834 THE BELL SYSTEM TECHNICAL JOURNAL, OCTOBER 1951 



Symbols in Section 4 — Physical Electron Streams 

 This section returns to the symboHsm of the general theory; and the 

 capital letters jE, t/, Q and I indicate total quantities. It also uses the 

 following special symbols: 

 V Actual electron velocity 



U Average of v 



N Mass density of electrons 



n Partial density in a range dv 



P Momentum density of electrons 



K Kinetic energy density of electrons. 



2. The General Solution for an Electron Stream 



The present theory of electron streams is a solution of two partial dif- 

 ferential equations, in which electron velocity U and electric intensity E 

 appear as the dependent variables. ^- 



The equation for electron velocity U is based on an idealism that is com- 

 monly used in vacuum tube theory. It assumes that the electron velocity 

 is a single-valued function of space and time or, stated in other words, it 

 assumes that all electrons in any plane normal to the ic-axis have the same 

 velocity. The variable U may then be regarded as a continuous function of 

 X and t, which is everywhere equal to the velocities of the individual elec- 

 trons. The differential equation for U follows at once from the fundamental 

 mechanics of electron motion, which states that for any individual electron 



where 17 is the acceleration constant of an electron, and the small relativity 

 terms are neglected. Then, since U is regarded as a continuous function of 

 Xy its total derivatives may be written in terms of partial derivatives, and 



vf + %^-r,E (2) 



dx dt 



which is here regarded as the fundamental equation for electron velocity. 

 It is of course based on an idealizing assumption that imposes limitations 

 on the general theory, and these limitations are discussed in a concluding 

 section of the article, where it is shown that the idealized velocity is an ap- 

 proximation for the average velocity in physical electron streams. 



The differential equation for the electric intensity E is given by the 

 theory of electromagnetism. It follows from this fundamental theory that 



