THERMIONIC ELECTRON EMISSION 473 



The equation for coaxial cylinders only applies to an equipotential 

 cathode. Ordinarily the cathode is a filament and the potential varies 

 along its length because of the heating current. It is of interest to 

 examine the modification of the equation due to this effect. It can 

 be shown that for Vp > F/ 



X [l-3IV4F,+ 3/24(7//Fp)2---], (83) 



where Vp is the applied potential between anode and negative end of 

 the filament and Vf is the total potential drop along the filament. 

 For the case in which Vp < Vf 



i = (2V2/9)(g/m)K2F/^V5i?/32F/) = 5.92 X lO-HWi^F/. (84) 

 For concentric spheres: 



i = (4V2/9)(e/m)KFV«') = 2.96 X 10-'V'/a\ (85) 



where a^ is a function of R/ro and has been tabulated by Langmuir and 

 Blodgett.^5 a^ increases with R/ro. For R/ro = 5.0, a^ = 1.141; for 

 R/ro = 10, a^ = 1.777; for R/ro = 100, a^ = 3.652. 



Electrons Emitted with Maxwellian Velocity Distribution 

 For infinite parallel plates the space charge limited current is given 

 by 



i = (V2/97r)(e/w)K(F- F„)V(^ - ^mY) 



X [l + 2.66(^r/(F- Vr,^)eyq 

 = 2.33 X 10-«((F- Vm)V{x - XmY) 



X [1 + 2.48 X 10-2(r/(F - FJ)^]. (86) 

 In this equation 



Vm = (- 2.3Tk/e) log (iji) = - 1.98 X IQ-^T log {iji) (87) 



and Xm = 1.092 X lO-^r^'tiA''. (88) 



where F is the potential applied between cathode and anode corrected 

 for the contact potential; Vm is the value of the potential maximum 

 measured with respect to the zero of potential previously defined; x 

 the distance between cathode and anode; x„ the distance from the 

 cathode to the potential maximum; T the temperature of the cathode; 

 and is the value of the saturation electron emission, f 1 is a function 

 of In (is/i). Table VI gives a few values of fi as a function of In 



