442 BELL SYSTEM TECHNICAL JOURNAL 



where P^ is the potential energy when 2 = oo and F = 0; e = 4.774 

 X 10"^''. Curve 2 is the potential energy Pp due to the patch or con- 

 denser along a line normal to the surface at the center of the condenser. 

 R has been taken as 4 X 10"* cm. The equation of this curve is 



Pp = UOOTallz/iz^ + R^)^. (63) 



In the derivation of this formula it has been assumed that either 

 R:^ I or else that z:^ I; for our case the first of these assumptions 

 will always be fulfilled so that the formula is applicable when z is 

 equal to or larger than /; it is not applicable for z less than /. 



Curve 3 is the algebraic sum of P values for curves 1 and 2. It 

 represents the potential energy along the central normal due to the 

 image and patch fields. Curve 4 represents the potential energy due 

 to an applied field of 6000 volts/cm. Curve 5 is the sum of curves 1 

 and 4; curve 6 that of 3 and 4. 



The effect of the applied field is to reduce the critical distance Zc 

 and the work function. A given applied field will reduce Zc more for a 

 clean surface than for one with the patch; but the converse is true 

 for the work function. The reduction in the work function is equal 

 to the reduction in the value of Pmle. This consists of three parts as 

 indicated in the figure for curve 6. APi/e is the decrease in P due to 

 the image forces from z = Zc to z = co ; APple is the decrease due to 

 the patch field from z = Zctoz = 00 ; Fzc is the decrease in P due to the 

 applied field from z = to 2 = Zc. These quantities can be evaluated 

 after one has determined the value of Zc as follows: 



The peak or maximum in curve 6 occurs at a value of s = Zc at 

 which 



dPIdz = dPildz + dPpldz = Fe. (64) 



P^rom equation (62) dPildz = 3.58 X \0-^Jz\ 



and from equation (63) 



^= - {12007ral)e 

 dz 



1 _+ '' 1 



(22+i?'-')i ' (22+i?2)! 



nOOirale—, 



that 



F = ^^^V^ + U00..1 



i?2 



{z'+R')i 



(65) 



From this equation /*' is plotted for various values of 2. For any value 

 of F a value of z can be read off. This value of z will be the critical 



