Black-body radiation: Stefan-Boltzmann and Wien

The big idea: Push a poker into a fire and watch it glow dull red, then orange, then white — a hot object pouring out light across all the wavelengths is behaving as a black body, a perfect absorber and emitter of radiation.

A star, or a glowing iron bar, is a good model of one.

Its brightness across the wavelengths makes a single humped curve called the black-body spectrum.

Spot it: Hotter → taller and bluer. As the temperature rises the curve gets taller (more total power) and its peak shifts to a shorter wavelength (towards blue).

That's why a heated bar glows dull red, then orange, then white as it gets hotter.

Two given equations turn that picture into numbers. The first is the Stefan-Boltzmann law — the total power a black body radiates (its luminosity):

Why the T⁴ matters: Because power depends on T to the fourth power, doubling the kelvin temperature multiplies the radiated power by 2⁴ = 16.

Always put T in kelvin (K), never °C: kelvin = °C + 273.

The second is Wien's displacement law — it locates the peak of the curve (the brightest wavelength):

Read Wien as 'they trade off': λ_max and T multiply to a constant, so they move opposite ways: a hotter body (bigger T) has a smaller peak wavelength (bluer light). Rearrange to λ_max = 2.9 × 10⁻³ ÷ T.

How this is tested — black-body radiation can appear on either paper:

The classic trap: Leaving T in °C instead of kelvin, or forgetting the power is T⁴ (not T).

Comparing two bodies: When two black bodies are compared, write L = σAT⁴ for each and divide one by the other — σ cancels, leaving a clean ratio of areas and temperatures. For a sphere the area is A = 4πr², so the area ratio is the radius ratio squared.