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The Maxwell's equations imply that the electric potential \(\phi\) obeys Poisson equation \(\nabla^2 \phi = -\rho/\epsilon_0,\) where \(\rho\) is the charged density.
The Maxwell's equations for electrostatics are
\begin{eqnarray}\label{EQ01} \nabla \cdot \vec E &=& \frac{\rho}{\epsilon_0},
\\ \label{EQ02} \nabla \times \vec E &=& 0. \end{eqnarray}The second Maxwell's equation implies the existence of potential \(\phi\), such that \(\vec E =-\nabla \phi.\) substituting this expression for \(\vec E\) in \eqref{EQ01}, we see that the potential obeys the Poisson equation
\begin{equation} \nabla^2 \phi = - \frac{\rho}{\epsilon_0}. \end{equation}
In regions of space where the charge density vanishes, the potential satisfied the Laplace equation \begin{equation} \nabla^2 \phi = 0. \end{equation} In presence of conductors, the solution for potential can be obtained if the charge density \(\rho\) and boundary conditions are given. There are two kinds of boundary conditions on conducting surfaces. Either the potential of a conductor, or the total charge needs to be specified.
