Wien Displacement Law
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Direct answer
Wien's displacement law states that the wavelength at which a black body radiates most strongly is inversely proportional to its absolute temperature: λm T = b, with the Wien constant b = 2.898 × 10⁻³ m K (equivalently 2898 micrometre-kelvin). Heat a body and the emission peak slides toward shorter wavelengths — red hot moves through orange to white-blue — which is how astronomers take a star's temperature from its colour: the Sun, peaking near 500 nm, sits at about 5800 K. The law pairs naturally with Stefan's fourth-power law in JEE problems: a hotter body both shifts its peak and radiates enormously more per unit area, and the two facts together answer most black-body comparisons.
What you must remember
- The law: λm T = b = 2.898 × 10⁻³ m K; if the peak is quoted in micrometres, T = 2898/λm(μm) kelvin — the two-digit form that solves most numericals in one line.
- Sun's numbers: peak near 500 nm (visible green, where the eye is most sensitive — no coincidence in evolution), giving T ≈ 5800 K, the standard anchor value.
- Human body peak: at 310 K the peak sits at 2.898 × 10⁻³/310 ≈ 9.3 μm, deep infrared — the band night-vision devices and pit-viper organs actually sense.
- Colour logic: a blue star is hotter than a red one; as T rises the peak moves from infrared (lukewarm) through red (about 1000 K, dull glowing) to blue-white (above 10⁴ K).
- Pairing with Stefan: radiated power per unit area is σT⁴; doubling T halves λm but multiplies power per unit area by 16 — Wien shifts the spectrum, Stefan inflates it.
- Wavelength window shifts: most of a room-temperature body's emission is infrared around 10 μm, which is why CO2 and water vapour (strong IR absorbers) matter for the greenhouse effect while visible passes freely.
- Microwave background cameo: the 2.7 K cosmic background peaks near 1 mm, a Wien calculation that has appeared as a general-knowledge blend in JEE Advanced passages.
Reading temperatures by colour
A star's spectrum peaks at 290 nm, roughly the ultraviolet. Wien hands over the temperature: T = 2.898 × 10⁻³/(290 × 10⁻⁹) = 10⁴ K. Against the Sun's 5800 K, this star is not merely hotter — by Stefan, each square metre of it radiates (10⁴/5800)⁴ ≈ 8.8 times the Sun's already fierce output, and its blue-white colour is the visible tip of a spectrum that has marched out of the yellow-green into the violet. Astronomers run this computation continuously: measure the peak, read the temperature, classify the star — and JEE packages the same three steps as a two-mark numerical.
Run the law downward in temperature. An electric iron at 400 K peaks at 2.898 × 10⁻³/400 ≈ 7.2 μm — invisible infrared, which is why an iron can be dangerously hot while looking exactly like a cold one; the glow everyone associates with heat (around 1000 K, peak near 2.9 μm with a visible red tail) begins only when the Wien tail spills appreciably into the visible band. And at body temperature the peak is 9.3 μm, squarely in the far infrared where atmosphere is partly transparent — the physical reason thermal cameras see people through darkness but not through glass.
Where students slip
Unit handling sinks most numericals: b in metre-kelvin with λm in nanometres requires the 10⁻⁹ conversion, and the alternative form 2898 μm·K invites micrometre answers that students mislabel as nanometres. The second trap is direction — Wien says hotter means shorter peak wavelength, and options always include the reversed reading for students operating on folk intuition ("hotter = more red"?). When paired with Stefan, remember the two laws answer different questions: the peak position and the total output; a question about brightness (with sizes equal) is Stefan, about colour is Wien. A conceptual favourite: the Sun's peak at 500 nm matches the eye's sensitivity maximum — JEE has framed this as an assertion ("the human eye evolved to the solar peak") to be judged with its reason, and the correct handling of such evolutionary framing is the actual test. Finally, λm is the intensity maximum of the wavelength spectrum; the frequency-spectrum peak sits elsewhere, a subtlety occasionally probed in Advanced multi-correct items.
Frequently asked questions
What is Wien's displacement law?
The product of a black body's peak wavelength and its absolute temperature is constant, λm T = 2.898 × 10⁻³ m K, so hotter bodies peak at shorter wavelengths.
How is the Sun's surface temperature estimated?
From its emission peak near 500 nm: T = b/λm = 2.898 × 10⁻³/(5 × 10⁻⁷) ≈ 5800 K.
What happens to the emitted spectrum as a body gets hotter?
The peak shifts toward shorter wavelengths (infrared to red to blue-white) while the total radiated power per unit area grows as T⁴ — the spectrum both slides and swells.
Why can thermal cameras see people but the eyes cannot?
The human body near 310 K peaks at about 9.3 μm in the far infrared, outside the visible band, so special infrared sensors are needed to register that radiation.
How do Wien's and Stefan's laws differ in what they tell you?
Wien's law locates the spectrum's peak (colour information), while Stefan's law integrates the whole spectrum into total power (brightness information) — most comparison problems need both.