Activity: Magnetic Vector Potential Due to a Spinning Charged Ring

Students work in groups of three to use the superposition principle \[\vec{A}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert}\, d\tau^{\prime}\] to find an integral expression for the magnetic vector potential, \(\vec{A}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{A}(\vec{r})\) either on the axis or in the plane of the ring, for either small or large values of the relevant geometric variable. Add an extra half hour or more to the time estimate for the optional extension.

What students learn
  • to perform a magnetic vector potential calculation using the superposition principle;
  • to decide which form of the superposition principle to use, depending on the dimensions of the current density;
  • how to find current from total charge \(Q\), period \(T\), and the geometry of the problem, radius \(R\);
  • to write the distance formula \(\vec{r}-\vec{r'}\) in both the numerator and denominator of the superposition principle in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
  • Media
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The Magnetic Vector Potential Due to a Spinning Ring of Charge
  • Find the magnetic vector potential everywhere in space due to a spinning charged ring with radius \(R\), total charge \(Q\), and period \(T\).
  • Evaluate your expression for the special case that \(\vec{r}\) is on the \(z\)-axis.
  • Find a series expansion for the electrostatic potential at these special locations:
    1. Near the center of the ring, in the plane of the ring;
    2. Near the center of the ring, on the axis of the ring;
    3. Far from the ring on the axis of symmetry;
    4. Far from the ring, in the plane of the ring.

Instructor's Guide

Introduction

Students should be assigned to work in groups of three and given the following instructions using the visual of a hula hoop or other large ring:

Prompt: "This is a ring with radius \(R\) and total charge \(Q\) and spinnging with period \(T\). Find a formula for the magnetic vector potential \(\vec{A}\) due to this ring that is valid everywhere in space".

Student Conversations

This activity is part of a sequence (the Ring Cycle Sequence) of four electrostatics activities involving a ring of charge: \(V\), \(\vec{E}\), \(\vec{A}\), \(\vec{B}\). They are arranged so that the mathematical complexity of the problems increases in a natural way. If you are doing this activity as a standalone, please see the Student Conversations section of the previous activities (Electrostatic Potential Due to a Ring of Charge, Electric Field Due to a Ring of Charge) for further advice.

Part I - Finding the potential everywhere in space

The new idea in the magnetic vector potential case is to find the linear current density (current) in the ring. Many students will have learned that current is “charge per time” or the derivative of charge with respect to time. Neither of these resources about current will be helpful to them here. They will need to know that current density is charge density times velocity: \begin{align} \vec{I}&=\lambda \vec{v}\\ &=\frac{Q}{2\pi R}\, \frac{2 \pi R}{T} \hat{\phi} \end{align} Be watchful. Many students will get the correct answer on dimensional grounds, but will not be able to justify their answer in a way that will extend to other problems.

Part II (Optional) - Series expansions

  • With the charged ring in the \(x,y-\)plane, students will make the power series expansion for either near or far from the plane on the \(z\) axis or near or far from the \(z\) axis in the \(x,y-\)plane. Once all students have made significant progress toward finding the integral from part I, and some students have successfully determined it, then the instructor can quickly have a whole class discussion followed by telling students to now create a power series expansion. The instructor may choose to have the whole class do one particular case or have different groups do different cases.
  • If you are doing this activity without having had students first create power series expansions for the electrostatic potential due to two charges, students will probably find this portion of the activity very challenging. If they have already done the Discrete Charges activity, or similar activity, students will probably be successful with the \(y\) axis case without a lot of assistance because it is very similar to the \(y\) axis case for the two \(+Q\) point charges. However, the \(y\) axis presents a new challenges because the ”something small” is two terms. It will probably not be obvious for students to let \(\epsilon = {2R\over{r}}\cos\phi'+\) \(R^2\over{r^2}\) (see Eq. 17 in the solutions) and suggestions should be given to avoid having them stuck for a long period of time. Once this has been done, students may also have trouble combining terms of the same order. For example the \(\epsilon^2\) term results in a third and forth order term in the expansion and students may not realize that to get a valid third order expansion they need to calculate the \(\epsilon^3\) term.

Wrap-up

  • Discuss which variables are variable and which variables are held constant - Students frequently think of anything represented by a letter as a "variable" and do not realize that for each particular situation certain variables remain constant during integration. For example students do frequently do not see that the \(R\) representing the radius of the ring is held constant during integrating over all space while the \(r\) representing the distance to the origin is varying. Understanding this is something trained physicists do naturally while students frequently don't even consider it. This is an important discussion that helps students understand this particular ring problem and also lays the groundwork for better understanding of integration in a variety of contexts.
  • Maple representation of elliptic integral - After finding the elliptic integral and doing the power series expansion, students can see what electric potential "looks like" over all space by using Maple.

Author Information
Corinne Manogue, Leonard Cerny
Learning Outcomes