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# Scattering of Electromagnetic Waves by Many Nano-Wires

Department of Mathematics, Kansas State University, Manhattan, KS 66506-2602, USA

Received: 10 May 2013 / Revised: 11 July 2013 / Accepted: 11 July 2013 / Published: 18 July 2013

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# Abstract

Electromagnetic wave scattering by many parallel to the*z*

*−*axis, thin, impedance, parallel, infinite cylinders is studied asymptotically as

*a*

*→*0. Let

*D*

*be the cross-section of the*

_{m}*m*

*−*th cylinder,

*a*be its radius and ${\text{x ^}}_{\text{m}}{\text{= (x}}_{\text{m1}}{\text{, x}}_{\text{m2}}\text{)}$ be its center, 1

*≤*

*m*

*≤*

*M*,

*M*=

*M*(

*a*). It is assumed that the points, ${\hat{x}}_{m}$ , are distributed, so that $N\left(\Delta \right)=\frac{1}{2\pi a}\underset{\Delta}{\int}N\left(\hat{x}\right)d\hat{x}[1+o(1\left)\right]$ where

*N*(∆) is the number of points, ${\hat{x}}_{m}$ , in an arbitrary open subset, ∆, of the plane,

*xoy*. The function, $N\left(\hat{x}\right)\text{}\ge 0$, is a continuous function, which an experimentalist can choose. An equation for the self-consistent (effective) field is derived as

*a*

*→*0. A formula is derived for the refraction coefficient in the medium in which many thin impedance cylinders are distributed. These cylinders may model nano-wires embedded in the medium. One can produce a desired refraction coefficient of the new medium by choosing a suitable boundary impedance of the thin cylinders and their distribution law.

*Keywords:*metamaterials; refraction coefficient; EM wave scattering

*This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.*

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**MDPI and ACS Style**

Ramm, A.G. Scattering of Electromagnetic Waves by Many Nano-Wires. *Mathematics* **2013**, *1*, 89-99.

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