The Mathematical Theory of Electricity and Magnetism (5th ed, 1927) — part 11 of 39

both sides. In the space left, after the interiors of such closed surfaces have been excluded, the functions u, v, w are continuous. We may accordingly apply Green's Theorem, and obtain

fdu . dv . dw\ 3cc

///'

• =- + ~- J dxdydz =-% ll(lu + mv + nw) dS

-2'jj(lu + mv + nw)dS (98),

where 2 denotes summation over the closed surfaces by which the original space was limited, and 2' denotes summation over the new closed surfaces which surround surfaces of discontinuity of u, v, w. Now corresponding to any element of area dS on a surface of dis- continuity there will be two elements of area of the enclosing surface. Let the direction-cosines of the two normals to dS be lx, mx, nx and l2, m2i n2, so that lx = — 12, mx = — m2, and nx = — ?v Let these direction-cosines be those of normals drawn from dS to the two sides of the surface, which we shall denote by 1 and 2, and let the values of u, v, w on the two sides of the surface of discontinuity at the element dS be ui> Vi> wi and u2, v2) w2. Then clearly the two elements of the enclosing surface, which fit against the element dS of the original surface of discontinuity, will contribute to

Fig. 54.

V I j(lu + mv + nw) dS

an amount

or

dS [(lxiix + m$x + nxwx) + (l2u2 + m2v2 + n2w2)] [li (ux - u2) + mx (vx -v^ + rh (wx — w2)) dS.

the form

Thus the whole value of 1' 1 1 (lu + mv + nw) dS may be expressed in form

2" 1 1 {h («i - u2) + m, (vx - v2) + nx (wx - w2)} dS,

where the integration is now over the actual surfaces of discontinuity. Thus Green's Theorem becomes

= — 2 I {lu + mv + nw) dS

• ^"\{k (Ui ~ «0 + ™i (vi ~ %) + *h (Pi ~ w«)] dS (99).

160 General Analytical Theorems [ch. vh

Special Form of Green's Theorem.

181. An  important  case  of  the  theorem  occurs  when  u,  v,  w  have  the 
     

special values

U = ^dx~> dy'

where <I> and M* are any functions of x, y and z. The value of (lu + mv + w) is now

dV d^ &P\

By

*dn->

where s- denotes differentiation along the normal, of which the direction-

on

cosines are I, m, n. We also have

^> + ^=ika+M$alu

dx dy dz dx\ dx) dy { dy

dz\ dz)

8<£ 9^ d® dV d<& d^ , fd2X¥ d^V dHr\ dx dx dy dy dz dz \ dx2 dy2 dz* )

Thus the theorem becomes

This theorem is true for all values of <P and 'SP, so that we may inter- change <£> and y, and the equation remains true. Subtracting the equation so obtained from equation (100), we get

[jT(<£V*¥ - ^V2$>) dxdydz = - 2 !!( <5> |? - ¥ ^) dS (101).

Applications of Green's Theorem.

182. In  equation  (101),  put  <£>  =  1  and  ^  =  7,  where  F  denotes  the 
     

electrostatic potential. We obtain

[fjv*Vdxdydz = -zff^dS (102).

181-183] Greerts Theorem 101

Let us divide the sum on the right into Ilt the integral over a single closed surface enclosing any number of conductors, and 72, the integrals over the surfaces of the conductors. Thus

'-//!>

where r- denotes differentiation along the normal drawn into the surface.

on

dV . Thus — -=— is equal to the component of intensity along this normal, and

therefore to — iV, where M is the component along the outward normal. Hence

I^-ffNdS.

dV At the surface of a conductor -^— = — 4nro-, so that

an

//'

J2= 47rS I IotZaS over conductors

= 4-7T x total charge on conductors. If there is any volume electrification, V2 V= — 4>irp, so that

1 1 ]V2Vdccdydz = — 4-7T 1 1 \pdxdydz,

and the integral on the right represents the total volume electrification. Thus equation (102) becomes

\NdS = 4nr x (total charge on conductors + total volume electrification),

so that the theorem reduces to Gauss' Theorem.

