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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.

In a magnetostatic equilibrium, inertia is absent and all forces balance. A force-free magnetic field is the limiting case in which the Lorentz force density vanishes:Thus the current density is parallel to the magnetic field, so for some scalar force-free parameter $\alpha (\mathbf{r})$,Using Ampère's law in magnetostatics givesTaking the divergence and using both the divergence of a curl is zero and $\nabla \cdot \mathbf{B}=0$ yieldssoThe force-free parameter is therefore constant along every magnetic field line.