Question

Figure shows a straight, long wire carrying a current i and a rod of length l coplanar with the wire and perpendicular to it. The rod moves with a constant velocity v 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.
Figure

Answer 1

Motional induced EMF is Negative of Product of Field, Speed and Current

Explanation:

• In this case, vector B varies  

• Hence considering a small element at the center of the rod of length dx at a dist x from the wire.

$\vec{B} = \frac {\mu_0i}{2\pi x}$

• So, de = $\frac {\mu_0i}{2\pi x} \times vxdx$

e = $\int_{0}^{e}de = \frac {\mu_0iv}{2\pi}$

= $\int_{x-t/2}^{x+t/2}\frac {dx}{x}$

= $\frac {\mu_0iv}{2\pi}[ln (x+l/2)-ln (x - l/2)]$

= $\frac {\mu_0iv}{2\pi}ln [\frac {x + l/2}{x - l/2}]$  

= $\frac {\mu_0iv}{2x}ln [\frac {2x + l}{2x - l}]$