Poisson's and Laplace's equations

Advanced

Differential equations for potential.

Key relation∇²V = −ρ/ε₀

∇²V = −ρ/ε₀ (Poisson) relates potential to charge density; with ρ = 0 it is Laplace's equation. Solutions are unique given boundary conditions.

Worked example

Used in numerical field calculations.

Common mistake

Potential has no local maxima in empty space.

In more detail

Poisson's equation relates the electric potential to the charge density, ∇²V = −ρ/ε0. Laplace's equation is the special case with no charge, ∇²V = 0.

Solving electrostatic problems numerically in engineering software.
Modelling electric fields in semiconductors and plasmas.
Engineers use Poisson's equation to compute the potential in complicated geometries such as semiconductor devices.

Why are these equations important?

They determine the potential everywhere given boundary conditions.

When is Laplace's equation used?

In regions without charge, such as between electrodes.

What do the two equations mean?

Poisson's equation relates the potential to the charge density, and Laplace's equation is the special case with no charge.