Derive the equation of torque on a magnetic needle in a uniform magnetic field.

Explanation

A compass needle of magnetic moment m is placed in a uniform magnetic field as shown in figure.

`NS=` magnetic length `=2l`
`q_m=` strength of each pole
Magnetic dipole moment `m= q_(m) (2l)`
According to figure in right angle `theta NDS`,
`ND=2 l sin theta`
Force on S pole `=F_S = - q_m` B
Force on N pole `=F_N = q_m B`
These equal and unlike forces form a couple which tend to rotate the needle clockwise.
{The torque on the needle} = {magnitude of either force `xx` perpendicular distance between the two forces.}
`tau = q_m B xx ND`
`tau = q_m B xx 2l sin theta`
`tau = q_m (2l) B sin theta `
`therefore tau = m B sin theta ` where `q_m (2l) =` magnetic dipole moment `m`
`therefore overset(to) (tau) = overset(to) (m) xx overset(to) (B) ( because m and B` are vectors `)`