Question

Show that the displacement current inside a parallel-plate capacitor can be written $\mathrm{C}\left(\frac{{\mathrm{dVC}}_{\mathrm{c}}}{\mathrm{dt}}\right)$.

Short answer

Displacement current inside a parallel plate capacitor can be written as:$I_d=C\frac{dVc}{dt}$

Step-by-step solution

Given Information:

We have been given a parallel plate capacitor and we need to write the equation of displacement current for it.

Solving

The general equation for displacement current for a parallel plate capacitor is given by:

$I_d={\epsilon }_o\left(\frac{d{\Phi }_e}{dt}\right)$equation 1

Also, it is known:

${\Phi }_e=\frac{Q}{{\epsilon }_o}$

localid="1649863179586" ${\Phi }_e{\epsilon }_o=Q=V_{C}C$equation 2

Combining the values of ${\Phi }_e$in equations 1 and 2.

$I_d=\frac{d}{dt}\left({\epsilon }_o{\Phi }_e\right)$

$I_d=C\frac{dVc}{dt}$

Hence, Proved.