## Activity: Wavelength of peak intensity

Contemporary Challenges 2022 (3 years)
This very short lecture introduces Wein's displacement law.

This lecture is needed if Wein's law homework is used.
We saw previously that the spectral intensity can be expressed with respect to wavelength: \begin{align} S_{\lambda}(\lambda) &= \frac{2\pi h c^2}{\lambda^5}\frac1{e^{\frac{hc}{\lambda k_BT}}-1} \end{align} I mentioned that the peak intensity shifts to lower wavelengths at higher temperature. We can solve for the peak in the spectral intensity by taking a derivative. The resultin equation can

but the result is a non-linear equation that is a bit of a pain. So it's convenient to just have an equation. The result is known as Wien's displacement law, and states that \begin{align} \lambda_{\text{peak}} &= \frac{b}{T} \end{align} where $b=2.9\times 10^{-3}\text{ m K}$ is called Wien's displacement constant.

• computer Blackbody PhET

computer Computer Simulation

30 min.

##### Blackbody PhET
Contemporary Challenges 2022 (4 years)

Students use a PhET to explore properties of the Planck distribution.
• group Thermal radiation at twice the temperature

group Small Group Activity

10 min.

##### Thermal radiation at twice the temperature
Contemporary Challenges 2022 (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 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 Spring Force Constant

assignment Homework

##### Spring Force Constant
Energy and Entropy 2021 (2 years) The spring constant $k$ for a one-dimensional spring is defined by: $F=k(x-x_0).$ Discuss briefly whether each of the variables in this equation is extensive or intensive.
• group A glass of water

group Small Group Activity

30 min.

##### A glass of water
Energy and Entropy 2021 (2 years)

Students generate a list of properties a glass of water might have. The class then discusses and categorizes those properties.
• face Energy and Entropy review

face Lecture

5 min.

##### Energy and Entropy review
Thermal and Statistical Physics 2020 (3 years)

This very quick lecture reviews the content taught in Energy and Entropy, and is the first content in Thermal and Statistical Physics.
• group Optical depth of atmosphere

group Small Group Activity

30 min.

##### Optical depth of atmosphere
Contemporary Challenges 2022 (4 years) In this activity students estimate the optical depth of the atmosphere at the infrared wavelength where carbon dioxide has peak absorption.
• group Work By An Electric Field (Contour Map)

group Small Group Activity

30 min.

##### Work By An Electric Field (Contour Map)

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.
• face Entropy and Temperature

face Lecture

120 min.

##### Entropy and Temperature
Thermal and Statistical Physics 2020

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.
• 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.

Learning Outcomes