Blackbody Radiation & Planck's Law: Birth of Quantum Theory

The ultraviolet catastrophe

A blackbody absorbs every wavelength that falls on it and, by Kirchhoff's law, is therefore also the most efficient possible emitter. The classical approach counted the standing electromagnetic modes in a cavity and assigned each an average energy of k_BT via the equipartition theorem — but the density of modes grows as frequency squared, so the Rayleigh-Jeans law predicts energy density rising without limit at short wavelengths:

Rayleigh-Jeans (fails at short λ):
B(λ,T) = 2c·k_B·T / λ⁴   →  diverges as λ → 0

This "ultraviolet catastrophe" implied any warm object
should instantly radiate infinite energy.

Planck's quantum hypothesis

Planck found the correct interpolating formula by reverse-engineering the entropy the cavity oscillators had to have. He showed it could only arise if each oscillator of frequency f could hold energy only in integer multiples of E = hf, where h = 6.626×10⁻³⁴ J·s is now called Planck's constant:

B(λ,T) = (2hc²/λ⁵) · 1/(exp(hc/λk_BT) − 1)

At long λ → reduces to Rayleigh-Jeans (classical limit)
At short λ → exponential term suppresses emission → resolves the catastrophe

Wien's law, Stefan-Boltzmann, and real applications

Two practical laws fall directly out of Planck's distribution. Wien's displacement law, λ_max·T = 2.898×10⁻³ m·K, says the emission peak shifts to shorter (bluer) wavelengths as temperature rises — a body at 300 K peaks in the far infrared, the Sun near 5778 K peaks at ~502 nm visible light, and a 20,000 K blue-white star peaks in the ultraviolet. Stefan-Boltzmann's law, P = σT⁴, means doubling temperature radiates sixteen times more power per unit area.

These aren't just historical curiosities: stellar spectral classification reads a star's surface temperature directly off its colour via Wien's law; infrared thermal cameras detect the 8–14 μm peak of objects near room temperature; and the cosmic microwave background is the most perfect blackbody spectrum ever measured, at 2.725 K, with deviations smaller than one part in 10,000 — powerful evidence for the Big Bang.

Frequently asked questions

Try it live

Everything above runs in your browser — open Blackbody Radiation and drag the temperature slider from 500 K to 30,000 K to watch Wien's peak shift and the total emitted power grow with T⁴.