DIELECTRIC PROPERTIES OF INSULATING MATERIALS 653 



and by other expressions — which in some cases may approximate 

 either to F = E or to F = E + (47r/3)P — for still other materials. 

 The quantities F and 5, are vectors, but for isotropic materials 5 is in 

 the same direction as F. 



If, following the method employed by Lorentz, we write an equation 

 of the form (10) for each charged particle in a physically small volume 

 8 (such as the cubes of Fig. 2), multiply each equation by e, add the 

 equations for all of the particles in 8, and divide by the volume 8, 

 we obtain 



mP -{- rP +fP = ne^F, (11) 



where P = (l/5)X!e5 and n is the number of charged particles charac- 

 terized by the constants m, r and / per unit volume. The volume 8 

 may be considered to be that of one of the cubes in Fig. 2. As indi- 

 cated earlier it should contain a sufftcient number of molecules to give 

 a good mean value for P, the polarization per unit volume, but at the 

 same time it should be small enough not to mask significant spatial 

 variations in P. 



\\'hen the impressed field E is varying sinusoidally with the time at 

 the frequency aj/27r, the local or internal field F tending to displace 

 each charged particle in the dielectric will also vary sinusoidally with 

 the time, though in general out of phase with E, if F is given by equation 

 (3), and can be considered to be given by the real part of Foe'"'. Under 

 these conditions 



P = kFoe"^' 



solution of equation (10) for the steady state provided that 



k^j-. ""'' ,.,, ■ (12) 



k is the polarizability per unit volume and is a complex quantity, since 

 the term zVco in the denominator is an imaginary {i = V — 1). 



Equations (10), (11) and (12) apply to a dielectric having a single 

 type of polarization characterized by the constants /, r, m, n and e. 

 But in general an applied field induces several types of polarization 

 simultaneously in a dielectric, and if we assume that it induces w 

 types which are independent of each other, the total polarization per 

 unit volume is given by 



P = kiF + k.F -\- • • • KF. (13) 



The total polarizability is then the sum of the individual polarizabili- 

 ties, or 



w 

 k = Z k,. (14) 



3=- 1 



is a 



