What is displacement current? Obtain an expression of displacement current for a charged capacitor. Write Ampere Maxwell’s law.

Displacement current is a concept in electromagnetism that was introduced by James Clerk Maxwell as part of his formulation of Maxwell's equations. It is an additional term in Ampere's law that accounts for the changing electric field in the region where a current is flowing. Displacement current plays a crucial role in understanding electromagnetic waves and the behavior of capacitors.

To understand displacement current, let's consider a charged capacitor. A capacitor consists of two parallel conducting plates separated by a dielectric material. When a voltage is applied across the plates, one plate accumulates positive charge while the other accumulates an equal amount of negative charge.

The displacement current arises from the changing electric field between the capacitor plates as the charge accumulates or depletes. It is not a flow of charges but rather a time-varying electric field. This changing electric field induces a magnetic field according to Faraday's law of electromagnetic induction.

The expression for displacement current through a charged capacitor can be obtained by considering the rate of change of electric flux passing through the surface bounded by the plates. Mathematically, it is given by:

I_d = ε₀ * d(Φ_E)/dt

Where:

I_d represents the displacement current.
ε₀ is the permittivity of free space (vacuum), approximately 8.854 × 10^(-12) F/m.
d(Φ_E)/dt is the rate of change of electric flux passing through the surface.
Now, let's write Ampere-Maxwell's law, which incorporates the displacement current term:

∇ × B = μ₀ * (J + ε₀ * dE/dt)

Where:

∇ × B represents the curl of the magnetic field.
μ₀ is the permeability of free space, approximately 4π × 10^(-7) T·m/A.
J is the conduction current density (current flowing through conductors).
ε₀ is the permittivity of free space.
dE/dt is the rate of change of the electric field.
This modified form of Ampere's law, with the addition of the displacement current term (ε₀ * dE/dt), ensures that the laws of electromagnetism are consistent with the conservation of charge and the behavior of electromagnetic waves.