Poisson's And Laplace's Equation

In vector notation, the electric field is expressed as a negative gradient of electric potential. Here, the gradient of the potential always points toward the steepest decrease in potential, and its solution gives a vector—an electric field.

According to the differential form of Gauss's law, the gradient of an electric field is equal to the ratio of the enclosed volume charge density to the free space permittivity.

Substituting the gradient of potential in Gauss's law gives an expression for Poisson's equation. Here, the divergence of the gradient of the scalar potential is the Laplacian operator, resulting in a scalar function.

Laplacian is analogous to the second-order differentiation of the scalar quantities. For the electric potential it is lower when near the local minima, and higher when near the local maxima.

When the enclosed volume charge is zero, the Poisson’s equation reduces to Laplace's equation.

The Laplace’s equation has a unique solution in a given volume if the potential at the boundary of the surface is specified, given by the first uniqueness theorem.