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 induced in the rotating rod OA is given by e = 1 2 B a^2 . The net emf driving current through the circuit is E - e , assuming the battery assists the rotation against gravity. The current in the circuit is I = E - 1 2 B a^2 R . The magnetic force on an element dr of the rod at distance r from O is dF_m = I B dr . The magnetic torque about the pivot O is _m = ₀^a r (I B dr) = I B a^2 2 . The torque due to gravity about O opposes the magnetic torque. Since the angle is measured from the horizontal,