WAVE PIJOPAOATIOX ALOXO AX KLl^rTHOX liKAM 403 



niatioii, the ac fields can l)t' cxaluatcd at the iiii|)cilini)('(l particle 

 l)()sit ion. 



Not all (tf the ac fields need to appear in the toi-ce ('(illations, however. 

 Keference to the field ('(illations shows that the (•t)iitril)uti()n.s of the ac 

 nuijiiietic fields to the force components arc smaller than those due to 

 the electric fields hy a factor of the order of (uo/r)" or smaller (where c is 

 the velocity of light), and hence may he neglected, in addition, the force 

 exerted by Ke i^ <>f the same order as that due to II r , and may be neg- 

 lected tt)(). 



Omitting the factor exp j\o:t - 7(^(1 + "oO] for brevity from all ac 

 terms, we can write the equations of motion as follows: 



? - (/•„ + 7){d, + 'df = -r][-dV,/dr + Kr + (aj + r)ie, + e)B,] (19) 



(/•u +r)l + 2Hdo -\-'d) = -n^'Bo (20) 



I = -7,7?., (21) 

 These equations may be simplified with the aid of (12): 



7J dVo/dr = (ro + l-)dl (22) 



and by recalling that the dots may be replaced by multiplication by 

 j(aj — yuo). We obtain, finally: 



r = -nEr/ioi — yuof (23) 



9 = (24) 



z = 77i?,/(co - 7''o)' (25) 



Although the foregoing equations deal with the dynamics of individual 

 electrons, the assumption that the beam behaves like a smoothed-out 

 "fluid" of charge, with a single \'el()city at each point, enables us to 

 assign values of velocity and all other ac quantities, to fixed positions in 

 space, (r, 6, z). In these coordinates, the do velocity is given by: 



vo = (0, rd, , Wo) (26) 



and the ac velocity by: 



V = (r, re, i) ^ ^27) 



= j(w - yuo)[{r, re, z)] 



Although theac quantities are defined at n , they may be taken to be the 

 same at r, to a Unear approximation. 



