When light encounters particles or density fluctuations much smaller than its wavelength, the electric field can drive the charges in the material into oscillation. Those oscillating charges reradiate electromagnetic waves in other directions. In the small-scatterer limit this is called Rayleigh scattering.

A defining feature is its strong wavelength dependence. Under the ideal Rayleigh approximation, the scattered intensity scales approximately as

$$\boxed{I_{\rm scatt}\propto\frac{1}{\lambda^4}}.$$

Shorter wavelengths are therefore scattered much more strongly than longer wavelengths.

Comparing two wavelengths

If every other relevant quantity is unchanged, the ratio of scattered intensities at wavelengths $\lambda_1$ and $\lambda_2$ is

$$\frac{I_1}{I_2}=\left(\frac{\lambda_2}{\lambda_1}\right)^4.$$

Worked example

Compare idealized scattering at

$$\lambda_{\rm blue}=450,\mathrm{nm}$$

and

$$\lambda_{\rm red}=650,\mathrm{nm}.$$

Then

$$\frac{I_{\rm blue}}{I_{\rm red}} =\left(\frac{650}{450}\right)^4 \approx4.35.$$

So in this simplified comparison, 450-nm light is scattered more than four times as strongly as 650-nm light.

Why the daytime sky is blue

Direct sunlight contains a broad visible spectrum. As it passes through the atmosphere, molecules and small-scale density fluctuations scatter shorter visible wavelengths more strongly. An observer looking away from the Sun therefore receives scattered light enriched in blue wavelengths from many directions across the sky.

The sky is not violet simply because the $1/\lambda^4$ law favors still shorter visible wavelengths. The solar spectrum, atmospheric transmission and the wavelength sensitivity of human vision also matter.

Why sunsets are redder

Near sunrise or sunset, direct sunlight travels through a much longer atmospheric path before reaching an observer. Preferential scattering removes more of the shorter-wavelength light from the direct beam, leaving the transmitted sunlight relatively enriched in red and orange wavelengths.

The blue sky and reddened sunset are therefore two views of the same wavelength-selective process: one observes the scattered light, while the other observes a direct beam after substantial short-wavelength scattering out of it.

Polarization of scattered light

Rayleigh scattering also depends on direction and polarization. Light scattered at roughly right angles to an incoming unpolarized beam can be strongly linearly polarized because an oscillating electric dipole does not radiate equally in every direction.

This provides another connection between atmospheric scattering and polarization measurements of the sky.

When the approximation applies

The simple $1/\lambda^4$ scaling is appropriate when the scattering structures are much smaller than the wavelength and absorption is not dominant. Larger particles such as cloud droplets require different scattering models and can scatter visible wavelengths much more evenly, helping clouds appear white or gray.

Rayleigh scattering is therefore a general wave-matter interaction, not a rule specific to the color of the sky. Its characteristic signature is the strong preferential scattering of shorter wavelengths by sufficiently small scatterers.