Force-free magnetic field

A force-free magnetic field obeys $(\nabla \times \mathbf{B})\times \mathbf{B}=0$, so its electric current is locally parallel to the magnetic field.

The general ideal magnetic-energy balance has outward-surface contribution ${\mu }_0^{-1}\oint [(u\cdot B)B-B^{2}u]\cdot dS$ and volume contribution $-\int u\cdot (J\times B)dV$. The latter vanishes for a force-free magnetic field. For an axisymmetric boundary rotating tangentially, $u\cdot B=\Omega f(\psi )$ and the directed flux of a thin tube is $2\pi d\psi$. Pair its exit and entry endpoints to obtain the displayed energy injection, counting each boundary endpoint once. Equal endpoint angular velocities inject no magnetic energy.

Writing $\nabla \times \mathbf{B}=(4\pi /c)\alpha \mathbf{B}$ defines the force-free parameter $\alpha$. Since the divergence of a curl vanishes and $\nabla \cdot \mathbf{B}=0$, it satisfies $\mathbf{B}\cdot \nabla \alpha =0$.

A cylindrical force-free magnetic field depends only on cylindrical radius and has current everywhere parallel to its magnetic field. Regularity on the axis often forces its radial component to vanish.