Question

A conducting circular loop is placed in a uniform magnetic field of induction B tesla with its plane normal to the field. Now, the radius of the loop starts shrinking at the rate $\left(\frac{dr}{dt}\right)$ . Then, the induced emf at the instant when the radius is r, is

• A. $\pi rB\left(\frac{dr}{dt}\right)$
• B. $2\pi rB\left(\frac{dr}{dt}\right)$
• C. $\pi r^2\left(\frac{dr}{dt}\right)$
• D. ${\left(\frac{\pi r^2}{2}\right)}^{2}B\left(\frac{dr}{dt}\right)$

Answer 1

Answer: (B)

Induced emf is given by $e=-\frac{d\varphi }{dt}$ If the radius of loop is r at a time t, then the instantaneous magnetic flux is given by
$\varphi =\pi r^{2}B$
$\therefore$ $e=-\frac{d}{dt}(\pi r^{2}B)$
$e=-\pi B\left(\frac{2rdr}{dt}\right)$
$e=-2\pi Br\frac{dr}{dt}$
Numerically, $e=2\pi Br\left(\frac{dr}{dt}\right)$