Wien's Displacement Law

Anything above absolute zero radiates, and the spectrum it radiates has a peak whose wavelength slides inversely with temperature. Multiply the peak wavelength by the absolute temperature and you always get the same number, $b = 2.898 \times 10^{-3}$ m·K. Hotter means bluer, in a strictly reciprocal way: double the temperature and the peak moves to half the wavelength. The constant is not an independent measurement — differentiate Planck's spectral distribution with respect to wavelength, set the derivative to zero, and $b$ drops out as $hc/(4.965\,k_B)$, where 4.965 is the numerical root of $x = 5(1 - e^{-x})$. Wien's law is a corollary of Planck's, and its constant is built from $h$, $c$ and $k_B$.

The Sun's photosphere at 5772 K gives $2.898 \times 10^{-3}/5772 = 502$ nm, in the green, near the centre of human visual sensitivity — which is not a coincidence but is also not quite the tidy story it is usually told as, for reasons in the last paragraph. A human body at 310 K peaks at 9.35 µm, in the mid-infrared band that thermal cameras are built around. A tungsten filament at 2800 K peaks at 1035 nm, in the near infrared, which is the quantitative reason an incandescent bulb is mostly a space heater with a visible by-product. And the cosmic microwave background at 2.725 K peaks at 1.06 mm. Astronomers read a star's surface temperature straight off its colour with this single division.

Wilhelm Wien derived the displacement law in 1893, seven years before there was any quantum to derive it from. His argument was thermodynamic: imagine slowly compressing a cavity full of radiation and track what the Doppler shift at the moving walls does to the spectrum. It won him the 1911 Nobel Prize. His separate distribution law, published the following year, fitted the short-wavelength end of the blackbody curve and failed badly at long wavelengths — and it was one of the two expressions Planck was interpolating between in 1900 when he arrived at the formula that required energy quanta. The distribution law was superseded; the displacement law was absorbed intact and is now read straight off Planck's.

The trap here is genuine and catches professionals: the peak depends on what you plot against. Spectral radiance per unit wavelength peaks, for the Sun, at 502 nm. The same physical spectrum plotted per unit frequency peaks at $3.40 \times 10^{14}$ Hz, which corresponds to a wavelength of 882 nm — deep in the infrared, and nowhere near 502 nm. Neither curve is wrong. They are different functions, related by a factor of $\lambda^2$ from the change of variable, and there is no reason their maxima should coincide. The constant on this page is the wavelength form; the frequency form has its own constant, $5.879 \times 10^{10}$ Hz/K. Quoting a peak from one convention and computing in the other is the standard error. Two lesser cautions: the temperature must be absolute, in kelvin, and the law describes an ideal blackbody, so a surface with strongly wavelength-dependent emissivity will peak somewhere else. And the peak is not where the energy is — the distribution is broad and lopsided with a long infrared tail, and more than half the Sun's radiated power arrives at wavelengths longer than the green peak.