142
6 Longitudinal Magnetic Field Effect
Finally, we discuss the velocity component v 2 in (6.57). The first term that contains
v 2 is transformed to ρ ff (J − J c )z. The resistivity ρ ff in this case has been experimentally confirmed to be same as the value ρ f in the transverse magnetic field [20]. Thus,
v 2 can be estimated with this relationship.
6.5 Completion of Theory
It can be said that the framework of this theory of electromagnetic phenomena in the
longitudinal magnetic field is almost complete except for two points. One of them is
the problem of Josephson’s theory [7], which predicts that the force-free state is stable
without stabilization by flux pinning interactions. The other is that the pinning force
does not appear in the force-balance equation resulting in the force-free condition,
as shown by (6.2), whereas the flux pinning strength plays an important role in the
determination of the critical current density.
Here, we discuss the first point. Josephson assumed that the work done by the
external power source is equal to the variation in the inner free energy in the equilibrium state in a pin-free superconductor. We assume that the magnetic flux density
and current density in the superconductor are B and J, respectively. This situation is
described as
−
V
J · δAdV = 0,
(6.58)
where δA is a variation in the vector potential and V represents the region occupied
by the superconductor. Josephson assumed that relation given by
δA = δu × B
(6.59)
holds by choosing a suitable gauge. In the above δu is the displacement of flux lines
corresponding to the variation. If this holds, the left-hand side of (6.58) is written as
−
V
J · (δu × B)dV =
V
δu · (J × B)dV .
(6.60)
The condition given by
J × B = 0
(6.61)
is required so that (6.58) is fulfilled for arbitrary δu. Thus, the force-free state is
derived. This is the result of Josephson’s theory. Yet, the assumption of (6.59) is
6 Longitudinal Magnetic Field Effect
Finally, we discuss the velocity component v 2 in (6.57). The first term that contains
v 2 is transformed to ρ ff (J − J c )z. The resistivity ρ ff in this case has been experimentally confirmed to be same as the value ρ f in the transverse magnetic field [20]. Thus,
v 2 can be estimated with this relationship.
6.5 Completion of Theory
It can be said that the framework of this theory of electromagnetic phenomena in the
longitudinal magnetic field is almost complete except for two points. One of them is
the problem of Josephson’s theory [7], which predicts that the force-free state is stable
without stabilization by flux pinning interactions. The other is that the pinning force
does not appear in the force-balance equation resulting in the force-free condition,
as shown by (6.2), whereas the flux pinning strength plays an important role in the
determination of the critical current density.
Here, we discuss the first point. Josephson assumed that the work done by the
external power source is equal to the variation in the inner free energy in the equilibrium state in a pin-free superconductor. We assume that the magnetic flux density
and current density in the superconductor are B and J, respectively. This situation is
described as
−
V
J · δAdV = 0,
(6.58)
where δA is a variation in the vector potential and V represents the region occupied
by the superconductor. Josephson assumed that relation given by
δA = δu × B
(6.59)
holds by choosing a suitable gauge. In the above δu is the displacement of flux lines
corresponding to the variation. If this holds, the left-hand side of (6.58) is written as
−
V
J · (δu × B)dV =
V
δu · (J × B)dV .
(6.60)
The condition given by
J × B = 0
(6.61)
is required so that (6.58) is fulfilled for arbitrary δu. Thus, the force-free state is
derived. This is the result of Josephson’s theory. Yet, the assumption of (6.59) is