183. Next put O and "^ each equal to V. Then equation (100) becomes

Take the surfaces now to be the surfaces of conductors, and a sphere of

1 o "rr

radius r at infinity. At infinity V is of order - , so that -*- is of order

r on

1 oV

— , and hence V-~- , integrated over the sphere at infinity, vanishes (§ 178).

• (Jit

The equation becomes

• 4tt fjfpVdxdydz 4- jjJR'dxdydz - 4tt ffVadS = 0. j. 11

162 General Analytical Theorems [ch. vn

The first and last terms together give — 47r x SeT7", where e is any element of charge, either of volume-electrification or surface-electrification. Thus the whole equation becomes

\JLeV= IIItt- dxdydz,

shewing that the energy may be regarded as distributed through the space

7?2

outside the conductors, to the amount 5— per unit volume — the result

07T

already obtained in § 168.

184. In  Green's  Theorem,  take 
     

V ox 2/ = 3> if —

—(*£)■

Here K is ultimately to be taken to be the inductive capacity, which may vary discontinuously on crossing the boundary between two dielectrics. We accordingly suppose u, v, w to be discontinuous, and use Green's Theorem in the form given in § 180. We have then

{ox dx oy oy oz oz) u

Hi -*//»(«S+-5+-S)->-

-%SKK^+K^W)dS (io3)-

where r— , z— have the meanings assigned to them in $ 140.

If we put <E> = 1, ^ = V, in this equation, it reduces, as in § 130, to

f f r)V

\K^-dS = — 4>7r x total charge inside surface,

so that the result is that of the extension of Gauss' Theorem. Again, if we put <& = "^ = V, the equation becomes

KR*

dxdydz = ^XeV}

8tt and the result is that of § 1G9.

183-187] Uniqueness of Solution 163

Greens 'Reciprocation Theorem.

185. In equation (101), put <£= V, V=V', where V is the potential of one distribution of electricity, and V is that of a second and independent distribution. The equation becomes

fff(pV'-P'V)dxdydz + zJf(*V'-<T'V)dS = Q,

which is simply the theorem of § 102, namely

XeV' = Xe'V (104).

If we assign the same values to <f>, ^ in equation (103), we again obtain equation (104), which is now seen to be applicable when dielectrics are present.

Uniqueness of Solution.

186. We can use Green's Theorem to obtain analytical proofs of the theorems already given in § 99.

Theorem. If the value of the potential V is known at every point on a number of closed surfaces by which a space is bounded internally and externally, there is only one value for V at every point of this intervening space, which satisfies the condition that V2V either vanishes or has an assigned value, at every point of this space.

For, if possible, let V, V denote two values of the potential, both of which satisfy the requisite conditions. Then V — V=0 at every point of the surfaces, and V2(F' — V) = 0 at every point of the space. Putting <& and rF each equal to V — Fin equation (100), we obtain

and this integral, being a sum of squares, can only vanish through the vanishing of each term. We must therefore have

5<V-r>-4<7-F)-5<F'.-F)-0 (105),

or V — V equal to a constant. And since V — V vanishes at the surfaces, this constant must be zero, so that V=V everywhere, i.e. the two solutions V and V are identical : there is only one solution.

dV .

187. Theorem. Given the value of ^ at every point of a number of

closed surfaces, there is only one possible value for V (except for additive constants), at each point of the intervening space, subject to the condition that V2V = 0 throughout this space, or has an assigned value at each point.

11—2

164

General Analytical Theorems

[CH. VII

The proof is almost identical with that of the last theorem, the only difference being that at ever}' point of the surfaces we have

J;<r-F)-«.

instead of the former condition V — V = 0. We still have

xfj(V'-V)?-n(V'-V)dS = 0,

so that equation (105) is true, and the result follows as before, except that V and V may now differ by a constant.

188. Theorems  exactly  similar  to  these  last  two  theorems  are  easily 
     

seen to be true when the dielectric is different from air.

For, let V, V be two solutions, such that

at all points of the space, and at the surface either V — V = 0, or ) (V-V') = 0.

dn

By Green's Theorem

d(v-v')y fi{V-V)y \d(V-V)

dx

By

dz

dxdydz

=-///<r- F'> [l{Kl (y- r>l +£{* 4 (f- r>}

+ai*5<F-^]

dxdydz

dS

= 0 by hypothesis.

