80 Week 2: Continuous Charge and Gauss’s Law
like the field of a point charge and Coulomb’s Law and so on are actuallyconsequencesof Gauss’s
Law (or consistently equivalent to Gauss’s Law) rather than the other way around. So basically,
everything else we do with the electrostatic field this semester will bea “use” of Gauss’s Law.
2.3: Using Gauss’s Law to Evaluate the Electric Field
One of the first and most important applications of Gauss’s law for our current purposes will be to
easily evaluate the electric field for certain symmetric charge distributions that we’d otherwise have
to integrate over, painfully. There are preciselythreesymmetries we can manage in this way:
- point (spherical symmetry)
- infinite line (cylindrical symmetry)
- infinite plane (planar symmetry)
That’s it! No more. For charge distributions that are spherically symmetric, cylindrically sym-
metric, or planarly symmetric, we can do the flux integral in Gauss’s lawonce and for allfor the
symmetry. As we’ll see, all that remains for us to be able to easily obtain the field from algebra is
for us to evaluate the total charge inside a Gaussian surface for any given symmetric distribution.
Here’s the recipe:
a) Draw a closedGaussian Surfacethat has the symmetry of the charge distribution. The various
pieces that make up the closed surface shouldeitherbeperpendicular to the field(which should
also be constant on those pieces) orparallel to the field(which may then vary but which
produces no flux through the surface).b) Evaluate the flux through this surface. The flux integral will have exactly the same form for
every problem with each given symmetry, so we will do this once and for all for each surface
type and be done with it!c) Compute thetotal charge inside this surface. This is the only part of the solution that is
“work”, or that might be different from problem to problem. Sometimes it will be easy,
adding it up on fingers and toes. Sometimes it will be fairly easy, multiplying a constant
charge per unit volume times a volume to obtain the charge, say. At worst it will be a problem
in integration if the associated density of charge is a function of position.d) Set the (once and for all) flux integral equal to the (computed per problem) charge inside the
surface and solve for|E~|. That’s all there is to it!Now, you don’t want to bememorizingthese steps, you want to belearningthem, so please use
exactly these stepsandshow all of your work doing theminevery homework problemthat requires
using them. If you use them five or six times in a row, in slightly differentcontexts, it will get quite
easy! At the very least, even if you get a problem where you can’t “do” (say) an integral to find
the charge inside a given surface, you’ll get most of the credit for laying out the precisely correct
method except for an integral you can’t quite do.
Note Well:Youcannotuse Gauss’s Law to e.g. evaluate the field of a ring of charge, or a disk
over charge, or a line segment of charge or any other continuous distribution that does not have the
symmetry of sphere, infinite cylinder, or infinite plane. Sorry, that’s just the way it is. It isn’t that
it isn’t true for these distributions, it is that we cannot compute theflux integral. Let’s do some
examples, at least one for each symmetry.