Answer to the Question 11/97
The question was:A variable capacitor is connected to two terminals of a battery of electromotive force E. The capacitor initially has a capacitance C(0) and charge q(0). The capacitance is caused to change with time so that current I is constant. Calculate the power supplied by the battery, and compare it with the time rate-of-change of the energy of the capacitor. Account for any difference.
(11/97) This problem has been solved correctly by Oded Farago (e-mail farago@orion.tau.ac.il) from Tel Aviv University and by Tom Snyder (e-mail tsnyder@mail.fgi.net) from Lincoln Land Community College, Springfield, Illinois. This problem was published in the "Graduate Problems in Physics" by J.A.Cronin, D.F. Greenberg and V.L. Telegdi, where a short solution can be found.
Here is (a slightly editted version of) the solution submitted by Snyder:
The battery supplies the constant power P1=EI. The electrostatic energy of the capacitor, U=qE/2, is changing at a rate
dU/dt = (E/2)(dq/dt)=EI/2.
Thus the battery is doing twice as much work as being stored in the capacitor. The difference is a work done by the capacitor on external agent that is causing the capacitance to change. To find an expression for the rate at which the agent delivers energy to the capacitor we imagine the situation in which the capacitor has been given a charge Qo and then isolated from the source. Thus Q will remain fixed at Qo. Now suppose the external agent varies the capacitance. Writing the potential energy U of the capacitor as Qo2/(2C) we find
dU/dt = - (Qo/C)2(C'/2) = - (C'/2)V2
where V is the instantaneous value of the potential difference between the plates and C' is the rate of change of capacitance. Since the capacitor is isolated from the source of emf, dU/dt here must represent the rate at which energy is supplied to the capacitor by the external agent alone. Denoting this rate by P2, we have P2 = -C'V2/2. Although derived for the case where the capacitor is isolated, this expression for P2 must be true in general since it involves only the instantaneous value of V and the instantaneous rate at which the agent is varying the capacitance, C'. So we may apply this expression for P2 to the original situation. In this case V = E, a constant. Then
P2 = -C'E2/2 = -d(CE)/dt * E/2 = -dQ/dt * E/2 = -IE/2
The total rate at which energy is delivered to the capacitor is then P1 + P2 = IE - IE/2 = IE/2 which is just the rate at which energy is being stored in the capacitor, dU/dt.
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