VACUUM TUBE ELECTRONICS 63 



equations is of paramount importance, a few words of explanation and 

 repetition will be necessary, 



II. Fundamental Relations 



For the development of the fundamental relations existing between 

 the two parallel planes, we have the classical equations of the electro- 

 magnetic theory which may be set down in the following form; 



(1) 



where E is the electric intensity, V the potential, P the charge density, 

 / the total current density consisting of conduction and displacement 

 components, and U is the charge velocity. These equations apply to 

 frequencies such that the time which would be taken by an electro- 

 magnetic wave in traveling between the two planes is inappreciable 

 when compared with the period of any alternating-current frequency 

 considered. Ordinarily this limitation will become of importance only 

 at frequencies higher even than those in the centimeter wave-length 

 range where the time of electron transit is of great importance, al- 

 though the time of passage of an electromagnetic wave is still negligibly 

 small. 



An electron situated between the two parallel plates will be acted 

 upon by a force which determines its acceleration. The resulting 

 velocity is a function both of the distance, x, from the cathode and 

 the time, /, so that in terms of partial derivatives, the equation 

 expressing the relation between the force and acceleration is 



f+C/|^=^£. (2) 



dt dx m 



From (1) and (2) may readily be obtained 



17'7"\t7" at / 't\ 



di dx j m ' 



In this equation we have a relation between the velocity and the total 

 current density. The advantage of this form of equation for a starting 

 point lies in the fact that the total current density / is not a function 



