposed to be constant, this force must be equilibrated by another of equal magnitude.
The current is thus analogous to the uniform motion of a body sliding on a rough surface.
The retarding force on a unit of electricity is found to be proportional to the intensity of the current, that is, its strength per unit of area.
Thus we have
-d— = k- ds a
where A; is a coefficient depending on the material of the wire, <r denotes the area of its section, and i the strength of the current. If we integrate the equation above, we get
Fi - F2 = If k and a be constant, this becomes
F.-F.-hf-* (i)
kl where / denotes the length of the wire, and It = — . The
quantity JR is termed the resistance of the wire.
If Fi - F2, the difference between the values of the potential at the extremities of the wire, be denoted by E, equation (1) may be written
E = Ri. (2)
112 Electric Currents.
This expresses what is called Ohm's Law. E is termed the electromotive force, and may be defined as the difference in potential between the extremities of the wire, or, more generally, as the line integral taken along the wire of the electromotive intensity.
The term ' electromotive force ' applied to this integral seems highly objectionable, but is sanctioned by long usage.
213. Solenoids. — If a wire be bent into the form of a circle, not quite closed, be carried on for a short distance at right angles to the plane of the circle, bent into another circle equal and parallel to the first, carried on again, and so on, and finally brought back in a straight line perpendicular to the planes of the circles and close to the connecting portions of wire between them ; and if an electric current be sent through the wire, we obtain what is termed a solenoid. As the portion of the current which is perpendicular to the planes of the circles consists of two parallel parts close together and flowing in opposite directions, it produces no attraction on a magnet-pole, and the solenoid may be regarded as being composed of a number of equal circles whose planes are perpendicular to a straight line passing through their centres.
it is found that at distances which are large compared with the diameter of one of the circles, the solenoid exercises the same action as a linear magnet.
If a denote the area of one of the circles, <$ the perpen- dicular distance between two of them, / the length of the solenoid, and i the strength of the current, it is found that the magnetic moment of the solenoid is expressed by
The magnetic moment of a linear magnet of equal length, made up of small magnets having each a magnetic moment /u
and an axial length h, is expressed by 7 /, The axial length h
n
is the distance between the centres of two of the small magnets of which the linear magnet is composed. If the moment of the solenoid be equal to that of the magnet, and if we suppose h = 8, we get i<r = fi.
Solenoids.
As the equivalence of the solenoid to the magnet holds fo°ndude thearr ^ ^^^ °f ^^ ™ the solenoid> we
A small circular current is equivalent to a small magnet whose centre coincides with that of the circle, whose axis is perpendicular to the plane of the circle, and whose moment **** ^ ^^ multiPlied b^ the strength of
The equivalence of a solenoid to a linear magnet holds good equally well if another plane curve be substituted for a circle, and becomes more rigorously true according as the diameter of the curve is diminished, compared with the distance of the magnet on which the solenoid acts. Hence we conclude that,
The magnetic action of an infinitely small electric circuit is equivalent to that of a magnetic particle whose axis is surrounded by the circuit and is perpendicular to its plane and whose magnetic moment is equal to the area of the circuit multiplied by the strength of the current.
i 2»4 ^qul.va>enfe «f Electric Circuit to Magnetic Trr~~r a.smgle-sheeted surface be described of which an electric circuit is the boundary, and a network of lines be drawn on this surface dividing it into a number of small
lements, the electric current is equivalent to a current of equal strength circulating in its direction (clockwise or counter-clockwise) round each of these elements. This is obvious if we remember that along the boundary line
tetween two adjacent elements there are two currents in opposite directions, one for each element. As these currents are equal, they neutralize each other; and the only current which remains uncompensated is that in the outer boundary -By increasing the number of lines in the network, the size ot each element can be diminished without limit.
From Art. 213 it appears that the electric circuit embracing the element dS of the surface is equivalent to a magnetic particle whose axis is perpendicular to dS and whose moment is MS, where * denotes the strength of the current.
114 Electric Currents.
Hence, the total electric circuit is equivalent to the assemblage of small magnets, normal to the surface S, whose moments are the areas of the elements surrounding the magnets multiplied by the strength of the current, that is, to the magnetic shell whose surface is S and whose strength is i.
215. Magnetic Potential of Electric Circuit.—
Since the magnetic action of an electric circuit is the same as that of a magnetic shell boutided by the circuit, the magnetic potential of an electric circuit whose strength is t at a point P is expressed by i"O, where Q denotes the solid angle subtended by the circuit at P. This potential is continuous everywhere except at the circuit itself.
For any closed curve not passing through the space surrounded by the circuit the potential is acyclic.
For a curve passing through this space and embracing the circuit the potential is cyclic, and the value of the cyclic constant is 4ni.
These characteristics of the potential show that in moving a magnet-pole round a closed curve which does not embrace the circuit no work is done, but that in moving the unit pole round a curve embracing the circuit and passing through its interior, if the direction of motion be opposed to the force, work is done represented by 4iri.
If we imagine a person to stand on the positive side of a shell equivalent to the current, that is, on the side towards which the north poles point, the current as seen by him will circulate counter-clockwise, and if a person is placed lying along the current which enters at his feet and goes out at his head, the motion of a north magnetic pole moved by the current round his body will as seen by him be counter- clockwise.
The first of these statements is deducible from the experiments made on solenoids; the second follows from the equivalence of the current to the magnetic shell.
216. Magnetic Force of Currents. — Since an electric circuit is equivalent to a magnetic shell, the components of force due to the current are in space outside the shell the same as a, /3, 7, the components of magnetic force due to the shell.
Energy due to presence of Current. 115
Outside the shell «, 0, 7 are the same as a, b, c, the com- ponents of induction due to the shell. At the shell a 3 are discontinuous, Art. 186; but since the magnetization 'of the shell is normal to its surface, a, b, c are continuous, Art. 187. The force-components of the current are everywhere continuous except at the current itself. Hence we conclude that for all space outside the current, the components of its magnetic force are expressed by a, b, c, beinsr the same as the components of induction of the equivalent magnetic shell.
217. Energy due to presence of Electric Current in Independent Magnetic Field.— Let «', j3 , 7 denote the components of magnetic force ; «', b', c' those of induction due to a magnetic shell 8 equivalent to the current; a, £, 7 the components of magnetic force ; «, b, c those of induction due to the field @, and A, £, C the components of its mag- netization. Let U denote the energy due to the presence ot the shell in the field, and W that due to the presence of the current.
By Art. 216 and (9), Art, 189, we have V = -l(a'A + p'£ + 7'C)d<&, W = - J (a' A + b'B + c'C) rf@.
Except at the surface of the shell, a' = a\ b' = /3', c = 7' ; but at 8 we have «V@ = a'd& + ^ildS, where I denotes the direction-cosine of the normal to S, with similar equations for b and c'.
Hence W = U - ±iri J (I A + mB + nC) dS. Again, by (22}, Art. 198,
and therefore
W = - t / [la + m/3 + ny + 4* (IA + mB + nC)}dS = - i / (la + mi + we) rf& 12
116
Electric Currents.
218. Force-Components of Current expressed a* Integrals.— If «, t>, c denote the components of magnetic induction due to a shell equivalent to the current, by Art. 193, and (24), Art. 199, we have
dR dO _ d_ f «fr[ _ d_ tidtf_ ~dy"~dz~ dy} T dz] r '
where x' ', /> s' denote the coordinates of a point on the circuit, and r the distance between this point and the point if, y, s, and the integrals are taken round the entire circuit.
d i 1 y-y
Sinoe d * = " ? ~~r~
dy z-z' <&' y -
we get
with similar expressions for b and c ; and if F}, Ftt Ft denote the components of force exercised by a circuit of strength i on a magnet-pole of strength m, situated at the point x, y, z, we have
z-z' dz' y-y^
(4)
dx' z-z'd8f
r da' T I ra
219. Force exerted by Element of Current on magnet-Pole.— The components of force given by (4) are the sums of the components of force contributed by the various elements of the circuit.
Hence, the circuit acts as if the force-components due to a single element dtf of a current whose strength is », acting on a magnet-p<>le of strength m, were expressed by
i» ids' Idy' z-z' dz' y- y'\ ~^~\d7 ~r~"d7 I )' imds' /dz'
x-x
\ds' ~T imds' (dx' y-y' "r7" \M ~r
d^_ z-x'
d* Y dy' x-ai ~
Force exerted by Element of Current on Magnet-Pole. 117
That these are the actual force-components due to a current element is shown at the end of this Article. In the above equations,
dx' dyf . dz
-J7, yy, and — , ds ds ds
are the direction-cosines of the current element ds', and 5=£, *Z£', and L±
those of r. Hence, if 61 denote the angle between ds' and r, and 3-j, 3*2, 33 the direction-angles of a perpendicular to their plane, the force-components due to the current element are expressed by
im sin Ods' im sin Bds' im sin Bds'
COS 9i, COS $2, COS $3.
Hence the force which a current element ds' of strength / exerts on a magnet-pole of strength m is perpendicular to the plane containing the pole and the current element, and tends to make the pole move in a counter-clockwise direction round the current element, along which the observer is sup- posed to be situated with the current entering at his feet and going out at his head. The magnitude of the force is
im sin Bds'
This result can be proved directly from the expression for the magnetic potential of the circuit.
If we suppose an element ds' of the circuit to be free to undergo a displacement under the action of a magnetic pole m, the work done by the force in this displacement will be equal to the loss of potential energy of the system.
The potential energy W oi the system is denoted by imQ., where Q, is the solid angle subtended at m by a surface S bounded by the circuit.
Let ds receive three displacements : one, S£, along ds' itself ; one, 8»j, perpendicular to ds' in the plane of ds' aud r ; and one, d%, perpendicular to the two former. S£ does not alter the surface S. The displacement 8»j by the motion of ds'
118 Electric Current*.
generates an increment of the surface S, but the element of surface so generated is in a plane containing r ; and, as its normal is perpendicular to r, it subtends no solid angle at m. The displacement S£ alters S by the amount <fe'3g, and (he normal to this element of surface lies in the plane ofr and r/s', and is perpendicular to the latter. Hence, if 0 denote the angle between dn' and r, the angle between r and the normal
is - - 6. Accordingly, the element of solid angle subtended at m by the element of surface is
and therefore 8 W = im Sfl
Hence the force between m and (fa' is in the direction of the displacement S£, and is expressed by
ini sin Qdst
The direction in which the force exerted by ds' on m tends to move the latter is in the direction in which the solid angle a at m is diminishing. Thus, we arrive at the results already stated.
220. Energy due to mutual action of two Electric Circuits. — Since the action of each circuit in space outside itself is the same as that of a magnetic shell, if W denote the energy due to the mutual action, by (25), Art. 199, we have
(5)
It is here assumed that the strength of each current is maintained constant.
221. Forces between two Electric Circuits.—
If X, Y, Z denote the components of the force acting on a current element in consequence of the mutual action between the circuits, for any system of small displacements we have 2 (Xfc + 1% + ZSz) = -SW.
Forces between Two Ekctric Circuits. 119
In order to determine the variation of JFwe must express cos £ in terms of r and its differential coefficients.
If x, y, z denote the coordinates of an element, ds of one current, and x', //, z those of an element, ds' of the other, and r the distance between these elements, remembering that .r, y, z are functions of st and #', y', z' of s', and that s and s' are independent of each other, we have
r2 = (x -x'Y + (>/ - yj + (s - s')3,
rf« ' ,ls
,/r * <ft Ate £+ « ) = _oos
</$ ds cferfis \fl?s «V ffe rfs as ds
Substituting for cos s in (5) we get d*r 1 dr dr ds ds' r ds ds
The first term under the integral sign can be integrated round either circuit, and, as the circuits are closed, it vanishes. Hence
JJ ( r\ds ds' ds' ds J ds ds' r2 j
If we integrate by parts the first two terms, since the circuits are closed, the single integrals vanish, and we have
, ff (1 dr dr d il. dr \ d (I dr\) »
F = - ' ' JJ |? * ^' + * (; £) + &d sjj * "s "s .„[[{_» *+i**j
JJ ( r ds ds r2 as as ,
3 dr dr } _
- — — , t or ds ds
r2 ds ds }
£ + | cos 0 cos 0' J gr rfs ^', (6)
where 6 and 0' denote the angles which r makes with s and s'.
"JJ5
120 Electric Ctin-entn.
Hence S (XBx + Tty
^ (cos £ + I cos 0 cos 0' J ds ds'Sr. (7)
Accordingly, the forces due to the mutual action of the circuits are equivalent to a system of forces acting in the lines joining the elements of one circuit to those of the other. If R denote the magnitude of the force acting in the line joining the elements ds and ds', by (7), we have
R = - cos e + cos 6 cos & ds ds'. (8)
The negative sign shows that the force between the elements is attractive when the currents are both approaching the shortest distance between their lines of direction.
The magnitude of It was discovered by Ampere. He assumed that the direction of the force between two current elements is the line joining them.
In the investigation above, nothing has been assumed ; but it has been shown that two closed currents act on each other as if there were a force JR along each line joining an element of one current to an element of the other.
So far as this investigation goes there may be other forces acting between each pair of elements, but these forces must be such as to produce no effect on the total action between two closed currents.
If Ult UZt and U3 denote three functions of s and *', in addition to E acting along r, there might be three forces :
d~Ui
•^rr ds M parallel to the axis of x,
dl7t
-p- ds ds' parallel to that of y, and
dU3
-TT its ds' parallel to that of s,
due to the action of ds' on ds.
Force on Current Element in Magnetic Field. 121
In this case the total force parallel to the axis of x acting on ds, resulting from these forces, would be
J ((IU* i' ds —7- 08 ,
ds
taken round the closed circuit s', and this would be zero.
As the expression for the force between two elements must be symmetrical with respect to these elements, the force exercised by ds on ds' parallel to the axis of x would, in this case, be
— r— ds ds' ; ds
and as this must be equal and opposite to the force exercised by ds' on ds, we have
__
ds = dx' '
Again, as Uis a function of s and s',
rfD; -££* + ££' *•-!£'<*-*•>.
dv ds ds
Hence — — ' is a function of s - s'. and therefore ds
In like manner,
F, -,«-«' U3
222. Force on Current Element in Magnetic Field. — If A, /j., v denote the direction-cosines of a current element ds, we have seen, Art. 219, that the components of the force which a magnet-pole exerts on ds are
(py - v/3) ids, (Va - Ay) ids, and (A/3 - /m) «&i
where a, (3, y denote the components of the magnetic force due to the magnet-pole.
122 Electric Current*.
If there be any number of magnet-poles, the components of force acting on ds are, therefore,
{ fj. (71 + 72 + 7s + &c.) - i' (/3i + & + ft, + &o.) ) ids, &c.
, if X, Y, ^denote the components of forc magnetic field whose force components ar
- v/3) ds, Y = i (va - A7) (k, Z = i (X/3 -
Hence, if X, Y, ^denote the components of force acting on ds in a magnetic field whose force components are a, |3, y, we have
(9)
223. Force exercised by Closed Circuit on Kleuient of another.— The closed circuit s' is equivalent to a magnetic shell, and the components of its magnetic force are a', b', c', the components of induction due to the shell. Hence, if X, Yt Z denote the components of the force exerted by &' on ds, we have
X = i(pcf-vb')d8, Y=i(v(i'-\cf)dy, Z= i(\b'-fia'}ds. (10)
If F', G', H' denote the components of the vector potential of *', by Art. 193, we have
dH' d& ,
a = — — , &c. ;
dy dz
whence by substitution we obtain
'dQ' dP\ (dP dH'
= i 1 \ — 4 dG> | (IH> (\d d +
I dx dx dx \ dx dy dz
Accordingly,
. ( . dF' dG' dH' dP ) X = 1 j A -7— + fj. + v — | ds,
F.*|x^+/*^+»^-^j^, y (ii)
( dy r dy dx ds |
„ . (' dP dG' dH' dH' j ,
£ = ^ (A + IL 4- w 9 ' ^/S
I ^2 dz dz ds I
Influence of Medium. 123
CHAPTER XI.
