## Activity: Travelling wave solution

Contemporary Challenges 2022 (4 years)
Students work in a small group to write down an equation for a travelling wave.
• Media

This activity follows Introducing quantum mechanics.

Write a function $y(x,t)$ that describes the motion of this string: A wave with an amplitude of 0.1 m, it travels to the right, and has a wavelength of 8 m. In one second it travels half a meter to the right.

You may put the origin of your coordinate system wherever you want. In your final expressions, do not use any symbolic variables except for $y$, $x$ and $t$. Please do remember (as always!) to include any units that are associated with constant numbers.

First try to construct this wave on your big whiteboards and then you can check your answer by putting it into Desmos.

##### Extra fun
Show that your answer is a solution to the wave equation: \begin{align} \frac{\partial^2 y}{\partial t^2} &= v^2 \frac{\partial^2 y}{\partial x^2} \end{align}

• assignment Series Convergence

assignment Homework

##### Series Convergence

Power Series Sequence (E&M)

Static Fields 2022 (5 years)

Recall that, if you take an infinite number of terms, the series for $\sin z$ and the function itself $f(z)=\sin z$ are equivalent representations of the same thing for all real numbers $z$, (in fact, for all complex numbers $z$). This is not always true. More commonly, a series is only a valid, equivalent representation of a function for some more restricted values of $z$. The technical name for this idea is convergence--the series only "converges" to the value of the function on some restricted domain, called the “interval” or “region of convergence.”

Find the power series for the function $f(z)=\frac{1}{1+z^2}$. Then, using the Mathematica worksheet from class (vfpowerapprox.nb) as a model, or some other computer algebra system like Sage or Maple, explore the convergence of this series. Where does your series for this new function converge? Can you tell anything about the region of convergence from the graphs of the various approximations? Print out a plot and write a brief description (a sentence or two) of the region of convergence. You may need to include a lot of terms to see the effect of the region of convergence. Keep adding terms until you see a really strong effect!

Note: As a matter of professional ettiquette (or in some cases, as a legal copyright requirement), if you use or modify a computer program written by someone else, you should always acknowledge that fact briefly in whatever you write up. Say something like: “This calculation was based on a (name of software package) program titled (title) originally written by (author) copyright (copyright date).”

• group Heat capacity of N$_2$

group Small Group Activity

30 min.

##### Heat capacity of N2
Contemporary Challenges 2022 (4 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 Power from the Ocean

assignment Homework

##### Power from the Ocean
heat engine efficiency Energy and Entropy 2021 (2 years)

It has been proposed to use the thermal gradient of the ocean to drive a heat engine. Suppose that at a certain location the water temperature is $22^\circ$C at the ocean surface and $4^{o}$C at the ocean floor.

1. What is the maximum possible efficiency of an engine operating between these two temperatures?

2. If the engine is to produce 1 GW of electrical power, what minimum volume of water must be processed every second? Note that the specific heat capacity of water $c_p = 4.2$ Jg$^{-1}$K$^{-1}$ and the density of water is 1 g cm$^{-3}$, and both are roughly constant over this temperature range.

• assignment Heat capacity of vacuum

assignment Homework

##### Heat capacity of vacuum
Heat capacity entropy Thermal and Statistical Physics 2020
1. Solve for the heat capacity of a vacuum, given the above, and assuming that photons represent all the energy present in vacuum.
2. Compare the heat capacity of vacuum at room temperature with the heat capacity of an equal volume of water.
• group Hydrogen emission

group Small Group Activity

30 min.

##### Hydrogen emission
Contemporary Challenges 2022 (5 years)

In this activity students work out energy level transitions in hydrogen that lead to visible light.
• group Flux through a Cone

group Small Group Activity

30 min.

##### Flux through a Cone
Static Fields 2022 (4 years)

Integration Sequence

Students calculate the flux from the vector field $\vec{F} = C\, z\, \hat{z}$ through a right cone of height $H$ and radius $R$ .
• face Introducing entropy

face Lecture

30 min.

##### Introducing entropy
Contemporary Challenges 2022 (4 years)

This lecture introduces the idea of entropy, including the relationship between entropy and multiplicity as well as the relationship between changes in entropy and heat.
• 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.
• 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.
• group Grey space capsule

group Small Group Activity

30 min.

##### Grey space capsule
Contemporary Challenges 2022 (4 years)

In this small group activity, students work out the steady state temperature of an object absorbing and emitting blackbody radiation.

Learning Outcomes