The big idea: A black body is a perfect absorber and emitter — it soaks up every wavelength that hits it and, when hot, radiates over every wavelength too.
A hot object (a star, 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):
- luminosity — total power radiated (W)
- Stefan-Boltzmann constant, 5.67 × 10⁻⁸ W m⁻² K⁻⁴ (given)
- surface area of the body (m²)
- absolute surface temperature (K — kelvin)
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):
- wavelength of peak (brightest) emission (m)
- absolute surface temperature (K — kelvin)
Read Wien as 'they trade off': λmax 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.
Worked example — peak wavelength of a star
A star has a surface temperature of 5.0 × 10³ K. Find the wavelength at which it radiates most strongly.
Solution
- Start with the given Wien law:
- Rearrange for the peak wavelength:
- Put in T = 5.0 × 10³ K:
- Work it out — keep the unit:
Final answer
λmax = 5.8 × 10⁻⁷ m (580 nm) — in the visible (yellow) range.
Stop wasting time on topics you know
Our AI identifies your weak areas and focuses your study time where it matters. No more overstudying easy topics.
How this is tested: Black-body radiation can be tested on either paper.
- Paper 1A: a quick MCQ — how the curve changes when T is raised or lowered (peak moves, height changes), or a one-step Stefan-Boltzmann / Wien sum. - Paper 2: 'determine' questions — compare two black bodies with L = σAT⁴ (e.g. find a star's radius), or find the peak wavelength of the Sun from its temperature.
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.
IB-style question — find the radius of a second star
Two stars behave as black bodies. Star P and star Q radiate the same total power. Star P has surface temperature 6.0 × 10³ K and radius 7.0 × 10⁸ m. Star Q is cooler, at 3.0 × 10³ K. Determine the radius of star Q. (Treat each star as a sphere, area A = 4πr².)
Solution
- Start with the given Stefan-Boltzmann law for each star:
- Equal power means LP = LQ. The σ and 4π cancel:
- Rearrange for rQ:
- Put in the numbers (TP / TQ = 6.0 × 10³ ÷ 3.0 × 10³ = 2.0):
- Work it out — keep the unit:
Final answer
rQ = 2.8 × 10⁹ m — the cooler star must be larger to radiate the same power, because L ∝ r²T⁴ and its T is halved (T⁴ falls by 16×).