Question

A flexible wire 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 as shown in the figure.

The graph of magnitude of induced emf in the coil is represented by

• A.
• B.
• C.
• D.

Answer 1

The correct option is B

Magnitude of induced emf, $\begin{array}{}\mid E\mid =\mid \frac{d\varphi }{dt}\mid =\mid \frac{d(B.A)}{dt}\mid =\mid B\frac{d\left(A\right)}{dt}\mid & \text{(1)}\end{array}$
Since for a-b and c-d, radius$\left(r\right)$ remains constant with time and thus area will remain constant. Hence for segment a-b and c-d emf induced will be zero according to equation $\left(1\right)$.
For segment b-c, radius $r\propto t$
Hence, $\frac{dr}{dt}=$ constant $=k$
$\Rightarrow \frac{dA}{dt}=\frac{d\left(\pi r^2\right)}{dt}=2\pi r\frac{dr}{dt}=2\pi rk$
Substituting this in $\left(1\right)$
$\mid E\mid =\mid B\frac{d\left(A\right)}{dt}\mid =B\times 2\pi rk$
$\mid E\mid \propto r$
$\mid E\mid \propto t$ $(\because r\propto t)$
Hence option b is correct.