Download Exercise Sheet 11 + Solutions

Universität Bonn
Physikalisches Institut
Prof. Dr. Johann Kroha, Johannes Rentrop and Ammar Nejati
21.01.2015
http://www.kroha.uni-bonn.de/teaching/lectures-and-seminars/wt2014/tcmp
Condensed Matter Theory I — WS14/15
Exercise 11
(Please return your solutions before 12:00 h Tue. 27.01.2014 [room PI 1.055].)
11.1 Meißner effect in a thin superconducting slab
(10 points)
Consider a thin superconducting slab of thickness d, which is infinitely extended in the y- and zdirections, immersed in a uniform magnetic field B parallel to the slab surface. The superconductor
is described by the London equations given in the lecture.
a) Obtain the magnetic field distribution H(x) in the slab. What is the London penetration
depth, λL ?
b) Obtain the average magnetization of the slab in terms of its thickness.
11.2 Critical field and critical current of a type-I superconducting wire
(10 points)
a) Critical field. Obtain the critical field Hc of a macroscopic sample of type-I superconductor,
using the Ginzburg-Landau theory.
b) Critical current. The current flowing through a superconducting wire generates a magnetic
field B around the wire. When the field at the surface of the wire reaches the critical value
Hc , the superconductivity will collapse and the wire returns to the normal state. Therefore
the critical field and the critical current depend on the geometry of the wire.
Let’s consider a cylindrical wire of radius R.
b-1) Show that the magnetic field lines form concentric circles around the wire and calculate the
field strength B around the wire surface. Determine the critical current in the wire at which
the superconductivity breaks down.
b-2) Obtain the current density j(r) in the wire and show that the current flows only in layer of
thickness λL underneath the surface of the wire.
b-3) Typical values for the London penetration depth and for the critical field are λL = 500 Å
and Hc = 500 Oe, respectively. What is the current density averaged over a surface layer of
thickness λL ?