DIELECTRICS.
224. Influence of Medium. — Faraday discovered that if one coating of a Leyden jar be raised to a given potential, and the other coating be at potential zero, the charges •the two coatings depend on the insulating medium ini between them.
The theory that electrical action is merely action at a distance, independent of the intervening medium, had there- fore to be abandoned, and it became necessary in studying electrical phenomena to take into account the changes in the non-conducting media, or dielectrics, interposed between conductors.
The primary medium is space devoid of matter but supposed to be occupied by what is called the luminiferous ether. Such a space is called a vacuum. In order to explain the observed phenomena Faraday originated, and Maxwell completed, a theory which regards the dielectrics interposed between conductors as the primary seat of electrical action, and looks upon apparent action at a distauce as a result of changes in the intervening medium.
A complete mechanical explanation of electrical pheno- mena, or a full and consistent theory of the nature of the ether, does not seem to have been reached as yet.
It is therefore necessary to start with assumptions, as to the electrical properties of dielectrics, based on observation.
These assumptions are statements of supposed facts which enable us to explain observed phenomena, but which them- selves await a further and more complete explanation resting on the nature of the luminiferous ether.
124 Dielectric*.
225. Electric Displacement or Polarization. —
When a conductor is electrically excited the conductors in the vicinity become electrically excited also, and a change is produced in the intervening medium or dielectric whereby at each point a directed or vector quantity is brought into existence in the medium.
This directed quantity is called by Maxwell the electric, displacement, and by Professor J. J. Thomson the electric polarization. The latter term is no doubt scientifically the more correct ; but the word ' polarization' is used so frequently , especially in the theory of light, that Maxwell's term is in practice the more convenient.
In order to bring about this change in the dielectric the expenditure of work is required. If the electric displacement per unit of volume be denoted by D, and its components *>y/>0>A, the expression for the total work §U per unit of volume, required to increase D by BD, is of the form
The quantities by which S/, Sg, and SA are multiplied in this expression are called the components of the electromotive intensity R.
It will be shown that the vector quantity thus defined has properties for the most part the same as those which belong to the electromotive intensity in the theory of action at a distance.
Since Xtydxdydz represents an element of work, XSfdxdy is of the nature of a mechanical force. Hence, if X be regarded as of the same nature as the force acting on the unit of electricity, fdxdy may be regarded as a quantity of electricity, and / as a surface-density.
In an isotropic dielectric whose properties are the same in every direction, the electromotive intensity is co-directional with, and proportional to, the electric displacement. Hence for such a dielectric we may write
4vf=kX, 47r</ = U', 47TA-AZ (1)
The constant k depends on the nature of the dielectric, and is called its specific inductive capacity.
Since / is of the nature of an electric surface-density, by (5), Art. 29, k must be a numerical quantity.
Energy due to Electric Displacement. 125
226. Energy due to Electric Displacement 1 1 U
denote the energy per unit of volume due to an electric displacement, by Art. 224, we have
Substituting for X, Y, Z, from (1) we get, by integration,
Yg + Zh\ (2)
Hence the total energy W, stored up in an isotropic dielectric @ in consequence of an electric displacement, is given by the equations
W = y f DV@ = ~ I" IPd® = i- I RDd®. (3)
The second of the expressions for W given by (3) differs from that in Art. 77 only by containing the factor k.
227. Conductors and Currents. — A permanent electric displacement cannot be set up in a conductor, but passes away immediately if not renewed. A displacement which is con- tinually passing on and being continually renewed constitutes an electric current. The intensity of a current is the rate of change of the corresponding displacement. When a conductor in electric equilibrium is situated in a dielectric in which there is a displacement, it constitutes a boundary to the dielectric ; and the surface integral of the normal component of the displacement taken over the conductor constitutes what is called the charge on the conductor.
228. Soleuoidal Distribution of Displacement.—
If a closed curve be drawn in a dielectric, and through each of its points a line be drawn in the direction of the electric displacement, we have what is called a tube of induction, or, in the language of Professor J. J. Thomson, a Faraday tube. Such a tube terminates at each end on a conductor, and, whatever be the electric charge at one end, an equal and opposite charge is found at the other. In an isotropic medium
126 Diekctrics.
tubes of induction are in the same direction as tubes of force, and are therefore at right angles to the surface of a conductor in electric equilibrium. Hence, if the tube be small, the positive displacement over the normal section directed into the tube at one end is equal in magnitude to the negative displacement directed into the tube over the normal section at the other end. Hence if Si and 22 denote the two normal sections, and D, and Dz the two displacements in the positive direction of the line of induction, we have -D,2i = Dt^t-
We conclude that, for any small tube of induction drawn in the dielectric, the product of the displacement and the normal section is constant.
From this it follows that, if any closed surface S be drawn whose interior is occupied continuously by the dielec- tric, and if /, m, n denote the direction-cosines of the normal, we have
/(//+ mg + n/i)dS = 0.
For, if i// be the angle which a line of induction makes with the normal to the surface at any point,
D2 = D cos i/, dS = (If + mg
and, as every tube of induction is cut twice, or some other even number of times by the closed surface,
J (If + mg + nh} dS = J D cos $dS = 0. (4)
If the volume enclosed by S be the element dx dy dz, we obtain
fdy dz - (f + .£ dx\ dy dz + g dz dx ~(ll + j- dy\ dz dx
+ h dx (hj - I h + — dz I dx dy = 0 ;
that is, + ' + ',0. (5)
do- dy dz
This equation expresses a fundamental property of the electric displacement, and is analogous to the condition fulfilled by the components of velocity in an incompressible
•fl 1 1 1 A *
Constancy of Charge on Insulated Conductor. 127
In the case of a conductor, /, g, h cannot exist except in the form
but the soleuoidal condition is still fulfilled, so that for a conductor we have
da dh
229. Constancy of Charge on Insulated Con- ductor. — If a conductor be insulated, its bounding surface, or surfaces, remains unchanged, and throughout the conductor by (6) we have
( d
Multiplying by dx dy dz, and integrating, throughout the conductor we get
df dg dh di + mdi + H dt
that is, — \(lf+ mg + nh }dS = 0.
flfeJV /
Hence / (If + mg + nh) dS, taken over the surface or surfaces of the conductor, is constant.
When a conductor is touched by another conductor, the bounding surface of the space through which the integration is effected is altered, and there is no longer any ground for asserting the constancy of the charge.
230. Displacement due to Electrified Sphere.—
If a conducting sphere, of radius «, placed in an isotropic medium, be uniformly electrified, the lines of force and of induction are perpendicular to its surface and pass through its centre, since there is perfect symmetry round this point.
128 Dielectrics.
Hence the sphere is in electric equilibrium, and over any concentric sphere of radius r the displacement D is uniformly distributed ; and if D0 denote the displacement at the surface of the sphere of radius a, we have 4rrr*D = 47rrrZ>0 = e, where f denotes the total charge on the electrified sphere. Hence
If a be sufficiently small, we may regard the sphere as an electrified particle.
The electromotive intensity R is given by the equation
and we have the result, that in an isotropic medium the force due to an electrified particle varies directly as the charge on the particle and inversely as the square of the distance.
231. Energy due to two Small Electrified Spheres. — Let the radii of the spheres be denoted by a and |3, and the spheres themselves by A and B. The electro- motive intensity due to the sphere A, on which there is a charge e^ is by Art. 229, on the hypothesis that the charge is
uniformly distributed, T—J, where rt denotes the distance A/'i
from the centre of the sphere. The electromotive intensity
due to the sphere B is in like manner — ^
ki'j
It is plain that the resultant force may be derived from a potential function V, where
If JFbe the energy due to the spheres, wo have, then,
Energy due to two small Electrified Spheres. 129
The surface integral is to be taken over a sphere of infinite radius and over the spheres whose radii are a and 8
At the surface 8, of the sphere A, if « be sufficiently small, ^ + a
a c
where c denotes the distance between the centres of A and B, and d
By Art. 26, = 0>
and, as V is constant at 8^ we have
In like manner,
The integral over the sphere of infinite radius is zero also V2F=0 throughout the field. Hence
and
If the sphere ^4 were alone in the field, the expression above would ^ become ^— • Similarly, if B were alone, it would be -~ • Hence the energy due to the mutual action
of the two spheres is -J-?« kc
130 Dielectric*.
232. Force between Electrified Particles.— If W
denote the energy due to the mutual action of two electrified particles, by Art. 231 we have
where r denotes the distance between them. Hence, if F be the mutual force which they exercise on one another,
dW I e,e,
F= ~ Wt~
Accordingly, the force between two electric particles acts in the line between them, and varies directly as the product of the quantities of electricity and inversely as the square of the distance.
Also, by (7), Art. 230, the electromotive intensity due to an electric particle is equal to the force which it exercises on the unit of electricity.
233. Irrotational Distribution of Electromotive Intensity. — The components X, Y, Z of the electromotive intensity, due to a permanent statical distribution of elec- tricity, must be the differential coefficients of an acyclic function of the coordinates.
For, if we draw any closed circuit and suppose it occupied by a conducting wire,
taken round the circuit, must be zero, as otherwise a perma- nent electric current would be set up in the wire without any expenditure of energy, which is impossible.
Hence J (Xdz + Ydy + Zdz) between two points must be independent of the path, and therefore
X<la> + Ytly + Zds = - dV, and V must be acyclic.
Distribution of Electricity on Conductors. 131
234. Distribution of Electricity on Conductors.—
If a conductor be in electric equilibrium, there can be no electromotive force acting in it, and therefore the potential is constant throughout. In the surrounding dielectric,
47T/ = - k (~, &c.,
and, accordingly, from (5) we have V2 V = 0. The poten- tial V is therefore determined in the same manner as on the hypothesis of action at a distance.
The charge on a conductor is /(//* + tncj + nlb)dS; and, by (1), Art. 225, this is equal to
-T-\^d8>
47rJ dv
where D and v are both drawn into the dielectric surrounding the conductor.
{ dV Hence, if the total charge be given, so also is -— dS.
235. Conditions at Boundary between two Dielec- trics.— If two dielectrics, whose specific inductive capacities are ki and k^ be in contact, at the boundary between them in passing from one to the other, Fis continuous, as otherwise the electromotive intensity perpendicular to the boundary would be infinite.
Again, the normal component of the displacement must be the same in one medium as in the other. To prove this, let us suppose two small tubes of induction resting on the same element of the boundary surface and drawn one in each medium. Let A and D2 denote the displacements, Si and 22 orthogonal sections of the tubes drawn close to the boundary surface 6T, and fa and fa the angles between the lines of displacement and the normal to S. Then, by Art. 228, we have DiSi = D2S2 ; but Si = f/»Scosi/>i, 22 = dS cos fa, and therefore DI cos fa = Dz cos fa.
Thdljonditions stated above give the equations
K 2
132 Dielectric*.
If the positive direction be that of the normal drawn into the medium whose inductive capacity is ki, the second equation above may be written /u JV"i = ktNt ; whence
(11)
If we suppose k2 to be greater than A'i, we see that a dielectric of greater inductive capacity, relatively to one of less, behaves like a conductor on which there is a charge of
7. _ /.
density * . 1 JV2. In the case of a conductor, we must 4irki
suppose kz infinite, then from (10) Nz = 0. In what precedes, Ni and N2 denote the components of electromotive intensity normal to the boundary.
236. Attraction on Dielectric in Field of Force. —
If a body composed of dielectric material be placed in a medium of different specific inductive capacity, the body in general behaves like a conductor in tending to move.
To see the reason of this we must remember that, in general, if a conductor or a dielectric of different inductive capacity be introduced into a medium occupying a field of force, the total energy of the field is altered ; and, unless the field be uniform, the alteration is different according to the part of the field into which the conductor or dielectric is introduced.
If a small change in the position of the conductor diminishes the total energy of the field, the conductor will have a tendency to move in the direction, producing a change of position whereby the total energy of the field is diminished.
The same thing holds good in the case of a body com- posed of dielectric material differing in inductive capacity from the medium by which the field of force is occupied.
237. Crystalline Dielectric. — In a crystalline, or anisotropic, dielectric different directions differ in their electric properties, and the electromotive force is not necessarily co-directional with the displacement.
Crystalline Dielectric. 133
In this case, the components of the one are linear functions of those of the other, so that we have
47r/ = knX + k19Y + kl3Z, \
4-n-g = knX + knY + k23Z, I (12)
47rh = knX + #3, F + A-sa^. J
If there be a function U of the components of force, representing per unit of volume the energy due to the displacement, we have
§U = X$f + YSg + ZBh. ( 1 3)
Substituting from (12) for S/, &o., in (13), and arranging, we get
but 8^.81.8^
dX dy <fc
and tlierefore
fiyi = 7t'12, /l*32 = /»'23> "%1 = »13 >
and
+ 2knYZ+ 2/U3 XZ; (14) also,
rfZT- rf«7 rftT
/=^' ^ = ^T' A'^'
By transformation of coordinates, 8-7r?7 can be reduced to the form
Wlien Z7" is reduced to this form, the coordinate axes are the principal axes of electric displacement, and /Cj, #2> /*3 denote the principal inductive capacities of the dielectric.
134 Dielectric*.
For the components of electric displacement we then, the equations
4irf=/nX, 4ir(/ = /,-2Y, 4ir/i = k3Z. (16)
If we take any point P of the dielectric as origin and draw the ellipsoid whose equation referred to the principal axes is
it is plain that if we draw a line through P in the direction of the electromotive intensity, and draw a tangent plane to the ellipsoid at the point in which it is met by this line, the perpendicular on this tangent plane is in the direction of the electric displacement.
238. Differential Equation for Potential in Crys- talline Medium. — If we express the principal components of displacement in terms of the electromotive intensity by (16), equation (5) becomes
. dX , dY , dZ
*•* +*3jr + *S5-0'
and therefore
239. Distribution of Electricity on Conductors.—
As there is no electromotive intensity in the substance of a conductor in electric equilibrium, V must be constant at the surface.
In the dielectric outside the surface, V must satisfy the equation
Also, the product of V and its differential coefficient integrated over a sphere of infinite radius must vanish.
This appears from the consideration that If + mg + nh integrated over a sphere of infinite radius is finite. Hence,
Distribution of Electricity on Conductors. 13&
if R denote the radius of the sphere, / must be of the order •jp ; but /, g, h are of the same order as the differential coefficients of F. Accordingly, V is of the order -^, and V — , &c., are of the order -=j-
Finally, if the charge on each conductor be assigned,
. tdV , dV dV\
kl I —r- + #8*» —p- + fan —— do dx dy dz j
is given for each conductor.
There is only one function Fwhich satisfies these conditions.
If there were two, let $ be the difference between them. Take the expression
df/ 1
and integrate it by parts throughout the whole of space — the first term with respect to x, the second with repect to ?/, and the third with respect to z ; then we get
From what precedes, it is plain that each term on the right- hand side of this equation is zero ; and, as k}, k2j and k3 are always positive, we have
dj > = d± = d± =
dx dy dz
and therefore $ is constant, and consequently zero for the whole of space.
