Question

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

Answer 1

When a capacitor is charging (or discharging), current flows in the circuit. However, there is no actual charge transfer in the insulated region between capacitor which is contradictory to the flow of current. Hence, displacement current is the current in the insulated region due to the changing electric flux.

For a charged capacitor, electric field between the plates is given by:

$E=\frac{Q}{{\epsilon }_{o}A}$

$Q={\epsilon }_o{\varphi }_E$

Displacement current is given by:

$i_d=\frac{dQ}{dt}={\epsilon }_o\frac{d{\varphi }_E}{dt}$

According to Maxwell's Ampere circuital Law, the line integral of magnetic field along a closed path is equal to ${\mu }_o$ times the total current.

$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_o(i_c+i_d)$

where $i_c:$ Conduction current

$i_d:$ Displacement current

$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_o(i_c+{\epsilon }_o\frac{d\varphi }{dt})$