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6 Longitudinal Magnetic Field Effect
Here, we realize the force-free state in a superconducting slab occupying 0 ≤ y ≤
2d . It is assumed that we apply the external magnetic field H e along the z-axis, and
then, the current I is applied in the same direction. The self-field along the x-axis
due to the current is denoted by H I . From symmetry we focus on half of the slab,
0 ≤ y ≤ d . The magnetic flux density inside the slab has no y-component, and we
can assume
B = (Bsinθ, 0, Bcosθ ),
(6.8)
where θ is the angle of the magnetic flux density measured from the z-axis. It can
be assumed that the spatial variation occurs only along the y-axis. Then, the current
densities along the x and z-axes are
J x =
1
μ 0
·
∂
∂y
Bcosθ =
1
μ 0
∂B
∂y
cosθ − Bsinθ
∂θ
∂y
,
(6.9a)
J z = −
1
μ 0
·
∂
∂y
Bsinθ = −
1
μ 0
∂B
∂y
sinθ + Bcosθ
∂θ
∂y
.
(6.9b)
If the current density is described as
J = (J sinθ, 0, J cosθ),
(6.10)
it is parallel to B in (6.8). In this case the current components in (6.9a) and (6.9b) are
J sinθ =
1
μ 0
∂B
∂y
cosθ − Bsinθ
∂θ
∂y
,
(6.11a)
J cosθ = −
1
μ 0
∂B
∂y
sinθ + Bcosθ
∂θ
∂y
.
(6.11b)
Eliminating J , we have
∂B
∂y
= 0.
(6.12)
This shows that the magnetic flux density is constant in the superconductor. This
value is equal to that on the superconductor surface
B = μ 0
H
2
e + H
2
I
1/2 .
(6.13)
This leads to the outcome that the current density depending on the magnetic flux
density is also constant inside the superconductor. Substituting (6.12) into (6.11a)
and (6.11b), we have