ULTRA-HIGH-FREQUENCY VACUUM TUBES 635 



ejm = — 1.77 X 10^ coulombs per gm.,* 

 k = 10~^ is the ratio of dyne-cm. to joules, 

 a is electron acceleration, cm. per sec.^ 



In this equation the effect of a magnetic field is disregarded. This is 

 thoroughly justified until electron velocities approach that of light or 

 the spacing between the two parallel planes becomes comparable with 

 the wave-length of any alternating field considered. A more detailed 

 discussion is given by Benham.^ 



A third fundamental equation is 



div eE = p, 



which, for the parallel planes now considered, becomes 



dE 



From (1), (2) and (3) is obtained 



el _ da ,.. 



kme dt 



The total current, /, may be considered to be composed of two parts, 

 the first being a constant component and the second being a function 

 of time only. On this basis we can write 



^^ = K + .'"W, (5) 



where K is the constant part, and (p"'{t) is the variable part, the primes 

 denoting derivatives with respect to the argument in parentheses. 

 Inserting (5) into (4) and integrating once with respect to time, we find : 



a ^ K{t- ta) -f <p"(t) - ip"{ta) -f aa + «(/«), (6) 



where ao + a{ta) is the acceleration when t = ta and fla is independent 



of ta. 



Another integration gives 



U = X^^-^^ + <p'{t) - <p'{ta) - {t- ta)^"{Q 



+ {t - ta)aa -\- (t - ta)a(ta) + Wa + mC^u), (7) 



where Ua + m(0 is the velocity when t = ta and Ua is independent of ta- 



* This value is based on deflection measurements (which are applicable to 

 vacuum tube analysis) rather than on spectroscopic measurements which give 1.76 

 X 10*. Compton and Langmuir « use the spectroscopic figure. For a compre- 

 hensive discussion of values of physical constants, see Birge, Phys. Rev. Supp., Vol. 

 1, July, 1929. 



