The net outward flux of per unit volume is given by This result is the differential form of Gauss' Law and is known as Maxwell's first equation. 1.6 Poisson's equation . If we write then Maxwell's first equation becomes Writing as this result becomes This result is known as Poisson's equation. The problem of finding the fields due to distributions of static charge reduces to solving this partial differential equation for the appropriate boundary conditions. If the charge density is zero in some region then Poisson's equation reduces to which is known as Laplace's equation.
Direct solution is often difficult.
However, if by symmetry or intuition we are able to find a solution that satisfies both Poisson's equation and the boundary conditions, then we can rest assured that this is the correct and only solution. This result is known as the uniqueness theorem . The method of images. Suppose we have a charge distribution and a certain boundary conditions. We wish to find the electrostatic potential for this problem a certain region.