Which of the following correctly relates frequency to Rayleigh scattering intensity?

• A. It is proportional to frequency raised to the fourth power
• B. It is proportional to frequency squared
• C. It is independent of frequency
• D. It is inversely proportional to frequency

Answer: (A)

Rayleigh scattering intensity scales with the fourth power of frequency because a small particle acts like an oscillating dipole when struck by light. The incident electric field induces a dipole moment p in the particle, and the radiated power from an oscillating dipole is proportional to the square of its acceleration. For a driving field at frequency ω, the dipole’s acceleration introduces a dependence on ω^4, and since p is proportional to the field amplitude, the scattered intensity ends up proportional to ω^4 (which is the same as proportional to f^4, with f the frequency). Equivalently, since f and wavelength λ are related by f = c/λ, this means the scattered intensity goes as 1/λ^4. This explains why shorter wavelengths scatter more strongly and why the sky appears blue. The other options don’t fit this behavior: a dependence on f^2 is too weak, independence on frequency ignores the observed color dependence, and an inverse dependence would imply red light scatters most, which isn’t observed.