Question

A circular ring of radius $a$ and resistance $R$is placed (fixed) in the x-y plane with its centre at the origin as shown in the figure. A magnetic field $\overrightarrow{B}=\left(\alpha x\right)\hat{i}+\left(\beta y\right)\hat{j}+\left(\gamma t\right)\hat{k}$ is switched on at $t=0$. Then,

• A. current induced in the ring is $\frac{\pi a^2\gamma }{R}$
• B. current induced in the ring is $\frac{\pi a^2\gamma }{2R}$
• C. the magnetic force acting on the ring is $\frac{{\pi }^2{a}^4(\alpha +\beta )\gamma }{2R}$
• D. the magnetic force acting on the ring is $\frac{{\pi }^2{a}^4(\alpha +\beta )\gamma }{R}$

Answer 1

Answer: (A), (D)

The correct options are
A current induced in the ring is $\frac{\pi a^2\gamma }{R}$
D the magnetic force acting on the ring is $\frac{{\pi }^2{a}^4(\alpha +\beta )\gamma }{R}$

Consider a current element on the circular ring at an angle with the positive x-axis as shown in the figure.


The current induced in the ring,
$i=\frac{\pi a^2}{R}\left(\frac{d{B}_z}{dt}\right)$
$\Rightarrow i=\frac{\pi a^2\gamma }{R}........\left(i\right)$

Now, $dF=i\overrightarrow{dl}.\overrightarrow{B}$
$=\left(iad\theta \right)\left(\alpha x\cos \theta \right)+\left(iad\theta \right)\left(\beta y\sin \theta \right)$
$\therefore F=4i\alpha a^2{\int }_0^{\pi /2}{\cos }^2\theta d\theta +4i\beta a^2{\int }_0^{\pi /2}{\sin }^2\theta d\theta$
$(\because x=acos\theta ,y=asin\theta )$

$\Rightarrow F=\pi i{a}^2(\alpha +\beta )$
$\Rightarrow F=\frac{{\pi }^2{a}^4(\alpha +\beta )\gamma }{R}\left[\text{using equation (i)}\right]$