Question

There is a uniform (in spatial districution) magnetic filed B in a circular region of radius R as shown in the figure whose magnitude varies uniformly a the rate $\beta$ w.r.t. time. The emf induced across the ends of a circular concentric conducting arc of radius $R_1$ having an agle $\theta$ as shown $(<OA{O}^{\prime }=\theta )$ is

• A. $\frac{\theta }{2\pi }{R}_1^2.\beta$
• B. $\frac{\theta }{2}{R}^2.\beta$
• C. $\frac{\theta }{2\pi }{R}^2.\beta$
• D. Zero

Answer 1

The correct option is C $\frac{\theta }{2\pi }{R}^2.\beta$

$Emf=\frac{d\varphi }{dt}$

where

$\frac{dB}{dt}=\beta$

$\varphi =\int B.ds$

$=B\frac{1}{2}{r}^2.\frac{\theta }{\pi }$

$=\frac{B{r}^2\theta }{2\pi }$

$Emf=\frac{dB}{dt}\frac{r^2\theta }{2\pi }$

$=\frac{\beta r^2\theta }{2\pi }$

$Emf=\frac{\theta }{2\pi }{r}^2.\beta$