"Poisson's equation is a partial differential equation of elliptic type with broad utility in electrostatics, mechanical engineering and theoretical physics. Commonly used to model diffusion, it is named after the French mathematician, geometer, and physicist Siméon Denis Poisson."
The equation is of the form: ∇²ρ = ƒ
"One of the cornerstones of electrostatics is setting up and solving problems described by the Poisson equation. Finding ρ for some given ƒ is an important practical problem, since this is the usual way to find the electric potential for a given charge distribution described by the density function."
Poisson's equation for electrostatics is derived as follows.
Step One: Gauss's Law
Start with Gauss's Law (aka Gauss's Flux Theorem for Electricity), one of Maxwell's Equations. Gauss's Law states that: "The net electric flux through any closed surface is equal to 1⁄ε times the net electric charge enclosed within that closed surface."