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## Electrostatic Potential Due to a Charged Ring: Instructor's Guide

### Main Ideas

• Electrostatic potential
• Spherical and cylindrical coordinates
• Superposition principle
• The analytical and geometric meaning of $\frac{1}{|\Vec r - \Vec r'|}$
• Integration as “chopping and adding”
• Linear charge density
• 3-dimensional geometric reasoning
• Power series expansion
• Symmetry

Estimated Time: 40 min; Wrap-up: 10 min

1. 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: “This is a ring with total charge $Q$ and radius $R$. Find the electrical potential due to this ring in all space.” Students do their work collectively with markers on a poster-sized sheet of whiteboard at their tables.
2. Students determine the power series expansion to represent the electrostatic potential due to the charged ring along a particular axis.
• Note: students should not be told about part II until they have completed part I.

### Prerequisite Knowledge

This activity is may be used as the second in the Ring Sequence, following the Electrostatic potential due to two points activity, or may be used on its own. Students will need understanding of:

• Spherical and cylindrical coordinates. You may want to consider using the Vector Differential activity, in which students use geometric arguments to derive the form of $d\Vec r$ in these curvilinear coordinate systems.
• Integration as chopping and adding
• Linear charge density (see Acting Out Charge Densities activity)

### Activity: 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: “This is a ring with total charge Q and radius R. Find the electrical potential due to this ring in all space.”

### Activity: Student Conversations

#### Part I - Finding the potential everywhere in space: Creating an elliptic integral

• The first concept students need to understand is linear charge density. Given that the ring has a charge $Q$ students will need a few minutes to realize that the charge density $\lambda = {Q\over{2\pi r}}$. In general students come up with this on their own without help.
• Students will grapple with how the linear density relates to the chopping and adding' aspect of integration. Students frequently leave math classes understanding integration as the area under a curve'. This activity pushes students to transform their understanding of integration to focus on chopping and adding.' Students may reach a correct representation on their own in a few minutes or the instructor may assist by using a hula hoop as a prop to help students in describing the chopped' bits of hoop.
• 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 should be guided to using curvilinear coordinates. Most students will choose to do this problem in cylindrical coordinates, but an interesting problem for groups who finish early is to redo the problem in spherical coordinates.
• Putting the whole thing together requires three dimensional geometric understanding. One of the big advantages to doing this problem in class as opposed to homework is that the instructor can interact with student making 3-dimesional arguments. 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.
• This activity also gives students the opportunity to use curvilinear coordinates and then realize that they cannot successfully integrate without transforming them using rectangular basis vectors. Understanding that $|\Vec{r} - \Vec r'|$ cannot be integrated by simply using $r'$ in curvilinear coordinates is an important realization. Unlike linear coordinates where $x - x'$ always refers to components of vectors that point in the same direction, this is not the case for curvilinear coordinates where $\Vec r$ and $\Vec r'$ can be oriented in different directions at any angle. Solving this problem entirely in rectangular coordinates from the beginning is overly cumbersome, but the curvilinear coordinates that successfully simplify the problem can lead one to incorrectly think that using $|\Vec{r} - \Vec r'|$ in curvilinear coordinates can be successfully integrated.
• The final component is that students need to recognize an elliptic integral and what to do when they run into one. Most commonly students have never seen such unsolvable' integrals in their calculus classes. In our case we had students do the power series expansion before the integral (see below).

#### Part II - Finding the potential along an axis: Power series expansion

• 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 emphasizing some of the points above, 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 Electrostatic potential due to two points 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 $\epsilon = {2R\over r}\cos\phi' + {R^2 \over r^2}$ (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.

### Activity: 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 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 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. For more information on this topic, see Students understanding of variables and constants.

### Extensions

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 Maple or Mathematica worksheet.

This activity is a part of the Ring Sequence, which uses a sequence of activities with similar geometries to help students learn how to solve a hard activity by breaking it up into several steps (A Master's Thesis about the Ring Sequence). The other activities in the sequence are:

• Prerequisite Activity:
• Electrostatic Potential Due to Two Point Charges: Students use what they learned about finding the distance between two points and apply what they know about power series approximations to find a general expression and asymptotic solution to the electrostatic potential due to two point charges. Though the charge distribution students consider is elementary, the ideas and techniques used are applied later on to the more difficult ring of charge.
• Follow-up Activities:
• Electric Field Due to a Ring of Charge: Now that students have had experience calculating the electrostatic potential due to a ring of charge, another layer of complexity is added to the problem. In this small group activity, students use the definition of the electric field as $\vec{E}(\vec{r})=\frac{1}{4\pi\epsilon_0}\int_{ring}{\frac{\lambda(\vec{r'})(\vec{r}-\vec{r'})|d\vec{r'}|}{|\vec{r}-\vec{r'}|^3}}$ in order to work out the electric field due to a ring of charge. Students must break this problem into multiple steps which develops a template for solving similar problems. This activity exemplifies the approach the Paradigms uses of “potential first”. Using techniques from the previous activity on the electrostatic potential, students tend to have an easier time transitioning from scalar fields to vector fields.
• Magnetic Vector Potential Due to a Spinning Ring of Charge: In many ways, this small group activity is simply the magnetic analogy to Electrostatic Potential Due to a Ring of Charge. Students are likely to recognize the problem is headed toward an integral which they will be approximated using power series. However, the set up for this problem is more complex because students now must think about current densities and the inherent vector nature of the magnetic vector potential. Again, in order to solve this problem, students practice breaking the problem into multiple steps, allowing them to develop a template approach for solving such problems.
• Magnetic Field Due to a Spinning Ring of Charge: Building from their experience with the previous exercises, students in this small group activity use techniques from the previous activities to obtain an expression for the magnetic field due to a spinning ring of charge.

This activity is part of the sequence of activities addressing Representations of Scalar Fields in the context of electrostatic potentials.

• Prerequisite Activities:
• Electrostatic Potential due to a Point Charge: This small whiteboard question typically results in an algebraic expression of one variable: the distance from the origin to the point charge. Discussions which will likely arise include notation of the distance from the origin to the point charge, the constants in the equation, and the dimensions of the equation. The representation used by students is predictably algebraic in form, however, the discussion can include other representations of a one-dimensional electrostatic potential.
• Drawing Equipotential Surfaces: This small group activity encourages students to work in the plane of four point charges arranged in a square to find level curves of equipotential. Students construct a contour plot of the electrostatic potential in the plane of the four charges and explore the constructed scalar field close to the charges, far from the charges, and at important points in the field. Most students are familiar with the elementary equation of the electrostatic potential but few reconcile the equation with the geometry of a scalar field. This small group activity forces students to explicitly work out the geometry of the potential of a quadrupole, allowing them to realize what's “scalar” about the electrostatic potential.
• Visualizing Electrostatic Potentials: Students begin by brainstorming ways in which to represent three-dimensional scalar fields in two-dimensions and then use a Mathematica notebook to explore various representations for a distribution of point charges. This activity allows students to check their solutions to Drawing Equipotential Surfaces as well as explore other representations. Students recognize that the electrostatic potential is a function of three spatial variables which requires an alternative way to represent the potential such as the use of color and plotting equipotential surfaces.

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