A conducting rod of length 1 m is pivoted at one end and rotated with a constant angular velocity of 10 rad s ⁻¹ in a uniform magnetic field of 2 T directed perpendicular to the plane of rotation. The free end and the pivot are connected to a stationary resistor of 5 . The external torque required to maintain the constant angular velocity of the rod is _______ N m . (Assume the resistance of the rod and connections t

Step-by-step solution

The motional EMF induced in a rod rotating about one end in a perpendicular magnetic field is: E = 1 2 B l² The induced current in the circuit is: I = E R = B l² 2R The magnetic force on a small element of length dr at a distance r from the pivot is dF = I B dr . The torque due to this force about the pivot is: d = r dF = I B r dr Integrating over the length of the rod to find the total opposing magnetic torque: _ m = ₀^ l I B r dr = 1 2 I B l² Substituting the expression for current I : _ m = 1 2 ( B l² 2R ) B l²