= Rayleigh limit
In <Rayleigh scattering>, a small bound system develops an induced <electric dipole moment> $\mathbf p(t)=\operatorname{Re}[\alpha(\omega)\mathbf E_0e^{-i\omega t}]$. Well below its binding resonances, the <electric polarizability> is approximately its static value $\alpha(0)$, independent of <frequency>. The mean radiated <power> and incident <light intensity> are
$$
\langle P\rangle=\frac{\omega^4|\alpha E_0|^2}{12\pi\epsilon_0c^3},\qquad
\langle S\rangle=\frac12\epsilon_0c|E_0|^2.
$$
Dividing gives the total scattering <cross section>
$$
\boxed{\sigma_R=\frac{\omega^4|\alpha(0)|^2}{6\pi\epsilon_0^2c^4}
\propto\omega^4\propto\lambda^{-4}.}
$$
This <Rayleigh scattering wavelength dependence> comes from the two time derivatives of an oscillating <electric dipole moment>, whose radiated field is proportional to $\omega^2p_0$. It requires the same scatterer with approximately constant low-frequency <electric polarizability>; resonances and strong <optical dispersion> invalidate the simple fourth-power law.
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