Question

The magnetic field $\overrightarrow{B}$ at all points within a circular region of radius $R$ is uniform in space and directed into the plane of the page, as shown in the figure. If the magnetic field is increasing at a rate of $dB/dt$, what is the magnitude of force on a stationary positive charge $q$ located at point $P$?

• A. $\frac{5qr}{2}\left|\frac{dB}{dt}\right|$
• B. $\frac{3qr}{2}\left|\frac{dB}{dt}\right|$
• C. $\frac{qr}{3}\left|\frac{dB}{dt}\right|$
• D. $\frac{qr}{2}\left|\frac{dB}{dt}\right|$

Answer 1

The correct option is D $\frac{qr}{2}\left|\frac{dB}{dt}\right|$

From Faraday's Law of Electromagnetic Induction,

$\oint \overrightarrow{E}.d\overrightarrow{l}=-\frac{d{\varphi }_B}{dt}$

Inside the circular region at distance $r$,

$E\left(2\pi r\right)=\left(\pi r^2\right)\left|\frac{dB}{dt}\right|$

$\Rightarrow E=\frac{r}{2}\left|\frac{dB}{dt}\right|$

So,

$F=qE=\frac{qr}{2}\left|\frac{dB}{dt}\right|$

Hence, option $\left(D\right)$ is the correct answer.