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