848 the bell system technical journal, october 1951 



4. Physical Electron Streams 



[This section returns to the symbolism of the general theory; and the 

 capital letters, F, -E, Z7, Q and / now indicate total values.] 



The preceding general solution for an electron stream is based on ideal- 

 ism, namely, the assumption that the electron velocity is a single-valued 

 function of space and time. The stream then obeys the differential equations 

 (2) and (5), and the theory is a general solution of these fundamental 

 equations. But it is well known that the velocity in a physical electron stream 

 is not single valued :^^^^ Electrons are emitted from their original source 

 with shghtly different velocities; and some electrons acquire energy from 

 the high-frequency electric field and overtake their slower neighbors. These 

 factors cause the velocity to be a multi-valued function of space, and the 

 electrons have various velocities in any plane normal to the axis of the 

 diode. The present section derives the differential equations for a multi- 

 velocity stream, and compares them with the idealized equations (2) and 



(5). 



For this purpose, the actual velocity of an electron is indicated as v. It is 

 also convenient to develop the equations in terms of mass, so we let A^ equal 

 the mass density of electrons at any coordinate x. The fractional mass 

 density of electrons with velocities lying in any range from u to u + dv may 

 likewise be expressed as ndv^ where w is a function of u, x and /; and it fol- 

 lows that 



■"L 



+00 



ndv. (65) 



The momentum density P of the electrons is then given by 



P = / nvdv (66) 



J— 00 



and their kinetic energy density K is 



K= r^dv (67) 



J— 00 ^ 



It also follows that the average electron velocity U is given by 



U = ? (68) 



TTiis is the new mechanical variable in the theory of physical electron 

 streams. 



The differential equation for the electric intensity is now easily derived 

 from the fundamental electromagnetic equation 



