Do extended effective-medium formulas scale properly?

Abstract

The original Bruggeman and Maxwell Garnett formulas for the average dielectric function of a composite medium satisfy a scaling law: if the dielectric functions of both components are scaled by the same factor, the average dielectric function scales by this factor. These formulas are independent of the size of polarizable elements. Extended formulas explicitly account for size, but at least one formula fails to satisfy the scaling law, a likely failing of all such formulas.