Q65P - Fundamentals Of Physics

Question

A parallel-plate capacitor with circular plates of radius $\mathit{R}$ is being discharged. The displacement current through a central circular area, parallel to the plates and with radius $\raisebox{1ex}{$\mathbf{R}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{2}$}\right.$, is  $\mathbf{\text{2.0 A}}$. What is the discharging current?

Step-by-Step Solution

Answer

The discharging current is  $i_d=\text{8.0 A}$.

Listing the given quantities:

Discharging current, $I_d=\text{2.0 A}$

Radius of the circular plates of the parallel plate capacitor is $R$.

Radius of the central circular plates through which the current flows is $\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$.

Understanding the concepts of displacement current:

Use the formula of the displacement current through capacitor and the current for charging capacitor to find the displacement current.

Formulae: 

The area is,

$A=\pi r^2$ 

The displacement current is,

$i_d={\epsilon }_{0}A\frac{dE}{dt}$ 

Here, ${\epsilon }_0$ is the permittivity of free space, $E$ is the electric field, and $t$ is time.

Calculations of the discharging current:

For circular plates of the parallel plate capacitor of radius , 

$A=\pi R^2$ 

And for central circular plates of radius $\raisebox{1ex}{$R$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$. Thus, the area will be,

$A_c=\pi \frac{R^2}{4}$

For displacement current, 

$i_d={\epsilon }_{0}A\frac{dE}{dt}$                                                                                                                  ..... (1)

And for charging capacitor, 

$i={\epsilon }_0{A}_c\frac{dE}{dt}$                                                                                                                     ..... (2)

Take a ratio of equation (1) and (2), and you have

$$\begin{gathered} \frac{i_d}{i}=\frac{\pi R^2}{\pi \frac{R^2}{4}} \\ {i}_d=4i \end{gathered}$$

$\begin{array}{rcl}{i}_d& =& 4\times \left(\text{2.0 A}\right)\\ & =& 8.\text{0 A}\end{array}$

Hence, the discharging current is $\begin{array}{rcl}& & 8.\text{0 A}\end{array}$.