## Activity: Visualizing Plane Waves

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector $\vec{k}$ and is asked to calculate the value of $\vec{k} \cdot \vec {r}$ for each point on the grid and to draw the set of points with constant value of $\vec{k} \cdot \vec{r}$ using rainbow colors to indicate increasing value.

What students learn Geometric understanding of what's planar about plane waves.

On your whiteboard, there should be a 5x5 square grid of dots. The instructor will draw a specific vector $\vec{k}$ on your grid.

For your $\vec{k}$, connect dots with the same value of $\vec{k} \cdot \vec{r}$.

## Instructor's Guide

Each small group of 3-4 students is given a white board or piece of paper with a square grid of points on it.

Each group is given a different two-dimensional vector $\vec{k}$ and is asked to calculate the value of $\vec{k} \cdot \vec{r}$ for each point on the grid and to draw the set of points with constant value of $\vec{k} \cdot \vec{r}$ using rainbow colors to indicate increasing value.

### Prerequisites

Students should know both the rectangular component and the geometric definitions of the dot product. Some of this activity can only be used if students know a little about complex number algebra, including Euler's formula.

### Student Conversations

The group part of this activity should be quite quick, 5-10 minutes.

• Go around to each group and mark an origin on the grid and a vector from the origin $\vec{k}$. Choose simple vectors, not longer than a few grid points. Choose a variety of vectors so that the compare and contrast wrap-up will work well.
• Some groups will have trouble getting started, particularly if they try to use the geometric definition of the dot product. Ask them if they know anything else about the dot product. This activity works well if preceded by Dot Product Review which prompts students to think about various definitions and representations of the dot product.
• Some groups will have trouble understanding what is meant by the position vector. Do not leave them stuck here. It is usually sufficient to draw one. If necessary, ask what the components are of the vector you have drawn.
• Prompt the students to connect the points with constant value of $\vec{k} \cdot \vec{r}$. Remind them to color code these connections with rainbow colors. Don't give away the punchline by asking the to draw the “lines” of constant $\vec{k} \cdot \vec{r}$.

### Wrap-up

This is a compare and contrast activity. Ask each group to present. They should show their white board, show their vector $\vec{k}$, and their curves of constant $k$.

Points that should arise:

• The points of constant $\vec{k} \cdot \vec{r}$ are straight lines.
• These lines are perpendicular to $\vec{k}$.
• The spacing between the lines is smaller when $\vec{k}$ is longer.

Questions to ask students:

1. Why are the lines of constant $\vec{k} \cdot \vec{r}$ perpendicular to $\vec{k}$?

Usually someone in the class can come up with the explanation that all the position vectors $\vec{r}$ that have the same projection onto $\vec{k}$ have the same value of $\vec{k} \cdot \vec{r}$.

This projection is easiest to interpret when I think about the dot product as $\vec{k} \cdot \vec{r} = |\vec{k}|\color{blue}{|\vec{r}| \cos\theta}$, where $|\vec{r}| \cos\theta$ is the projection of $\vec{r}$ onto $\vec{k}$.

This is a good time to remind the students that they have had to use two different representations of the dot product to completely understand this problem.

2. What if the grid of points was made a 3D cube of points? What would constant $\vec{k} \cdot \vec{r}$ look like?

Constant $\vec{k} \cdot \vec{r}$ are planes.

3. What does $\cos(\vec{k} \cdot \vec{r})$ look like?

Planes that vary in value from -1 to 1 sinusoidally.

4. What does $E_0\cos(\vec{k} \cdot \vec{r})$ look like?

Planes that vary in value from -$E_0$ to $E_0$ sinusoidally.

5. What does $E_0\cos(\vec{k} \cdot \vec{r}-\omega t)$ look like?

Planes that move in the direction of $\vec{k}$ at a rate of $\omega/k$.

• assignment Divergence through a Prism

assignment Homework

##### Divergence through a Prism
Static Fields 2022 (5 years)

Consider the vector field $\vec F=(x+2)\hat{x} +(z+2)\hat{z}$.

1. Calculate the divergence of $\vec F$.
2. In which direction does the vector field $\vec F$ point on the plane $z=x$? What is the value of $\vec F\cdot \hat n$ on this plane where $\hat n$ is the unit normal to the plane?
3. Verify the divergence theorem for this vector field where the volume involved is drawn below. (“Verify” means calculate both sides of the divergence theorem, separately, for this example and show that they are the same.)

• assignment Current from a Spinning Cylinder

assignment Homework

##### Current from a Spinning Cylinder
A solid cylinder with radius $R$ and height $H$ has its base on the $x,y$-plane and is symmetric around the $z$-axis. There is a fixed volume charge density on the cylinder $\rho=\alpha z$. If the cylinder is spinning with period $T$:
1. Find the volume current density.
2. Find the total current.
• assignment Flux through a Plane

assignment Homework

##### Flux through a Plane
Static Fields 2022 (3 years) Find the upward pointing flux of the vector field $\boldsymbol{\vec{H}}=2z\,\boldsymbol{\hat{x}} +\frac{1}{x^2+1}\boldsymbol{\hat{y}}+(3+2z)\boldsymbol{\hat{z}}$ through the rectangle $R$ with one edge along the $y$ axis and the other in the $xz$-plane along the line $z=x$, with $0\le y\le2$ and $0\le x\le3$.
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keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• assignment Current in a Wire

assignment Homework

##### Current in a Wire
Static Fields 2022 (3 years) The current density in a cylindrical wire of radius $R$ is given by $\vec{J}(\vec{r})=\alpha s^3\cos^2\phi\,\hat{z}$. Find the total current in the wire.
• keyboard Electrostatic potential of a square of charge

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120 min.

##### Electrostatic potential of a square of charge
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Students write python programs to compute the potential due to a square of surface charge, and then to visualize the result. This activity can be used to introduce students to the process of integrating numerically.
• group The Hillside

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
• group Vector Surface and Volume Elements

group Small Group Activity

30 min.

##### Vector Surface and Volume Elements
Static Fields 2022 (3 years)

Integration Sequence

Students use known algebraic expressions for vector line elements $d\vec{r}$ to determine all simple vector area $d\vec{A}$ and volume elements $d\tau$ in cylindrical and spherical coordinates.

This activity is identical to Scalar Surface and Volume Elements except uses a vector approach to find directed surface and volume elements.

• assignment Magnetic Field and Current

assignment Homework

##### Magnetic Field and Current
Static Fields 2022 (3 years) Consider the magnetic field $\vec{B}(s,\phi,z)= \begin{cases} 0&0\le s<a\\ \alpha \frac{1}{s}(s^4-a^4)\, \hat{\phi}&a<s<b\\ 0&s>b \end{cases}$
1. Use step and/or delta functions to write this magnetic field as a single expression valid everywhere in space.
2. Find a formula for the current density that creates this magnetic field.
3. Interpret your formula for the current density, i.e. explain briefly in words where the current is.
• assignment Flux through a Paraboloid

assignment Homework

##### Flux through a Paraboloid
Static Fields 2022 (5 years)

Find the upward pointing flux of the electric field $\vec E =E_0\, z\, \hat z$ through the part of the surface $z=-3 s^2 +12$ (cylindrical coordinates) that sits above the $(x, y)$--plane.

Learning Outcomes