Fundamentals of Plasma Physics

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)