6.6 Comparison with Electromagnetic Phenomena in the Transverse Magnetic Field
149
avoid any confusion. It is sometimes useful, however, to clarify v to find essential
points such as the rotational motion.
In this section we treated only the simple process of applying a longitudinal
magnetic field and then applying a current, or changing the direction of the field to
realize the phenomena arising from the longitudinal magnetic field effects. There are
various cases where the longitudinal magnetic field is changed later, or the longitudinal magnetic field and current are simultaneously changed, etc. It is necessary to
clarify the distribution of the pinning energy to the force balance and torque balance
in each case. It is also necessary to clarify if the energy dissipation really takes on
its minimum value.
6.7 New Electromagnetism
Traditional electromagnetism was systematized by Maxwell, and its framework was
shown in Chap. 2. The fundamental principles are the Coulomb force (including
Coulomb’s law), the Lorentz force (including the Biot-Savart law), the displacement
current, and Faraday’s law for induction. The empirical Ohm’s law for the resistivity
is included to describe practical phenomena. Thus, the magnetic force is only the
Lorentz force. The quasi-static electromagnetic phenomena in superconductors in the
transverse magnetic field can be described by replacing Ohm’s law by the relationship
between E and J in the critical state theory, which is no longer a phenomenological
model. In the resistive state, the phenomenological dynamic critical state model that
assumes the flux flow resistivity is used.
In the superconductor in the longitudinal magnetic field, another magnetic general
force, the force-free torque, appears in addition to the Lorentz force. This torque was
not known in the traditional framework of Maxwell’s theory, although the existence
of this torque is not denied by Maxwell’s theory. In practice, the force-free torque
is derived using Maxwell’s theory. This is similar to the point that the electromagnetic phenomena in superconductors can be described by Maxwell’s theory, although
superconductivity was not discovered when Maxwell’s theory was completed. The
state in which the force-free torque appears is the state with a finite static magnetic
helicity. This state can be realized only in superconductors with the flux pinning
effect. For this reason, the discovery of the force-free torque was delayed. The new
framework of Maxwell’s theory is shown in Fig. 6.24. The current flow in the static
condition is determined by the critical state theory, taking into account the distribution of the pinning energy, and that in the dynamic condition is determined by the
phenomenological model extended to the resistive state.
The force-free torque appears only in superconductors and under the special condition of a longitudinal magnetic field. This is the reason why the discovery of the
force-free torque was delayed. There is another big reason. This can be understood
from the fact that this torque is an internal torque, as can be seen in Fig. 6.11. That is,
its value is zero when averaged inside the superconductor. The flux lines in the former
plane receive the torque shown by the arrows, while those in the next plane receive
149
avoid any confusion. It is sometimes useful, however, to clarify v to find essential
points such as the rotational motion.
In this section we treated only the simple process of applying a longitudinal
magnetic field and then applying a current, or changing the direction of the field to
realize the phenomena arising from the longitudinal magnetic field effects. There are
various cases where the longitudinal magnetic field is changed later, or the longitudinal magnetic field and current are simultaneously changed, etc. It is necessary to
clarify the distribution of the pinning energy to the force balance and torque balance
in each case. It is also necessary to clarify if the energy dissipation really takes on
its minimum value.
6.7 New Electromagnetism
Traditional electromagnetism was systematized by Maxwell, and its framework was
shown in Chap. 2. The fundamental principles are the Coulomb force (including
Coulomb’s law), the Lorentz force (including the Biot-Savart law), the displacement
current, and Faraday’s law for induction. The empirical Ohm’s law for the resistivity
is included to describe practical phenomena. Thus, the magnetic force is only the
Lorentz force. The quasi-static electromagnetic phenomena in superconductors in the
transverse magnetic field can be described by replacing Ohm’s law by the relationship
between E and J in the critical state theory, which is no longer a phenomenological
model. In the resistive state, the phenomenological dynamic critical state model that
assumes the flux flow resistivity is used.
In the superconductor in the longitudinal magnetic field, another magnetic general
force, the force-free torque, appears in addition to the Lorentz force. This torque was
not known in the traditional framework of Maxwell’s theory, although the existence
of this torque is not denied by Maxwell’s theory. In practice, the force-free torque
is derived using Maxwell’s theory. This is similar to the point that the electromagnetic phenomena in superconductors can be described by Maxwell’s theory, although
superconductivity was not discovered when Maxwell’s theory was completed. The
state in which the force-free torque appears is the state with a finite static magnetic
helicity. This state can be realized only in superconductors with the flux pinning
effect. For this reason, the discovery of the force-free torque was delayed. The new
framework of Maxwell’s theory is shown in Fig. 6.24. The current flow in the static
condition is determined by the critical state theory, taking into account the distribution of the pinning energy, and that in the dynamic condition is determined by the
phenomenological model extended to the resistive state.
The force-free torque appears only in superconductors and under the special condition of a longitudinal magnetic field. This is the reason why the discovery of the
force-free torque was delayed. There is another big reason. This can be understood
from the fact that this torque is an internal torque, as can be seen in Fig. 6.11. That is,
its value is zero when averaged inside the superconductor. The flux lines in the former
plane receive the torque shown by the arrows, while those in the next plane receive