## Activity: Electrostatic Potential Due to a Ring of Charge

Static Fields 2022 (7 years)

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.

• This activity is used in the following sequences
What students learn
• The electrostatic potential $V$ from a distribution of charges can be found, via the superposition principle, by adding up the contribution from many small chunks of charge;
• For round problems, the superposition should be performed as an integral over round coordinates;
• The analytical and geometric meaning of the distance formula $\vert\vec{r} - \vec{r}^{\prime}\vert$;
• How to calculate linear charge density from a total charge and a distance;
• How to use power series expansions to approximate integrals.
• Media
The Electrostatic Potential Due to a Ring of Charge
1. Find the electrostatic potential everywhere in space due to a charged ring with radius $R$ and total charge $Q$.
2. Evaluate your expression for the special case of the potential on the $z$-axis.
3. Find a series expansion for the electrostatic potential in these special regions:
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

Part I - Finding the potential everywhere in space

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 $Q$. Find a formula for the electrostatic potential $V$ due to this ring that is valid everywhere in space".

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

With the charged ring in the $x,y$-plane, student groups are asked to 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. The instructor may choose to have the whole class do one particular case or have different groups do different cases, in a Compare and Contrast strategy (Compare and Contrast Activities).

### Student Conversations

Part I - Finding the potential everywhere in space

• The expression that students are trying to find is an elliptic integral. Most commonly students have never seen such "unsolvable" integrals in their calculus classes and will be surprised to be asked to find a definite integral that they cannot evaluate in terms of the functions that they already know. Let them try to evaluate the integral briefly, but not so long tha they get frustrated. Eventually, explain what is happening and tell them that they should stop when they have an expression that Mathematica could evaluate to find the potential at a given point, i.e. \begin{equation} V(r, \theta, z)=\frac{1}{4\pi\epsilon_0}\frac{Q}{2\pi R}\int_0^{2\pi} \frac{1}{\sqrt{r^2+r^{\prime 2}-2rr^{\prime}\cos(\phi-\phi^{\prime})+z^2}}\, R d\phi^{\prime} \end{equation} Knowing how to recognize when to stop is an important learning goal of the activity.
• Students will need a few minutes to realize that the charge density is given by the total charge divided by the circumference of the ring $\lambda = \frac{Q}{2\pi R}$. Watch for those students who try only dimensional arguments (who will not get the factor of $2\pi$), those who chant "charge per distance" but don't know what to do with those words, or those who try to use a formula that charge density is the derivative of charge (who will not make progress at all).
• Students must use an appropriate coordinate system to take advantage of the symmetry of the problem. Students attempting to do the problem in rectangular coordinates can be given a few minutes to struggle and see the problems that arise and then, if necessary, guided to using curvilinear coordinates. Most students will choose to do this problem in cylindrical coordinates, but spherical coordinate work equally well.
• Students often muddle the primed and unprimed variables, so it is important to ask them to clarify their notation (e.g. what variable are you integrating over? which $\vec{r}$ is this? which direction does $\vec{r} - \vec{r}'$ point?, etc.) The convention we use is that $\vec{r}$ points from the origin to the point where the field is being evaluated and $\vec{r}^{\prime}$ points from the origin to the source.
• Students frequently leave math classes understanding integration primarily as "the area under a curve". This activity pushes students to generalize their understanding of integration to focus on "chop" (the region of space into small pieces), "multiply" (the integrand by a differential--the small chopped length), "add" (the contributions from each chopped piece)".
• Students may reach a correct figure (chopped pieces, an origin, and labeled position vectors) on their own in a few minutes or they may need help. This is a three-dimensional problem. Either a hoop or a ring drawn on the table can be used to ask students about the potential at points in space that are outside the plane of the ring.
• Students will grapple with how the linear density relates to the "multiply" step of the "chop, multiply, add" aspect of integration.
• The fact that the integral cannot be evaluated is a great opportunity for a mini-lecture transition about why to do power series expansions. If students do the power series expansion in the integrand, it is then possible to do the integration term-by-term, (see Part II, below).

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

• 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 Electrostatic potential due to two points Electrostatic Potential Due to a Pair of Charges (with Series) 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 $y$ 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 $\epsilon = \frac{2R}{r}\cos\phi^{\prime} + \frac{R^2}{r^2}$ 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 quantities 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 quantities remain constant during integration. For example students frequently do not see that the $R$ representing the radius of the ring is held constant during the integration over all space while the r representing the distance to the origin is varying. Understanding this is something professional physicists do automatically 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. For more information on this topic, see [[whitepapers:variables:start|Students understanding of variables and constants]].
• Emphasize that while one may not be able to perform a particular integral, the power series expansion of that integrand can be integrated term by term.
• It is very helpful to end this activity with a way to visualize the value of the potential everywhere in space. Maple/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 a activities:guides:vfvring.mw|Maple or activities:guides:vfvring.nb|Mathematica worksheet.
• 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 (7 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 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 (5 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 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 (6 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.

• assignment Linear Quadrupole (w/o series)

assignment Homework

##### Linear Quadrupole (w/o series)
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?

• assignment Linear Quadrupole (w/ series)

assignment Homework

##### Linear Quadrupole (w/ series)

Power Series Sequence (E&M)

Static Fields 2022 (5 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 $\vec{r}$ in the $xy$-plane at a distance $s$ from the center of the quadrupole. The formula for the electrostatic potential $V$ at a point $\vec{r}$ due to a charge $Q$ at the point $\vec{r'}$ is given by: $V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert}$ Electrostatic potentials satisfy the superposition principle.
2. Assume $s\gg D$. Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

• format_list_numbered Ring Cycle Sequence

format_list_numbered Sequence

##### Ring Cycle Sequence
Students calculate electrostatic fields ($V$, $\vec{E}$) and magnetostatic fields ($\vec{A}$, $\vec{B}$) from charge and current sources with a common geometry. The sequence of activities is arranged so that the mathematical complexity of the formulas students encounter increases with each activity. Several auxiliary activities allow students to focus on the geometric/physical meaning of the distance formula, charge densities, and steady currents. A meta goal of the entire sequence is that students gain confidence in their ability to parse and manipulate complicated equations.
• 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 (5 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.
• keyboard Electrostatic potential of four point charges

keyboard Computational Activity

120 min.

##### Electrostatic potential of four point charges
Computational Physics Lab II 2022

Students write python programs to compute and visualize the potential due to four point charges. For students with minimal programming ability and no python experience, this activity can be a good introduction to writing code in python using numpy and matplotlib.
• face Time Evolution Refresher (Mini-Lecture)

face Lecture

30 min.

##### Time Evolution Refresher (Mini-Lecture)
Central Forces 2023 (3 years)

Quantum Ring Sequence

The instructor gives a brief lecture about time dependence of energy eigenstates (e.g. McIntyre, 3.1). Notes for the students are attached.
• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.

Author Information
Corinne Manogue, Leonard Cerny
Learning Outcomes