If there is a static spherically symmetric Gaussian charge density ρ ρ f ( r ) = Q σ σ 3 2 π π 3 e − − r 2 / ( 2 σ σ 2 ) , where Q is the total charge, then the solution φ (r) of Poisson's equation ∇ ∇ 2 φ φ = − − ρ ρ f ε ε is given by φ φ ( r ) = 1 4 π π ε ε Q r erf ⁡ ⁡ ( r 2 σ σ ) , where erf(x) is the error function.