Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 314 / 1 / a / Solution

A magnetostatic force-free magnetic field is in equilibrium when its electric current exerts no Lorentz force density:Thus $\mathbf{J}$ is pointwise parallel to $\mathbf{B}$, so for some scalar $\alpha (\mathbf{r})$,Using the magnetostatic Ampère's law, $\nabla \times \mathbf{B}=(4\pi /c)\mathbf{J}$, givesTaking the divergence and using $\nabla \cdot \mathbf{B}=0$ yieldsThe force-free parameter is therefore constant along each magnetic field line.