TRANSVERSE MOTION OF ELECTRONS 623 



In terms of -v and y as usually defined 



8 = X -\- jy 

 equation (13.38) becomes 



x[(x'- -f-+ /-) - 2y(y + 6)] = (13.39) 



(y + bXx-' - / + /2) _^ 2x'y +1 = (13.40) 



From the x = solution of (13.39) we obtain 



X = (13.41) 



h = -^, - y. (13.42) 



y" - P 



It is found that this solution obtains for large and small values of b. For 

 very large and very small values of b, either 



y = -b (13.43) 



or 



y = H (13.44) 



The wave given by (13.43) is a circuit wave; that given by (13.44) repre- 

 sents electrons travehng down the tube and oscillating with the cyclotron 

 frequency in the magnetic field. 



In an intermediate range of 6, we have from (13.39) 



X = ±\/2y{y-\- b) - (P - y^) (13.45) 



and 



b = -2y ± \/p - l/2y. (13.46) 



For a given value of /- we can assume values of y and obtain values of b. 

 Then, x can be obtained from (13.45). In Figs. 13.5-13.10, .v and y are plotted 

 vs. b for/- = 0, .5, 1, 4 and 10. It should be noted that .Vi, the parameter 

 expressing the rate of increase of the increasing wave, has a maximum at 

 larger values of b as/ is increased (as the magnetic focusing field is increased). 

 Thus, for higher magnetic focusing fields the electrons must be shot into the 

 circuit faster to get optimum results than for low fields. In Fig. 13.11, the 

 maximum positive value of .v is plotted vs. /. The plot serves to illustrate the 

 effect on gain of increasing the magnetic field. 

 Let us consider an example. Suppose 



X = 7.5 cm 



D = .03 



