Derive the expression for the displacement current term in the modified Ampere's circuital law. Define displacement current and state its $SI$ unit.

(N/A) Ampere's circuital law states that $\oint \vec{B} \cdot d\vec{l} = \mu_0 I$. However,this law is inconsistent for time-varying electric fields,such as inside a charging capacitor.
Consider a parallel plate capacitor with plate area $A$ and charge $Q$. The electric field $E$ between the plates is given by $E = \frac{\sigma}{\epsilon_0} = \frac{Q}{A \epsilon_0}$.
The electric flux $\Phi_E$ through a surface $A$ inside the capacitor is $\Phi_E = E A = \left( \frac{Q}{A \epsilon_0} \right) A = \frac{Q}{\epsilon_0}$.
Therefore,$Q = \epsilon_0 \Phi_E$.
Differentiating with respect to time $t$,we get $\frac{dQ}{dt} = \epsilon_0 \frac{d\Phi_E}{dt}$.
Since the conduction current $I_c = \frac{dQ}{dt}$,we define the displacement current $I_d$ as $I_d = \epsilon_0 \frac{d\Phi_E}{dt}$.
Definition: Displacement current is the current that arises due to the time-varying electric field in a region,even in the absence of actual charge carriers.
$SI$ Unit: The $SI$ unit of displacement current is the Ampere $(A)$.