Planck's Law Calculator

Compute blackbody spectral radiance / exitance for any temperature and wavelength. Includes Wien's displacement, Stefan-Boltzmann, and interactive graph.

Planck's Law (spectral exitance): M(λ,T) = (2πhc²)/λ⁵ · 1/(e^{hc/(λkT)}−1)

λ: wavelength (μm), T: temperature (K), h, c, k: fundamental constants

Kelvin (K), must be > 0
micrometers (μm), range 0.01–1000 typical
Sun (5778 K) Tungsten (2856 K) Room temp (300 K) CMB (2.725 K) Red hot (1000 K)
UV (0.2 μm) Violet (0.4 μm) Green (0.55 μm) Red (0.7 μm) NIR (1 μm) IR (10 μm)
Wavelength span for the plot
Computing spectrum...

Understanding Planck's Law

Max Planck derived this law in 1900, marking the birth of quantum mechanics. It describes the spectral density of electromagnetic radiation emitted by a black body in thermal equilibrium at a given temperature T.

Mathematical formulation (spectral exitance):

M(λ,T) = \frac{2\pi h c^2}{\lambda^5} \frac{1}{e^{hc/(\lambda k T)}-1}

where M is in W·m⁻²·m⁻¹ (per meter). To obtain W·m⁻²·μm⁻¹, divide by 10⁶.

Key Constants & Derived Laws

Constant / Law Symbol / Formula Value / Relation
Planck constant h 6.62607015×10⁻³⁴ J·s
Speed of light c 2.99792458×10⁸ m/s
Boltzmann constant k 1.380649×10⁻²³ J/K
Wien's displacement λ_max·T = b b ≈ 2898 μm·K
Stefan‑Boltzmann M_total = σT⁴ σ ≈ 5.670374×10⁻⁸ W/m²/K⁴
First radiation constant (c₁) 2πhc² 3.741771×10⁻¹⁶ W·m²
Second radiation constant (c₂) hc/k 1.438777×10⁻² m·K

Limiting Behaviors

1

Wien approximation (short wavelengths): M ≈ (2πhc²/λ⁵) e^{-hc/(λkT)}

2

Rayleigh–Jeans law (long wavelengths): M ≈ (2πc kT)/λ⁴

Applications

  • Astrophysics: stellar spectra, cosmic microwave background
  • Thermal imaging: sensor design, remote sensing
  • Lighting: color temperature of lamps
  • Climate science: Earth's radiation budget
  • Metrology: high-temperature fixed points

Calculator features:

  • Accurate computation using CODATA 2019 constants
  • Automatic peak wavelength and total exitance
  • Interactive graph with adjustable range
  • Units: W·m⁻²·μm⁻¹ (spectral exitance) and W·cm⁻²·μm⁻¹ on demand

Frequently Asked Questions

Spectral radiance L(λ) is power per unit area per unit solid angle per wavelength. Exitance M(λ) is radiance integrated over hemisphere: M = πL for a Lambertian source. This calculator gives exitance.

Wien's displacement law: λ_max ∝ 1/T. Hotter objects emit peak radiation at shorter wavelengths (e.g., blue stars vs red stars).

The "Auto" range centers the plot around the Wien peak. For solar temperatures (5778 K), the peak is at 0.5 μm (visible). For room temperature, peak is ~10 μm (infrared).

Yes, the calculator shows both W/m²/μm and W/cm²/μm. For radiance, multiply exitance by 1/π.

We use the 2019 CODATA recommended values (exact, since SI redefinition). The calculator is suitable for teaching and research.