[NOTES/EM-07018] Poisson Equation for Vector Potential.

Substituting \(\vec{B}=\nabla \times \vec{A}\) in Maxwell's equation  \begin{equation} \nabla \times \vec{B} = \mu_0 \vec{j},
\end{equation} gives \begin{eqnarray} \nabla\times(\nabla\times\vec{A})=\mu_0\vec{j}\\  \nabla(\nabla\cdot\vec{A})- \nabla^2\vec{A}= \mu_0\vec{j}. \end{eqnarray} We shall assume, the Coulomb gauge, \(\nabla\cdot\vec{A}=0\). Thus we get  \begin{equation} \nabla^2\vec{A}=-\mu_0\vec{j}. \end{equation} These are three equations, one for each component of \(\vec{A}\) \begin{equation} \nabla^2A_x=-\mu_0 j_x, \quad \nabla^2A_y=-\mu_0 j_y, \quad \nabla^2A_z=-\mu_0 j_z.\end{equation}