What does Poissons equation really mean

The Poisson's equation (after Siméon Denis Poisson) describes a boundary value problem in which the derivatives of a vector field on the surface of a volume are given. This is used, for example, in electrostatics (Gauss law). One can also determine the gravitational potential of a given mass distribution.

The Poisson equation is a partial differential equation of the second order:

or shorter


ΔΦ = f

i.e. in the Poisson's equation the Laplace operator Δ applied to a function Φ becomes equal f set.

The homogeneous form of Poisson's equation is the Laplace equation.


Since the electric field is a conservative field, it can be over the gradient of a potential be expressed with


Using another Nabla operator results in


However, according to the first Maxwell equation, the following also applies


in which the charge density and ε0 which are permittivity.

It follows that for the Poisson equation of the electric field


The gravitational acceleration results from the law of gravitation


The flow through the surface of any volume is then


in which is the normal vector. The following applies in spherical coordinates


From which follows

From one by one mass density The mass distribution described results in the total mass


So it follows


With Gauss's theorem, however, we also get for the integral


and thus


Since the shape of the volume is arbitrary, the integrands must be equal, so that

is. The gravity represents a conservative force field, so the relationship

applies. This gives the Poisson equation for gravity


the minus sign disappears.

Category: Electrostatics