1364 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1954 



do the other ac quantities. We can write 



i'=--E. (A7) 



at in 



Here e/m, the charge-to-mass ratio of the electron, is taken as a posi- 

 tive quantity. 



In (A7), dv/dt is the rate of change of v with respect to t for a single 

 electron; that is dv/dt observed riding along with the electron. If we ride 

 along with the electron for a time dt we move along distance dz 



dz = {uq -f- v) dt 



For small signals we neglect v in this expression and write 



dz = Ua dt 



Hence, the total change dv in the velocity of the electron in the time dt 

 is 



dv = — dt -{- — Uodt 

 dt dz 



Hence, we find that 



dv 



,^ = i(a) - l3uo)v (A8) 



dt 



Using (A7) and (A8), we see that 



m 



(co — /3uo) 



(A9) 



We can combme (A9) with (A6) and write for the convection current 

 density 



. ^ ~^'£"^"^- (AlO) 



Let us now consider a special, hypothetical case in which the electric 

 field is in the z direction only, so that there are no transverse electric 



