Appendix
181
where (4.17) was used. Hence, the momentum of a superconducting electron is given
by
p s = m
∗ v sθ i θ = −
r
i θ ,
(A.3.3)
where i θ is the azimuthal unit vector. It is assumed that the quantized flux line flows
with velocity v along the x-axis, as discussed in Sect. 4.3. If the velocity of the
superconducting electron is v s , its gauge-invariant momentum is given by
m
∗ v s = p s + 2eA.
(A.3.4)
If we approximate that the magnetic flux density B is almost uniform in the area
where the current flows around the normal core of the quantized flux line, we have
A = (Br/2)i θ . The force on the superconducting electron is denoted by f e . Then,
the equation of motion of the superconducting electron is
m
∗ dv s
dt
= f e .
(A.3.5)
When the velocity of the quantized flux line is small enough, we can approximate as
d/dt ∼ = −(v · ∇) = −v(∂/∂x) and (A.3.5) leads to
f e = v
∂
∂x
r
− eBr
i θ .
(A.3.6)
Since this force is given by the electric field, the electric field is
e = −
f e
2e
= −v
∂
∂x
φ 0
2π r
−
Br
2
i θ .
(A.3.7)
The relationships between the two-dimensional Cartesian coordinates (x, y) and
the cylindrical coordinates (r, θ) are
x = rcosθ, y = rsinθ
(A.3.8)
and the unit vectors in the cylindrical coordinates are written as
i r = i x cosθ + i y sinθ, i θ = −i x sinθ + i y cosθ.
(A.3.9)
Using these relationships, we have
∂
∂x
(ri θ ) = i y ,
∂
∂x
1
r
i θ
=
1
r 2 (−i θ cosθ + i r sinθ ).
(A.3.10)