Planck Radiation Law

Equivalent forms

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

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

One equation contains the ultraviolet catastrophe's solution, Wien's law, the Rayleigh-Jeans limit, and Stefan-Boltzmann — the seed of quantum theory.

Symbol	Name	SI unit	Dimension	Range
$B_λ$	Spectral radianceoutput Power per area per wavelength per solid angle	W·m⁻³·sr⁻¹	M·L⁻¹·T⁻³	0 – 1000000000000000
$λ$	Wavelength Emission wavelength	m	L	1e-9 – 0.001
$T$	Temperature Absolute temperature of blackbody	K	Θ	100 – 10000

Unit systems

Symbol	SI	CGS	Imperial
$B_λ$	W·m⁻³·sr⁻¹	erg·s⁻¹·cm⁻³·sr⁻¹	BTU·s⁻¹·ft⁻³·sr⁻¹
$λ$	m	cm	ft
$T$	K	K	°R

Where it holds

Exact for an ideal blackbody in thermal equilibrium. Real surfaces multiply by spectral emissivity

\varepsilon (\lambda

,T) < 1.

Dimensional analysis

$[B_\lambda ] = (M\cdot L^{2}\cdot T^{-2}\cdot s \cdot L^{2}\cdot T^{-2}) / L^{5} = M\cdot L^{-1}\cdot T^{-3}$ (per steradian implicit $) \checkmark$

Discovery

Max Planck · 1900

Planck assumed energy was quantized (E = hν) as a 'desperate mathematical trick' to fit the blackbody spectrum. He inadvertently birthed quantum mechanics — winning the 1918 Nobel Prize.

Try this

Why doesn't the inside of an oven emit infinite UV light?

Compute the spectral radiance of a 3000 K blackbody at λ = 500 nm.

Research status: stable

Real-world applications

CMB cosmology and the standard model of the early universe
Stellar atmospheres and spectroscopy
Pyrometry and remote temperature sensing
Solar cell design (matching absorber bandgap to solar spectrum)

Common misconceptions

Planck initially viewed quantization as a mathematical trick, not real — Einstein took it literally in 1905
$B_\lambda and B_\nu$ have peaks at DIFFERENT positions because $d\lambda \neq d\nu$ (Jacobian $\lambda ^{2} /c)$
Real materials are NOT blackbodies; spectral emissivity matters

Experimental verification

COBE/FIRAS measured the CMB as a 2.7255 K Planck spectrum to 50 parts per million — the most precise blackbody ever measured. Laboratory cavity radiation also matches across 6 orders of magnitude in wavelength.

Derivation

Treat the EM field in a cavity as a set of quantized harmonic oscillators with energy

E_n = nh\nu

.

Bose-Einstein occupation gives ⟨n⟩

= 1/(\exp (h\nu /k_BT) - 1)

.

Density of modes per volume per frequency is

8\pi \nu ^{2}/c^{3}

.

Multiplying gives spectral energy density

u(\nu

,T), and

B_\lambda

follows by converting variables.

Limiting cases

hc/(\lambda k_BT)

≪ 1 (long

\lambda )

⟶

B_\lambda \to 2ck_BT/\lambda ^{4}

(Rayleigh-Jeans)Classical limit — recovers the divergence-prone form that Planck fixed.

hc/(\lambda k_BT)

≫ 1 (short

\lambda )

⟶

B_\lambda \to (2hc^{2}/\lambda ^{5})\cdot \exp (-hc/\lambda k_BT)

(Wien)Wien's approximation — exponentially suppresses UV emission.

Integrate over

all \lambda

⟶

\int B_\lambda d\lambda = \sigma T^{4}/\pi

Recovers Stefan-Boltzmann law.

What if…

What if Planck's constant were zero?

You'd recover Rayleigh-Jeans at all wavelengths $\to UV$ catastrophe → infinite emitted power. Quantization saves the universe.

What if you double the temperature?

Peak shifts to half the wavelength (Wien's law) and total emitted power increases by $16\times$ (Stefan-Boltzmann).

What if the cavity is filled with a dielectric (n > 1)?

Spectrum is modified: $c \to c/n$ in the formula, increasing density of modes and shifting the spectrum.

1

B_λ at 500 nm, T = 3000 K

Given ·

λ:: 5e-7
T:: 3000

Find ·

B_\lambda

Steps

Compute $x = hc/(\lambda k_BT) = 9.59$
Prefactor $2hc^{2}/\lambda ^{5} = 2(6.626e-34)(3e_{8})^{2}/(5e-7)^{5} \approx 3.82e16$
Divide by $e^9.59 - 1 \approx 14613$
$B_\lambda \approx 3.82e16 / 14613 \approx 2.61e12 W\cdot m^{-3}\cdot sr^{-1}$

Result ·

hc/(\lambda k_BT) = (6.626e-34 \times 3e_{8})/(5e-7 \times 1.381e-23 \times 3000) \approx 9.59

.

B_\lambda = (2hc^{2}/\lambda ^{5})\cdot 1/(e^9.59 - 1) \approx 3.83e12 W\cdot m^{-3}\cdot sr^{-1}

2

Rayleigh-Jeans check at radio wavelength

Given ·

λ:: 0.01
T:: 300

Find ·

B_\lambda

Steps

Compute $x = (6.626e-34)(3e_{8}) / (0.01)(1.381e-23)(300) \approx 4.8e-3$
Since x ≪ 1, use Rayleigh-Jeans limit
$B_\lambda \approx 2ck_BT/\lambda ^{4} = 2(3e_{8})(1.381e-23)(300)/(0.01)^{4} \approx 2.49e-9 W\cdot m^{-3}\cdot sr^{-1}$

Result ·

x = hc/(\lambda k_BT) \approx 4.8e-3

≪

1 \to use 2ck_BT/\lambda ^{4} \approx 2.49e-9 W\cdot m^{-3}\cdot sr^{-1}

Stefan-Boltzmann Law→Wien's Displacement Law→Boltzmann Entropy→