GENERAL CIRCUIT CONSIDERATIONS 225 



In the latter case, if resonators had parallel walls of very fine mesh normal 

 to the direction of electron motion there would be substantially no trans- 

 verse electric field. All the electric field representing stored energy would 

 act on the electron stream. In this case, we would have 



W 



= \l E'dZ (5.17) 



Here dH is an elementary area normal to the direction of propagation. W 

 given by this expression is the total electric and magnetic stored energy 

 per unit length. Where E is less than its peak value, the magnetic energy 

 makes up the difference. 



In evaluating £-/o in (5.16) we will have as an effective value 



(£/o)eff = JoJEd^ (5.18) 



Hence, we will have for the gain parameter C 



Jo j E' dl 



C = 



(^)^ (^ / FJ dz) z,(8Fo) 



/ T \ 1/3 



(5.19) 



C = 



4 - eVg Vo 



It is of interest to put this in a slightly different form. Suppose Xo is the 

 free-space wavelength. Then 



^ = ^^ (5.20) 



V Xo V 



where c is the velocity of light 



c = 3 X 10^° cm/sec - 3 X 10^ m/sec 



Further, we have for synchronism between the electron velocity m 

 and the phase velocity v 



i^ = 2r,Vo (5.21) 



Also 



c = l/V^ 



e = l/cVJ^e (5.22) 



r 



\ ix/f: = 377 ohms 



