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By the end of this section, you will be able to:
  • Explain what spherical, cylindrical, and planar symmetry are
  • Recognize whether or not a given system possesses one of these symmetries
  • Apply Gauss’s law to determine the electric field of a system with one of these symmetries

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 E is parallel to n ^ everywhere on the surface, then E · n ^ = E . (If E and n ^ are antiparallel everywhere on the surface, then E · n ^ = E . ) Gauss’s law then simplifies to

Φ = S E · n ^ d A = E S d A = E A = q enc ε 0 ,

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

E ~ q enc ε 0 area .

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 E . Here is a summary of the steps we will follow:

Problem-solving strategy: gauss’s law

  1. Identify the spatial symmetry of the charge distribution . This is an important first step that allows us to choose the appropriate Gaussian surface. As examples, an isolated point charge has spherical symmetry, and an infinite line of charge has cylindrical symmetry.
  2. Choose a Gaussian surface with the same symmetry as the charge distribution and identify its consequences . With this choice, E · n ^ is easily determined over the Gaussian surface.
  3. Evaluate the integral S E · n ^ d A over the Gaussian surface, that is, calculate the flux through the surface . The symmetry of the Gaussian surface allows us to factor E · n ^ outside the integral.
  4. 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.
  5. Evaluate the electric field of the charge distribution . The field may now be found using the results of steps 3 and 4.

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.

Practice Key Terms 3

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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