A circular loop of radius r carrying a current i is maintained at the centre of a second circular loop of radius R( r) which carries a current I . The plane of the smaller loop is oriented at an angle of 30^ relative to the plane of the larger loop. What is the minimum magnitude of a single force that must be applied at a point on the periphery of the smaller loop to hold it fixed in this position?

Step-by-step solution

The magnetic field produced by the larger loop at its centre is given by: B = ₀ I 2 R The magnetic moment of the smaller loop is: M = i A = i r^2 The angle between the plane of the smaller loop and the plane of the larger loop is 30^ . Therefore, the angle between their area vectors (or between the magnetic moment M and the magnetic field B ) is also = 30^ . The magnitude of the torque exerted by the magnetic field on the smaller loop is: = M B Substituting the values, we get: = (i r^2) ( ₀ I 2 R ) 30^ = ₀ i I r^2