500 Chapter 17. Dusty plasmas
Using Gauss’s law to prescribe the radial electric field at the dust grain surface and also
choosingdso thatψ=1atr ̄= ̄rogives
ψ 1 ( ̄r)=
α ̄a^3
3
+
(
1 −
1
̄ri
αa ̄^3
3
)
r ̄
r ̄
. (17.80)
In region 2, which is governed by Eq.(17.76), the effective dependent variable isψ+α
which has growing and decaying Yukawa-like solutions ∼exp(± ̄r)/ ̄r.It is convenient to
write the region 2 solution as
ψ 2 ( ̄r)=
r ̄o
(
1
Z ̄+α
)
cosh( ̄r− ̄ro)+Bsinh( ̄r− ̄ro)−αr ̄
̄r
(17.81)
where the coefficients have been chosen so thatψ= 1/Z ̄atr ̄o,the as-yet undetermined
location of the interface between regions 2 and 3. The coefficientBis also undetermined
for now, and it should be noted thatψis independent ofBat ̄r= ̄ro.
Region 3, governed by Eq.(17.77), has a solution which can be expressed as
ψ 3 ( ̄r)=
r ̄ 0
Z ̄r ̄exp
(
−
√
αZ ̄+1 ( ̄r− ̄r 0 )
)
(17.82)
where the coefficients have been chosen to giveψ = 1/Z ̄atr ̄o,the interface between
regions 2 and 3.
The condition that∂ψ/∂r ̄is continuous at ̄rogives
B=α−
r ̄o
Z ̄
√
αZ ̄+1 (17.83)
and this result completes the matching process atr ̄osince bothψand∂ψ/∂r ̄have now
been arranged to be continuous atr ̄o. Sinceψ 1 = 1has already been arranged at ̄ri, in
order to have continuity ofψatr ̄iit is necessary to setψ 2 =1at ̄ri, i.e.,
̄ri= ̄ro
(
1
Z ̄+α
)
cosh( ̄ri−r ̄o)+Bsinh( ̄ri−r ̄o)−α ̄ri. (17.84)
All that remains to be done is to arrange for continuity of∂ψ/∂ ̄rat ̄r= ̄ri.Sinceψ
has already been arranged to be continuous at ̄r= ̄ri,continuity of∂( ̄rψ)/∂ ̄rat ̄r= ̄ri
implies continuity of∂ψ/∂r. ̄This means that continuity of∂ψ/∂r ̄at ̄r= ̄riis attained
by equating the derivatives of the numerators of the right hand sides of Eqs.(17.80) and
(17.81) at ̄ri, i.e., continuity of∂ψ/∂ ̄rat ̄r= ̄riis attained by setting
1 −
1
r ̄i
α ̄a^3
3
= ̄ro
(
1
Z ̄+α
)
sinh( ̄ri−r ̄o)+Bcosh( ̄ri− ̄ro)−α. (17.85)
SinceZ>> ̄ 1 andαis of order unity, Eqs.(17.84) and (17.85) can be approximated as
(1+α) ̄ri = α ̄rocosh( ̄ri− ̄ro)+αsinh( ̄ri−r ̄o) (17.86)
1+α−
1
̄ri
α ̄a^3
3
= α ̄rosinh( ̄ri− ̄ro)+αcosh( ̄ri−r ̄o). (17.87)