136 Dielectric*.
240. Energy expressed as Surface Integral. — If V
denote the potential, and JFthe energy due to the electric displacement, by an integration similar to that employed in the last Article, by (14), Art. 236, we obtain
Hence, by (16) and (17), we have
2 W = 2 / V(lf + my + nh) dS. (18)
Since V is constant at the surface of each conductor, nnd since / (If + nig + nh) dS denotes the total charge on the conductor, equation (18) may be written
11W = 2>F. (19)
241. Energy due to Electrified Particle in Electric
Field. — Let us suppose the field to be due to a single con- ductor, whose surface may be denoted by Si, on which there is a charge <v Let a conductor, whose surface may be denoted by S2, on which there is a charge e2t be introduced into the field. Let V denote the potential at any part of the field before the introduction of $2, and V + v the potential after- wards; also, let JPand W + IP denote the total energy of the field before and after the introduction of Sz. Then we have
2 W = Vielt 2(W + tc) = ( F, + ri) el + ( K2 + r») ft.
If we now suppose & and ft to be infinitely small, so also is t>, and the term i\et is of the second order, and therefore negligible. Hence we have
Energy in Electric Field. 137
On the hypothesis that & and e% are infinitely small, we have Vie* = v\e\ ; for, if we integrate the expression
K-
dV dv 1 dV dv 1 dV dv\ >l ~7~ T" + - ~T~ ~r 4 3 ~7T T )
! since F'and v each satisfy equation (17), we get
(20) ! j \ oo? «// «c y \ /
At the surface Si the potential V is constant, and
dx ~ d;/
is zero, since the introduction of S2 does not alter the total charge on Si. Again, before the introduction of the con- ductor Sz the total charge on the space surface 82 was zero, and therefore
K
„ dv , dv 7 dv
«*i ~r + m'^ T + nlc* -j 1 dx dy dz
Hence, as Sz is infinitely small,
dv dv 7 dv
1- WKZ — + nka —
dx at/ dz
cannot differ from - 4vr F2e2 by more than an infinitely small quantity of the second order. Accordingly the left-hand side of (20) is equal to - 4?r Fift.
Again, as V+v and Fare each constant at Si, so also is i?, and therefore
138 Dielectrics.
Also, at the surface & we have
J(
, dV , dV , tfV
//U — + W/,'2 —r- 4 Ufa ——
ax ay dz
and, since V+ v is constant, and Fcan vary only by an infi- nitely small quantity, the variation of v must be infinitely small, and
v ( Iki -7- + w/i-j -r- + nks —j- } dSz, } \ dx di/ ftz J
taken over the infinitely small surface Si, cannot differ from zero by more than an infinitely small quantity of the second order. Accordingly the right-hand side of (20) is equal to
We have, then, F2?2 = rid ; and therefore 2w Hence, by bringing an electric particle et to a point where the potential of the field is Fa, the energy produced is F2^.
It is obvious that the result arrived at above can be extended to an electric field due to any number of conduc- tors, so that in general Vc denotes the energy produced by bringing a small body having a charge of electricity e to a point where the potential is V.
If, instead of supposing a small charged conductor intro- duced into the field, we suppose the charge on one of the conductors St, already in the field, increased by the amount Set, we can show in a manner similar to that employed above that S IF, the increase of energy, is given by the equation
In fact,
and 2SJT= F,&>, + ^ST, + csSF2 + &c. ;
but Pi 8* = e,g F, + r,8 F, + *3S F, + &c.,
as may be shown in the following manner.
System of Charged Conductors. 139
Let the original potential at any point of the field be denoted by V, and the increase of potential due to the intro- duction of &?i by v, then, by an integration similar to that already employed, we have
+ k,n dSl + dS2 + &c.
d_V_ f dV dV\( \
dx dy dz J \ ')
At each of the conductors Fis constant, and also V+ r, and therefore v.
Again, at each conductor, except the first,
dv dv^
dy sH dz,
is zero, and at the first this integral is - 4ir<fct. Also, at each conductor,
dZ. + jfm (— + kn <IV\
(Y dv v*v- ,.„ ,
JV ' Tx + '*m~-+ »W--J(
hence the equation above becomes
4?r ViSei = 4ir (e^^ + e-,rz that is,
(21)
Hence we obtain
$W= Vfo, (22)
and therefore we conclude that under any circumstances the energy produced by bringing a small quantity of electricity e to a point where the potential is V is denoted by Ve.
242. System of Charged Conductor*. — It is now
easy to see that, if there be a system of charged conductors in a crystalline dielectric, the equations which hold good
140 Dielectrics.
between the charges and potentials are of the same form as those belonging to an isotropic medium, that is,
Fi = p\\e\ + pizd + piaCa + &C.,
V-i = puei +pnt* + p™ea + &G., F3 = piaei + j0z3<?2 + py*e3 + &c.,
&c., &c.
In fact, every step in the process by which these equations are proved in Art. 128 holds good here.
For, from equation (16), it appears that if each component of displacement be altered in the same ratio, so also are the differential coefficients of the potential and the total dis- placement. Accordingly, if the mode of distribution of the displacement be assigned, the potential at any one point varies as the displacement to whicli it is due. Also two systems of displacement which are each in equilibrium may be superposed without disturbing the equilibrium.
243. Force on Electric Particle in Electric Field. — We have seen that the energy due to an electric particle e in an electric field is Ve.
If the particle receive a displacement whose components are &r, £//, and Ss, the energy of the field is increased by
(dV « dV . dV * \
<?—£»+ — Sy + -— S* . \rf<s dy • dz )
This is the work done against the forces of the field which must therefore be
dV dV dV
~e^ -*7-y> aud "'A'
Accordingly, the force acting on an electric particle per unit of mass is the same as the electromotive force of the field.
244. Potential due to Spherical Conductor.— If a
charged spherical conductor whose radius is a be alone in the field, the potential Vis constant at the surface of the sphere, and in the space outside satisfies equation (17). If we assume
A/*I £ = z> \/h n = //, and v/^3 £ = s, (17) becomes <?V tfV <PV
Force due to Spherical Conductor. 141
and, at the surface of the sphere,
7u£2 + &„* + W = a2. (24)
We have therefore to find a function of £, »}, and £ whicli satisfies (23), and which is constant when £, rj, £ satisfy (24). This is the same px-oblem as to find the potential of an ellip- soidal charged conductor. Hence the form of Fis given by Ex. 3, Art. 75.
If ki > kz > /i'3, we have
dx
(25) where X is the greatest root of the equation
r2 + x^T2 + x^7"L
The constant C is determined from the equation
dV dV dV\
?! — + wkz — + nki — — j dS = — 4-rre, dx dy dz J
where e denotes the charge on the surface S of the conductor.
245. Force due to Spherical Conductor. — Diffe- rentiating (26) we obtain
The quantity inside the bracket on the right-hand side may be denoted by — , and we get
a\= PZZ d£ X(X2-&2)* In like manner we have
142 Dielectrics.
Again,
accordingly, if -X", 7, Z denote tlie components of force, we have
~^~7*T * A'-*'
Y=-liV- =.°_ ;- --*-, - \
" X
246. Force due to Spherical Particle.— In the
oase of a particle, a becomes infinitely small, and so also do k and // ; then
A' = A2 - /*' = A2 - F,
and p* = A2 = f2 + nj + ^-
Accordingly, we have
If we substitute for £, »j, and £ in terms of .r, //, and 2, we obtain
Cx C,j Cz
X = 'k7\^ r=^' z = w>
where
Hence we conclude that the force exercised by a spherical particle at a point P is not in the direction of the line joining P to the centre of the particle, and does not vary inversely as the square of the distance of P from the centre of the sphere.
Force fine to Spherical Partick. 143
From (16) and (27) we have
/=-?l?, <j = —J, h = —3~, (28)
where r denotes the distance of P from the origin.
Hence, the direction at any point of the displacement due to a spherical particle passes througli the centre of the particle, but the magnitude of the displacement does not vary inversely as the square of the distance.
144 Electromayuelic 'ihconj of Light.
CHAPTER XII.
ELKCTKOMAGNET10 THEORY OF LIGHT.
247. Introductory. — The electromagnetic theory of light cannot be considered part of the theory of Attraction ; but it is so intimately connected with the properties of dielectrics, and with those of electricity and magnetism which have been explained in the foregoing chapters, that some account of Maxwell's great investigations does not seem out of place here.
248. Energy of Current in Magnetic Field.— From the identity of the action of an electric current with that of a magnetic shell, in Art. 217 it was concluded that the potential energy JFof a current in an independent magnetic field is given by the equation
JF= - tj(fo 4 wb + nc)dS. (1)
If i assume the infinitely small value S/, equation (1) becomes
8 W = - St / (la + mb + nc) dS. (2)
This equation holds good whether the field be independent or not, as a change in the integral due to an infinitely small value of i must be infinitely small, and when multiplied by Si becomes evanescent. We cannot, however, regard the energy due to the presence of an electric current as potential, because the current is not a permanent natural agent whose action varies merely with its position. The current may cease, and, if so, its energy disappears.
We must therefore consider the energy due to an electric current as kinetic.
Energy and Electromotive Force. 145
In a dj'namical system, if work be done against the natural motion of the system, the energy, if potential, is increased, but the energy, if kinetic, is diminished.
Hence we conclude that, if the energy of an electric current be kinetic, the expression for the variation of energy due to a variation of i must have the opposite sign from that which it would have if the energy were potential. Therefore, if T denote the kinetic energy due to the presence of an electric current in a magnetic field, we have
gr = &'/(/« + mb + no) dS. (3)
dT
Since §T - — &', we obtain di
l^ = $(la + mb + nc)dS. (4)
249. Energy and Electromotive Force. — The con- nexion between variation of energy and force is given by Lagrange's Equations, Dynamics, Art. 207.
In the present case of the dynamical system consisting of electric currents in a magnetic field, the position of the system is specified by the geometrical coordinates of the various magnets and electric circuits, and in the case of each current, by the distances along the circuit which the electric molecules have travelled at any time since a definite epoch. If s denote the distance along the circuit which a
molecule of electricity has travelled, its velocity is — .
dt
Again, if p denote the density of the electricity, and a the area of a section of the circuit, the quantity of electricity
which passes the section in the unit of time is pj — , but this, Art. 212, is i the strength of the current. Hence,
ft Cf fa Cs
idt = \ pa — dt = \ pa ds = iL (s) - \L (s0)} <0 JV dt J*o
rt since p is constant, and <r a function of s. Hence, if i dt
J 'o
146 Electromagnetic Theory of Light.
and *0 be assigned so also is *. Accordingly, instead of specifying the position of a molecule of electricity by s we
may do so by the coordinate »j, where TJ = i dt. Again,
J ^0
since i is uniform throughout the circuit, >/ is the same for all the molecules of electricity.
If now X', Y', Z' denote the components of the total electric force at any point of the circuit, Lagrange's equation of motion, corresponding to the coordinate i), is
dt <ty dti
and -if X, Y, Z denote the components of electromotive intensity, the corresponding forces X", Y", Z" are given by the equations
X." = ptrXd*, Y" = p<rYd*t Z" = p<rZ<t*, also dr\ = ifdt = pad*.
Hen oe
From Art. 212 it appears that when a current is passing the electromotive force is opposed by the resistance of the circuit, so that the generalized component of force tending to increase TJ is not E but E - Ri.
Again, the kinetic energy T does not depend on »/ but
i rp
on 77 or i. Hence — is zero always, and Lagrange's
equation of motion corresponding to the generalized co- ordinate »j becomes
Maxwell's Theory of Light. 147
This equation may be written
If T remain unchanged, (6) becomes Ohm's equation (2), Art. 212.
If T vary in consequence of a change in the electro- magnetic field, the electromotive force keeping up the
current is diminished by — — . If this be negative, the dt di
electromotive force is increased.
This property of currents is abundantly confirmed by experiment. It is indeed on this property that almost all the modern applications of electricity depend. It was originally discovered by observation ; but its exact mathe- matical expression as given above is due to Maxwell.
A simple case of this phenomenon is exhibited if two* currents which repel one another be made to approach. An additional electromotive force is then developed in each circuit tending to increase the current.
This still holds good if E be originally zero in one circuit. A current is then produced tending to oppose the motion. Such currents are called ' induction currents.' It is on their existence that the whole theory of light as an electromagnetic phenomenon depends.
The general principle exemplified in the production of induction currents may be expressed by the statement
In any circuit contained in an electromagnetic field every variation in the strength of the field produces an electromotive force which tends to diminish the variation.
250. Maxwell's Theory of Light. — Maxwell supposes the entire universe to be filled with a dielectric called the lumiuiferous ether.
If there be a variable electric displacement iu any part of this dielectric, the variation of the displacement constitutes an electric current which produces an electromagnetic field. The variation of the current produces an electromotive force in all
L2
148 Electromagnetic Theory of Lii/Jit.
the surrounding circuits. These electromotive forces produce currents which again give rise to other electromotive forces and currents, and so the original variable displacement is propagated througli space.
In the case of light, the original displacement is vibratory ; that is, it begins in a certain direction, increases in that direction up to a certain amount, and afterwards takes place in the opposite direction till it reaches the same amount as before, only in the opposite direction, when it is again reversed ; and this process is repeated so long as the light remains steady.
The displacement is therefore quantitatively the same as the distance moved through by a vibrating particle, and may be represented by an expression of the form
a sin — t.
T
The whole phenomenon may therefore be termed an electric vibration ; and, when propagated through space, may be called an electric wave.
From the results already arrived at, the laws which govern this propagation may be deduced, as will be shown in the following Articles.
In the study of an electric vibration we have to do with five vector quantities : the displacement, the electromotive intensity, the current intensity, the magnetic force, and the magnetic induction.
Let
/, g, h denote the components of electric displacement ;
X, Y, Z those of electromotive intensity ;
*/, v, w those of current intensity ;
a, /3, 7 those of magnetic force ;
a, b, c those of magnetic induction.
We seek to determine differential equations for the com- ponents of one of the vectors which will enable us to arrive at the laws of its propagation.
Magnetic Induction and Electromotive Intensity. 149
251. Magnetic Induction and Electromotive In- tensity.— We have seen, Art. 249, that for any circuit s, if X, Y) Z denote the components of electromotive intensity due to current induction,
r vT" 9 r} \ J /lr\
,, ,. - - ^ -r- + .r — - + Z — }ds. (7)
at dt J \ f/s </s fltey
If we imagine a surface-sheet /S filling up the circuit s, by Stokes's theorem, Art. 192, the right-hand side of (7) is equal to
n.fdz dY\ \
\]l[j T- ) + &c. rfiS';
J ( V<?y rfs / )
and, by (4),
dT f
— = (la + mb + nc) dS.
Hence,
— (la + mb + nc) dS
JdT dZ\ fdZ dX fdX rfF\) ,
\-j --- T ) + m I-, --- 3- + 'M —j --- r it <*&• \dz (I// J \dx dz \di/ ds J)
In the case of an electric disturbance in a continuous medium, this equation holds good for every circuit which can be drawn ; and therefore we have
da_dY_dZ db _ dZ _ dX clc_ _ dX _ dY ~di~~dz~dy* df.~dx~~az' di ~ ~dy ~ dx ' ('
252. Current Intensity and Magnetic Force.— If
we suppose a surface-sheet S drawn in the dielectric, the total current passing across it is denoted by
/ (In + mv + nw) dS.
