computer Computer Simulation

30 min.

Approximating Functions with Power Series
Static Fields 2022 (7 years)

Taylor series power series approximation

Power Series Sequence (E&M)

Students use prepared Sage code or a prepared Mathematica notebook to plot \(\sin\theta\) simultaneously with several terms of a power series expansion to judge how well the approximation fits. Students can alter the worksheet to change the number of terms in the expansion and even to change the function that is being considered. Students should have already calculated the coefficients for the power series expansion in a previous activity, Calculating Coefficients for a Power Series.

assignment Homework

Series Convergence

Power Series Sequence (E&M)

Static Fields 2022 (4 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 Small Group Activity

30 min.

Calculating Coefficients for a Power Series
Static Fields 2022 (5 years)

Taylor series power series approximation

Power Series Sequence (E&M)

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

\[c_n={1\over n!}\, f^{(n)}(z_0)\]

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a \(\sin\theta\) (around both the origin and \(\frac{\pi}{6}\)). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

assignment Homework

Bottle in a Bottle
irreversible helium internal energy work first law Energy and Entropy 2021 (2 years)

The internal energy of helium gas at temperature \(T\) is to a very good approximation given by \begin{align} U &= \frac32 Nk_BT \end{align}

Consider a very irreversible process in which a small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated. What is the change in temperature when this process is complete? How much of the helium will remain in the small bottle?

assignment Homework

Translating Contours
Energy and Entropy 2021 (2 years)

Consider the diagram of \(T\) vs \(V\) for several different constant values of \(p\).

  1. Translate this diagram to a \(p\) vs \(V\) w/ constant \(T\) graph, including the point \(A\). Complete your graph by hand and make a fairly accurate sketch by printing out the attached grid or in some other way making nice square axes with appropriate tick marks.

  2. Are the lines that you drew straight or curved? What feature of the \(TV\) graph would have to change to change this result?

  3. Sketch the line of constant temperature that passes through the point \(A\).

  4. What are the values of all the thermodynamic variables associated with the point A?

group Small Group Activity

30 min.

Ideal Gas Model

Ideal Gas surfaces thermo

Students should be familiar with the basic definitions of pressure, volume, temperature, and internal energy.

assignment_ind Small White Board Question

10 min.

Partial Derivatives from a Contour Map
Static Fields 2022 (3 years) Students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?

assignment Homework

Yukawa
Central Forces 2021

In a solid, a free electron doesn't see” a bare nuclear charge since the nucleus is surrounded by a cloud of other electrons. The nucleus will look like the Coulomb potential close-up, but be screened” from far away. A common model for such problems is described by the Yukawa or screened potential: \begin{equation} U(r)= -\frac{k}{r} e^{-\frac{r}{\alpha}} \end{equation}

  1. Graph the potential, with and without the exponential term. Describe how the Yukawa potential approximates the “real” situation. In particular, describe the role of the parameter \(\alpha\).
  2. Draw the effective potential for the two choices \(\alpha=10\) and \(\alpha=0.1\) with \(k=1\) and \(\ell=1\). For which value(s) of \(\alpha\) is there the possibility of stable circular orbits?

group Small Group Activity

30 min.

Quantifying Change (Remote)

Thermo Derivatives

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.

assignment Homework

Approximating a Delta Function with Isoceles Triangles
Static Fields 2022 (4 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 \(\epsilon \rightarrow 0\).

assignment Homework

Gibbs entropy is extensive
Gibbs entropy Probability Thermal and Statistical Physics 2020

Consider two noninteracting systems \(A\) and \(B\). We can either treat these systems as separate, or as a single combined system \(AB\). We can enumerate all states of the combined by enumerating all states of each separate system. The probability of the combined state \((i_A,j_B)\) is given by \(P_{ij}^{AB} = P_i^AP_j^B\). In other words, the probabilities combine in the same way as two dice rolls would, or the probabilities of any other uncorrelated events.

