ELECTRON STREAMS IN A DIODE 851 



When U is set equal to U, the first equations in the two sets are identical ; 

 and in this respect the theory of the idealized stream corresponds to that 

 of the physical stream. But the second equation for the physical stream then 

 differs from its analogue by the inclusion of an additional term 



The bracketed quantity in this term is the difference between the actual 

 kinetic energy density and the kinetic energy density calculated as if the 

 electrons were all moving with their average velocity U. It is often a small 

 term that can be neglected, and the physical stream is then approximately 

 described by the idealized equations (2) and (5). 



It is, however, rather obvious that there are cases in which this approxi- 

 mation cannot be made. It is invalid in the region between a thermionic 

 cathode and its voltage minimum, where the electrons are travehng in both 



Nm 

 directions along the a:-axis, and cause K and — — to have appreciably dif- 

 ferent values. So, when the first plane of the diode is a space-change-Hmited 

 cathode, the idealized theory can apply only in the region beyond the 

 voltage minimum. This difficulty is usually overcome by considering the 

 virtual cathode as the first plane of the diode. In all other regions the 

 electrons are normally traveling in one direction only, and the idealized 

 equations are then an approximation for the physical stream over a wide 

 range of conditions. 



The nature of this approximation is seen more clearly by considering the 

 electrons to be uniformly distributed over a velocity range s, where 5 is a 

 function of x and /. Then the mechanical equation (82) is 



Under the usual conditions encountered in electronic tubes, - is small 



compared to — , and its gradient may be neglected in comparison to that 



of g^ 



2 • 



The approximation can also be considered in a more rigorous manner as 

 follows: The velocity spread s may be expressed in electron volts by the 

 relation 



