Fundamentals of Plasma Physics

492 Chapter 17. Dusty plasmas

The equations are combined by taking the divergence of Eq.(17.31) and then substituting
Eq.(17.32) to obtain
∂^2 ni 1
∂t^2

=

ZκTe

mi

∇^2 ni 1 (17.33)

which is a wave equation with wave velocityc^2 s=ZκTe/mi.
This method can now be generalized to a plasma consisting of ions with charge+e,
free electrons with charge−e, and dust grains with charge−Zde.The free electron density
is assumed to be small so thatP >> 1 in which case the configuration is on the left of
Fig.17.2. The wave phase velocity is assumed to be much faster than the dust grain thermal
velocity, but much slower than the ion/electron thermal velocities so the respective electron,
ion and dust grain equations of motion are

0 = −neeE−∇(neκTe) (17.34)

0 = +nieE−∇(niκTi) (17.35)

ndmd

dud

dt

= −ndZdeE. (17.36)

Adding the above three equations and invoking quasineutrality (i.e.,ni =ndZd+ne)
results in

ndmd

dud

dt

=−∇(niκTi+neκTe). (17.37)

In analogy to the conventional ion acoustic wave, here the ion and electron pressures couple
to the dust via the electric field. Linearization of Eq.(17.37) gives

nd 0 md

∂ud 1

∂t

=−κTi∇ni 1 −κTe∇ne 1 (17.38)

while linearization of Eqs.(17.34) and (17.35) gives

0 = −ne 0 eE 1 −κTe∇ne 1 (17.39)

0 = +ni 0 eE 1 −κTi∇ni 1. (17.40)

EliminatingE 1 between these last two equations shows that

κTe∇ne 1 =−

ne 0

ni 0

κTi∇ni 1 (17.41)

which can be integrated to give

ne 1 =−

ne 0 Ti

ni 0 Te

ni 1. (17.42)

Inserting Eq.(17.42) into the linearized quasineutrality expressionni 1 = nd 1 Zd+ne 1
gives

ni 1 =

1

1+

ne 0 Ti

ni 0 Te

Zdnd 1 (17.43)

and hence

ne 1 =−

ne 0 Ti/ni 0 Te

1+

ne 0 Ti

ni 0 Te

Zdnd 1. (17.44)