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and Maxwell's equations,
∇×
B
=
μ
0
J
∇×
E
=−
∂
B
/∂
t
∇·
B
=
0
∇·
E
=
ρ
c
/ε
0
where the vacuum displacement current has been ignored due to the high conduc-
tivity and low frequency. These equations constitute the single-fluid description
of a magnetized plasma or of conducting fluids such as the element mercury and
the earth's molten core.
If we neglect the displacement current, we can also set
∇·
E
=
0, since
ρ
c
=
0.
Whatever electric fields are present, they change in space only when
t
is
nonzero or the reference frame is changed. These equations constitute an MHD
approximation, and to them must be added a relationship between the fields and
currents. For a true conducting fluid this is just
∂
B
/∂
J
=
σ
E
since the conductivity is isotropic. As before,
E
is the electric field in the frame
moving with the fluid. In some other frame in which the plasma velocity is
V
and the electric field is
E
, we have
E
E
−
J
,
B
B
for
V
=
V
×
B
and
J
=
=
c
.
J
=
σ
E
and
Thus,
J
=
J
=
σ(
E
+
V
×
B
)
(2.55c)
with
J
and
B
all measured in the second frame. The conductivity of a
plasma is not isotropic, so it is not clear that the MHD fluid approach should
work. However, for a collisionless plasma,
,
E
,
V
,
σ
0
is so high parallel to
B
that in the
frame of reference moving with the plasma the electric field is zero. This is also
true in an infinitely conducting isotropic fluid. Thus, in the limit that
σ
0
goes to
infinity, the plasma will behave like an infinitely conducting fluid, even though
it is an anisotropic material.
For
large,
E
≈
0 and (2.55c) is not very useful. Instead, for the perpendicular
component of
J
, we solve (2.55b) for
J
σ
by taking the cross product with the
⊥
magnetic field. Then,
+
ρ
g
B
d
V
dt
B
B
2
×∇
ρ
B
2
×
B
⊥
=
−
ρ
×
J
(2.56)
B
2
which is independent of
E
.
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