Question

In a cylindrical region of radius R, there exists a time varying magnetic field B such that $\frac{dB}{dt}=k(>0)$. A charged particle having charge q is placed at the point P at a distance d (> R) from its centre O. Now, the particle is moved in the direction perpendicular to OP (see figure) by an external agent upto infinity so that there is no gain in kinetic energy of the charged particle. Work done by external agent in the process is :

• A. $\frac{q\pi R^2}{4}k$
• B. $\frac{q\pi R^2}{8}k$
• C. $\frac{q\pi R^2}{6}k$
• D. $\frac{q\pi R^2}{3}k$

Answer 1

The correct option is A $\frac{q\pi R^2}{4}k$

$\int \overrightarrow{E}.d\overrightarrow{l}=A\frac{dB}{dt}$
$E2\pi \sqrt{x^2+d^2}=\pi R^{2}k$
$E=\frac{\pi R^{2}k}{2\sqrt{x^2+d^2}}$
$W_{ext}={\int }_0^{\infty }q\overrightarrow{E}.dx=\frac{q\pi R^2}{4}k$