The line integral of the magnetic force, taken round a circuit s, bounding the surface S, is due altogether to the current passing across /S', since for magnetic forces due to
150 Electromagnetic Theory of Light.
currents not embraced by s this line integral is zero. Hence, by Art. 215, we have
r / j*, j.,, A* \
</|
3 _ da
" dy
and, since this equation holds good for every circuit and corresponding surface which can be drawn in the medium, we have
dy dQ (fa dy . dQ da ._.
4iru = -7- - -£-, 4irv = — - -f-9 4irw = -£- - — • (9) dy dz dz dx dx dy
253. Relation between Magnetic Force and Induction. — We have seen, Art. 201, that in a body magnetically isotropic, in which there is no permanent magnetism, the components of magnetic induction are in a constant ratio to those of magnetic force, so that
a = zja, b = w/9, c = zay. (10)
In what follows, we shall always suppose the medium to be magnetically isotropic.
254. Equations of the Electromagnetic Field and of Propagation of Disturbance. — In the general case of a dielectric electrically crystalline, collecting the results given by (16), Art. 237, by Art, 227, and by (9), (10), and (8) of the present Chapter, we have the following group of equations holding good in the electromagnetic field : —
da
|
4n/-jr,x, |
4irg-- |
= Ktl |
4- |
rh = & |
•$••• |
(11) |
||
|
S.i |
'• * |
-*, |
dh di |
- |
?r. |
(12) |
||
|
§rfy dy |
-*' 4™ |
da |
dy dx1 |
4mc |
-«. dx |
da |
(13) |
|
|
a = zsa |
, b = |
w/3, |
c = |
Vy. |
(14) |
|||
|
dY ~dz |
dZ db dy * dt |
dZ dx |
dX |
dc di |
-' |
dX dy ~ |
dY dx ' |
(15) |
Solution of Equation of Propagation. 151
By differentiation from (12) and by (13), &c., we have
that is,
dY
dt \d{i dz « \ at/ <(t dz
H*.((!x _<*¥}_ d (<tz _
ts\dy\ di/ dx ) 'dz \ dv
y If we assume
we get
In the case of an isotropic medium,
A*- = IT- = 6'2 - F2, and we have
255. Solution of Equation of Propagation. —
Equations (16) and (17) are very general in their character; and to obtain a solution suitable for the present investigation we must consider some of the characteristics of a ray of light. When light emanating from a point passes through a lens whose focus is at the luminous point, a cylindrical beam is obtained whose parallel sections are planes having similar characteristics in reference to the beam. We may assume therefore that one of the vibrations which constitutes the light is propagated so that its direction remains parallel
152 Electromagnetic Theory of Light.
to a line fixed in space, and that at all points of a section of the beam parallel to a certain definite direction the vibrations are in parallel directions, and in a similar state or phase. Consequently, if D denote one of the displacements whose vibrations constitute the light, the direction of D is constant, and the direction in which D is propagated through space is also constant.
If X, JJL, v denote the direction-cosines of D, we may there- fore assume that X, ft, v are constant for all positions of Dy and we have
Equation (17) assumes its simplest form when /is a function of one coordinate; and, as a particular case of (17), we may write
%-"&
By Art. 53, the solution of (18) is
This expression for/ indicates a variable quantity whose magnitude at a given point is continually altering and whose every state or phase advances through space in the direction of z with a velocity V.
This is obvious, because
${F(/ + O-(* + O) =<t>(Vt-z)t provided fT-s';
and, accordingly, the value of / at the point z at the time t is the same as the value of / at the point z + z at the time t + t'.
If $ be a periodic function, the disturbance in the medium is called a wave.
The distance between two points on the line of propaga- gation at which the disturbance is in the same state is called the wave-length.
If T denote the period of the disturbance, that is, the length of time in which the disturbance at a fixed point P
Direction of Displacement in Isotropic Medium. 153
passes through all its phases and returns to its original state, the wave-length is equal to FT. For, during the period T, the original disturbance reaches a point Q whose distance from P is VT, and the disturbance at P has during the same time returned to its original state. Hence, at the end of the period r the disturbance at Q is in the same state or phase as that in which it is at P, and consequently PQ is a wave-length.
When a wave is passing through a medium, the locus of the points at which the disturbance is in the same phase is called the wave-front.
If the wave-front be a plane parallel to a plane fixed in space, the wave is called a plane wave.
In the case of a plane wave, the direction of propagation is the normal to the wave-front, and the direction of vibration is parallel to a line fixed in space.
We can now generalize the solution of (18) so as to satisfy (17), and to represent the propagation of a plane wave of electric displacement through the dielectric.
We may assume
/= XZ), g = ^D, h = vl), D=<t>\Vt- (Ix + my + ws)} , (19)
where /, m, n denote the direction-cosines of a line fixed in space.
Then V2D = (/' + mz + n2)<f>" = <j>", and
and, accordingly, (J7) and the corresponding equations for g and h are satisfied, also D represents the displacement in a plane wave whose line of propagation is in the direction /, m, n.
256. Direction of Displacement in Isotropic Medium. — The expressions for ^, &c., given by ^12) and (13), show that
±W^dg dh\
dt \dx dt/ dz )
154 Electromagnetic Theory of Light.
If there be an electric displacement in the medium before the disturbance takes place, by (5), Art. 228,
£+4 + ^-0. (20)
dx dy dz
Hence this equation always holds good ; but /=A0{F* -(& + »*// + MS)},
with corresponding equations for g and /<, and therefore by (20),
(A/ + fim + vn)<f>' = 0, and accordingly
A/ + fjLtn + vn = 0,
and we learn that in a plane wave the disturbance is per- pendicular to the wave-normal, and is therefore in the wave-front.
This is often expressed by saying that the disturbance is in the plane of the wave.
257. Magnetic Force in Isotropic Medium. —By
(14), &c., we have
dt ~ -a (It
Integrating with respect to /*, we obtain
a = 4ir V(mv - nfji)D + constant.
As we are concerned only with the magnetic force due to the disturbance, the constant may be uegle<3ted, and we have
a = 4irVD(mv- Wju), \
- /v), (21)
- m\). J
Crystalline Medium.
Hence the magnetic force is in the plane of the wave and perpendicular to the displacement, and its magnitude H i& given by the equation
H = 4-rrVD. (22)
258. Crystalline Medium.— The solution found, Art, 255, for (17) holds good for (16) with some modifications.
In fact, if we assume equations (19) and substitute in (16) and the two corresponding equations, we get
F2A = Az\ - l(Azl\ + £zmfj. + Clnv), ]
Vzfi = .5V - m (Azl\ + Wmp + C"W), I (23)
F2v = C-v - n (A'l\ + Rnifi + Cznv). J
In the solution of (17) V is given, and we find that A, ju, v are indeterminate, provided they fulfil the condition
/A + nifj. + vn = 0.
In the present case, when /, m, n are given, equations (23) determine F2 and A, p, v. If we eliminate A, /u, v from (23), we get a cubic equation to determine F"2. The absolute term of this equation is
If we call this determinant Q, we have lz-l
I
tn n
m~ - 1
m
n9-!
lz-l m*
I2 mz n* - 1
0 mz nz 0 w2-! «2 0 mz nz-l
156 Electromagnetic Theory of Light.
Hence one value of V1 is zero. The corresponding values of X, fji, v are proportional to
but they have no physical import, as the displacement to which they belong is not propagated through the dielectric.
For each of the values of V* which are not zero there is a corresponding set of values of X, /u, v, indicating two possible directions of displacement with a given wave-front.
If we multiply the first of equations (23) by /, the second by m, and the third by n, and add, we get
F'(/X + mn + nv) = (^2/X + ffmfjL + C*vn)(l -I1- w2 - M') = 0.
Hence,
l\ + mfi + nv = 0, (24)
and we infer that the two directions of displacement corre- sponding to a given plane wave-front lie in the plane of the wave.
If we multiply the first of equations (23) by X, the second by fji, the third by v, and add, we get
= A*X* + By + <?V - (A* IX + Knifi + C*nv)(l\ + MH + «w), and therefore, by (24), we have
F2 = A*X* + £y + <? V. (25)
If Xi, /ai, i'j ; Xj, /tij, i/2 denote the direction-cosines of the two displacements perpendicular to /, m, n, and V\ and Vi the corresponding velocities of propagation, we have
(A* - V?} X: = 1(A*IX, + £'»!/«, + C'nvi),
with two corresponding equations.
Multiplying the first by X,, the second by /ua, the third by i/j and adding, since /Xa + m^ + nvt = 0, we get
F,2 (A,X2
Wave- Surf ace. 157
In like manner, we have
F22(\|A2 + ^tijU2 + VlVz) = -42AiA2 + B^fJLifJLz + CZVi\'z.
Consequently, unless Fi = F"2, we obtain
A,A2 + fruz + i/ii/2 = 0, (26)
^2A,A2 + IPnifti + <?Vn>2 = 0. (27)
Hence we learn that the two directions of displacement belonging to the same wave-plane are perpendicular to each other in the plane of the wave, and are also conjugate in the ellipsoid whose equation is
-4V + BY + Cz*2 = constant.
Since these two directions are perpendicular and conjugate to each other, they are axes of the section of this ellipsoid made by the wave-plane.
259. "Wave - Surface. — If a vibratory disturbance emanate from a point 0 and spread in all directions through a medium surrounding 0, the locus of points at which at any time the disturbance is in the same state or phase is called the wave-surface.
If the medium surrounding 0 be isotropic, the disturbance is propagated with equal volocities in all directions, and the wave-surface is a sphere having 0 as centre.
If the medium be not isotropic, we may suppose a number of small plane waves to start simultaneously from 0 in all possible directions. Each of these is propagated with a velocity corresponding to the direction of its normal. The envelope at any time of all these plane wave-fronts is the wave-surface corresponding to the medium.
260. Construction for Wave-Surface of Crystalline Medium. — When an electric disturbance takes place in a crystalline medium, the equations of Art. 258 enable us to give a construction by which the wave-surface may be obtained.
158 Electromagnetic Theory of Light.
If we take any period of time tf,, and assume
a = Atlt b = Btly c = Ctlt the ellipsoid, whose equation is
xz . //> s*
is that which Fresnel called the ' ellipsoid of elasticity,' and may be termed Fresnel's ellipsoid.
Let an electric disturbance emanate from the centre 0 of this ellipsoid, and let OP be the direction of the electric dis- placement in a plane wave due to the disturbance. Draw a tangent plane to Fresnel's ellipsoid perpendicular to OP ; let Q be its point of contact, and draw OP" perpendicular to the plane POQ.
Then OT and OQ are conjugate; and, being also at right angles to each other, are the axes of the section of Fresnel's ellipsoid.
Let the direction-cosines of OP, OT, and OQ be denoted by A|, pi, vi ; A2, juj, v2 ; A', fS, v ; then Au /«„ 1-1 are proportional to
A' L • v
and therefore, since X'A2 + fj.'fj.z + v'i>2 = 0, we have
that is, ^!2AiAz + J^/ui/u, + C78vii/, = 0.
Also, \\ + /U!^u2 + I/,)-, = 0.
Hence, by Art. 258, OY must be the second possible direction of displacement in the wave-plane corresponding to OP, and this wave-plane must be POY.
Draw OS in the plane POQ perpendicular aud equal to OP ; then OS is the wave-normal, and its length is the distance through which the wave-front has advanced in the time ti. If OZ'be drawn in the plane QOP perpendicular and equal to OQ, the locus of Tfor all possible positions of OP
Construction for Wave- Surf ace.
159
is a surface which touches at T the wave-frout perpendicular to OS.
Fin. 1.
To prove this, take on the tangent to Fresnel's ellipsoid, QP, a point Q' infinitely near Q, and in the plane QOP draw OT' perpendicular to OQ' ; then OT = OQ', and if a plane be drawn perpendicular to OT', it passes through OQ', and one axis of the section of Fresuel's ellipsoid made by this plane is infinitely near OQ' and, being an axis, is there- fore equal to OQ' and consequently to OT'. Accordingly T' must be a point on the locus surface, and TT a tangent to this surface.
Again, draw TT" parallel to OF, and take on it T" infinitely near T. Then, since TT" is perpendicular to 02\ we have OT" equal to OT. Again, since OQ is perpen- dicular to the plane TOT", the plane perpendicular to OT" passes through OQ, and the axis of the section of Fresnel's ellipsoid made by this plane, being infinitely near to OQ, is equal to it,' and therefore to Ol'and OT". Hence T" is a point on the locus surface, and Tl ' a tangent to this surface.
Accordingly, the plane STT" is a tangent-plane to the locus surface ; but this plane is the position of the wave-front at the time ^. Hence the locus-surface is the envelope of all possible wave-fronts at the time tif and is therefore the wave- surface.
160 Electromagnetic Theory of Liyht.
261. Equation of Wave-Surface. — It is now easy to find, in the manner of MacCullagh, the equation of the wave- surface.
If r denote the length of any radius-vector of Fresnel's ellipsoid, a sphere, having 0 as centre and r as radius, meets the ellipsoid in the cone whose equation is
A tangent plane to this cone meets the ellipsoid in a section in which two consecutive radii vectores are equal to r. Hence the line of contact is an axis of this section, and the extremity of an intercept equal to r on the perpendicular to the tangent-plane to the cone is a point on the wave- surface. If r be regarded as constant, the equation of the cone reciprocal to the cone of intersection of the sphere and ellipsoid is
0. (28)
.
fll - r* b* - r* c* - r?
The coordinates of a point on the wave-surface whose distance from 0 is r satisfy this equation. Hence, if
*•' = z2 + if + z>,
equation (28) becomes the equation of the wave-surface.
Rejecting the factor r2, and getting rid of fractious, we i have
- r2) (a2 - r2)
-rj ft'-r3 = 0.
Arranging in powers of r, and dividing by r2, we get, finally,
(a1 tf + b* y* + c2 s2) r» - a2 (&» + c') x* - i2 (c2 + «2) //2
0. (29)
The surface whose equation we have obtained was discovered by Fresnel, and is known as Fresnel's wave- surface.
Mayni'tic Force. 161
262. Magnetic Force.— From Art. 254, we have
dt CT
Integrating with respect to z1, we get
Hence we have
/3 = (w^X.-
7 = 1 (/^3^ -
From (30), we see that
la + m +
also,
= 0.
Accordingly, the magnetic force is in the wave-plane, and perpendicular to the displacement; that is, its direction coincides with the second possible direction of displacement in the wave-plane.
162 Electromagnetic Theory of Light.
Hence if H denote the magnetic force, we have
with two other corresponding equations. Multiplying first by Az, second by /u2, third by v», and adding, we get
H =
= 1^ (A*\S + *V + 0'".') = ^ F2 = 4;r PZ>. (31)
263. Electromotive Intensity. — If F denote the resultant electromotive intensity, and 0i, 02, and 03 its direction-angles, we have
with two similar equations ; then F is in the direction of OQ, fig. 1 ; and if ^ denote the angle between the displacement and the resultant electromotive intensity, we have
F cos x = F (\! cos 0i + /ui cos & + in cos 03)
and
F = 4irwP\D sec X- (32)
264. Conditions at a Boundary. — When a disturbance passes from one medium into another, six conditions must be fulfilled at the boundary ; but of these six, only four are independent.
