## Circle Vector, Version 2

• assignment Tetrahedron

assignment Homework

##### Tetrahedron
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Using a dot product, find the angle between any two line segments that join the center of a regular tetrahedron to its vertices. Hint: Think of the vertices of the tetrahedron as sitting at the vertices of a cube (at coordinates (0,0,0), (1,1,0), (1,0,1) and (0,1,1)---you may need to build a model and play with it to see how this works!)

• assignment Divergence through a Prism

assignment Homework

##### Divergence through a Prism
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

• assignment Directional Derivative

assignment Homework

##### Directional Derivative
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Imagine you're standing on a landscape with a local topography described by the function $f(x, y)= k x^{2}y$, where $k=20 \mathrm{\frac{m}{km^3}}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot $(3~\mathrm{km},2~\mathrm{km})$ and there is a cottage located at $(1~\mathrm{km}, 2~\mathrm{km})$. At the spot you're standing, what is the slope of the ground in the direction of the cottage? Plot the function $f(x, y)$ and also its level curves in your favorite plotting software. Does your result makes sense from the graph?

• 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
AIMS Maxwell AIMS 21 Static Fields Winter 2021

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.

• assignment Center of Mass for Two Uncoupled Particles

assignment Homework

##### Center of Mass for Two Uncoupled Particles
Central Forces Spring 2021

Consider two particles of equal mass $m$. The forces on the particles are $\vec F_1=0$ and $\vec F_2=F_0\hat{x}$. If the particles are initially at rest at the origin, find the position, velocity, and acceleration of the center of mass as functions of time. Solve this problem in two ways, with or without theorems about the center of mass motion. Write a short description comparing the two solutions.

• assignment Cone Surface

assignment Homework

##### Cone Surface
AIMS Maxwell AIMS 21 Static Fields Winter 2021

Using integration, find the surface area of a cone with height $H$ and radius $R$. Do this problem in both cylindrical and spherical coordinates.

• group Energy and Angular Momentum for a Quantum Particle on a Ring

group Small Group Activity

30 min.

##### Energy and Angular Momentum for a Quantum Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring using three different notations: Dirac bra-ket, matrix, and wave function. After calculating the angular momentum and energy measurement probabilities, students compare their calculation methods for notation.
• assignment Sum Shift

assignment Homework

##### Sum Shift
Central Forces Spring 2021

In each of the following sums, shift the index $n\rightarrow n+2$. Don't forget to shift the limits of the sum as well. Then write out all of the terms in the sum (if the sum has a finite number of terms) or the first five terms in the sum (if the sum has an infinite number of terms) and convince yourself that the two different expressions for each sum are the same:

1. $$\sum_{n=0}^3 n$$
2. $$\sum_{n=1}^5 e^{in\phi}$$
3. $$\sum_{n=0}^{\infty} a_n n(n-1)z^{n-2}$$

• face Quantum Reference Sheet

face Lecture

5 min.

##### Quantum Reference Sheet
Central Forces Spring 2021 Central Forces Spring 2021
• 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.
• AIMS Maxwell AIMS 21 Static Fields Winter 2021

Learn more about the geometry of $\vert \vec{r}-\vec{r'}\vert$ in two dimensions.

1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
2. Make a sketch of the graph $$\vert \vec{r} - \vec{a} \vert = 2$$

for each of the following values of $\vec a$: \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}

3. Derive a more familiar equation equivalent to $$\vert \vec r - \vec a \vert = 2$$ for arbitrary $\vec a$, by expanding $\vec r$ and $\vec a$ in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
4. Write a brief description of the geometric meaning of the equation $$\vert \vec r - \vec a \vert = 2$$