622 THE BELL SYSTEM TECHNICAL JOURNAL, MAY 1956 



Of course this presumes that the approximation contained in (C18) is 

 valid down to very small values of time. This assumption is well founded 

 as the transient does vanish after a rather short time. 

 Inserting (C19) and (C20) in (C18) then gives us 



p = NM exp {R/r) + N[N - M exp iR/r)]e-'"''">' (C21) 



in which only 771 remains to be determined. 



Substitution of (C21) into (C16), recalling the definitions of M and L, 

 shows that it already satisfies (C16) for any time, t. Thus (C16) cannot 

 be used for determining 771 . 



On the other hand we note from (C21) that as soon as r becomes of 

 order, R, p becomes almost independent of r, being given 



p = N{N + (N - M)e-'"'''''} (C22) 



Since L is of the order lOR or greater, this means that throughout most 

 of the volume, l/N (in fact throughout 0.999 1/A^) p is independent 

 of r. Effectively, the entire volume 1/iV has been drained of ions, i.e., 

 they have been trapped. The total ion content at time t, may then be 

 taken as the product of p, given by (C22), with 1/iV, that is, 



N + (N - M)e-'""'''' (C23) : 



The time rate of change of this content must be given by the flux Airr J*. 



^[Ar+ (iV - M)e-''^'^°'] 



, , (C24)l 



= -mDo(N - M)e-'"^°' = 47rrV*(r, t) 



= -AwRN^D^e-''"'''' 



in which (C21) has been substituted into (10.16) to pass from the third 

 to the fourth expression. Comparing the second and fourth term of (C24) 

 reveals 



or 



1 KkTjN - M) 



"" ~ 771'Do ~ Wn'Do 



the value quoted in (10.25). 



(C26) 



