Question

A uniform but time varying magnetic field exists in a cylindrical region as shown in the figure. The direction of magnetic field is into the plane of the paper and its magnitude is decreasing at a constant rate of $2\times {10}^{-3}T/s$. A particle of charge $1\mu C$ is moved slowly along a circle of radius $1m$ by an external force as shown in figure. The plane of the circle lies on the plane of the paper and it is cocentric with the cylindrical region. The work done by the external force in moving this charge along the circle will be :

• A. zero
• B. $2\pi \times {10}^{-9}J$
• C. $\pi \times {10}^{-9}J$
• D. $4\pi \times {10}^{-6}J$

Answer 1

The correct option is B $2\pi \times {10}^{-9}J$

$El=\frac{d\varphi }{dt}=S\frac{dB}{dt}$
$\therefore E\left(2\pi r\right)=\left(\pi r^2\right)\frac{dB}{dt}$
$E=\frac{r}{2}\frac{dB}{dt}$
$F=qE=\frac{qr}{2}\frac{dB}{dt}$
$W=F.d;d=\left(2\pi r\right)$
$=\pi r^{2}q\left(\frac{dB}{dt}\right)=\left(\frac{22}{7}\right){\left(1\right)}^2\left({10}^{-6}\right)(2\times {10}^{-3})=2\pi \times {10}^{-9}J$