Magnetic vector potential

In magnetostatics, the magnetic vector potential ($\vec{A}$) is defined as:

$$\vec{B}=\nabla \times \vec{A}$$

A key property of the vector potential is that its divergence is zero at infinity. This is motivated by a few key assumptions, mainly that our definition above only defines the curl.

$$\nabla \cdot \vec{A}=0$$

With this property, Ampère’s law becomes a variant of Poisson’s equation:

$${\nabla }^2\vec{A}=-{\mu }_0\vec{J}$$