A smooth pair of thick metallic rails is connected across a battery of emf E with negligible internal resistance. A wire ab having length l and resistance r can slide smoothly on these rails. The whole system is situated in a horizontal plane and placed in a uniform vertical magnetic field B . At a certain instant t , the wire is imparted a small velocity v towards the right. After some time, the wire ab will slide w

Step-by-step solution

Let the velocity of the wire at any instant be v' . The motional emf induced across the ends of the wire due to its motion in the magnetic field is e = B l v' . The current in the circuit is given by the net emf divided by the resistance: I = E - B l v' r The magnetic force acting on the current-carrying wire is: F = I l B = ( E - B l v' r ) l B The wire will achieve a constant velocity when its acceleration is zero, which means the net magnetic force acting on it must be zero. Setting F = 0 gives: E - B l v' = 0 v