Planck's Blackbody Radiation Law

Equivalent forms

u_\nu=\frac{8\pi h\nu^3}{c^3}\frac{1}{e^{h\nu/k_BT}-1}

One formula spans a campfire, the Sun, and the 2.7 K cosmic microwave background — and it forced h into physics for the first time.

Symbol	Name	SI unit	Dimension	Range
$B_\lambda$	Spectral radianceoutput Power per area per solid angle per wavelength	Wm^-3sr^-1	ML^-1T^-3	—
$\lambda$	Wavelength Wavelength of emitted radiation	m	L	1e-7 – 0.000003
$T$	Temperature Absolute temperature of the body	K	Θ	1000 – 8000
$h$	Planck constant Quantum of action	J*s	ML^2T^-1	—
$c$	Speed of light Speed of light in vacuum	m/s	L*T^-1	—
$k_B$	Boltzmann constant Links temperature to energy	J/K	ML^2T^-2*Θ^-1	—

Where it holds

Ideal (perfectly absorbing) blackbody in thermal equilibrium. Real surfaces multiply by an emissivity < 1.

Dimensional analysis

B_\lambda \to [M*L^-1*T^-3]

per steradian

2hc^{2}/\lambda ^{5} \to [M*L^2*T^-1][L*T^-1]^{2}/[L^{5}] = [M*L^-1*T^-3]

Both sides are power per area per wavelength per solid angle.

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Discovery

Max Planck · 1900

Planck interpolated between Wien's law (good at short wavelengths) and Rayleigh's (good at long ones), then spent months finding a physical justification. On 14 December 1900 he presented it to the German Physical Society, forced to assume oscillators exchange energy only in quanta E = hf. He called it 'an act of desperation' and spent years trying to rid physics of the quantum he had invented.

Research status: stable

Real-world applications

Pyrometry: reading a furnace or star's temperature from its color
The 2.7 K cosmic microwave background
Incandescent-bulb and infrared-camera design

Common misconceptions

Planck did not believe photons were real particles; Einstein took that step in 1905
The peak wavelength shifts with temperature (Wien's law), but the whole curve rises, not just the peak
'Ultraviolet catastrophe' is a later name — Planck was fitting data, not resolving a paradox

Experimental verification

Lummer and Pringsheim's 1899 cavity measurements matched Planck's curve; the COBE satellite (1990) confirmed the CMB is a 2.725 K blackbody to better than 0.01%.

Derivation

Count electromagnetic standing-wave modes in a cavity: mode density is

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

unit frequency.

Classically each mode gets k_BT (equipartition

) \to

Rayleigh–Jeans, which diverges

as \nu \to \infty

.

Planck instead gives each mode the average energy of a quantized oscillator, ⟨E⟩

= h\nu /(e^{h\nu /k_BT}-1)

, from the geometric-series Bose factor.

Multiply mode density by ⟨E⟩ to get the spectral energy density; convert to radiance with a factor

c/4\pi

.

What if…

What if h were zero?

The Bose factor becomes $k_BT/h\nu and$ you recover Rayleigh–Jeans — the ultraviolet catastrophe returns and every warm object would blast infinite UV.

What if you double T?

The peak wavelength halves (Wien) and the total power rises $16\times$ (Stefan–Boltzmann, $T^{4})$ .

1

Peak wavelength of the Sun

Given ·

T:: 5778 K

Find · wavelength of maximum emission

Steps

Use Wien' $s law \lambda _\max T = 2.898e-3 m\cdot K$
$\lambda _\max = 2.898e-3 / 5778 \approx 5.0e-7\,\mathrm{m}$

Result ·

\lambda _\max = b/T = 2.898e-3 / 5778 \approx 502\,\mathrm{nm}

(green-blue)

Wien's Displacement Law→Stefan-Boltzmann Law→Photon Energy (Planck-Einstein Relation)→rayleigh jeans law