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:

In a capacitor, the displacement current compensates for the conduction current inside the plates. This ensures that the total current remains continuous in the circuit.

Answer 1

Sum of Conduction and Displacement Current in a Parallel Plate Capacitor

When an AC source charges a parallel plate capacitor, both conduction current (\( I_c \)) and displacement current (\( I_d \)) are present. We will demonstrate that the sum of these currents is constant throughout the circuit.

Step 1: Conduction Current (\( I_c \)) in the Circuit

The conduction current (\( I_c \)) flows through the external circuit as the capacitor is charged by the AC source. By Ohm's Law, this current is given by:

\[ I_c = \frac{V}{Z} \]

where \( V \) is the applied AC voltage and \( Z \) is the circuit's impedance. Consequently, the conduction current is also an alternating current with the same frequency as the source.

Step 2: Displacement Current (\( I_d \)) in the Capacitor

Within the capacitor, a displacement current (\( I_d \)) is generated by the changing electric field between the plates. This current is defined as the time rate of change of the electric field:

\[ I_d = \epsilon_0 A \frac{dE}{dt} \]

Here, \( \epsilon_0 \) is the permittivity of free space, \( A \) is the area of the capacitor plates, and \( \frac{dE}{dt} \) is the rate of change of the electric field. The electric field \( E \) between the plates is related to the voltage \( V \) across them by:

\[ E = \frac{V}{d} \]

where \( d \) is the plate separation. Substituting this into the displacement current equation yields:

\[ I_d = \epsilon_0 A \frac{d}{dt} \left( \frac{V}{d} \right) = \epsilon_0 A \frac{dV}{dt} \]

Step 3: Relationship Between Conduction and Displacement Currents

The conduction and displacement currents are interconnected due to the capacitor's charging behavior. The alternating voltage across the capacitor creates a changing electric field, which in turn produces the displacement current. The rate at which charge accumulates on the capacitor plates is equivalent to this displacement current.

For an ideal capacitor, the conduction current (\( I_c \)) and the displacement current (\( I_d \)) are equal in magnitude and opposite in direction. Therefore, when considering the capacitor as a system, the total current entering and leaving it is the sum of the conduction current in the external circuit and the displacement current within the capacitor. This implies:

\[ I_c = I_d \]

Consequently, the sum of the conduction current (\( I_c \)) and the displacement current (\( I_d \)) remains constant at all points in the circuit.

Conclusion:

Hence, for a parallel plate capacitor being charged by an AC source, the sum of the conduction current (\( I_c \)) and the displacement current (\( I_d \)) is uniform throughout the circuit.