Magnetic scalar potential ($\mathbf{B}=-\nabla \psi$)

In a simply connected current-free region of constant magnetic permeability, Ampère's circuital law makes the magnetic field curl-free, so it admits a scalar potential. The absence of magnetic monopoles then gives Laplace's equation for that potential. Some conventions define the potential for $\mathbf{H}$ instead of $\mathbf{B}$, differing by the constant permeability. Boundary conditions, including imposed fields at infinity, determine the potential up to an additive constant.

For an isolated razor-thin disk, rescale its exterior magnetic scalar potential to satisfy $\sqrt{4\pi G/{\mu }_0}\mathbf{B}=-\nabla {\Phi }_M$. Comparing the normal magnetic boundary derivative with ${\partial }_z\Phi =2\pi G\Sigma$ identifies the displayed equivalent signed surface density. Both exterior potentials satisfy Laplace's equation. The equivalent surface density is a representation of the magnetic boundary data, rather than physical negative mass.