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A flat conducting sheet A is suspended by an insulating thread at distance s from both inner surfaces of a bent conducting sheet B. The entire gap between the bent sheet B is filled with a dielectric of constant K. Sheets A and B are oppositely charged and maintained at a constant potential difference delta_V. This produces a force F pulling A inward. If the magnitude of work done by an external agent to slowly increase the inserted depth by delta_y equals P * (delta_V)² * b * K * epsilon0 * delta_y / s, where b is the width of the sheets, find P.

1. 1/2
2. 1
3. 2
4. 4

Correct answer: 1/2

Solution

At constant voltage, the energy stored in the capacitor changes as the overlap increases. Work by external agent + work by battery = change in stored energy. Work by battery = (delta_V) * delta_Q = (delta_V)² * delta_C. Change in energy = (1/2)*(delta_V)² * delta_C. So work by external agent = delta_U - W_battery = (1/2)(delta_V)² * delta_C - (delta_V)² * delta_C = -(1/2)(delta_V)² * delta_C. The magnitude = (1/2)(delta_V)² * (K * epsilon0 * b / s) * delta_y. Comparing with P*(delta_V)² * b * K * epsilon0 * delta_y / s gives P = 1/2.

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