Permeabilities of metamaterials

Scattering of electromagnetic (EM) waves by many small particles, embedded in a given 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. An analytical formula for the permeability $\mu(x)$ of the limiting medium is given. Analysis of this formula allows one to find out the range of the values of the permeability in the material, obtained by embedding many small particles.