Hence, (3.4)

becomes

where is the charge density. (3.9) is Poisson's equation. However, when there are no charges so that , this reduces to Laplace's equation

- We will simply state the general solution to (3.9).
Firstly, remember that the potential for a point charge is

If there are more charges we simply add the potentials together. Thus, it is not too surprising that when there is a continuous distribution instead of adding we integrate over the volume containing the charge. Thus,

where and . Thus,

- The flux of across a closed surface containing charges is

by the divergence theorem, and using (3.4) we get

Hence, we have that the flux of the electric field across the surface is

- If and , (3.9) becomes in
spherical coordinates,

Integrating once we obtain

where is a constant. Integrating a second time gives the electrostatic potential as

and the radial electric field component is

or substituting for from (3.13) we get

Thus,

in agreement with (3.6).