Torque Acting on a Bar Magnet in Uniform Magnetic Field

Place a bar magnet of pole strength $q_m$ and length $2l$ in a uniform field $\vec B$, tilted at angle $\theta$ to the field. Each pole feels a force of magnitude $q_mB$, but the two forces ($\vec F_N = q_m\vec B$ at the north pole, $\vec F_S = -q_m\vec B$ at the south pole) are equal and opposite, so the net force is zero -- a uniform field never translates a dipole, it only rotates it. Being equal, opposite, and not collinear, these two forces form a couple. Taking moments about the centre $O$, with each pole a perpendicular distance $l\sin\theta$ from the line of action of the other pole's force, the magnitude of the net torque is

using $p_m = q_m(2l)$. In vector form,

$\boxed{\vec\tau = \vec p_m \times \vec B}$ …