WAVE PICTURE OF MICROWAVE TUBES 1365 



fields and no transverse displacement current. Then the total ac current 

 density it is the sum of the convection current density and the displace- 

 ment current density, or, 



ii = i -\- ju€E^ 



It = 



(All) 



Let us use a quantity Wp , which was long ago named the plasma fre- 

 quency (radian frequency) 



e 

 m^' (A12) 



Using cop , (AlO) can be written as 



According to Maxwell's equations the divergence of the total current 

 is zero. Both components of it vary mth z. If, as we have assumed, there 

 is no current normal to the z direction, then it must be zero. If this is 

 to be so, we must have 



(co - /3wo)' = 



^ = -±-. (AM) 



■Uo Wo 



In actual electron beams there is transverse electric field away from 

 the beam, and hence it is not zero. It is found, however, that when ojp is 

 small compared with co, we can write quite accurately 



^^-d.''-^ (A15) 



Here Wg , which is knowai as the effective plasma frequency, is smaller than 

 cop . As CO is raised, so that the wavelength of the space-charge Avaves 

 becomes smaller compared with the diameter of the electron beam, the 

 electric field tends to become largely longitudinal and w, approaches Up . 



