MEASUREMENT OF DIELECTRIC AND MAGNETIC PROPERTIES 447 



Integration of the electric or magnetic field over the volume of the 

 cavity yields the stored energy in the empty cavity at resonance for one 

 of the two circularly polarized modes 



W''' = 0.2387 ^° • f^' B'a'L (A8) 



4 kc^ 



The magnetic energy Wj'^ in the disc is found by integrating (A4) 

 over the volume of the disc and assuming that the field is constant over 

 the thickness t of the disc. We obtain: 



Wj" = 0.2387 "^ f- BVtixmR, ± kU.} (A9) 



The two functions Ri and Rt depend on the ratio of disc radius to cavity 

 radius 



R, = 4.1893 ^ ^{J,{Kn)Y + [l- -^}j (J,(/c,ro))'] (AlO) 



Rt = 2.4720 (Ji(/Ccro))' (All) 



It is interesting to note that these two functions are approximately 

 equal (Fig. 4) if the disc radius is less than half the cavity radius. In 

 this region the field in the cavity is essentially circularly polarized, 

 whereas elliptical polarization exists near the wall of the cavity. Inserting 

 (A8) and (A9) into (Al) we find the desired relationship between the 

 two frequency shifts associated with positive and negative circular 

 polarization and the tensor components Xm and k. 



2, 



ojo 2 L^ 



Equations (Al) and (A12) hold for complex Xm and k if a complex 

 frequency shift is introduced as follows: 



c?(y = CO — ojo + j{cx — ao) (A13) 



The attenuation constant a may be defined in terms of the internal Q 

 of the cavity, a = §co/Q and the internal Q is defined as 



^ _ w (energy stored in circuit) 

 average power loss 



Thus, the imaginary part of the frequency shift may be expressed as 

 the difference between (l/Q) of the perturbed cavity at the new reso- 

 nance frequency w and (1/Qo) referring to the empty cavity at wo . 



A(l/Q) = ^ - ^ = 2 ^^LZL^ (A14) 



We note that the imaginary part of the right hand side of (Al) does 

 indeed represent the power dissipation in the perturbing sample over 

 the stored energy times co. Hence, taking the imaginary part of (A12) 



