A metal disc of radius $R$ rotates with angular velocity $\omega$ in a uniform magnetic field $B$ perpendicular to its plane. The potential difference between the center and the rim is?

Solution

1. Consider a small radial element of length $dr$ at a distance $r$ from the center of the disc.
2. The motional electromotive force $d\epsilon$ induced in this element as it rotates is given by $d\epsilon =Bvdr$, where $v=\omega r$.
3. Integrate this expression from the center ($r=0$) to the rim ($r=R$) to find the total potential difference: ${\int }_0^{R}B\omega rdr$.
4. Evaluating the integral yields $\frac{1}{2}B\omega R^2$.
5. Hence the answer is (B).