A hot cavity glows through a small hole with a spectrum that depends only on its temperature T. This sim plots two laws for that spectrum, the classical Rayleigh–Jeans law and Planck's quantum law, on one fixed frame so you can compare them directly.

What's on screen

• Planck curve (solid cyan): the energy density u(f) from Planck's law.
• Rayleigh–Jeans curve (dashed amber): the classical law. It is cut off at the top of the frame.
• f peak (violet marker): where the Planck curve is highest.
• η inset (top right): both laws against η = hf/kT, with the peak at η = 2.82 marked.
• Readouts: the peak frequency f_max and the total energy density, both for the Planck curve.

Press ⌖ on a row to see where that control sits in the sim.

Fixed: the plot frame (f up to 1×10¹⁵ Hz, u up to 2×10⁻¹⁵ J m⁻³ Hz⁻¹), so curves at different T compare directly. The sim has no motion or time: T is its only input.

Both laws build the spectrum the same way: the number of standing-wave modes per unit volume per unit frequency, times the average energy each mode holds.

The count, 8πf²/c³ (two polarisations included), grows as f² and is shared by both laws. Classically, equipartition gives every mode the same average energy kT, whatever its frequency:

So the dashed curve rises as f² with no turnover. This is the ultraviolet catastrophe. It leaves the top of the frame at 1.61×10¹⁴ Hz for T = 6000 K, and at 2.28×10¹⁴ Hz for 3000 K.

In Planck's law a mode of frequency f can hold energy only in whole steps of hf. At temperature T its average energy is then:

• When hf ≪ kT the steps are tiny next to kT, and the average tends to kT: the classical value.
• When hf ≫ kT even one step is rarely occupied, and the average falls toward zero.
• The mode count keeps rising as f², so the product rises and then falls. The peak is at η = hf/kT = 2.82, the root of η = 3(1 − e^−η), not at η = 1.

Divide both laws by 8π(kT)³/h²c³ and f and T appear only through η = hf/kT: Planck's law becomes η³/(e^η − 1) and the classical law becomes η². The two limits of the bracket in Planck's law:

• The ratio (classical)/(Planck) is (e^η − 1)/η: 1.05 at η = 0.1, 1.72 at η = 1, 5.6 at the peak.
• At the right edge of the frame η = 8.0 at 6000 K, where the classical value is about 370 times too high, and η = 16.0 at 3000 K, where it is about 5.5×10⁵ times too high.
• Raising T lowers η at any given f, so the laws agree over a wider range of f. At η = 0.1 that range ends at f = 1.25×10¹³ Hz for 6000 K.

Because the η inset has no T in it, it does not move when T changes.

The peak of the Planck curve is at η_p = 2.8214. Because η_p is a pure number, the peak frequency is proportional to T:

• Doubling T doubles f_max.
• The height at the peak goes as T³, so doubling T raises it eightfold.

The total energy density is the area under the Planck curve. Substituting η = hf/kT pulls (kT)⁴ out of the integral, so the total is proportional to T⁴:

• Doubling T multiplies the total by 2⁴ = 16: from 0.0613 J/m³ at 3000 K to 0.9805 J/m³ at 6000 K.
• The classical curve has no total. Its integral grows without limit as the upper frequency is raised, so the readout shows Planck's value only.

This is energy per unit volume inside the cavity. The power per unit area leaving the hole is (c/4)aT⁴ = σT⁴, with σ = 5.67×10⁻⁸ W m⁻² K⁻⁴.

• The Sun has an effective temperature of 5772 K. Its per-frequency peak is at 3.39×10¹⁴ Hz, in the near infrared.
• An incandescent filament at about 2800 K, treated as an ideal black body, puts only about 8% of its power into visible light. Most is infrared.
• The cosmic microwave background at 2.725 K peaks at 1.6×10¹¹ Hz, in the microwave band. The COBE satellite found its spectrum matches a black body to better than 1 part in 10⁴ of the peak.

Key points

• The classical law rises as f² without limit and has no total.
• Planck's law turns over at η = 2.82, so f_max = 5.879×10¹⁰ Hz K⁻¹ × T.
• The total energy density is aT⁴.
• The classical law is the η → 0 limit of Planck's law.

Common mix-ups

• "There are fewer high-frequency modes, so the curve falls." The count keeps growing as f². The fall comes from the average energy per mode.
• "The peak is at hf = kT." The peak is at η = 2.82, where the rising count and the falling energy per mode balance.
• "A hotter cavity only makes the curve taller." The peak also moves to higher frequency, in proportion to T.

Things to try on your own

• Read f_max at 3000 K and at 6000 K, and check that the ratio is 2.
• Read the total at 3000 K and at 6000 K, and check that the ratio is 16.
• Move T slowly and watch where the dashed curve leaves the top of the frame. Compare 3000 K with 6000 K.