Equation (105) now follows as before, so that the result is proved.

Comparisons of different fields.

189. Theorem, i/ any number of surfaces are fixed in position, and a given charge is placed on each surface, then the energy is a minimum when the charges are placed so that every surface is an equipotential.

Let V be the actual potential at any point of the field, and V the potential when the electricity is arranged so that each surface is

187-190] Comparisons of different Fields 165

an equipotential. Calling the corresponding energies W and W, we have

If we put 3>= 7, ¥ = V- V, in equation (100), we find that the last integral becomes

47rJJ \dn dn or, since V is by hypothesis constant over each conductor,

and this vanishes since each total charge \ cr'dS is the same as the corre- sponding total charge 1 1 adS. Thus

,r-w'-s///K-«),+ ••■(***•

This integral is essentially positive, so that W is greater than W, which proves the theorem.

If any distribution is suddenly set free and allowed to flow so that the surface of each conductor becomes an equipotential, the loss of energy W — W is seen to be equal to the energy of a field of potential V — V at any point.

190. Theorem. The introduction of a new conductor lessens the energy of the field.

Let accented symbols refer to the field after a new conductor 8 has been introduced, insulated and uncharged. Then

W— W = — II j B?dxdydz through the field before S is introduced o HI R'2dxdydz through the field after S is introduced = q- II I B?dxdydz through the space ultimately occupied by #

• g- \{B;2-R'2) through the field after S is introduced.

166 General Analytical Theorems [ch. vii

The last integral and this, as in the last theorem, is equal to

kilj^-j^)'+--}d°:dyd°

. ♦£WZ(£-©« ,

where 2 denotes summation over all conductors, including S. This last sum of surface integrals vanishes, so that

W-W'= ~ Iff R2 dx dy dz through S

• -Q~llj(^ -5 — ) +•••}■ dxdydz through the field after

$ has been introduced.

Thus W— W is essentially positive, which proves the theorem.

On putting the new conductor to the earth, it follows from the preceding theorem that the energy is still further lessened.

191. Theorem. Any increase in the inductive capacity of the dielectric hetween conductors lessens the energy of the field.

Let the conductors of the field be supposed fixed in position and in- sulated, so that their total charge remains unaltered. Let the inductive capacity at any point change from K to K + 8K, and as a consequence let the potential change from V to V+SV, and the total energy of the field from W to W+BW.

If E1} E2,... denote the total charges of the conductors, V1} V2i... their potentials, and p the volume density ab any point,

W = %XEV+\ fffpV dxdydz,

so that, since the E's and p remain unaltered by changes in K, we have

hW=\ZEhV+%[\LhV dxdydz (106).

We also have so that

*--isi!«m+m^)hd^

190-192] Earnshavfs Theorem 167

By Green's Theorem, the last line

the summation of surface integrals being over the surfaces of all the conductors,

= jfjpBVdxdydz + 2 ffaSVdS

• ffjpBVdxdydz + XEBV

= 28 IF by equation (106). Thus equation (107) becomes

BW = -L fjJR'BKdxdydz - 28W,

so that BW = -~ IjJR2 BKdcc dy dz.

Thus BW is necessarily negative if 8K is positive, proving the theorem.

It is worth noticing that, on the molecular theory of dielectrics, the increase in the inductive capacity of the dielectric at any point will be most readily accomplished by introducing new molecules. If, as in Chap, v, these molecules are regarded as uncharged conductors, the theorem just proved becomes identical with that of § 190.

Earnshaw's Theorem.

192. Theorem. A charged body placed in an electric field of force cannot rest in stable equilibrium under the influence of the electric forces alone.

Let us suppose the charged body A to be in any position, in the field of force produced by other bodies B, B', First suppose all the elec- tricity on A, B, B', ... to be fixed in position on these conductors. Let V denote the potential, at any point of the field, of the electricity on B, B' , — Let x, y, z be the coordinates of any definite point in A, say its centre of gravity, and let x + a, y + b, z + c be the coordinates of any other point. The potential energy of any element of charge e at x + a, y + b, z + c is e V, where V is evaluated at x + a, y + b, z + c. Denoting e V by w, we clearly have

since V is a solution of Laplace's equation.

