Blackbody Radiation Calculator
Calculate peak wavelength, total emitted power, and spectral radiance using Planck, Stefan-Boltzmann, and Wien's Laws.
Enter the temperature, surface area, emissivity, and a wavelength of interest to compute all key blackbody radiation quantities instantly.
Blackbody Radiation Calculator
Calculate peak wavelength, total emitted power, and spectral radiance using Planck, Stefan-Boltzmann, and Wien's Laws.
About the Blackbody Radiation Calculator
A blackbody is an idealised object that absorbs all incoming electromagnetic radiation and re-emits it purely as a function of its temperature, with no reflection or transmission. Although no perfect blackbody exists in nature, many objects approximate it closely: the Sun, incandescent lamp filaments, stars, and even the human body all emit radiation that can be modelled to a useful degree with blackbody formulae.
The cornerstone of blackbody theory is Planck's radiation law, published in 1900, which gives the spectral radiance (power emitted per unit area, per unit solid angle, per unit wavelength) as a function of temperature and wavelength: B(λ,T) = 2hc²/λ⁵ × 1/(e^(hc/λk_BT) − 1), where h = 6.626 × 10⁻³⁴ J·s is the Planck constant, c = 2.998 × 10⁸ m/s is the speed of light, k_B = 1.381 × 10⁻²³ J/K is Boltzmann's constant, λ is wavelength, and T is absolute temperature in kelvin. Planck's derivation, which required quantising the electromagnetic field in discrete energy packets (photons), marked the birth of quantum mechanics.
Wien's displacement law states that the wavelength of peak emission is inversely proportional to temperature: λ_max = b/T, where b = 2.898 × 10⁻³ m·K is Wien's displacement constant. For the Sun (T ≈ 5778 K), this gives λ_max ≈ 501 nm — right in the middle of the visible green spectrum, which is no coincidence: human vision evolved to be most sensitive at the peak output of our star. For the Earth (T ≈ 288 K), λ_max ≈ 10.1 μm — deep infrared, which is why thermal cameras can image objects at room temperature.
The Stefan–Boltzmann law gives the total power radiated per unit area: M = εσT⁴, where σ = 5.670 × 10⁻⁸ W·m⁻²·K⁻⁴ is the Stefan–Boltzmann constant and ε is the emissivity of the surface. For a perfect blackbody ε = 1; for a grey body 0 < ε < 1; for a perfect mirror ε = 0. The total power emitted by a surface of area A is P = εσAT⁴.
The calculator computes all these quantities simultaneously for a given temperature and optional surface properties. The spectral radiance at a user-specified wavelength uses the full Planck formula, allowing you to explore how the spectrum shifts with temperature — the principle underlying the colour temperature of light sources, the greenhouse effect, stellar classification, and remote sensing of planetary surfaces.
Practical applications span a wide range: lighting engineers use blackbody spectra to specify colour rendering indices; astronomers use Wien's law to estimate stellar surface temperatures from colour; climate scientists model planetary energy balance with the Stefan–Boltzmann law; and industrial furnace operators control temperatures by monitoring thermal emission spectra.
Blackbody Radiation Examples
Click any example button to load the parameters into the calculator.
| Parameters | Key results | Source / context |
|---|---|---|
| T=5778 K, A=1 m², λ=500 nm, ε=1 | λ_max ≈ 501.6 nm, P ≈ 6.32 × 10⁷ W, B ≈ 2.64 × 10⁴ W·m⁻²·sr⁻¹·nm⁻¹ | Sun's photosphere |
| T=288 K, A=1 m², λ=10000 nm, ε=0.98 | λ_max ≈ 10063 nm, P ≈ 382 W, B ≈ 7.96 × 10⁻³ W·m⁻²·sr⁻¹·nm⁻¹ | Earth's average surface |
| T=2700 K, A=0.001 m², λ=700 nm, ε=0.9 | λ_max ≈ 1073 nm, P ≈ 2712 W, B ≈ 316 W·m⁻²·sr⁻¹·nm⁻¹ | Tungsten filament (incandescent) |
How to Use the Blackbody Radiation Calculator
- Enter the temperature in kelvin (K). Use 5778 K for the Sun, 288 K for Earth's surface, or 2700 K for a typical incandescent bulb filament.
- Enter the surface area in square metres (m²). Use 1 m² to get per-square-metre values, or the actual area of your emitter.
- Enter the wavelength of interest in nanometres (nm). For visible light use 380–700 nm; for mid-infrared use 3000–10000 nm.
- Enter the emissivity (0–1). Use 1 for an ideal blackbody, 0.9–0.95 for most non-metallic surfaces, or 0.02–0.1 for polished metals.
- Click Calculate to see the peak wavelength (Wien's law), total radiated power (Stefan-Boltzmann), spectral radiance at your wavelength (Planck's law), and radiant exitance.
Frequently Asked Questions
What is the difference between a blackbody and a grey body?
A perfect blackbody has emissivity ε = 1 and absorbs all incident radiation. A grey body has 0 < ε < 1 and emits a fixed fraction of the blackbody power at all wavelengths. Real surfaces often have wavelength-dependent emissivity and are therefore neither, but the grey-body approximation is useful for many engineering calculations.
Why does the peak wavelength shift to blue as temperature increases?
Wien's displacement law λ_max = b/T shows a direct inverse relationship between peak wavelength and temperature. Higher temperatures correspond to higher photon energies, which correspond to shorter (bluer) wavelengths. A red-hot piece of metal emits mostly infrared with some deep red; a white-hot piece emits across the visible spectrum.
What is emissivity and how does it affect the result?
Emissivity ε is the ratio of radiation emitted by a surface to the radiation emitted by an ideal blackbody at the same temperature. It ranges from 0 (perfect reflector) to 1 (perfect absorber). The total power scales linearly with ε: doubling the emissivity doubles the emitted power. It does not affect the peak wavelength, which depends only on temperature.
How accurate is Wien's law compared to Planck's formula?
Wien's approximation (ignoring the −1 in Planck's denominator) is accurate to within 1% for wavelengths well below the peak (hc/λk_BT ≫ 1) but overestimates at longer wavelengths. For the exact peak wavelength, Wien's displacement law is precise. This calculator uses Planck's full formula for spectral radiance and Wien's displacement constant for the peak wavelength.
Can I use this to find the colour temperature of a light source?
Yes. Colour temperature is defined as the temperature of a blackbody that would emit light of a matching colour. Incandescent bulbs are around 2700 K (warm white), halogen lamps 3200 K, daylight is approximately 6500 K, and clear blue sky can exceed 10000 K. Enter the temperature and observe the peak wavelength and spectrum shape.
What is the Stefan-Boltzmann constant?
The Stefan-Boltzmann constant σ = 5.670 × 10⁻⁸ W·m⁻²·K⁻⁴ relates the total power radiated per unit area of a blackbody to the fourth power of its temperature: M = σT⁴. It can be derived from fundamental constants as σ = 2π⁵k_B⁴/(15h³c²). It plays a central role in stellar physics, climate science, and thermal engineering.