WAVES IN ELECTRON STREAMS AND CIRCUITS 



631 



if the current ai of the electron stream, where a is the area of electron flow, 

 changes with distance, a current / will flow into the line per unit length as 

 shown in Fig. 1.1 where 



di 



J = -(T 



dz 



If V is the voltage on the line and / is the current in the line we write 



^ = -C — - - 

 dz dt ^ dz 



— = -Z — 

 dz dt 



(1.9) 



(1.10) 

 (1.11) 



-Ir 



CURRENT DENSITY 



I^ .^_^TRKKK5M^yM5WO^JMM^ L PER UNIT LENGTH 



C PER UNIT LENGTH 



+Z 



Fig. 1.1 — A transmission line with an electron beam very close to it. 



We can eliminate / by differentiating 



(1.12) 



dzdt L dz" dt^ 



We can further identify the field acting on the electrons as 



dz 



In (1.8), let us replace E by means of (1.13), and let us differentiate with 

 respect to z and again with respect to /. We obtain 



(1.13) 





{^i\^o ^v^'»^a- ^^v^^^ 



(^\ = 



e d'V 



— Po 

 m 



dz'df- 



