Question

In (A parallel plate capacitor is charged by an ac source. Show that the sum of conduction current Ic and the displacement current Id has the same value at all points of the circuit.), is Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.

Hint:

Kirchhoff's first rule is based on the conservation of charge, and it applies in all circuits, including those involving capacitors with displacement currents.

Answer 1

Validity of Kirchhoff's First Law at Capacitor Plates 

Problem Statement: 

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

Explanation:

1. What is Conduction Current and Displacement Current?

• Conduction Current \( I_c \): The usual electric current due to flow of charges in a conductor (e.g., wires).
• Displacement Current \( I_d \): A current-like term introduced by Maxwell in the dielectric (non-conducting) region between capacitor plates, given by:
  \( I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \), where \( \Phi_E \) is the electric flux.

2. AC Source with Capacitor:

In an AC circuit, the current \( I_c \) continuously changes direction. Charges accumulate and deplete on the capacitor plates, creating a time-varying electric field between them. This changing field gives rise to a displacement current \( I_d \) in the dielectric.

3. Continuity of Current:

Maxwell showed that:
\( I_c = I_d \)

This equality ensures that the current appears continuous throughout the entire circuit, including the space between the capacitor plates where no actual charge carriers move.

Answer:

Yes, Kirchhoff’s first law (junction rule) is valid at each plate of the capacitor, because the sum of the conduction current \( I_c \) and the displacement current \( I_d \) is the same at all points in the circuit.

Reason: The displacement current ensures there is no accumulation of charge at any point in the circuit. Therefore, current continuity is maintained, and the junction rule:

\( \sum I_{\text{in}} = \sum I_{\text{out}} \)

holds true even at the surfaces of the capacitor plates.

Conclusion:

The concept of displacement current bridges the gap in the dielectric region of a capacitor, thereby upholding Kirchhoff’s current law universally, even in time-varying (AC) circuits.