A conducting rod ab of length l, mass m and resistance R slides on a smooth, thick pair of metallic rails. The plane of the rails makes an angles `theta` with the horizontal. A magnetic field B acts along the perpendicular to the horizontal plane in upward direction. If the rod slides down on the rails at a constant speed v, then show that `B=sqrt((mgRsintheta)/(l^(2)vcos^(2)theta))`.

Explanation

During the sliding of the rod with constant speed, the net force acting on the rod is zero.
`therefore F_(m)costheta=mgsintheta`
or, `BIlcostheta=mgsintheta`
or, `mgsintheta=B((Blvcostheta)/(R))lcostheta`
or, `mgsintheta=(B^(2)l^(2)vcos^(2)theta)/(R)`
or, `B^(2)=(mgRsintheta)/(l^(2)vcos^(2)theta)`
`therefore B=sqrt((mgRsintheta)/(l^(2)vcos^(2)theta))`