Wien's Displacement Law

Equivalent forms

\lambda_{\max} = \frac{b}{T}

b = 2.8978 \times 10^{-3} \text{ m·K}

One constant b = 2.898 mm·K tells you the color of any glowing object — from interstellar dust to the Big Bang's CMB.

Symbol	Name	SI unit	Dimension	Range
$λ_max$	Peak wavelengthoutput Wavelength of maximum spectral intensity	m	L	1e-9 – 0.001
$T$	Temperature Absolute temperature of blackbody	K	Θ	100 – 10000
$b$	Wien displacement constant 2.8978e-3 m·K	m·K	L·Θ	0.0028978 – 0.0028978

Unit systems

Symbol	SI	CGS	Imperial
$λ_max$	m	cm	ft
$T$	K	K	°R
$b$	m·K	cm·K	ft·°R

Where it holds

Strictly valid for ideal blackbody emitters. Real bodies with emissivity < 1 follow the law approximately if spectral emissivity is roughly flat near the peak.

Dimensional analysis

$[\lambda _\max ][T] = (L)(\Theta ) = L\cdot \Theta = [b] \checkmark$

Discovery

Wilhelm Wien · 1893

Wien derived this scaling law from thermodynamic arguments before Planck's full radiation formula. Earned him the 1911 Nobel Prize.

Try this

Why is the Sun yellow but a coal stove red?

The Sun's surface is at 5778 K. At what wavelength does it emit most intensely?

Research status: stable

Real-world applications

Stellar temperature determination (color → spectral type)
Thermal imaging cameras $(\lambda \approx 10 \mu m$ for 300 K objects)
Pyrometry — non-contact temperature measurement
Cosmology — CMB confirms early universe was blackbody

Common misconceptions

There are two 'Wien's laws' — the displacement law (peak vs T) and the older approximation $B_\lambda \approx (2hc^{2}/\lambda ^{5})\cdot \exp (-hc/\lambda k_BT)$ , which fails at long wavelengths
The peak wavelength differs depending on whether you plot intensity per unit $\lambda or$ per unit $\nu$ — different x positions in the spectrum
Most stars are NOT perfect blackbodies; metal-line absorption shifts effective peak

Experimental verification

COBE/FIRAS satellite measured the CMB spectrum as a 2.7255 K blackbody, with peak at 1.06 mm — agreeing with Wien's law to 5 decimal places.

Derivation

Take Planck's spectral radiance

B_\lambda (T) = (2hc^{2}/\lambda ^{5}) / [\exp (hc/\lambda k_BT) - 1]

.

Differentiate

dB_\lambda /d\lambda = 0

to find the peak.

Let x = hc/(\lambda k_BT)

; the resulting transcendental equation

x = 5(1 - e^{-}

ˣ) has solution

x \approx 4.965

.

So \lambda _\max \cdot T = hc/(4.965\cdot k_B) \approx 2.898\times 10^{-3} m\cdot K

.

Limiting cases

T \to \infty

⟶

\lambda _\max \to 0

Very hot objects emit predominantly in X-ray / gamma range.

T \to 0

⟶

\lambda _\max \to \infty

Cold objects glow in far-infrared / radio.

T = 2.725\,\mathrm{K} (CMB)

⟶

\lambda _\max \approx 1.06\,\mathrm{mm}

Cosmic microwave background — fossil radiation from Big Bang.

What if…

What if you double the temperature?

Peak wavelength halves — object shifts from red-hot to blue-hot (e.g., $1500\,\mathrm{K} \to 3000\,\mathrm{K}$ halves $\lambda _\max )$ .

What if you observe radiation

at \lambda _\max = 1\,\mathrm{mm}

?

Source temperature $\approx 2.9\,\mathrm{K}$ — that's the cosmic microwave background, leftover from 380,000 years after the Big Bang.

What if you plot intensity per unit frequency instead?

The peak frequency satisfies a different relation: $\nu _\max /T \approx 5.879\times 10^{10} Hz/K$ — same physics, different spectral coordinate.

1

Peak wavelength of the Sun

Given ·

T:: 5778

Find ·

\lambda _\max

Steps

Apply $\lambda _\max = b/T$
$\lambda _\max = 2.8978\times 10^{-3} / 5778$
$\lambda _\max \approx 5.015\times 10^{-7} m = 502\,\mathrm{nm}$ (peak in visible green)

Result ·

\lambda _\max = 2.8978e-3 / 5778 = 5.015e-7\,\mathrm{m} \approx 502\,\mathrm{nm}

(green-yellow)

2

Temperature of human body's peak emission

Given ·

T:: 310

Find ·

\lambda _\max

Steps

$\lambda _\max = b/T = 2.8978\times 10^{-3} / 310$
$\lambda _\max \approx 9.35 \mu m$
This is why thermal IR cameras detect humans at $8-12 \mu m$

Result ·

\lambda _\max = 2.8978e-3 / 310 \approx 9.35e-6\,\mathrm{m} = 9.35 \mu m (mid-IR)

Stefan-Boltzmann Law→Planck Radiation Law→Ideal Gas Law→