Question

A time varying electric field exists in a certain space. The magnitude of displacement current varies with time as shown in the diagram. What is the magnitude of electric field in that region at $t=10\text{sec}$, if the electric field at $t=0\text{sec}$ is zero.
(Take area of the space, $A=100{\text{cm}}^2$ and ${ϵ}_o=8.85\times {10}^{-12}\text{SI units}$)

• A. $0.01\text{A}$
• B. $1\text{A}$
• C. $0.1\text{A}$
• D. Zero

Answer 1

The correct option is D Zero


Slope of the line is,

$m=\frac{10-0}{0-5}=-2$

The equation of the given straight line is,

$I_D=-2t+10$

Now, the displacement current is given by,

$I_D={ϵ}_o\frac{d{\varphi }_E}{dt}$

$\therefore -2t+10={ϵ}_o\frac{d{\varphi }_E}{dt}$

Now, $\frac{d{\varphi }_E}{dt}=\frac{d}{dt}\left(EA\right)=A\frac{dE}{dt}$

$\therefore -2t+10=A{ϵ}_o\frac{dE}{dt}$

$\Rightarrow dE=\frac{1}{A{ϵ}_o}(-2tdt+10dt)$

The expression for electric field is,

$E=\int dE=\frac{1}{A{ϵ}_o}\left(\int -2tdt+\int 10dt\right)$
$=\frac{1}{A{ϵ}_o}(-t^2+10t+C)$

At $t=0,E=0\Rightarrow C=0$

$\therefore E=\frac{1}{A{ϵ}_o}(-t^2+10t)$

At $t=10\text{sec}$,
$E=\frac{1}{A{ϵ}_o}(-(10{)}^2+10\left(10\right))=0$

Hence, $\left(D\right)$ is the correct answer.