170
7 Concluding Remarks
minimize the energy dissipation, and most pinning energy is allotted for the torque
balance. This explains why the pinning force does not appear in the force balance,
resulting in the force-free state, whereas the critical current density is determined
by the flux pinning strength. All the electromagnetic phenomena of the longitudinal
magnetic field effect, including the negative electric field, can be generally explained
by the rotational flux motion driven by the force-free torque. The magnetic helicity
is retained, even in the situation of rotational flux motion. It is interesting to compare
the behavior of flux lines in the superconductor in a longitudinal magnetic field with
that of flux lines in a plasma that also has magnetic helicity. The electromagnetic
phenomena in the longitudinal magnetic field are essentially different from those
in the superconductor in the transverse magnetic field. The magnetic helicity in the
latter case is zero. Although the force-free torque appears only in the superconductor
in the longitudinal magnetic field, it surely adds a new page to Maxwell’s theory.
In addition, the characteristic force-free state appears only under the influence
of flux pinning. In other words, the longitudinal magnetic field effect is an effect in
which the flux pinning plays a dominant role and it is essentially irreversible. The
flux pinning effect has attracted attention in vortex physics as one of the factors that
determine the phase of the flux line system. This is the first case where the flux
pinning effect is dominant in the field of fundamental science.
The electromagnetic phenomena in the transverse magnetic field are described
by the phenomenological critical state model. It was shown that the assumption in
this model that the pinning force makes the maximum effort to prevent a variation
in the magnetic flux distribution does not contradict the principle of irreversible
thermodynamics of minimum energy dissipation. There are two kinds of balance in
determining the current direction with two degrees of freedom under a longitudinal
magnetic field, and it was shown again that the pinning energy is distributed so as
to minimize the energy dissipation, as assumed in the principle. It is an important
topic to prove theoretically that this principle holds generally, even in a nonlinear
dissipation system, as discussed in Appendix A.10. Detailed analyses of various
electromagnetic phenomena in the longitudinal magnetic field are not sufficient as
yet, and research by many scientists is needed in the future. Thus, the physics of the
longitudinal magnetic field effect is a new science, in which electromagnetism and
irreversible thermodynamics merge together. Efforts to deepen this field are expected.
Coffee break (7)
Poynting’s vector
One of most useful concepts besides the continuity equation of flux lines in this
book is Poynting’s vector. The energy that penetrates the superconductor while the
external magnetic field is changing can be estimated using Poynting’s vector. As
shown in this book, the variation in the internal magnetic energy is different from
this energy, indicating the possibility of the appearance or disappearance of energy.
This is the work done by the driving force. The force-free torque in the longitudinal
magnetic field effect is also derived using Poynting’s vector.
7 Concluding Remarks
minimize the energy dissipation, and most pinning energy is allotted for the torque
balance. This explains why the pinning force does not appear in the force balance,
resulting in the force-free state, whereas the critical current density is determined
by the flux pinning strength. All the electromagnetic phenomena of the longitudinal
magnetic field effect, including the negative electric field, can be generally explained
by the rotational flux motion driven by the force-free torque. The magnetic helicity
is retained, even in the situation of rotational flux motion. It is interesting to compare
the behavior of flux lines in the superconductor in a longitudinal magnetic field with
that of flux lines in a plasma that also has magnetic helicity. The electromagnetic
phenomena in the longitudinal magnetic field are essentially different from those
in the superconductor in the transverse magnetic field. The magnetic helicity in the
latter case is zero. Although the force-free torque appears only in the superconductor
in the longitudinal magnetic field, it surely adds a new page to Maxwell’s theory.
In addition, the characteristic force-free state appears only under the influence
of flux pinning. In other words, the longitudinal magnetic field effect is an effect in
which the flux pinning plays a dominant role and it is essentially irreversible. The
flux pinning effect has attracted attention in vortex physics as one of the factors that
determine the phase of the flux line system. This is the first case where the flux
pinning effect is dominant in the field of fundamental science.
The electromagnetic phenomena in the transverse magnetic field are described
by the phenomenological critical state model. It was shown that the assumption in
this model that the pinning force makes the maximum effort to prevent a variation
in the magnetic flux distribution does not contradict the principle of irreversible
thermodynamics of minimum energy dissipation. There are two kinds of balance in
determining the current direction with two degrees of freedom under a longitudinal
magnetic field, and it was shown again that the pinning energy is distributed so as
to minimize the energy dissipation, as assumed in the principle. It is an important
topic to prove theoretically that this principle holds generally, even in a nonlinear
dissipation system, as discussed in Appendix A.10. Detailed analyses of various
electromagnetic phenomena in the longitudinal magnetic field are not sufficient as
yet, and research by many scientists is needed in the future. Thus, the physics of the
longitudinal magnetic field effect is a new science, in which electromagnetism and
irreversible thermodynamics merge together. Efforts to deepen this field are expected.
Coffee break (7)
Poynting’s vector
One of most useful concepts besides the continuity equation of flux lines in this
book is Poynting’s vector. The energy that penetrates the superconductor while the
external magnetic field is changing can be estimated using Poynting’s vector. As
shown in this book, the variation in the internal magnetic energy is different from
this energy, indicating the possibility of the appearance or disappearance of energy.
This is the work done by the driving force. The force-free torque in the longitudinal
magnetic field effect is also derived using Poynting’s vector.