Superconductivity from Formulation to the Meissner Effect
By Erik V. McGrath
Faculty Advisor: Dr. Edward Deveney
Meissner Effect
Two high temperature
superconducting disks (HTS).
The one on the left is
YBa2Cu3O7 (Tc=93K), the one
on the left is Bi2Sr2Can1CunO2n+4+x, with n=1,2,3
(Tc=95-108K),
The Meissner effect is the expulsion of a
magnetic field from a superconductor during its
transition to the superconducting state. This
remarkable phenomenon is the physical basis for
magnetic levitation, or maglev, which is being
exploited to create faster, more fuel efficient
bullet trains.
This can be easily demonstrated in the
undergraduate lab using simple procedures and
precautions and the equipment shown here.
Mathematical Treatment of the
Expulsion of the Magnetic Field Lines
The same material disks as
above but these are encased in
thick brass slugs and connected
to electrical leads in order to
facilitate taking data. The leads
themselves form a four-point
probe which, when attached to
a voltmeter and the proper
thermocouple, permits
extremely detailed
measurements with common,
low cost equipment.
A permanent magnet locked
above a surface in the
superconducting state. The
maximum height of the magnet
depends on the strength of the
magnetic field excluded. Image
courtesy of Wikipedia.
The mathematics and physics required to
derive the London equations are very advanced,
but the demonstration of the key part of the
Meissner effect is well within the capabilities of a
third year physics undergraduate student.
Using the London Equations and identities
included on this presentation as well as a solid
knowledge and proficiency with vector calculus the
manipulations below follow relatively simply.
The London Equations
js is the superconducting current
ns is a phenomenological constant
unique to the material
Insulated aluminum cryostat for
safely cooling the individual disks
left. It is filled with insulating
beads to allow liquid nitrogen to
safely boil without spillige or
sputtering as the disks cool.
The London Equations relate the
superconducting current to the magnetic
and electric fields.
Useful Identities
B is the magnetic field
E is the electric field
A is the vector-potential
V is the scalar potential
K is an arbitrary constant.
By applying these known
quantities and vector identities to
the London Equations we can show
mathematically that the magnetic
field inside a superconducting
material is identically zero beyond
many skin depths.
The characteristic skin depth is
left to a more advanced treatment of
these equations and methods.
It is this final line that is so remarkable. No other
quantitative theory can yet explain this peculiar
property of materials in the superconducting state.
All graphics created by Erik McGrath or obtained through images in the public domain or under creative commons licenses.
These are not superconducting
materials, rather they are
thermocouples made of the same
primary metals in the HTS in
order to minimize systematic
error when taking data in the lab.
Dissimilar materials create an
additional thermocoupling effect
that distorts the voltages read
across the leads of the four-point
probe.
Diagram of the Meissner effect.
Magnetic field lines,
represented as arrows, are
excluded from a
superconductor when it is
below its critical temperature.
Image and description courtesy
of Wikipedia.