12 BELL SYSTEM TECHNICAL JOURNAL 



Now let us integrate with respect to s, around the closed curve; 

 we get 



J rids = J Eds — ico J Ids J cos (s, s')\(s, s')ds' 



= V — i(j} J lids, 



thus defining the impressed voltage V, and the inductance per unit 

 length I. Finally, if we assume that this current variation along the 

 conductor is negligibly small, we get 



I J rds = V — ic^I J Ids, 



which may be written as 



RI + ic^LI = V, (14) 



which is the usual form of the equation of circuit theory for a closed 

 loop. 



In deducing (14) from (10) there is one important point which should 

 be noticed. The assumption that the variation in the current I 

 along the conductor is sufficiently small to justify passing from (13) to 

 (14) does not by any means imply that the effect of the distributed 

 charge, which is absent in (14), is negligible. The term (d/ds)^ 

 vanishes in passing from (12) to (13) because the integration is carried 

 around a closed path. Actually comparing the terms icoA and (d/ds)^, 

 we see that their ratio involves the factor (co/c)^ which is an exceedingly 

 small quantity even at very high frequencies. Consequently extremely 

 small variations in the current are sufficient to establish charges 

 which can and do profoundly modify the resultant electric field. 

 These, in the case of a closed circuit, are eliminated from explicit 

 consideration by integrating around a closed curve. 



This may be illustrated by brief consideration of a second case 

 where the conductor is not closed but is terminated in the plates of a 

 condenser at s = Si and s = S2 respectively. Making the same as- 

 sumption as above, after integrating (11) from s = Si to s = 52, we get 



RI + icoLI + $2 - $1 = V, (15) 



where $2 — ^1 is the difference in $ between the condenser plates. 

 Assuming these very close together, $2 — *i is approximately propor- 

 tional to the charge on the condenser, that is, to 



/ 



Idt = i /, 



too 



