310 BELL SYSTEM TECHNICAL JOURNAL 



Figure 3 shows the variation in field strength along the cathode for the 

 typical operating point, Vg = —2 and Vp = 100 volts. It is to be noted 

 that for d\/a less than 0.6, the electric field actually changes sign as we move 

 from a point immediately below a grid wire to the midpoint between two 

 grid wires. In other words a part of the cathode will not emit at all in these 

 cases while the remainder emits in a non-uniform manner. In the rather 

 extreme case of di/a = 0.4 only about a quarter of the cathode is emitting. 

 It is worth noting how relatively rapid the "shadow" or "island" formation 

 increases between di/a = 0.64 and 0.5 as compared to the increase in the 

 interval from 0.5 to 0.4. 



If the equation for m is solved for C{x) and the result substituted back in 

 the expression for E^ at the cathode we find : 



''' d, + (d, + d,)/^. ^'^^ 



where here of course m varies with x. This is identical with the expression 

 derived by Benham^® from Maxwell's approximate solution except that in 

 the latter case m was a constant. Our colleague, Mr. L. R. Walker, has 

 pointed out that the equation follows directly from the assumption of small 

 grid wires without explicit solution for the potential function. Since the 

 charge density Cc on the cathode is proportional to the field strength (the 

 factor of proportionality in MKS units is the dielectric constant e of vacuum 

 or 9.854 X 10"^" farads/meter), Maxwell's capacity coefficients Cg„ and 

 Cpc may be calculated from 



O-j = eEy = —{CgeVg + CpcVp) (25) 



The minus sign is used here because we are taking the ratio of charge to 

 voltage at the negative plate of the condenser consisting of cathode, grid and 

 anode surfaces. Hence 



di + (^1 + <^)/m 



C = ^-l^ (27) 



^' dr + (d, + d,)/^. ^"-'^ 



Since m is variable, an integration is required to determine the total capaci- 

 tance. From the periodicity of m with grid spacing it is possible to express 

 the result in terms of the average values of Cgc and Cpc over an interval of 

 length a along a direction parallel to the grid plane and multiply these 

 values by the total area of cathode surface. 



