Recall the use of the superposition principle to find the potential due to any charge distribution, starting from the electric potential due to a point charge, namely \begin{eqnarray*} V(\rr) = {1\over 4\pi\epsilon_0} {q\over|\rr-\rr'|} \end{eqnarray*} resulting in \begin{eqnarray*} V(\rr) = {1\over 4\pi\epsilon_0} \int {\rho(\rrp)\,d\tau'\over|\rr-\rrp|} \end{eqnarray*}

There is a similar expression starting with current density $\JJ$ instead of charge density $\rho$, namely \begin{eqnarray*} \AA(\rr) = {\mu_0\over 4\pi} \int {\JJ(\rrp)\,d\tau'\over|\rr-\rrp|} = {\mu_0\over 4\pi} \int {\rho(\rrp)\,\vv\,d\tau'\over|\rr-\rrp|} \end{eqnarray*} We call $\AA$ the (magnetic) vector potential. The constant $\mu_0$ is called the permeability of free space, and plays a similar role for magnetic fields as does $\epsilon_0$, the permittivity of free space, for electric fields.

Similar expressions hold for other current distributions. For a surface current, we have \begin{eqnarray*} \AA(\rr) = {\mu_0\over 4\pi} \int {\sigma(\rrp)\,\vv\,dA'\over|\rr-\rrp|} = {\mu_0\over 4\pi} \int {\KK(\rrp)\,dA'\over|\rr-\rrp|} \end{eqnarray*} and for a linear distribution we have \begin{eqnarray*} \AA(\rr) = {\mu_0\over 4\pi} \int {\lambda(\rrp)\,\vv\,ds'\over|\rr-\rrp|} = {\mu_0\over 4\pi} \int {\II(\rrp)\,ds'\over|\rr-\rrp|} \end{eqnarray*}