10 BELL SYSTEM TECHNICAL JOURNAL 



the electrons form a hollow cylmdrical beam in an axially symmetrical cir- 

 cuit, a case of some practical importance. 



Besides these assumptions, it is assumed in this section that the electrons 

 are displaced by the a-c field in the axial direction only. This may be ap- 

 proximately true in many cases and is essentially so when a strong magnetic 

 focusing field is used. The efTects of transverse motion will be discussed in 

 Chapter XIII. 



In this chapter an approximate relation suitable for electron speeds small 

 compared to the velocity of light is used in computing interaction between 

 electrons and the circuit. 



A more general relation between impressed current and circuit field, valid 

 for faster waves, will be given in Chapter VI. Non-relativistic equations of 

 motion will, however, be used throughout the book. With whatever speed 

 the waves travel, it will be assumed that the electron speed is always small 

 compared with the speed of light. 



We consider here the interaction between an electric circuit capable of 

 propagating a slow electromagnetic wave and a stream of electrons. We can 

 consider that the signal current in the circuit is the result of the disturbed 

 electron stream acting on the circuit and we can consider that the disturbance 

 on the electron stream is the result of the fields of the circuit acting on the 

 electrons. Thus the problem naturally divides itself into two parts. 



2.3 The Field Caused by an Impressed Current 



We will first consider the problem of the disturbance produced in the 

 circuit by a bunched electron stream. In considering this problem in this sec- 

 tion in a manner valid for slow waves and small electron velocities, we will 

 use the picture in Fig. 2.3. Here we have a circuit or network with uniformly 



I M M M i i ^-^^'A 

 * T T T T T T T 



Fig. 2.3 — E(|uivalent circuit of a traveling-wave tube. The distributed inductance 

 and capacitance are chosen to match the jihase velocity and field strength of the field act- 

 ing on the electrons. The impressed current due to the electrons is —dj/dz, where / is the 

 electron convection current. 



distributed series inductance and shunt capacitance and with current / and 

 voltage V. The circuit extends infinitely in the z direction. An electron con- 

 vection current i flows along very close to the circuit. The sum of the dis- 

 placement and convection current into any little volume of the electron 

 beam must be zero. Because the convection current varies with distance in 



II 



