Question

A cylindrical space of radius $R$ is filled with a uniform magnetic induction $B$ parallel to the axis of the cylinder. If $B$ changes at a constant rate, the graph showing the variation of induced electric field with distance $r$ from the axis of cylinder is

• A.
• B.
• C.
• D.

Answer 1

The correct option is A

According to the question,

$\frac{dB}{dt}=\text{constant}=B_0\text{(say)}$

$\left(i\right)$ When $r<R:$

Let at a distance $r$ the induced electric field is $E$. As,

$E=|\oint \overrightarrow{E}\cdot d\overrightarrow{l}|=|-\frac{d\varphi }{dt}|........\left(1\right)$

$E\cdot 2\pi r=|-\frac{d[B\cdot (\pi r^2\left)\right]}{dt}|$

$\Rightarrow E=\frac{r}{2}\frac{dB}{dt}=\frac{B_{0}r}{2}$

$\therefore E\propto r......\left(2\right)$


$\left(ii\right)$ When $r>R:$

Using equation $\left(1\right)$, we have

$E\cdot 2\pi r=\frac{d\left(B\pi R^2\right)}{dt}$

$\Rightarrow E=\frac{R^2}{2r}\cdot \frac{dB}{dt}=\frac{B_0{R}^2}{2r}$

$\therefore E\propto \frac{1}{r}......\left(3\right)$


Using equations $\left(2\right)$ and $\left(3\right)$, the graph of variation of the electric field with distance $r$ is drawn as shown below.


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