Page 225 - Electromagnetics

and thus
2
ω
c p
˜ (ω) = 0 1 − ,
2
ν + ω 2
which matches (4.76) as expected.
4.6.2 Complex dyadic permittivity of a magnetized plasma
When an electron plasma is exposed to a magnetostatic field, as occurs in the earth’s
ionosphere, the behavior of the plasma is altered so that the secondary current is no longer
aligned with the electric field, requiring the constitutive relationships to be written in
terms of a complex dyadic permittivity. If the static field is B 0 , the velocity field of the
plasma is determined by adding the magnetic component of the Lorentz force to (4.71),
giving
˜
−q e [E + ˜ v × B 0 ] = ˜ v( jωm e + m e ν)
or equivalently
q e q e
˜
˜ v − j ˜ v × B 0 = j E. (4.78)
m e (ω − jν) m e (ω − jν)
Writing this expression generically as
v + v × C = A, (4.79)
we can solve for v as follows. Dotting both sides of the equation with C we quickly
establish that C · v = C · A. Crossing both sides of the equation with C, using (B.7), and
substituting C · A for C · v,wehave

v × C = A × C + v(C · C) − C(A · C).
Finally, substituting v × C back into (4.79) we obtain
A − A × C + (A · C)C
v = . (4.80)
1 + C · C
Let us first consider a lossless plasma for which ν = 0. We can solve (4.78) for ˜ v by
setting
0 ω 2
ω c p
˜
C =− j , A = j E,
ω ωNq e
where
q e
ω c = B 0 .
m e
Here ω c = q e B 0 /m e =|ω c | is called the electron cyclotron frequency. Substituting these
into (4.80) we have

0 ωω 2 0 ω 2 0 ω 2
˜
˜
˜
2 2 p p ω c p
˜
ω − ω v = j E + ω c × E − j ω c · E.
c
Nq e Nq e ω Nq e
˜ s
Since the secondary current produced by the moving electrons is just J =−Nq e ˜ v,we
have
2 2 2
0 ω 0 ω 0 ω
˜ s p ˜ p ˜ ω c p ˜
J = jω − E + j ω c × E + ω c · E . (4.81)
2
2
2
2
2
ω − ω 2 ω(ω − ω ) ω ω − ω 2
c c c
© 2001 by CRC Press LLC