A small magnetic needle is set into oscillations in a magnetic field B obtain an expression for the time period of oscillation.

Explanation

Expression for time period of oscillation :
(1) A small compass needle (magnetic dipole) of known magnetic moment m and moment of Intertia I is placing in uniform magnetic field B and allowing it to oscillate in the magnetic field.
(2) This arrangement is shown in Figure.
(3) The torque on the needle is `tau = m xx B`.
(4) In magnitude `tau = mB sin theta`.
Here `tau` is restoring torque and `theta` is the angle between m and B.
(5) Therefore, in equilibrium `I (d^(2) theta)/(d t^(2)) = -mB sin theta`. Negative sign with `mB sin theta` implies that restoring torque is in opposition to deflecting torque.

(6) For small values of `theta` in radians, we approximate `sin theta ~~ theta` and get `I (d^(2) theta)/(d t^(2)) - m B theta`
`(d^(2) theta)/(d t^(2)) ~~ (-mB)/(I)theta` - (I)
(7) From defination of simple harmonic motion, we have `(d^(2) theta)/(d t^(2)) = - omega^(2) theta` - (II)
From equation (I) and (II), we get `rArr omega^(2) = (mB)/(I)`
`:. omega = sqrt((mB)/(I))`
(8) Therefore, the time period is `T = (2pi)/(omega) = 2pisqrt((I)/(mB))`