Problem Set 7
Physics 140A
Due Thursday February 26, 2009
Late HW accepted until class on Tuesday March 3
Do Myers 6.3, plus the following:
1. An extremely pure aluminum piece has length 20 mm and cross-sectional area 2 mm2. One end is at
a temperature of 0.008 K, the other at 0.050 K. In this temperature range phonons dominate the
thermal conductivity of aluminum. (It’s because the aluminum is superconducting, but that’s for
next quarter.) The thermal conductivity of the phonons is κ= 0.3λT 3W/cm2K4, where λis the
phonon mean free path.
a) What is λ? (Hint: what are the consequences of the aluminum’s being “extremely pure” and the
temperature’s being low?)
b) If the system is in steady state, so that the heat flow is constant, find the temperature T(x) at
all spots along the aluminum piece.
2. Find the free electron density of states N(E) in two dimensions.
3. Find the total kinetic energy of a three-dimensional gas of Nfree electrons at zero temperature. (Note
that free electrons don’t have potential energy, by definition, so the entire energy is kinetic.)
4. The atom 3He has spin 1
2, so it’s a fermion. The density of liquid 3He is 0.081 g/cm3near T= 0.
a) Calculate the Fermi energy and Fermi temperature.
b) If one liter of 3He gas at room temperature costs about $200, calculate the cost of one liter of
liquid 3He. (This has nothing to do with the rest of the problem set, but it gives you some
practice thinking about relevant units and produces a jaw-dropping number.)
