A parallel plate air capacitor is connected to a battery. The plates are pulled apart at uniform speed v . If x is the separation between the plates at any instant, then the time rate of change of electrostatic energy of the capacitor is proportional to x^ , where is _____.

Step-by-step solution

The electrostatic energy of a parallel plate capacitor is given by U = 1 2 C V^2 . For a parallel plate capacitor, the capacitance is C = ₀ A x , where x is the separation between the plates. Since the capacitor remains connected to the battery, the potential difference V across the plates is constant. Substituting C into the energy equation gives: U = ₀ A V^2 2x Differentiating U with respect to time t to find the time rate of change of electrostatic energy: dU dt = d dt ( ₀ A V^2 2x ) = - ₀ A V^2 2x^2 dx dt Given