In the figure shown a coil of single turns is would on a sphere of radius R and mass m .The palne the coil is parallel to the planes and lies in the equatorial plane of the sphere . Current in the coil is i. The vaule of B if the sphere is in equailibrum is

Options

A. `(mg)/( pi r)`
B. ` (mg sin theta)/(pi i)`
C. `(mgr sin theta)/(pi i)`
D. none of these

Answer

A

Explanation

The gravitational torque must be counter balanced by the magnetic torque about 0, for equilibrium of the sphere. The gravitational torque `= tau_(gr) = |overline(mg) xx vec(r )|`
`implies tau_(gr) = mgr = sin theta`
The magnetic torque `vec(tau_(m)) = vec(mu) xx vec(B)`
Where the magnetic moment of the coil `= mu = (i pi r^(2))`
`implies tau_(m) = pi i r^(2) B sin theta`
`pi i r^(2) B sin theta = mgr sin theta implies B = mg//pi i r`