IIIGII-FREQUENCY AMPLIFIER 45 



further assumed that only those a-c. quantities are allowed which have no 

 azimuthal variation, that is, — = 0. Fig. 6 shows the electron beam. 



Outside the beam the appropriate Maxwell's equations are 

 r dr rjo 



^ ^ irH,) = j-E, (A-1) 



(A-2) 

 (A-3) 



where 



(A-4) 



(A-5) 



Inside the beam, equations (A-2) and (A-3) remain the same, but instead 

 of equation (A-1) we have 



- I- (rH^) ^ j~E.+ q, + q, - (A-6) 



r dr 7/n 



where ^i and q^ are the first order a-c. convection current densities of the two 

 streams. These quantities can be calculated from the force equation and 

 the equation for the conservation of charge. Assuming that all a-c. quanti- 

 ties vary as expj(cot — /3z), the force equation is (for stream number one, say) 



juvi — jlSuiVi = — (e/ni)Ez (A-7) 



and the equation for the conservation of charge is 



jl3poiVi + jlSuipi = +iwpi (A-8) 



Equations (A-7) and (A-8) can be solved for vi and pi : 



— {e/m)E2^ 



(A-7a) 



(A-8a) 

 where 



Vl 



