734 BELL SYSTEM TECHNICAL JOURNAL 



In other words 



^^= -VEy. (9) 



This relation will also be approximately correct remote from the axis in an 

 axially symmetrical tube. Here we let y represent a displacement in the r 

 direction. 



We may also define a quantity a so that 



Ey=jaE, (10) 



-{A exp QTy) - B exp (-;Ty)) ,^^. 



Ay exp {jVy) + B exp {-jVy) 



For an active mode, such as the one we consider, the chief component of 

 jV is a positive real number. Hence, for large positive values of y, the 

 quantity a approaches a value 



a = 1 . (12) 



This is characteristic of a plane symmetrical field far from the axis and also 

 of an axially symmetrical field far from the axis. 

 Using (9) and (12) we rewrite (4) 



P* = l-E*fe - j«*(r*/oy + qy)] (13) 



We see from this that, according to our assumptions, for the mode considered, 



£. = {q, - ia*(r* /oy + qy)) ^^^^^l'_ ^2^^ • (14) 



We will henceforward assume that a and ^ are so nearly real that we can 

 regard them as real quantities, giving 



E, = \q. ~ jaiT* 7„ y + qy)] ^^^^^I[_ pg) • ( l-'^) 



This is, then, the circu't equation which we will use. 

 2. Electronics Equations 



We will assume an unperturbed motion of velocity uq in the z direction, 

 parallel to a uniform magnetic field of strength B. Products of a-c quantities 

 will be neglected. 



In the X direction, perpendicular to the y and z direction 



Assume that :t = at y = 0. Then 



X = -riBy. (17) 



J 



