5.2 Critical State Model
85
F v = −η
B
φ 0
v,
(5.42)
where η is the viscous coefficient. When (5.41) is divided by the magnetic flux density
B and v is eliminated using (4.41), we have
J − J c −
η
φ 0 B
E = 0.
(5.43)
This can be rewritten to the E-J characteristics of (4.48). Hence, the viscosity is
related to the flow resistivity as
η =
φ 0 B
ρ f
.
(5.44)
As discussed above, the critical state model forms the analytic system in which
B and v are used as variables, and the force balance equation and the continuity
equation for flux lines are used for analysis. It is also possible to use the electric
field E and Maxwell’s (2.49) instead of v and the continuity equation for flux lines.
The present method using v is beneficial, however, because of the analogy with
mechanical systems.
5.3 Reversible Flux Motion
The critical state model correctly describes irreversible electromagnetic phenomena
in superconductors, as shown in the last section, although the flux pinning phenomena
originate from energetic interaction between flux lines and defects, and such
individual interactions are in principle reversible. In fact, detailed observations
reveal some reversible phenomena. Such reversible phenomena, i.e., electromagnetic
phenomena associated with reversible flux motion, are introduced in this section. This
seems to be contradictory to the principle of the critical state model. The reason why
the irreversibility occurs will be discussed in Sect. 5.4.
When the external magnetic field is decreased after increasing to some strength,
the critical state model assumes that the current keeps its critical current density J c ,
but sharply changes its direction across the branching point of the magnetic flux
distribution, as shown in Fig. 5.10. This is a rough approximation. If we look at
the magnetic flux distribution in more detail, however, the distribution is found to
be slightly different from the description by the model. Since magnetic flux lines
elastically repel each other through their magnetic interaction, the magnetic flux
distribution does not change sharply at the branching point but changes gradually
in space. The current also gradually changes its density from J c to −J c around the
branching point. The characteristic length of this spatial variation is typically on the
order of 1 μm, and if this length is negligible in comparison with the size of the
superconductor, the macroscopic description by the critical state model is sufficient.
85
F v = −η
B
φ 0
v,
(5.42)
where η is the viscous coefficient. When (5.41) is divided by the magnetic flux density
B and v is eliminated using (4.41), we have
J − J c −
η
φ 0 B
E = 0.
(5.43)
This can be rewritten to the E-J characteristics of (4.48). Hence, the viscosity is
related to the flow resistivity as
η =
φ 0 B
ρ f
.
(5.44)
As discussed above, the critical state model forms the analytic system in which
B and v are used as variables, and the force balance equation and the continuity
equation for flux lines are used for analysis. It is also possible to use the electric
field E and Maxwell’s (2.49) instead of v and the continuity equation for flux lines.
The present method using v is beneficial, however, because of the analogy with
mechanical systems.
5.3 Reversible Flux Motion
The critical state model correctly describes irreversible electromagnetic phenomena
in superconductors, as shown in the last section, although the flux pinning phenomena
originate from energetic interaction between flux lines and defects, and such
individual interactions are in principle reversible. In fact, detailed observations
reveal some reversible phenomena. Such reversible phenomena, i.e., electromagnetic
phenomena associated with reversible flux motion, are introduced in this section. This
seems to be contradictory to the principle of the critical state model. The reason why
the irreversibility occurs will be discussed in Sect. 5.4.
When the external magnetic field is decreased after increasing to some strength,
the critical state model assumes that the current keeps its critical current density J c ,
but sharply changes its direction across the branching point of the magnetic flux
distribution, as shown in Fig. 5.10. This is a rough approximation. If we look at
the magnetic flux distribution in more detail, however, the distribution is found to
be slightly different from the description by the model. Since magnetic flux lines
elastically repel each other through their magnetic interaction, the magnetic flux
distribution does not change sharply at the branching point but changes gradually
in space. The current also gradually changes its density from J c to −J c around the
branching point. The characteristic length of this spatial variation is typically on the
order of 1 μm, and if this length is negligible in comparison with the size of the
superconductor, the macroscopic description by the critical state model is sufficient.