156
7 Concluding Remarks
to release the distortion. In fact, the force-free torque, independent of the Lorentz
force, is derived as the general driving force from the penetration of energy during the
virtual displacement to introduce the distortion. This is quite similar to the derivation
of the Lorentz force that corresponds to the distortions of the magnetic structure in
the usual transverse magnetic field (see Appendix A.8). Since the observed critical
current density depends on the flux pinning strength, similarly to that in the transverse
magnetic field, it is considered that the minimization of the total free energy density,
including the pinning energy, is essential. Thus, it is reasonable that the equation of
torque balance to determine the critical current density is derived from first principles. As a result, if the increase in energy by introducing the force-free distortion
cannot be stored as the pinning energy, the distorted state cannot be stabilized. The
solution of rotational motion of flux lines driven by the excess of the force-free torque
over the pinning torque is derived from the continuity equation for flux lines, and
the electric field structure on the surface in the resistive state is explained. Thus, the
longitudinal magnetic field effect, which is quite different from the electromagnetic
phenomena in the transverse magnetic field, is generally explained. As shown above,
the essential mechanism that causes the longitudinal magnetic field effect is the flux
pinning that stabilizes the force-free distortion. It has been believed that the theory
of electromagnetism was completed in the 19th century, but a new door has been
opened now by the introduction of the longitudinal magnetic field effect.
Zero electrical resistivity is a specific character of superconductors, and application of this property to various devices is expected. In many cases superconductors
are used in a high transverse magnetic field, and flux lines penetrate the superconductor in the form of quantized magnetic flux. The central area of each quantized
magnetic flux line must be in the normal state, however, so as to prevent the superconducting current density from diverging due to the singularity in the phase of the
order parameter. As a result, if flux lines are driven by the Lorentz force when a
current is applied to the superconductor, energy dissipation takes place, and electrical resistance appears. If the flux motion can be stopped by introducing defects,
we can apply a transport current without appearance of the electrical resistance.
This is the principle of flux pinning and is the origin for various applications of
superconductivity.
In the static state, there is no energy dissipation caused by flux pinning interactions. Under varying conditions, however, energy dissipation occurs, and electromagnetic phenomena in the superconductor become irreversible. This irreversibility
can be explained by the critical state model, which assumes that the flux pinning
interactions minimize the variation in the magnetic flux distribution inside the superconductor. The energy dissipation in this case is of the hysteretic type, such as iron
loss, and its value does not depend on the resistivity, but rather on the critical current
density, although the mechanism of energy dissipation is the same as for copper
loss. Its dependence on the frequency in the case of alternating variation is also
different: It is proportional to the frequency, while the copper loss is proportional
to the second power of the frequency. The irreversibility and hysteretic nature of
the pinning loss can be explained by the theory of the flux pinning mechanism that
assumes that there is a many body interaction between flux lines and randomly
7 Concluding Remarks
to release the distortion. In fact, the force-free torque, independent of the Lorentz
force, is derived as the general driving force from the penetration of energy during the
virtual displacement to introduce the distortion. This is quite similar to the derivation
of the Lorentz force that corresponds to the distortions of the magnetic structure in
the usual transverse magnetic field (see Appendix A.8). Since the observed critical
current density depends on the flux pinning strength, similarly to that in the transverse
magnetic field, it is considered that the minimization of the total free energy density,
including the pinning energy, is essential. Thus, it is reasonable that the equation of
torque balance to determine the critical current density is derived from first principles. As a result, if the increase in energy by introducing the force-free distortion
cannot be stored as the pinning energy, the distorted state cannot be stabilized. The
solution of rotational motion of flux lines driven by the excess of the force-free torque
over the pinning torque is derived from the continuity equation for flux lines, and
the electric field structure on the surface in the resistive state is explained. Thus, the
longitudinal magnetic field effect, which is quite different from the electromagnetic
phenomena in the transverse magnetic field, is generally explained. As shown above,
the essential mechanism that causes the longitudinal magnetic field effect is the flux
pinning that stabilizes the force-free distortion. It has been believed that the theory
of electromagnetism was completed in the 19th century, but a new door has been
opened now by the introduction of the longitudinal magnetic field effect.
Zero electrical resistivity is a specific character of superconductors, and application of this property to various devices is expected. In many cases superconductors
are used in a high transverse magnetic field, and flux lines penetrate the superconductor in the form of quantized magnetic flux. The central area of each quantized
magnetic flux line must be in the normal state, however, so as to prevent the superconducting current density from diverging due to the singularity in the phase of the
order parameter. As a result, if flux lines are driven by the Lorentz force when a
current is applied to the superconductor, energy dissipation takes place, and electrical resistance appears. If the flux motion can be stopped by introducing defects,
we can apply a transport current without appearance of the electrical resistance.
This is the principle of flux pinning and is the origin for various applications of
superconductivity.
In the static state, there is no energy dissipation caused by flux pinning interactions. Under varying conditions, however, energy dissipation occurs, and electromagnetic phenomena in the superconductor become irreversible. This irreversibility
can be explained by the critical state model, which assumes that the flux pinning
interactions minimize the variation in the magnetic flux distribution inside the superconductor. The energy dissipation in this case is of the hysteretic type, such as iron
loss, and its value does not depend on the resistivity, but rather on the critical current
density, although the mechanism of energy dissipation is the same as for copper
loss. Its dependence on the frequency in the case of alternating variation is also
different: It is proportional to the frequency, while the copper loss is proportional
to the second power of the frequency. The irreversibility and hysteretic nature of
the pinning loss can be explained by the theory of the flux pinning mechanism that
assumes that there is a many body interaction between flux lines and randomly