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

Answer 1

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/ε0

From an identity of vector calculus we get the following:

∇2φ = -ρ/ε0, which is Poisson's equation with f = -ρ/ε0.