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BELL SYSTEM TECHNICAL JOURNAL 



expression (15) was obtained by integrating a supposedly exact expression. 

 What does this inequahty mean? 



The answer is that for the current to have the limiting value, electrons 

 of all allowable velocities must approach each part of the spot from all angles 

 lying within the cone of half angle 6. When the average current density in 

 the spot is less than the limiting current density, the possibihties are 



(a) Electrons are approaching each point in the beam from all angles, but 

 along some angles only electrons which left the cathode with greater than 

 zero velocity can reach the spot. 



(b) Electrons leaving the cathode with all velocities can reach the spot, 

 but at some portions of the spot electrons don't come in at all angles within 

 the cone angle 6. 



Fig. 5- 



-Relation between nearness of approach to limiting current density and fraction 

 of current utilized. 



Thus, we can have less than the limiting current either because electrons 

 do not reach the spot with all allowable velocities or from all allowable 

 angles. Of course both factors may operate. 



We can easily see how lens aberrations, which we know are present in all 

 electron-optical systems, can prevent our attaining the limiting current 

 density. There is a more fundamental limitation, however. It can be 

 shown that even with perfect focusing, we must sort out and throw away 

 part of the current in order to approach the limiting current density, and 

 we can even derive a theoretical curve for the case of perfect focusing re- 

 lating the fraction of the limiting current density which is attained to the 

 fraction of the cathode current which can reach the spot. Figure 5 shows 

 such a curve which applies for voltages higher than, say, 10 volts. 



Usually, the failure to approach the limiting current density is chiefly 

 caused by aberrations, and in ordinary cathode ray tubes the current 

 density in the spot may be only a small fraction of the limiting value. A 

 very close approach to the limiting current density has been achieved in a 



