Analytical Solutions

Analytical expressions for the electric field or, equivalently, the electric potential exist only for simple systems, especially those with high symmetry. For example, the electric field generated by a thin charged plate with surface charge density $\sigma$ is $2 \pi \sigma$ (in vacuum), with direction normal and outward from the plate for positive charge density $\sigma$, i.e. a positive test charge is driven away from the plate with a force independent of distance from the plate. For a charged cylinder with linear charge density $\lambda$, the electric field outside the cylinder points radially outward for positive $\lambda$ and falls off as $2\lambda/R$ with radial distance from the center of the cylinder R. An insightful discussion of elementary problems in electrostatics may be found in the Feynman lectures [7].

As the symmetry of the system decreases, the difficulty of analytically solving the differential equation which corresponds to Gauss' Law increases rapidly. One must then resort to numerical methods. In principle such calculations are straightforward. For example, in vacuum, the electric potential generated by a point charge q is q/r, where r is the distance from the charge, and the total electrical potential anywhere in space is the sum of the potentials generated by each point charge. However, because the potentials are long ranged and diverge at the locations of the point charges, numerical evaluation using this approach may be poorly convergent. Solution of the Poisson Eq., (6), which can be done using lattice methods described below, often provides a preferred route.