30 ELECTRICITY [ELECTEOSTATICAL THEORY.
- <?> + 8Q =
_dQ (30), (81). Referring to the second of the expressions in (27), we see that this may be written
- - - i 2" S E,!,-^ 2
From this it is evident that in similarly electrified states of the same system the force tending to produce a given displacement varies as the square of the electrification. It is important to remark that in the present case the sys tem tends to move so that its potential energy is decreased. Secondly, let us suppose that the potentials of the dif ferent conductors are kept constant during any displace ment, energy being supplied from without. We shall suppose the change made in two steps. First, we shall suppose the given displacement to take place while the charges remain constant. On this supposition the force exerted will,. to the first order of small quantities, be the same as that exerted when we suppose the potential not to vary ; hence Next, supply energy from without so that the potentials become again VD V" 2 ,&c., . . . and the charges Ej + 5Ej , E 2 + SE 2 , &c. The final result will be the same, to first order of small quantities, as if the two changes had been made simultaneously. Now, applying the theorem of mutual potential energy to the two states of our system, J V we have hence and EUEj + JEj V 2(E 1 2ESV = -2VSE therefore * - 42E j- - 4 By (27) this may be written
- = i 2 2 V.V.
r-l ^1 The energy supplied from without is j-=~(V const.) dq r , (32); . (33). = !25EV-2E5V= - 2E5 V = 2*8<J> = 25Q, by (32). In other words, when the potentials of a system are kept constant by supply of energy from without, the system tends to move so as to increase the potential energy of electrical separation, and the amount of energy supplied from without is double this increase. If we suspend side by side two balls, each connected with the positive pole of a battery, the other pole of which is connected with the ground, the balls will tend to separate, and in separating they will gain with reference to gravity a certain amount SQ of potential energy ; the charges on the balls will also in crease to an extent representing an increase of electrical potential energy SQ, and the batteries will be drawn upon for an amount of 28Q.
- ases The problem of electrical equilibrium has been com-
rhere pletely solved in very few cases. We proceed to give a iroblem gh ort s k e tch of what has been done in this way, which may olved 0n iudicate to the reader what is known on this head, illips id ^ e can Deduce the distribution and potential in the case of an ellipsoid from known propositions about the attrac tions of ellipsoidal shells of gravitating matter. Consider an ellipsoidal shell, the axes of whose bounding sur faces are (a, b, c) (a + da, b + db, c + dc), where f = ^ = = u. The a c potential of such a shell at any internal point is constant, and the equipotential surfaces for external space are ellipsoids coufocal with (a, b, c). (See Thomson and Tait, 519 sgq.) Hence if we dis tribute electricity on an ellipsoid (a, b, c) such that its density at every point is proportional to the thickness of the shell formed by the similar ellipsoids (a,b,c) (a + dn,b + db,c + dc), the distribution will be in equilibrium. Thus if <r = A0p, where is the thickness at any point and p the volume density of the shell ; then the quantity of electricity on any element dS> is A times the mass of the correspond ing element of the shell; and if Q be the whole quantity of elec tricity on the ellipsoid, Q = A times the whole mass of the shell. The mass of the shell is %Trpd(abc) = iir/j.abcp, therefore Q = A Also 6 = pp where p is the perpendicular from the centre of the ellipsoid on the tangent plane. Whence we get 4-irabc (34); that is, the density at any point varies directly as the distance of the tangent plane at that point from the centre. Returning again to our ellipsoidal shell, we know that the result ant force at any external point P due to this shell is to that due to a " confocal shell 1 passing through the point in the ratio of the masses. Let the volume density in the two be p, and let the perpendicular on the tangent plane at P to the coufocal (Va 2 + A, VPTX, V C TA.) through P be . Then the thickness of the shell at P is /io>, and the force at P due to the shell 4irp/j.<a. Hence the force due to the original shell is dV . abc ^. =s 47TOUCi) "i~ "" ~^~ --.-.-- .---- v T :n f fl) dv V( a s + A)(& 2 + A)(c 2 + A) dv being an element of the normal at P. Now if x,y,z be the co ordinates of P, we have, by differentiation of 2xdx %ydy 2 i x "" ;,i! i i " , ^K Suppose we take dx, dy, dz in the direction of the normal, then dx = di/ , &c., and the last equation reduces to d = 2u)C?/ . Hence from (a) we get -dV--f- Integrating this from A to oo , and remembering that the potential vanishes at an infinite distance, we get We pass from this to the electrical case by putting for which is the mass of the shell, Q, which represents the quantity of electricity on the ellipsoid. "We thus get V " f V vV (35) which gives the potential due to a charge Q on an isolated ellipsoid abc at any point on the confocal ( Ja 2 + A, *Jb- + A, v/c 2 + A) . It is obvious that, of the three confocals at P, that is meant which belongs to the same family as (a, b, c), e.g., if (a,b,c) be an ellipsoid, as opposed to a hyperboloid of one or two sheets, then ( Ja~ + A > >Jb 2 + A , J c 2 -.{ A) must be an ellipsoid. If we put A = 0, we get the value of the potential V at the surface. Now is what we have defined above as the capacity of the ellipsoid; we get therefore in the recipro cal of the integral 1 /- d 2/0 VP (36), an expression for the capacity of an isolated ellipsoid. In the particular case of an ellipsoid of revolution, the Plane- above integral, which is in general an elliptic integral, tl ^7 can be found in finite terms. In the case of a planetary clll l )SOld ellipsoid, a = b>c ; and we find for the capacity (37), where e is the least angle whose tangent is , a 8 . If we make c = 0, then e = ; and the planetary ellipsoid C irculai reduces to a circular disc, the capacity for which is there- d isc - 2ci 1 fore - , that is, -- that of a sphere of the same radius n- 1-571 1 This demonstration was suggested by that given by Thomson
(Reprint of Papers, p. 10) to establish a slightly different formula.