Question

A uniform magnetic field $B$ fills a cylindrical volume of radius $R$. A metal rod of length $l$ is placed as shown. If $B$ is changing at the rate of $dB/dt$, the emf that is produced by the changing magnetic field is $\xi$. [Here, $p=\sqrt{R^2-(l/2{)}^2}]$. Find the induced emf in the rod.

Answer 1

$E=-\frac{d\varphi }{dt}=-A\frac{dB}{dt}$

$A=$ Area of the triangle sandwiched between ends of the rod and the radii connecting the centre with the ends

$A=\frac{1}{2}\times \text{base}\times \text{height}=\frac{1}{2}\times l\times \sqrt{R^2-(\frac{l}{2}{)}^2}$

$\therefore E=-\frac{1}{2}\sqrt{R^2-(\frac{l}{2}{)}^2}\times \frac{dB}{dt}=-\frac{dB}{dt}\times \frac{1}{2}\sqrt{R^2-(\frac{l}{2}{)}^2}$