134
6 Longitudinal Magnetic Field Effect
The practical magnetic flux distribution is formally given by (6.8) and (6.15), as
discussed before, whereas the variation rate of the angle α f is not given by (6.17) but
is a constant given by
α f =
μ 0 J c
B
.
(6.35)
In this case the angle of the magnetic flux density is
θ = θ 0 − α f y; 0 ≤ y < θ 0 /α f ,
= 0;
θ 0 /α f ≤ y ≤ d ,
(6.36)
where θ 0 is given by
θ 0 = tan
−1
H I
H e
.
(6.37)
The area from the surface (y = 0) to the depth θ 0 /α f is in the force-free state.
Here we denote the velocity of the flux lines as
v =
v x , v y , v z
.
(6.38)
In the above, only v y represents the penetration of magnetic flux contributing to
the variation in the magnetic flux density. From symmetry, we can assume that this
component does not depend on x and z. Since the flux motion in x-z planes does not
bring about any variation in the magnetic flux density, the divergence of v must be
zero in these planes. Thus, the condition
∂v x
∂x
+
∂v z
∂z
= 0
(6.39)
must be satisfied. Since the velocity v is defined to be normal to the magnetic flux
density B, the condition of (6.39) is written as
v x sinθ + v z cosθ = 0.
(6.40)
Using (6.39) and (6.40), the continuity equation for flux lines (5.34) is written as
(see Appendix A.14)
∂B
∂t
= −B
∂v y
∂y
(6.41)
and