Gauss's law

Gauss's law combines the concepts of symmetry and flux to provide a method to relate electic field to configurations of charge.

Three fundamental symmetries we will deal with concerning charge configurations are planar, cylindrical and spherical symmetry.

Flux typically denotes the flow of something. Electric flux is rather abstract, it helps to relate it to a more common kind of flux. For example, a good analogy is water flowing from a faucet with a screen. The flux of water is just the amount of water flowing through the screen. Electric flux is the amount of electric field passing through a surface. There is nothing moving, like with the water analogy, but the field is passing through the surface, just as the water is passing through the screen.

To find the amount of electric flux through a surface, we need to define an area vector A as perpendicular to the surface. The magnitude of A is equal to the magnitude of the surface area.

The electric flux through a surface is defined as the dot product of the electric field vector with the area vector. The magnitude of A is equal to the magnitude of the surface area.

We saw that the electric field vectors of a point charge go as 1/r² and radiate outward from a point charge. E field vectors of an infinite line of charge go as 1/r and radiate away from the line, and field vectors from an infinite plane of charge are perpendicular to the surface while the magnitude of the field is constant, with no dependance on r. The configuration of the field vectors reflects the symmetry of the charge distribution.

Gaussian surfaces

A Gaussian surface is a mathematical construct. It is a closed surface that encompasses a space. The area vectors are defined as pointing away from the outer surface. The lines of electric flux passing through a Gaussian surface indicate the charge enclosed by the surface.

Consider the Gaussian surface indicated by the green cube, as it relates to the dipole charges shown.

In this case, the charge enclosed by the Gaussian surface is positive, since the negative charge resides outside the cube.

In this case, the charge enclosed by the Gaussian surface is zero, since both the positive and negative charges reside inside the cube.

Gauss's law

Gauss's law provides a method for calculating the charge contained in a volume by analyzing the electric field fluxing through the surfaces of the volume. It states that the charge enclosed in a Gaussian volume determines the electric flux through the total surface.

This integral is done over a closed surface, the entire Gaussian surface.

As an illustration, we can calculate the electric flux of a single point charge enclosed by a Gaussian surface. We have taken advantage of the spherical symmetry to draw a Gaussian surface where the tangent planes to the surface are all parallel to the electric field lines. We let the areas of the tangent planes go to zero for our integral.

Since the electric field of a point charge is spherically symmetric, the value of E is the same at every point on the Gaussian surface. This allows us to take the electric field term out of the integral, as a constant. The closed surface integral is then just the surface area of the sphere.

The electric field falls off as the square of the distance and the surface area increases as the square of the radius, so the spatial terms cancel.

It is important to note that Gauss's law does not just apply for cases where the electric field is constant at the Gaussian surface. If the electric field varies over space, the electric field function can be kept inside the integral for evaluation of the flux.



Gauss's law is powerful. It basically says if you know the total electric flux through a closed surface, you know the charge enclosed. If you know the charge enclosed, you know the total flux. If the surface area of the Gaussian sphere increases, the strength of the electric field is smaller, but the surface area is larger by the same amount.



What about the electric field of a charge outside a Gaussian surface? Isn't there flux inside the closed surface? Yes, but the flux going into the surface equals the flux going out, so the net flux is zero.

Practice problems

1. A rectangular surface measures 2.0 m x 4.0 m and lies in the x-y plane. Find the electric flux (in Nm²/C) through the surface if the E field is given by

2. A point charge of 20 nC is at the center of a cube that measures 3.0 m on each edge. What is the electric flux (in Nm²/C) through the top face of the cube?

3. Which rectangular Gaussian surface has the greatest total flux?