information:extraction
Electrostatics

According to Maxwell's equations, an electric field (u, v) in two space dimensions that is independent of time satisfies

and

where ρ is the charge density. The first Maxwell equation is the integrability condition for the differential

so the electric potential φ may be constructed to satisfy

The second of Maxwell's equations then implies that

which is the Poisson equation. The Laplace equation can be used in three-dimensional problems in electrostatics and fluid flow just as in two dimensions.