Blackbody Radiation Calculator
Calculate peak wavelength, total radiated power per unit area, photon energy, and Planck spectrum for any temperature and emissivity. Visual spectrum chart included.
Ideal blackbody (ε ≈ 1)
Object Presets
Press Enter to calculate · Esc to reset · Inputs auto-saved
What Is the Blackbody Radiation Calculator?
This calculator computes the complete thermal emission profile of any blackbody (or grey body) from its temperature. It uses all three classical laws simultaneously, Planck, Stefan-Boltzmann, and Wien, and provides a visual Planck spectrum chart with the visible light band highlighted, so you can immediately see where the emission peaks relative to what the human eye can detect.
- ›Temperature in K, °C, or °F, enter in whichever unit you know; the calculator converts to Kelvin internally and shows all three units.
- ›Emissivity slider (0–1), adjust to model real materials: human skin ≈ 0.98, polished aluminium ≈ 0.05, brick ≈ 0.90.
- ›Surface area (optional), add your object's area in m² to get the absolute total radiated power in watts alongside the per-m² value.
- ›9 object presets, one click loads Sun, CMB, human body, incandescent bulb, hot lava, liquid nitrogen, and more.
- ›Planck spectrum SVG chart, auto-scales to the temperature range, marks the peak wavelength with a dashed line, and overlays a rainbow band for the 380–750 nm visible window.
- ›Photon energy at peak, shows both joules and electron-volts, useful for photovoltaic design and detector selection.
- ›Visible light fraction, the percentage of total emitted radiation that falls in the 380–750 nm band (numerically integrated).
- ›Step-by-step derivation, a collapsible panel shows every formula step with actual numbers substituted in.
Formula
Three fundamental laws of blackbody radiation
1. Stefan-Boltzmann Law, total radiated power
M = ε × σ × T⁴
M = radiated power per unit area [W/m²]
2. Wien's Displacement Law, peak wavelength
λ_max = b / T
λ_max in metres, T in Kelvin, b = 2.897 771 955 × 10⁻³ m·K
3. Planck's Law, full spectral radiance
B(λ,T) = (2hc²/λ⁵) × 1 / [exp(hc/λk_BT) − 1]
B = spectral radiance [W·m⁻²·sr⁻¹·m⁻¹]
Physical constants used
| Symbol | Quantity | Value | Units |
|---|---|---|---|
| σ | Stefan-Boltzmann constant | 5.670 374 419 × 10⁻⁸ | W m⁻² K⁻⁴ |
| b | Wien wavelength constant | 2.897 771 955 × 10⁻³ | m · K |
| b_f | Wien frequency constant | 5.878 925 757 × 10¹⁰ | Hz / K |
| h | Planck constant | 6.626 070 15 × 10⁻³⁴ | J · s |
| c | Speed of light | 2.997 924 58 × 10⁸ | m / s |
| k_B | Boltzmann constant | 1.380 649 × 10⁻²³ | J / K |
| C₁ | 2hc² (1st radiation const.) | 3.741 771 852 × 10⁻¹⁶ | W · m² |
| C₂ | hc/k_B (2nd radiation const.) | 1.438 776 877 × 10⁻² | m · K |
Connecting the three laws
σ = 2π⁵k_B⁴ / (15h³c²) ← Stefan-Boltzmann derived from Planck
Integrating B(λ,T) over all λ and hemispheres gives M = σT⁴
Setting dB/dλ = 0 (maximum) gives Wien: λ_max = hc/(4.965114 k_B T)
b = hc / (4.965114 k_B) = 2.897 772 × 10⁻³ m·K
How to Use
- 1Enter temperature: Type the temperature and choose the unit (K, °C, or °F). A live conversion line shows the equivalent in all three units.
- 2Set emissivity: Drag the slider or type a value between 0 and 1. For an ideal blackbody leave it at 1. For real surfaces, use measured emissivity values (e.g. 0.98 for human skin).
- 3Add area (optional): Click "Add surface area" to reveal the area input. Enter the surface area in m² to get total radiated power in watts alongside the per-m² value.
- 4Try a preset: Click any object preset, Sun, CMB, Human body, etc., to load a real-world example and see how the output changes across a vast temperature range.
- 5Press Enter or click Calculate: The Planck spectrum chart appears with the visible light band (rainbow overlay) and a dashed peak-wavelength marker. The stat grid shows power per area, peak wavelength, peak frequency, photon energy, spectral band, and visible fraction.
