15.1.The History of Superconductivity
Superconductivity was first discovered in 1911 by H.Kammerlingh Onnes at leiden. He observed that as a sample of mercury was cooled, its resistance disappeared abruptly and apparently completely at 4.2 K. In a more sensitive experiment using persistent current induced in a loop of superconducting wire, Kammerlingh Onnes estimated the resistance in the superconducting state to be at most 10-12 of the resistance in the normal state. In more recent experiments at the Massachusetts Institute of Technology, it was found that an induced current of several hundred amperes in a superconducting lead ring showed no change in the magnitude of the current for a period of at least one year, which provides strong evidance that the resistance in the superconducting state is indeed zero. The early experiments opened up a whole field of endeavor to characterize the new effect. It has been found that more than 20 elements and hundreds of alloys and intermetallic compounds are superconductors with transition temperatures ranging from subtantially less than 1K (e.g., 0.12K for hafnium) to about 23 K, as well as perovskite compounds containing copper-oxygen bonds (the latter materials having transition temperatures higher than 100 K in some cases). The transition, or critical, temperature is the temperature at which the transition from the normal state to the superconducting state takes place and is characteristic of the particular material being concidered. The critical temperature depends to some extent on both the chemical purity and crystalline perfection of the sample being tested. Actually, inhomogeneities in the purity and strain of the sample generally tend to broaden the temperature range of the transition between the normal and superconducting states. A pure well-annelead sample may have a transition-temperature range as small as 0.001 K.
Hc = H0[1-(T/Tc)2]
Where Hc is the critical field, T is the absolute (or Kelvin) temperature of observation, and Tc and H0 represent the sample characteristics (critical temperature at zero field and critical field at zero absolute temperature). In addition to broadening the transition, inhomogeneities may also have a marked effect on H0, sometimes increasing it by orders of magnitude. Such effects are of major importance in applications with high magnetic fields.
15.2.Perfect Conductivity and Perfect Diamagnetism of Superconductors
We noted in the preceding section that superconductors exhibit two unique properties. They have essentially infinite conductivity, and they also exclude magnetic flux as demonstrated by the Meissner-Ochsenteld experiment(so long as the magnetic field at the surface of the superconductor nowhere exceeds the critical field). These properties are independent in the sense that neither implies the other but, of course, both must and do emerge from what is meant by the independence of those two properties, we examine the behavior of a perfect conductor, but one that does not exhibit the Meissner effect in a magnetic field.
Consider a sphere whose conductivity can be switched from a finite value to infinity in some way, for example, by changing its temperature. When the conductivity is infinite the electric field is zero everywhere inside the material, and consequently its curl and are also zero. Thus, if the sphere is cooled (attaining perfect conductivity) in a uniform field B0, the flux density in the sphere remains B0, even if the applied field is reduced to zero, until the perfect conductivity is destroyed. On the other hand, if the sphere is cooled in zero field, the flux density remains zero until the perfect conductivity is switched back to normal conductivity. Thus perfect conductivity does not imply flux exclusion and consequently B = 0 is a postulate that must be introduced separately. Similiarly B = 0 does not imply perfect conductivity, for a material with susceptibility xm = -1 would always have B = 0, and this would not restrict the possible conductivity of the material.
15.3.Examples Involving Perfect Flux Exclusion
To reinforce the ideas presented in the preceding section, we will cosider two elementary examples: a superconducting sphere in an asymptotically uniform field and an infinitely long, current-carrying, superconducting cylinder. Both formulations of section 15.2 will be used to show explicitly that they are equivalent in these cases.
B = H outside
inside
15.4.The London Equation
In the preceding section, flux exclusion was discussed on the basis of a highly idealized representation of a superconductor. This representation reproduces many of the observed features of superconductivity but fails to account adequately for some of the readily observable details. A more sophisticated theory can be developed by starting from the concept of perfect conductivity and making an appropriate modification to include the Meissner effect.
In a perfect conductor (not a superconductor), charge carries would experience no retarding forces. Consequently, in an electric field E they would move according to
Mpv = qE
Where mp is the mass of the charge carrier and v is its accelerecation
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