• To practice finding a magnetic field from a current density
• To practice finding total current from a current density
• To make explicit symmetric arguments in applying Ampère's Law

Estimated Time: 45 minutes

To find the total current and the magnetic field due to radially varying current densities in infinitely-long cylindrical shells.

• Making symmetry arguments (see Gauss' Law activity)
• Current densities (see Current Density activity)
• Vector line integrals
• Algebraic introduction to Ampère's Law

• Tabletop Whiteboard with markers
• A Model Observer for each table
• Cylinder for each table
• A handout for each student

We start this activity with a lecture that covers:

• Introduction to algebraic form of Ampère's Law, total current
• Refresher about Proof by Contradiction (“Little Observer” arguments) that they did in Gauss' Law Activity
• Symmetry arguments for an infinite sheet of current

1. What units do $\alpha$ and $k$ have?
   • This is a good opportunity to point out that the arguments of special functions must be dimensionless which suffices to determine the dimensions of $k$. The dimensions of $k$ are different in each case. On the other hand, since the functions in the last three cases are inherently dimensionless, the dimensions of $\alpha$ are the same for these three cases. The function in the first case has dimensions $(length)^3$, so the dimensions of $\alpha$ are different for this case.
2. Find the total current flowing through the wire.
   • The total current flowing through the wire is different in each case. thus
   • The Current Density activity will have helped the students understand that total current is a flux. This is a great chance to get them to try to connect to that knowledge. If they are not sure of the surface to use, then, that makes a perfect class discussion!
3. Use Ampere's Law and symmetry arguments to find the magnetic field at each of the three radii given:
   1. $r_1 < a$
   2. $a < r_2 < b$
   3. $r_3 > b$.
   • Symmetry arguments – 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 here than in the Gauss' Law Activity, since the non-zero components and the functional dependencies are not with respect to the same variables.
     • Directional Components: magnetic field is purely in the $\hat{\phi}$ direction.
       • Magnetic field must be perpendicular to the current at all points (cross product in Biot-Savart Law): $\hat{J}=\hat{z} \Rightarrow \hat{B}\perp\hat{z}$
       • An observer stands on a point above the surface facing in the direction of the current ($\hat{z}$). If she were to close her eyes, reverse the direction of the current ($-\hat{z}$), rotate 180$^\circ$ and open her eyes, there would be nothing to indicate that she had turned. Therefore, every measurement that she takes must be the same in both orientations. If she were to first measure the magnetic field in the $+\hat{r}$ direction, go through the same process, and measure again, the magnetic field would measure in the $-\hat{r}$ direction since the current reversed direction. Since every measurement must be the same in both orientations, this is a contradiction and the magnetic field cannot have a component in the $\hat{r}$ direction.
     • Coordinate Dependence: magnetic field only depends on radius.
       • If the same observer were to close her eyes, move in the $\phi$ or $z$ direction, and open her eyes, the current distribution would look the same. Suppose she measures the magnitude of the magnetic field before and after this process. If the magnetic field depended on either $\phi$ or $z$, she would measure a different magnitude at different locations (e.g. $B=3$ at $\phi=0$ and $B=6$ at $\phi=\pi$). Since the current distribution looks the same regardless of $\phi$ and $z$ position, the magnitude of the magnetic field cannot depend on either.
   • Assuming what you want to prove – The students often assume the thing that they are trying to prove, namely that the magnetic field is zero inside the cylinder. Typically, they are using memory from an earlier class. Acknowledge that their memory is correct, but point out to them that they are nevertheless expected to prove the result here.
   • “By symmetry” – Sometimes, they get this result as a misapplication of a symmetry argument. Students tend to mimic the words “from symmetry” without really understanding what that means. They will draw the contribution to the magnetic field from two symmetrically placed current bits. Check, first, that they are getting their right hand rule right and that they really get those two bits canceling (or not canceling) correctly. Then ask them to think about contributions from other points and make sure they can get those correct. A few students can make this argument really work, but then make them write it down. Most students see that they can't really manage it and begin to appreciate the arguments above, which are also used in the lecture example and in the AJP Ampère's Law paper.
   • “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).
4. For $\alpha = 1$, $k = 1$, sketch the magnitude of the magnetic field as a function of $r$.
   • Notice that the magnitude of the magnitude of the vector field is continuous in every case. Discuss why this is true.
   • Make sure to discuss where the value of the total current appears in the graph. Since the total current flowing through the wire is different in each case, it slightly challenging to compare the graphs in question 4 across different cases. Of course, it would be possible to put a multiplicative constant out in front of each case that would adjust the total currents to agree. But there are many reasons not to do this: students are confused generally by parameters (see discussion of Constants v. Variables); the real world never comes with problems adjusted to be nice like this; and this is one of those places where the discussion that arises from a problem that is not stated perfectly smoothly actually aids the students' comprehension - where on the graph does the total current appear?

This is a compare and contrast activity. Each group has a different radial dependence and total current, thus it is difficult to compare graphs across cases. As each group reports, it is important to bring out the fact that the magnetic field is zero for $r<a$ in every case; like an infinitesimally thin wire for $r>b$ in every case; and smooth and continuous, but different in detail in the different cases for $a<r<b$.

• This activity is best if preceded by the Current Density activity and the Gauss' Law activity (last of a sequence of activities on the Geometry of Flux).
• Take the limit of the cylindrical shells such that the thickness goes to zero but the total current stays constant. What happens to the graph of the magnitude of the magnetic field?

This activity is the culminating activity within a sequence of activities addressing Ampere’s law. The following activities are additional activities which are done previously within this sequence.

• Preceding activities:
  • Gauss's Law: A small group activity which students actively make symmetry arguments by Proof by Contradiction to calculate the electric field due to a highly symmetric charge distribution using Gauss's law.
  • 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 and Symmetry Argument Lecture: This lecture serves as an introduction to the Ampere's Law activity by refreshing students about the flux of a current density from Acting Out Current Density, introducing Ampere's law, and discussing Proof by Contradiction in detail.