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: 

Given a parallel plate capacitor connected to an AC source, demonstrate that the sum of conduction current \( I_c \) and displacement current \( I_d \) remains constant throughout the circuit.

Explanation:

1. Definitions of Conduction and Displacement Current:

• Conduction Current \( I_c \): Standard electric current resulting from charge movement in conductors (e.g., wires).
• Displacement Current \( I_d \): A term introduced by Maxwell for the dielectric material between capacitor plates, representing a current-like effect. It is defined as:
  \( I_d = \varepsilon_0 \frac{d\Phi_E}{dt} \), where \( \Phi_E \) is the electric flux.

2. AC Source and Capacitor Interaction:

In AC circuits, the current direction reverses periodically. As charges accumulate and deplete on the capacitor plates, a fluctuating electric field is established between them. This varying electric field generates a displacement current \( I_d \) within the dielectric material.

3. Current Continuity Principle:

Maxwell's findings indicate:
\( I_c = I_d \)

This equality ensures that current flow is continuous across the entire circuit, including the dielectric region between the capacitor plates where no physical charge carriers are present.

Answer:

Yes, Kirchhoff’s first law (junction rule) is applicable to each capacitor plate, as the combined value of conduction current \( I_c \) and displacement current \( I_d \) is invariant at all points in the circuit.

Rationale: The presence of displacement current prevents charge buildup at any circuit point. This maintains current continuity, allowing the junction rule:

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

to be valid even at the capacitor plate surfaces.

Conclusion:

The concept of displacement current resolves the discontinuity in the dielectric region of a capacitor, thus universally validating Kirchhoff’s current law in time-varying (AC) circuits.