By Art. 228, the normal component of electric displace- ment must be continuous. Hence, if /, m, n denote the direction-cosines of the normal to the boundary, /, 17, // the components of displacement on one side of the boundary- surface, and /', /, // those on the other, we have
I (f -./') + m(g-g')+ n (h - //) = 0. (33)
Conditions at a Boundary. 163
Again, the tangential components of electromotive intensity are continuous. In fact, each component of electromotive intensity must be continuous in a direction perpendicular to its own, as otherwise, by (15), there would be an infinite rate of change in the magnetic induction.
Accordingly, if X, F, Z, and X', Y', Z' denote the components of electromotive intensity at the two sides of the boundary-surface, and Ai, m, Vl ; A2, /u«, v2 the direction- cosines of two mutually perpendicular tangents to the surface, we have
|_ ££iSlZ£-nI-(*-J>i£J (34)
As the magnetic induction fulfils the solenoidal condition, each of its components must be continuous in a direction coinciding with its own, and therefore the component normal to the surface must be continuous. Hence we have
I (a - a') + m (b - b') + n(c- c") = 0. (35)
Also, by (13), the components of magnetic force tangential to the surface are continuous, and therefore
Ai (a - a') + fj.i(p - /3') + vi (7 - 7') = 0,
i (36) A, (a -a') + /«»(j3-/3') + 1-2(7-7') = 0.
Equation (35) follows from equations (34), as may be shown in the following manner : —
By (15) we have da dn\ fdb db'\ fdc dc'
dy
<ty\ ) dx
M2
164 Electromagnetic Theory of Light.
From equations (34) it appears that X-X', Y- Y', and Z-Z' are proportional to (p\vi- vi/uz), &c., that is, to /, in, it ; or, if U= 0 be the equation of the boundary-surface to
dU dU , dU
Hence, if A denote an undetermined function of the coordinates, and Q be put for
we obtain
d A dU d 4 dU\ (d dU d A rfi
— A -. -T- A -7- ) + »* (-T A -r- - 3- A —
</s f/y </y flfe / \^ </2 (/s d.
(d dU d dU\
+ » ( y A — r A -r- 1
\</y dx (to ay )
iy dz dy dz ) \ dz dx dz dx
1 A\ I _ *,_
V dx dy dx dy
dx \ dy dz dy dz J dy \ dz dx dz dx J
i | j f =0
dz \ dx dy dx dy J j
In a similar manner, from equations (12), (13), and (36) we get
l^f(f-f) + »»-£ (g - g') + n 1 (k - //) = o.
Propagation of Light. 165
By integration, we obtain
1 (/-/') + »i (9 ~ g'} + n (/* - //) = constant, l(a - a') + m (b - b') + n(c - c') = constant ;
but as we are here considering only the results of the disturbance, we must suppose f,g,h; f, g', h' ; a, &c., to be all initially zero, and therefore we get (33) and (35).
265. Propagation of Light. — If we suppose each point of a plane area S to be a centre of disturbance, and draw the wave-surfaces of which these points are the centres, and which all correspond to the same period of time ti, a plane 2', parallel to S, which touches one of these surfaces will touch them all ; and if we draw straight lines from the boundary of S to the points of contact with S' of the surfaces whose centres are on this boundary, the area S' enclosed by this cylinder is made up of points at which the disturbances are all in this plaue, parallel to one another, and in the same phase. Consequently, 3" is the wave-plane at the time ^. Outside the cylinder the plane 2' does not touch any of the wave-surfaces, and the disturbances due to wave-surfaces corresponding to a period different from ^ are not in the plane 2', nor parallel to one another, so that instead of strengthening they interfere with each other. Thus the sensible effect is limited to the area within the cylinder passing through the boundary of S ; accordingly, the light is propagated in a straight line, and the direction of the cylindrical beam or ray is that of a line drawn from the centre of one of the wave-surfaces to its point of contact with 2'.
If the medium be isotropic, the wave-surfaces are spheres, and the ray of light is perpendicular to the wave-plane.
If the medium be not isotropic, the ray is in general not perpendicular to the wave-plane.
266. Reflexion and Refraction.— When a disturbance advancing through a medium reaches the boundary of another adjoining medium, the continuity of propagation is inter- rupted. The most general hypothesis we can make is, that disturbances, starting from the boundary, are set up in both media. A small portion of the boundary between the two media may be regarded as a plane area, and we may suppose
166
Electromagnetic Theory <f L></ht.
a cylindrical ray of disturbance to reach this area. The plane containing the wave-normal of the incident ray and the normal to the boundary is called the plane of incidence.
All the plane sections of the cylindrical ray which are parallel to the plane of incidence have a common perpendicular lying in the tangent plane to the boundary.
We shall suppose at first that each medium is isotropic.
Fio. 2.
Let AB be the line in which the boundary-surface is met by that plane of incidence which contains the longest of the parallel chords of the cylindrical beam of light. Let IA and JB be the lines of intersection of this plane with the cylindrical boundary of the beam.
Draw AP perpendicular to IA. When the disturbance reaches A, wave-surfaces start from A in each medium ; and when the disturbance at P reaches B, the wave-surface start- ing from A is a sphere having A as centre, and a radius equal to PB. There are corresponding wave-surfaces having their centres at all the points of the beam which lie on the plane
Reflexion and Refraction. 167
boundary of the two media. If we draw through B a perpen- dicular to the plane of incidence, a plane, through this line, touching the sphere having A as centre, touches all the wave- surfaces, and is therefore the wave-front of the reflected beam. A perpendicular to this plane will be in the direction of the reflected ray.
If £Q be a tangent to the section of the wave-surface starting from A, the reflected ray is in the direction of AQ.
We see, then, from the equality of the triangles AQB and £PA, that the incident and reflected rays make equal angles with the normal to the boundary-surface. We have seen above that the reflected ray lies in the plane of incidence. We have thus the two laws of reflexion in an isotropic medium. The direction of the refracted ray is obtained by a method similar to that employed for the reflected.
Describe, with A as centre, the wave-surface belonging to the second medium and corresponding to the period of time required by the incident ray to travel from P to B. If the velocity of propagation in the second medium is less than in the first, tlie sphere in the second medium will have a radius AQ' less than PB; and if * and i\ be the angles which the incident and refracted rays make with the normal to the boundary-surface, we see that the refracted ray is in the plane of incidence., and that
sin it _ AQ' _ V\ ~^7 = £P~V
where F and Fi denote the velocities of propagation in the first and in the second medium.
The ratio ~ is called ' the index of refraction of the two
V\ media' ; and if we denote it by /n, we have sin t = /z sin h>
When the second medium is crystalline, its wave-surface will have two sheets, and two tangent planes can be drawn passing through the perpendicular at B to the plane of incidence. The corresponding directions of displacement are obtained by means of Art. 260, and the lines from A to the points of contact of the tangent planes are the directions of the rays. In a crystalline medium there is thus double refrac- tion, and a single ray of light becomes, in general, two rays.
168 Electromagnetic Theory of Light.
267. Common Light and Polarized Light.— In an
isotropic medium the direction of displacement may be any whatever perpendicular to the ray. In the case of common light, the direction of displacement is not fixed, but after a few hundred vibrations passes into another direction in the wave-plane. In the case of light, some billions of vibrations are completed during a second, so that in any appreciable length of time we may consider that there are as many vibrations in any one direction in the wave-plane as in any other. When light is polarized, all the vibrations belonging to a given ray are in the same direction.
We have seen that when light passes into a crystalline medium it necessarily becomes polarized. In fact, when the direction of the ray is given, the tangent-plane to the wave- surface at the point where it is met by the ray is the wave- front, and the line in which this plane is met by the plane containing the ray and the wave-normal is the direction of vibration.
268. Intensity of Light. — The ultimate measure of the intensify of light is its effect on the eye, but indirectly we can ascertain how it depends on the displacement producing the light and obtain its mathematical expression.
It is ascertained experimentally that if light emanate from a constant source, the intensity of the illumination of a small plane area perpendicular to the direction of the light varies inversely as the square of the distance from the source.
We conclude from this that the intensity of light varies as the energy of the disturbance per unit of volume. In fact, if a disturbance emanates from a source 0 in an isotropic medium and spreads equally in all directions, the mean total kinetic energyremains constant,andthedisturbance atany time occupies the space between two spheres whose radii differ by a wave-length. Since the wave-length is very small, the space occupied by the disturbance is represented by 4m 2A, where A denotes the wave-length. Hence if thekiuetio energy be denoted
fTJ
by T, the energy per unit of volume is - — pr ; and this varies
*
inversely as the square of the distance from the source.
Intensity of Light. 169
The simplest form of expression for a periodic disturbance producing a plane wave whose front is perpendicular to the axis of x is a cos ^, where
and V denotes the velocity of wave-propagation. The corresponding velocity v of vibration is
27TF
-- r — a sin <•/>. A
Hence the mean value of t>2 is
47T2 1 f27r 27T*
— — a* — - sin20 d$ ; that is, — «».
Accordingly, the density of the medium being constant, the kinetic energy per unit of volume varies as — «z, or as the
square of the amplitude if T be assigned. If we now suppose a small plane surface to be illuminated by two similar sources of light, the rays from which are approximately perpendicular to the surface, and whose distances from it are equal, the disturbance due to one of these sources may be represented in any direction perpendicular to the ray by a cos 0, and that due to the other by a cos (0 + e) .
The total disturbance is, then, 2a cos |e cos ($ + |e). In a short period e passes from 0 to 2?r, and the mean value of the square of the amplitude is
which is equal to 2«2. Hence, if we suppose that the intensity of light is measured by the energy per unit of volume due to the disturbance, we find that the illumination given by two similar sources of light is double that given by one. Thus the conclusion already arrived at is confirmed.
170 Electromagnetic Theory of Light.
269. Energy due to Electromagnetic Disturb- ance.— We have seen (Art. 248) that if T be the kinetic energy of a system of currents in an electromagnetic field,
j tr\
— r = /(/« + wfl + nc)dS.
Since TIB a homogeneous quadratic function of the strengths of the currents,
«•-*£, also -^-f,*.. di dy ds
Substituting in (4), and applying Stokes's theorem, we have
.(IT . |Ynrfe ~dy ,,d*\. j . -rr = i F -r- + G-+ + H — } dn ; rft J\ rfs rf« rfyl
. r/«
but J — = all,
wliere a is the orthogonal section of the current, and i< the component of its intensity, and ads = rf@. Hence we get
where the integral is to be taken throughout the whole of space. Now, by (13),
</7 r//3 - 4nu = — -- -, &o. ; rf^r ds
whence, substituting, we have
= / { F(my - w/8) + (? (,,o - /y) + II (//3 - #wa)} dS
Energy due to Electromagnetic Disturbance. 171
where the volume-integral is to be taken throughout the whole of space, and the surface-integral over botli sides of every surface separating two media, and over a sphere whose radius is infinite.
Since a, /3, y are each at infinity of the order — , where
M is infinite, the surface-integral at infinity is zero. Again, tny - >?/3 is the magnetic force in the plane of yz perpendicular to the normal to the surface S. By (13), such a force, being tangential, is continuous in passing from one side of the surface to the other, and therefore the corresponding surface- integral, when taken over both sides of <S, vanishes. Hence, if we substitute for
dH dG
their equivalents «, b, c, we obtain
8irT= $(aa + Jj3 + Cy) r/@ = -a J(a' + |38 + 72) rf3.
Substituting for a2 + j32 + 72 its value from (31), we get
87rT=16
whence T= 2*
In addition to the kinetic energy of the electric currents, the disturbance produces potential energy W due to the electric displacement.
If we substitute ™A2, ^B\ and zsC* for
1 1 1
*' K' and K,
by equations (14) and (16), Art. 237, we get W = VTT Jw (A2f* + Bz(f + CW) f/@
= 27r / v (A*\* + B'-f.c + G'V) Z>V/8 = 2;r J * V*D*<i<5.
(38)
Hence, if E denote the total energy per unit of volume, we have
\ (39)
172 Electromagnetic Theory of Light.
270. Quantities to be determined in Reflexion and Refraction. — When light passes from one medium into another, there are four quantities to be determined by means of the equations holding good at the boundary. These quantities differ according to the nature of the media.
When light passes from one isotropic medium into another, the direction, intensity, and line of displacement of the incident ray being given, the directions of the reflected and refracted rays are known by Art. 266, and we have to determine their intensities and lines of displacement.
When the first medium is isotropic and the second crys- talline, the directions and lines of displacement of the two refracted rays are determined by Arts. 266 and 260, and also the direction of the reflected ray. We have, then, to find the intensity and line of displacement of the reflected ray, and the intensities of the two refracted rays.
Similarly, when light passes from a crystalline into an isotropic medium, we have to determine the intensities of the two reflected rays, and the intensity and line of displacement of the refracted.
Lastly, when botli media are crystalline, we have to determine the intensities of the two reflected, and of the two refracted rays.
271. Reflexion and Refraction. Isotropic Media.— Polarized light passes from one isotropic medium into another : determine the intensities and directions of electric displacement of the reflected and refracted rays.
Let D, D', and Dl denote the displacements belonging to the incident, reflected, and refracted rays ; then we may put
-D = acos^, !>' = &' cos $', DI = aj cos^, and we may assume that at the surface separating the media
* - f - f .. Again, if we put
/ = x/2^ Fa, 1' = yi^n Fu', 7, = y^, F,a,,
where Fand FI denote the velocities of propagation of the incident and refracted rays, we have I1 = ZTTZJ F2a2 = mean; value of E, by Art. 269.
Reflexion and Refraction. Isotropic Media. 173
Hence P, 7'2, and 7? express the intensities of the incident reflected, and refracted rays.
In the case of most dielectric?, w is sensibly the same so that we may assume •esl = -a.
Let the normal to the separating surface drawn into the second medium be the axis of X, and the plane of incidence the plane of XY. Then, by Art, 266, the axis of Zis the line of intersection of the three wave-planes.
FIG.
Let 01 be a wave-normal or ray, and the plane OZDSM the corresponding wave-plane, OZTthe direction of displace- ment^ and OM, perpendicular to 02), the corresponding direction of magnetic force.
Equations (21) show that the magnetic force is perpen- dicular to the wave-normal and to the displacement, and is so directed that seen from it the wave-normal must be turned counter-clockwise in order to coincide with the displacement.
Let the displacements make angles 0, 6', and Ql with OZ, and let the wave-normals make angles i, i', and i\ with OX; then, by Art. 266, wo have
Vi .
i = TT - /, sin ii = — sin i,
where V and Vl denote the velocities of wave- propagation in the first and in the second medium.
174 Electromagnetic Theory of Light.
If X, Y,Z-, X', Y',Z'; Xi, F,, Z, denote the compo- nents of electromotive intensity, and a, /3, 7 ; «', /3', 7' ; «i> /3i, 71 those of magnetic force corresponding to the three rays, by Art. 264 wo have
F+F'=F,, Z+Z' = Zlt /3 + 0' = /3lf 7 + 7' = 7,. (40) By Arts. 257, 254, we have
also, from fig. 3, we see that
F= - F sin 0 cos ?', F'= F'sin 0' cos/, FI = - Fj sin #1 cos A,}
Z = F cos 0, Z' = F' cos 0', Zi = Ft cos 0,.