168 General Analytical Theorems [ch. vii

Let W be the total energy of the body A in the field of force from B, B', .... Then W=Xw, and therefore

d2w d2w d*w A

1 1 = 0

dx* ^ dy2 dz* '

i.e. the sum W = Sw satisfies Laplace's equation, because this equation is satisfied by the terms of the sum separately. It follows from this equation, as in § 52, that W cannot be a true maximum or a true minimum for any values of x, y, z. Thus, whatever the position of the body A, it will always be possible to find a displacement — i.e. a change in the values of x, y, z — for which W decreases. If, after this displacement, the electricity on the con- ductors A,B,B', ... is set free, so that each surface becomes an equipotential, it follows from § 189 that the energy of the field is still further lessened. Thus a displacement of the body A has been found which lessens the energy of the field, and therefore the body A cannot rest in stable equilibrium.

One physical application of Earnshaw's Theorem is of extreme importance. The theorem shews that an electron cannot rest in stable equilibrium under the forces of attraction and repulsion from other charges, so long as these forces are supposed to obey the law of the inverse square of the distance. Thus, if a molecule is to be regarded as a cluster of electrons and positive charges, as in § 151, then the law of force must be some- thing different from that of the inverse square.

There seems to be no difficulty about the supposition that at very small distances the law of force is different from the inverse square. On the contrary, there would be a very real difficulty in supposing that the law l/?-2 held down to zero values of r. For the force between two charges at zero distance would be infinite ; we should have charges of oppo- site sign continually rushing together and, when once together, no force would be adequate to separate them. Thus the universe would in time consist only of doublets, each consisting of permanently interlocked positive and negative charges. If the law 1/r2 held down to zero values of r, the distance apart of the charges would be zero, so that the strength of each doublet would be nil, and there would be no way of detecting its presence. Thus the matter in the universe would tend to shrink into nothing or to diminish indefinitely in size. The observed permanence of matter precludes any such hypothesis.

Earnshaw's Theorem accordingly limits us to two alternatives. Either the molecule does not consist of a cluster of electrons in relative rest, or else the law of the inverse square fails at molecular distances.

Eecent experimental investigations decide very definitely against the second alternative and in favour of the first. Recent experiments on the deflection of the positively charged a-particles by matter indicate that the law of the inverse square holds down to distances of the order of 10~u cms., a distance which is less than a thousandth part of the radius of the hydrogen atom, and a large mass of other evidence suggests, with a probability approximating to certainty, that the electrons in an atom or molecule must be in rapid orbital motion. Thus the problem of the structure of the molecule is removed from the province of Earnshaw's Theorem.

192, 193] Stresses in the Medium 169

Stresses in the Medium.

193. Let us take any surface S in the medium, enclosing any number of charges at points and on surfaces 8lt S2,

Let I, m, n be the direction-cosines of the normal at any point of Slt S2, ... or S, the normal being supposed drawn, as in Green's Theorem, into the space between the surfaces.

The total mechanical force acting on all the matter inside this surface is compounded of a force eR in the direction of the intensity acting on every point charge or element of volume-charge e, and a force 2ira2 or ^aR per unit area on each element of conducting surface. If X, Y, Z are the com- ponents parallel to the axes of the total mechanical force,

X = ZeX + % UaXdS

= fffpX dxdydz + 2 (Uo-XdS,

where the surface integral is taken over all conductors Si, S2, ... inside the surface S, and the volume integral throughout the space between S and these surfaces. Substituting for p and <r,

1 [f[fd*V d2V d2V\ dV , , ,

x= ^jj){w+w + w)tedxdydz

By Green's Theorem,

///

|I |? dxdydz = hfffl (^j dxdyd*

"

— isili^),«w-i//i^),«.

IHw IF ■""** = -JSSw h (^) dxdydt

dy2 dx y J J J dy dy

fly ""^ J J " " fa fiy

xjl^vwu-fU*™*.

