THERMIONIC ELECTRON EMISSION ' 443 



distance Zc. To obtain A^, Zc is substituted in the equation, 



3.58 X 10-8 



eA<^ = ^Pi + ^P, + Fz, = P^- 



Zc 



-^rn0^..li^l-J^^^,-\-Fz^|• (66) 



This A((0 is the decrease in the work function for a region near the center 

 of the patch. For other regions on the patch A(^ will be smaller; for 

 regions on the uncovered portion of the surface A(^ will be still smaller 

 until at large distances from the patch A(^ will correspond to the A^ 

 appropriate for the image law. 



To obtain the effect of the patch on the log i vs. F or log i vs. V F 

 curve it is necessary to divide the entire surface into small regions, 

 compute A(p for each and substitute these in equation (56) ; the values 

 of io in this equation are the same for all regions of equal area since at 

 large distances P^ has the same value over all regions. The values of i 

 are then added up for all regions and log i is plotted vs. V F. Since 

 this process is very tedious, and since in most thermionic experiments 

 one is not likely to deal with a single patch, it is not worth while to 

 make such an exact computation. It is, however, instructive to make 

 some further computations based on simplifying avSsumptions. 



Suppose we assume: (1) That for all regions on the patch, Av? has the 

 same value as for the central region, and (2) that the current from the 

 patch is large compared to the current from the uncovered portions 

 of the surface. These assumptions approximate the true conditions 

 for some cases and the errors due to the first assumption tend to 

 balance out those due to the second. If " a " is the area of the patch 

 then 



log ai = log a + log C/+ 2 log T - ^ejl.SkT + {A<p)el2.3kT 



= log aio + A<pel2.3kT, (67) 



where aio is the current from the patch area when F = 0. 



Hence log ilU = {A^ej2.3kT). (68) 



Values of Aipe obtained from equation (66) are substituted in equation 

 (68) and log ilia is plotted as a function of V F. Figures 9 and 10 

 show such plots. 



Figure 9 shows the effect of varying the radius R of the patch while 

 the charge density a is kept constant. The value of a is so chosen that 



