Appendix
197
(c) the structure of electric fields in the resistive state, including the negative field
that appears on the surface,
(d) the appearance of a radial electric field component in the resistive state, and
(e) the resistivity that follows the Bardeen-Stephen model.
(a) and (d) can be explained only by movement of the longitudinal magnetic flux
component and cannot be explained by the flux cutting model, which assumes only the
motion of the transverse component. (e) shows that the energy dissipation occurs in
the whole length of the normal core in each flux line, while the energy dissipation may
occur only in the region of cutting in the flux cutting model, resulting in appreciably
smaller electrical resistivity than in the theoretical prediction of the Bardeen-Stephen
model.
Other phenomena that cannot be explained by the flux cutting model are
(f) the occurrence of the longitudinal magnetic field effect, even when the external
magnetic field and current are simultaneously increased so that the angle of the
total magnetic field on the surface does not change, as shown in Fig. 6.22, and
(g) the helical flow of current to produce the longitudinal magnetic field in the
resistive state when only the current is applied, as shown in Fig. 6.21.
Since the angle of the total magnetic field is not changed, the condition of the
flux cutting is not fulfilled. In more detail, the process is quite opposite to the flux
cutting, i.e., the appearance of flux lines with different angles from flux lines directed
in the same direction, must occur to explain the phenomena in (f). It is also possible
to point out as follows:
(h) The reason why the force-free state is realized cannot be explained, even though
the flux pinning governs the phenomena.
These points, (f), (g), and (h), are explained by the principle of irreversible thermodynamics, in which the direction of current is determined so as to minimize the
energy dissipation.
A.13 Magnetic Helicity in the Force-Free State
In the force-free state where the current density J is parallel to the magnetic flux
density B, it can be expressed as J = kB, using a scalar function k. In this case we
have
∇ × J = k∇ × B − B × ∇k = μ 0 k
2 B − B × ∇k.
(A.13.1)
This quantity is not zero, since the first term is parallel to B, and the second term
is normal to B. In fact, we have ∇ × J =
α
2
f /μ 0
B from (6.10) and (6.14). On
the other hand, the static electric field must satisfy ∇ × E = 0. Hence, the electric
resistivity must be zero. Thus, the static force-free state can be achieved only in
superconductors.
197
(c) the structure of electric fields in the resistive state, including the negative field
that appears on the surface,
(d) the appearance of a radial electric field component in the resistive state, and
(e) the resistivity that follows the Bardeen-Stephen model.
(a) and (d) can be explained only by movement of the longitudinal magnetic flux
component and cannot be explained by the flux cutting model, which assumes only the
motion of the transverse component. (e) shows that the energy dissipation occurs in
the whole length of the normal core in each flux line, while the energy dissipation may
occur only in the region of cutting in the flux cutting model, resulting in appreciably
smaller electrical resistivity than in the theoretical prediction of the Bardeen-Stephen
model.
Other phenomena that cannot be explained by the flux cutting model are
(f) the occurrence of the longitudinal magnetic field effect, even when the external
magnetic field and current are simultaneously increased so that the angle of the
total magnetic field on the surface does not change, as shown in Fig. 6.22, and
(g) the helical flow of current to produce the longitudinal magnetic field in the
resistive state when only the current is applied, as shown in Fig. 6.21.
Since the angle of the total magnetic field is not changed, the condition of the
flux cutting is not fulfilled. In more detail, the process is quite opposite to the flux
cutting, i.e., the appearance of flux lines with different angles from flux lines directed
in the same direction, must occur to explain the phenomena in (f). It is also possible
to point out as follows:
(h) The reason why the force-free state is realized cannot be explained, even though
the flux pinning governs the phenomena.
These points, (f), (g), and (h), are explained by the principle of irreversible thermodynamics, in which the direction of current is determined so as to minimize the
energy dissipation.
A.13 Magnetic Helicity in the Force-Free State
In the force-free state where the current density J is parallel to the magnetic flux
density B, it can be expressed as J = kB, using a scalar function k. In this case we
have
∇ × J = k∇ × B − B × ∇k = μ 0 k
2 B − B × ∇k.
(A.13.1)
This quantity is not zero, since the first term is parallel to B, and the second term
is normal to B. In fact, we have ∇ × J =
α
2
f /μ 0
B from (6.10) and (6.14). On
the other hand, the static electric field must satisfy ∇ × E = 0. Hence, the electric
resistivity must be zero. Thus, the static force-free state can be achieved only in
superconductors.