WAVE PROPAGATION ALONG AN ELECTRON BEAM 401 



^^^j_dE._j<^j^ (2) 

 7 dr 7" 



H, ^'^Er- ^- Jr (3) 



TE WAVE 



r dr \ dr / r dr 



H, = I (^Jh + J.) (5) 



y \ dr 



7 y \ dr I 



Here (r, 0, 2) are the polar cjdindrical coordinates, co the angular driving 

 frequency, e the dielectric constant and /x the permeability, of free space, 

 in ]MKS units. The ac ampUtudes of the electric and magnetic fields, and 

 the convection-current density, respectively, are represented by the 

 components of E, H, and J. All ac quantities have been assumed to vary 



as exp j(<^^-7^)- 



When the assumption is made that the convection current density in 

 the beam is of the same order of magnitude as the displacement current 

 density, equations (2) and (6) reduce to the following: 



E, = i ?f' (7) 



7 dr 



E, = -^ ?p (8) 



7^ dr 



In order to evaluate the components of / in the beam, it is necessary 

 to determine the velocity and charge distributions, first in the unmodu- 

 lated, and then in the ac modulated beam. 



The focusing of long cyhndrical electron beams by axial magnetic 

 fields of moderate strength has been fully described by Brillouin^ and 

 Samuel^ This type of electron motion, called "Brillouin flow", can be 

 established when a parallel electron beam abruptly enters a suitable 

 magnetic field. The electrons thereupon acquire an angular ^•clocity 

 component which leads to a balance of radial forces in the beam. 



The equations of motion of electrons in an axial magnetic field Bo are 

 as follows: 



r - r^ = v(dVo/dr - rdBo) (9) 



rd + 2rd = vrBo (10) 



z = rj-dVo/dz (11) 



