A circular loop of radius a is situated in a vertical plane within a uniform magnetic field B directed perpendicular to the plane of the loop. A rod OA of length a and mass m is pivoted at the loop's centre O , with its other end touching the loop at A . Between O and a point C on the loop, a battery of emf E and a variable resistance R are connected. The connecting wires, the rod, and the loop have negligible resist

Step-by-step solution

The motional emf e induced in the rotating rod OA is given by e = 1 2 B a^2 . Since the rod and loop have negligible resistance, the total resistance of the circuit is R (from the wire OC ). The induced current in the circuit is I = e R = B a^2 2R . The magnetic force on an element of the rod of length dr at a distance r from O is dF_m = I B dr . The magnetic torque about the pivot O opposes the rotation: _m = ₀^a r dF_m = ₀^a I B r dr = I B [ r^2 2 ]₀^a = I B a^2 2 Substituting the value of I : _m = ( B a^2 2R ) B