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Electromagnetic wave scattering by many small particles and creating materials with a desired permeability

Scattering of electromagnetic (EM) waves by many small particles (bodies) embedded in a homogeneous medium is studied. Physical properties of the particles are described by their boundary impedances. The limiting equation is obtained for the effective EM field in the limiting medium, in the limit $a\to 0$, where $a$ is the characteristic size of a particle and the number $M(a)$ of the particles tends to infinity at a suitable rate. The proposed theory allows one to create a medium with an inhomogeneous permeability. The main new physical result is the explicit analytical formula for the permeability $μ(x)$ of the limiting medium. While the initial medium has a constant permeability $μ_0$, the limiting medium, obtained as a result of embedding many small particles with prescribed boundary impedances, has a non-homogeneous permeability which is expressed analytically in terms of the density of the distribution of the small particles and their boundary impedances.