6.3 Applying gauss’s law

Gauss’s law is very helpful in determining expressions for the electric field, even though the law is not directly about the electric field; it is about the electric flux. It turns out that in situations that have certain symmetries (spherical, cylindrical, or planar) in the charge distribution, we can deduce the electric field based on knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has constant magnitude. Furthermore, if $\overrightarrow{\text{E}}$ is parallel to $\widehat{\text{n}}$ everywhere on the surface, then $\overrightarrow{\text{E}}·\widehat{\text{n}}=E.$ (If $\overrightarrow{\text{E}}$ and $\widehat{\text{n}}$ are antiparallel everywhere on the surface, then $\overrightarrow{\text{E}}·\widehat{\text{n}}=\text{−}E.$ ) Gauss’s law then simplifies to

where A is the area of the surface. Note that these symmetries lead to the transformation of the flux integral into a product of the magnitude of the electric field and an appropriate area. When you use this flux in the expression for Gauss’s law, you obtain an algebraic equation that you can solve for the magnitude of the electric field, which looks like

The direction of the electric field at the field point P is obtained from the symmetry of the charge distribution and the type of charge in the distribution. Therefore, Gauss’s law can be used to determine $\overrightarrow{\text{E}}.$ Here is a summary of the steps we will follow:

Basically, there are only three types of symmetry that allow Gauss’s law to be used to deduce the electric field. They are

• A charge distribution with spherical symmetry
• A charge distribution with cylindrical symmetry
• A charge distribution with planar symmetry

To exploit the symmetry, we perform the calculations in appropriate coordinate systems and use the right kind of Gaussian surface for that symmetry, applying the remaining four steps.