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

Static Fields 2022 (5 years)

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.

• 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
• 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 $T$.
• Evaluate your expression for the special case that $\vec{r}$ is on the $z$-axis.
• Evaluate your expression for the special case that $\vec{r}$ is on the $x$-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 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.

• group Magnetic Vector Potential Due to a Spinning Charged Ring

group Small Group Activity

30 min.

##### Magnetic Vector Potential Due to a Spinning Charged Ring
Static Fields 2022 (4 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

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.

• 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.

• group Static Fields Equation Sheet

group Small Group Activity

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##### Static Fields Equation Sheet
Static Fields 2022 (3 years)
• computer Visualization of Quantum Probabilities for a Particle Confined to a Ring

computer Mathematica Activity

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##### 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.
• accessibility_new The Distance Formula (Star Trek)

accessibility_new Kinesthetic

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##### The Distance Formula (Star Trek)
Static Fields 2022 (4 years)

Ring Cycle Sequence

A short improvisational role-playing skit based on the Star Trek series in which students explore the definition and notation for position vectors, the importance of choosing an origin, and the geometric nature of the distance formula. $\vert\vec{r}-\vec{r}^\prime\vert=\sqrt{(x-x^\prime)^2+(y-y^\prime)^2-(z-z^\prime)^2}$
• assignment Events on Spacetime Diagrams

assignment Homework

##### Events on Spacetime Diagrams
Special Relativity Spacetime Diagram Simultaneity Colocation Theoretical Mechanics 2021 (2 years)
1. Which pairs of events (if any) are simultaneous in the unprimed frame?

2. Which pairs of events (if any) are simultaneous in the primed frame?

3. Which pairs of events (if any) are colocated in the unprimed frame?

4. Which pairs of events (if any) are colocated in the primed frame?

1. For each of the figures, answer the following questions:
1. Which event occurs first in the unprimed frame?

2. Which event occurs first in the primed frame?

• group Events on Spacetime Diagrams

group Small Group Activity

5 min.

##### Events on Spacetime Diagrams
Theoretical Mechanics 2021

Students practice identifying whether events on spacetime diagrams are simultaneous, colocated, or neither for different observers. Then students decide which of two events occurs first in two different reference frames.
• group Electrostatic Potential Due to a Pair of Charges (with Series)

group Small Group Activity

60 min.

##### Electrostatic Potential Due to a Pair of Charges (with Series)
Static Fields 2022 (4 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Students work in small groups to use the superposition principle $V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}$ to find the electrostatic potential $V$ everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.
• assignment Linear Quadrupole (w/o series)

assignment Homework

Static Fields 2022 (3 years) Consider a collection of three charges arranged in a line along the $z$-axis: charges $+Q$ at $z=\pm D$ and charge $-2Q$ at $z=0$.
1. Find the electrostatic potential at a point $\vec{r}$ on the $x$-axis at a distance $x$ from the center of the quadrupole.

2. A series of charges arranged in this way is called a linear quadrupole. Why?

Author Information
Corinne Manogue, Leonard Cerny
Keywords
magnetic fields current Biot-Savart law vector field symmetry
Learning Outcomes