14 BELL SYSTEM TECHNICAL JOURNAL 



We may think of (i,. as the phase constant of a disturbance traveUng with 

 the electron velocity. 



We have another equation to work with, a relation which is sometimes 

 called the equation of continuity and sometimes the equation of conserva- 

 tion of charge. If the convection current changes with distance, charge 

 must accumulate or decrease in any small elementary distance, and we see 

 that in one dimension the relation obeyed must be 



^i = -^^ (2.17) 



dz dt 



Again we may proceed as before and solve for the a-c charge density p 



-Ti = -joip 



p= zJIi (2.18) 



CO 



The total convection current is the total velocity times the total charge 

 density 



-/n+ i = («o+ f)(po+ p) (2.19) 



Again we will linearize this equation by neglecting products of a-c quanti- 

 ties in comparison with products of a-c quantities and a d-c quantity. This 

 gives us 



i = pqv + Uop (2.20) 



We can now substitute the value p obtained from (2.18) into (2.20) and solve 

 for the convection current in terms of the velocity, obtaining 



Using (2.15) which gives the velocity in terms of the voltage, we obtain 

 the convection current in terms of the voltage 



iVoU^e - r)^ 



2.5 OVKKAI.L ClkCUlT AND ElKCTKOMC EQUATION 



In (2.22) we have the convection current in terms of the voltage. In (2.10) 

 we have the voltage in terms of the convection current. Any value of F for 

 which both of these equations are satisfied represents a natural mode of 



