23.4: Gauss's Law: Problem-Solving
Gauss's law helps determine electric fields even though the law is not directly about electric fields but electric flux. In situations with certain symmetries (spherical, cylindrical, or planar) in the charge distribution, the electric field can be deduced based on the knowledge of the electric flux. In these systems, we can find a Gaussian surface S over which the electric field has a constant magnitude. Furthermore, suppose the electric field is parallel (or antiparallel) to the area vector everywhere on the surface. In that case, the flux integral transforms into the product of the electric field magnitude and an appropriate area. Thus, the equation representing Gauss's law simplifies to the following:
When this flux is used in the expression for Gauss's law, an algebraic equation is obtained, which can be solved to find the magnitude of the electric field.
To summarize, when applying Gauss's law to solve a problem, the following steps are followed:
- Identify the spatial symmetry of the charge distribution. This is an important first step that allows the choice of the appropriate Gaussian surface. For example, an isolated point charge has spherical symmetry, whereas an infinite line of charge has cylindrical symmetry.
- Choose a Gaussian surface with the same symmetry as the charge distribution, and identify its consequences. With this choice, the electric flux can be easily determined over the Gaussian surface.
- Evaluate the flux through the surface. The symmetry of the Gaussian surface allows for factoring the electric field outside the integral.
- Determine the amount of charge enclosed by the Gaussian surface. This is an evaluation of the right-hand side of the equation representing Gauss's law. It is often necessary to perform an integration to obtain the net enclosed charge.
- Evaluate the electric field of the charge distribution.