Chapter 24: Gauss's Law

Learning Goals

1. Understand the concept of Electric Flux.
2. Understand the formal definition of Gauss's Law
3. Apply Gauss's law, and understand how to utilize symmetry.
4. Understand how to use Gauss's law to determine important facts about conductors.

Gauss's Law

The fact that we call this a law means that it is very fundamental. In fact, it has the same relationship to electromagnetic theory as Newton’s laws do to mechanics: it is one of Maxwell's equations.

In words, Gauss’s law states that the total outward electric flux through a closed surface is proportional to the net charge enclosed by the surface. In symbols:

Applications of Gauss’s Law

Although it is a powerful theoretical tool, Gauss’s law is difficult to solve for all but the most symmetric situations. When the charge distribution is very symmetric, Gauss’s law becomes easy to apply. The steps to follow when solving Gauss’s law for the electric field due to a symmetric charge distribution are:

1. Choose a Gaussian surface with the same symmetry as your charge distribution.
2. Evaluate the scalar product and field magnitude on your surface and pull out any constants within the integrand.
3. It should now be the case that the surface integral evaluates to the area of a simple geometric surface.
4. Evaluate the q_in and solve for the explicit form of the field magnitude.

Conductors

A conductor with excess charge carriers all at rest (no currents) is said to be in electrostatic equilibrium. Conductors in electrostatic equilibrium have the following properties (which you should be able to demonstrate):

• The electric field inside the conductor is zero everywhere, regardless of any excess or nearby electric charge.
• Any excess charge placed on the conductor resides entirely on its surface.
• The electric field just outside the conductor’s surface is perpendicular to that surface with magnitude equal to s/e₀ , where s is the charge per unit area at that point.
• On an irregularly shaped conductor, charge tends to accumulate at sharp points and move away from valleys.