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Then using
∂
B
/∂
t
=−∇×
E
,
B
=
μ
0
H
, and the vector identity
∇·
(
E
×
B
)
=
B
·
(
∇×
E
)
−
E
·
(
∇×
B
)
, we have
∂
W
B
/∂
t
=−
∇·
(
E
×
H
)
dV
−
E
·
(
∇×
H
)
dV
Using
J
H
and applying the divergence theorem to
convert the first volume integral to a surface integral gives
=
σ(
E
+
V
×
B
)
and
J
=∇×
J
B
·
J
dV
∂
W
B
/∂
t
=−
(
E
×
H
)
·
d
a
−
/σ
−
V
×
S
where the area element
d
a
points outward from the surface of the volume. Finally,
rearranging this equation yields
J
2
V
dV
·
J
B
dV
∂
W
B
/∂
t
=−
(
E
×
H
)
·
d
a
−
/σ
−
×
(2.62)
S
In words, the change in stored magnetic energy in a volume equals the energy
flux across the surface into the volume in the form of the Poynting flux (
E
H
),
minus the resistive energy loss inside the volume, minus the mechanical work
done against the
J
×
B
force inside the volume.
In a truly closed magnetosphere with the surface an equipotential, no energy
crosses the surface (
E
is everywhere normal to an equipotential), and there could
be no internal circulation (convection), no dissipation by ionospheric currents,
and no storage of magnetic energy for later release (e.g., in what are called
magnetic substorms). Two sources for generating a component of
E
parallel to
the magnetopause, and thus a net Poynting flux inward, are viscous interaction
and reconnection. Both of these processes thus can result in a net flow of energy
into the magnetosphere. We return to these matters in Chapter 8.
×
2.6 Are Ionospheric Electric Fields Real?
Vasyliunas and Song (2005) have argued that the paradigm used in ionospheric
physics is based on a false premise: that ionospheric electric fields cause the
plasma motions. They argue that the plasma velocity is fundamental and that it
generates the electric field rather than the other way around. Such an argument
is clearly valid in the case of the solar wind, which is driven by intense pressure
and has no electric field in the wind frame. In the frame of the earth, however,
there is an electric field equal to
B
in a region that is connected to the
polar ionosphere by magnetic field lines. Is this field real, or is it generated by
the motion of the polar cap plasma?
−
V
×
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