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:
- 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
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.
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