Fundamentals of Plasma Physics

(C. Jardin) #1
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=−


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)
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