GREEN’S THEOREM

In the notation of the differential calculus this result is expressed by the equation dQe = P, (4) dE in which it is to be remembered that all the charges but one are maintained constant. 34.] Returning to equation (2), we have already shewn that Q = 12 ∑(EP ) and Q′ = 12 ∑(E ′ P ′ ); (5) we may therefore write equation (2) 1 (E ′ P ′ ) = 12 ∑(EP ) + 12 ∑(E ′ P ′ − EP + E ′ P − EP ′ ). 2∑ (6)

Cutting out from the equation the terms which destroy each other, we ob- tain ′ ′ (7) ∑(EP ) = ∑(E P ), or in words,

Theorem 3

35.] In the first place we may write, as before, 1 {(E ′ − E)(P ′ + P )} = 12 ∑(E ′ P ′ − EP + E ′ P − EP ′ ); 2∑ (8) adding and subtracting the equal quantities of equation (7), 0 = ∑(EP ′ − E ′ P ), (9)RECIPROCITY OF POTENTIALS. 33 and the right-hand side becomes 1 (E ′ P ′ − EP − E ′ P + EP ′ ), 2∑ (10) or 1 {(E ′ − E)(P ′ + P )} = Q′ − Q = 12 ∑{(E ′ + E)(P ′ − P )}, 2∑ (11) or in words,

Theorem 4

36.] If all the conductors but one are maintained at constant potentials (which may be done by connecting them with voltaic batteries of constant electromotive force), equation (11) is reduced to

or Q′ − Q = 12 (E ′ + E)(P ′ − P ),(12) Q′ − Q 1 ′ = 2 (E + E). P′ − P(13)

If the increment of the potential is taken successively smaller and smaller, till it ultimately vanishes, E ′ becomes at last equal to E and the equation may be interpreted thus:—

The rate of increase of the electrical energy due to the increase of potential of one of the conductors at a rate unity is numerically equal to the charge of that conductor. In the notation of the differential calculus this result is expressed by the equation dQp = E, (14) d

PRECIPROCITY OF POTENTIALS. 34 in which it is to be remembered that all the potentials but one are maintained constant. 37.] We have next to point out some of the results which may be deduced from Theorem III. If any conductor, as At , is insulated and without charge both in the initial and the final state, then Et = 0 and Et ′ = 0, and therefore Et Pt ′ = 0 and Et ′ Pt = 0, (15) so that the terms depending on At disappear from both members of equation (7). Again, if another conductor, say Au , be connected with the earth both in the initial and in the final state, Pu = 0 and Pu ′ = 0, so that Eu Pu ′ = 0 and Eu ′ Pu = 0; so that, in this case also, the terms depending on Au disappear from both sides of equation (7). If, therefore, all the conductors with the exception of two, say Ar and As , are either insulated and without charge, or else connected with the earth, equation (7) is reduced to the form Er Pr ′ + Es Ps ′ = Er ′ Pr + Es ′ Ps . (16) Let us first suppose that in the initial state all the conductors except Ar are without charge, and that in the final state all the conductors except As are without charge. The equation then becomes Er P r ′ = E s ′ P s , or [If, therefore, or in words, Ps P′ = r′, Er Es Er = E s ′ , Ps = Pr ′ ], (17)RECIPROCITY OF POTENTIALS AND CHARGES. 35 Theorem V.

In a system of fixed insulated conductors, the potential (Ps ) produced in As by a charge E communicated to Ar is equal to the potential (Pr ′ ) produced in Ar by an equal charge E communicated to As . This is the first instance we have met with of the reciprocal relation of two bodies. There are many such reciprocal relations. They occur in every branch of science, and they often enable us to deduce the solution of new electrical problems from those of simpler problems with which we are already familiar. Thus, if we know the potential which an electrified sphere produces at a point in its neighbourhood, we can deduce the effect which a small electrified body, placed at that point, would have in raising the potential of the sphere. 38.] Let us next suppose that the original potential of As is Ps and that all the other conductors are kept at potential zero by being connected with the walls of the room, and let the final potential of Ar be Pr ′ , that of all the others being zero, then in equation (7) all the terms involving zero potentials will vanish, and we shall have in this case also Er P r ′ = E s ′ P s . If, therefore, Pr ′ = Ps , Er = Es ′ , (18) (19) or in words, Theorem VI.

In a system of fixed conductors, connected, all but one, with the walls of the room, the charge (Er ) induced on Ar when As is raised to the potential Ps is equal to the charge (Es ′ ) induced on As when Ar is raised to an equal potential (Pr ′ ). 39.] As a third case, let us suppose all the conductors insulated and without charge, and that a charge is communicated to Ar which raises its potential to Pr and that of As to Ps . Next, let As be connected with the earth, and let a charge Er ′ on Ar induce the charge Es ′ on As .GREEN’S THEOREM ON POTENTIALS AND CHARGES. 36 In equation (16) we have Er = 0 and Ps ′ = 0, so that the left-hand member vanishes and the equation becomes 0 = Er ′ Pr + Es ′ Ps , or (20) Ps E′ = − r′ . Pr Es Hence, if Ps = nPr , Er ′ = −nEs ′ , (21) or in words,