24.11: Poisson's And Laplace's Equation
The electric potential of the system can be calculated by relating it to the electric charge densities that give rise to the electric potential. The differential form of Gauss's law expresses the electric field's divergence in terms of the electric charge density.
On the other hand, the electric field is expressed as the gradient of the electric potential.
Combining the above two equations, an electric potential is expressed in terms of the electric charge density.
This equation is called Poisson's equation. When the electric charge density is zero, that is, there is no charge present in the given volume, then the above equation can be reduced to Laplace's equation.
The divergence of the gradient of a function, which is a mathematical operation, is called the Laplacian. Depending upon the symmetry of the system, the Laplacian operator in different symmetries can be used. For example, the Laplacian in spherical polar coordinates can be used for a charge distribution with spherical symmetry.
The potential is a scalar quantity, and, thus, calculating the potential from electric charge densities is easier. Additionally, once the potential of the system is known, then the corresponding electric fields can be estimated by using the gradient of the potential.