The magnetic stress tensor in magnetized matter

We derive the form of the magnetic stress tensor in a completely general, stationary magnetic medium, with an arbitrary magnetization field $\vec M(\vec r)$ and free current density $\vec j(\vec r)$. We start with the magnetic force density $\vec f$ acting on a matter element, modelled as a collection of microscopic magnetic dipoles in addition to the free currents. We show that there is a unique tensor ${\bf T}$ quadratic in the magnetic flux density $\vec B(\vec r)$ and the magnetic field $\vec H(\vec r)=\vec B-4\pi\vec M$ whose divergence is $\nabla\cdot{\bf T}=\vec f$. In the limit $\vec M=0$, the well-known vacuum magnetic stress tensor is recovered. However, the general form of the tensor is asymmetric, leading to a divergent angular acceleration for matter elements of vanishing size. We argue that this is not inconsistent, because it occurs only if $\vec M$ and $\vec B$ are not parallel, in which case the macroscopic field does indeed exert a torque on each of the microscopic dipoles, so this state is only possible if there are material stresses which keep the dipoles aligned with each other and misaligned with the macroscopic field. We briefly discuss the consequences for the stability of strongly magnetized stars.