These lecture notes covering week 8 of https://paradigms.oregonstate.edu/courses/ph441 include a small group activity in which students derive the Carnot efficiency.

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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.

A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?

In our week on radiation, we saw that the Helmholtz free energy of a box of radiation at temperature \(T\) is \begin{align} F &= -8\pi \frac{V(kT)^4}{h^3c^3}\frac{\pi^4}{45} \end{align} From this we also found the internal energy and entropy \begin{align} U &= 24\pi \frac{(kT)^4}{h^3c^3}\frac{\pi^4}{45} V \\ S &= 32\pi kV\left(\frac{kT}{hc}\right)^3 \frac{\pi^4}{45} \end{align} Given these results, let us consider a Carnot engine that uses an empty metalic piston (i.e. a photon gas).

1. Given \(T_H\) and \(T_C\), as well as \(V_1\) and \(V_2\) (the two volumes at \(T_H\)), determine \(V_3\) and \(V_4\) (the two volumes at \(T_C\)).

2. What is the heat \(Q_H\) taken up and the work done by the gas during the first isothermal expansion? Are they equal to each other, as for the ideal gas?

3. Does the work done on the two isentropic stages cancel each other, as for the ideal gas?

4. Calculate the total work done by the gas during one cycle. Compare it with the heat taken up at \(T_H\) and show that the energy conversion efficiency is the Carnot efficiency.

1. Show that for a reversible heat pump the energy required per unit of heat delivered inside the building is given by the Carnot efficiency: \begin{align} \frac{W}{Q_H} &= \eta_C = \frac{T_H-T_C}{T_H} \end{align} What happens if the heat pump is not reversible?

2. Assume that the electricity consumed by a reversible heat pump must itself be generated by a Carnot engine operating between the even hotter temperature \(T_{HH}\) and the cold (outdoors) temperature \(T_C\). What is the ratio \(\frac{Q_{HH}}{Q_H}\) of the heat consumed at \(T_{HH}\) (i.e. fuel burned) to the heat delivered at \(T_H\) (in the house we want to heat)? Give numerical values for \(T_{HH}=600\text{K}\); \(T_{H}=300\text{K}\); \(T_{C}=270\text{K}\).

3. Draw an energy-entropy flow diagram for the combination heat engine-heat pump, similar to Figures 8.1, 8.2 and 8.4 in the text (or the equivalent but sloppier) figures in the course notes. However, in this case we will involve no external work at all, only energy and entropy flows at three temperatures, since the work done is all generated from heat.

• Work as area under curve in a \(pV\) plot
• Heat transfer as area under a curve in a \(TS\) plot
• Reminder that internal energy is a state function
• Reminder of First Law

This is the first activity relating the surfaces to the corresponding contour diagrams, thus emphasizing the use of multiple representations.

Students work in small groups to interpret level curves representing different concentrations of lead.

Students work in small groups 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.

Students work in small groups to use the Biot-Savart law \[\vec{B}(\vec{r}) =\frac{\mu_0}{4\pi}\int\frac{\vec{J}(\vec{r}^{\,\prime})\times \left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the magnetic field, \(\vec{B}(\vec{r})\), due to a spinning ring of charge.

In an optional extension, students find a series expansion for \(\vec{B}(\vec{r})\) 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.

Students work in small groups to use Coulomb's Law \[\vec{E}(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})\left(\vec{r}-\vec{r}^{\,\prime}\right)}{\vert \vec{r}-\vec{r}^{\,\prime}\vert^3} \, d\tau^{\prime}\] to find an integral expression for the electric field, \(\vec{E}(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(\vec{E}(\vec{r})\) 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.

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

This small group activity is designed to help students visualize the cross product. Students work in small groups to determine the area of a triangle in space. The whole class wrap-up discussion emphasizes the geometric interpretation of the cross product.

