# 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 ε_{0} 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

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