Question

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

Hint:

Kirchhoff's first rule is valid as long as the currents entering and leaving a junction (in this case, the plates of a capacitor) are balanced, including the contribution from the displacement current.

Answer 1

Kirchhoff's First Rule at the Plates of a Capacitor 

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. This rule is based on the conservation of electric charge.

How Kirchhoff's First Rule Applies to a Capacitor:

At each plate of the capacitor, there are two types of currents:

• Conduction Current (I_c): This current enters the plate of the capacitor through the external circuit.
• Displacement Current (I_d): This current exits through the dielectric between the plates of the capacitor, causing a change in the electric field.

Since the displacement current is equal in magnitude to the conduction current in the circuit, the total current entering and leaving each plate of the capacitor is balanced. Therefore, Kirchhoff's first rule holds at each plate because the total current entering (conduction current) equals the total current leaving (displacement current).

Conclusion:

Thus, Kirchhoff's junction rule is valid at each plate of the capacitor, as the conduction and displacement currents balance each other, ensuring the total current entering and leaving the plates remains equal.

Answer 2

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

Kirchhoff's First Rule states that the algebraic sum of currents entering a junction is equal to the algebraic sum of currents leaving the junction. In other words, the total current entering a point (junction) must be zero. This is based on the principle of conservation of charge.

2. Applying Kirchhoff's Junction Rule at the Plates of a Capacitor:

The capacitor has two plates, and these plates store charge. When a voltage is applied across the plates, a current flows onto one plate and off the other. However, at each plate, the current flowing onto the plate is not the same as the current flowing off the plate, because the capacitor is storing charge on its plates.

The current entering the plate results in the accumulation of charge on the plate. So, when a current enters the plate, it does not leave immediately but instead gets "stored" as charge. This stored charge results in the plate having a voltage across it, which increases as the charge accumulates. The relationship between the charge \( Q \), the capacitance \( C \), and the voltage \( V \) across the capacitor is:

\[ Q = C \cdot V \]

Since the current entering a plate of the capacitor leads to the storage of charge, the current cannot "leave" the plate in the traditional sense (as it would at a junction in a simple resistor network). Thus, Kirchhoff's First Rule does not apply in the conventional form at the plates of the capacitor because the current does not pass directly through a junction. Instead, the current causes charge to accumulate on the plates.

3. Conclusion:

Kirchhoff's First Rule (junction rule) is not valid at the plates of the capacitor because the current entering the plate results in the accumulation of charge rather than the direct flow of current through a junction. The rule is based on the assumption that the current entering and leaving a junction is balanced, but at a capacitor plate, the current entering causes charge storage, not an immediate current flow out of the junction.