Question

A copper rod of length 'l' rotates at an angular velocity ${}^{\prime }{\omega }^{\prime }$ in a uniform magnetic field B as shown in figure. What is the induced emf across its ends?

Answer 1

The rod is supposed to be the combination of a number of infinitesimal elements. Speed of each element is different. Consider an element at a distance l from O. Speed of this element is $\omega l$, and is perpendicular to its length.

$\Rightarrow \epsilon =\int vBdl,$ As $v=\omega l,\epsilon =\omega B{\int }_0^{L}ldl=\frac{1}{2}\omega B{L}^2$

'O' turns out to be ${}^{\prime }{+}^{\prime }$ and P the ${}^{\prime }{-}^{\prime }$ve terminal.