Dilute spherical-inclusion conductivity ($a^{\ast }=a_{0}I+p\beta I+o(p)$)

Well-separated shrinking spheres respond independently at leading order, so their polarization contributions add according to volume fraction. Clusters kept at fixed relative internal separation can have a different leading response; the dilute limit alone does not guarantee independent-sphere polarizability.

Matching the far dipole of a spherical composite with the sum of isolated spherical-inclusion dipoles gives this approximation. Its dilute expansion is $a_0+p\beta +p^2{\beta }^2/(3a_0)+O(p^3)$. The rational expression is a useful closure, not the exact conductivity of every spherical-inclusion microstructure.

If the first exciting-field correction at inclusion $i$ is $-\beta {\Lambda }_{i}E/p_i$, its revised polarization is $\beta {\chi }_i(E-\beta {\Lambda }_{i}E/p_i)$. Averaging weights it by $p_i$ and gives the displayed second-reflection approximation. Further dipole reflections and finite-size multipoles are neglected.