= Magnetic dipole force
{title2=$\mathbf F=\nabla(\mathbf m\cdot\mathbf B)$}
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 $I\,d\boldsymbol\ell\times\mathbf B$ 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 $F_z=m_z\,dB_z/dz$. 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.
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