A square-shaped coil of edge a having  $n$ turns is rotated with a uniform angular velocity $\mathrm{\omega }$  about one of its diagonals which is kept fixed in a horizontal position. A uniform magnetic field   exists in the vertical direction. Find the emf induced in the  coil.
 

The area of the square frame is $\mathrm{A}={\mathrm{a}}^2$ . If the normal to the frame makes an angle  with the magnetic field at t =0, this angle will become $\mathrm{\theta }=\omega t$ after some time . 

The flux of the  magnetic field at this time is .$\mathrm{\Phi }=\mathrm{nBAcos}\mathrm{\theta }={\mathrm{nBa}}^2\cos \mathrm{\omega t}$
The induced emf is $\mathrm{\epsilon }=-\frac{\mathrm{d\Phi }}{\mathrm{dt}}={\mathrm{nBa}}^2\mathrm{\omega sin}\mathrm{\omega t}$
Thus, an alternating emf is induced in the coil.