Question

A uniform but time varying magnetic field is present in a circular region of radius $R$. The magnetic field is perpendicular and into the plane of the field, and is increasing at a constant rate $\alpha$. There is a straight conducting rod of length $2R$ placed as shown in the figure. The magnitude of induced emf across the rod is given by $\frac{\pi R^2\alpha }{x}$. Find $x$.

Answer 1

Considering a square loop, emf across the loop
$e=\oint \overrightarrow{E}.\overrightarrow{dl}=-\frac{d\varphi }{dt}$
$\Rightarrow e=-\pi R^2\frac{dB}{dt}=-\pi R^2\alpha$
$\therefore$ magnitude of emf across one side $=\frac{\pi R^2\alpha }{4}$