224 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations
Since the energy density stored in the vacuum electric field is
wE=ε 0 |E ̃|^2
4(7.80)
and the energy density stored in the vacuum magnetic field is
wB=|B ̃|^2
4 μ 0(7.81)
the change in particle kinetic energy density associated with bringingthe wave into exis-
tence is
w ̄part=ε 0
4E ̃∗·
[
∂
∂ω(
ω←→
K(ω))
−
←→
I
]
·E ̃. (7.82)
Although this result has been established for the general case of the dielectric ten-sor
←→
K(ω)of a cold magnetized plasma, in order to appreciate its meaning it is use-
ful to consider the simple example of high frequency electrostatic oscillations in an un-
magnetized plasma. In this simplest caseS =P = 1−ω^2 pe/ω^2 andD = 0so that
←→
K(ω)=
(
1 −ω^2 pe/ω^2)←→
I.Since the oscillations are electrostatic,wB=0.The energy
density of the particles is therefore
w ̄part =
ε 0 |E ̃|^2
4[
∂
∂ω{
ω(
1 −
ω^2 pe
ω^2)}
− 1
]
=
ε 0 |E ̃|^2
4[
2
ω^2 pe
ω^2− 1
]
=
ε 0 |E ̃|^2
4(7.83)
where the dispersion relation 1 −ω^2 pe/ω^2 =0has been used. Thus, for this simple case,
half of the average wave energy density is contained in the electric fieldwhile the other half
is contained in the coherent particle motion associated with the wave.
7.6 Finite-temperature plasma wave energy equation
The dielectric tensor does not depend on the wavevectorkin a cold plasma, but does
in a finite temperature plasma. For example, the electrostatic unmagnetized cold plasma
dielectricP(ω)=1−ω^2 pe/ω^2 becomesP(ω,k)=1−(1+3k^2 λ^2 De)ω^2 pe/ω^2 in a warm
plasma. The analysis of the previous section will now be generalized to allow for the
possibility that the dielectric tensor depends onkas well as onω.In analogy to the method
used in the previous section for treating complexω, herekwill also be assumed to have
a small imaginary part. In this case, Taylor expansion of the dielectric tensor and then
extracting the anti-Hermitian part shows that the anti-Hermitian part is
←→
Kah=iωi[
∂
∂ω←→
K(ω,k)]
ω=ωr,k=kr+iki·[
∂
∂k←→
K(ω,k)]
ω=ωr,k=kr