Now

///^ h (S) dxdydz = /(/* I Q2 dxdydz

170 General Analytical Theorems

so that the last equation becomes

[CH. VII

— m

Mi.

s* dVdV dx dy ) dVdV

and there is a similar value for

d*vdv

dxdydz.

— m

dx dy

\dS dS,

dz* dx Substituting these values, equation (108) becomes

_iq*

lj/

Z

?ZY-(— Y-

8F9F aF3F) ox dy ox 02

dx J \dy J Since we have at every point of the surface of a conductor

d_V dV d_V

dx _ dy dz

I m n

it follows that the integral over each conductor vanishes, leaving only the integral with respect to dS, which gives

•(109),

X =

(lPxx + mPzy + nPxz)d$,

where

%, = ■£- (x*-Y*-z*)t

07T

1

If we write also

P = — TZ

the resultant force parallel to the axis of Y will be

Y - - jj(lPxy + mPyy + nPyz) dS,

and there is a similar value for Z. The action is therefore the same (cf. § 159) as if there was a system of stresses of components

P P P P P P

■Lxxi ■Lyyy *zz> Jyz> xzx> J-xy>

given by the above equations : i.e. these may be regarded as the stresses of the medium.

193, 194] Stresses in the Medium 171

194. It remains to investigate the couples on the system inside S. If L, M, N are the moments of the resultant couple about the axes of x, y, z, we have

L = Jffp (yZ-zY) dxdydz + \t tt* (yZ-zY) dS

1 [f[fdV , dV , d*V\ ( dV dV\ . _ _

Now SSSd^{yd^-zw)dxdydz

=-\\diL{yYz-zd^)dxdydz

^ffjdVf dV dV,„ ffjdVf dV dV\ 7e

-%)\lTx\y^-z^)dS-\lte\y^-2^)d*>

4>7rl]]\dx dx V dz Z -by) dy dy \V dz Z dy )

dV d ( dV dV], , 7

so that

L

dz dz V dz dy Jj -^t!!{ldx- + m^ + n^){^~Zdy-)dS

i fff7dv dV dV( dV dv\ ;o ,„„x

The first term in this expression

j_ r/rf tdv dv . dv dv bv&t

47T

[Hi (dv^X_ w&v_ dVd*v\

i]j\y [dee dxdz + dy dydz + dz dz2 )

U dxdydz

'dV d-V dVd2V dV d"V

J)x dxdy dy dy* dz dydz J)

=-L\{yd^-zd-w)dxdydz

= 1- 2 ff(ynR* - zmR*) dS + -L ff(ynIP - zm&) d$ (111).

The second term in expression (110) for L may, in virtue of the relations (109), be expressed in the form

• i- 2 IkynK - zmR2) dS,

which is exactly cancelled by the first term in expression (111).

172 General Analytical Theorems [ch. vii

We are accordingly left with

-if/H— >-«('S^-5F+-©('E-')}

= - \{y (lP*z + mPyz + nP„) - z {lPxy + mPyy + nPy2)} dS,

verifying that the couples are also accounted for by the supposed system of ether-stresses.

195. Thus the stresses in the ether are identical with those already found in Chapter vi, and these, as we have seen, may be supposed to

consist of a tension 3— per unit area across the lines of force, and a

07T

pressure ^— per unit area in directions perpendicular to the lines of force.

07T

Mechanical Forces on Dielectrics in the Field.

196. Let us begin by considering a field in which there are no surface charges, and no discontinuities in the structure of the dielectrics. We shall afterwards be able to treat surface-charges and discontinuities as limiting cases.

Let us suppose that the mechanical forces on material bodies are 3, H, Z per unit volume at any typical point x, y, z of this field.

