24.11:
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.
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.
Suggested Reading
- Ellingson, Steven W. (2018) Electromagnetics, Vol. 1. Blacksburg, VA: VT Publishing. https://doi.org/10.21061/electromagnetics-vol-1 CC BY-SA 4.0. Section 5.15, pages 115-116