(41)
/3 = - TTcos 0 cos e, /3' = 7f 'cos 0' cos /, /3i = - J7i cos 0j cos *',, }
7 = - H sin 0, 7' = - H' sin 0', 7! = - TTj sin 0,. )
(42)
As stated above, we may assume TO, = TO, and if in equations (40) the members of the first two be divided each by 2v/<!7rTO, and the members of the last two be multiplied
each by , these equations by (41) and (42), when
<6 y &TT
—. — r is substituted for — , become
61111, Pi
sin i cos i (/sin 9-1' sin #') = /! sin ^ cos A sin 0M sin t'(7cos 0 + /' cos 0') = II sin t\ cos 0,, cos i (I cos 0 - lf cos ^) = 7, cos t\ cos 0U
7 sin 0 + 1' sin 0' = /i sin 0,. From the first and last of equations (43) we obtain
sin 2« + sin 2t\ _ 27 £in0= — - 7,
sm 2t
(44) OT, . , sin2f-sin2i, .
27 sin 0 = — 7, sin ft.
sin 2t
Reflexion and Refraction. 175
And from the second and third we have
(45)
If the displacement of the incident ray be perpendicular to the plane of incidence, 0 = 0, and by (44) we have /! sin 0! = 0, whence I, = 0, or 0X = 0 ; but if we adopt the former alternative, by (45) we have I cos 0 = 0, which is impossible. Hence 0i = 0. In like manner, we get 0' = 0 ; and we learn that in this case all the displacements are perpendicular to the plane of incidence. Again from (45) we have
T _ sin 2i sin («,-»)
/'-shTCTTTr/' 7 ^InT/TTT)7- (46)
If the displacement of the incident ray be in the plane of incidence, 0 = ?, and from (45) we see that
a TT „, TT 1 = 2' 2;
as otlierwise, by 44, we should have 1=0. Hence in this case all the displacements are in the plane of incidence. Also, by (44), we have
/ - 2 sin 2i sin 2i - sin 2t\
sin 2i + sin 2t'i ' ~ sin 2i + sin 2«,
In this case, if sin 2<\ = sin 2i, we have /' = 0 ; that is, there is no reflected ray. When 2*\
,
«i = n - I, and — — r = tan i : 2 sin i ,
that is, if // be the index of refraction, i = tan'1 ^ ; and we learn that if the tangent of the angle of incidence be equal to the index of refraction, there is no reflected ray when the displacement of the incident ray is in the plane of incidence. The displacements belonging to common light may be resolved each into two components, one in the plane of incidence, and the other perpendicular to that plane.
176
Electromagnetic Theory of Light.
The whole of the reflected light is produced by the latter displacements when the angle of incidence is tan'1 ju. This light is therefore polarized, and, if it be made to impinge on a second reflecting surface so that the second plane of incidence is perpendicular to the first, there is no reflected ray when the tangent of the angle of incidence at the second reflecting surface is equal to the corresponding index of refraction.
The discovery of polarized light was partly based on the observation of the phenomenon stated above, and common light, when reflected at the angle of incidence tan"1 /u, was said to be polarized in the plane of incidence.
The plane of polarization as thus specified is perpendicular to the direction of the electric displacement which produces the light.
272. Reflexion and Refraction. Crystalline Medium. — Polarized light passes from an isotropic into a crystalline medium : find the intensity and direction of dis- placement of the reflected ray, and the intensities of the two refracted rays.
Adopting a notation similar to that employed in the last Article, and putting D, D' , />„ and Dt for the displacements belonging to the four rays, we may assume that at the boundary-surface <£ = <f>' = ^j = 02.
As before, take the plane of incidence for the plane of
Fio 4.
XY, and the normal to the boundary-surface as the axis of
Reflexion and Refraction. 177
Then all the -wave-planes pass through the axis of Z, and all the wave-normals lie in the plane of XY.
Let ZODS be the wave-plane, 01 the wave-normal, and OD the direction of displacement corresponding to one of the refracted rays. Then, by Art. 260, the plane 10 D contains OQ, the line of direction of the electromotive intensity, and QOD is the angle denoted by x in Arts. 260 and 263.
Equations (34) and (36) become, in this case,
F + r = Yl + F2, Z+Z' = Zi + Z2)\
0 + /3' = ft + ft, 7+7' = 7i + 7*-j
Also, by Art. 263, we have F, = 4:inzV'\Di sec Xi-
Now F, = FlcosQiF; but (fig. 4) from the spherical triangle Qi^F we have
cos Qi Y = cos Qi /! cos II Y + sin Qi /\ sin /! Y cos QJr F,
7T 7T 7T
Hence cos Qi Y = - sin xi sin i\ - cos Xi cos i\ sin 0t.
A similar equation holds good for Q>Y. Substituting in the value of Fi, given above, and using 1, 1', I1} and /2 as in Art. 271, we get, instead of Fi, the expression
- /! sin ii sec x\(sin xi sin i\ + cos xi cos i\ sin 0i) ; that is, - /i (sin t\ cos i\ sin 0i + sin2/! tan xO-
Again, Z± = Fi cos QiZ = Fi cos xi cos 0i.
Hence, instead of Z\, we get /i sin ^ cos 0lt
The expressions to be substituted for the magnetic forces are similar to those made use of in the case of isotropic media. Thus equations (48) become sin i cos i (/sin 9-1' sin 0')
= /i (sin ii cos i\ sin 0! -f sin2 ii tan xO
+ /2 (sin iz cos iz sin 02 + sin2 ia tan x? sin i(lGOsO + lf cos0') = /, sin?! cos0! + /2 sin i2 cos02, •cos ?(/cos0 - /' cos0') = /i cosz'i cos0, + /2 cos ?2 cos I /sin 0 + /' sin 0' = /! sin 0, + /2 sin 02.
N
178 Electromagnetic Theory of Light.
273. Uniradial Directions. — When the angle of incidence is given there are two directions of the displace- ment belonging to the incident ray for which there is only one refracted ray.
To find one of these directions we may suppose 72 zero in equations (49), and determine 0 in terms of i, i\, and 0t.
Making I2 equal to zero, and eliminating I' sin & from the first and last of equations (49), we get
/sin 2i sin 6 = I{ {sin (i + 1\) cos (* - t'j) sin 0t + sin2 1\ tan Y_, ) •
(50)
In like manner, from the second and third we obtain
/ sin 2» cos 0 = /i sin (i + i,) cos 0^ Hence, by division, we get
tan 0 = cos (i - f,) tan Oi + . " <! . . tan Y,. (51) sin (t -f «,)
The second value of tan 6 is obtained by putting 02 and /, for 0| and ii in (51).
274. L nia\al Crystals. — In the case of what are called uniaxal crystals, Fresnel's ellipsoid is a surface of revolution. If we suppose c = b in the equation of the wave-surface (29), Art. 261, that equation becomes
that is, (f*-P)(aV+#
But (52) is the equation of the surface composed of sphere whose equation is r" = i*, and the ellipsoid of revo- lution whose semi-axis of revolution is b, and whoso other semi-axis is a.
If a > b, and Fresnel's ellipsoid is prolate, the ellipsoid forming part of the wave-surface is oblate.
ITniaxal Crystals. jyg
These conditions hold good in the case of a crystal of calcium Carbonate commonly called Iceland spar This crystal is very celebrated in the history of science "s observations of its behaviour led to the discovery of double refraction and of polarized light.
The axis of revolution of Fresnel's ellipsoid is coincident with the line which is called the axis of the crystal This me is the axis of symmetry, and can be determined from the geometrical form of the crystal.
In the case of an uniaxal crystal, all rays inside the crystal whose directions of electric displacement are perpen- dicular to the axis are propagated with the same velocity Ilm appears from (25) by making C = B and A = 0 • then' M2 + v = 1, and V2 = B\ Conversely, if
we have (A2 - £2)cos2S = 0, and therefore S = ~. For these rays the wave-surface is a sphere.
Again, if the wave-surface be an ellipsoid of revolution since the normal to a surface of revolution meets the axi«' the ray, the wave-normal, and the axis must be in the same plane ; but the plane containing the ray and the wave-normal by Art. 260, contains the direction of electric displacement' Hence this direction is in the plane containing the ray and the axis.
When light passing through an isotropic medium is refracted at the surface of an uniaxal crystal, one refracted ray is refracted in the same manner as "if the crystal were isotropic, since the wave-surface of this ray is a sphere. I his ray is called, therefore, the ordinary ray. The other refracted ray, whose wave-surface is an ellipsoid of revolution, is called the extraordinary ray.
Both rays are polarized, and as a result of experiment it is said that the ordinary ray is polarized in the principal plane. By the principal plane is meant the plane passing through the refracted ray and the axis of the crystal. Hence we see again that the direction of electric displacement is perpendicular to the plane of polarization.
N2
180 Elccti-uiiKKjm-lic Theory of L'ujht.
275. Uniaxal Crystal. Reflexion and Itefrac- tlon. — In the case of an uniaxal costal, since \\ = 0, equutioiis (49) beeomu
sin » cos i (I sin 9-1' sin ff) = J, sin i\ cos n sin 6^
+ It (siu fa cos i2 siu02 + Biii*/8 tan x*)> sin /(7cos0 + 7'cos0') = I\ sin?', cos0i + /2 sinfjcos cos /(/cos0 - 1' cos 61') = /i cos?'i cos0i + 72 cos/2 cos 92, I em 6 + I' sin 0' = J, sin 9t + 72 sin 02.
As an example of the use of these equations, we may suppose light to fall on the surface of an uuiaxal crystal cut perpendicular to the axis.
In this case, since the axis of the crystal is the normal to the surface, the plane of incidence contains the axis and the wave-normal of the extraordinary ray, and, consequently, the ray itself. Hence both refracted rays are in the plane of incidence ; and, by Art. 274, we have
Accordingly, equations (53) become
sin i cos i(I sin 9-1' sin 6") \
= /2(sin ?2 cos it + siiiV2 tan ^2),
sin i(Icioad + /' cosfl'j = 7, sin i,, v (54)
cos » (7cos0 - 7' cos0') = Ii cos t\, 7 siu 0 + I' siu 0' = 72.
If we now suppose that the incident light is polarized in the plane of incidence, 0 = 0; and from the first and last of equations (54) we have
It (sin i cos i -f sin it cos i, -f siiiV2 tan \,) = 0.
Un'mial Crystal. 181
Since the expression by which 72 is multiplied cannot be zero, we get 72 = 0, and therefore sin 9' = 0. The second and third of equations (54) become, then,
sin / (7 + /') = 7, sin ?„ cos i (I- I'} = I, cos t\ • whence we get
27 sin / cos i = ^ sin (t\ + i), 21' sin i cos i = £ sin (i\ - i). Finally, we obtain
L=I sin2' r _ r si" ('. - i)
sin (*•+!,)' sTnTTwo'
Again, if the incident light be polarized in a plane perpendicular to the plane of incidence, 9 = ?, and the second and third of equations (54) become
le sin i cos 9' = 7X sin /\, - /' cos i cos Q' = l^ cos i\. Hence we obtain 7, sin (i + i\) = 0, and therefore ^ = 0 ; whence also cos 9' = 0, and 9' = - •
From the first and last of equations (54} we have, then, sin i cos i (I- 1'} = 72 'sin i2 cos H + sin2/3 tan ^2), I + I' = 72. "Whence
,, _ j sin * cos z ~ (sin 4 cos 4 + sin2 /2 tan ^2 sin i cos a" + sin i\ cos ?'2 + sin2 «2 tan ^2
r _ r 2 sin z cos «
sin e cos « + sin ?'2 cos ?2 + sin2 «2 tan ^3 '
These expressions can be put into a simpler form.
The angle x is the angle between the directions of electro- motive force and electric displacement, and is measured from the former towards the latter in the same direction as the line of displacement is turned in order to become the wave- normal. This appears from the figures and formulae of
182
Electromagnetic Theory of Liy/tt.
Arts. 260 and 272. What has been said amounts to this — that in equations (53) ^ is to be regarded as positive when the direction of the electromotive force does not lie between those of the displacement and the wave-normal, and conse- quently the ray does occupy this position. In the present case the axis-minor of the wave-ellipse is the normal to the surface, and the positive angular direction is from it to the refracted wave-normal. The refracted ray lies farther from the axis than the normal, and consequently does not lie between the electric displacement and the wave-normal. Hence in (56; the angle %? is negative.
An expression for tan x can be found by the geometry of the ellipse.
FIG. 5.
In the figure OX represents the axis of the crystal, OQ the line of electromotive force, OP that of electric displacement, 01 the extraordinary wave-normal. Then v is the angle QOP ; but QOP = R01, and
— = UI' OI = 2 triangle ROI 01 = OP 01* ~
Now, if pi and jh be the focal perpendiculars on the tangent, and ti and U the intercepts on the tangent between
their feet and the point of contact, — = — , and therefore
(p<+p,)(t,-t,) - (/>, -
Ui/id.i'fil Crt/sfa/.
183
but £(;;, + jh)((\ - tz) is double the area of the triangle ROI, and | Q0i -Jfc)(& + &) is double the area of the right-angled triangle whose sides are ^/((i2 - bz) sin t'2 and v/(rt2 - I'} cose'z, the angle XOI being «,. Hence, if Of be denoted by ;j, we have
In the present case ^2 is negative, and we have
(a2 - i*) sin2 /2 \ sin e, cos it + sin- /2 tan v2 = sin ?2 cos Ml- —
\ fl2sin2/2 + 62cos2?2y
J2 sin i-i cos /2 jB2 sin «2 cos «'2
az sin2 ?2 + i2 eos2 it A2 sin2 /2 + B* cos2 /2
Again, if Fdenote the velocity of propagation in the external medium,
F2 sin2 /2 = (^42 sin2 ?2 + ^2 cos2 ^2) sin2 / ; whence
sin2 h + B2 cos2/2
F2 -
Hence
B2 sin /2 cos /2 5 sin t
^'Bin'fc + .B'oos8/,
and
sin/ cost + sin /2 cos/2 + siu3'/2 tan ^2
sin e ( F2 cos i + B </ F2 - A2 sin2 /j Vi
siiu'cos? - (sin/2 cos/2 + si
sin? { F2 cost - B </ F2 - A2 sin2?! F2
(58)
184 Electromagnetic Tli<or>/ of Liyht.
Accordingly,
F'oos/- 7? y/CF2- ^4' si n2?) F2 cos > + £ v/( F' - ^4* wn»/) '
2 F8 cos i
7, = 7
F8 cos i + 7V( V* - A* si
If the value of i be such that I' - 0, the reflected ray, when common light falls on the crystal, is polarized in the plane of incidence. This value of i is called the polari/ing angle of the crystal when cut perpendicular to its axis.
Making 7' = 0 in (59), we have
F4(l - sin'i) - &( V* - A* sin'*) ;
whence
F;(F8-7J2)
F* -^8/y2
(60)
276. Reflexion and Refraction at Interior Surface
of Crystal. — When light passes from the interior of an uuiaxal crystal into an isotropic medium, there are, in gem- ral, two reflected rays ; and when the incident ray is nn ordinary ray, we have
sin /i cos f\ (/, sin Oi - 1\ sin + (sin i\ cos i\ sin W*
tan \z) l
= 73 sin ?3 co8/s sin 0., sin f\(7i cos Oi + I\ cos 0',) + 7'a sin i\ cos 0'2,
= 73 sin /s cos ^3, cos ;',(/i cos 0i - 7'i cos 0'i) + 7'j cos ?"2 cos 0'a.
= 73 cos /3 cos 03, 7, sin 0! 4 /', sin 0\ + 7', sin 0'a = 7, sin 08.
