6.3 Derivation of the Force Free Torque
131
When the angle θ 0 is sufficiently small, expansion as sinθ 0 ∼ = θ 0 − θ
3
0 /6 of sinθ 0 in
the brackets in the above equation leads to
p =
B
2
6μ 0
θ 0
∂θ 0
∂t
.
(6.30)
Thus, the energy density that penetrates the superconductor during the increase in
the angle of the external magnetic field from 0 to θ m is given by
w =
pdt =
B
2
6μ 0
θ m
0
θ 0 dθ 0 =
B
2
12μ 0
θ
2
m .
(6.31)
The force-free torque density that works to release the introduced distortion is [14]
=
∂w
∂θ m
=
B
2
6μ 0
θ m =
1
6
JBy 0 .
(6.32)
Thus, the force-free torque density is proportional to the magnetic flux density and
the current density, both of which correspond to the strength of the distortion. This
is similar to the Lorentz force.
When the force-free current flows, the magnetic energy does not change, since
the magnetic flux density does not change. The energy associated with the force-free
distortion shown in Fig. 6.11 penetrates the superconductor, however. This must be
the work done by the torque, which is similar to the work done by the Lorentz force in
the usual transverse magnetic field (see Appendix A.7). Under practical conditions,
this additional energy is absorbed as an increase in the pinning energy, i.e., a kind
of thermodynamic energy. As a consequence, the distorted flux line structure is
stabilized by the pinning interaction. If there is no pinning interaction, the distorted
structure cannot be stabilized, resulting in a resistive state. Thus, the static critical
state is determined by the torque balance between the force-free torque density and
the pinning torque density p [14]
+ p = 0.
(6.33)
This result is derived by minimizing the Gibbs free energy and is similar to the derivation of the force-balance (5.6) (see Appendix A.8). Equation (6.33) is consistent with
the experimental results showing that the critical current density depends on the flux
pinning strength. On the other hand, the above theoretical treatment essentially denies
the mechanism of flux cutting [15]. That is, flux cutting is a mechanism of magnetic
interaction between flux lines, and hence, it is independent of the flux pinning. In
addition, the energy that was introduced as work done by the driving torque cannot be
stored as a pinning energy. Thus, the principle of energy conservation is not fulfilled.
See the details in (2) in Appendix A.12.
131
When the angle θ 0 is sufficiently small, expansion as sinθ 0 ∼ = θ 0 − θ
3
0 /6 of sinθ 0 in
the brackets in the above equation leads to
p =
B
2
6μ 0
θ 0
∂θ 0
∂t
.
(6.30)
Thus, the energy density that penetrates the superconductor during the increase in
the angle of the external magnetic field from 0 to θ m is given by
w =
pdt =
B
2
6μ 0
θ m
0
θ 0 dθ 0 =
B
2
12μ 0
θ
2
m .
(6.31)
The force-free torque density that works to release the introduced distortion is [14]
=
∂w
∂θ m
=
B
2
6μ 0
θ m =
1
6
JBy 0 .
(6.32)
Thus, the force-free torque density is proportional to the magnetic flux density and
the current density, both of which correspond to the strength of the distortion. This
is similar to the Lorentz force.
When the force-free current flows, the magnetic energy does not change, since
the magnetic flux density does not change. The energy associated with the force-free
distortion shown in Fig. 6.11 penetrates the superconductor, however. This must be
the work done by the torque, which is similar to the work done by the Lorentz force in
the usual transverse magnetic field (see Appendix A.7). Under practical conditions,
this additional energy is absorbed as an increase in the pinning energy, i.e., a kind
of thermodynamic energy. As a consequence, the distorted flux line structure is
stabilized by the pinning interaction. If there is no pinning interaction, the distorted
structure cannot be stabilized, resulting in a resistive state. Thus, the static critical
state is determined by the torque balance between the force-free torque density and
the pinning torque density p [14]
+ p = 0.
(6.33)
This result is derived by minimizing the Gibbs free energy and is similar to the derivation of the force-balance (5.6) (see Appendix A.8). Equation (6.33) is consistent with
the experimental results showing that the critical current density depends on the flux
pinning strength. On the other hand, the above theoretical treatment essentially denies
the mechanism of flux cutting [15]. That is, flux cutting is a mechanism of magnetic
interaction between flux lines, and hence, it is independent of the flux pinning. In
addition, the energy that was introduced as work done by the driving torque cannot be
stored as a pinning energy. Thus, the principle of energy conservation is not fulfilled.
See the details in (2) in Appendix A.12.