Question

A flexible wire is bent in the form of a circle is placed in a uniform magnetic field perpendicular to the plane of the coil. The radius of the coil changes with time as shown in figure. The graph of induced emf versus time in the coil is represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

We know that, emf induced in the wire is,

$e=|-\frac{d\varphi }{dt}|$

Here, $\varphi =BA\cos 0^{\circ }=B\left(\pi r^2\right)$

$\Rightarrow e=2\pi Br\frac{dr}{dt}......\left(1\right)$

$\text{Case I:}$ During the time interval, $t=0\text{s}$ to $t=1\text{s}$, $r=\text{constant}$

$\Rightarrow \frac{dr}{dt}=0$

Using equation $\left(1\right)$, $e=0$

$\text{Case II:}$ During the time interval $t=$ to $t=1\text{s}$, $r$ is increasing at constant rate. i.e,

$\frac{dr}{dt}=\text{constant}$

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

$e=2\pi r\times \left(\text{constant}\right)$

$\therefore e\propto r$

$\therefore$ The graph will be a straight line.

$\text{Case III:}$ Beyond the time, $t=2\text{s}$,

$r=\text{constant}\Rightarrow e=0$

Only graph $B$ follows all the three situation.


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