Effective Maxwell Equations in a Geometry with Flat Rings of Arbitrary Shape
Agnes Lamacz, Ben Schweizer · SIAM Journal on Mathematical Analysis · 2013
We analyze the time-harmonic Maxwell equations in a complex geometry: many (order $\eta^{-3}$) small (order $\eta^1$), thin (order $\eta^2$), and highly conductive (order $\eta^{-3}$) metallic objects are distributed in a domain $\Omega\subset \mathbb{R}^3$. We determine the effective behavior of this metamaterial in the limit $\eta\searrow 0$. For $\eta>0$, each single conductor occupies a simply connected domain, but the conductor closes to a ring in the limit $\eta\searrow 0$. This change of topology allows for an extra dimension in the solution space of the corresponding cell-problem. Even though both original materials (metal and void) have the same positive magnetic permeability $\mu_0>0$, the effective Maxwell system exhibits, depending on the frequency, a negative magnetic response.