1140 THE BELL SYSTEM TECHNICAL JOURNAL, SEPTEMBER 1954 



gives an eciiiation for (3, while any other set, with n ^ m, gives the 

 excitation of the n^^ mode. Substituting in (7) we have 



+ (/i3 — IJLl)H,moi^zm] dS ) .4„ = jLA, 



(8a) 



and 





+ (63 - e,)E,m,E.„] dS Bm ^ jMB 



(8b) 



Ignoring squares and products of small ciuantities, one then has 



5^ = 1^ (L + .1/). (9) 



The first example to be considered is the effect on the propagation in a 

 circularly cylindrical waveguide, when a coaxial, magnetized pencil of 

 ferrite of very small radius is introduced. The guide radius is n and that 

 of the pencil is Vi . Before the ferrite is introduced, mi == /^o and ei = eo , 

 where mo and eo are the free space values. The unperturbed fields are 

 those of the usual TE and TM modes in round guide. It is necessary to 

 calculate first the zero order electric and magnetic fields within the 

 magnetized pencil ; it will be sufficient to work out the magnetic case and 

 deduce the electric one by analogy. Since the cross-section of the pencil 

 is very small and transverse propagation effects consequently negligible, 

 the internal field may be calculated by solving a static problem. The 

 transverse magnetic field before the pencil is inserted is Htm and it is 

 assumed that the pencil is so small that over a circle with a few times 

 the radius of the latter, Htm is essentially uniform. We must now solve 

 Laplace's equation for a cylindrical rod immersed in a magnetic field 

 which is to be uniform at large distances. Within the rod, Bt = iiHt 

 — jiiHt*, and at its surface the usual boundary conditions pre^•ail. 

 Hereafter we write m for jU2 . 



The fields are derivable from potentials $out , ^in , which are of the 

 form 



'I'in = (Htmo-r), 



^ fu ^\ ^ (^■^) 



'S'out = {Htm-r) + -— - , 



