Planck Radiation Law

Equivalent forms

B(\nu, T) = \frac{2h\nu^3}{c^2}\cdot\frac{1}{e^{h\nu/(k_B T)} - 1}

u(\nu, T) = \frac{8\pi h\nu^3}{c^3}\cdot\frac{1}{e^{h\nu/(k_B T)} - 1}

One formula spans the cosmic microwave background, stellar spectra, and the glow of a fireplace.

Symbol	Name	SI unit	Dimension	Range
$B$	Spectral radianceoutput Power emitted per area, per solid angle, per wavelength	W·m⁻³·sr⁻¹	M·L⁻¹·T⁻³	0 – 100000000000000
$\lambda$	Wavelength Emission wavelength	m	L	1e-8 – 0.001
$T$	Temperature Absolute temperature of the body	K	Θ	1 – 10000000
$h$	Planck constant Quantum of action	J·s	M·L²·T⁻¹	6.62607015e-34 – 6.62607015e-34
$c$	Speed of light Speed of light in vacuum	m/s	L·T⁻¹	299792458 – 299792458
$k_B$	Boltzmann constant Energy per kelvin	J/K	M·L²·T⁻²·Θ⁻¹	1.380649e-23 – 1.380649e-23

Unit systems

Symbol	SI	CGS	Imperial
$B$	W·m⁻³·sr⁻¹	erg·s⁻¹·cm⁻³·sr⁻¹	—
$\lambda$	m	cm	in
$T$	K	K	°R
$h$	J·s	—	—
$c$	m/s	—	—
$k_B$	J/K	—	—

Where it holds

Exact for radiation in thermal equilibrium with matter (blackbody). Real surfaces with emissivity

\varepsilon (\lambda )

< 1 emit

\varepsilon \cdot B

. Quantum corrections (QED, photon-photon scattering) negligible until extreme fields

\sim 10^{18} V/m

.

Dimensional analysis

$[2hc^{2}/\lambda ^{5}] = (J\cdot s)(m/s)^{2}/m^{5} = J/(m^{3}\cdot s)$ → power/(volume $\times$ steradian $) \checkmark$

Discovery

Max Planck · 1900

To fit blackbody data, Planck made a 'desperate' assumption: oscillators in the cavity walls absorb and emit energy only in discrete units E = hν. He thought it was a mathematical trick — until Einstein took the quanta seriously in 1905 and quantum mechanics was born.

Try this

Why don't hot ovens emit infinite ultraviolet light?

Classical physics predicted that any hot object should emit unbounded energy at short wavelengths — the 'ultraviolet catastrophe'. Yet real ovens emit a finite, peaked spectrum. What's the right formula?

Research status: stable

Real-world applications

Cosmic microwave background — direct evidence of the Big Bang's hot, dense early phase.
Pyrometry — non-contact temperature measurement of hot industrial furnaces.
Stellar classification — spectral type (O,B,A,F,G,K,M) maps to blackbody temperature.
Thermal imaging cameras — detect IR blackbody emission from skin $(\sim 310\,\mathrm{K})$ and machinery.

Common misconceptions

Blackbody peak depends on whether you plot $B(\lambda )$ or $B(\nu )$ — Wien' $s \lambda _\max \cdot T$ differs from $\nu _\max /T$ by a factor $of \sim 1.76$ .
Real objects aren't blackbodies — but most are 'gray bodies' with emissivity 0.5–0.95 across IR.
Planck didn't believe quanta were physical at first — he treated h as a fitting parameter for years.

Experimental verification

CMB measured by COBE/WMAP/Planck satellites — the most perfect blackbody known

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

to

10^{-5}

. Cavity radiation experiments since 1900 match Planck's curve across 7 decades in wavelength. Stellar spectra obey it (with line features).

Derivation

Count modes in a cubic cavity: density of states

g(\nu ) = 8\pi \nu ^{2}/c^{3}

.

Assign each mode an average energy from Bose-Einstein statistics: ⟨E⟩

= h\nu /(e^(h\nu /kT) - 1)

.

Energy density

u(\nu ) = g(\nu )

⟨E⟩.

Convert to spectral radiance via

B = (c/4\pi )\cdot u

, then change variables to wavelength using |

d\nu /d\lambda | = c/\lambda ^{2}

.

Limiting cases

hc/(\lambda k_B T)

≪ 1 (long wavelength)⟶

B \to 2ck_B T/\lambda ^{4}

(Rayleigh-Jeans)Classical limit — recovers the 19th-century formula valid at radio/IR wavelengths only.

hc/(\lambda k_B T)

≫ 1 (short wavelength)⟶

B \to (2hc^{2}/\lambda ^{5})\cdot \exp (-hc/\lambda k_B T)

(Wien)Quantum suppression kills the UV catastrophe; Wien's original empirical formula.

T \to 0

⟶

B \to 0

for

all \lambda

Cold bodies don't glow — though quantum vacuum fluctuations persist (Casimir effect).

What if…

What if h were exactly zero?

We'd recover the classical Rayleigh-Jeans formula and the UV catastrophe: total emitted power diverges. Stars couldn't exist; matter couldn't be stable.

What if the CMB temperature were 100 K instead of 2.7 K?

Background sky would glow in $far-IR (\sim 30 \mu m$ peak), bathing planets in $10^{6}\times$ more background power — life would face a hot sky everywhere.

1

Peak wavelength of the Sun

Given ·

T:: 5778
h:: 6.62607015e-34
c:: 299792458
k B:: 1.380649e-23

Find · \lambda_max

Steps

Wien: $\lambda _\max \cdot T = 2.898e-3 m\cdot K$
$\lambda _\max = 2.898e-3 / 5778 \approx 5.02e-7\,\mathrm{m} = 502\,\mathrm{nm}$
Solar spectrum peaks in visible green — fortuitously where human eyes are most sensitive.

Result ·

\lambda _\max \approx 502\,\mathrm{nm}

(green-blue) via Wien's displacement law

2

Spectral radiance at 500 nm for T = 5778 K

Given ·

\lambda:: 5e-7
T:: 5778
h:: 6.62607015e-34
c:: 299792458
k B:: 1.380649e-23

Find · B

Steps

Numerator: $2hc^{2}/\lambda ^{5} = 2\times 6.626e-34\times (3e_{8})^{2}/(5e-7)^{5} \approx 3.82e15$
Exponent: $hc/(\lambda k_B T) = (6.626e-34\times 3e_{8})/(5e-7\times 1.381e-23\times 5778) \approx 4.98$
Denominator: $e^4.98 - 1 \approx 144.4$
$B \approx 3.82e15 / 144.4 \approx 2.65e13 W\cdot m^{-3}\cdot sr^{-1}$

Result ·

B \approx 2.6 \times 10^{13} W\cdot m^{-3}\cdot sr^{-1}

Wien's Displacement Law→Stefan-Boltzmann Law→Rydberg Formula→