Blackbody Radiation

Start at the Sun’s 5 778 K. Look at where the two curves sit on the right of the plot: in the infrared they lie on top of each other, and the classical theory is not merely close, it is correct. Now follow them leftwards towards shorter wavelengths. The Planck curve turns over and dives. The classical one keeps climbing, leaves the top of the frame, and never comes down.

Then drag the temperature. Watch the amber Wien marker slide left as things get hotter, and watch the visible share in the panel: 8% at a light-bulb filament’s 2 800 K, 44% at the surface of the Sun, and back down to 39% for Sirius, whose peak has already run past the short end of the visible band into the ultraviolet. That share peaks at about 47%, near 7 000 K — drag across it and see. The Sun sits within a couple of points of the best a hot object can do for an eye like yours, which is not a coincidence about the Sun.

Treat the radiation in a hot cavity as a set of standing waves, give each one the kT of energy that classical equipartition demands, and count the waves. Short wavelengths fit into the box in far more ways than long ones — the number of modes grows without limit as the wavelength shrinks. Every one of them gets its kT.

The result is the Rayleigh–Jeans law, 2ckT/λ⁴, and its total energy is the integral of that from zero upwards, which does not converge. Classical physics predicts that a warm cavity contains infinite energy and that opening it would sterilise the room with ultraviolet. Ehrenfest named it the ultraviolet catastrophe. It was not a rounding error; it was a theory returning ∞ for a quantity you can measure with a thermometer.

In 1900 Planck found a formula that fitted the measured curve, and then spent two months working out what would have to be true for it to come out of the physics. The answer was that a cavity wall can only exchange energy with a mode in whole multiples of hf.

That single assumption kills the catastrophe by making short wavelengths expensive. A mode at short λ has a large hf, so at temperature T the thermal energy available is not enough to buy even one quantum, and the mode sits empty instead of collecting its classical kT. The exponential in the denominator of Planck’s law is that suppression. At long wavelengths hf is small compared with kT, quanta are cheap, and the formula collapses back into Rayleigh–Jeans exactly — which is why the two curves coincide on the right of the plot.

Planck called it “an act of desperation” and spent years trying to derive his own result without it. He never could. This is the moment quantum mechanics starts.

Wien’s displacement law. The peak sits at λ = b/T with b = 2.898 × 10⁻³ m·K, so double the temperature and the peak halves its wavelength. That is the whole of astronomical colour: a star’s hue is a thermometer reading. Sirius peaks at 292 nm, already ultraviolet, and looks blue-white because we only see the long tail of it.

Stefan–Boltzmann. Integrate the whole curve and the total power comes to σT⁴ — the fourth power, so a doubling in temperature is a sixteen-fold jump in output. Both laws only exist because the integral converges. Try to derive a Stefan–Boltzmann constant from the classical curve and you get infinity, which is why σ has an h in it: σ = 2π⁵k⁴/15c²h³.

Set the slider to the 2 800 K of an incandescent filament and read the visible share. Only 8% of the output lands between 380 and 750 nm; the rest is infrared, which is to say heat. A real bulb does worse still — tungsten is not quite a blackbody and the glass absorbs some of what does get out — but the ceiling is set by this curve, not by the engineering. You cannot fix it by improving the bulb, because the spectrum follows the filament’s temperature and tungsten melts at 3 695 K. The only way out was to stop using a hot object as a light source at all, which is what an LED is.

The same curve explains why a thermal camera works in the dark (at 310 K your peak is around 9 µm, and the camera is simply an infrared eye), and why the cosmic microwave background is the best blackbody ever measured — its spectrum matches Planck’s law to a few parts in 100 000, at 2.7 K, from radiation that last scattered 13.8 billion years ago.