Electrostatics: Gauss's Law & Superposition Principle

Gauss’s Law is a useful tool for solving electrostatics problems. Instead of computing hard nasty integrals, we can use Gauss’s Law to simplify the computation for nice symmetric cases.

Gaussian Surfaces

To start, we’ll define a Gaussian surface. A Gaussian surface is a CLOSED surface that contains some electric charge. For instance, below we have an electron represented by the minus sign and a Gaussian surface represented by dotted lines.

The Gaussian surface isn’t necessarily geometrically constricted; however, choosing highly symmetric surfaces helps with problem solving. If the problem scenario is the same from all angles (i.e. no matter which angle you look at the problem, it is the same) spherical coordinates and spherical Gaussian surfaces can help reduce problem complexity. If the problem scenario has circular symmetry along an axis, cylindrical coordinates and surfaces can help.

Superposition Principle

The superposition principle claims that, in a linear system, the net response of two or more stimuli is the sum of the responses from those stimuli individually.

Example 1: (Warmup)

A solid insulating sphere with charge +q and radius a is surrounded by a conducting shell of unknown charge and radius b where b > a. The electric field outside the shell is 0. Find the electric field inside the shell and the charge density of the shell. Assume the +q charge is uniformly distributed on the volume of the sphere.

First, we’ll start with the easier part of the problem. We can tell that the problem has spherical symmetry (they are uniform spheres after all). Therefore, we can draw a spherical Gaussian surface of radius r > b .

We also know the electric field outside the shell is 0 as given by the problem statement (how convenient…). Therefore, the integral becomes: