Question

Figure (38-E26) shows a straight, long wire carrying a current i and a rod of length l coplanar with the wife and perpendicular to it. The rod moves with a constant velocity u in a direction parallel to the wire. The distance of the wire from the centre of the rod is x. Find the motional emf induced in the rod.

Answer 1

In this case $\overrightarrow{B}$ varies, so we consider a small element at centre of rod of length dx at a distance x from the wire.

$\overrightarrow{B}=\frac{{\mu }_{0}i}{2\pi x}$

So, $dE=\frac{{\mu }_{0}i}{2\pi x}\times vxdx$

$\therefore E={\int }_0^{\theta }dE$

$=\frac{{\mu }_{0}iv}{2\pi }\left[ln\right(x+\frac{1}{2})-ln(x-\frac{1}{2}\left)\right]$

$=\frac{{\mu }_{0}iv}{2\pi }ln\left[\frac{x+\frac{l}{2}}{x-\frac{l}{2}}\right]$