Question

A parallel plate capacitor is charged by an ac source, Kirchhoff's first rule (junction rule) valid at each plate of the capacitor? Explain.

Hint:

Kirchhoff’s current law is valid even in circuits with capacitors, as displacement current ensures the continuity of charge flow.

Answer 1

In a parallel plate capacitor charged by an AC source, we need to consider the behavior of currents and voltages at the plates and whether Kirchhoff's first rule (junction rule) is valid at each plate of the capacitor. 

1. Kirchhoff's First Rule (Junction Rule):

Kirchhoff's first rule (also known as the junction rule) states that the sum of currents entering a junction is equal to the sum of currents leaving the junction. Mathematically, this is expressed as:

\[ \sum I_{\text{in}} = \sum I_{\text{out}} \]

This rule is based on the principle of conservation of charge, as no charge can be lost or created at a junction.

2. Behavior of a Parallel Plate Capacitor with an AC Source:

When a parallel plate capacitor is connected to an AC source, the voltage across the plates varies sinusoidally. As the AC current flows through the circuit, the current entering the capacitor from the AC source leads to the accumulation of charge on the plates, and the charge on the plates oscillates in response to the alternating voltage.

In this setup:

• The current from the AC source causes a displacement of charge on the capacitor plates, resulting in an alternating electric field between the plates.
• The plates themselves do not allow direct current (DC) flow between them because they are isolated by the dielectric (insulating) material between them. However, the alternating current leads to a time-varying electric field that causes a charging and discharging effect on the plates.
• Because of the dielectric material, the current does not "flow" directly through the capacitor, but rather the current is responsible for charging and discharging the plates during each AC cycle.

3. Is Kirchhoff's Junction Rule Valid at the Plates of the Capacitor?

At first glance, it might seem that Kirchhoff's first rule wouldn't apply directly to the capacitor plates since the current doesn't actually "flow" through the dielectric between the plates. However, Kirchhoff's junction rule still holds in the following way:

• The current entering and leaving each plate of the capacitor must still obey the junction rule, but we must consider the displacement current in the capacitor plates, which is associated with the changing electric field between the plates. The displacement current is defined as:
• Thus, the current entering and leaving each plate of the capacitor can still be considered in terms of displacement current, which is due to the time-varying electric field in the dielectric material.
• At each plate of the capacitor, the total current entering and leaving the plate (including the displacement current) must satisfy Kirchhoff’s first rule. The sum of the currents entering the junction (which is the plate) must equal the sum of the currents leaving it. This includes both the conduction current supplied by the AC source and the displacement current caused by the changing electric field in the capacitor.

4. Conclusion:

Yes, Kirchhoff's first rule (junction rule) is valid at each plate of the capacitor in an AC circuit, as long as we account for the displacement current in the capacitor. The displacement current is responsible for the charging and discharging of the plates and ensures that the current entering and leaving the capacitor plates obeys the junction rule, just as it would for any other junction in a circuit.