Question

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

• A. $2\pi (R_0+t)B$ clock wise
• B. $\pi (R_0+t)B$ clock wise
• C. $2\pi (R_0+t)B$ anti-clock wise
• D. zero

Answer 1

The correct option is C $2\pi (R_0+t)B$ anti-clock wise

Given:
$R=R_0+t$

According to Faraday's law induced emf
$\left|\begin{array}{c}e\end{array}\right|=\frac{d\varphi }{dt}=\frac{d}{dt}(\overrightarrow{B}.\overrightarrow{A})(\because \varphi =\overrightarrow{B}.\overrightarrow{A})$

$\left|\begin{array}{c}e\end{array}\right|=B\times \frac{dA}{dt}=B\frac{d}{dt}\left(\pi R^2\right)(\because A=\pi R^2)$

$\left|\begin{array}{c}e\end{array}\right|=B\times \pi \times \frac{d(R_0+t{)}^2}{dt}$

$\left|\begin{array}{c}e\end{array}\right|=2B\pi (R_0+t)$

To get the direction we use lenz's law, $e=-\frac{d\varphi }{dt}$

$\therefore e=2B\pi (R_0+t)\circlearrowleft$

Hence, option (c) is the correct answer.