Find an expression for the torque acting on an magnetic dipole (Bar Magnet) suspended in a uniform magnetic field.

Explanation

Consider a magnetic dipole placed in uniform magnetic field `vec(B)` such that angle between the direction of magnetic dipole moment `(vec(m))` and direction of magnetic field is `theta`. Let Magnetic length of magnet= 2I
Pole strength of each pole= `q_(m)`
Force acting on North Pole = `q_(m)B` along direction of `vec(B)`
Force acting in south pole `=q_(m)B` opposite of the direction of `vec(B)`
Torque acting on a bar magnetic or magnetic dipole is given by `tau`= Magnitude of force X Perpendicular distance between forces i..e, `tau= q_(m) B xx A N` ...(i)
in rt `angled Delta ANS`
`sin theta= (AN )/(SN)`
`AN = SN sin theta`
`AN = 2l sin theta` ...(ii)
`therefore` From (i) & (ii) `tau= q_(m) B xx 2l sin theta`
`tau = q_(m) xx 2l B sin theta`
`tau= m B sin [because m= q_(m) xx 2l]`
where m = magnetic dipole moment
In vector from `vec(tau)= vec(m) xx vec(B)`