# 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 or

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.

### Electrostatics

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 ### Gravity

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