Macroscopic permittivity of dielectric mixtures with application to microwave attenuation of rain and hail

A theory is presented for calculating the macroscopic effective permittivity of mixtures that are composed of a background medium and inhomogeneous scatterers. The scatterers are two-component spheres, where both the spherical core and the spherical shell that forms the surface of the scatterer are homogeneous with a constant permittivity. The effective permittivity is calculated as the function of the constituent permittivities and their volume fractions. The analysis is quasistatic and therefore the results only apply to mixtures where the sizes of the scatterers are considerably smaller than the wavelength of the operating RF field. The results are applied to calculating the microwave attenuation of melting hail, at a frequency of 1 GHz and where the hydrometeors are assumed to be in the form of a spherical ice core with a spherical water coating shell. There is a peak in the attenuation of hail as it falls through a warm layer. The attenuation is greater with a thin water coating (the fraction of water from the hail particle is 0.09) than it is when the hail particle has completely melted. This resembles the same phenomenon of increased extinction which is observed in the radar backscattering from melting hail.