Let us displace the material bodies in the field in such a way that the point x, y, z comes to the point x + Bx, y + By, z + 8z. The work done in the whole field will be

= -[j[(BSx + my+Z8z)dxdydz (112),

and this must shew itself in an equal increase in the electric energy. The electric energy W can be put in either of the forms

W = W1 = ±fffPVdxdydz,

When the displacement takes place, there will be a slight variation in the distribution of electricity and a slight alteration of the potential. There is also a slight change in the value of K at any point owing to the motion of the dielectrics in the field. Thus we can put

BW^SWi^iBW^ +(BW1)y,

BW = BW2 = (BWi)K+(BW2)v,

where (BW^p denotes the change produced in the function W[ by the varia-

194-196] Mechanical Forces on Dielectrics 173

tion of electrical density alone, (BW^y that produced by the variation of potential alone, and so on.

We have

(SWJr = k fffp 8 V dxdydz,

By Green's Theorem, the last expression transforms into

^-c///^e(is+4('al)+£('©}**

= I \p8V dxdydz, so that 2(Wx)v = {hW2)r.

We accordingly have

8W= 28W, -8W2 = 2 (8W,), - (BW2)K, the variation produced by alterations in V no longer appearing.

Now (8W,), = \ [[ftp Vdxdydz,

so that 8W= ft f{v8p-^8K\ dxdydz (113).

The change in p is due to two causes. In the first place, the electrifica- tion at'x, y, z was originally at x — 8x,y — 8y, z — 8z, so that 8p has as part of its value

-&«-!*-ifc <114>-

Again, the element of volume dxdydz becomes changed by displacement into an element

TX + dx ^ dx\ \dy + dy ^ dy\ \dz + dz ^ dz\ *

7 7 , /, d8x d8y d8z\ ,,,-x

«•<«• (1 + -5T + ^ + 17,) <115>-

so that, even if there were no motion of translation, an original charge pdxdydz would after displacement occupy the volume given by expression (115), and this would give an increase in p of amount

-'(£+?+© <116>-

174 General Analytical Theorems [ch. vii

Combining the two parts of Bp given by expressions (114) and (115), we find

.~g<P0 + 4G>*> + !((.8O}.

The change in K is also due to two causes. In the first place the point which in the displaced position is at x, y, z was originally at x — Bxty — By, z—Bz. Hence as part of the value in BK we have

dK , dK . dK fi

Also, with the displacement, the density of the medium is changed, so that its molecular structure is changed, and there is a corresponding change in K. If we denote the density of the medium by t, and the increase in t produced by the displacement by Bt, the increase in K due to this cause will be

OT

and we know, as in equation (116), that

'dSx dBy dBz\ dx dy dz J '

~. UViV VOL OOZ\

or = - r (^—

We now have, as the total value of BK,

SEr dK & dK . dK «

dx dy J dz

dK fdBx dBy d_Bz\ dr V dx dy dz J '

and hence, on substituting in equation (113) for Bp and BK,

W — jJf V^P^ + d^ + dJ^^dxdydz [[[ R (dK B dK* dK e \ , , .

+jjl 8^{-dx-8x+dy-By+-dz- **) dxdydz

[f[R* dK fdBx dBy dBz, . ,

• j]j^T^{-dx- + -dJ + Wjdxdyd2-

Integrating by parts, this becomes

sw= \^p*x+\y:phy+%phz)dxdydz

196-198]

Stresses in Dielectrics

175

or, rearranging the terms,

BW =

dV B? fdK

\dx)

" dx 8-7T V dx J dx \8-rr ' dr J

d(^r

dK\

Sx +

J [(I

Comparing with expression (112), we obtain

dVB^dK d_(R

8y +

i-1 P

dK

T

dx 8ir dx ' dx \8tt ' dr etc., giving the body forces acting on the matter of the dielectric.

Bzvdxdydz.

.(117),

197. This  may  be  written  in  the  form 
     

R^d_K d_(R?_ dK\ 8tt dx dx V87T dr )

Thus in addition to the force of components (pX, pY, pZ) acting on the charges of the dielectric, there is an additional force of components

_R?_dK R^dK _&d_K

87r dx ' 8rr dy ' 8ir dz

arising from variations in K, and also a force of components

dx \8tt T dr )' dy V8tt T dr ) ' dz W T dr ) '

which occurs when either the intensity of the field or the structure of the dielectric varies from point to point.

Stresses in Dielectric Media.

