## Activity: Proportional Reasoning

Static Fields 2023 (3 years)
In this small group activity, students calculate a (linear) function to represent the charge density on a one-dimensional rod from a description of the charge density in words.
What students learn
• how to distinguish the two different uses of the word “linear” in a linear charge density that varies linearly;
• some of the words for describing functional variation: linear, quadratic, exponential, falls off like ..., proportional to the square, etc.
• how to “name the thing you don't know” with an algebraic symbol so that it can appear in an equation.

Consider a rod of length $L$ lying on the $z$-axis. Find an algebraic expression for the mass density of the rod if the mass density at $z=0$ is $\lambda_0$ and at $z=L$ is $7\lambda_0$ and you know that the mass density increases linearly.

## Instructor's Guide

### Student Conversations

This activity is surprisingly difficult for some students. This activity is partly an opportunity to introduce language for functional dependence and partly an exercise in modeling physical behavior with a function with parameters that must be determined. Once students get the idea, it's pretty straightforward, but it helps solidify the concept if the exercise recurs a few times as part of future in-class activities and homework.

• Make sure to discuss lots of different language for functional dependence in the wrap-up: linear, quadratic, exponential, falls off like ..., proportional to the square, etc.
• Some students will not realize that they need to start with an equation with one or more unknown parameters in it. In the case of the given prompt: $\lambda(z)=\beta+\alpha z$. Lots of times, you can't write an equation unless you give an algebraic name to the unknowns. We call this strategy “Name the thing you don't know.”
• Be particularly aware of the fact that some students incorrectly believe the word “exponentially” means “has an exponent” as in $x^2$ rather than correctly believe that it means varies like the exponential function $e^{\alpha x}$. Make sure that this topic comes up in the conversation.

• assignment Mass Density

assignment Homework

##### Mass Density
Static Fields 2023 (4 years) Consider a rod of length $L$ lying on the $z$-axis. Find an algebraic expression for the mass density of the rod if the mass density at $z=0$ is $\lambda_0$ and at $z=L$ is $7\lambda_0$ and you know that the mass density increases
• linearly;
• like the square of the distance along the rod;
• exponentially.
• assignment Gauss's Law for a Rod inside a Cube

assignment Homework

##### Gauss's Law for a Rod inside a Cube
Static Fields 2023 (4 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $x,y$-plane. The charge density $\lambda_0$ is constant. Find the total flux of the electric field through a closed cubical surface with sides of length $3L$ centered at the origin.
• 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 2023 (6 years)

Power Series Sequence (E&M)

Ring Cycle Sequence

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.

• assignment Electric Field from a Rod

assignment Homework

##### Electric Field from a Rod
Static Fields 2023 (5 years) Consider a thin charged rod of length $L$ standing along the $z$-axis with the bottom end on the $xy$-plane. The charge density $\lambda$ is constant. Find the electric field at the point $(0,0,2L)$.
• assignment Line Sources Using Coulomb's Law

assignment Homework

##### Line Sources Using Coulomb's Law
Static Fields 2023 (6 years)
1. Find the electric field around a finite, uniformly charged, straight rod, at a point a distance $s$ straight out from the midpoint, starting from Coulomb's Law.
2. Find the electric field around an infinite, uniformly charged, straight rod, starting from the result for a finite rod.
• accessibility_new Acting Out Charge Densities

accessibility_new Kinesthetic

10 min.

##### Acting Out Charge Densities
Static Fields 2023 (7 years)

Integration Sequence

Ring Cycle Sequence

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.
• accessibility_new Acting Out Current Density

accessibility_new Kinesthetic

10 min.

##### Acting Out Current Density
Static Fields 2023 (6 years)

Integration Sequence

Ring Cycle 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.
• assignment Charge on a Spiral

assignment Homework

##### Charge on a Spiral
Static Fields 2023 (3 years) A charged spiral in the $x,y$-plane has 6 turns from the origin out to a maximum radius $R$ , with $\phi$ increasing proportionally to the distance from the center of the spiral. Charge is distributed on the spiral so that the charge density increases linearly as the radial distance from the center increases. At the center of the spiral the linear charge density is $0~\frac{\textrm{C}}{\textrm{m}}$. At the end of the spiral, the linear charge density is $13~\frac{\textrm{C}}{\textrm{m}}$. What is the total charge on the spiral?
• assignment Derivatives from Data (NIST)

assignment Homework

##### Derivatives from Data (NIST)
Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
1. Find the partial derivatives $\left(\frac{\partial {S}}{\partial {T}}\right)_{p}$ $\left(\frac{\partial {S}}{\partial {T}}\right)_{V}$ where $p$ is the pressure, $V$ is the volume, $S$ is the entropy, and $T$ is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
2. Why does it take only two variables to define the state?
3. Why are the derivatives above different?
4. What do the words isobaric, isothermal, and isochoric mean?
• group Applying the equipartition theorem

group Small Group Activity

30 min.

##### Applying the equipartition theorem
Contemporary Challenges 2021 (4 years)

Students count the quadratic degrees of freedom of a few toy molecules to predict their internal energy at temperature $T$.

Author Information
Corinne Manogue
Learning Outcomes