An electrical circuit consists of a resistance R, inductance L and capacitance C is series. If a charge is put on the capacitor at some instant, determine the condition that VC, the voltage across the capacitor, is subsequently oscillatory. Assuming that this condition is satisfied, derived an expression for the time T for the amplitude of VC to drop by a factor of e. An external voltage source of variable frequency is introduced into the circuit in series with the other components. Show that, for small R, the width of the resonance (defined as the angular frequency range for which the amplitude of VC is greater than 1/√2 of its resonance value) is approximately 2/T