Question

The magnetic field $\overrightarrow{B}$ at all points within a circular region of radius $R$ is uniform in space and directed into the lane of the page in figure. If the magnetic field is increasing at a rate $dB/dt$, what are the magnitude and direction of the force on a stationary positive point charge $q$ located at points $a,b,c$? (Point $a$ is a distance $r$ above the centre of the region, point $b$ is a distance $r$ to the right of the centre, and the point $c$ is at the centre of the region.)

Answer 1

Inside the circular region at distance $r$,
$El=\frac{d\varphi }{dt}=s\left(\frac{dB}{dt}\right)$
$\therefore E\left(2\pi r\right)=\left(\pi r^2\right)\cdot \frac{dB}{dt}$
$\therefore E=\frac{r}{2}\frac{dB}{dt}$
$F=qE=\frac{qr}{2}\frac{dB}{dt}$
At points $a$ and $b$, distance from centre is $r$
$\therefore$ $F=\frac{qr}{2}\frac{dB}{dt}$
At point $C$, distance $r=0$
$\therefore$ $F=0$
$⨂$ magnetic field is increasing. hence induced current in an imaginary loop passing through $a$ and $b$ should be anti-clockwise. Force on positive charge is in the direction of induced current. Hence force at $a$ is towards left and force at $b$ is upwards.