952 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1954 



where ei = eo(l — (f)- Suppose now that the applied field is weak, so that 

 CO ^loe "^ CO Moci '^ /8 . Then, the cotangents may be expanded and by 

 equating admittances one obtains 



J^ 



r?/3 - - 



J^^i 



(co-Moei - (3-)(a;o - .Ti) 



or 



^ = 1 + — ' - " 



^ ^"^^ -ei + ( '7(8 - — ) (a:o - Xi) 



(12) 



Since e — ei is of the second order in a this equation may be solved by 

 substituting the unperturbed value of /?, ± co\/juo€j, in the right hand 

 side. It is clear that although the system is non-reciprocal, it will be 

 so only to third order in a. This system, therefore, illustrates the fact 

 pointed out in Section 1.1 that even when the fields are plane polarized 

 in the absence of a dc magnetic field, non-reciprocity may arise, although 

 it may be very small in weak fields. 



The third example to be considered is one which has been referred to 

 in Section 1.1, namely that in which a strip of ferrite is placed across 

 the short dimension of rectangular wave guide, see Fig. 1(a). In view 

 of the fact that this problem has been discussed with great thoroughness 

 by Lax, Button and Roth, we shall, after deriving the characteristic 

 equation, consider only the case of a very thin strip. Let the thickness of 

 the strip be 2xo and the distance from its two faces to the nearest guide 

 wall be Xi and x^. respectively. The admittances at the two faces are then 



— cot ViSo- - jS' xi 



and 



-J 



2 



— cot V/3o'- - /S^ X2 



COjUo 



1-, = -[<^W - /?'] = -(/3/ - ^'), 



respectively, where /So = co juoco . Inside the ferrite 

 and 



2n -rr dEv 



ax 



