Planck's Radiation Law
Planck's radiation law describes the spectral distribution of electromagnetic radiation from a black body.
Learn with examples.
The Formula
Planck's radiation law gives the spectral radiance of a black body at any wavelength for a given temperature. It describes how the intensity of emitted radiation varies across the electromagnetic spectrum.
German physicist Max Planck derived this formula in 1900, marking the birth of quantum physics. To make the mathematics work, he had to assume that energy is emitted in discrete packets (quanta) of size E = hf. This revolutionary idea contradicted classical physics and eventually led to the development of quantum mechanics.
The formula predicts that each temperature has a peak wavelength where emission is strongest. This peak shifts to shorter wavelengths at higher temperatures, which is described by Wien's displacement law. At room temperature, the peak is in the infrared range, while at the Sun's surface temperature, it peaks in the visible light range.
Variables
| Symbol | Meaning |
|---|---|
| B(λ,T) | Spectral radiance (W/m²/sr/m) |
| h | Planck's constant (6.626 × 10⁻³⁴ J·s) |
| c | Speed of light (3 × 10⁸ m/s) |
| λ | Wavelength of radiation (meters) |
| k | Boltzmann constant (1.381 × 10⁻²³ J/K) |
| T | Temperature of the black body (kelvin, K) |
Example 1
At what wavelength does a black body at 5,778 K (the Sun) emit most intensely? (Using Wien's law derived from Planck's law)
Wien's displacement law: λ_max = b/T where b = 2.898 × 10⁻³ m·K
λ_max = 2.898 × 10⁻³ / 5778
λ_max ≈ 501 nm (green-yellow light, near the peak sensitivity of human vision)
Example 2
A red-hot metal piece has a peak emission wavelength of 1,200 nm. What is its temperature?
Rearrange Wien's law: T = b/λ_max = 2.898 × 10⁻³ / 1.2 × 10⁻³
T ≈ 2,415 K (about 2,142 degrees Celsius, consistent with glowing red-hot metal)
When to Use It
Use Planck's radiation law to determine the spectral emission of any object based on its temperature.
- Analyzing stellar spectra to determine star surface temperatures
- Designing infrared sensors and thermal cameras
- Calculating the color temperature of light sources
- Understanding the greenhouse effect and Earth's energy balance