Derive an expression for magnetic potential energy .for a magnetic dipole kept in a uniform magnetic field.

Explanation

As shown in figure a bar magnet is held at an angle `theta` with the direction of a uniform magnetic field `overset(to) (B)`.

The potential energy of this bar magnet is indicated by `U_(m)`,
`U_(m) = int tau (theta) d theta`
`= int m B sin theta d theta`
`= - m B cos theta`
`U= - overset(to) (m). overset(to)(B)...(1)`
Special cases :
(1) If bar magnet makes an angle `0^@` with magnetic field `overset(to) (B)`.
U = - mB which is the minimum value of potential energy. The magnet is in most stable position.
(2) If bar magnet makes an angle 180° with magnetic field `overset(to) (B)`.
U = + mB which is the maximum value of potential energy. The magnet is in most unstable position.
(3) If bar magnet is perpendicular to `overset(to)(B)`,
`U= m B cos 90^@`
`U=0`