Question

A planar circular coil, of radius a, has N turns and it can be made to rotate about its diameter as axis. The coil has a resistance R and is present in a region where a uniform magnetic field, B, directed perpendicular to the axis of rotation of the coil, exists. If the coil has to be kept rotating with a constant frequency v, (assuming friction and other dissipative effects to be negligible)

(1) an external torque, of magnitude 2πvNB sin² (2πvt) must be made to act on it

(2) an external torque, of magnitude 2πvN [πa²B sin (2πvt)] must be made to act on it

(3) an external torque, of magnitude 2πvN [πa²B]² sin (2πvt) must be made to act on it

(4) it just needs to be given the necessary ‘spin’ at start, it would then keep on rotating at its initial rate, due to its ‘rotational inertia’.

Answer 1

(2) an external torque, of magnitude 2πvN [πa²B sin (2πvt)] must be made to act on it

Let the coil rotate about a diameter, taken as the z–axis. The magnetic field isdirected along the x–axis. The flux, linked with the coil, at any instant, is

The magnetic moment, associated with this induced current,

M = i.(πa²)

The magnetic field exerts a torque, which has an instantneous value \(\tau\) = MB sinθ

The external torque is needed to oppose this torque on the coil due to the induced current. 

Hence external torque = \(\tau\) = MB sinθ = 2πvNπa² Bsin(2πvt)