Clausius–Mossotti relation

Remarks

Put $s = N \alpha / (3 \varepsilon_0)$. From $P = N \alpha (E + P / (3 \varepsilon_0))$ one gets $P = 3 \varepsilon_0 s E / (1 - s)$, so $\chi_e = 3 s / (1 - s)$. Then $\varepsilon_r = 1 + \chi_e = (1 + 2s)/(1 - s)$, and solving for $s$ gives the relation. The local field formula is the Lorentz field: the field at the centre of a small spherical cavity in a uniformly polarized medium, where the nearby dipoles cancel. That cancellation holds for molecules on a cubic lattice or in random positions, so the relation is a model result for such materials, mostly gases and simple liquids. Example: for a dilute gas, $s$ is small and the relation gives $\varepsilon_r \approx 1 + N \alpha / \varepsilon_0$.

Going deeper

For a student

The relation follows from the definitions of polarization density and permittivity together with the Lorentz local field. It predicts that $(\varepsilon_r - 1)/(\varepsilon_r + 2)$ is proportional to the density, which agrees well with measurements on gases. Its optical form, with $\varepsilon_r$ replaced by the square of the refractive index, is the Lorentz–Lorenz equation. The assumption $N \alpha < 3 \varepsilon_0$ is needed; as $N \alpha$ approaches $3 \varepsilon_0$ the model predicts $\varepsilon_r \to \infty$, the so-called polarization catastrophe.