HIGH-FREQ UENC Y A MPLIFIER 



37 



V and p in two extreme cases. If the wavelength in the stream, X, , is 

 very short (fi large), so that transverse a-c. fields are negligible, then from 

 Poisson's equation we have 



P = coiS" F (14) 



If, on the other hand, the wavelength is long compared with the tube radius 

 (jS small) so that the fields are chiefly transverse, the lines of force running 

 from the beam outward to the surrounding tube, we may write 



p = CV (15) 



^^«— 



Fig. 1 — A "circuit" curve for a narrow electron stream in a tube. The ratio of the a-c. 

 charge density p to the a-c. voltage V produced bj' the charge density is plotted vs. a 

 parameter /3/|3o , which is inversely proportional to the wavelength X, in the flow. Curve 1 

 holds for very large values of /3//3o ; curve 2 holds for very small values of i3//3o , and curve 

 3 over-all shows approximately how p/V varies for intermediate values of /3//3o . 



Here C is a constant expressmg the capacitance per unit length between the 

 region occupied by the electron flow and the tube wall. 



We see from (14) and (15) that if at some particular frequency we plot 

 p/V vs. /3//3o for real values of /3, p/V will be constant for small values of jS 

 and will rise as /S^ for large values of /3, approximately as shown in Fig. 1. 

 For another frequency, /3o would be different and, as p/7 is a function of /?, 

 the horizontal scale of the curve would be different. 



Now, we have assumed that the charge is produced by the action of the 

 voltage, according to the ballistical equation (10). This relation is plotted 

 in Fig. 2, for a relatively large value of Jq/uqVq (curve 1) and for a smaller 



value of Jq/uqVq (curve 2). There are poles at /3//3o 



1 ± - , and a mini- 



mum between the poles. The height of the minimum increases as Jo/uqVo 

 is increased. 



A circuit curve similar to that of Fig. 1 is also plotted on Fig. 2. We see 



