RADIAL BEAM VACUUM TUBE 195 



at a point of zero potential. In this case a real electron optical image of the 

 cathode is obtained. 



Neglecting the distortion of the field in the vinicity of the cathode, the 

 force equation for the electrons is 



d^x V 



m — = e — 



df R 



where V is the maximum anode potential, R is the radius of the anode struc- 

 ture and X is measured in the direction of the fields. Since the acceleration 

 is uniform the transit time t, neglecting space charge effects, may be obtained 

 from the expression 



1 /^\ .2 ^ 



2\dty 



R 



R 



Combining these equations we get 



t = 



The condition for focus is that the electrons make one revolution around the 



lines of force in time t. The angular velocity of the electrons is given by the 



well-known expression 



He 



CO = — 



m 



Setting oit = 2ir we get 



^--rV 



or in practical units 



^ R — 



Since the effect of the magnetic field on the space charge has not been 

 evaluated, we can only estimate the order of magnitude of the increase of 

 transit time due to the space charge. On the assumption that this increase 

 introduces a factor of 3/2* the above expression with space charge is 



rr _ 7.1V7 



^--R~ 



This formula has been found to check well experimentally. 



* The factor of 3/2 is the ratio of the transit times in a plane parallel diode with and 

 without space charge. See for example JNIillman and Seely, "Electronics," Chapt. 7, 

 p. 231. 



