Fundamentals of Plasma Physics

17.3 Dust charge 487

Equation (17.11) can then be written in terms of the dust grain surface potential as

α=−

4 πε 0 rdnd 0

ni 0 e

φd. (17.14)

It is now convenient to introduce the dimensionless variable

ψ=−

eφ

κTi

(17.15)

so thatαbecomes
α=4πnd 0 rdλ^2 diψd (17.16)
where

λdi=

√

ε 0 κTi

ni 0 e^2

(17.17)

is the ion Debye length. Largeψdmeans that ions fall into a potential energy well much
deeper than their thermal energy.
It is also useful to introduce the Wigner-Seitz radiusa, a measure of the nominal spacing
between adjacent dust grains. This spacing is defined by dividing the total volume of the
systemVby the total number of dust grainsNto find a nominal volume surrounding each
dust grain. It is then imagined that this nominal volume is spherical with radiusaso that

N

4 πa^3

3

=V (17.18)

in which case

nd 0 =

3

4 πa^3

(17.19)

and

α=

3 rdλ^2 di

a^3

ψd. (17.20)

It is also convenient to normalize lengths to the ion Debye length so

α=Pψd (17.21)

where

P=3

r ̄d

̄a^3

(17.22)

and the bar means normalized to an ion Debye length.
The dust charge can also be expressed in a non-dimensional fashion using Eq.(17.13)
to give
Zd
4 πni 0 λ^3 di

= ̄rdψd. (17.23)

Thefloating conditionIi+Ie= 0shows that the equilibrium dust potential is given
by

0=ni 0 vTi

(

1 −

eφd

κTi

)

−ne 0 vTeexp(eφd/κTe) (17.24)

which can be re-arranged as

(1+ψd)

√

meTi

miTe

exp(ψdTi/Te)=1−α. (17.25)