How do you derive Poisson's equation from Maxwell's equations? - Answers

Start with the differential form of Gauss's law:∇ ● E = ρ/ε0, the divergence of the electric field is equal to the total charge density divided by the permittivity of free space.Make the following substitution, assuming electrostatic charges:E = -∇φ, the electric field at a point is equal to the negative gradient of the scalar electric potential.This gives:∇ ● ∇φ = -ρ/ε0From an identity of vector calculus we get the following:∇2φ = -ρ/ε0, which is Poisson's equation with f = -ρ/ε0.