Magnetic dipole potential energy ($U=-\mathbf{m}\cdot \mathbf{B}$)

In an external field that varies slowly across a localized magnetic dipole,

The magnetic dipole moment determines the coupling to the external magnetic field.

For a small fixed magnetic dipole moment $\mathbf{m}$ in a slowly varying external magnetic field, $U=-\mathbf{m}\cdot \mathbf{B}$ gives the displayed force. For a compact current loop, integrate $Id\mathbf{\ell }\times \mathbf{B}$ to first order in the field's spatial variation to get the same result. For a coaxial small loop with $m_z=NI\pi a^2$, this becomes $F_z=m_{z}d{B}_z/dz$. A spatially uniform field can exert torque but gives zero net force; a field gradient is essential. The approximation assumes the field does not appreciably vary over the dipole's dimensions.

The orientation-dependent interaction of two distant localized magnetic dipole moments. Its $r^{-3}$ scale makes close approach important in a concentrated magnetic-particle dispersion. Depending on geometry, it can favor chains, collective alignment or frustrated configurations.