The derivation of Poisson's equation under these circumstances is straightforward. Substituting the potential gradient for the electric field, ∇ ∇ ⋅ ⋅ E = ∇ ∇ ⋅ ⋅ ( − − ∇ ∇ φ φ ) = − − ∇ ∇ 2 φ φ = ρ ρ f ε ε , directly produces Poisson's equation for electrostatics, which is ∇ ∇ 2 φ φ = − − ρ ρ f ε ε .