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)