A capacitor having capacitance C is initially provided with a charge Q . At t = 0 , it is linked to an ideal battery of emf E via a resistance R . Determine the charge on the capacitor at time t .

Step-by-step solution

Applying Kirchhoff's loop rule for the circuit at time t : E - iR - q C = 0 Substituting i = dq dt : R dq dt = E - q C = C E - q C Rearranging and integrating with the initial condition q(0) = Q : _ Q ^ q dq C E - q = ₀^ t dt RC [- (C E - q)]_ Q ^ q = t RC ( C E - q C E - Q ) = - t RC C E - q = (C E - Q) e^ - t RC q = C E (1 - e^ - t RC ) + Q e^ - t RC