These are notes, essentially the equation sheet, from the final review session for https://paradigms.oregonstate.edu/courses/ph441.

As discussed in class, we can consider a black body as a large box with a small hole in it. If we treat the large box a metal cube with side length \(L\) and metal walls, the frequency of each normal mode will be given by: \begin{align} \omega_{n_xn_yn_z} &= \frac{\pi c}{L}\sqrt{n_x^2 + n_y^2 + n_z^2} \end{align} where each of \(n_x\), \(n_y\), and \(n_z\) will have positive integer values. This simply comes from the fact that a half wavelength must fit in the box. There is an additional quantum number for polarization, which has two possible values, but does not affect the frequency. Note that in this problem I'm using different boundary conditions from what I use in class. It is worth learning to work with either set of quantum numbers. Each normal mode is a harmonic oscillator, with energy eigenstates \(E_n = n\hbar\omega\) where we will not include the zero-point energy \(\frac12\hbar\omega\), since that energy cannot be extracted from the box. (See the Casimir effect for an example where the zero point energy of photon modes does have an effect.)

Note

This is a slight approximation, as the boundary conditions for light are a bit more complicated. However, for large \(n\) values this gives the correct result.

1. Show that the free energy is given by \begin{align} F &= 8\pi \frac{V(kT)^4}{h^3c^3} \int_0^\infty \ln\left(1-e^{-\xi}\right)\xi^2d\xi \\ &= -\frac{8\pi^5}{45} \frac{V(kT)^4}{h^3c^3} \\ &= -\frac{\pi^2}{45} \frac{V(kT)^4}{\hbar^3c^3} \end{align} provided the box is big enough that \(\frac{\hbar c}{LkT}\ll 1\). Note that you may end up with a slightly different dimensionless integral that numerically evaluates to the same result, which would be fine. I also do not expect you to solve this definite integral analytically, a numerical confirmation is fine. However, you must manipulate your integral until it is dimensionless and has all the dimensionful quantities removed from it!

2. Show that the entropy of this box full of photons at temperature \(T\) is \begin{align} S &= \frac{32\pi^5}{45} k V \left(\frac{kT}{hc}\right)^3 \\ &= \frac{4\pi^2}{45} k V \left(\frac{kT}{\hbar c}\right)^3 \end{align}

3. Show that the internal energy of this box full of photons at temperature \(T\) is \begin{align} \frac{U}{V} &= \frac{8\pi^5}{15}\frac{(kT)^4}{h^3c^3} \\ &= \frac{\pi^2}{15}\frac{(kT)^4}{\hbar^3c^3} \end{align}

(modified from K&K 4.6) We discussed in class that \begin{align} p &= -\left(\frac{\partial F}{\partial V}\right)_T \end{align} Use this relationship to show that

1. \begin{align} p &= -\sum_j \langle n_j\rangle\hbar \left(\frac{d\omega_j}{dV}\right), \end{align} where \(\langle n_j\rangle\) is the number of photons in the mode \(j\);

2. Solve for the relationship between pressure and internal energy.

Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.

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.

Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.

This is really just a handout.

9 Rules for Professional Typography in Physics

1. Avoid: \(2kg\)
   Instead use: \(2 \text{ kg}\)
   There is a space between the number and the unit (the number and the unit are separate “words”). Units are not italicized. This helps distinguish between \(2\text{ kg}\) (two kilograms) and \(2kg\) (\(2\) times \(k\) times \(g\)).
2. Avoid: 10^12, or 1E12
   Instead use: \(10^{12}\)
   In scientific writing, you have to use superscript. The only time for 1E12 is computer coding. The only time for 10^12 is email programs that don't have a superscript option.
3. Avoid: VLED or V_LED or \(V_{LED}\)
   Instead use: \(V_{\text{LED}}\)
   In scientific writing, you have to use subscripts. The only time for V_LED is email programs that don't have a subscript option. Note that text such as “LED” is not italicized. In LaTeX you can code this as V_{\text{LED}}.
4. Avoid: wavelength=d*sin(theta)
   Instead use: \(\lambda=d\sin\theta\)
   Algebraic variables are italicized. There are spaces on either side of the equals sign. The sine function is not italic. LaTeX and Microsoft Equation Editor will manage much of this for you.
5. Avoid: 10 Ohm
   Instead use: 10 \(\Omega\)
   In Microsoft word you can use the font called "Symbol" to get Greek letters. Alternatively, Latex and Microsoft Equation Editor also take care of Greek letters by typing (\Omega).
6. Avoid: Voltage (v)
   Instead use: Voltage (V)
   Units are case-sensitive. The symbol for the volt unit is capital V.
7. Avoid: \(\theta = 0.674740942\)
   Instead use: \(\theta = 0.67\) or \(\theta = 0.675\)
   It is unlikely your solution will require more than 1% accuracy.
8. Algebraic variables are defined in the text the first time they are used.
9. Use a consistent font size for equations and text.

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.

These notes from the fourth week of https://paradigms.oregonstate.edu/courses/ph441 cover blackbody radiation and the Planck distribution. They include a number of small group activities.

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.

1. Consider a system that may be unoccupied with energy zero, or occupied by one particle in either of two states, one of energy zero and one of energy \(\varepsilon\). Find the Gibbs sum for this system is in terms of the activity \(\lambda\equiv e^{\beta\mu}\). Note that the system can hold a maximum of one particle.

2. Solve for the thermal average occupancy of the system in terms of \(\lambda\).

3. Show that the thermal average occupancy of the state at energy \(\varepsilon\) is \begin{align} \langle N(\varepsilon)\rangle = \frac{\lambda e^{-\frac{\varepsilon}{kT}}}{\mathcal{Z}} \end{align}

4. Find an expression for the thermal average energy of the system.

5. Allow the possibility that the orbitals at \(0\) and at \(\varepsilon\) may each be occupied each by one particle at the same time; Show that \begin{align} \mathcal{Z} &= 1 + \lambda + \lambda e^{-\frac{\varepsilon}{kT}} + \lambda^2 e^{-\frac{\varepsilon}{kT}} \\ &= (1+\lambda)\left(1+e^{-\frac{\varepsilon}{kT}}\right) \end{align} Because \(\mathcal{Z}\) can be factored as shown, we have in effect two independent systems.