ABSTRACT
In order to describe the Meissner effect1, F. and H. London2 modified a basic law of electrodynamics, the Ohm’s law. Maxwell’s equations always remain valid. To do this, they consider the superconductor as a two-fluid system, a fluid consisting of normal electrons and a fluid formed by superconducting electrons (idea originally introduced by Gorter and Casimir in 19343). According to this two-fluid model, the first fluid consists of ‘normal’ electrons, of number density nN, and these behave in exactly the same way as the free electrons in a normal metal. They are accelerated by an electric field E, but are frequently scattered by impurities and defects in the ion lattice and by thermal vibrations of the lattice. The scattering limits the speed of the electrons, and they attain a mean drift velocity:
vN = – et E (2.1)
Where t is the mean time between scattering events for the electrons and m is the electron mass. The current density JN due to flow of these electrons is:
EvJ NN m
en N
en N
2 (2.2)
The second fluid is formed by superconducting electrons with number density nS. The superconducting electrons are not scattered by impurities, defects or thermal vibrations, so they are freely accelerated by an electric field. If the velocity of a superconducting electron is vS, its equation of motion is:
E vS
em dt
d (2.3)
Combining this equation with the expression for the current density:
JS = – nSevS (2.4)
the equation becomes:
E vSJS
m
en en
t
t (2.5)
which is different to Equation 2.2. Scattering of the normal electrons leads to a constant current in a constant electric field, whereas the absence of scattering of the electrons in a superconductor means that the current density would increase progressively in a constant electric field. However, if one considers a constant current flowing in the superconductor, then:
0JS
t (2.6)
So:
E = 0 (2.7)
Therefore the normal current density must be zero, all of the steady current in a superconductor is carried by the superconducting electrons. Of course, with no electric field within the superconductor, there will be no potential difference across it, and so it has zero resistance. It was argued above that a material that just had the property of zero resistance, a perfect conductor rather than a superconductor, would maintain a constant magnetic field in its interior, and would not expel any field that was present when the material became superconducting. It shall now be shown how that conclusion follows from an application of Maxwell’s equations to a perfect conductor. One can then see what additional assumptions are needed to account for the Meissner effect in a superconductor. If it is assumed that the electrons in a perfect conductor (or a proportion of them) are not scattered, and therefore the current density is governed by Equation 2.5. However, the subscript ‘p’ shall be used (for perfect conductor) here to indicate that it is not superconductor that is being dealt with. The interest is in the magnetic field in a perfect conductor, so Maxwell’s equations will be applied to this situation. Faraday’s law, valid in all situations, is given by:
t
B Ecurl (2.8)
and if one substitutes E using Equation 2.5, one obtains:
t
BJP curl
m
en
t
Looking now at the Ampère-Maxwell law:
t
D JFHcurl (2.10)
it shall be assumed that our perfect conductor is either weakly diamagnetic or weakly paramagnetic, so that μ ~_ 1 and H ~_ B/μ0 are very good approximations. Maxwell’s term, ∂D/∂t shall also be omitted, since this is negligible for the static, or slowly-varying, fields that shall be considered. With these approximations, the Ampère-Maxwell law simplifies to Ampère’s law:
curl B = m0 JP (2.11)
where use of the subscript ‘p’ for the current density reminds us that the free current JF is carried by the perfectly-conducting electrons. One now uses this expression to eliminate JP from Equation 2.9:
t
B
t
B curlcurl
m
A standard vector identity from inside the back cover to rewrite the lefthand side of this equation:
t
B
t
B grad
t
B curlcurl 2div (2.13)
The no-monopole law, div B = 0, means that the first term on the right-hand side of this equation is zero, so Equation 2.12 can be rewritten as:
t
B
t
B
m
0 (2.14)
This equation determines how ∂B/∂t varies in a perfect conductor. One shall look for the solution of Equation 2.14. For the simple geometry shown in Fig. 2.1, a conductor has a boundary corresponding to the plane z = 0, and occupies the region z > 0, with a uniform field outside the conductor given by B0 = B0 ex. The uniform external field in the x-direction means that the field inside the conductor will also be in the x-direction, and its strength will depend only on z. So, Equation 2.14 reduces to the one-dimensional form:
t
t)( ,zB x
t
t)(z,B x
z 2
(2.15)
where the equation has been simplified, for reasons that will soon become clear, by writing:
m
en
(2.16)
The general solution of this equation is:
t
t)( ,zB x (2.17)
Where a and b are independent of the position. The second term on the right-hand side corresponds to a rate of change of field strength that continues to increase exponentially with distance from the boundary; since this is unphysical, one sets b = 0. The boundary condition for the field parallel to the boundary is that H// is continuous, and since it is being assumed that μ ~_ 1 in both the air and the conductor, this is equivalent to B// being the same on either side of the boundary at all times. This means that ∂B/∂t is the same on either side of the boundary, so:
t
a (2.18)
and the field within the perfect conductor satisfies the equation:
t
( )tB 0
t
t)( ,zB x (2.19)
This indicates that any changes in the external magnetic field are attenuated exponentially with distance below the surface of the perfect conductor. If the distance λp is very small, then the field will not change within the bulk of the perfect conductor. Note that this does not mean the magnetic field must be expelled: flux expulsion requires B = 0, rather than just ∂B/∂t = 0. So how does one modify the description that has been given of a perfect conductor so that it describes a superconductor and leads to a prediction that B = 0? In order to explain the Meissner effect, the London brothers proposed that in a superconductor, Equation 2.9 is replaced by the more restrictive relationship:
BJScurl m
(2.20)
This equation, and Equation 2.5 which relates the rate of change of current to the electric field, are known as the London equations. It is important to note that these equations are not an explanation of superconductivity.
