868 



THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1957 



>' + 'V (I, + -. 



r J. /M - 3F 







J- + 'v(^^ + ^ 



dd 



Ve = 0, 



^i)^ , dv, . dv. 



-\- Vr -\- U 



dt dr dz 



V 



j^h + (V 



dr 



a7 



dz' 



(8b) 



(9a) 



(9b) 

 (10a) 

 (10b) 



An expression for the ac space-charge density, jo, can be obtained in 

 terms of its steady-state counterpart, po , by means of the charge-con- 

 servation equation : 



dp 



dt 



— Vo-grad p — 5"grad po — po div v — p div ^o 

 — Vo-grad po — po div Vo . 



(11) 



As the beam diameter changes slowly, the dc space-charge density at 

 each plane is taken to be inversely proportional to the square of the 

 radius, b: 



grad PC I = 



dpo 

 ~dz 



2podb^ 

 b dz~ 



2tv 

 ^lr 



Po 



div Vo = — - 



yo-grad po 



•'■"' + "' (I + 7, 



■po 



po 



'1 d 



2Vr 



r 



(rvr) - jl3v, ( 1 - j i-- , 

 r dr \ (311 r /_ 



(12) 

 (13) 

 (14) 



At low frequencies (the UHF region), 



dr r 



as {yrf « 1. This inequality is also true of other ac quantities pro- 

 portional to V, such as V. and p, and with a small error can be assumed 

 to be true for Vr . When the operator d/dr is omitted from (8), (9), (10), 

 and (14), it is possible to solve explicitly for p in terms of po and T". 



