628 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



Here wo is a constant component and v is sl small fluctiiatin? or a-c. com- 

 ponent 



charge density = p + Po 



where again po is the average or d-c. component, which will of course be 

 negative, and p is the a-c. component 



convection current density 



i — lo 



Here /o is the average or a c. current density and, as the electrons are 

 assumed to move in the +2 direction, the current density in the +2 direc- 

 tion is taken as — /o . In other work I have used i and /o as current rather 

 than as current density; I hope that this will cause no confusion. 



It is assumed that there is no average field. It is assumed that there is an 

 a-c. field in the z direction only, and this is called E. 



We have two equations to work with. One is 



div -i- uq) e ^ 

 _ = — _ /^ 



at m 



Here e/m, the charge-to-mass ratio of the electron, is taken as a positive 

 quantity. The time derivative is that moving with an electron. We can in- 

 stead take derivatives at a fixed point 



d{v + «o) d{v -\- Uq) . f . . div + uo) 



-r. = — -\- \v + Uq) 



dt dt dz 



which gives 



dv , dv 

 dt + '^dz 



+ ^+(. + «o)^ (1.1) 



dt dz 



. dv e -, 



-{- V— = --E 

 dz m 



The terms on the second line are zero because dm/dt = 0, du^/dz = 0. 

 Further, let us consider a series of solutions of (1.1) for fields in which E 

 has the same form in time and space, but varies in magnitude. As E is made 

 smaller and smaller, v will become smaller and smaller, and the term vdv/dz, 

 which is a product of two a-c. quantities, will become relatively smaller 



