526 BELL SYSTEM TECHNICAL JOURNAL 



This equation shows that specifying the values of r and (eo — €«,) 

 gives as much information as specifying 7^ and (eo — €00). In some 

 applications there are advantages in using r but in other applications 

 greater simplicity of description is gained by using 700. 



There are several convenient ways of calculating the relaxation- 

 time. The more familiar ones depend upon the position of maxima 

 which occur in certain dielectric properties when they are plotted 

 against the frequency: there are maxima in the loss factor vs. frequency 

 curve, in the tangent of the loss angle vs. frequency curve and in 

 the power factor vs. frequency curve. As these maxima occur at 

 different frequencies, the corresponding expressions for the relaxation- 

 time are also different. They are listed in Table II. It will be ob- 



TABLE II 

 List of Formulae for Calculating the Relaxation-Time (t) 



1. The frequency at which the maximum in loss factor (e" or 



«' tan 6) occurs is Wmax(l) T = l/'«Jmax(l) 



2. The frequency at which the maximum in loss angle (e"/e' or 



..\ • /*«> 1 • 

 tan 5) occurs is Wmax(2) '^ — \ 



\ eo C0max(2) 



3. The frequency at which the maximum in power factor 



e"/(e'' + t"'yl^ occurs is a,n,ax(3) T = V2 J^ -^— 



' to Wmax(3) 



4. The quantity «"/(«' "■ «<») is a linear function of co ''' ~ Jf ('"/(*' ~«oo)) 



5. The relaxation-time is proportional to the ratio of the 

 absorptive part («o — ««>) of the static dielectric constant to 



the infinite-frequency conductivity (700) r = (eo — eQo)/47r7oo 



Note: An example of the application of these formulae is provided by the curves of 

 Fig. 4. The value of t for ice at — 2.6°C is 25.8 microseconds as calculated from the 

 position of the maximum in e", 24.6 microseconds as calculated from (eo — e«i)/^T'"Y<x), 

 where 7t» is in e.s.u., and 23.1 microseconds as calculated from the slope of e"/(e' — eoo). 



served that the simplest of these formulae for the relaxation-time is the 

 one involving the maximum in the loss factor. 



The function e"/{e' — e^) is a linear function of co with slope equal 

 to T. This property provides an alternative method of calculating the 

 relaxation-time. An example of its application to an actual material 

 is provided by the data for ice plotted in Fig. 4. 



It is interesting that e"/{e — e^) has no maximum while e"Je' has a 

 maximum. The physical basis of this is that in subtracting e„ from e' 

 we remove the contribution to e' made by optical polarizations. What 

 is left represents only the dielectric constant due to the polarization 

 responsible for anomalous dispersion. Consequently the tangent of 

 the loss angle of the polarization current responsible for anomalous 



