Question

A parallel plate capacitor with circular plates of radius $R$ is being charged as shown. At the instant shown, the displacement current in the region between the plates enclosed between $\frac{R}{2}$ and $R$ is given by,

• A. $\frac{3}{4}i$
• B. $\frac{i}{4}$
• C. $3i$
• D. $\frac{4}{3}i$

Answer 1

The correct option is A $\frac{3}{4}i$

We know that the displacement current is given by

$i_d={\epsilon }_{0}A\frac{dE}{dt}$

Here, $i_d=i$

$[\because \text{conduction current = displacement current}]$

and $A=\pi R^2$

$\Rightarrow i={\epsilon }_0\pi R^2\frac{dE}{dt}$

The displacement current in the region between the plates enclosed between $\frac{R}{2}$ and $R$ is given by

$i_d^{\prime }={\epsilon }_0{A}^{\prime }\frac{dE}{dt}$

Here, $A^{\prime }=\pi R^2-\pi {\left(\frac{R}{2}\right)}^2=\frac{3}{4}\pi R^2$

$\Rightarrow i{{}_d}^{{}^{\prime }}={\epsilon }_0\times \frac{3\pi R^2}{4}\frac{dE}{dt}$

$\Rightarrow i{{}_d}^{{}^{\prime }}=\frac{3}{4}\left[{\epsilon }_0\pi R^2\frac{dE}{dt}\right]=\frac{3i}{4}$

Hence, option $\left(A\right)$ is correct.