This small group activity introduces students to constrained optimization problems. Students work in small groups to optimize a simple function on a given region. The whole class wrap-up discussion emphasizes the importance of the boundary.

This small group activity using surfaces combines practice with the multivariable chain rule while emphasizing numerical representations of derivatives. Students work in small groups to measure partial derivatives in both rectangular and polar coordinates, then verify their results using the chain rule. The whole class wrap-up discussion emphasizes the relationship between a directional derivative in the \(r\)-direction and derivatives in \(x\)- and \(y\)-directions using the chain rule.

This small group activity is designed to provide practice with the chain rule and to develop familiarity with polar coordinates. Students work in small groups to relate partial derivatives in rectangular and polar coordinates. The whole class wrap-up discussion emphasizes the importance of specifying what quantities are being held constant.

This small group activity using surfaces introduces a geometric interpretation of partial derivatives in terms of measured ratios of small changes. Students work in small groups to identify locations on their surface with particular properties. The whole class wrap-up discussion emphasizes the equivalence of multiple representations of partial derivatives.

This small group activity is designed to help students visual the process of chopping, adding, and multiplying in single integrals. Students work in small groups to determine the volume of a cylinder in as many ways as possible. The whole class wrap-up discussion emphasizes the equivalence of different ways of chopping the cylinder.

Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). Different groups are assigned different arrangements of charges and different regions of space to consider: either on the axis of the charges or in the plane equidistant from the two charges, for either small or large values of the relevant geometric variable. Each group is asked to find a power series expansion for the electrostatic potential, valid in their group's assigned region of space. The whole class wrap-up discussion then compares and contrasts the results and discuss the symmetries of the two cases.

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

In this activity students work out energy level transitions in hydrogen that lead to visible light.

Students work in a small group to write down an equation for a travelling wave.

Students work in small groups to use the superposition principle \[V(\vec{r}) = \frac{1}{4\pi\epsilon_0}\sum_i \frac{q_i}{\vert\vec{r}-\vec{r}_i\vert}\] to find the electrostatic potential \(V\) everywhere in space due to a pair of charges (either identical charges or a dipole). This activity can be paired with activity 29 to find the limiting cases of the potential on the axes of symmetry.

These lecture notes for the first week of https://paradigms.oregonstate.edu/courses/ph441 include a couple of small group activities in which students work with the Gibbs formulation of the entropy.

Students work in small groups to use completeness relations to change the basis of quantum states.

There are two versions of this activity:

As a whole class activity, the instructor cuts a pumpkin in order to produce a small volume element \(d\tau\), interspersing their work with a sequence of small whiteboard questions. This version of the activity is described here.

As a small group activity, students are given pineapple rounds and pumpkin wedges to explore area and volume elements in cylindrical and spherical coordinate systems. In this version of the activity, the fruit is distributed to the students with appropriate children's pumpkin cutting equipment, as part of activities Vector Differential--Curvilinear, Scalar Surface and Volume Elements, or Vector Surface and Volume Elements.

Students work in groups to measure the steepest slope and direction at a given point on a plastic surface and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).

This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.

Students are asked to find eigenvalues, probabilities, and expectation values for \(H\), \(L^2\), and \(L_z\) for a superposition of \(\vert n \ell m \rangle\) states. This can be done on small whiteboards or with the students working in groups on large whiteboards.

Students then work together in small groups to find the matrices that correspond to \(H\), \(L^2\), and \(L_z\) and to redo \(\langle E\rangle\) in matrix notation.

Students work in small groups to use the superposition principle \[V(\vec{r}) =\frac{1}{4\pi\epsilon_0}\int\frac{\rho(\vec{r}^{\,\prime})}{\vert \vec{r}-\vec{r}^{\,\prime}\vert} \, d\tau^{\prime}\] to find an integral expression for the electrostatic potential, \(V(\vec{r})\), everywhere in space, due to a ring of charge.

In an optional extension, students find a series expansion for \(V(\vec{r})\) 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.