**Physics 798S**

**Superconductivity**

**Spring 2004**

**Homework 1**

**Due ****February 10, 2004**

Homework Policy

Your grade will be based on homework and a paper. In exchange for not giving exams, I ask that you do the homework. You may work on homework together, but not doing the homework will imperil your grade--I am willing to give bad grades if homework is not done.

Please hand in your homework on time. I will not accept late homework, unless a valid excuse (such as illness) is given, preferably before the homework is due.

Please do not do integrals by Mathematica when they can be done analytically. It is fine to use Mathematica for plotting functions, or checking the results of your calculations.

**1**. Screening in a superconducting slab. Solve the _{0} is applied
parallel to both surfaces. Find both the
magnetic field and the supercurrent inside the slab. As examples, plot the current and magnetic
field for a thickness 2t = l, and 2t =
10l.

**2**. Drude
insulator. Following the discussion of
the Drude conductor in class, consider a system of
independent charges q, each subjected to a force

a) Using the engineer’s time convention e^{i}^{wt},
find the frequency-dependent complex conductivity s(w). Sketch this conductivity (real and imaginary
parts) as a function of frequency.

b) What simple lumped-element circuit has an admittance A=1/Z with the same
frequency dependence?

c) Evaluate the
integral (s = s_{1} – i s_{2})

in the case where k = 0 (Drude metal). The residue theorem is useful. BEWARE: the algebra is messy; you will receive nearly complete credit for properly setting up the integral and explaining clearly how to proceed.

**3**. Two-fluid model. A
more realistic model for a superconductor assumes that there is a density n_{n} of normal electrons which obey a Drude-like equation

as well as a density n_{s} of superelectrons which obey a

a) Using the
engineer’s time convention e^{i}^{wt}, find the frequency-dependent
complex conductivity s(w).
Assume that each “fluid” responds independently to the electric field.

b) What simple lumped-element circuit has an admittance A=1/Z with the same frequency dependence?

c) Show that, in the low-frequency
limit, the normal-fluid response is purely ohmic,
while the superfluid response is purely inductive. In this limit, plot s_{1}(T) and s_{2}(T) vs
T using the empirical relationships

where n_{o} is the density
of electrons in the material. The
expression for n_{s}(T) is a fairly good
approximation for the superfluid density in a clean metal, but the second
expression is flawed: n_{s}(T)
+ n_{n}(T) is not equal to the total electron
density.