Effective dielectric constant of a two-component material with shape distribution

The effective dielectric constant of a two-component material is investigated. By considering the shape distribution of the components, we derive Maxwell-Garnett-type approximation, which has been established based on the reciprocity theorem (del Rio J A et al 1998 Solid State Commun. 106 183). Based on a self-consistent condition on the electric field (Bruggeman-type effective medium approximation), we obtain the well-known Lichtenecker's mixture formulae with alpha = 1/2. Moreover, we derive the differential effective medium approximation by taking into account the shape distribution. All these formulae are checked through the spectral density function and compared with previous bounds.