Forces and Torques
The total force acting on a magnetic body with magnetisation \(\mathbf{M}\) due to an external field \(\mathbf{B'}\) can be divided into the sum of a volume and surface integral:
\[
\mathbf{F} = -\int_V (\nabla \cdot \mathbf{M}) \mathbf{B'} \: dv + \int_S (\mathbf{M} \cdot \mathbf{\hat{n}}) \mathbf{B'} \: ds
\]
which can be rewritten as
\[
\mathbf{F} = \int_V \rho_m \mathbf{B'} \: dv + \int_S \sigma_m \mathbf{B'} \: ds
\]
where \(\rho_m = -\nabla \cdot \mathbf{M}\) and \(\sigma_m = \mathbf{M} \cdot \mathbf{\hat{n}}\) are the volume and surface charge densities.
For a uniformly magnetised body, \(\rho_m = -\nabla \cdot \mathbf{M} = 0\), thus the total magnetic force acting on a magnet due to an external field \(\mathbf{B'}\) can be deduced using the surface charge alone:
\[
\mathbf{F} = \int_S \sigma_m \mathbf{B'} \: ds
\]
Similarly, the torque on a uniformly magnetised body can be determined using
\[
\mathbf{T} = \int_S \sigma_m (\mathbf{r} \times \mathbf{B'}) \: ds
\]
where \(\mathbf{r}\) is the vector from the point about which the torque is computed, usually the centre of mass.