group Small Group Activity

30 min.

##### Covariation in Thermal Systems
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.

face Lecture

30 min.

##### Review of Thermal Physics
These are notes, essentially the equation sheet, from the final review session for Thermal and Statistical Physics.

• Found in: Thermal and Statistical Physics course(s)

group Small Group Activity

30 min.

##### Total Charge
In this small group activity, students integrate over non-uniform charge densities in cylindrical and spherical coordinates to calculate total charge.

• Found in: Static Fields, AIMS Maxwell course(s) Found in: Integration Sequence sequence(s)

None

##### Eigen Spin Challenge
Consider the arbitrary Pauli matrix $\sigma_n=\hat n\cdot\vec \sigma$ where $\hat n$ is the unit vector pointing in an arbitrary direction.
1. Find the eigenvalues and normalized eigenvectors for $\sigma_n$. The answer is: $\begin{pmatrix} \cos\frac{\theta}{2}e^{-i\phi/2}\\{} \sin\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix} \begin{pmatrix} -\sin\frac{\theta}{2}e^{-i\phi/2}\\{} \cos\frac{\theta}{2}e^{i\phi/2}\\ \end{pmatrix}$ It is not sufficient to show that this answer is correct by plugging into the eigenvalue equation. Rather, you should do all the steps of finding the eigenvalues and eigenvectors as if you don't know the answer. Hint: $\sin\theta=\sqrt{1-\cos^2\theta}$.
2. Show that the eigenvectors from part (a) above are orthogonal.
3. Simplify your results from part (a) above by considering the three separate special cases: $\hat n=\hat\imath$, $\hat n=\hat\jmath$, $\hat n=\hat k$. In this way, find the eigenvectors and eigenvalues of $\sigma_x$, $\sigma_y$, and $\sigma_z$.
• Found in: Quantum Fundamentals course(s)

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##### Distance Formula in Curvilinear Coordinates

The distance $\left\vert\vec r -\vec r\,{}'\right\vert$ between the point $\vec r$ and the point $\vec r'$ is a coordinate-independent, physical and geometric quantity. But, in practice, you will need to know how to express this quantity in different coordinate systems.

1. Find the distance $\left\vert\vec r -\vec r\,{}'\right\vert$ between the point $\vec r$ and the point $\vec r'$ in rectangular coordinates.
2. Show that this same distance written in cylindrical coordinates is: $$\left|\vec r -\vec r\,{}'\right| =\sqrt{s^2+s\,{}'^2-2ss\,{}'\cos(\phi-\phi\,{}') +(z-z\,{}')^2}$$
3. Show that this same distance written in spherical coordinates is: $$\left\vert\vec r -\vec r\,{}'\right\vert =\sqrt{r'^2+r\,{}^2-2rr\,{}' \left[\sin\theta\sin\theta\,{}'\cos(\phi-\phi\,{}') +\cos\theta\cos\theta\,{}'\right]}$$
4. Now assume that $\vec r\,{}'$ and $\vec r$ are in the $x$-$y$ plane. Simplify the previous two formulas.

• Found in: E&M Ring Cycle Sequence sequence(s) Found in: Static Fields, AIMS Maxwell course(s)

None

##### Electric Field and Charge
Consider the electric field $$\vec E(r,\theta,\phi) = \begin{cases} 0&\textrm{for } r<a\\ \frac{1}{4\pi\epsilon_0} \,\frac{Q}{b^3-a^3}\, \left( r-\frac{a^3}{r^2}\right)\, \hat r & \textrm{for } a<r<b\\ 0 & \textrm{for } r>b \\ \end{cases}$$
1. Use step and/or delta functions to write this electric field as a single expression valid everywhere in space.
2. Find a formula for the charge density that creates this electric field.
3. Interpret your formula for the charge density, i.e. explain briefly in words where the charge is.
• Found in: AIMS Maxwell, Static Fields course(s)

face Lecture

30 min.

face Lecture

120 min.

##### Ideal Gas
These notes from week 6 of Thermal and Statistical Physics cover the ideal gas from a grand canonical standpoint starting with the solutions to a particle in a three-dimensional box. They include a number of small group activities.

• Found in: Thermal and Statistical Physics course(s)