  In a $3D$ bounded and $C^{1, \alpha}$-smooth domain $\Omega$, $\alpha \in (0,
1)$, we distribute a cluster of nanoparticles enjoying moderately contrasting
relative permittivity and permeability which can be anisotropic.
  We show that the effective permittivity and permeability generated by such
cluster is explicitly characterized by the corresponding electric and magnetic
polarization tensors of the fixed shape. The error of the approximation of the
scattered fields corresponding to the cluster and the effective medium is
inversely proportional to the dilution parameter $c_r:=\frac{\delta}{a}$ where
$a$ is the maximum diameter of the nanoparticles and $\delta$ the minimum
distance between them. The constant of the proportionality is given in terms of
a priori bounds on the cluster of nanoparticles (i.e. upper and lower bounds on
their permittivity and permeability parameters, upper bound on the dilution
parameter $c_r$, the used incident frequency and the domain $\Omega$).
  A key point of the analysis is to show that the Foldy-Lax field appearing in
the meso-scale approximation, derived in \cite{AM}, is a discrete form of a
(continuous) system of Lippmann-Schwinger equations with a related effective
permittivity and permeability contrasts. To derive this, we prove that the
Lippmann-Schwinger operator, for the Maxwell system, is invertible in the
H\"{o}lder spaces. As a by-product, this shows a H\"{o}lder regularity property
of the electromagnetic fields up to the boundary of the inhomogeneity.

Analysis of PDEs