A wire loop enclosing a semicircle of radius $R$ is located on the boundary of a uniform magnetic field $B$. At the moment $t = 0$, the loop is set into rotation with a constant angular acceleration $\alpha$ about an axis $O$ coinciding with a line of vector $\overrightarrow{B}$ on the boundary. Find the emf induced in the loop as a function of time. 

Emf is induced in the loop because area inside the magnetic field is continually charging. 

From $\theta =0\text{ to }\pi ,2\pi \text{ to }3\pi ,4\pi \text{ to }5\pi$, the loop begins to enter in the magnetic field. Thus the magnetic field passing through the loop is increasing. Hence, current in the loop is anticlockwise, and for $\theta =\pi \text{ to }2\pi ,3\pi \text{ to }4\pi ,5\pi \text{ to }6\pi$ etc. magnetic field passing through the loop is decreasing. Hence current in the loop is clockwise.

Angular acceleration of the loop is constant, therefore angle turned in time $t$ is $\theta =1/2\alpha t^2$ and time taken, $t=\sqrt{2\theta /\alpha }$.

$t_1=\text{time taken to rotate angle }\pi =\sqrt{\frac{2\pi }{\alpha }}$

From $t=0$ to $t=t_1$, emf is negative 

$t_2=\text{time taken to rotate angle }2\pi =\sqrt{\frac{4\pi }{\alpha }}$

From $t=t_1$ to $t=t_2$, emf is positive …. …… ……..

$t_n=\text{time taken to rotate angle }n\pi =\sqrt{\frac{2n\pi }{\alpha }}$

Area inside the field is 

$A=\frac{1}{2}{R}^2\theta$

or, $A=\frac{1}{4}{R}^2\alpha t^2$

so flux passing through the loop, 

$\varphi =BA=\frac{1}{4}B{R}^2\alpha t^2$

$E=\left|\frac{d\varphi }{dt}\right|=\frac{1}{2}B{R}^2\alpha t$

$e\propto t$