Question

1. 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.
2. In case (a) above, is Kirchhoff’s first rule (junction rule) valid at each plate of the capacitor? Explain.

Answer 1

a. The electric field between the capacitor plates is given by:

E = `sigma/epsilon_0 = q/(epsilon_0A)`

where q is the charge accumulated on the positive plate.

The electric flux through this plate is:

Φ_E = EA

= `q/(epsilon_0A) * A`

= `q/epsilon_0`

∴ Displacement current (I_D) = `epsilon_0 (dphi)/dt`

= `epsilon_0 d/dt [q/epsilon_0]`

= `(dq)/dt`

But `(dq)/dt` = the rate at which charge flows to the positive plate through the conducting wire.

Hence, I_D = I_C

i.e., Displacement current between capacitor plates = Conduction current in connecting wires.

b. Kirchhoff’s first rule (junction rule) applies to each plate of the capacitor. The junction rule states that the total current entering and leaving a junction must be equal.

Capacitors require balanced conduction and displacement currents. This guarantees that the current entering and leaving the capacitor is equal, maintaining circuit continuity.