Question

A conducting loop of radius $R$ is present in a uniform magnetic field $B$ perpendicular to the plane of the ring. If radius $R$ varies as a function of time $t$, as $R=R_0+t$. The e.m.f induced in the loop is

• A. $2\pi (R_0+t)B,\text{clockwise}$
• B. $\pi (R_0+t)B,\text{clockwise}$
• C. $2\pi (R_0+t)B,\text{anticlockwise}$
• D. Zero

Answer 1

The correct option is C $2\pi (R_0+t)B,\text{anticlockwise}$

Given:
$R=(R_0+t)$

Flux linked with the loop is,

$\varphi =\overrightarrow{B}.\overrightarrow{A}$

$=BA\cos 0^{\circ }[\because \theta =0^{\circ }]$

$=B\times \pi R^2=\pi B(R_0+t{)}^2$

$\therefore$ Emf induced in the loop is,

$E=\left|\frac{d\varphi }{dt}\right|=\left|\frac{d}{dt}\right[\pi B(R_0+t{)}^2]|$

$=\pi B\times 2(R_0+t)=2\pi B(R_0+t)$

As the radius of the loop is increasing, the area of the loop will also increase with time. Due to this, the inward flux linked with the coil also increases with time.

According to Lenz's law, current will flow anti-clockwise in the loop to oppose the increase in magnetic flux.

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