Question

Write an expression for the displacement current.

Answer 1

Hint: It was well known that magnetic fields are produced by an electric current. But, it was shown by Maxwell that a changing electric field must also produce a magnetic field. Maxwell noticed a term missing in the Ampere’s circuital law. He modified this law and the new term added was displacement current.

Complete step-by-step answer:
Ampere’s Circuital law states that the integral of magnetic field density along an imaginary closed loop is the same as the product of current enclosed by this imaginary path and absolute permeability of the medium.
Ampere’s Circuital law is given by
$\oint{\mathbf{B}.d\mathbf{l}={{\mu }_{0}}I(t)}$
Maxwell noticed a term missing in the Ampere’s circuital law. According to him, displacement current is a missing term in Ampere’s circuital law. He showed that to generalize Ampere’s law, we are required to add another term which is $\epsilon_0$ times the rate of change of electric flux through the same loop and named it as displacement current. The current produced because of flow of charges is called conduction current. But current which is due to changing electric field (or electric displacement) is known as displacement current or more precisely Maxwell’s displacement current.
The generalized Ampere’s circuital law is given by
$\oint{\mathbf{B}.d\mathbf{l}={{\mu }_{0}}\left( {{I}_{c}}+{{I}_{d}} \right)}$
Where ${{I}_{d}}$ is the displacement current and is given by
${{I}_{d}}={{\epsilon }_{0}}\dfrac{d{{\phi }_{E}}}{dt}$

Note: Magnetic field is not only produced by the conduction electric current due to flowing charges. It can also be produced due to the varying electric field.
The physical effects produced by the displacement current are the same as produced by the conduction current.