  Arrays of cylindrical metal micro-resonators embedded in a dielectric matrix
were proposed by Pendry, et. al., in 1999 as a means of creating a microscopic
structure that exhibits strong bulk magnetic behavior at frequencies not
realized in nature. This behavior arises for H-polarized fields in the
quasi-static regime, in which the scale of the micro-structure is much smaller
than the free-space wavelength of the fields. We carry out both formal and
rigorous two-scale homogenization analyses, paying special attention to the
appropriate method of averaging, which does not involve the usual cell
averages. We show that the effective magnetic and dielectric coefficients
obtained by means of such averaging characterize a bulk medium that, to leading
order, produces the same scattering data as the micro-structured composite.
