Maxwell's equations are a system of coupled differential equations for the electric field $\EE$ and the magnetic field $\BB$. In traditional language, they take the form \begin{align} \grad\cdot\EE &= 4\pi\rho \\ \grad\cdot\BB &= 0 \\ \grad\times\EE + \dot\BB &= 0 \\ \grad\times\BB - \dot\EE &= 4\pi\JJ \end{align} where $\rho$ is the charge density, $\JJ$ is the current density, and dots denote time derivatives. Taking the divergence of the last equation, and using the first, leads to the continuity equation \begin{equation} \grad\cdot\JJ + \dot\rho = 0 \end{equation} and making the Ansatz \begin{align} \BB &= \grad\times\AA \\ \EE &= -\grad\Phi - \dot\AA \end{align} automatically solves the middle two (source-free) equations.