Rayleigh scattering

Remarks

Check: a sphere in a uniform field $\mathbf{E}_0$ gets the dipole moment $\mathbf{p} = 4\pi\varepsilon a^3 \dfrac{\varepsilon_r - 1}{\varepsilon_r + 2} \mathbf{E}_0$, where $\varepsilon$ is the permittivity of the surrounding medium. An oscillating dipole radiates power $\dfrac{c k^4 \abs{\mathbf{p}}^2}{12\pi\varepsilon}$ with $c$ the speed of light in the medium, and dividing by the incident intensity $\frac{1}{2} c\, \varepsilon \abs{\mathbf{E}_0}^2$ gives the formula above. For unpolarized incident light the scattered intensity varies with the scattering angle $\theta$ as $1 + \cos^2\theta$, and at $\theta = 90^\circ$ the scattered light is fully polarized. Example: for blue light at $450$ nm and red light at $700$ nm, the ratio of cross-sections is $(700/450)^4 \approx 5.9$. Non-example: cloud droplets have radii of several micrometres, so $ka$ is much larger than $1$. Their scattering is described by the Mie solution instead. It depends only weakly on wavelength, which is why clouds are white.