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Gauss's Law
Total Charge
Symmetry Arguments
Estimated Time: 60 minutes
Students are asked to work in groups to find the electric field using Gauss's Law for either a spherically or cylindrically symmetric charge density. Students must make explicit symmetry arguments using Proof by Contradiction as part of their solution.
Students should know how to integrate non-constant charge densities to find total charge (see
Total Charge Activity).
Students should have some knowledge of electric fields
We start this activity with a mini lecture about Gauss's Law.
Gauss's law relates the charge and electric field - specifically the electric flux through a closed surface and the charge enclosed by the surface.
Finding the electric field using Gauss's Law is an inverse problem and can only be done when there is a sufficient amount of symmetry in the charge distribution so that the field be pulled out of the flux integral.
Proof by Contradiction (“Little Observer” arguments)
In order to pull the field out of the integral, one must use the symmetry to make assumptions about the components and the dependence of the field (e.g. that there is only a radial component and it is only dependent on the radius). To justify these assumptions, we introduce the idea of proof by contradiction, where one assumes the opposite of what we want to show and demonstrates a contradiction.
The idea is to make arguments about the functional dependence and direction of the electric field by comparing the assumptions made about the electric field and what an observer placed near the charge density in various locations and facing various directions would expect to see based on how she perceives the charge distribution to change.
Example: Infinite plane of charge
It is helpful to do the explicit example of Gauss's law for a sheet of charge that extends infinitely in the $x$ and $y$ directions and has finite thickness in the $z$ direction.
Students have a lot of difficulty imagining the correct Gaussian surface in this case. They want it to extend from $z=0$ to $z=h$ rather than symmetrically from $z=-h$ to $z=h$.
In order to use Gauss' law, one needs to show that there is no component of the field tangent to the surface using a proof by contradiction:
If a person standing on a point above the surface were to close her eyes and rotate 180$^\circ$ and open her eyes, there would be nothing to indicate that she has turned. Therefore, every measurement that she takes must be the same in both orientations.
If we assume that there is a tangential component to the field that points straight out from her feet and she closes her eyes, turns, and measures again, the direction of the field would now be behind her instead of in front of her.
Since every measurement she makes must be the same in both orientations, there cannot be a tangential component to the field.
One must similarly show that the field is only dependent on the z component (e.g. by moving the little person to some other point at the same z-value to demonstrate that there is no non-z dependence).
“By symmetry” Students have seen instructors use symmetry arguments to simplify problems. As professionals, we use the words “by symmetry” as a shorthand for explaining how the symmetries of the problem lead to a simplification of the solution. However, students tend to treat this as an incantation that magically simplifies the answer without going through the argument of how (sort of like a proof by inspection). Students should be instructed that they must be able to make an explicit symmetry argument using Proof by Contradiction. It is really common for at least one student in the class to be totally resentful of having to make an explicit symmetry argument for Gauss's law since the symmetries are so obvious in this case. The Ampère's law activity often convinces them.
Component vs. Dependence Students often don't realize the distinction between a vector field having a radial component and having the magnitude depend on the radial coordinate. Students should be reminded that each of the three components of the vector field can in principle be functions of all three coordinates. Students must argue away both components and functional dependencies separately. Some students will find it easier to make this distinction in the Ampère's law activity, when the non-zero components and the functional dependencies are not with respect to the same variables.
“E=0 because q=0” Most students will conclude that, for a Gaussian surface entirely inside the shell of charge, the electric field will be zero since the charge enclosed is zero. This is an opportunity to remind them that, just because the value of a definite integral is zero does not mean that the integrand is zero. It is only the fact that the integrand is
constant and the integral is zero that allows one to make a conclusion about the value of the integrand. Drawing the graph of a sine function and asking them what the integral is can help. If you have done the
Visualizing Flux activity, remind them that the electric field for a charge located outside the box was not zero, even though the charge enclosed by the box and the total flux through the six sides of the box were both zero.
“Little Observer” orientation In order for the “little observer” arguments to work on curved surfaces, the person must be standing perpendicular to the surface. When students draw 2D representations of the surface on their boards, they will sometimes neglect the 3D nature of the surface and try to reason with the little observer standing on the whiteboard (i.e. “laying” on the surface”).
Charge Enclosed A few students may continue to struggle to relate the enclosed charge to the given charge density. When students attempt to calculate the electric field, they may think that the enclosed charge is just a given value which is constant. If you have done the
Total Charge activity, remind students that they have performed previous calculations to find the total charge in similar geometries.
This is a compare and contrast activity. Some groups do cylindrical shell examples and some do spherical shell examples. As each group reports, it is important to bring out the fact that all examples have zero field inside the shell, different answers within the shell, and answers that depend only on the total charge on the shell for points outside the shell. A nice discussion of symmetry ensues.
We were surprised one year to ask the students about the analogous calculation with gravitational fields. They had been totally unimpressed that the electric field was zero inside the shell and were total impressed that the gravitational field would be zero inside the shell. Several students exclaimed: “Wow, you'd be floating!” Gravitational fields are much more intuitive and geometrically meaningful for many students than electric fields.
This activity is the final activity of a sequence of activities on the Geometry of Flux.
Preceding activities:
Concept of Flux: A kinesthetic activity in which students use rulers to represent a vector field while the instructor uses a hula hoop to represent a surface which is followed by a class discussion on what contributes to the flux through a surface. There is also a
narrative of this activity that describes it in detail for a specific class.
Calculating Flux: A small group activity in which students calculate the flux of a simple but non-constant vector field through a cone.
Visualizing Flux: A Mathematica (or Maple) activity which allows students to work in small groups to explore the flux of the electric field due to a point charge through the surface of a cube, a Gaussian surface.
This activity is included within a sequence of activities addressing Ampere’s law. The following activities are additional activities which are included within this sequence.
Follow-up activities:
Acting Out Current Density: This kinesthetic activity prompts many discussions about the definitions of various current densities by having students each represent a charge and move as a class to demonstrate various current densities.
Ampere's Law: A small group activity which uses similar skills as those developed in the
Gauss's Law activity by extending symmetry arguments using
Proof by Contradiction in Ampere's law with highly symmetric current densities.
This activity is included within a sequence of activities addressing Gauss’s law in integral form. The following activities are part of this sequence and can be used as preparation for this activity.
Preceding activities:
Acting Out Charge Densities: A kinesthetic activity in which students act as individual charges and move about the classroom to demonstrate linear, surface, and volume charge distributions.
Total Charge: In this small group activity, students calculate the total charge in spherical or cylindrical dielectric shells from charge densities which vary in space.
The Geometry of Flux Sequence: This sequence of activities addresses the geometry of flux as well as allowing students ample practice in the mathematics which is used to calculate flux through various surfaces.