- 6Expand step-by-step: Click "Show step-by-step derivation" to see the Stefan-Boltzmann, Wien, and frequency calculations written out with your exact numbers.
Example Calculation
Sun's photosphere: T = 5778 K, ε = 1
Step 1: Total power per unit area (Stefan-Boltzmann)
M = 1 × 5.670374×10⁻⁸ × (5778)⁴
= 5.670374×10⁻⁸ × 1.11574×10¹⁵
= 6.326×10⁷ W/m² (63.26 MW/m²)
Step 2: Peak wavelength (Wien)
λ_max = 2.897772×10⁻³ / 5778 = 5.014×10⁻⁷ m
λ_max = 501 nm (green-yellow, visible)
Step 3: Peak frequency
f_max = 5.878926×10¹⁰ × 5778 = 3.397×10¹⁴ Hz
f_max = 339.7 THz
Step 4: Photon energy at peak
E = 6.626×10⁻³⁴ × 3.397×10¹⁴ = 2.250×10⁻¹⁹ J
E = 1.406 eV
Why the Sun appears white, not green
Wien's law places the Sun's peak at ~501 nm (green), but the human eye perceives the Sun as white or pale yellow. The reason: a blackbody at 5778 K emits strongly across the entire visible range, blue, green, and red simultaneously. The mix of all colours appears white to the eye. Atmospheric Rayleigh scattering removes some blue on its way to the surface, shifting the perceived colour toward yellow-white. You can confirm this by observing the Sun from space, it is distinctly white.
Human body: T = 310 K (37 °C), ε = 0.98
M = 0.98 × 5.670374×10⁻⁸ × (310)⁴
M = 513.5 W/m²
λ_max = 2.897772×10⁻³ / 310
λ_max = 9.35 μm (thermal infrared, invisible to naked eye)
With area ≈ 1.7 m²: Total ≈ 873 W emitted by the whole body surface. Net loss is lower because surroundings also radiate back.
Understanding Blackbody Radiation
What Is Blackbody Radiation?
A blackbody is a theoretical object that absorbs every photon that hits it, perfectly, at all wavelengths, and re-emits radiation that depends only on its temperature, not on what it is made of. No real object is a perfect blackbody, but many come close: the Sun, the cosmic microwave background, and the interior of a furnace all behave nearly like ideal blackbodies over their relevant wavelength ranges.
The radiation a blackbody emits is purely thermal, produced by the random jiggling of charged particles (electrons and atomic nuclei) that increases with temperature. At room temperature this thermal radiation is mostly mid-infrared, invisible to the human eye but detectable by thermal cameras. Heat a piece of metal to 700°C (973 K) and it begins to glow visibly red. Heat it further and it becomes orange, yellow, and eventually white as shorter wavelengths are added.
The Stefan-Boltzmann Law, Why Doubling Temperature Multiplies Power by 16
The total power emitted per unit area, the radiant exitance M, scales as the fourth powerof absolute temperature:
M = ε σ T⁴
If T doubles: M → ε σ (2T)⁴ = 16 × εσT⁴ = 16 × M
This is why stellar luminosity differences are enormous. A blue supergiant at 30 000 K emits about (30 000/5778)⁴ ≈ 720 times more power per square metre than the Sun's photosphere. Stars that appear bright in the night sky are often intrinsically much hotter or larger than the Sun, often both.
Wien's Displacement Law, Colour as a Thermometer
Wien's law tells us that the wavelength of peak emission is inversely proportional to temperature. Hotter objects peak at shorter (bluer) wavelengths; cooler objects peak at longer (redder) wavelengths. This single law underpins some of the most powerful measurement techniques in physics:
- ›Stellar spectroscopy, the colour of a star's continuous spectrum reveals its surface temperature to within a few percent, even from thousands of light-years away.
- ›Non-contact thermometry, an infrared thermometer measures the peak wavelength of emitted IR and converts it to temperature using Wien's law.
- ›CMB cosmology, the cosmic microwave background peaks at ~1.06 mm, corresponding to 2.725 K, the temperature of the universe 380 000 years after the Big Bang, red-shifted over 13.8 billion years.
- ›Material processing, furnace temperatures are read optically by comparing the colour of glowing metal to known blackbody colours.
Planck's Law and the Birth of Quantum Physics
The Planck radiation formula was derived in 1900 when Max Planck, struggling to fit the observed blackbody spectrum, proposed a radical idea: energy is not emitted continuously, but only in discrete packets (quanta) of size E = hf. This was the founding equation of quantum mechanics.
