Electrodynamics/Laplace's Equation

Laplace's Equation

If we use the Laplacian operator on the electric potential function, we get a special equation called Laplace's Equation.

${\mathrm{\nabla }}^2\phi =\frac{{\partial }^2\phi }{\partial x^2}+\frac{{\partial }^2\phi }{\partial y^2}+\frac{{\partial }^2\phi }{\partial z^2}=0$

This equation is a second-order partial differential equation. Solutions to Laplace's Equation represent the shape of the electric field.