134 BELL SYSTEM TECHNICAL JOURNAL 



value of A' at any one point and the values of p at all intermediate 

 points are preassigned. Thus let Ao represent the preassigncd value 

 of A' at .v=0, and Xj represent the value of A' at .v=(/; wc have 



Aj=-4jr/'pr/.v + A'o. (10) 



•'0 



Consequently the P.D. between an\- two points is also determined; 

 that between a=0 and x=d is 



Vi- Vo= -i^ f (ix Tp dx + Xo(L (11) 



Jo Jo 



Now we introduce the further assumption that the electric charge 

 is concenlraled upon corpuscles (electrons or charged atoms) of 

 one kind, of ec|ual charge E and mass ni, of which there are iidv in a 

 very small volume dv at .v; n is a function of .v. Then 



«£ = p. (12) 



Assume finally that the corpuscles are moving with speed u, identical 

 for all corpuscles having the same A;-coordinate, but depending on x; 

 represent the current-density by i; we have 



ttEu = i (13) 



and consequently 



p = i/u. (14) 



I 



Now consider the llow of current between two parallel jjlanes, from 

 one electrode at x = to the t)lhcr at x=d. If the current is borne 

 by corpuscles of one kind, and the assumption last made is true; and if 

 we know the speed of the corpuscles at every point between the plates, 

 and the field strength at someone point; then we can calculate the field 

 strength everywhere between the plates, and the potential-difTerence 

 between them. 



The customary convention about the field strength is to assume 

 it to be xero at the electrode from which the corpuscles start, so that 

 Xo = in (11). Rewriting (11) to take account of (14), we have 



Vd - To = - 4 TTt / dx f'dx/u ( 1 o) 



Jo Jo 



as the general equation. 



Krudiciit iiiiplif!) (xjsilivf space-charge; iinirumi field implies zero space-charge. 

 It is instructive lo exaniiiie mappings of fielcl-distribiition with this principle in 

 mind: such mappings, tor example, aslhose in Kig. '>. The uniform fu-ld in a current- 

 carrying wire nieans th.it |K>sitive and negative charges are distril)ute(l everywhere 

 in the metal with equ.d density — a conclusion one might forget, but for these more 

 general cases. 



