Question

What is meant by displacement current? A capacitor is being charged by a battery. Show that Ampere-Maxwell law justifies continuity and constancy of the current flowing in the circuit.

Answer 1

Displacement Current and Ampere–Maxwell Law
Displacement Current:
Displacement current is the current associated with a time-varying electric field.
It does not involve actual movement of free charges through a conductor, but it produces a magnetic field just like conduction current.
Mathematically, displacement current is given by:
I_d = ε₀ (dΦ_E / dt)
where ε₀ is the permittivity of free space and Φ_E is the electric flux.

Charging of a Capacitor
Consider a capacitor connected to a battery. When the switch is closed:
- Electrons flow through the connecting wires → this is conduction current.
- Charges accumulate on the plates of the capacitor.
- Between the plates, there is no physical movement of charges across the gap.
- However, the electric field between the plates increases with time.
This changing electric field gives rise to displacement current in the space between the plates.

Ampere–Maxwell Law
The modified form of Ampere’s law is:
∮ B · dl = μ₀ (I_c + I_d)
where:
I_c = conduction current
I_d = displacement current

Justification of Continuity and Constancy of Current
If we apply Ampere’s law to a loop enclosing the wire:
- We detect conduction current I_c.
If we stretch the loop between the plates of the capacitor:
- There is no conduction current.
- But there is displacement current I_d due to changing electric field.
For a charging capacitor:
I_d = ε₀ (dΦ_E / dt)
It can be shown that:
I_d = I_c
Thus, the displacement current between the plates equals the conduction current in the wires.
Therefore:
I_total = I_c + I_d
Since I_d equals I_c during charging, the current remains continuous and constant throughout the circuit.

Conclusion:
Displacement current ensures continuity of current in circuits containing capacitors. Ampere–Maxwell law proves that even in the absence of conduction current (between capacitor plates), displacement current maintains a continuous and constant current in the entire circuit.