*assignment*Photon carnot engine*assignment*Homework##### Photon carnot engine

Carnot engine Work Energy Entropy Thermal and Statistical Physics 2020In 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).

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 \(T_C\)).

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?

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

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.

*assignment*Linear Quadrupole (w/ series)*assignment*Homework##### Linear Quadrupole (w/ series)

Static Fields 2023 (6 years)Consider a collection of three charges arranged in a line along the \(z\)-axis: charges \(+Q\) at \(z=\pm D\) and charge \(-2Q\) at \(z=0\).

Find the electrostatic potential at a point \(\vec{r}\) in the \(xy\)-plane at a distance \(s\) from the center of the quadrupole. The formula for the electrostatic potential \(V\) at a point \(\vec{r}\) due to a charge \(Q\) at the point \(\vec{r'}\) is given by: \[ V(\vec{r})=\frac{1}{4\pi\epsilon_0} \frac{Q}{\vert \vec{r}-\vec{r'}\vert} \] Electrostatic potentials satisfy the superposition principle.

Assume \(s\gg D\). Find the first two non-zero terms of a power series expansion to the electrostatic potential you found in the first part of this problem.

- A series of charges arranged in this way is called a linear quadrupole. Why?

*assignment*Quantum harmonic oscillator*assignment*Homework##### Quantum harmonic oscillator

Entropy Quantum harmonic oscillator Frequency Energy Thermal and Statistical Physics 2020Find the entropy of a set of \(N\) oscillators of frequency \(\omega\) as a function of the total quantum number \(n\). Use the multiplicity function: \begin{equation} g(N,n) = \frac{(N+n-1)!}{n!(N-1)!} \end{equation} and assume that \(N\gg 1\). This means you can make the Sitrling approximation that \(\log N! \approx N\log N - N\). It also means that \(N-1 \approx N\).

Let \(U\) denote the total energy \(n\hbar\omega\) of the oscillators. Express the entropy as \(S(U,N)\). Show that the total energy at temperature \(T\) is \begin{equation} U = \frac{N\hbar\omega}{e^{\frac{\hbar\omega}{kT}}-1} \end{equation} This is the Planck result found the

**hard**way. We will get to the easy way soon, and you will never again need to work with a multiplicity function like this.

*assignment*Free energy of a harmonic oscillator*assignment*Homework##### Free energy of a harmonic oscillator

Helmholtz free energy harmonic oscillator Thermal and Statistical Physics 2020A one-dimensional harmonic oscillator has an infinite series of equally spaced energy states, with \(\varepsilon_n = n\hbar\omega\), where \(n\) is an integer \(\ge 0\), and \(\omega\) is the classical frequency of the oscillator. We have chosen the zero of energy at the state \(n=0\)

*which we can get away with here, but is not actually the zero of energy!*To find the true energy we would have to add a \(\frac12\hbar\omega\) for each oscillator.Show that for a harmonic oscillator the free energy is \begin{equation} F = k_BT\log\left(1 - e^{-\frac{\hbar\omega}{k_BT}}\right) \end{equation} Note that at high temperatures such that \(k_BT\gg\hbar\omega\) we may expand the argument of the logarithm to obtain \(F\approx k_BT\log\left(\frac{\hbar\omega}{kT}\right)\).

From the free energy above, show that the entropy is \begin{equation} \frac{S}{k_B} = \frac{\frac{\hbar\omega}{kT}}{e^{\frac{\hbar\omega}{kT}}-1} - \log\left(1-e^{-\frac{\hbar\omega}{kT}}\right) \end{equation}

This entropy is shown in the nearby figure, as well as the heat capacity.

*face*Thermal radiation and Planck distribution*face*Lecture120 min.

##### Thermal radiation and Planck distribution

Thermal and Statistical Physics 2020Planck distribution blackbody radiation photon statistical mechanics

These notes from the fourth week of Thermal and Statistical Physics cover blackbody radiation and the Planck distribution. They include a number of small group activities.*assignment*Centrifuge*assignment*Homework##### Centrifuge

Centrifugal potential Thermal and Statistical Physics 2020 A circular cylinder of radius \(R\) rotates about the long axis with angular velocity \(\omega\). The cylinder contains an ideal gas of atoms of mass \(M\) at temperature \(T\). Find an expression for the dependence of the concentration \(n(r)\) on the radial distance \(r\) from the axis, in terms of \(n(0)\)*on*the axis. Take \(\mu\) as for an ideal gas.*group*Energy radiated from one oscillator*group*Small Group Activity30 min.

##### Energy radiated from one oscillator

Contemporary Challenges 2021 (4 years) This lecture is one step in motivating the form of the Planck distribution.*face*Guide to Professional Typography in Physics*face*Lecture5 min.

##### Guide to Professional Typography in Physics

Contemporary Challenges 2021 (4 years) This is really a handout, which gives students guidelines on how to type up physics content.*group*Fourier Transform of a Plane Wave*accessibility_new*Using Arms to Represent Time Dependence in Spin 1/2 Systems*accessibility_new*Kinesthetic10 min.

##### Using Arms to Represent Time Dependence in Spin 1/2 Systems

Quantum Fundamentals 2023 (2 years) Students, working in pairs, use their left arms to demonstrate time evolution in spin 1/2 quantum systems.-
Energy and Entropy 2021 (2 years)
Find the differential of each of the following expressions; zap each of the following with \(d\):

\[f=3x-5z^2+2xy\]

\[g=\frac{c^{1/2}b}{a^2}\]

\[h=\sin^2(\omega t)\]

\[j=a^x\]

- \[k=5 \tan\left(\ln{\left(\frac{V_1}{V_2}\right)}\right)\]