We approximate iron as a blackbody. A plate with an area of 1 cm² is heated from 1000 K to 2000 K, and we examine its light spectrum and radiated power.
Adjust conditions:
The sliders adjust the values
1000 Kの黒体を2000 Kに熱すると、放射電力は何倍か？
見える光（380〜780 nm）の放射電力は何倍か？
The equations explain the phenomenon
1We count the light modes in a cavity.Number of modes per unit volume=(8πν²c³) dν……
2We calculate the energy density of light using classical physics.u(ν)=(8πν²c³) k T……
3We sum over all frequencies.∫ u(ν) dν=∞……
4We restrict energy to discrete values.E=n h ν……
Eₙ = nhν is the energy of the nth state, where n = 0, 1, 2, ….wₙ = e−Eₙ/kT is the statistical weight of that state, given by the Boltzmann factor.
5We calculate the average energy per mode.⟨E⟩=h νehν/kT − 1……
6We obtain Planck's law.u(ν)=(8π h ν³c³)/(ehν/kT − 1)……
7We find the peak wavelength.λmax T=2.898×10⁻³ m·K……
8We calculate the radiated power per 1 m² from the energy density inside the cavity.PS=(c4) ∫0∞ u(ν) dν……
9We set x = hν/kT to factor the temperature dependence out of the integral.PS=(2πk⁴T⁴c²h³)∫0∞x³ex − 1 dx……
10We evaluate the remaining integral to find σ.PS=σ T⁴……
11We integrate over the visible range only.Visible-light ratio=∫visible uλ(λ,T2)dλ/ ∫visible uλ(λ,T1)dλ……
Further study: From blackbody radiation to quantum mechanics
12This leads to quantum mechanics.The energy of a photon is E=hν.……
What this chapter has shown
The light spectrum of a blackbody is determined by its temperature. The peak of the distribution per unit wavelength shifts to shorter wavelengths as temperature rises (λmax T = 2.898×10⁻³ m·K). Light at the short-wavelength end of the visible range also becomes stronger, changing the appearance from reddish to whitish. The perceived color depends on the distribution across the entire visible range, as well as the peak position.
16 timesRatio of radiated powers
Visible light alone increases by a factor of about 2.4 times ten thousand from 1000 K to 2000 K.
In everyday lifeThe heating element in an electric heater glows red, but most of the energy it emits is invisible infrared radiation.
Detailed summary
In classical physics, counting the modes of light in a box and assigning kT to them 1 by 1 led to an infinite total energy for the light. Restricting energy to quanta of hν suppresses the energy in high-frequency modes and yields Planck's law, which agrees with the observed distribution. The wavelength at the peak of the distribution is inversely proportional to temperature (Wien's displacement law), and the power radiated by a blackbody is proportional to σT⁴. Doubling the temperature increases the radiated power by a factor of 16, but the visible light in the tail of the distribution increases by a factor of about 24054. Heated iron changes from red to white because the peak shifts toward visible wavelengths, and blue light becomes much stronger relative to red light.