Black-body radiation: spectrum and temperature

Every body with a temperature above absolute zero emits electromagnetic waves — this is thermal radiation. A black body is a body that absorbs all radiation falling on it at every wavelength. Its own emission depends only on temperature, not on the material, which makes the black body the reference for thermal radiation. A good model of a black body is a closed cavity with a small hole: a ray that gets inside is reflected many times and is almost completely absorbed. Soot, glowing coals in a furnace and stars are close to black bodies.

The black-body spectrum is given by Planck’s law: the spectral radiant exitance r_λ tells how much energy 1 m² of the surface emits per second in a narrow wavelength interval around λ. The r_λ(λ) curve has a single peak. Move the temperature slider and watch three laws at once. Wien’s displacement law: λmax = b/T — the hotter the body, the shorter the peak wavelength. The Stefan–Boltzmann law: the total radiant exitance (the area under the curve) is R = σT⁴ — make the body twice as hot and it radiates 16 times more. The third is colour: at 300 K the whole spectrum lies in the infrared and the body looks black, a dull red glow appears at about 800 K, an incandescent lamp (≈2800 K) shines yellow, the Sun (≈5800 K) white, and stars hotter than 10,000 K bluish.

The measurements panel shows λmax, R = σT⁴, the peak value of r_λ and the shares of energy in the ultraviolet, visible and infrared ranges. Look at an incandescent lamp: only about 10% of its energy goes into visible light, the rest is heat — which is why such lamps were replaced by LEDs. The “Remember curve” button keeps a spectrum on the graph for comparison, and the “Thermal camera” mode shows that even a cold body radiates — only in the infrared, invisible to the eye. The photons in the scene are emitted with the wavelength distribution given by Planck’s law, and their number grows with temperature as T³.

Turn on the Rayleigh–Jeans curve — the prediction of classical physics. At long wavelengths it agrees with experiment, but as λ decreases it shoots off to infinity: the body would have to emit infinite energy in the ultraviolet. This paradox was called the “ultraviolet catastrophe”. In 1900 Max Planck resolved it by assuming that light is emitted in portions — quanta with energy E = hν = hc/λ. Short-wavelength quanta are too “expensive” and the body can hardly emit them, so the curve falls off. This is how quantum physics was born. The “Check yourself” box asks you to predict a result first and then test it on the model.