220 Chapter 7. Waves in inhomogeneous plasmas and wave energy relations
and expressing this result as
∂w
∂t
+∇·P=0 (7.53)
where
∂w
∂t
=E·J+ε 0 E·∂E
∂t+
1
μ 0B·
∂B
∂t(7.54)
and
P=E×B
μ 0(7.55)
is called the Poyntingflux. The quantitiesPand∂w/∂tare respectively interpreted as the
electromagnetic energyflux into the system and the rate of change of energy density of the
system. The energy density is obtained by time integration and is
w(t) = w(t 0 )+∫tt 0dt{
E·J+ε 0 E·∂E
∂t+
1
μ 0B·
∂B
∂t}
= w(t 0 )+∫tt 0dtE·J+[
ε 0
2E^2 +
B^2
2 μ 0]tt 0(7.56)
wherew(t 0 )is the energy density at some reference timet 0.
The quantityE·Jis the rate of change of kinetic energy density of the particles. This
can be seen by first dotting the Lorentz equation withvto obtain
mv·
dv
dt=qv·(E+v×B) (7.57)or
d
dt
(
1
2
mv^2)
=qE·v. (7.58)Since this is the rate of change of kinetic energy of a single particle, the rate of change of
the kinetic energy density of all the particles, found by summing over all theparticles, is
d
dt
(kinetic energy density) =∑
σ∫
dvfσqσE·v=
∑
σnσqσE·uσ= E·J. (7.59)
This shows that positiveE·Jcorresponds to work going into the particles (increase
of particle kinetic energy) whereas negativeE·Jcorresponds to work coming out of the
particles (decrease of the particle kinetic energy). The latter situation is obviously possible
only if the particles start with a finite initial kinetic energy. SinceE·Jaccounts for changes
in the particle kinetic energy density,wmust be the sum of the electromagnetic field density
and the particle kinetic energy density.
The time integration of Eq.(7.59) must be done with great care ifEandJare wave
fields. This is because writing a wave field asψ=ψ ̃exp(ik·x−iωt)must always be un-
derstood as a notational convenience which should never be taken to mean thatthe actual
physical wave field is complex. The physical wave field is always real and soit is always
understood that the physically meaningful variable isψ=Re
[
ψ ̃exp(ik·x−iωt)]
.This