(a) A small compass needle of magnetic moment 'm' is free to turn about an axis perpendicular to the direction of uniform magnetic field 'B'. The moment of inertia of the needle about the axis is 'I'. The needle is slightly disturbed from its stable position and then released. Prove that it executes simple harmonic motion. Hence, deduce the expression for its time period .
(b) A compass needle , free to turn in a vertical orients itself with its axis vertical at a certain place on the earth . Find out the values of (i) horizontal component of earth's magnetic field and (ii) angle of dip at the place.

Explanation

(b) (i) When a compass needle, free to turn in a vertical plane, orients itself with its axis vertical at a certain place on the earth, it means that magnetic field is along vertical direction at the place. Consequently, the horizontal component of earth.s magnetic field is zero at that place.
(ii) As tan `delta = (B_V)/(B_H) and B_H` is zero, `tan delta = oo and ` so `delta = 90^@` . Thus , of dip `delta` at given place is `90^@`.