76
5 Flux Pinning Phenomena
flux invades the superconductor from both surfaces. The magnetic flux density inside
the superconductor is also the z-component. It can also be assumed that no quantity
varies along the y- or z-axes. Hence, the current density has a y-component only:
J = −
1
μ 0
·
∂B
∂x
i y .
(5.10)
The Lorentz force is of the y-component, and (5.6) is reduced to
B
∂B
∂x
+ δμ 0 F p = 0,
(5.11)
where δ is a sign factor and takes the values +1 and −1, when the Lorentz force
is directed along the positive or negative x-axis, respectively. In the present initial
condition, since the flux lines are driven in the positive direction by the Lorentz force,
δ = 1. Here, we assume the simple B-dependence of F p as
F p = α p B.
(5.12)
This is called Bean’s model [2]. In this case we have
J c = α p (const.).
(5.13)
Under the boundary condition B(0) = μ 0 H 0 , (5.11) can be solved easily as
B(x) = μ 0 (H 0 − J c x).
(5.14)
For H 0 < J c d , the magnetic flux does not reach the center of the superconducting
slab, as shown in Fig. 5.8, and (5.14) holds for 0 ≤ x ≤ H 0 /J c . The magnetic flux
density is zero in the region H 0 /J c < x ≤ d . The penetration of the magnetic flux
induces the current, and its distribution is given by
J = J c ; 0 ≤ x ≤ H 0 /J c ,
= 0; H 0 /J c < x ≤ d .
(5.15)
This current distribution is shown in the lower panel in Fig. 5.8.
For H 0 > J c d , the magnetic flux penetrates the entire region of the superconducting slab, and the current with density J c flows throughout the whole area. The
distributions of the magnetic flux and current in this case are shown in Fig. 5.9.
The external magnetic field at which the magnetic flux reaches the center of the
superconductor is given by
H 0 = J c d ≡ H p .
(5.16)
5 Flux Pinning Phenomena
flux invades the superconductor from both surfaces. The magnetic flux density inside
the superconductor is also the z-component. It can also be assumed that no quantity
varies along the y- or z-axes. Hence, the current density has a y-component only:
J = −
1
μ 0
·
∂B
∂x
i y .
(5.10)
The Lorentz force is of the y-component, and (5.6) is reduced to
B
∂B
∂x
+ δμ 0 F p = 0,
(5.11)
where δ is a sign factor and takes the values +1 and −1, when the Lorentz force
is directed along the positive or negative x-axis, respectively. In the present initial
condition, since the flux lines are driven in the positive direction by the Lorentz force,
δ = 1. Here, we assume the simple B-dependence of F p as
F p = α p B.
(5.12)
This is called Bean’s model [2]. In this case we have
J c = α p (const.).
(5.13)
Under the boundary condition B(0) = μ 0 H 0 , (5.11) can be solved easily as
B(x) = μ 0 (H 0 − J c x).
(5.14)
For H 0 < J c d , the magnetic flux does not reach the center of the superconducting
slab, as shown in Fig. 5.8, and (5.14) holds for 0 ≤ x ≤ H 0 /J c . The magnetic flux
density is zero in the region H 0 /J c < x ≤ d . The penetration of the magnetic flux
induces the current, and its distribution is given by
J = J c ; 0 ≤ x ≤ H 0 /J c ,
= 0; H 0 /J c < x ≤ d .
(5.15)
This current distribution is shown in the lower panel in Fig. 5.8.
For H 0 > J c d , the magnetic flux penetrates the entire region of the superconducting slab, and the current with density J c flows throughout the whole area. The
distributions of the magnetic flux and current in this case are shown in Fig. 5.9.
The external magnetic field at which the magnetic flux reaches the center of the
superconductor is given by
H 0 = J c d ≡ H p .
(5.16)