  1. Show that the entropy of the combined system \(S_{AB}\) is the sum of entropies of the two separate systems considered individually, i.e. \(S_{AB} = S_A+S_B\). This means that entropy is extensive. Use the Gibbs entropy for this computation. You need make no approximation in solving this problem.
  2. Show that if you have \(N\) identical non-interacting systems, their total entropy is \(NS_1\) where \(S_1\) is the entropy of a single system.

Note
In real materials, we treat properties as being extensive even when there are interactions in the system. In this case, extensivity is a property of large systems, in which surface effects may be neglected.

group Small Group Activity

10 min.

Survivor Outer Space: A kinesthetic approach to (re)viewing center-of-mass
Central Forces 2021 A group of students, tethered together, are floating freely in outer space. Their task is to devise a method to reach a food cache some distance from their group.

assignment Homework

Tetrahedron
Static Fields 2022 (4 years)

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!)

group Small Group Activity

30 min.

Electric Potential of Two Charged Plates
Students examine a plastic "surface" graph of the electric potential due to two changes plates (near the center of the plates) and explore the properties of the electric potential.

assignment Homework

Derivatives from Data (NIST)
Energy and Entropy 2021 (2 years) Use the NIST web site “Thermophysical Properties of Fluid Systems” to answer the following questions. This site is an excellent resource for finding experimentally measured properties of fluids.
  1. Find the partial derivatives \[\left(\frac{\partial {S}}{\partial {T}}\right)_{p}\] \[\left(\frac{\partial {S}}{\partial {T}}\right)_{V}\] where \(p\) is the pressure, \(V\) is the volume, \(S\) is the entropy, and \(T\) is the temperature. Please find these derivatives for one gram of methanol at one atmosphere of pressure and at room temperature.
  2. Why does it take only two variables to define the state?
  3. Why are the derivatives above different?
  4. What do the words isobaric, isothermal, and isochoric mean?

accessibility_new Kinesthetic

10 min.

Acting Out Charge Densities
Static Fields 2022 (4 years)

density charge density mass density linear density uniform idealization

Ring Cycle Sequence

Integration Sequence

Students, pretending they are point charges, move around the room acting out various prompts from the instructor regarding charge densities, including linear \(\lambda\), surface \(\sigma\), and volume \(\rho\) charge densities, both uniform and non-uniform. The instructor demonstrates what it means to measure these quantities. In a remote setting, we have students manipulate 10 coins to model the prompts in this activity and the we demonstrate the answers with coins under a doc cam.

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 Homework

Bottle in a Bottle 2
heat entropy ideal gas Energy and Entropy 2021 (2 years)

Consider the bottle in a bottle problem in a previous problem set, summarized here.

A small bottle of helium is placed inside a large bottle, which otherwise contains vacuum. The inner bottle contains a slow leak, so that the helium leaks into the outer bottle. The inner bottle contains one tenth the volume of the outer bottle, which is insulated.

The volume of the small bottle is 0.001 m23 and the volume of the big bottle is 0.01 m3. The initial state of the gas in the small bottle was \(p=106\) Pa and its temperature \(T=300\) K. Approximate the helium gas as an ideal gas of equations of state \(pV=Nk_BT\) and \(U=\frac32 Nk_BT\).

  1. How many molecules of gas does the large bottle contain? What is the final temperature of the gas?

  2. Compute the integral \(\int \frac{{\mathit{\unicode{273}}} Q}{T}\) and the change of entropy \(\Delta S\) between the initial state (gas in the small bottle) and the final state (gas leaked in the big bottle).

  3. Discuss your results.

face Lecture

120 min.

Entropy and Temperature
Thermal and Statistical Physics 2020

paramagnet entropy temperature statistical mechanics

These lecture notes for the second week of Thermal and Statistical Physics involve relating entropy and temperature in the microcanonical ensemble, using a paramagnet as an example. These notes include a few small group activities.

group Small Group Activity

30 min.

Gravitational Force

Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics

Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.