^
Reflexion and Refraction. 185
When the incident ray is an extraordinary ray, the equations at the refracting surface become
(sin it cos /2 sin 02 + sin2 ?2 tan ^2) J2 1
+ I' \ sin i\ cos i'\ sin O't \ + (sin /'2 cos i'z sin 0'2 + sin2«'2 tan x'2) 7'2
= /3 sin 4 cosz'g sin 0,3,
/2 sin /2 cos 02 + I\ sin ?'i cos 0\ + 7'2 sin A cos 0'z <. (62)
s 73 sin /3 cos 0 /2 cos /2 cos 02 + I' i cos A cos B'i + I'z cos A cos B'z
— /3 C(^S ?3 COS 0
/2 sin 02 + l'\ sin 0'j + /'2 sin 0r2 = /3 sin 03. j
When the crystal is cut perpendicularly to its axis,
In this case, the first and last of equations (61) become (sin <"2 cos i'z + siir/'2 tan ^'2) 7'2 = /:3 sin 03 sin ?'3 cos /j,
/'a = /3 sin 03. Hence
/3 sin 03 {sin /3 cos ^ - (sin A cos i'z + sin2 A tan x'z)} = 0 ;
but. the multiplier of 73 sin 03 in this equation is nor, in general, zero, and therefore we have
sin 03 = 0, I', = 0.
Consequently there is no extraordinary reflected ray, and the refracted ray is polarized in the plane of incidence. From the second and third of equations (61) we then obtain
sin (>, - i.) si'i_ /63)
186 Electromagnetic Theory of Light.
If the ray incident on the interior surface of the crystal bo the extraordinary ray, and the crystal, as before, he cut perpendicularly to its axis, since all the rays and wave- normals are in the plane of incidence which cuts the wave- ellipsoid in an ellipse whose axis-minor is the axis of the crystal and also the normal to the surface, we have
i'a = TT - i,, x's = - X« »
whence
sin i't cos i't + sinYa tan x'a = - (sin it cos /2 + sin2/2 tan xJ-
In this case, the second and third of equations (62) become
/' 'i sin /'i = Is sin /3 cos #,, I'\ cos ?', = I3 cos /3 cos 03,
whence I3 cos 03 sin (/3 - i'i) = 0 ; but siu(i3-i'i) cannot be zero, and therefore cos03 = 0, and 7', = 0. Consequently, there is no ordinary reflected ray, and the refracted ray is polarized in a plane perpendicular to the plane of incidence. The first and last of equations (62) now become
(sin it cos it + sin2 it tan x8) ( ?t ~ 1'*) - fa giu '» cos ''»>
It + I't = /a- Hence
sin i2 cos it + sin2 /2 tan x^ - sin /3 cos /3 2 sin it cos it + sin2/2 tan xa + gin '3 cos /3 '
2 (sin it cos it + sin'/a tan x«)
sin it cos it + smV2 tan xa + sin h cos »3
By reductions similar to those effected in the ca-e of -equations (56) we get.
,2 - A* sin*/,) + F32 cos /, '
a2 - A1 sin2/3) - Fs" cos /3
-r j & L* *y y r a — •** Dill 13;
2 « '/T^32-^28in2/3)-f F82oos/3'
Singularities of the Wave-Surface. 187
If an uniaxal crystal, bounded by faces parallel to each other and perpendicular to the axis, be placed in an isotropic medium and a ray of light polarized in a plane perpendicular to the plane of incidence be transmitted through the crystal the incident and emergent rays are parallel, and the plane of polarization remains unchanged. Then ,', = ,', and in virtue of equations (59) equations (64) become
(65)
When the incident ray falls on the first surface of the crystal at the polarizing angle, we have
l' = 0, 7'2 = 0, and J3 = 72 = /.
In this case, the incident light passes through the crystal unchanged in intensity, direction of electric displacement, and direction of propagation,
277. Singularities of the Wave-Surface — The
equation of the wave-surface, Art. 261, may be put in the form
<«V 4 by + cV - «V)0*;S + if + s2 - V) - (<? - V)(b* - c2)//2 = 0.
From this equation it appears that if the point of intersection of the three surfaces
« V + I? if + cV - rt»e» = 0, x2 + i' + z* - b2 = 0, // = 0
be taken as origin, the lowest terms in the equation of the wave-surface are of the second degree, and therefore that the origin is a double point on the wave-surface at which there is a tangent cone of the second degree.
188 Electromagnetic Throry of Light,
If we seek for the coordinates of the points of intersection of the three surfaces, we have
«V + cV = rtV, x* + z> = b\ >/ = 0 ;
whence we obtain for the coordinates of the point the
expressions
,.j M ;,» _ rt
*-'£$< **-°> *' = a^- (fi6)
The equation of the circular sections of Fresnel's ellipsoid
whence, if w,, tstt and za3 denote the direction-cosines of the perpendicular to a plane of circular section, we have
,fi~
From (66) and (6?) it appears that a singular point on the wave-surface is on a perpendicular to the plane of & circular section of Fresnel's ellipsoid at a distance l> from the origin.
The existence of such points follows readily from the mode of generation of the wave-surface described in Art. 260. Fiona thence it appears that the perpendicular to each section of Fresjiel's ellipsoid meets the wave-surface in two points whose distances from the centre are equal to the principal semi-axea of the section.
If the section be circular, every axis is a principal axis, and all the corresponding points on the wave-surface coalesce into one.
The perpendiculars on the corresponding tangent-planes of the ellipsoid are, however, not in the same plane ; and thus corresponding to the one ray going from the centre to the singular point there are an infinite number of wave-front* that is, an infinite number of tangent-planes to the wave- surface meeting at the singular point.
Singularities of the Wave- Surf nee. 189
As the wave-normals and velocities of propagation are different for these fronts, when the ray reaches the surface of the crystal it is refracted into an infinite number of rays, forming a cone, and the phenomenon exhibited is termed conical refraction.
From the consideration of the ellipsoid reciprocal to Fresnel's ellipsoid, it is easy to see that the wave-surface must possess singularities of another kind in addition to those mentioned above.
From Art. 260, it appears that the perpendicular to each section of the reciprocal ellipsoid is perpendicular to two tangent-planes of the wave-surface, and meets them in points whose distances from the centre are the reciprocals of the semi-axes of the section. If the section be a circular section, every axis is a principal axis, and all the corresponding feet of perpendiculars on tangent-planes to the wave-surface coalesce into one.
The central radii of the reciprocal ellipsoid are co-direc- tional with perpendiculars on tangent-planes of Fresnel's ellipsoid, which are the reciprocals of the radii, so that all the perpendiculars to tangent-planes of Fresnel's ellipsoid which lie in a circular section of the reciprocal ellipsoid are equal to the mean semi-axis of Fresnel's ellipsoid, and corre- spond to a single tangent-plane to the wave-surface. The corresponding radii of Fresnel's ellipsoid do not, however, lie in the same plane, and are not equal, so that there are an infinite number of rays corresponding to the same wave-front which must therefore touch the wave-surface all along a curve. To find the nature of this curve, we may proceed thus.
Let p denote the length of the central perpendicular on a tangent-plane of Fresnel's ellipsoid, and a, /3, 7 its direction- .angles.
If /; lie in the circular section of the reciprocal ellipsoid, we have p = b, and therefore
<iz cos2a + b~ cos2/3 + c2cos27 = £2(cos2a + cos2/3 + cos2^) ; that is, («2-&2) C082a - (62-C2) C082y = 0.
Also, cos2 a + cos2 7 = siu2/3 ;
b"1 — c2 a2 — b~
whence cos2a = — -, siu2/3, cos2 7 = 2 _ ^ sin'/j (68)
190 Electromagnetic Theory of Light.
Let r denote the central radius of Fresnel's ellipsoid to tlie point of contact of the tangent-plane perpendicular to j), then
(i* cos2 a + i4 cos2/3 + c* co.-r-y ~V~
and if p denotes the distance of this point of contact from the foot of the perpendicular, /o8 = r9 - ;A In the present case, p = b, and we have
a* cos2 n + b* cos2 /3 + <••* cos2 y - 4*
p* - —j-r-
_ (r«4 - &') cos2 « - (6* - /•') cos? Y P
Substituting for cos2a and cos27 their values from (68), we get
It is plain, from the construction in Art. 260, that p is the distance from the foot of the perpendicular on the tangent- plane to the wave-surface to its point of contact, and that this distance is parallel to the corresponding direction of displace- ment in the wave-plane. In the present case the wave-plan! contains the axis of y, and /3 is the angle which the electric displacement makes with this axis. Hence j3 is the angle which the line from the foot of the perpendicular to the point of contact of the wave-front with the wave-.surt;i( .- makes with a parallel to the axis of y in the wuve-front.
Accordingly,
-*•>*. ft (69,
is the equation of the curve along which the wave-front touches the wave-surface. This curve is therefore a circle which touches the parallel to the axis of y at the foot of the perpendicular from the centre, and whose diameter is denoted by the expression
Total Reflexion. 19j
Corresponding to the wave-plane we have been considering there are an infinite number of rays which meet the wave- iront along its circle of contact with the wave-surface. All these rays have the same wave-normal, and are propagated with the same normal velocity. Hence, when they are re- fracted at the surface of the crystal, the emergent rays are parallel and form a cylinder. Unless the wave-normal be normal to the surface, the section of this cylinder made by the plane bounding the crystal is an ellipse.
The remarkable phenomena described above were foretold by Hamilton as consequences of properties of Fresnel's wave- surface discovered by him. They were realized experimentally nrst by Lloyd, and long afterwards by Fitzgerald.
278. Total Reflexion.— When light passes from a denser into a rarer medium, if the angle of incidence exceed sin"1 -,
where ^ denotes the relative index of refraction of the medm there is no refracted ray. In fact, under these circumstances^ a refracted wave-plane is impossible, as it would in the case of an isotropic medium, be a tangent-plane to a sphere drawn through a line lying inside the sphere. If both media be isotropic, equations (43) seem impossible to satisfy ; for, if we suppose /! zero, these equations cannot be satisfied unless we make I and /' each zero.
Mathematically it is possible to give a solution of equa- tions (43), which in its final result is physically satisfactory ; but it seems impossible to obtain a sound physical basis for the equations themselves.
The mathematical solution is as follows : — Assume D = ae-<4>, I)' = a'e-'p', D1 = aj e-«i>i,
where < = v/- 1 , and (/> = -^ { Vt - (Ix + my 4 ws) j,
A
'
AI
then the differential equations of wave -propagation are satisfied, and D, &c., are periodic.
192 Electromagnetic Theory of Light.
If we now suppose the incident light polarized in the jilane of incidence, since
Fi 1 V 1
A7 = ^' T = ^' and ri = T'
at the origin, where x, //, and x are all zero, we have
f - f' ~* * f» J
and as equations (43) mathematically hold good, we have sin 1 1 cos i - cos », sin t r sin f , cos i + cos t\ sin «
But sin »', = /z sin i, cos ix = t -v/O*2 sia8' ~ !)•
and therefore
/' fi sin t cos « - i sin » v/(/u2 sin2 i - 1) 1 - t tan e / a sin i cos t + t sin & *S ($£ sin2 * — 1) 1 + i tun t
v/^sii.2/-!) = (cos c + ( sin e s = e"2", where tan e = — *• — .
fj. COS /
Hence D' = a r2'* <r"J>' = a e-'(<?>'-i-2«),
and, accordingly, the intensity of the reflected light is equal to tiiat of the incident ; but its phase is increased by 2e. Again,
and since the axis of x is normal to the surface separating the media, and the axis of z perpendicular to the plane of incidence, we have
/j = cos i, = t </(fj? siu't - 1), nil = AI sin i, N! = 0,
2iir,¥,
(70)
Absorption of Light. 193
nt ™r-hHS ^^f1?"' ^e .P°*er of e whose index is real is P^10.d^; and since A, is very small when * is of sensible magnitude this factor tends to become very small Hence at any sensible distance from the boundary A is very small and there is no visible refracted ray.
In other cases of total reflexion a similar mode of treat- ment may be employed The results obtained above satisfy the mathematical conditions holding good when the reflexion is not total, and the final result is consistent with the observed phenomena; but the whole investigation can scarcely be regarded as having any physical validity.
279. Absorption of tight.— When a medium is not a perfect insulator, an electromotive force produces not only a change 01 electric displacement but also a conduction-current
i± G be the electric conductibility of the medium, the istance of an element of unit section parallel to the axis
of x is — , and the electromotive force for this element is
Xdx. Hence, if i\ denote the intensity of the conduction- current parallel to x, we have i\ = CX.
The total current is made up of the conduction-current and that due to a change of the electric displacement; accordingly, we have
and as X = ~f, we obtain u =/+ ^
•**
ic
Substituting for u in terms of the components of magneti force, and for the latter in terms of those of displacement, by means of equations (13), (15), and (11), we get
f v./ + . (7J)
K -&K dxdx d v '
The last term in this equation is zero; and if we take the normal to the plane of the wave as the axis of z, the displacement /is a function of z only, and (71) becomes
K
( >
194 Electromagnetic Theory of Light.
If U denote the velocity of wave-propagation when there is no absorption, we have
and putting ir-nC = k, we get Thus (72) becomes
To solve this equation, we may assume /= a «'0>< -»«*),
where /-— 2?r
i = v/-l, n = — ,
and m is a quantity to be determined so as to satisfy (73). We have, then,
- n*
that is, «2
-^-4*««. (74)
Assume m-q- ip, then
Eliminating g, we get /hence
;>'' =
Absorption of Light. 195
As/> is real, p2 must be positive, and therefore
Here A; is of the same order as C, which is of the order
K 1
or
Hence -— is of the form v^2, where v is a numerical coefficient depending on C and on the units selected, T the time of vibration in the wave of light, and ^the unit of time. In order that C should have any sensible magnitude, V must be enormously great compared with r. Hence WU' jg a
^j2
small quantity, whose square may be neglected in the expansion of the square root, and we have
2U* 2n* Substituting q - tp for m, we get
/ = a, ft* €'(»*- **L (75)
As the wave is advancing in the direction of 2 positive, q is positive ; and since pq is positive, p must be positive. Hence
2ir^CU, (76)
also
_ 2kn _ n _ '2Tr
q = IT ~ 1J = Ur' and
2i7T
- (Ut- Z\
f = ae-f'eVr • (77)
196 Electromagnetic Theory of Light.
It follows, from the expression obtained for f, that the velocity of wave-propagation is U, and is therefore unaltered by absorption. In consequence of the factor e~ps, the amplitude of /diminishes as z increases. Since ;; varies as C, unless C be very small, the amplitude of /diminishes rapidly, and the medium is practically opaque.
280. Electrostatic and Electromagnetic measure. —
The reader of the foregoing pages may have been struck by an apparent inconsistency between the present Chapter and Chapter XI.
In Chapter XI. the specific inductive capacity k is of the nature of a numerical quantity. In the present Chapter, the specific inductive capacity TTis regarded as the reciprocal of the square of a velocity. The apparent inconsistency results from the fact that in Chapter XI. the various quantities are supposed to be expressed in electrostatic measure, whereas in the present Chapter they are supposed to be expressed in electromagnetic.
We must consider the hypotheses on which the two modes of measurement are based, and how it is that in reference to space, time, and mechanical force, the expression for the same physical quality of a body is in one mode of expression a quantity of a nature different from what it is in the other.
