TRAVELING-WAVE TUBES 57 



£.4 = B,K,{yr)e'''''-'" (9) 



outside radius a. Omitting the factor g'^"^"^'^ the radial and circumferential 

 components associated with these, obtained by applying the curl equation, 

 are, inside radius a, 



(10) 

 (11) 



(12) 

 7 



£.3 = bJ^ hiyr) (13) 



and outside radius a 



H^i= -B,^—K,(yr) (14) 



7 



Hr2 - -Bj-^Kiiyr) (15) 



E,,= B,^^K,{yr) (16) 



7 



£,4 = -Bj-^Kiiyr). (17) 



7 



The boundary conditions which must be satisfied at the cylinder of radius 

 a are that the tangential electric field must be perpendicular to the helix 

 direction 



E,3 sin ^ + £^1 cos ^ - (18) 



£,4 sin >^ + £^2 cos ^ = 0, (19) 



the tangential electric field must be continuous across the cylinder 



£23 = £.-4 (and £^1 = £^2), (20) 



and the tangential component of magnetic field parallel to the helix direc- 

 tion must be continuous across the cylinder, since there can be no current 

 in the surface perpendicular to this direction. 



H^i sin ^ + //d,3 cos "^ = H^1 sin ^ 



(21) 

 + H^i cos ^. 



