A parallel-plate air capacitor remains connected to an ideal battery while the plates are pulled apart at a uniform speed. If x is the instantaneous plate separation and the magnitude of the time rate of change of electrostatic energy is proportional to xᵅ, what is α?
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Question

A parallel-plate air capacitor remains connected to an ideal battery while the plates are pulled apart at a uniform speed. If x is the instantaneous plate separation and the magnitude of the time rate of change of electrostatic energy is proportional to xᵅ, what is α?

Options

G1 Physics · Accepted Answer

Because the capacitor remains connected to a battery, its voltage V is constant. The capacitance is C = ε₀A ÷ x. Therefore the stored energy is U = (1 ÷ 2)CV² = ε₀AV² ÷ (2x). Differentiate with respect to time: |dU ÷ dt| = (ε₀AV² ÷ 2)(1 ÷ x²)|dx ÷ dt|. The plates are pulled apart at uniform speed, so |dx ÷ dt| is constant. Hence, |dU ÷ dt| ∝ x⁻². Therefore, α = -2. Correct option: A.