## Activity: Vector Integrals (Contour Map)

What students learn
• Recall the relationship between the sign of the dot product and the orientation of the vectors.
• Use graphical methods to estimate the value of a vector line integral.
• Lay the groundwork for thinking about conservative and non-conservative vector fields.
• Media
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

Students will estimate the work done by a given electric field. They will connect the work done to the height of a plastic surface graph of the electric potential.
• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Quantifying Change (Remote)

group Small Group Activity

30 min.

##### Quantifying Change (Remote)

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• assignment_ind Vector Differential--Rectangular

assignment_ind Small White Board Question

10 min.

##### Vector Differential--Rectangular
AIMS Maxwell AIMS 21 Vector Calculus II Fall 2021 Vector Calculus II Summer 21 Static Fields Winter 2021

Integration Sequence

In this introductory lecture/SWBQ, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in rectangular coordinates for coordinate equals constant paths.

This activity can be done as a mini-lecture/SWBQ as an introduction to Vector Differential--Curvilinear where students find the vector differential in cylindrical and spherical coordinates..

• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• assignment Paramagnet (multiple solutions)

assignment Homework

##### Paramagnet (multiple solutions)
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021 We have the following equations of state for the total magnetization $M$, and the entropy $S$ of a paramagnetic system: \begin{align} M&=N\mu\, \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}}\\ S&=Nk_B\left\{\ln 2 + \ln \left(e^{\frac{\mu B}{k_B T}}+e^{-\frac{\mu B}{k_B T}}\right) +\frac{\mu B}{k_B T} \frac{e^{\frac{\mu B}{k_B T}} - e^{-\frac{\mu B}{k_B T}}} {e^{\frac{\mu B}{k_B T}} + e^{-\frac{\mu B}{k_B T}}} \right\} \end{align}
1. List variables in their proper positions in the middle columns of the charts below.

2. Solve for the magnetic susceptibility, which is defined as: $\chi_B=\left(\frac{\partial M}{\partial B}\right)_T$

3. Using both the differentials (zapping with d) and chain rule diagram methods, find a chain rule for:

$\left(\frac{\partial M}{\partial B}\right)_S$

4. Evaluate your chain rule. Sense-making: Why does this come out to zero?  • group Gravitational Potential Energy

group Small Group Activity

60 min.

##### Gravitational Potential Energy

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.
• assignment Free Expansion

assignment Homework

##### Free Expansion
Energy and Entropy Fall 2020 Energy and Entropy Fall 2021

The internal energy is of any ideal gas can be written as \begin{align} U &= U(T,N) \end{align} meaning that the internal energy depends only on the number of particles and the temperature, but not the volume.*

The ideal gas law \begin{align} pV &= Nk_BT \end{align} defines the relationship between $p$, $V$ and $T$. You may take the number of molecules $N$ to be constant. Consider the free adiabatic expansion of an ideal gas to twice its volume. “Free expansion” means that no work is done, but also that the process is also neither quasistatic nor reversible.
1. What is the change in entropy of the gas? How do you know this?

2. What is the change in temperature of the gas?

• group Squishability'' of Water Vapor (Contour Map)

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

Students determine the “squishibility” (an extensive compressibility) by taking $-\partial V/\partial P$ holding either temperature or entropy fixed.
• group Changes in Internal Energy (Remote)

group Small Group Activity

30 min.

##### Changes in Internal Energy (Remote)

Students consider the change in internal energy during three different processes involving a container of water vapor on a stove. Using the 1st Law of Thermodynamics, students reason about how the internal energy would change and then compare this prediction with data from NIST presented as a contour plot.

## Vector Integrals

What Students Learn:

• Recall the relationship between the sign of the dot product and the orientation of the vectors.
• Use graphical methods to estimate the value of a vector line integral.
• Lay the groundwork for thinking about conservative and non-conservative vector fields.

Time Estimate: 15 minutes

Equipment

• Dry-erase markers & erasers
• Whiteboard for each group
• Student handout for each student

Introduction

• This activity is a warmup activity and students do not need an introduction.

Whole Class Discussion / Wrap Up:

• This activity is a warmup to two other activities: Counting Paths and Work by an Electric Field. We highly recommend doing this activity before either of these activities.

• Because there are multiple possible answers for the first part of the activity, and multiple possible strategies for evaluating the integral in the second part, this activity is particularly well suited to a whole-class discussion in which different groups of students are asked to share their answers and discuss the advantages and disadvantages of each.

The Sign of Work: For the force vector shown at left, draw three paths such that the work done by $\vec{F}$ is positive, zero, and negative for small displacements along each path.

Goal: This gets at the dot product relationship students will need to perform the vector path integral. It does not get at the ideas that you have to chop the whole path into small pieces to do the integral (see next question).

Follow-Up: - “Is there only one correct path for each?” Students usually draw the parallel/antiparallel/perpendicular paths and do not consider other options.

Estimate Vector Line Integral: For each segment of the path on the vector field $\vec{F}$ shown, estimate the value of the integral:

$\int_{\textrm{path}} \vec{F} \cdot d\vec{r}$

where the side of each square is 1 cm and the length of the longest arrow is 10 units (appropriate for the field $\vec{F})$.

Discussion: We particularly recommend students use the Chop, Multiply, Add strategy, but there are other reasonable methods here.

Units: Students might ask about the units of $\vec{F}$. This is a great place to talk about how for an integral, the units of the result are the units of the integrand times the units of the infinitesimal (students often ignore the units of the infinitesimal). Also, for E&M, students are going to do a similar integral with the electric field - which does not have units of force - so being mindful about the units of the field is important.

Answer: For the leftward path, students need to be able to chop the path into segments (we recommend 4 segments), find the value of the dot product for that path, and then add up those four values to get the line integral.

For the upward path, the path is always perpendicular to the vector field, so the line integral is zero.

For the diagonal path (the hardest path), students need to chop up the path and consider the different magnitude at each point, but they also need to account for the nontrivial angle between the path and the vector field. This can be done in several ways---students will often try to find an explicit form for $\vec{dr}$, but they can also use a geometric argument to use only the vertical component of the path.

Keywords
E&M Path integrals
Learning Outcomes