Black-Body Radiation and Wien's Displacement Law

Black-Body Radiation and Wien's Displacement Law

The ideal black body

A black body is an idealised surface (introduced in Section 10.11) that absorbs the entirety of the

electromagnetic radiation falling on it, at every single wavelength, reflecting none of it and

transmitting none of it, so that its absorptive power is exactly $a = 1$ across the whole spectrum. By

Kirchhoff's law (Section 10.12), a black body is therefore also the most efficient possible emitter of

radiation at every wavelength, with $e = 1$ throughout -- so the radiation a black body emits depends

only on its own absolute temperature, and not at all on what particular material it happens to be made

of. In the laboratory, a very good practical approximation to an ideal black body is obtained not from any

single material but from a shape: a hollow, opaque enclosure (a cavity) with only a small hole cut into

one wall. Radiation entering through the hole is reflected repeatedly off the inside walls of the cavity,

losing a little of its energy to absorption at each reflection, so that by the time (if ever) it manages

to find its way back out through the same small hole, it has effectively been almost completely absorbed

-- making the hole itself behave, from the outside, almost exactly like an ideal black surface.

The black-body spectrum and its single peak

If the radiant energy a black body emits is measured separately at each wavelength and plotted as a

curve of "energy emitted per unit wavelength interval" (spectral radiance) against wavelength $\lambda$,

for one fixed absolute temperature, the resulting curve has a very characteristic shape: it rises from

almost zero at very short wavelengths, climbs smoothly to a single rounded maximum at one particular

wavelength, and then falls away again, more gradually, towards longer wavelengths. Repeating this

measurement at a higher temperature produces a curve of the same general single-humped shape, but with

its peak shifted to a shorter wavelength, and with a markedly greater height (and greater total area

under the curve) than the cooler curve -- the accompanying figure shows three such curves, for three

different temperatures, superimposed on the same axes.

Wien's displacement law

Wien's displacement law is the precise statement of how the wavelength of peak emission, $\lambda_m$,

shifts as a black body's absolute temperature $T$ changes: $\lambda_m$ is found to be inversely proportional to $T$,

where $b$ is a universal constant, Wien's constant, $b \approx 2.898\times10^{-3}\ \text{m K}$. In

words: the hotter a black body is, the shorter the wavelength at which it radiates most intensely --

the peak of its emission curve is "displaced" towards shorter wavelengths (and correspondingly higher

frequencies) as temperature rises, which is exactly the trend visible in the three-curve figure

accompanying this section.

A familiar illustration and an application

This displacement is exactly why a piece of metal, heated to progressively higher temperatures in a

furnace, changes colour in a predictable sequence -- first glowing a dull, deep red as its peak emission

first enters the visible range from the infrared side, then brightening through orange and yellow, and …