Chapter 5, Problem 5.30P

Consider a composite material made up of two species: having number densities N₁ and N₂ molecule s/m³, respectively. The two materials are uniformly mixed, yielding a total number density of N = N₁ +N_2- The presence of an electric field E induces molecular dipole moments p₁ and p₂ within the individual species, whether mixed or not. Show that the dielectric constant of the composite material is given by ${\epsilon }_r=f{\epsilon }_{r1}+\left(1-f\right){\epsilon }_{r2}$, where f is the number fraction of species 1 dipoles in the composite, and where ${\epsilon }_{r1}$ and ${\epsilon }_{r2}$ are the dielectric constants that the unmixed species would have if each had number density N.