Lot e and E denote quantities of electricity expressed in electrostatic and electromagnetic measure, JCand 3£ the corre- sponding electromotive intensities, and / and f the displace- ments. Let L denote a linear magnitude, and T a portion of time, and let us use the symbol = to mean that two expressions denote quantities of the same nature.
Electrostatic measure is based on the assumption that the product of two quantities of electricity divided by the square
e* of a line denotes a mechanical force, that is, — = mechanical
force.
Electromagnetic measure is based on the assumption that the product of the strengths of two magnetic poles divided by the square of a line denotes a mechanical force, that is, if m denote the strength of a magnetic pole,
«»« e»
yj- = mechanical force «= — ; whence e = m.
Electrostatic and Electromagnetic Measure. 197
Again, i denoting the strength of a current, E= Ti; but f = ,/, where/ denotes the strength of a magnetic shell, and JD = magnetic moment = mL ; wlience
The electromotive intensity multiplied by a quantity of electricity denotes in either system of measurement a mechanical force ; according^, eX = E3, ; but
T T
E = -e, and therefore £ = —X.
-Li 1
Again, f I? = E, and fD = e ; whence
I-*,, also /-£-*,
and therefore
T T2 1
where F denotes a velocity. Thus k is a numerical quantity, but K the reciprocal of the square of a velocity.
The magnitude of the unit of electricity differs very much in the two systems of measurement.
In the electromagnetic system, two units at the unit of distance apart act on each other with the unit force.
In the electromagnetic system, two magnet-poles of unit strength, at the unit distance apart, act on each other with the unit force.
A circular current of unit strength acts on a unit magnet- pole at its centre with a force which is 2?r times the unit of force, provided the radius of the circle be of unit length.
The quantity of electricity which passes through a section of this circuit in the unit of time is the unit quantity of electricity expressed in electromagnetic measure.
The quantity of electricity contained in the electro- magnetic unit is n times the quantity contained in the electrostatic.
198 Electromagnetic Theory of Light.
If, then, E and e denote the same absolute quantity of electricity expressed, one in electromagnetic, the other in electrostatic units, and if L and T denote the units of length and of time, we have
but E1S, = eX, where 3c and X denote the electromotive force corresponding to the quantity of electricity denoted by E and e ; whence
Then,
E = ^-e = ~ UkX = ISk-^r- nL nL M* L2
whence
When the second and centimetre are taken as the units of time and length,
n = '3 x 1010 approximately.
Note on Thomson and Dirichlet's Theorem. 199
NOTE ON THOMSON AND DIEICHLET'S THEOREM,
ARTICLE 70.
When the number of given surfaces is reduced to one, this theorem is proved by Gauss in the following manner : —
(1) On a given surface 8 a homogeneous distribution of a given quantity of mass is always possible, such that J VadS is a minimum. For this distribution, F"is constant for all occupied parts of the surface, and there is no part unoccupied.
If r denote the longest distance between any two points of S, and M the total mass, it is obvious that at any point
of S the potential cannot be less than — , since the distri-
bution is homogeneous, that is, composed of mass having everywhere the same algebraical sign. Hence J VadS cannot
be less than — • r
Consequently \VadS cannot be diminished without limit, and there must be a distribution such that J VadS cannot be made less. In this distribution V must be constant. For, if for an occupied portion Si of the surface, Vi be everywhere greater than A, and for another equal portion S2 of the surface Vz be everywhere less than A, at each point of 2i let §<T = - v, and at each point of S2 let 8<r = + v, then the total mass remains unaltered, and
since SF is the distribution resulting from So-, and therefore, by Art. 51, we have
200 Note on Thomson and Dirichlet's Theorem.
Accordingly, J VadS has received a variation which is essentially negative, nnd consequently cannot be a minimum for the distribution <r. Hence, when the integral is a mini- mum, V is constant for the occupied part of the surface. If there were a part unoccupied by Art. 66, V would be less for this part than for the occupied part, and hence as before the integral could be made less. Accordingly, in the distribution for which jVadS is least, there is no part of the surface unoccupied.
(2) If U be a given function of the coordinates, a homogeneous distribution of given mass over S is possible, such that J(F- 2U)adS is a minimum. For this distri- bution V- U is constant for all occupied portions of the surface.
If V be the largest value of U on S, it is clear that / ( V- 2 U) <j(tS cannot be less than
and therefore that there must be a distribution such that J (V '- 2U)adS cannot be made less. For this distribution V - U is constant at all occupied parts of the surface.
Let JF= J ( F - 2 Z7) odS, then
8 W= J $r*dS + $ (V- 2ET) falS = 2l(V
If V- U be greater than A at every point of an occupied portion of surface 2,, and less than A at every point of an equal portion 2* °f surface, as in (1), SJP can be made negative, and therefore W cannot be the least possible.
In this case, if part of the surface S be unoccupied, V- U may be greater on this part than it is on the occu- pied part, and therefore in this case we cannot show that the whole surface must be occupied.
(3) Suppose now three distributions of mass on S.
1. A distribution whose surface-density is <TO and potential Fo, such that J VadS is the least possible, the total mass being M.
Gauss's Method. 201
2. A distribution whose surf ace-density island potential F, such that l(V-^U} odS is the least possible, the total mass being M, and e being a given constant
3. , A distribution whose density is <r2, and potential Ftf such that
then the total mass is zero, and
but this is constant for all parts of the surface occupied by <r,.
If £ be diminished without limit, the distribution cr, passes into <TO, and in this case there is no finite portion of the surface S left unoccupied.
Hence, when £ is diminished without limit, Fi - U is constant for the whole surface S.
Let us now superpose on <r2 the distribution whose density is a<rQ, where a is constant. Then
V= a V, + V,, and V- U= a V, + F2 - U.
By a proper determination of a the right-hand member of this equation can be made zero at all points of S.
Accordingly for a single surface Thomson and Dirichlet's Theorem is proved.
This theorem in its most general form can readily be deduced from the properties of fluid motion.
Suppose that the given surfaces <S\, S2, &c., are sur- rounded by liquid, or incompressible fluid, of unit density, extending to infinity. Apply to the liquid at each surface an impulsive pressure which at each surface is equal to the given value of Thomson's function for that surface. The liquid begins to move irrotationally, and the velocity potential of the motion is the same as the impulsive pressure, and is equal at each surface to the given value of Thomson's function, and satisfies Laplace's equation for the whole of
202 Note on Thomson and Dirichkfs Theorem.
Thus the truth of Thomson and Dirichlet's Theorem is established.
It is easy to show from Green's theorem that, if there be a given quantity of mass on each of a number of surfaces, this mass may be so distributed that the potential shall be constant over each surface.
Let 2fF=2f FarfS. Then
- +£>a - f \(*r\\ (*T)'+ f^
dv dv J }\\dx ) \dijj \dz
- - z
Hence W is essentially positive, and cannot therefore be diminished without limit, and there must be a distribution of mass such that W cannot be made less. For this distribution V must be constant for each surface. For if V be not con- stant, W may be made to receive a variation which is negative by transferring positive mass from points on the surface where Fis greater than A to points where it is less than A. In this case it is not necessary that the part of the surface from which the transfer is made should be occupied. On the other hand, if V be constant for each surface, any change of distribution increases W. For, let v be the change in F, then
dv
dv
where the surface-integrals are to be taken over both sides of the surfaces ; but as Fis constant for each surface, and
) — dS zero, dv
since the total mass is constant, we have
since V2/y = 0 throughout the field. Hence the change in W is essentially positive, and JFis least when Fis constant over each surface.
Gauss's Method. 203
The property of the potential made use of in (1) to show that the whole surface must be occupied is perhaps more readily established by the method of Gauss than by that employed in Art. 66.
Gauss's method is as follows : —
If there be no mass outside a surface S on which the potential is everywhere positive, its value at a point 0, outside S, is positive, and less than A its greatest value on S.
For if the potential P at 0 be greater than A, draw lines in all directions from 0 ; they meet the surface S or go to infinity, and the potential on any one of them passes from P to A, or to some value less than A. Hence on every line there is a point at which the value of the potential is B, lying between P and A. All these points form a closed surface at which the potential is constant ; and as there is no mass inside it, the potential has the same value throughout the interior of the surface, Art, 61. Hence the value of the potential at 0 is B, and cannot consequently be P as was supposed. If P were negative, we could show in like manner that the potential at 0 must lie between P and zero, and could not therefore have the supposed value.
Again, the potential at 0 cannot be A or zero. For if it has either of these values, describe a sphere round 0 as centre. At no point of the surface of this sphere can the potential be greater than A or less than zero. Hence its mean value on this surface cannot be A or zero, unless it have this value for the whole surface of the sphere, in which case it would have the same value for the whole of space external to S, which is impossible.
If the potential be everywhere negative on the surface o, its value at a point 0 outside S is negative, and less in absolute magnitude than its greatest negative value on /S'. This is proved in a manner precisely similar to that adopted in the case of the positive potential.
INDEX.
\_Thef.gures refer to the pages. ~\
Absorption of light, 193.
Analogues of tesseral harmonics, 47,
56. Anisotropic ellipsoid in magnetic field,
106.
dielectric, 132. magnetic medium, 105. Attraction, Earth's tangential com- ponent of, 14. on dielectric body, 132. Axis of magnet, 95.
Boundary between two dielectrit media, conditions at, 131, 162.
Clairaut's theorem, 12. Complete solid harmonics, 26.
spherical harmonics, 3, 23. Conductors, charge on, 127.
distribution of electricity on, 131,
134.
Conductors, system of charged, 139. Conical refraction, 189. Crystalline dielectric, 132.
distribution of electricity on con- ductors in, 134. electromagnetic disturbance in,
155. energy in, expressed as surface
inte'gral, 136. energy due to electrified particle
in, 136.
force on electric particle in, 140. force due to spherical conductor
in, 141. force due to spherical particle in,
142.
potential in, 134. potential due to spherical con- ductor in, 140. wave-surface in, 157, 160.
Current, electric, energy of, in mag- netic field, 115, 144.
in presence of a second, 118.
intensity and magnetic force, 149. Cylindrical refraction, 191.
Dielectrics, 124.
Dielectric body, attraction on, 132.
Dirichlet, 199".
Displacement, electric, 124.
direction of, in plane- wave, 153. distribution of, 125. due to electrified sphere, 127. energy due to, 125, 132. Distribution of given mass on surfaces for potential constant for each surface, 202.
Disturbance, electromagnetic, propa- gation of, 150.
Earth, figure of the, 11.
Earth's magnetic force, locality of
source of, 108. Earth's magnetic poles, 109.
potential, 107. Electric current, energy due to
presence of, 115, 144. force on current element exercised
by, 122.
magnetic force of, 114, 116. magnetic potential of, 114. magnetic shell equivalent to, 113. properties of, 110. Electric current element, force on, in
magnetic field, 121. magnetic force of, 116. Electric currents, energy due to
mutual action of, 118. mutual forces between, 118.
206
Index.
Electric displacement, 124. distribution of, 125. due to electrified sphere, 127. energy due to, 125. Electricity, distribution of, on con- ductors, 131.
Electrified particle in crystalline me- dium, force on, 140. force due to, 142. Electrified particles, force between,
130.
Electrified spheres, energy dur to, 128. Electromagnetic disturbance, energy
of, 170. Electromagnetic field, equations of,
160.
Electromagnetic theory of light, 144. Electromotive force, 112. Electromotive intensity. 130, 149,
162. Electrostatic and electromagnetic
measure, 196.
Ellipsoid, anistropic, in field of mag- netic force, 106. components of attraction of, 77. isotropic in magnetic field, 105. potential of, 78, 79. Ellipsoids in general, 60. of revolution, 39. oblate, 54. prolate, 39. Ellipsoidal harmonics, 39, 60.
expressed as functions of Carte- sian coordinates, 66. identity of terms in equal series
of, 70.
surface integral of product of, 69. vanishing at infinity, 65. Energy and electromotive force, 145. Energy due to electromagnetic dis- turbance, 170.
Faraday tube, 125. Fitzgerald, 191.
Fluid, homogeneous mass of revolv- ing, 10. Focaloid, attraction of, 77.
for ellipsoid in general, 77.
for oblate ellipsoid, 58.
for prolate ellipsoid, 52. Fresnel, 158. Fresnel's ellipsoid. 158, 178.
wave-surface, 160, 178.
Gauss, 107, 199, 203.
Hamilton, 191.
Harmonics, complete spherical, 3, 23.
ellipsoidal, 39, 60, 73.
incomplete, 34.
of imaginary degree, 38.
reduction of function to spherical, 24.
solid, 1, 26, 74.
spherical, 2, 18.
applications of, 7. Homocoid for ellipsoid in general, 7'2.
for oblate ellipsoid, 68.
for prolate ellipsoid, 52.
Induction, magnetic, 87, 88, 102,
149, 150.
Intensity of light, 168. Interior surface of; crystal, reflexion
and refraction at, 184. Isotropic medium, 154.
Lagrange's equations, 145. Laplace, coefficients of, 2, 20. Legendre, coefficients of, 3, 15. Light, absorption of, 193.
common, 168.
electromagnetic theory of, 144.
intensity of, 168.
Maxwell's theory of 147.
polarized, 168.
Lloyd
propagation of, 165. 'd, 107,
191.
MacCullagh, 160. Magnetic axis of body, 95.
force, 87, 88, 149, 150, 154, 101.
induction, 87, 88, 102, 149, 150. vector potential of, 90, 92.
moment of element of body, 81.
of body, 96.
particle, vector potential of, 95.
shell, 97.
energy due to, 100, 101.
potential of, 97.
system, energy due to, 90. Magnetism, induced, 102.
distribution of, 103.
terrestrial, 107.
207
Magnetization, intensity of, 81. Magnetized ellipsoid, 84.
body, potential of, 82, 86. sphere, 83. Magnets, constitution and action of,
81.
energy due to, 89. Maxwell's method of forming solid
harmonics, 28. theory of electric displacement,
124.
light, 147.
Medium, dielectric influence of, 123. Moment, magnetic, of body, 95.
Oblate, ellipsoid of revolution, 54, 50.
Poisson's equation for magnetized body, 83.
Polarization, electric, 124.
Polarized light, 168.
Potential, expansion of, in solid har- monics, 1.
of surface distribution on pro- late ellipsoid, 50. on oblate ellipsoid, 56.
Prolate, ellipsoid of revolution, 39, 50.
Propagation of disturbance in electro- magnetic field, 150.
Reflexion and refraction of light, 165.
from crystalline medium, 176.
at interior surface of crystal, 184.
in isotropic media, 172.
quantities to be determined in, 172.
for uniaxal crystal, 180. Reflexion, total, 191. Revolving fluid, 10.
Solenoids, 112. Solid Harmonics, 1, 26. Spherical harmonics, 2, 3, 18, 23, -'4 34, 38.
applications of, 7. Spheroid, potential of homogeneous, 8.
heterogeneous 8. Stokes, 13, 91. Surface distribution, potential of, 7
50, 56, 71. Sylvester, 29.
Tesseral harmonics, 20.
Thomson, J. J., 124, 125.
Thomson and Dirichlet's theorem,
199. Total reflexion, 191.
Uniaxal crystals, 178.
Fresnel's ellipsoid for, 178. wave-surface for, 178.
reflexion and refraction for, Uniradial directions, 178.
Vector, potential, of magnetic induc- tion, 90, 92. for magnetic particle, 95.
Wave, 152.
-front, 153. -length, 152. -plane, 153. -surface, 157.
in crystalline medium, l'>7.
equation of, 160, 178.
singularities of, 187.
THE END.