3.1 Electricity in a Conductor and Magnetism in a Superconductor
37
In this case, the calculation of the magnetic flux density can also be done with the
magnetic potential instead of the vector potential. It is found from Fig. 3.7 that the
magnetic potential decreases monotonically with motion along the direction of the
magnetic flux density. Since the magnetic flux lines are closed, however, the magnetic
potential does not take the initial value when going back to the initial position. That
is, the magnetic potential is a multi-valued function, the value of which changes by
the same amount when circulating once. On the other hand, the vector potential takes
a constant value on the closed flux lines in Fig. 3.7, and the equi-vector potential
surfaces are of the same form as the equipotential surfaces in Fig. 3.6.
3.2 Prediction of Superconductivity
As is seen in the last section, the electric phenomena in a conductor showing E = 0
and the magnetic phenomena in a superconductor showing B = 0 are analogous to
each other. This is an outstanding analogy in the present E-Banalogy. If this analogy
was believed, someone might have happened to predict the superconductor, a material
showing perfect diamagnetism, even just after the formulation of Maxwell’s theory
in the 19th century. People know now that the resistivity of this material is zero. It
could be easily shown that the resistivity of a material with perfect diamagnetism
must be zero. That is, the prediction of a perfectly diamagnetic material is equivalent
to the prediction of superconductivity.
Here, the proof of the zero-resistivity is treated. It may be supposed that the zeroresistivity can be easily proved, since the current must continue to flow to shield
the magnetic field produced by the outer current, as in the case of Fig. 3.7. We
have to consider the point, however, that such shielding can be accomplished by the
diamagnetism of the material itself, as will be mentioned later. Since this type of
material was not discovered in the 19th century, it was not possible to attribute the
diamagnetic response to the shielding by the current or the intrinsic diamagnetism.
Hence, we have to consider the case in which a current is directly applied to a
diamagnetic material. In this case the current must flow on the surface of the material,
as shown in (3.6). As shown in Fig. 3.8, here, we suppose that rectangle C is located
inside the material in such a way that one side is placed on the surface and parallel to
the current. The current density is integrated on C. The integrated value is not zero
on the side on the surface, while the contribution from other three sides is zero. That
is, we have
C
i · ds = 0.
(3.23)
If Ohm’s law E = ρ r i holds, (3.23) leads to
37
In this case, the calculation of the magnetic flux density can also be done with the
magnetic potential instead of the vector potential. It is found from Fig. 3.7 that the
magnetic potential decreases monotonically with motion along the direction of the
magnetic flux density. Since the magnetic flux lines are closed, however, the magnetic
potential does not take the initial value when going back to the initial position. That
is, the magnetic potential is a multi-valued function, the value of which changes by
the same amount when circulating once. On the other hand, the vector potential takes
a constant value on the closed flux lines in Fig. 3.7, and the equi-vector potential
surfaces are of the same form as the equipotential surfaces in Fig. 3.6.
3.2 Prediction of Superconductivity
As is seen in the last section, the electric phenomena in a conductor showing E = 0
and the magnetic phenomena in a superconductor showing B = 0 are analogous to
each other. This is an outstanding analogy in the present E-Banalogy. If this analogy
was believed, someone might have happened to predict the superconductor, a material
showing perfect diamagnetism, even just after the formulation of Maxwell’s theory
in the 19th century. People know now that the resistivity of this material is zero. It
could be easily shown that the resistivity of a material with perfect diamagnetism
must be zero. That is, the prediction of a perfectly diamagnetic material is equivalent
to the prediction of superconductivity.
Here, the proof of the zero-resistivity is treated. It may be supposed that the zeroresistivity can be easily proved, since the current must continue to flow to shield
the magnetic field produced by the outer current, as in the case of Fig. 3.7. We
have to consider the point, however, that such shielding can be accomplished by the
diamagnetism of the material itself, as will be mentioned later. Since this type of
material was not discovered in the 19th century, it was not possible to attribute the
diamagnetic response to the shielding by the current or the intrinsic diamagnetism.
Hence, we have to consider the case in which a current is directly applied to a
diamagnetic material. In this case the current must flow on the surface of the material,
as shown in (3.6). As shown in Fig. 3.8, here, we suppose that rectangle C is located
inside the material in such a way that one side is placed on the surface and parallel to
the current. The current density is integrated on C. The integrated value is not zero
on the side on the surface, while the contribution from other three sides is zero. That
is, we have
C
i · ds = 0.
(3.23)
If Ohm’s law E = ρ r i holds, (3.23) leads to