possibilities of Future Development 115 



Here $ is the potential on the axis produced by the electrodes 

 alone, H and p are the values of the magnetic field and space 

 charge density along the axis, the primes denote differentiation 

 with respect to the axis z, and R is 



1 

 R^r^^ (42) 



It can be seen that the magnetic field is compensated by the 

 space charge if Hull's value pu is substituted. Therefore, if 

 Hull's were the true or only possible distribution of charges in a 

 static magnetron, the diverging lens as proposed would be an 

 impossibility. But there are good reasons to doubt this. 



The author has carried out a theoretical investigation of the 

 problem of stationary distribution of electrons in combined elec- 

 tric and magnetic fields, to be published elsewhere,* which starts 

 from more general assumptions. Instead of considering equi- 

 librium orbits, the theory assumes that the electron trajectories 

 inside the accessible space are so long, so complicated, and so 

 irregular, that statistical considerations can be applied to them. 

 This leads to a distribution, illustrated in figure 42, which is 

 essentially different from Hull's solution. In figure 42 it is as- 

 sumed that the cathode is not placed exactly into a position of 

 zero magnetic flux, so that the electrons cannot reach the axis, 

 and the accessible space becomes annular, bounded inside and 

 outside by two radii, ri and Vo. It can be easily shown from the 

 momentum theorem (2) that these will be the two roots of the 

 equation 



where Aq and ro are the values of A and r at the cathode. Statisti- 

 cal mechanics gives the result, that between these two limits 

 the electron density will distribute itself in inverse ratio to the 

 distance from the axis, i.e., according to p = const./r. The con- 



* D. Gabor, Stationary Electron Swarms in Electromagnetic Fields, 

 Proc. Roy. Soc. A., 183, 436 (1945). 



