5.7 Critical State Theory
107
F m =
B
2
0
2μ 0
−2
∂u
∂ξ
+
∂u
∂ζ
2
+
∂u
∂ξ
2
.
(5.97)
The condition needed to minimize the free energy density is given by Euler’s equation
(see Appendix A.6):
∂F
∂u
− i ξ
∂
∂ζ
∂F
∂(∂u/∂ζ )
+
∂
∂ξ
∂F
∂(∂u/∂ξ )
= 0.
(5.98)
The first term is written as
∂U p
∂u
= −F,
(5.99)
where F is the pinning force density. The second term is reduced to
−i ξ
B
2
0
μ 0
∂
2 u
∂ζ 2 +
∂
2 u
∂ξ 2
.
(5.100)
The current density is given by
J =
1
μ 0
∇ × b =
1
μ 0
B 0 ∇ ×
∂u
∂ζ
− ∇ × B 0
∂u
∂ξ
= i η
B 0
μ 0
∂
2 u
∂ζ 2 +
∂
2 u
∂ξ 2
,
(5.101)
where i η is a unit vector given by
i η = i ζ × i ξ .
(5.102)
The Lorentz force is written as
F L = J × B 0 = i ξ
B
2
0
μ 0
∂
2 u
∂ζ 2 +
∂
2 u
∂ξ 2
,
(5.103)
which is equal to (5.100) with the opposite sign. Hence, we obtain the force balance
equation [19]:
F L + F = 0.
(5.104)
The Lorentz force is rewritten as
F L = i ξ
C 11
∂
2 u
∂ξ 2 + C 44
∂
2 u
∂ζ 2
,
(5.105)