Fundamentals of Plasma Physics

486 Chapter 17. Dusty plasmas

wherevT=

√

2 κT/mis the thermal velocity. The incident particles will be collisionless
as they travel inside thelmfpsphere. Integration of Eqs.(17.5) and (17.6) gives

Iattractive = q

∫∞

0

πr^2 d

(

1 −

2 qφd

mv^2

)

vf(v)4πv^2 dv

= 2π^1 /^2 r^2 dn 0 qvT

∫∞

0

(

1 −

2 qφd

mvT^2 x

)

e−xxdx

=

2

π^1 /^2

(

1 −

qφd

κT

)

n 0 qvTσgeometric (17.8)

and

Irepulsive = 2π^1 /^2 r^2 dn 0 qvT

∫∞

qφd/κT

(

1 −

qφd

κTx

)

e−xxdx

=

2

π^1 /^2

e−qφd/κTn 0 qvTσgeometric. (17.9)

17.3 Dust charge

As discussed above, a neutral dust grain inserted into a plasma will become negatively
charged becausevTe>>vTi. The electron currentIeis thus a repulsive-type current and
the ion currentIiis an attractive-type current. As the dust grain becomes more negatively
charged,|Ie|decreases and|Ii|increases untilIi+Ie=0at which time dust grain charging
ceases.
The presence of negatively charged dust grains affects the quasi-neutrality condition.
Assuming singly charged ions and a charge ofZdon the negatively charged dust grains,
the quasineutrality condition generalizes to

ni 0 =ne 0 +Zdnd 0. (17.10)

Since the ion density is unaffected by the charging process, it is convenient to normalize all
densities to the ion density and define

α=

Zdnd 0

ni 0

(17.11)

so that n
e 0
ni 0
=1−α (17.12)

wherene 0 ,ni 0 ,andnd 0 are the volume-averaged electron, ion, and dust grain densities.
Furthermore, ifrdis small compared to the effective shielding lengthλd, then the dust
grain chargeZdis related to the dust grain surface potential and the dust grain radiusrdvia
the vacuum Coulomb relationship, i.e.,

φd = −

Zdeexp(−rd/λd)

4 πε 0 rd

≃−

Zde

4 πε 0 rd

ifrd<<λd. (17.13)