Torque on a Magnetic Dipole in a Uniform Magnetic Field

No net force, but a net torque. A bar magnet placed in a UNIFORM external magnetic field $\vec{B}$ experiences a force $q_mB$ on its N pole (in the direction of $\vec{B}$) and an equal and opposite force $q_mB$ on its S pole (opposite to $\vec{B}$) -- so the NET FORCE on the magnet as a whole is exactly zero, and the magnet does not accelerate bodily in any direction. These two equal, opposite forces, however, act at the two ENDS of the magnet rather than along one common line, forming a couple that produces a net turning effect -- a torque -- even though the net force vanishes.

Deriving the torque. Suppose the dipole moment $\vec{m}$ makes angle $\theta$ with $\vec{B}$. The perpendicular distance between the parallel lines of action of the two forces is $2l\sin\theta$ (the component of the pole separation perpendicular to $\vec{B}$). The torque produced by a couple equals (force) $\times$ (perpendicular distance between the forces):

using $m=q_m(2l)$ from Section 1.4. In full vector form, since the torque acts perpendicular to the plane containing both $\vec{m}$ and $\vec{B}$,

exactly the magnetic counterpart of the electric-dipole torque $\vec{\tau}=\vec{p}\times\vec{E}$.

Potential energy. The torque always acts to rotate $\vec{m}$ toward alignment with $\vec{B}$, so work must be done AGAINST the torque to increase $\theta$; this work is stored as potential energy. Taking the (arbitrary but conventional) reference $U=0$ at $\theta=90^\circ$, integrating the torque from $90^\circ$ to $\theta$ gives

$U(\theta) = -mB\cos\theta = -\vec{m}\cdot\vec{B}$ …