TRANSVERSE MOTION OF ELECTRONS 617 



Here 5 and b have their usual meanings; a is the ratio between the transverse 

 and longitudinal field strengths, and /is proportional to the strength of the 

 magnetic focusing field. 



In case of a purely transverse field, a new gain parameter D is defined. 

 D is the same as C except that the longitudinal a-c field is replaced by the 

 transverse a-c field. In terms of D, b and 5 are redefined by (13.36) and 

 (13.37), and the final equation is (13.38). Figures 13.5-13.10 show how the 

 x's and 3''s vary with b for various values of / (various magnetic fields) and 

 Fig. 13.11 shows how Xi , which is proportional to the gain of the increasing 

 wave in db per wavelength, decreases as magnetic field is increased. A nu- 

 merical example shows that, assuming reasonable circuit impedance, a 

 magnetic field which would provide a considerable focusing action would 

 still allow a reasonable gain. 



The curves of Figs. 13.6-13.10 resemble very much the curves of Figs. 

 8.7-8.9 of Chapter VIII, which show the effect of space charge in terms of 

 the parameter QC. This is not unnatural; in one case space charge forces 

 tend to return electrons which are accelerated longitudinally to their un- 

 disturbed positions. In the other case, magnetic forces tend to return elec- 

 trons which are accelerated transversely to their undisturbed positions. In 

 each case the circuit field acts on an electron stream which can itself sustain 

 oscillations. In one case, the oscillations are of a plasma type, and the re- 

 storing force is caused by space charge of the bunched electron stream; in 

 the other case the electrons can oscillate transversely in the magnetic field 

 with cyclotron frequency. 



Let us, for instance, compare (7.13), which applies to purely longitudinal 

 displacements with space charge, with (13.38), which applies to purely 

 transverse fields with a longitudinal magnetic field. For zero loss (d = 0), 

 (7.13) becomes 



1 = (j8 - 6) (62 + 4QC) 

 While 



1 = 0'5- bW+f) (13.38) 



describes the transverse case. Thus, if we let 



4QC=P 



the equations are identical. 



When there is both a longitudinal and a transverse electric field, the equa- 

 tion for 8 is of the fifth degree. Thus, there are five forward waves. For an 

 electron velocity equal to the circuit phase velocity (b = 0) and for no at- 

 tenuation, the two new waves are unattenuated. 



If there is no magnetic field, the presence of a transverse field component 

 merely adds to the gain of the increasing wave. If a small magnetic field is 



