182 



DIELECTRIC CONSTANT. ABSORI'TION AND SCATTERING 



lens;tli in air of the radiation. The luoasiiruineiit of 'I'lii; l>i;so.\ atoi; (J il irnioo''"' 



x„ th.ns yields 8;. Simihirly, one finds that 



tanhS. = |£'mi„/£„ax|. (89) 



2. Calculation of ilick'clric cunsiaui and less fac- 

 tor from tcnninatiiKj iiii pedancc. The intrinsic im- 

 ])t'danee of the dielectric-filled portion of the guide 

 is found to be 



Z(0) = Z„2 tanh To d, 



where 



Z„, 



T'2 



(90) 



(91) 



The suliseript 3 refers to tlie dielectric medium, ju 

 is its permeability and y.^ is the propagation constant 

 of dielectric-filled section of the guide; for the TE 

 waves, Z„, is the impedance of the dielectric medium 

 itself. Using ecpuitions (iiO) and (01), one gets 



tanh72rf Z(0) 



Ml 



72rf 



Zai d 71 M2 



(92) 



The propagation constant y., determines finally the 

 complex dielectric constant tc through the funda- 

 mental relation 



'0-- „=„..]*, <») 



where the cutoff wavelength Ac is determined by the 

 geometry of the guide and the type of wave. In the 

 air-filled section of the guide 



72 = 



71 



= .(S)'""''^"'] 



(94) 



Consequently, from equations (03) and (94), the 

 complex dielectric constant of the material under 

 study becomes 



where 



(l/\,)^-(72/27r)'^ 

 (l/X,)2+(i/Xi)2 ' 



71 



.2t 



for free space. Finally 



-'"& 



(95) 



(96) 



(97) 



The solution of equation (93) can be found from 

 charts. It is claimed that with this method materials 

 with very low dielectric losses can be investigated 

 satisfactorily. 



Here the procedure consists in measuring the 

 change of resonant fi'ecjuency of a closed cylindrical 

 resonator upon the insertion, along the axis, of a rod 

 of the dielectric matei'ial in question. By observing 

 the change of Q value resulting from the insertion of 

 similarly dimensioned specimens of difl^erent mate- 

 rials, it is i^ossible to obtain comparative loss tangent 

 values. The relevant theoretical relations are sum- 

 marized below. 



By definition the (J \alue of a resonator system is 

 (jivcn in convenient form hv the relation 



Q = 27r 



energy stored 



energy loss per half cycle 



Both the energy stored and the energy loss can be 

 computed from the field distributions within the 

 resonator, and these are given, for a TM wave, as 



Hi = A ./i iyp), 



E, 



A, 



a + ,/a)f r 



■•/u (yp), 



(98) 



(99) 



where Eg is the tangential magnetic field strength 

 in amperes per meter. E, is the axial electric field 

 strength in volts per meter, p is the distance of the 

 point in question from the cylinder axis, y is the propa- 

 gation constant 



7" = jU6,a)" — ,](^(^P, 



(1) is the angular frequency, p. the permeability in 

 henrys per meter, and A is a constant determined by 

 the strength of the exciting source. In the formulas 

 (08) and (99) it was assumed that the walls of the 

 resonator are of infinite conductivity so that no elec- 

 tric intensity exists in them. This requires that 



EAp= a) = ./„ (70.) = 0, 



(100) 



where a denotes the radius of the resonator. This 

 equation has an infinite number of real roots, the low- 

 est being yft = 2.4048, and this determines the fun- 

 damental resonant frequency and wavelength An. If cr, 

 the conductivity of the dielectric, is neglected in com- 

 jjarison with er<a, the propagation constant becomes 



— 2,r 

 y = CO ^J per = — . 

 X 



(101) 



£r is the dielectric constant of the material filling the 

 resonator taken relative to air. Since A can be meas- 



