Poisson’s and Laplace’s equations

Poisson’s equation. Differential form of Gauss’s law states that the divergence of electric field E at any point is equal to 1/ε₀ times the charge density at that point. That is

        Δ . E = p/ ε₀

where p is the volume density of charge.

If E and V are the electric field and electric potential at any point respectively, then we have the relation
            E = - Δ V

Eq. (1) becomes,     Δ . Δ V = - p/ ε₀
or            Δ² V    = - p/ ε₀

This equation is known as Poisson’s equation. It expresses a relation between the potential V and the charge density p at any point in an electric field in space. Any static electric field must satisfy this relationship.

Laplace’s equation. In a region where there are not free charges (p = O), the posson’s equation reduces to
            Δ² V = 0

This is called Laplace’s equation. This equation applies to the particular case where all the charges are distributed on surface of conducting bodies so that the volume charge density is zero at all points. So, the electrostatic potential function in the space between the conductors and outside is found by this equation.