. The Bell System technical journal . ollisions, which may actually makethe current density smaller. 310 BELL SYSTEM TECHNICAL JOURNAL of current density carried by electrons which have velocity componentsabout 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 zerothermal velocity from the cathode, it will attain the velocity given by theright side of (12) by falling through the potential drop V. As electronscannot have velocities smaller than this, we have (13) and (14). By integrating (11) with appropriate limits
. The Bell System technical journal . ollisions, which may actually makethe current density smaller. 310 BELL SYSTEM TECHNICAL JOURNAL of current density carried by electrons which have velocity componentsabout 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 zerothermal velocity from the cathode, it will attain the velocity given by theright side of (12) by falling through the potential drop V. As electronscannot have velocities smaller than this, we have (13) and (14). By integrating (11) with appropriate limits we obtain a more specializedbut very useful expression J < Jn = ]■ . /, , 11600V\ sin-6 (15) For usual values of voltage, unity in the parentheses is negligible, and wecan say that if all the electron paths approaching a given point in an electronbeam lie within a cone of half angle 6, the current density j at that pointcannot be greater than a limiting value jm which is proportional to the ELECTRONLENS DEFLECTINGPLATES CATHODE. Fig. 3—Parameters important in determining spot size in a cathode ray tube. cathode current density, to the vohage, to sin-^, and inversely proportionalto the cathode temperature. Let us see what this means in some practical cases. Figure 3 shows acathode ray tube. The electron stream has a width W at the final electronlens, and is focused on a screen a distance L beyond the lens. The halfangle 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 spotcurrent be i. Then from (15), 4i ^ . 1 + U^) ^W/2Lf i < 1 + HfL^) iW/2Lf (17) PHYSICAL LIMITATIONS IIV ELECTRON BALLISTICS 311 Thus if for a given spot size we want to increase the spot current, and if weare limited to a given cathode current density because of cathode Hfe,we must make 1 larger, W larger or L smaller. Making W larger increases both lens and deflection aberrations. MakingL smaller mean
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