198. Replacing  p  by  its  value,  as  given  by  Laplace's  equation,  we  obtain 
     

equation (117) in the form

•   1 
  

Kk^+IU^+Kk^)

2^

8tt [ dx \dx\ dxj ' dy\ dy J ' dz

j

d_K

dx

dihm+m

dz)

dx \ dr

1_

8tt

d_

dx

K

m^hm

dV d

' dV\ Kd_ (dVV dx dx\ dx J dx\dx)

K- (—

dx \dy

dx dy \ dy ndV d (TsdV\ Tjr d (dV*

• 2dx-dz{Kte) + KTx[-dF)

dx \ dr )\

176 General Analytical Theorems [ch. vii

8tt \

dx

«€J<h?r

If we put

<-xx

K f/3F* /dV' idV'\ R< dK 1R.

p--efa3y'et0 (119)'

dPcx . dixy . dPx2

this becomes B=W + ~dJ + dz '

Let us suppose that a medium is subjected to a system of internal

stresses Pxx, Pxy, etc.; and let it be found that a system of body forces

of components B', H', Z' is just sufficient to keep the medium at rest

when under the action of these stresses. Then from equation (79) we

must have

dPxx , dPxy . dPXi

g/ = _ l"_£Z+^B +

dx dy dz

Thus if Pxx, Pxy> etc. have the values given by equations (118) and (119),

we have

H' = -B, etc.

This shews that the mechanical force H, H, Z reversed would just be in equilibrium with the system of stresses Pxx, Pxy, etc. given by equations (118) and (119). In other words, the mechanical forces which have been found to act on a dielectric can exactly be accounted for by a system of stresses in the medium, these stresses being given by equations (118) and (119).

199. The system of stresses given by equations (118) and (119) can be regarded as the superposition of two systems :

I. A system in which

II. A system in which

'" " " Sir dr'

tXy = *yz = *ZX = U.

198-200] Stresses in Dielectric Media 177

The first system is exactly K times the system which has been found to occur in free ether, while the second system represents a hydrostatic pressure of amount)

& dK

8tt T dr '

(In general ■=- will be positive, so that this pressure will be negative, and must be interpreted as a tension.)

Hence, as in § 165, the system of stresses may be supposed to consist of:

ten2

(i) a tension -5 — per unit area in the direction of the lines of force ;

07T

(ii) a pressure -~ — per unit area perpendicular to the lines of force ; (iii) a hydrostatic pressure of amount — 5— t ^r— in all directions.

07T OT

The system of stresses we have obtained was first given by Helmholtz. The system

differs from that given by Maxwell by including the pressure - — r -=- . The neglect of

this pressure by Maxwell, and by other writers who have followed him, does not appear to be defensible. Helmholtz has shewn that still further terms are required if the dielectric is such that the value of K changes when the medium is subjected to distortion without change of volume.

200. This system of stresses has not been proved to be the only system of stresses by which the mechanical forces can be replaced, and, as we have seen, it is not certain that the mechanical forces must be regarded as arising from a system of stresses at all, rather than from action at a distance.

It may be noticed, however, that whether or not these stresses actually exist, the resultant force on any piece of dielectric must be exactly the same as it would be if the stresses actually existed. For the resultant force on any piece of dielectric has a component X parallel to the axis of x, given by

X = I lEidxdydz

= - IklPxx + mPxy + nPx2) dS

by Green's Theorem, and this shews that the actual force is identical with what it would be if these stresses existed (cf. § 193).

J. 12

178 General Analytical Theorems [ch. vii

Force on a charged conductor.

201. The mechanical force on the surface of a charged conductor immersed in a dielectric can be obtained at once by regarding it as produced by the stresses in the ether. There will be no stresses in the interior of the conductor, so that the force on its surface may be regarded as due to the tensions of the tubes of force in the dielectric. The tension is accordingly of amount

KB? E2 dK 87T 8-7T dr

per unit area, an expression which can be written in the simpler form

R2 d IV \

Force at boundary of a dielectric.

202. Let us consider the equilibrium of a dielectric at a surface of discontinuity, at which the lines of force undergo refraction on passing from one medium of inductive capacity Kl to a second of inductive capacity K2.

