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BELL SYSTEM TECHNICAL JOURNAL 



of current density carried by electrons which have velocity components 

 about Vx, Vy, V,, lying in the little range of velocity dvx dvy dvz. 



The reason for restriction (12) is that if an electron starts with zero 

 thermal velocity from the cathode, it will attain the velocity given by the 

 right side of (12) by falling through the potential drop V. As electrons 

 cannot have velocities smaller than this, we have (13) and (14). 



By integrating (11) with appropriate limits we obtain a more specialized 

 but very useful expression 



J < J'n = ]■ 



. /, , 11600V\ 



sin- 6 



(15) 



For usual values of voltage, unity in the parentheses is negligible, and we 

 can say that if all the electron paths approaching a given point in an electron 

 beam lie within a cone of half angle 6, the current density j at that point 

 cannot be greater than a limiting value jm which is proportional to the 



ELECTRON 

 LENS 



DEFLECTING 

 PLATES 



CATHODE 



Fig. 3— Parameters important in determining spot size in a cathode ray tube. 



cathode current density, to the vohage, to sin-^, and inversely proportional 

 to the cathode temperature. 



Let us see what this means in some practical cases. Figure 3 shows a 

 cathode ray tube. The electron stream has a width W at the final electron 

 lens, and is focused on a screen a distance L beyond the lens. The half 

 angle of the cone of rays reaching the screen cannot be greater than 



sin e= d = W/2L (16) 



Suppose the spot must have a diameter not greater than d. Let the spot 

 current be i. Then from (15), 



4i ^ . 



1' + 



U^) ^W/2Lf 



i < 





1 + HfL^) iW/2Lf 



(17) 



