5.7 Critical State Theory
109
In the above we used the condition that the electric field is zero at the center z = 0,
where the magnetic flux density does not change from symmetry. We assume that
the current density is increased from J to J + δJ in a short period δt. The power that
penetrates into the region between z and z + δz through a unit area in the x-y plane
in this period is calculated using Poynting’s vector as
δP =
1
μ 0
E y (z + δz)B x (z + δz) − E y (z)B x (z)
=
3μ 0 z
2
δz
2
· J
∂J
∂t
.
(5.110)
Thus, the energy that comes into this region during this period is determined as
δF =
3μ 0 z
2
δz
2
δt
0
J
∂J
∂t
dt =
3μ 0 z
2
δz
2
J δJ .
(5.111)
It should be noted that the internal magnetic energy changes during this period,
since the internal magnetic flux density changes. When the current density is J, the
magnetic energy at position z is
1
2μ 0
B
2
0 + B
2
x (z)
=
1
2μ 0
B
2
0 + μ
2
0 J
2 z
2
.
(5.112)
When the current density is increased by δJ , the increase in the magnetic energy in
a unit area of the x–y plane in the region between z and z + δz is given by
δF m =
μ 0
2
z+δz
z
(J + δJ )
2
− J
2
z
2 dz = μ 0 J δJz
2
δz.
(5.113)
The difference in the energy is given by
δW = δF − δF m =
μ 0
2
J δJz
2
δz,
(5.114)
which is the work done by the elastic restoring force when the magnetic structure
changes. The displacement of flux lines δu along the x-axis due to the restoring force
is obtained from the continuity equation for flux lines:
∂δu(z)
∂z
=
δB x (z)
B 0
,
(5.115)
which leads to