Question

A conducting disc of radius r rotates with a small but constant angular velocity ω about its axis. A uniform magnetic field B exists parallel to the axis of rotation. Find the motional emf between the centre and the periphery of the disc.

Answer 1

The angular velocity of the disc is ω. Also, the magnetic field of magnitude B is perpendicular to the disc.



Let us take a circular element of thickness da at a distance a from the centre.
Linear speed of the element at a from the centre, v = ωa
Now,
$$\begin{gathered} de=Blv \\ de=B\times da\times a\mathrm{\omega } \\ \Rightarrow e={\int }_0^r\left(B\omega a\right)da \\ e=\frac{1}{2}B\omega r^2 \end{gathered}$$