WAVES IN ELECTRON STREAMS AND CIRCUITS 629 



than the other two terms involving v. In our small signal theory we neglect 

 the term vdv/dZj and write 



t: + «o - = — E (1.2) 



dt dz m 



We note, then, that this approximates the true equation for small values of 

 E and v only. 



We have another equation 



~{i-h)= -I-Ap + ih,) (1.3) 



dz dt 



This is the equation of continuity, or of conservation of charge. If we 

 integrate it over a small distance As we obtain 



{i - Io)z+Az - {i - h)z = -Q^ [(P + Po)A2;l 



The quantity (p + po)Az is the charge per unit cross section in the distance 

 Az. Thus, the right-hand side is the rate at which charge in the distance Az 

 decreases. The quantity on the left is obviously the rate at which charge 

 per unit cross section is flowing out of the space Az long. 

 If we carry out the operations in (1.3) we obtain 



As dio/dz = 0, dpo/dt 



(1.4) 



(1.5) 



We need to add that the convection current is given by 



i — Iq= (p + Po)(t' + Wo) 



i — lo = PqV -{- puo + poUo + pv 



The term pqUq is a constant term and is to be identified with — /o 



— /o = PqUq (1.6) 



The term pv is a product of a-c. quantities. Suppose we solve all our equa- 

 tions neglecting pv. Then, the error caused by this approximation will be 

 less as p and v are less, that is, at small signal levels. Thus, we write 



i = pMo + vpo (1.7) 



