Question

A parallel plate capacitor is charged by an ac source. Show that the sum of conduction current \( I_c \) and the displacement current \( I_d \) has the same value at all points of the circuit.

Hint:

The displacement current in a capacitor is analogous to the conduction current, and both are equal in magnitude in an ac circuit.

Answer 1

In a parallel plate capacitor, the conduction current \( I_c \) is the current flowing through the plates, and the displacement current \( I_d \) is the current that is related to the changing electric field between the plates. The total current is the sum of both: \[ I = I_c + I_d \] For a capacitor, the conduction current \( I_c \) is given by: \[ I_c = \frac{Q}{t} \] where \( Q \) is the charge on the capacitor. The displacement current is related to the rate of change of the electric field between the plates: \[ I_d = \epsilon_0 A \frac{dE}{dt} \] where \( A \) is the area of the plates and \( E \) is the electric field between the plates. Since the displacement current is equivalent to the conduction current in terms of charge flow, we have: \[ I_c = I_d \] Thus, the sum of the conduction and displacement currents is the same at all points in the circuit.