## Activity: Magnetic Vector Potential Due to a Spinning Charged Ring

Static Fields 2022 (4 years)

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.

• This activity is used in the following sequences
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
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 spinning 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 - Evaluate on the $z$-axis and on the $x$-axis

(See solution.)

Part III (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 $z$ axis case without a lot of assistance because it is very similar to the $z$ axis case for the two $+Q$ point charges. However, the $x$ axis presents a new challenges because the ”something small” is two terms. It will probably not be obvious for students to let $\epsilon = {2R\over{s}}\cos\phi'+$ $R^2\over{s^2}$ (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 truly variable and which variables are held constant - Students frequently think of anything represented by a letter as a "variable" and do not realize that throughout while they are solving this problem the symbols that they want to think of a variable or constant change. While they are doing the integration, the primed variables are variable (i.e. changing) but the unprimed variables are constant. After the integration, the primed variables are gone and the unprimed variable become parameters, i.e. they can change.
• Mathematica 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 Mathematica, see solution.
• group Magnetic Field Due to a Spinning Ring of Charge

group Small Group Activity

30 min.

##### Magnetic Field Due to a Spinning Ring of Charge
Static Fields 2022 (5 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the Biot-Savart law $\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the magnetic field, $\vec{B}(\vec{r})$, due to a spinning ring of charge.

In an optional extension, students find a series expansion for $\vec{B}(\vec{r})$ 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.

• group Electrostatic Potential Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electrostatic Potential Due to a Ring of Charge
Static Fields 2022 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use the superposition principle $V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}$ to find an integral expression for the electrostatic potential, $V(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $V(\vec{r})$ 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.

• group Electric Field Due to a Ring of Charge

group Small Group Activity

30 min.

##### Electric Field Due to a Ring of Charge
Static Fields 2022 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in groups of three to use Coulomb's Law $\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}$ to find an integral expression for the electric field, $\vec{E}(\vec{r})$, everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for $\vec{E}(\vec{r})$ 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.

• assignment Total Charge

assignment Homework

##### Total Charge
charge density curvilinear coordinates

Integration Sequence

Static Fields 2022 (4 years)

For each case below, find the total charge.

1. A positively charged (dielectric) spherical shell of inner radius $a$ and outer radius $b$ with a spherically symmetric internal charge density $$\rho(\vec{r})=3\alpha\, e^{(kr)^3}$$
2. A positively charged (dielectric) cylindrical shell of inner radius $a$ and outer radius $b$ with a cylindrically symmetric internal charge density $$\rho(\vec{r})=\alpha\, \frac{1}{s}\, e^{ks}$$

• assignment Mass of a Slab

assignment Homework

##### Mass of a Slab
Static Fields 2022 (4 years)

Determine the total mass of each of the slabs below.

1. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho=A\pi\sin(\pi z/h).$$
2. A square slab of side length $L$ with thickness $h$, resting on a table top at $z=0$, whose mass density is given by $$\rho = 2A \Big( \Theta(z)-\Theta(z-h) \Big)$$
3. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose surface density is given by $\sigma=2Ah$.
4. An infinitesimally thin square sheet of side length $L$, resting on a table top at $z=0$, whose mass density is given by $\rho=2Ah\,\delta(z)$.
5. What are the dimensions of $A$?
6. Write several sentences comparing your answers to the different cases above.

• group Time Dependence for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Time Dependence for a Quantum Particle on a Ring
Central Forces 2022 (3 years)

Quantum Ring Sequence

Students calculate probabilities for energy, angular momentum, and position as a function of time for an initial state that is a linear combination of energy/angular momentum eigenstates for a particle confined to a ring written in bra-ket notation. This activity helps students build an understanding of when they can expect a quantity to depend on time and to give them more practice moving between representations.
• group Superposition States for a Particle on a Ring

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.
• format_list_numbered Quantum Ring Sequence

format_list_numbered Sequence

##### Quantum Ring Sequence
Students calculate probabilities and expectation values for a quantum mechanical particle confined to a circular ring in bra/ket, matrix, and wave function representations and compare the different calculation methods. Several different graphical representations of the time dependence for both states with special symmetry and arbitrary states are explored in a Mathematica notebook. Compared to the analogous particle-in-a-box, this quantum system has a new feature---degenerate energy eigenstates.
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

30 min.

##### Visualization of Quantum Probabilities for a Particle Confined to a Ring
Central Forces 2022 (2 years)

Quantum Ring Sequence

Students see probability density for eigenstates and linear combinations of eigenstates for a particle on a ring. The three visual representations: standard position vs probability density plot, a ring with colormapping, and cylindrical plot with height and colormapping, are also animated to visualize time-evolution.
• assignment Ring Table

assignment Homework

##### Ring Table
Central Forces 2022 (2 years)

Attached, you will find a table showing different representations of physical quantities associated with a quantum particle confined to a ring. Fill in all of the missing entries. Hint: You may look ahead. We filled out a number of the entries throughout the table to give you hints about what the forms of the other entries might be. pdf link for the Table or doc link for the Table

Author Information
Corinne Manogue, Leonard Cerny
Learning Outcomes