WAVE PICTURE OF MICROWAVE TUBES 1363 



UJ3 of dc and ac parts as follows:* 



charge density: — po + p 



con\'ection current densit}^: — /o + t 



velocit}^ : Vo + v 



Here po , h and ?/o are positive dc quantities. The quantities on the right 

 are the ac components. 



AVe have from the definition of con\'ection current 



(-/o + i) = (-P0 + p)(uo + v) (Al) 



In the case of very low level operation, we neglect products of ac 

 quantities in comparison with products of ac and dc quantities. Doing 

 this, we obtain from (Al) the dc and ac convection currents 



/o = poUo (A2) 



or 



i = —pov + vop (A3) 



i + po?' 



Uo 



(A4) 



We can apply the continuity equation, or, the equation of conserva- 

 tion of charge, to the ac convection current 



(A5) 



^i 



(A6) 



0) — /3mo 



Thus, if we have a wave with a given phase constant /3, and if we know 

 Po and cj, (A6) gives the convection current in terms of ac electron 

 velocity. How can we find what j8 will be? To find this we must consider 

 the effect of the electric field on the electrons. Consider an ac electric 

 field E^ in the z direction, which also varies with time and distance as 



* It will be convenient elsewhere to use — /o and i as currents rather than 

 current densities and — po and p as charge per unit length. 



