Poisson's equation

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The Poisson equation , named after the French mathematician and physicist Siméon Denis Poisson , is an elliptical partial differential equation of the second order that is used as part of boundary value problems in large parts of physics.

Mathematical formulation

The Poisson equation is general

Here designated

  • the Laplace operator
  • the solution you are looking for
  • a function. If the equation becomes Laplace's equation .

To solve Poisson's equation, additional information must be given, e.g. B. in the form of a Dirichlet boundary condition :

with open and restricted.

In this case, we construct a solution using the fundamental solution of Laplace's equation:

It denotes the area of ​​the unit sphere in the -dimensional Euclidean space .

The convolution gives a solution to Poisson's equation.

In order to meet the boundary condition can be the Green's function use

is a correction function that

Fulfills. It is generally dependent on and is easy to find only for simple areas.

If one knows , then a solution of the boundary value problem from above is given by

where the surface dimension denotes.

The solution can also be found with the help of the platform method or a variation approach .

Applications in physics

The Poisson equation, for example, is satisfied by the electrostatic potential and the gravitational potential , each with symbols . The function is proportional to the electrical charge density or to the mass density

If it is known everywhere, the general solution of Poisson's equation, which approaches zero for large distances, is the integral


In words: every charge at the location in the small area of ​​size contributes additively to the potential at the location with its electrostatic or gravitational potential:


Since the electrostatic field is a conservative field , it can be expressed in terms of the gradient of a potential :

Applying the divergence results in

with the Laplace operator .

According to the first Maxwell equation , however, also applies


  • the charge density
  • the permittivity .

It follows for the Poisson equation of the electric field

The special case for each location in the area under consideration is called the Laplace equation of electrostatics .

Electrodynamics of stationary currents

The emitter of a silicon solar cell , which can be described as purely two-dimensional to a good approximation, is considered here as an example . The emitter is located in the xy-plane, the z-axis points into the base. The lateral surface current density in the emitter depends on the z-component of the (volume) current density of the base occurring at the emitter , which is indicated by the continuity equation in the form

can be described (with the two-dimensional Nabla operator ). The surface current density depends on the local Ohm's law with the lateral electric field in the emitter together: ; here is the specific sheet resistance of the emitter assumed to be homogeneous . If you write (as discussed in the section on electrostatics) the electric field as a gradient of the electric potential, you get a Poisson equation for the potential distribution in the emitter in the form


Just like the electrostatic field


the gravitational field  g is also a conservative field:


It is

  • G is the gravitational constant
  • the mass density.

Since only the charges are replaced by masses and by , analogously to the first Maxwell equation applies


This results in the Poisson equation for gravity



Individual evidence

  1. Wolfgang Nolting: Basic course in theoretical physics . [Online excl. the] 8th [dr.]. 3. Electrodynamics. Springer, Berlin, ISBN 978-3-540-71252-7 .