Wien's & Stefan's Laws
Hotter bodies glow bluer and radiate far more.
Lesson at a glanceShow the key points
- Thermal radiation is electromagnetic; at 300 K it peaks near 10 µm; bodies glow red near 1100 K, white near 3000 K.
- Wien: λ_m T = b ≈ 0.29 cm K (28.5).
- Explaining the spectrum needed Planck's hypothesis E = nhν.
- Stefan–Boltzmann: u = σAT⁴ (28.6), σ = 5.67 × 10⁻⁸ W m⁻² K⁻⁴; other bodies u = eσAT⁴ (28.7), e = a.
- Net loss in surroundings at T₀: eσA(T⁴ − T₀⁴) (28.8).
Heat a poker: first it only feels warm, then glows dull red, then orange, then white. Two laws describe this: one for the colour, one for the power.
What thermal radiation is
Once emitted, thermal radiation is an electromagnetic wave like light. It obeys the laws of waves and, since its wavelength is small, travels in straight lines, casts shadows, reflects and refracts. A body emits a mixture of wavelengths, but most of the energy comes from a limited range.
At 300 K most of it is around 10 µm, deep in the infrared. Around 1100 K enough red is emitted for a body to glow red; around 3000 K enough shorter wavelengths for it to look white, though even then the peak is near 950 nm, still infrared.
Wien’s displacement law
As the temperature rises, the wavelength of maximum intensity falls, in inverse proportion:
A star’s light is most intense near 290 nm. Treating it as a blackbody, estimate its surface temperature.
The Stefan–Boltzmann law
The total energy radiated per second by a blackbody of area A at absolute temperature T is
Other bodies emit less, but still in proportion to T⁴:
Net loss to the surroundings
In surroundings at T₀ the body absorbs eσAT₀⁴ (what it would emit at T₀), so its net loss is
A blackbody of area 20 cm² at 227°C hangs in a room at 27°C. Find its initial net rate of heat loss by radiation.
- 1Thermal radiation is electromagnetic; it follows the rules of optics.
- 2Wien: λ_m T = b ≈ 0.29 cm K; hotter means bluer.
- 3Stefan–Boltzmann: u = σAT⁴ (blackbody), eσAT⁴ in general; emissivity e = absorptive power a.
- 4Net loss eσA(T⁴ − T₀⁴).
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