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Foi' convenience of notation the electrostatic problem of a powdered 

 dielectric in an electric tield will be treated. The results are trans- 

 lated into the magnetic case by substituting the permeability (x for 

 the dielectric constant e. 



Our problem is to calculate the eü'ective dielectric constant of a 

 powder under the above assumptions as to the smallness of the field 

 and the random distribution of the axes when the density of packing 

 of the powder and the dielectric constant of the material of the 

 powder are known. 



Definition of "effective dielectric constant" . 



Let us consider a portion of the powder which contains many 



particles and let us take the mean electric intensity and the mean 



electric displacenient throughout this portion. (The mean being taken 



with respect to volume). We define: "effective dielectric constant" = 



, mean electric displacement 



= 4jr ; — r : • 



mean electric intensity 



We presuppose that this definition is unique which implies that 

 the powder is sufficiently fine for otherwise it is not possible to 

 include a sufficient number of particles without making the portion 

 so large that the field would vary in it^^ (from point to point) if the 

 powder were replaced by a solid. 



Let us draw a spherical surface inside the powder. According to 

 the well known treatment of polarized media the electric intensity 

 inside the sphere is equal to the electric intensity due to charges 

 inside the sphere plus the intensity due to charges of polarization 

 on the surface of the sphere and plus the intensity due to charges 

 of polarization on the outer surface of the powder as well as that 

 due to charges outside and inside the powder. This means that the 

 electric intensity 



E = Ei + E^, + E, 

 where 



Ei = effect of charges inside the sphere 



^^, = effect of charges of polarization on the surface of the sphere 



E, = effect of charges of polarization on the outer surface of the 

 powder -\- external field 



where "external field" := field due to all real cliarges and the 

 charges of polarization not belonging to the powder. 



Since each individual particle is uncharged Ei is obtained by 

 summing the fields due to charges of polarization on the surfaces 

 of the particles inside the sphere. 



