Derive Poisson's and Laplace's equations

Derivation of Poisson's equation

Start with three basic relations of electrostatics:

1. Gauss's law, point form: ∇ · D = ρ_v
2. Constitutive relation: D = εE
3. Relation between field and potential: E = −∇V

Substituting (2) and (3) into (1):

∇ · (ε E) = ρ_v
∇ · (−ε ∇V) = ρ_v

For a homogeneous medium (ε constant):

∇ · ∇V = ∇²V = −ρ_v / ε            ← Poisson's equation

Laplace's equation

In a region with no free charge, ρ_v = 0, and Poisson's equation reduces to

∇²V = 0                              ← Laplace's equation

Forms of ∇²V

Cartesian:     ∇²V = ∂²V/∂x² + ∂²V/∂y² + ∂²V/∂z²

Cylindrical:   ∇²V = (1/ρ) ∂/∂ρ (ρ ∂V/∂ρ) + (1/ρ²) ∂²V/∂φ² + ∂²V/∂z²

Spherical:     ∇²V = (1/r²) ∂/∂r (r² ∂V/∂r) + (1/(r² sinθ)) ∂/∂θ (sinθ ∂V/∂θ)
                     + (1/(r² sin²θ)) ∂²V/∂φ²

Poisson's and Laplace's equations are used to find the potential in a region when boundary conditions (or charge distribution) are known, for example in capacitors.