Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 314 / 4 / d / Solution

The toroidal component of $(\nabla \times \mathbf{B})\times \mathbf{B}=0$ givesBoth $\psi$ and $r{B}_{\varphi }$ are therefore constant along each poloidal field line, soThe poloidal components of the same force-free equation then reduce to the Grad-Shafranov equation for a force-free magnetic fieldThe poloidal current density isAmpère's law around an azimuthal circle gives the enclosed poloidal electric current:up to the orientation sign. Thus $F$ is the enclosed-current function in units ${\mu }_0/(2\pi )$.