Question

There is a uniform magnetic field $B$ in a circular region of radius $R$ as shown in figure, whose magnitude changes at the rate of $dB/dt$. The emf induced across the ends of a circular concentric conducting arc of radius $R_1$ having angle $\theta$ as shown ($-OA{O}^{\prime }=\theta$) is

• A. $\frac{\theta }{2\pi }{R}_1^2\frac{dB}{dt}$
• B. $\frac{\theta }{2}{R}^2\frac{dB}{dt}$
• C. $\theta 2\pi R$
• D. None of these

Answer 1

The correct option is B $\frac{\theta }{2}{R}^2\frac{dB}{dt}$

Consider the region OAO'
Magnetic Flux
$\varphi =\int B.dA=BA=B\frac{\theta }{2\pi }\pi R^2$
$ϵ=-\frac{d\varphi }{dt}=-d(\frac{B\frac{\theta }{2\pi }\pi R^2)}{dt}=\frac{\theta R^2}{2}\frac{dB}{dt}$