Planck's law ( $B_{ν} (T), B_{λ} (T)$ )

Planck's law gives the spectral radiance of an ideal blackbody in thermal equilibrium at absolute temperature

T > 0

. It predicts how much power is carried in each small frequency or wavelength interval, per projected area and per solid angle. The whole curve is determined by

T

: no material-specific parameter enters this ideal Planck function.

Writing

ν

for frequency in hertz and

λ

for wavelength in metres, the two forms of the Planck function are

B_{ν} (T) = \frac{2 h ν ^{3}}{c ^{2}} \frac{1}{e ^{h ν / (k_{B} T)} - 1}, B_{λ} (T) = \frac{2 h c ^{2}}{λ ^{5}} \frac{1}{e ^{h c / (λ k_{B} T)} - 1} .

(1)

Here

h

is the Planck constant,

k_{B}

the Boltzmann constant, and

c

the speed of light in vacuum. Their exact values in the International System of Units are

h = 6.62607015 \times 1 0^{- 34} J s

k_{B} = 1.380649 \times 1 0^{- 23} J K^{- 1}

, and

c = 299792458 m s^{- 1}

. ^[ref]

The units matter:

B_{ν}

is measured in

W m^{- 2} s r^{- 1} H z^{- 1}

, whereas

B_{λ}

is measured in

W m^{- 2} s r^{- 1} m^{- 1}

. Both are spectral radiance, rather than radiant exitance or energy density. To report

B_{λ}

per micrometre, multiply its per-metre value by

1 0^{- 6}

; a micrometre-wide interval is

1 0^{- 6}

metres wide.

For a computed example, at

T = 6000 K

and

λ = 0.500 μ m

, the exponent is

h c / (λ k_{B} T) \approx 4.79592

. Substitution into Planck's law gives

B_{λ} \approx 3.17569 \times 1 0^{13} W m^{- 2} s r^{- 1} m^{- 1}

, or

3.17569 \times 1 0^{7} W m^{- 2} s r^{- 1}

per micrometre. At the same wavelength, the illustrative temperatures

3000 K

and

4500 K

give

2.60268 \times 1 0^{5}

and

6.37793 \times 1 0^{6}

in the latter units. These are ideal predictions for stipulated temperatures.

Figure 1.
Ideal Planck's law spectra at three temperatures
. The vertical axis is spectral radiance per micrometre, including the $1 0^{- 6}$ conversion from per metre. Increasing temperature raises the curve at every fixed wavelength and moves its marked peak toward shorter wavelengths. The peaks are approximately $966$ , $644$ , and $483 nm$ for $3000$ , $4500$ , and $6000 K$ , respectively; only $0.1$ to $3 μ m$ of each spectrum is shown. **Details**.

The key change of variables is

ν = c / λ

. Corresponding positive intervals carry the same power, so

B_{λ} d λ = B_{ν} ∣ d ν ∣

and

B_{λ} (T) = B_{ν} (T)_{ν = c / λ} \frac{c}{λ ^{2}} .

(2)

Equal-width wavelength intervals have unequal widths in frequency; the factor

c / λ^{2}

compensates for this. Merely replacing the horizontal-axis labels of

B_{ν}

c / ν

does not produce

B_{λ}

. This explains the frequency and wavelength peaks of Planck's law.

A short derivation uses the Boltzmann distribution for a single electromagnetic mode with thermal energies

nh ν

n = 0, 1, \dots

. Put

q = e^{- h ν / (k_{B} T)}

. The geometric series gives

Z = \sum_{n \geq 0} q^{n} = (1 - q)^{- 1}

and mean thermal energy

h ν q / (1 - q)

. The electromagnetic mode density in a box, including both independent polarization of an electromagnetic wave states of a photon, is

8 π ν^{2} / c^{3}

per volume per unit frequency. Multiplication gives spectral energy density

u_{ν} = 8 πh ν^{3} / [c^{3} (e^{h ν / (k_{B} T)} - 1)]

; isotropy gives

B_{ν} = c u_{ν} / (4 π)

. This counts thermal excitation energy, excluding zero-point energy. ^[ref]

Planck's law connects quantum mechanics to a measurable radiation curve. Its small-

h ν / (k_{B} T)

limit is the Rayleigh-Jeans law,

B_{ν} \approx 2 k_{B} T ν^{2} / c^{2}

; its large-

h ν / (k_{B} T)

limit is the Wien approximation,

B_{ν} \approx (2 h ν^{3} / c^{2}) e^{- h ν / (k_{B} T)}

. The suppression at high frequency makes the total energy density finite. Integrating the curve gives the Stefan–Boltzmann law, while locating its peak gives Wien's displacement law. Thus one formula predicts both the spectral shape and the total emission.

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Planck's law by

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Planck's law describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature. Formulated by Max Planck in 1900, it provides a theoretical foundation for understanding black body radiation. The law states that the intensity of radiation emitted at a specific wavelength is proportional to the wavelength and depends on the temperature of the black body.

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Planck's law by

Ciro Santilli  40 Updated 2025-07-16

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Used to explain the black-body radiation experiment.

Published as: On the Theory of the Energy Distribution Law of the Normal Spectrum by Max Planck (1900).

The Quantum Story by Jim Baggott (2011) page 9 mentions that Planck apparently immediately recognized that Planck constant was a new fundamental physical constant, and could have potential applications in the definition of the system of units (TODO where was that published):

Planck wrote that the constants offered: 'the possibility of establishing units of length, mass, time and temperature which are independent of speciﬁc bodies or materials and which necessarily maintain their meaning for all time and for all civilizations, even those which are extraterrestrial and nonhuman, constants which therefore can be called "fundamental physical units of measurement".'

This was a visionary insight, and was finally realized in the 2019 redefinition of the SI base units.

TODO how can it be derived from theoretical principles alone? There is one derivation at; en.wikipedia.org/wiki/Planck%27s_law#Derivation but it does not seem to mention the Schrödinger equation at all.

Video 1.

Quantum Mechanics 2 - Photons by ViaScience (2012)

Source. Contains a good explanation of how discretization + energy increases with frequency explains the black-body radiation experiment curve: you need more and more energy for small wavelengths, each time higher above the average energy available.

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Planck's law (Bν​(T), Bλ​(T))

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