Date: Dec 18, 1998 (Friday)
Time allowed: 120 minutes.
Answer all questions.
1. Current distribution in a superconducting slab Consider
a semi-infinite superconductor filling the space z<0. For simplicity,
let us assume the sample has a length of L in the y direction and 
as shown in the figure. The penetration depth of the superconductor is 
.
The superconductor is carrying a current I flowing in the +x direction.
(a) What is the magnetic field just outside the superconductor? (b) Use
your result in (a) as boundary condition, find the magnetic field inside
the superconductor by solving London equation. (c) Determine the current
distribution as a function of z (i.e. 
).
Solution: HTML Format PDF Format
2. Magnetism of rare earth ions The basic electronic configurations
of 
, 
, and 
are 
, 
,
and 
respectively. (a) Determine the ground
state of these three ions with Hund's rule. (b) Assume the same ion density
n=N/V, compare the susceptibility of a 
compound
to that of 
at the same temperature. (c) Assume
the same ion density n=N/V, compare the susceptibility of a 
compound
to that of 
at the same temperature. The g-factor
is given by the Lande equation:
.
Solution: HTML Format PDF Format
3. Doublet excited states Some organic molecules have
a doublet (
) excited state at an energy
above
a singlet (S=0) ground state. (a) Find an expression for the magnetic moment 
in a field B. (b) Show that the susceptibility for 
is approximately independent of 
. (c) Will
this system be cooled by adiabatic magnetization, or demagnetization? Explain
your answer.
Solution: HTML Format PDF Format
4. Free electron gas (a) Show that the kinetic energy
of a free electron gas at absolute zero is 
where 
is
the Fermi energy. (b) Derive expressions for the pressure 
and
the bulk modulus 
. (c) Estimate the contribution
of the conduction electrons to B for potassium and compare your answer
to the experimentally measured bulk modulus 
.
Potassium has a bcc structure of lattice parameter 5.225
.
Solution: HTML
Format PDF
Format
5. Josephson Tunneling Experimental data from a Josephson
junction is shown in the figure. (a) What is the critical current 
?
(b) When no RF power is applied, what is the phase difference of phase
across the junction at I=1.0mA? (c) When RF radiation is applied, Shapiro
steps are formed. Judging from the figure, what is the frequency of the
microwave radiation?
Solution: HTML Format PDF Format