## Activity: Magnetic Field Due to a Spinning Ring of Charge

Static Fields 2023 (7 years)

Students work in small groups 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.

• This activity is used in the following sequences
What students learn
• to perform a magnetic field calculation using the Biot-Savart Law;
• to decide which form of the Biot-Savart Law 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 perform the cross product in the numerator of the Biot-Savart Law using cyclindrical basis vectors;
• to write the distance formula $\vec{r}-\vec{r'}$ in both the numerator and denominator of the Biot-Savart Law in an appropriate mix of cylindrical coordinates and rectangular basis vectors;
The Magnetic Field Due to a Spinning Ring of Charge
1. 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 the magnetic field everywhere in space due to a spinning charged ring with radius $R$, total charge $Q$, and period $T$.
2. Evaluate your expression for the special case that $\vec{r}$ is on the $z$-axis.
3. Evaluate your expression for the special case that $\vec{r}$ is on the $x$-axis.
4. 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 small groups 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 field $\vec{B}$ 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, Magnetic Vector Potential Due to a Spinning Charged Ring) for further advice.

Part I - Finding the magnetic field everywhere in space

The new idea in the magnetic field case is the cross product in the numerator. Many students will find this cross product using rectangular basis vectors, which is correct. But they may NOT realize that curvilinear basis vectors AT A SINGLE POINT are also orthonormal and it is perfectly correct to find the cross product using them IF the curvilinear basis vectors are both primed OR both unprimed, but not otherwise.

Part II (Optional) - Power series expansion along an axis

• 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

If you are doing this activity as a standalone, please see the Wrap-Up section of the previous activities (Electrostatic Potential Due to a Ring of Charge, Electric Field Due to a Ring of Charge, Magnetic Vector Potential Due to a Spinning Charged Ring) for further advice.

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##### Magnetic Vector Potential Due to a Spinning Charged Ring
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Students work in small groups 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.

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Students work in small groups 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.

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##### Electrostatic Potential Due to a Pair of Charges (without Series)
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