CHAPTER XIII 

 TRANSVERSE MOTION OF ELECTRONS 



Synopsis of Ch.a.pter 



SO FAR WE HAVE taken into account only longitudinal motions of 

 electrons. This is sufficient if the transverse fields are small compared to 

 the longitudinal fields (as, near the axis of an axially symmetrical circuit) 

 or, if a strong magnetic focusing field is used, so that transverse motions are 

 inliibited. It is possible, however, to obtain traveling-wave gain in a tube in 

 which the longitudinal field is zero at the mean position of the electron beam. 

 For a slow wave, the electric field is purely transverse only along a plane. 

 The transverse field in this plane forces electrons away from the plane and 

 preferentially throws them into regions of retarding field, where they give up 

 energy to the circuit. This mechanism is not dissimilar to that in the longi- 

 tudinal field case, in which the electrons are moved longitudinally from their 

 unperturbed positions, preferentially into regions of more retarding field. 



Whatever may be said about tubes utilizing transverse fields, it is cer- 

 tainly true that they have been less worked on than longitudinal-field tubes. 

 In view of this, we shall present only a simple analysis of their operation 

 along the lines of Chapter II. In this analysis we take cognizance of the fact 

 that the charge induced in the circuit by a narrow stream of electrons is a 

 function not only of the charge per unit length of the beam, but of the dis- 

 tance between the beam and the circuit as well. 



The factor of proportionality between distance and induced charge can be 

 related to the field produced by the circuit. Thus, if the variation of V in the 

 X, y plane (normal to the direction of propagation) is expressed by a function 

 <l>, as in (13.3), the effective charge ps is expressed by (13.8) and, if y is the 

 displacement of the beam normal to the z axis, by (13.9) where $' is the de- 

 rivative of $ with respect to y. 



The equations of motion used must include displacements normal to the 

 z direction; they are worked out including a constant longitudinal magnetic 

 focusing field. Finally, a combined equation (13.23) is arrived at. This is 

 rewritten in terms of dimensionless parameters, neglecting some small terms, 

 as (13.26) 



62 ' (52 -\- P) ' 

 616 



