Page 221 - Electromagnetics

dense to be considered in the macroscopic sense, then there is no net field produced
by the gas and thus no electromagnetic interaction between the particles. We also
assume that the plasma is homogeneous and that the number density of the electrons
3
N (number of electrons per m ) is independent of time and position. In contrast to
this are electron beams, whose properties differ significantly from neutral plasmas
because of bunching of electrons by the applied field [148].
2. We ignore the motion of the positive ions in the computation of the secondary
current, since the ratio of the mass of an ion to that of an electron is at least as
large as the ratio of a proton to an electron (m p /m e = 1837) and thus the ions
accelerate much more slowly.
3. We assume that the applied field is that of an electromagnetic wave. In § 2.10.6
we found that for a wave in free space the ratio of magnetic to electric field is
√
|H|/|E|= 0 /µ 0 , so that

|B| 0 √ 1
= µ 0 = µ 0 0 = .
|E| µ 0 c
Thus, in the Lorentz force equation we may approximate the force on an electron
as

F =−q e (E + v × B) ≈−q e E

as long as v c. Here q e is the unsigned charge on an electron, q e = 1.6021 ×
10 −19 C. Note that when an external static magnetic field accompanies the field of
the wave, as is the case in the earth’s ionosphere for example, we cannot ignore the
magnetic component of the Lorentz force. This case will be considered in § 4.6.2.
4. We assume that the mechanical interactions between particles can be described
using a collision frequency ν, which describes the rate at which a directed plasma
velocity becomes random in the absence of external forces.

With these assumptions we can write the equation of motion for the plasma medium.
Let v(r, t) represent the macroscopic velocity of the plasma medium. Then, by Newton’s
second law, the force acting at each point on the medium is balanced by the time-rate of
change in momentum at that point. Because of collisions, the total change in momentum
density is described by
d℘(r, t)
F(r, t) =−Nq e E(r, t) = + ν℘(r, t) (4.69)
dt
where
℘(r, t) = Nm e v(r, t)

is the volume density of momentum. Note that if there is no externally-applied electro-
magnetic force, then (4.69) becomes
d℘(r, t)
+ ν℘(r, t) = 0.
dt
Hence
℘(r, t) = ℘ 0 (r)e −νt ,

and we see that ν describes the rate at which the electron velocities move toward a
random state, producing a macroscopic plasma velocity v of zero.

© 2001 by CRC Press LLC