The electric potential φ(x,y,z) generated by a electric charge distribution ρ(x,y,z) satisfies the equation
- – Δφ = ρ in a domain Ω, with
- φ = g given on the boundary of Ω,
with Δ the Laplacian. Same equation is valid for the gravitational potential generated by a mass distribution. See this 2d example with the potential = 0 on the boundary of a disc and 1 on a surrounding square and no charge in the domain between. See flow of (test) charges given by the force F = -∇φ = -(dφ/dx, dφ/dy). Motivate model. Change domain, charge and boundary conditions.