## Memorize Power Series

• This problem is used in the following sequences
• assignment Memorize $d\vec{r}$

assignment Homework

##### Memorize $d\vec{r}$
Static Fields 2022 (2 years)

Write $\vec{dr}$ in rectangular, cylindrical, and spherical coordinates.

1. Rectangular: $$\vec{dr}=$$
2. Cylindrical: $$\vec{dr}=$$
3. Spherical: $$\vec{dr}=$$

• assignment One-dimensional gas

assignment Homework

##### One-dimensional gas
Ideal gas Entropy Tempurature Thermal and Statistical Physics 2020 Consider an ideal gas of $N$ particles, each of mass $M$, confined to a one-dimensional line of length $L$. The particles have spin zero (so you can ignore spin) and do not interact with one another. Find the entropy at temperature $T$. You may assume that the temperature is high enough that $k_B T$ is much greater than the ground state energy of one particle.
• face Thermal radiation and Planck distribution

face Lecture

120 min.

##### Thermal radiation and Planck distribution
Thermal and Statistical Physics 2020

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.
• assignment Spin Fermi Estimate

assignment Homework

##### Spin Fermi Estimate
Quantum Fundamentals 2022 The following two problems ask you to make Fermi estimates. In a good Fermi estimate, you start from basic scientific facts you already know or quantities that you can reasonably estimate based on your life experiences and then reason your way to estimate a quantity that you would not be able guess. You may look up useful conversion factors or constants. Use words, pictures, and equations to explain your reasoning:
1. Imagine that you send a pea-sized bead of silver through a Stern-Gerlach device oriented to measure the z-component of intrinsic spin. Estimate the total z-component of the intrinsic spin of the ball you would measure in the HIGHLY improbable case that every atom is spin up.
2. Protons, neutrons, and electrons are all spin-1/2 particles. Give a (very crude) order of magnitude estimate of the number of these particles in your body.
• assignment Potential vs. Potential Energy

assignment Homework

##### Potential vs. Potential Energy
Static Fields 2022 (4 years)

In this course, two of the primary examples we will be using are the potential due to gravity and the potential due to an electric charge. Both of these forces vary like $\frac{1}{r}$, so they will have many, many similarities. Most of the calculations we do for the one case will be true for the other. But there are some extremely important differences:

1. Find the value of the electrostatic potential energy of a system consisting of a hydrogen nucleus and an electron separated by the Bohr radius. Find the value of the gravitational potential energy of the same two particles at the same radius. Use the same system of units in both cases. Compare and the contrast the two answers.
2. Find the value of the electrostatic potential due to the nucleus of a hydrogen atom at the Bohr radius. Find the gravitational potential due to the nucleus at the same radius. Use the same system of units in both cases. Compare and contrast the two answers.
3. Briefly discuss at least one other fundamental difference between electromagnetic and gravitational systems. Hint: Why are we bound to the earth gravitationally, but not electromagnetically?

• assignment Heat of vaporization of ice

assignment Homework

##### Heat of vaporization of ice
Vaporization Heat Thermal and Statistical Physics 2020 The pressure of water vapor over ice is 518 Pa at $-2^\circ\text{C}$. The vapor pressure of water at its triple point is 611 Pa, at 0.01$^\circ\text{C}$ (see Estimate in $\text{J mol}^{-1}$ the heat of vaporization of ice just under freezing. How does this compare with the heat of vaporization of water?
• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (3 years)

Students sketch the temperature-dependent heat capacity of molecular nitrogen. They apply the equipartition theorem and compute the temperatures at which degrees of freedom “freeze out.”
• assignment Series Notation 1

assignment Homework

##### Series Notation 1

Power Series Sequence (E&M)

Static Fields 2022 (4 years)

Write out the first four nonzero terms in the series:

1. $\sum\limits_{n=0}^\infty \frac{1}{n!}$

2. $\sum\limits_{n=1}^\infty \frac{(-1)^n}{n!}$
3. $$\sum\limits_{n=0}^\infty {(-2)^{n}\,\theta^{2n}}$$

• assignment Contours

assignment Homework

##### Contours

Static Fields 2022 (4 years)

Shown below is a contour plot of a scalar field, $\mu(x,y)$. Assume that $x$ and $y$ are measured in meters and that $\mu$ is measured in kilograms. Four points are indicated on the plot.

1. Determine $\frac{\partial\mu}{\partial x}$ and $\frac{\partial\mu}{\partial y}$ at each of the four points.
2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of $\mu(x,y)$ using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
3. Evaluate the gradient of $h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3$ at the point $(x,y)=(3,-2)$.

• assignment Calculation of $\frac{dT}{dp}$ for water

assignment Homework

##### Calculation of $\frac{dT}{dp}$ for water
Clausius-Clapeyron Thermal and Statistical Physics 2020 Calculate based on the Clausius-Clapeyron equation the value of $\frac{dT}{dp}$ near $p=1\text{atm}$ for the liquid-vapor equilibrium of water. The heat of vaporization at $100^\circ\text{C}$ is $2260\text{ J g}^{-1}$. Express the result in kelvin/atm.
• Static Fields 2022 (2 years)

Look up and memorize the power series to fourth order for $e^z$, $\sin z$, $\cos z$, $(1+z)^p$ and $\ln(1+z)$. For what values of $z$ do these series converge?