format_list_numbered Sequence

##### Integration Sequence
Students learn/review how to do integrals in a multivariable context, using the vector differential $d\vec{r}=dx\, \hat{x}+dy\, \hat{y}+dz\, \hat{z}$ and its curvilinear coordinate analogues as a unifying strategy. This strategy is common among physicists, but is NOT typically taught in vector calculus courses and will be new to most students.

group Small Group Activity

10 min.

##### Fourier Transform of a Gaussian
Periodic Systems 2022

Fourier Transforms and Wave Packets

assignment Homework

##### Icecream Mass
Static Fields 2023 (6 years)

Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).

assignment Homework

##### Inner Product Properties
None 2023 The properties that an inner product on an abstract vector space must satisfy can be found in: Definition and Properties of an Inner Product. Definition: The inner product for any two vectors in the vector space of periodic functions with a given period (let's pick $2\pi$ for simplicity) is given by: $\left\langle {f}\middle|{g}\right\rangle =\int_0^{2\pi} f^*(x)\, g(x)\, dx$
1. Show that the first property of inner products $\left\langle {f}\middle|{g}\right\rangle =\left\langle {g}\middle|{f}\right\rangle ^*$ is satisfied for this definition.
2. Show that the second property of inner products $\left\langle {f}\right|\Big(\lambda\left|{g}\right\rangle + \mu \left|{h}\right\rangle \Big) = \lambda\left\langle {f}\middle|{g}\right\rangle +\mu\left\langle {f}\middle|{h}\right\rangle$ is satisfied for this definition.

assignment Homework

##### Potential energy of gas in gravitational field
Potential energy Heat capacity Thermal and Statistical Physics 2020 Consider a column of atoms each of mass $M$ at temperature $T$ in a uniform gravitational field $g$. Find the thermal average potential energy per atom. The thermal average kinetic energy is independent of height. Find the total heat capacity per atom. The total heat capacity is the sum of contributions from the kinetic energy and from the potential energy. Take the zero of the gravitational energy at the bottom $h=0$ of the column. Integrate from $h=0$ to $h=\infty$. You may assume the gas is ideal.

group Small Group Activity

30 min.

##### Vector Integrals (Contour Map)

Students explore path integrals using a vector field map and thinking about integration as chop-multiply-add.

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!)

assignment Homework

##### Sphere in Cylindrical Coordinates
Static Fields 2023 (4 years) Find the surface area of a sphere using cylindrical coordinates. Note: The fact that you can describe spheres nicely in cylindrical coordinates underlies the equal area cylindrical map project that allows you to draw maps of the earth where everything has the correct area, even if the shapes seem distorted. If you want to plot something like population density, you need an area preserving map projection.

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.

assignment Homework

##### Approximating a Delta Function with Isoceles Triangles
Static Fields 2023 (6 years)

Remember that the delta function is defined so that $\delta(x-a)= \begin{cases} 0, &x\ne a\\ \infty, & x=a \end{cases}$

Also: $\int_{-\infty}^{\infty} \delta(x-a)\, dx =1$.

1. Find a set of functions that approximate the delta function $\delta(x-a)$ with a sequence of isosceles triangles $\delta_{\epsilon}(x-a)$, centered at $a$, that get narrower and taller as the parameter $\epsilon$ approaches zero.
2. Using the test function $f(x)=3x^2$, find the value of $\int_{-\infty}^{\infty} f(x)\delta_{\epsilon}(x-a)\, dx$ Then, show that the integral approaches $f(a)$ in the limit that $\epsilon \rightarrow 0$.

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.

assignment Homework

##### Flux through a Paraboloid
Static Fields 2021 (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.

keyboard Computational Activity

120 min.

##### Electrostatic potential and Electric Field of a square of charge
Computational Physics Lab II 2023 (2 years)

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 Small Group Activity

30 min.

##### Total Charge
Static Fields 2023 (6 years)

Integration Sequence

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

group Small Group Activity

30 min.

##### Superposition States for a Particle on a Ring

Quantum Ring Sequence

Students calculate probabilities for a particle on a ring whose wavefunction is not easily separated into eigenstates by inspection. To find the energy, angular momentum, and position probabilities, students perform integrations with the wavefunction or decompose the wavefunction into a superposition of eigenfunctions.

assignment Homework

##### Einstein condensation temperature
Einstein condensation Density Thermal and Statistical Physics 2020

Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.

Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.

assignment Homework

##### Cube Charge
charge density

Integration Sequence

Static Fields 2023 (6 years)
1. Charge is distributed throughout the volume of a dielectric cube with charge density $\rho=\beta z^2$, where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge inside the cube? Do this problem in two ways as both a single integral and as a triple integral.
2. On a different cube: Charge is distributed on the surface of a cube with charge density $\sigma=\alpha z$ where $z$ is the height from the bottom of the cube, and where each side of the cube has length $L$. What is the total charge on the cube? Don't forget about the top and bottom of the cube.

assignment Homework

##### Dimensional Analysis of Kets
dirac notation dimensions probability completeness relations

Completeness Relations

1. $\left\langle {\Psi}\middle|{\Psi}\right\rangle =1$ Identify and discuss the dimensions of $\left|{\Psi}\right\rangle$.
2. For a spin $\frac{1}{2}$ system, $\left\langle {\Psi}\middle|{+}\right\rangle \left\langle {+}\middle|{\Psi}\right\rangle + \left\langle {\Psi}\middle|{-}\right\rangle \left\langle {-}\middle|{\Psi}\right\rangle =1$. Identify and discuss the dimensions of $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
3. In the position basis $\int \left\langle {\Psi}\middle|{x}\right\rangle \left\langle {x}\middle|{\Psi}\right\rangle dx = 1$. Identify and discuss the dimesions of $\left|{x}\right\rangle$.

assignment Homework

##### Flux through a Plane
Static Fields 2023 (4 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 $0\le x\le3$.

assignment Homework

##### Vapor pressure equation
phase transformation Clausius-Clapeyron Thermal and Statistical Physics 2020 Consider a phase transformation between either solid or liquid and gas. Assume that the volume of the gas is way bigger than that of the liquid or solid, such that $\Delta V \approx V_g$. Furthermore, assume that the ideal gas law applies to the gas phase. Note: this problem is solved in the textbook, in the section on the Clausius-Clapeyron equation.
1. Solve for $\frac{dp}{dT}$ in terms of the pressure of the vapor and the latent heat $L$ and the temperature.

2. Assume further that the latent heat is roughly independent of temperature. Integrate to find the vapor pressure itself as a function of temperature (and of course, the latent heat).