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A theory of electromagnetic (EM) wave scattering by many small particles ofan arbitrary shape is developed. The particles are perfectly conducting orimpedance. For a small impedance particle of an arbitrary shape an explicitanalytical formula is derived for the scattering amplitude. The formula holdsas $a\to 0$, where $a$ is a characteristic size of the small particle and thewavelength is arbitrary but fixed. The scattering amplitude for a smallimpedance particle is shown to be proportional to $a^{2-\kappa}$, where$\kappa\in [0,1)$ is a parameter which can be chosen by an experimenter ashe/she wants. The boundary impedance of a small particle is assumed to be ofthe form $\zeta=ha^{-\kappa}$, where $h=$const, Re$h\ge 0$. The scatteringamplitude for a small perfectly conducting particle is proportional to $a^3$,it is much smaller than that for the small impedance particle. The many-bodyscattering problem is solved under the physical assumptions $a\ll d\ll\lambda$, where $d$ is the minimal distance between neighboring particles and$\lambda$ is the wavelength. The distribution law for the small impedanceparticles is $\mathcal{N}(\delta)\sim\int_{\delta}N(x)dx$ as $a\to 0$. Here$N(x)\ge 0$ is an arbitrary continuous function that can be chosen by theexperimenter and $\mathcal{N}(\delta)$ is the number of particles in anarbitrary sub-domain $\Delta$. It is proved that the EM field in the mediumwhere many small particles, impedance or perfectly conducting, are distributed,has a limit, as $a\to 0$ and a differential equation is derived for thelimiting field. On this basis the recipe is given for creating materials with adesired refraction coefficient by embedding many small impedance particles intoa given material.

Scattering of EM waves by many small perfectly conducting or impedance bodies