[UPSC CSE 2011] Electrostatics and Magnetostatics (Q153, 20M)

An electrical circuit consists of a resistance R, inductance L and capacitance C in series. If a charge is put on the capacitor at some instant, determine the condition that V_C, the voltage across the capacitor, is subsequently oscillatory. Derive an expression for the quality factor Q of the circuit by considering the decay of the oscillation, using the result that the amplitude falls by a factor of e in \left(\frac{Q}{\pi}\right) period.