Note that in this equation, epsilon

is the Dielectric Constant at the medium and Rho is the local charge density.

Now, there's another important theorem in electrostatics which relates

the divergence of a quantity, and it's known as the Divergence Theorem.

Now, this theorem converts a surface integral into a volume integral

through the emergence of a quantity called a divergence.

Now, this leads to an important relation for

the case that we are considering, that the electrostatic potential

must satisfy a well-known equation known as the Poisson Equation.

The Poisson Equation simply states that the Laplacian

of the Electrostatic Potential can be related to the local charge density rho

divided by the dielectric constant of the medium.

Now, let's consider a simple example.

A sphere with radius a, which has a total charge q now, due to spherical

symmetry of the problem, the electrostatic potential around the sphere is radium.

That is, Psi is simply a function of r.

Now, let's write down the Poisson Equation outside the sphere.