Geoscience Reference
In-Depth Information
Substituting this into one of Maxwell's equations,
∇×
B
=
μ
0
J
+
μ
0
ε
0
∂
E
/∂
t
gives
=
μ
0
nM
+
ε
0
B
2
∇×
B
/
∂
E
/∂
t
so we can define a dielectric constant via the expression
nM
B
2
nM
B
2
+
ε
0
=
ε
=
/
/
Now a required assumption necessary for using the guiding center approxi-
mation is that the time scale
τ>
−
i
. Hence, this dielectric constant should be
valid for electromagnetic waves with frequencies
f
<
i
. Indeed, the expression
for the phase velocity of an electromagnetic wave is
1
/
2
V
ph
=
(
1
/μ
0
ε)
Substituting the dielectric constant derived above yields
1
/
2
/
(
μ
0
ρ
)
V
ph
=
B
This velocity is the Alfvén speed, which is the velocity of an electromagnetic
wave propagating parallel to
B
in a magnetized plasma when its frequency sat-
isfies
f
i
(e.g., Spitzer (1962) or any elementary plasma text).
If the magnetic field is curved as in a dipole field, particles moving along
B
will feel a force given by
MV
2
R
F
=−
||
/
n
ˆ
where
is the particle velocity parallel to
B
, and
R
is the radius of curvature (see Fig. 2.13 for a sketch of the coordi-
nate system). Substituting this force into (2.55), we find that the particles drift
perpendicular to the field with velocity
n
is a unit vector pointed inward,
V
ˆ
||
MV
2
R
B
qB
2
W
D
=
||
/
׈
n
/
Similarly, if there is a gradient in the magnetic field with a gradient-scale length
large compared to a gyroradius, we can consider the force on a magnetic dipole
of moment
μ
, which is given by
F
=−
μ
∇
B
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