Let axes be taken so that the boundary is the plane of xy, while the lines of force at the point under consideration lie in the plane of xz. Let the components of intensity in the first medium be (Xx, 0, Zj), while the corresponding quantities in the second medium

are (X2, 0, Z2). The boundary conditions ob-

tained in § 137 require that

Xx = X2, K\Z^ = K2Z2 — 4nrh, where h is the normal component of polarisation.

In view of a later physical interpretation of the forces, it will be convenient to regard these forces as divided up into the two systems mentioned in § 199, and to consider the contributions from these systems separately.

As regards the contribution from the first system, the force per unit area acting on the dielectric from the first medium has components

*«. o, gw-zA

while that from the second medium has components

K K

4J X2Z2, 0, -J (Zi - Xi).

201, 202] Stresses in Dielectric Media 179

Since KXXXZX = K2X2Z2, it follows that the resultant force on the boundary is parallel to Oz — i.e. is normal to the surface. Its amount, measured as a tension dragging the surface in the direction from medium 1 to medium 2

which after simplification can be shewn to be equal to

X? 2irh*

X? 2-7r/i2.„ _.

This is always positive if KY > K2. Thus this force invariably tends to drag the surface from the medium in which K is greater, to that in which K is less — i.e. to increase the region in which K is large at the expense of the region in which K is small. This normal force is exactly similar to the normal force on the surface of a conductor, which tends to increase the volume of the region enclosed by the conducting surface.

On Maxwell's Theory, the forces which have now been considered are the only ones in existence, so that according to this theory the total mechanical force is that just found, and the boundary forces ought always to tend to increase the region in which K is large. This theory, as we have said, is incomplete, so that it is not surprising that the result just stated is not confirmed by experiment.

We now proceed to consider the action of the second system of forces — the system of negative hydrostatic pressures. There are pressures per unit area of amounts

Rl dK, R? dK%

87r 1 3tx ' 8ir 2 3t2

acting respectively on the two sides of the boundary. There is accordingly a resultant tension of amount

1 / dK, dK2

per unit area, tending to drag the boundary surface from region 1 to region 2. Thus the total tension per unit area, dragging the surface into region 1, is

fe+^J^'-^-sP^-^id (120)-

In § 139, in considering a parallel plate condenser with a movable dielectric slab, we discovered the existence of a mechanical force tending to drag the dielectric in between the plates. This force is identical with the mechanical force just discussed. But we have now arrived at a mechanical interpretation of this force, for we can regard the pull on the dielectric as the resultant of the pulls of the tubes of force at the different parts of the surface of the dielectric.

12—2

180 General Analytical Theorems [ch. vii

Let us attempt to assign physical interpretations to the terms of ex- pression (120) by considering their significance in this particular instance. Consider first a region in the condenser so far removed from the edges of the condenser and of the slab of dielectric, that the field may be treated

4<7r/t as absolutely uniform (cf. fig. 44, p. 124). We put K2=l, X1 — 0, Ry = -^r-

in expression (120) and obtain

2rf(¥-£i) <121>

as the force per unit area on either face of the dielectric, acting normally outwards.

The forces will of course act in such a direction that they tend to decrease the electrostatic energy of the field. Now this energy is made up

of contributions 27rA2 per unit volume from air, and ^=- per unit volume

from the dielectric. From the conditions of the problem h must remain unaltered. Thus the total energy can be decreased in either of two ways — by increasing the volume occupied by dielectric and decreasing that occupied by air, or by increasing the value of K in the dielectric. There will therefore be a tendency for the boundary of the dielectric to move in such a direction as to increase the volume occupied by dielectric, and also a tendency for this boundary to move so that K will be increased by the consequent change of density. These two tendencies are represented by the two terms of expression (121).

If — is negative, an expansion of the dielectric will both increase the

OT

volume occupied by the dielectric, and will also increase the value of K inside the dielectric. In this case, then, both tendencies act towards an expansion of the dielectric, and we accordingly find that both terms in expression (121) are positive.

dK

If -r— is positive, the tendency to expansion, represented by the first