Students integrate numerically to find the electric field due to a cone of surface charge, and then visualize the result. This integral can be done in either spherical or cylindrical coordinates, giving students a chance to reason about which coordinate system would be more convenient.
It is very common for students to want to create one function that returns the entire vector. This can actually work well, but most often does not. It often results in code that is inordinately slow (as it computes all three components, and then throws two out). Students also tend tend to struggle as in practice it requires returning a tuple of values, which is a data structure that students don't know about. So please try hard to get students to write three separate functions! Students frequently have difficulties taking components of the electric field. A surprisingly common mistake is to just “take the \(x\) stuff”, resulting in something like \begin{align} E_x(x,y,z) &= k\sigma\int \frac{x-x'}{\Big((x-x')^2\Big)^{\frac32}}dA' \end{align} When I encounter this, I try to talk with students about the meaning of taking a component. It is either the thing in front of \(\hat x\), or \(E_x = \vec E\cdot \hat x\).
assignment Homework
Einstein condensation temperature Starting from the density of free particle orbitals per unit energy range \begin{align} \mathcal{D}(\varepsilon) = \frac{V}{4\pi^2}\left(\frac{2M}{\hbar^2}\right)^{\frac32}\varepsilon^{\frac12} \end{align} show that the lowest temperature at which the total number of atoms in excited states is equal to the total number of atoms is \begin{align} T_E &= \frac1{k_B} \frac{\hbar^2}{2M} \left( \frac{N}{V} \frac{4\pi^2}{\int_0^\infty\frac{\sqrt{\xi}}{e^\xi-1}d\xi} \right)^{\frac23} T_E &= \end{align} The infinite sum may be numerically evaluated to be 2.612. Note that the number derived by integrating over the density of states, since the density of states includes all the states except the ground state.
Note: This problem is solved in the text itself. I intend to discuss Bose-Einstein condensation in class, but will not derive this result.
face Lecture
120 min.
Fermi level fermion boson Bose gas Bose-Einstein condensate ideal gas statistical mechanics phase transition
These lecture notes from week 7 of Thermal and Statistical Physics apply the grand canonical ensemble to fermion and bosons ideal gasses. They include a few small group activities.assignment Homework
assignment Homework
Use integration to find the total mass of the icecream in a packed cone (both the cone and the hemisphere of icecream on top).
assignment Homework
group Small Group Activity
30 min.
assignment Homework
assignment Homework
assignment Homework
assignment Homework
In this entire problem, keep results to first order in the van der Waals correction terms \(a\) and $b.
Show that the entropy of the van der Waals gas is \begin{align} S &= Nk\left\{\ln\left(\frac{n_Q(V-Nb)}{N}\right)+\frac52\right\} \end{align}
Show that the energy is \begin{align} U &= \frac32 NkT - \frac{N^2a}{V} \end{align}
Show that the enthalpy \(H\equiv U+pV\) is \begin{align} H(T,V) &= \frac52NkT + \frac{N^2bkT}{V} - 2\frac{N^2a}{V} \\ H(T,p) &= \frac52NkT + Nbp - \frac{2Nap}{kT} \end{align}