At t = 0 , a parallel-plate capacitor with plate-area A and plate separation d is connected to a battery having emf E and internal resistance R . The displacement resistance of the region between the plates is defined as R_d = V/i_d , where V represents the potential difference across the plates and i_d is the displacement current. Which of the following expressions represents the variation of R_d with time t (where

Step-by-step solution

The charge on the capacitor at any time t during charging is given by q = C E (1 - e^ -t/ ) . The potential difference across the plates of the capacitor is V = q C = E (1 - e^ -t/ ) . The displacement current i_d between the plates is equal to the conduction current i in the circuit. i_d = i = dq dt = E R e^ -t/ The displacement resistance R_d is defined as R_d = V i_d . Substituting the values of V and i_d : R_d = E (1 - e^ -t/ ) E R e^ -t/ R_d = R ( 1 - e^ -t/ e^ -t/ ) R_d = R(e^ t/ - 1)