A dying ember glows dull red; a gas flame burns blue; the hottest stars shine blue-white. Colour, it turns out, is a thermometer. Wien’s displacement law makes the link exact: the wavelength a body shines brightest at tells you its temperature — the trick that lets us take the temperature of a star we can never touch.
📚 What you need to know
Wien’s displacement law: the peak wavelengthλmax of a black body’s radiation is inversely proportional to its absolute temperature.
Measure λmax from a star’s spectrum → get its surface temperature; combine with Stefan–Boltzmann for luminosity or size.
Colour is a thermometer
Everyday life already hints at the rule. Turn up a hob and the element shifts from black to red to orange. A candle flame is cooler and yellow; a blowtorch is hotter and blue. The colour a hot object shows tracks its temperature — and for stars, that colour is our single most powerful clue to how hot they are.
A star’s colour maps directly to its surface temperature: cool stars glow red (long peak wavelength), hot stars blue (short peak wavelength), with the Sun’s yellow-white sitting in between.
It runs opposite to the way we usually talk. In everyday language “red hot” sounds like the extreme — but in physics red is the cool end and blue the hot one. A red star might be a few thousand kelvin; a blue-white one, tens of thousands. Colour and temperature are locked together, and blue wins.
The peak shifts with temperature
Look back at the black-body curve from the radiation page. Every object has one, but its peak sits at a wavelength set by temperature. Heat the object and the whole curve grows and its peak slides towards shorter wavelengths. That sliding of the peak is the “displacement” the law is named after.
As temperature rises, each black-body curve grows taller and its peak shifts to a shorter wavelength — the hotter body peaks furthest to the left.
Notice the two effects stacking up. A hotter body radiates more at every wavelength (that was Stefan–Boltzmann, the total under the curve), and its peak moves to shorter wavelengths (that’s Wien, the position of the peak). One law is about the area, the other about where the hump sits.
The inverse law
Put a number on it and the relationship is beautifully simple — a constant divided by temperature:
Wien’s displacement lawλmaxT = 2.9 × 10−3 m Kλmax = 2.9 × 10−3 / T
Peak wavelength against temperature: an inverse relationship. Double the temperature and the peak wavelength halves.
Because the two multiply to a constant, they trade off exactly: double the temperature and the peak wavelength halves; halve the temperature and it doubles. Rearranged as T = 2.9 × 10−3 / λmax, it becomes a thermometer — measure the peak wavelength of starlight and read off the temperature.
Worked examples
WE 1
The Sun’s surface temperature is 5800 K. Find the peak wavelength of its radiation.
λmax = 2.9 × 10−3 / T= 2.9 × 10−3 ÷ 5800λmax = 5.0 × 10−7 m = 500 nmRight in the middle of the visible band — little wonder the Sun looks yellow-white and our eyes evolved tuned to it.
WE 2
A star’s radiation peaks at 290 nm. Find its surface temperature.
Convert: λmax = 290 nm = 2.9 × 10−7 mT = 2.9 × 10−3 / λmax= 2.9 × 10−3 ÷ (2.9 × 10−7)T = 10 000 KThe peak is in the ultraviolet — a hot, blue-white star, far hotter than the Sun.
WE 3
A star’s radiation peaks at 480 nm and its radius is 8.0 × 108 m. Find (a) its surface temperature and (b) its luminosity. (σ = 5.67 × 10−8)
(a) T = 2.9 × 10−3 / λmax = 2.9 × 10−3 ÷ (480 × 10−9)T ≈ 6000 K(b) L = 4πr²σT⁴= 4π(8.0 × 108)² × 5.67 × 10−8 × 60424L ≈ 6.1 × 1026 WWien reads the temperature off the colour; Stefan–Boltzmann turns it into total power. The two laws work hand in hand.
Combine: peak → T (Wien) → L (Stefan–Boltzmann, L = σAT4).
peak wavelength from the colour
Wien
temperature T
Stefan– Boltzmann
luminosity L
Quick recap: Wien’s displacement law says a black body’s peak wavelength is inversely proportional to its absolute temperature: λmaxT = 2.9 × 10−3 m K. Hotter bodies peak at shorter wavelengths (bluer), cooler ones at longer wavelengths (redder). Rearranged, T = 2.9 × 10−3/λmax reads a star’s temperature from its colour — then Stefan–Boltzmann turns that into luminosity.
💡 Top tips
Metres and kelvin. Convert nm to metres (×10−9) and use absolute temperature.
Inverse, and the right way round. Hotter → shorter peak wavelength (bluer). Don’t flip it.
Trade-off: double T → half λmax.
It’s the peak. The law is about λmax, the wavelength of maximum intensity — not just any wavelength.
Chain it: colour → T (Wien) → L (Stefan–Boltzmann) → b (inverse-square).
⚠ Common mistakes
Leaving λmax in nanometres instead of metres
Using °C instead of kelvin for the temperature
Getting the inverse backwards — thinking hotter means a longer peak wavelength
Confusing Wien’s constant (2.9 × 10−3 m K) with the Stefan–Boltzmann constant σ
Forgetting it applies to the peak wavelength, λmax, specifically
And that closes the loop. Across these last pages you’ve built the astronomer’s whole toolkit for starlight: take a star’s temperature from its colour (Wien), turn that temperature and size into its total power (Stefan–Boltzmann, L = σAT4), and predict how bright it looks from Earth (b = L/4πd2). From particles jiggling in a solid to the light of distant suns — it’s all the same physics of thermal energy. These three radiation laws are exactly what underpins classifying stars on the Hertzsprung–Russell diagram, where astrophysics picks the story up next.
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