Black Body Radiation Calculator
A black body is an idealized object that absorbs all electromagnetic radiation and emits radiation purely based on its temperature. This calculator applies Wien's Displacement Law and the Stefan-Boltzmann Law to find peak wavelength and total emitted power.
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Formula
λ_max = b/T; P = σT⁴A
Wien's Displacement Law: λ_max = b/T, where b = 2,897,771 nm·K (Wien's constant) and T is absolute temperature in Kelvin. The result is peak wavelength in nanometers. Stefan-Boltzmann Law: P = σT⁴A, where σ = 5.6704×10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant and A is surface area. This gives total power radiated across all wavelengths.
How to use the Black Body Radiation Calculator
- 1
Enter your temperature
Value should be in K.
- 2
Enter your surface area
Value should be in m².
- 3
Read your results instantly
Results update in real time as you type.
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What is a black body?
A black body is an idealized physical object that absorbs all incident electromagnetic radiation regardless of frequency or angle. It's a perfect absorber and, by Kirchhoff's law of thermal radiation, also a perfect emitter — it radiates energy at the maximum possible rate for its temperature.
Real objects are 'gray bodies' — they emit a fraction ε (emissivity, 0–1) of the black body radiation. Human skin has ε ≈ 0.98. Polished aluminum has ε ≈ 0.05. The Stefan-Boltzmann law for real objects: P = εσT⁴A. This is why emergency 'space blankets' are reflective — low emissivity means low heat radiation, keeping you warm.
Wien's law and the color of stars
Wien's Displacement Law says hotter objects emit peak radiation at shorter wavelengths. The Sun's surface temperature is about 5,778 K; its peak emission is at about 501 nm — green light, right in the middle of the visible spectrum. Our eyes evolved to be most sensitive near this wavelength.
Cooler stars (3,000 K) peak in the infrared and appear red/orange. Hotter stars (10,000 K) peak in the UV and appear blue-white. A star's color directly tells you its surface temperature. A temperature of 2,898 K gives peak emission at exactly 1,000 nm (infrared). At 29,000 K, the peak is at 100 nm (extreme UV).
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The ultraviolet catastrophe and quantum mechanics
Classical physics (Rayleigh-Jeans Law) predicted that a black body would radiate infinite power at high frequencies — the 'ultraviolet catastrophe.' In 1900, Max Planck resolved this by proposing that electromagnetic energy is emitted in discrete packets (quanta) of energy E = hf, where h is Planck's constant.
This was the birth of quantum mechanics. Planck's distribution correctly describes black body radiation at all frequencies. At low frequencies (long wavelengths), it matches the classical result. At high frequencies, the quantization prevents the divergence. Einstein later used Planck's quantization to explain the photoelectric effect, further establishing quantum theory.
Tips & Insights
Temperature must be in Kelvin
K = °C + 273.15. Room temperature (20°C) = 293 K. The Sun's surface = 5,778 K. An incandescent bulb filament ≈ 2,700 K.
Human body radiation
At 37°C = 310 K: peak wavelength = 2,897,771/310 ≈ 9,347 nm (mid-infrared). This is why thermal cameras image people — we glow in the infrared.
Power scales as T⁴
Doubling temperature increases radiated power by 2⁴ = 16×. This extreme temperature dependence makes high-temperature materials radiate far more than low-temperature ones.
Worked Examples
The Sun
Peak wavelength = 2,897,771/5,778 ≈ 501 nm (green/yellow). Power per m² = σ×5,778⁴ ≈ 63.2 MW/m² (the Sun's total output is ~3.85×10²⁶ W).
Incandescent light bulb filament
Peak wavelength = 2,897,771/2,700 ≈ 1,073 nm (near-infrared). Most energy is heat, not light — why incandescents are inefficient.
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Frequently Asked Questions
What is Wien's Displacement Law?
λ_max = b/T. The peak emission wavelength of a black body is inversely proportional to its temperature. Hotter = shorter wavelength (bluer light).
What is the Stefan-Boltzmann Law?
P = σT⁴A. Total radiated power scales with the fourth power of absolute temperature. Double the temperature, emit 16× more power.
Why does the Sun appear yellow?
The Sun peaks at ~501 nm (green), but appears yellow-white because it emits across the full visible spectrum. Atmospheric scattering of blue light (Rayleigh scattering) makes the Sun appear more yellow.
What temperature is a red-hot object?
Steel becomes visibly red around 700-800°C (973-1,073 K). Orange: ~1,000°C. Yellow: ~1,200°C. White hot: ~1,400°C+. At these temperatures, peak emission is still in the infrared.
What is emissivity?
Emissivity ε (0–1) is the fraction of black body radiation an object emits. Perfect black body: ε = 1. Polished metal: ε ≈ 0.05. Human skin: ε ≈ 0.98. P = εσT⁴A for real objects.
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