532 BELL SYSTEM TECHNICAL JOURNAL 



APPENDIX 



The D-C Counterparts of Anomalous Dispersion 



Having developed the a-c characteristics of the model, the properties 

 of which were specified in the preceding paper, ^^ we now turn to the 

 direct-current characteristics of this model. This involves investigat- 

 ing the characteristics of the currents produced when a constant or 

 direct voltage Vi is applied to a condenser containing a dielectric hav- 

 ing properties which correspond in all respects essential to the discus- 

 sion to those of the model just described. Let the resistance of the 

 leads to the condenser be R and let the source of the electromotive 

 force Vi have a negligible internal resistance. These conditions re- 

 quire the following five equations to be satisfied simultaneously: ^^ 



dP 

 r"^ -]- fP - ne'-F = 0, (1) 



F = E + APt, (2) 



Pt = kiF + P, (3) 



D = E + Pt, (4) 



£ = E.-Ci?f. (5) 



(Rational e.s.u. are used in these equations for convenience but the 

 final relations are converted into electrostatic units. ^^) Pt denotes the 

 total polarization; it is the sum of the optical polarizations, given by 

 kiF, and the polarization P which forms comparatively slowly and 

 causes anomalous dispersion. The total displacement D, is given by 

 (4) . Co is the air capacitance of the condenser. V is the potential drop 

 over the condenser. The separation of the plates of the condenser is d. 

 When the whole drop in potential is concentrated over the condenser 

 the applied field strength £i is given by £i = Vi/d. Equation (5) 

 is obtained by equating the total drop in potential over the leads and 

 the condenser to the applied potential Vi. 



Equations (1) to (5) may be combined to give the following: 



d'^D dD 



(1 + 2Po)t'CR^ + {(1 - po)r' + CR{1 + 2po + 2Pr)]^ 



+ (1 - Po- pi)D - (1 + 2po + 2pi)Ei = 0. (6) 



1^ These properties are also outlined on page 513 of this paper. 



'^ Cf . P. Debye, " Polar Molecules," pp. 86-88, where an analogous case is discussed. 



1' For conversion factors see, for example. Mason and Weaver, Reference 2, page 

 370. The rational electrostatic unit of conductivity is smaller than the e.s.u.; the 

 ratio is 4ir. The dielectric constant is unaffected by changing from rational e.s.u. 

 to e.s.u. 



