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

Consider a current element on the circular ring at an angle $\theta$ with the positive x-axis as shown in the figure.
The current induced in the ring.
$i=\frac{\pi a^2}{R}\left(\frac{d{B}_z}{dt}\right)$ 
 
$\Rightarrow i=\frac{\pi a^2\gamma }{R}$ ……………..(i)
Now, $dF=\left(iad\theta \right)\left(\alpha x\cos \theta \right)+\left(iad\theta \right)\left(\beta y\sin \theta \right)$  
$\therefore F=4i\alpha a^2\underset{0}{\overset{\pi /2}{\int }}{\cos }^2\theta d\theta +4i\beta a^2\underset{0}{\overset{\pi /2}{\int }}{\sin }^2\theta d\theta$ 
$\Rightarrow F=\pi i{a}^2(\alpha +\beta )$ 
$\Rightarrow F=\frac{{\pi }^2{a}^4(\alpha +\beta )\gamma }{R}$ [using equation (i)]