Effective Maxwell equations in a geometry with flat rings of arbitrary shape

We analyze the time harmonic Maxwell's equations in a complex geometry. The homogenization process is performed in the case that many small, thin conductors are distributed in a subdomain of ${\mathbb{R}}^3$. Each single conductor is, topologically, a split ring resonator, but we allow arbitrary flat shapes. In the limit of large conductivities in the rings and small ring diameters we obtain an effective Maxwell system. Depending on the frequency, the effective system can exhibit a negative effective permeability.