200
Appendix
v x = r
μ 0
B
∂H I
∂t
cosθ 0 cosθ
1 − tanθ 0 α f (d − y)
,
v y =
μ 0
B
∂H I
∂t
sinθ 0 (d − y),
v z = −r
μ 0
B
∂H I
∂t
cosθ 0 sinθ
1 − tanθ 0 α f (d − y)
.
(A.15.2)
Then, we will directly check if the velocity given by (A.15.2) satisfies the continuity
equation of magnetic flux. The variation of the x-component with time is given by
∂B x
∂t
= −[rot(B × v)] x .
(A.15.3)
The left-hand side is written as
∂B x
∂t
=
∂B
∂t
sinθ + Bcosθ
∂θ
∂t
= μ 0
∂H I
∂t
cos(θ − θ 0 ).
(A.15.4)
After a tedious calculation the right-hand side is written as
−[rot(B × v)] x = −B
∂
∂y
v y sinθ
+ B
∂
∂z
(v x cosθ − v z sinθ )
= μ 0
∂H I
∂t
cos(θ − θ 0 ),
(A.15.5)
which is the same as (A.15.4). The variation of the z-component with time is given
by
∂B z
∂t
= −[rot(B × v)] z .
(A.15.6)
The left-hand side is
∂B z
∂t
=
∂B
∂t
cosθ − Bsinθ
∂θ
∂t
= −μ 0
∂H I
∂t
sin(θ − θ 0 )
(A.15.7)
and the right-hand side is
−[rot(B × v)] z = −B
∂
∂x
(v x cosθ − v z sinθ ) − B
∂
∂y
v y cosθ
= −μ 0
∂H I
∂t
sin(θ − θ 0 ),
(A.15.8)
which is the same as (A.15.7). Thus, (6.45) can be proved as the solution of the
continuity equation of magnetic flux.