Page 222 - Electromagnetics

The time derivative in (4.69) is the total derivative as defined in (A.58):
d℘(r, t) ∂℘(r, t)
= + (v ·∇)℘(r, t). (4.70)
dt ∂t
The second term on the right accounts for the time-rate of change of momentum per-
ceived as the observer moves through regions of spatially-changing momentum. Since
the electron velocity is induced by the electromagnetic field, we anticipate that for a
sinusoidal wave the spatial variation will be on the order of the wavelength of the field:
λ = 2πc/ω. Thus, while the first term in (4.70) is proportional to ω, the second term is
proportional to ωv/c and can be neglected for non-relativistic particle velocities. Then,
writing E(r, t) and v(r, t) as inverse Fourier transforms, we see that (4.69) yields
˜
− q e E = jωm e ˜ v + m e ν˜ v (4.71)
and thus
q e ˜
E
m e
˜ v =− . (4.72)
ν + jω
The secondary current associated with the moving electrons is (since q e is unsigned)
0 ω 2
˜ s p (ν − jω)E (4.73)
˜
J =−Nq e ˜ v =
2
ω + ν 2
where
Nq 2
2 e
ω = (4.74)
p
0 m e
is called the plasma frequency.
The frequency-domain Ampere’s law for primary and secondary currents in free space
is merely
˜
˜ s
˜ i
˜
∇× H = J + J + jω 0 E.
Substitution from (4.73) gives
2
0 ω ν ω 2
˜ ˜ i p ˜ p ˜
∇× H = J + 2 2 E + jω 0 1 − 2 2 E.
ω + ν ω + ν
We can determine the material properties of the plasma by realizing that the above
expression can be written as
˜
˜
˜ s
˜ i
∇× H = J + J + jωD
with the constitutive relations
˜
˜
˜ s
˜
J = ˜σE, D = ˜ E.
Here we identify the conductivity of the plasma as
2
0 ω ν
p
˜ σ(ω) = (4.75)
2
ω + ν 2
and the permittivity as
2
ω
p
˜ (ω) = 0 1 − .
2
ω + ν 2
© 2001 by CRC Press LLC