Exactly Solvable Microscopic Geometries and Rigorous Bounds for the Complex Dielectric Constant of a Two-Component Composite Material

Exact bounds for the complex, bulk, effective dielectric constant ${\epsilon }_e$ of a two-component macroscopic composite that depend on the available information about the composite are presented and discussed. Some of these bounds are readily ascribable to special, exactly solvable, microscopic geometries. As a consequence, it is shown that there can exist composites where the real part of ${\epsilon }_e$ diverges as $\omega \to 0$ while the dc conductivity ${\sigma }_e\ne 0$.