## Student handout: Blackbody PhET

Contemporary Challenges 2021 (4 years)
Students use a PhET to explore properties of the Planck distribution.

Google “phet blackbody spectrum”' and open the simulation.

1. At what wavelength is the peak in spectral intensity
• $\lambda_{\text{peak}}$ for a black rock on the Earth's surface,
• $\lambda_{\text{peak}}$ for the black walls of a pizza oven,
• $\lambda_{\text{peak}}$ for a light bulb,
• $\lambda_{\text{peak}}$ for the sun.
2. Check that the peak wavelength decreases with temperature following a $1/T$ relationship.
1. Use the numerical integration feature (the checkbox labelled “intensity” near the upper-right corner of the graph) to find the total intensity, in units of $\text{W/m}^2$, emitted by
• a black rock on the Earth's surface,
• the black walls of a pizza oven,
• the surface of a tungsten light bulb filament,
• the surface of the sun.
2. Check that these intensities are proportional to $T^4$. Note, the quick way to check involves ratios: Does $\frac{I_1}{I_2} = \left(\frac{T_1}{T_2}\right)^4$?
1. How cold should you make an object if you want zero thermal radiation emitted?
2. (Extra---if your group has time)
1. For an incandescent light bulb with a filament surface area of $A$, estimate how efficiently it converts electrical energy into visible photons. Hint: you will need to estimate the following ratio: \begin{align*} \frac{\text{Electromagnetic radiation in visible wavelengths}}{ \text{Total electromagnetic radiation} } = \frac{ A\int_{400\text{ nm}}^{700\text{ nm}} S_\lambda(\lambda, T)d\lambda }{ A\int_{0}^{\infty} S_\lambda(\lambda, T)d\lambda } \end{align*}
2. Estimate the filament surface area $A$ for a 60 W light bulb.

• face Wavelength of peak intensity

face Lecture

5 min.

##### Wavelength of peak intensity
Contemporary Challenges 2021 (3 years)

This very short lecture introduces Wein's displacement law.
• assignment Light bulb in a refrigerator

assignment Homework

##### Light bulb in a refrigerator
Carnot refridgerator Work Entropy Thermal and Statistical Physics 2020 A 100W light bulb is left burning inside a Carnot refridgerator that draws 100W. Can the refridgerator cool below room temperature?
• assignment Carbon monoxide poisoning

assignment Homework

##### Carbon monoxide poisoning
Equilibrium Absorbtion Thermal and Statistical Physics 2020

In carbon monoxide poisoning the CO replaces the $\textsf{O}_{2}$ adsorbed on hemoglobin ($\text{Hb}$) molecules in the blood. To show the effect, consider a model for which each adsorption site on a heme may be vacant or may be occupied either with energy $\varepsilon_A$ by one molecule $\textsf{O}_{2}$ or with energy $\varepsilon_B$ by one molecule CO. Let $N$ fixed heme sites be in equilibrium with $\textsf{O}_{2}$ and CO in the gas phases at concentrations such that the activities are $\lambda(\text{O}_2) = 1\times 10^{-5}$ and $\lambda(\text{CO}) = 1\times 10^{-7}$, all at body temperature $37^\circ\text{C}$. Neglect any spin multiplicity factors.

1. First consider the system in the absence of CO. Evaluate $\varepsilon_A$ such that 90 percent of the $\text{Hb}$ sites are occupied by $\textsf{O}_{2}$. Express the answer in eV per $\textsf{O}_{2}$.

2. Now admit the CO under the specified conditions. Fine $\varepsilon_B$ such that only 10% of the Hb sites are occupied by $\textsf{O}_{2}$.

• group Fourier Transform of the Delta Function

group Small Group Activity

5 min.

##### Fourier Transform of the Delta Function
Periodic Systems 2022

Fourier Transforms and Wave Packets

Students calculate the Fourier transform of the Dirac delta function.
• assignment Fourier Transform of Cosine and Sine

assignment Homework

##### Fourier Transform of Cosine and Sine
Periodic Systems 2022
1. Find the Fourier transforms of $f(x)=\cos kx$ and $g(x)=\sin kx$.
2. Find the Fourier transform of $g(x)$ using the formula for the Fourier transform of a derivative and your result for the Fourier transform of $f(x)$. Compare with your previous answer.
3. In quantum mechanics, the Fourier transform is the set of coefficients in the expansion of a quantum state in terms of plane waves, i.e. the function $\tilde{f}(k)$ is a continuous histogram of how much each functions $e^{ikx}$ contributes to the quantum state. What does the Fourier transform of the function $\cos kx$ tell you about which plane waves make up this quantum state? Write a sentence or two about how this makes sense.
• group Thermal radiation at twice the temperature

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2021 (4 years)

This small group activity has students reasoning about how the Planck distribution shifts when the temperature is doubled. This leads to a qualitative argument for the Stefan-Boltzmann law.
• group Guess the Fourier Series from a Graph

group Small Group Activity

10 min.

##### Guess the Fourier Series from a Graph
Oscillations and Waves 2023 (2 years) The students are shown the graph of a function that is a superposition of three harmonic functions and asked to guess the harmonic terms of the Fourier series. Students then use prewritten Sage code to verify the coefficients from their guess. The program allows the students to enter functions of their own choice as well as the one that is preset.
• face Chemical potential and Gibbs distribution

face Lecture

120 min.

##### Chemical potential and Gibbs distribution
Thermal and Statistical Physics 2020

These notes from the fifth week of Thermal and Statistical Physics cover the grand canonical ensemble. They include several small group activities.
• group Optical depth of atmosphere

group Small Group Activity

30 min.

##### Optical depth of atmosphere
Contemporary Challenges 2021 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
• assignment Circle Trigonometry

assignment Homework

##### Circle Trigonometry
trigonometry cosine sine math circle Quantum Fundamentals 2023 (3 years)

On the following diagrams, mark both $\theta$ and $\sin\theta$ for $\theta_1=\frac{5\pi}{6}$ and $\theta_2=\frac{7\pi}{6}$. Write one to three sentences about how these two representations are related to each other. (For example, see: this PHET)  Learning Outcomes