## Activity: Acting Out Charge Densities

Static Fields 2022 (5 years)
Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear $\lambda$, surface $\sigma$, and volume $\rho$ charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.
• This activity is used in the following sequences
What students learn
• Conceptual/geometric understanding of linear, surface and volume densities;
• The existence of mass, charge, and number densities;
• The distinction between uniform and non-uniform densities;
• The relationship between discrete densities, represented by bodies, and continuous densities represented by functions such as $\lambda(x)$.

### Introduction

Tell the students that they each represent a point charge. Do NOT specify that they are each the same charge unless they ask. Then initiate a conversation with the whole class by asking the prompts listed in "Student Conversations," below. Be flexible about the order of the prompts, responding to the ideas brought up by the students.

Note: It helps if the instructor stands on a chair or table so they are high enough to see all the students.

### Student Conversations

• Students usually line up in a straight line. An excellent follow-up question is: "Does a linear charge density need to be arranged in a straight line?" Answer: "No, a linear charge density just means that the charges are distributed in one dimension, which may be along a curve."
• A few students may interpret the word linear to mean that the number of charges in each interval is increasing linearly. This is an excellent opportunity to discuss the two different common uses of the term "linear" - as "one-dimensional" and as "$y=mx$”. Specify that in this context, we are choosing the first definition.
2. A next question might be: "Make a non-uniform linear charge density?"
• You can then discuss what "uniform" means. You may now need to specify that you intend for them to represent equal charges.
• Discuss that if they represented masses instead of charges, then it might be more appropriate to have them represent unequal masses. (Avoid fat shaming!!) For number density, of course, all students are equal.
3. At some point, typically about now, the idea of idealization should come up. What do we mean by a continuous distribution of charges described by a charge density? Discuss how to approximate the charge density by holding up a 2 meter stick at various places along the curve the students have formed and count how many students fall into the 2 meter range. Introduce the symbol $\lambda(u)$ and discuss the dimensions of linear charge density.
4. The next question might be: "Make a surface charge density."
• Sometimes students will spread themselves around the room, sometimes they will line up two-by-two. You can ask the students whether their distribution needs to be flat and whether it needs to be uniform.
• Introduce the symbol $\sigma(x,y)$ and, if appropriate to the level of the class, introduce the symbol $\sigma(u,v)$ and lightly introduce the idea of parameterization.
• Discuss the meaning of surface charge density, i.e. the number of students that fit into a square meter. Demonstrate the geometry of a square meter with meter sticks surrounding the students and describe the dimensions of surface charge density.
5. The final question might be: "Make a volume charge density." Be alert, some students may want to jump up onto unsafe furniture.

### Wrap-up

The instructor can wrap up by making an organized table of the notations used to describe the various types of charge densities ($\lambda$, $\sigma$, and $\rho$) and their dimensions. If there is time, get the students to help generate the table.

• accessibility_new Acting Out Current Density

accessibility_new Kinesthetic

10 min.

##### Acting Out Current Density
Static Fields 2022 (5 years)

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room so as to make an imaginary magnetic field meter register a constant magnetic field, introducing the concept of steady current. Students act out linear $\vec{I}$, surface $\vec{K}$, and volume $\vec{J}$ current densities. The instructor demonstrates what it means to measure these quantities by counting how many students pass through a gate.
• 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 (7 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

Warm-Up

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.

• assignment Gravitational Field and Mass

assignment Homework

##### Gravitational Field and Mass
Static Fields 2022 (4 years)

The gravitational field due to a spherical shell of matter (or equivalently, the electric field due to a spherical shell of charge) is given by: $$\vec g = \begin{cases} 0&\textrm{for } r<a\\ -G \,\frac{M}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ -G\,\frac{M}{r^2}\, \hat r & \textrm{for } r>b \\ \end{cases}$$

This problem explores the consequences of the divergence theorem for this shell.

1. Using the given description of the gravitational field, find the divergence of the gravitational field everywhere in space. You will need to divide this question up into three parts: $r<a$, $a<r<b$, and $r>b$.
2. Briefly discuss the physical meaning of the divergence in this particular example.
3. For this gravitational field, verify the divergence theorem on a sphere, concentric with the shell, with radius $Q$, where $a<Q<b$. ("Verify" the divergence theorem means calculate the integrals from both sides of the divergence theorem and show that they give the same answer.)
4. Briefly discuss how this example would change if you were discussing the electric field of a uniformly charged spherical shell.

• assignment Electric Field and Charge

assignment Homework

##### Electric Field and Charge
divergence charge density Maxwell's equations electric field Static Fields 2022 (3 years) Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• assignment Electric Field of a Finite Line

assignment Homework

##### Electric Field of a Finite Line

Consider the finite line with a uniform charge density from class.

1. Write an integral expression for the electric field at any point in space due to the finite line. In addition to your usual physics sense-making, you must include a clearly labeled figure and discuss what happens to the direction of the unit vectors as you integrate.Consider the finite line with a uniform charge density from class.
2. Perform the integral to find the $z$-component of the electric field. In addition to your usual physics sense-making, you must compare your result to the gradient of the electric potential we found in class. (If you want to challenge yourself, do the $s$-component as well!)

• 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 Electric Potential of Two Charged Plates

group Small Group Activity

30 min.

##### Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two charged plates (near the center of the plates) and explore the properties of the electric potential.
• group Total Charge

group Small Group Activity

30 min.

##### Total Charge
Static Fields 2022 (5 years)

Integration Sequence

In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.
• 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.