Theorem 7

If in a system of fixed conductors insulated and originally without charge a charge be communicated to Ar which raises its potential to Pr , unity, and that of As to n, then if in the same system of conductors a charge unity be communicated to As and Ar be connected with the earth the charge induced on Ar will be −n.

If, instead of supposing the other conductors At &c. to be all insulated and without charge, we supposed some or all of them to be connected with the earth, the theorem would still be true, only the value of n would be different according to the arrangement we adopted.

This is one of Green’s theorems. As an example of its application, let us suppose that we have ascertained the distribution of electric charge induced on the various parts of the surface of a conductor by a small electrified body in a given position with unit charge. Then by means of this theorem we can solve the following problem:—The potential at every point of a surface coinciding in position with that of the conductor being given, determine the potential at a point corresponding to the position of the small electrified body. Hence, if the potential is known at all points of any closed surface, it may be determined for any point within that surface if there be no electrified body within it, and for any point outside if there be no electrified body outside.

GREEN’S THEOREM ON POTENTIALS AND CHARGES.

Mechanical work done by the electric forces during the displacement of a system of insulated electrified conductors. 40.] Let A1 , A2 &c. be the conductors, E1 , E2 &c. their charges, which, as the conductors are insulated, remain constant. Let P1 , P2 &c. be their potentials before and P1 ′ , P2 ′ &c. their potentials after the displacement. The electrical energy of the system before the displacement is Q = 12 ∑(EP ). (22)

During the displacement the electric forces which act in the same direction as the displacement perform an amount of work equal to W, and the energy remaining in the system is Q′ = 12 ∑(EP ′ ). (23) The original energy, Q, is thus transformed into the work W and the final energy Q′ , so that the equation of energy is Q = W + Q′ ,(24) W = 12 ∑[E(P − P ′ )].(25) or This expression gives the work done during any displacement, small or large, of an insulated system. To find the force, we must make the displacement so small that the configuration of the system is not sensibly altered thereby. The ultimate value of the quotient found by dividing the work by the displacement is the value of the force resolved in the direction of the displacement.

Mechanical work done by the electric force during the displacement of a system of conductors each of which is kept at a constant potential. 41.] Let us begin by supposing each conductor of the system insulated, and that a small displacement is given to the system, by which the potentialMECHANICAL WORK DURING DISPLACEMENT. 38 is changed from P to P1 . The work done during this displacement is, as we have shewn, W = 12 ∑[E(P − P1 )]. (26) Next, let the conductors remain fixed while the charges of the conductors are altered from E to E1 , so as to bring back the value of the potential from P1 to P. Then we know by equation (7) that ∑(EP − E1 P1 ) = 0. (27) Hence, substituting for ∑(EP ) in (26), W = 12 ∑[(E1 − E)P1 ]. (28) Performing these two operations alternately for any number of times, and distinguishing each pair of operations by a suffix, we find the whole work W = W1 + W2 + &c. = 12 ∑[(E1 − E)P1 ] + 12 ∑[(E2 − E1 )P2 ] + &c. (29) (30)

By making each of the partial displacements small enough, the values of P1 , P2 &c. may be made to approach without limit to P, the constant value of the potential, and the expression becomes W = 12 ∑[(E1 −E)P ]+ 12 ∑[(E2 −E1 )P ]+&c.+ 12 ∑[(E ′ −En−1 )P ], (31) where E ′ is the value of E after the last operation. The final result is therefore W = 12 ∑[(E ′ − E)P ], (32) which is an expression giving the work done during a displacement of any magnitude of a system of conductors, the potential of each of which is main- tained constant during the displacement. We may write this result W = 12 ∑(E ′ P ) − 12 ∑(EP ), (33) 39 or W = Q′ − Q; (34) or the work done by the electric forces is equal to the increase of the elec- tric energy of the system during the displacement when the potential of each conductor is maintained constant. When the charge of each conductor was maintained constant, the work done was equal to the decrease of the energy of the system.

MECHANICAL WORK DURING DISPLACEMENT.

Hence, when the potential of each conductor is maintained constant during a displacement in which a quantity of work, W, is done, the voltaic batteries which are employed to keep the potentials constant must do an amount of work equal to 2W. Of this energy supplied to the system, half is spent in increasing the energy of the system, and the other half appears as mechanical work.