Classical physics predicted that the energy emitted at short wavelengths would diverge to infinity (the "ultraviolet catastrophe"), which clearly does not happen in reality. Planck's quantization of energy naturally suppresses emission at very short wavelengths, producing the characteristic bell-shaped Planck curve that matches experiment perfectly.
The ultraviolet catastrophe explained
Classical electromagnetic theory (Rayleigh–Jeans law) predicted that the spectral radiance B(λ, T) should grow without bound as λ → 0, implying infinite total power, absurd and physically impossible. Planck solved this by introducing the exp(hc/λk_BT) − 1 term in the denominator. At short wavelengths this exponential grows very fast, suppressing the spectrum and creating the left-side dropoff you see on the Planck chart. Quantum mechanics is literally baked into every blackbody curve.
Emissivity, How Real Objects Differ from Ideal Blackbodies
Real objects emit less radiation than an ideal blackbody at the same temperature. The ratio is called emissivity ε, which ranges from 0 (perfect reflector, no emission) to 1 (perfect blackbody). Kirchhoff's law states that emissivity equals absorptivity at the same wavelength, a good absorber is always a good emitter.
| Material | Emissivity ε | Notes |
|---|---|---|
| Human skin | 0.97–0.99 | Nearly ideal, regardless of skin colour |
| Matte black paint | 0.95–0.98 | Used to approximate blackbody in experiments |
| Water / ice | 0.96–0.98 | High emissivity in thermal IR |
| Brick / concrete | 0.85–0.95 | Building materials radiate efficiently |
| Oxidised iron | 0.64–0.80 | Rust increases emissivity vs polished iron |
| Polished aluminium | 0.03–0.10 | Very low, most energy reflected, not emitted |
| Gold / silver foil | 0.01–0.04 | Used in spacecraft thermal insulation (MLI) |
Blackbody Temperatures Across the Universe
| Object | Temperature | Power / m² | λ_max | Spectral band |
|---|---|---|---|---|
| CMB (universe) | 2.725 K | 3.16×10⁻⁶ W/m² | 1.06 mm | Far-infrared/microwave |
| Liquid nitrogen | 77 K | 2.01×10⁻² W/m² | 37.6 μm | Far-infrared |
| Human body | 310 K | 524 W/m² | 9.35 μm | Thermal IR |
| Earth (effective) | 255 K | 240 W/m² | 11.4 μm | Thermal IR |
| Incandescent bulb | 2700 K | 3.01×10⁶ W/m² | 1073 nm | Near-infrared |
| Sun surface | 5778 K | 6.33×10⁷ W/m² | 501 nm | Visible (green-yellow) |
| Blue supergiant star | 30 000 K | 4.59×10¹⁰ W/m² | 96.6 nm | Extreme UV |
Real-World Applications
- ›Astrophysics, determine stellar surface temperatures from spectral type; classify stars as O, B, A, F, G, K, M by temperature and colour.
- ›Climate science, model Earth's energy balance using Stefan-Boltzmann: absorbed solar power must equal re-emitted thermal IR at 255 K effective temperature.
- ›Infrared thermography, thermal cameras measure IR emission to find heat leaks in buildings, detect fever in humans, and inspect electrical equipment.
- ›LED and solar cell design, solar spectrum peaks near 500 nm; photovoltaic bandgaps are optimised to match the Planck curve of 5778 K.
- ›Medical imaging, MRI coil temperature monitoring, laser thermal therapy, and photodynamic therapy all involve controlled thermal radiation.
- ›Furnace and kiln control, optical pyrometers read furnace temperatures from the colour of glowing charge material without contact.
Frequently Asked Questions
What is a blackbody and does it actually exist?
A blackbody is an idealized emitter, perfectly absorbing and re-emitting all radiation based solely on temperature. It is a theoretical ideal, but many real objects come very close:
- ›The Sun, behaves as a 5778 K blackbody across most of its spectrum
- ›A furnace cavity, absorbs and re-emits so efficiently it approaches ε = 1
- ›The cosmic microwave background, the most perfect blackbody spectrum ever measured
- ›Human skin, ε ≈ 0.97–0.99 in the thermal infrared regardless of visible skin colour
Emissivity ε (0 to 1) quantifies the departure from the ideal. The actual emitted power is M = ε × σ × T⁴.
What is Wien's displacement law and how do I use it?
Wien's displacement law gives the wavelength at which a blackbody emits most intensely:
b = 2.897 771 955 × 10⁻³ m·K
Two common uses:
- ›Find the peak wavelength: λ_max = 2.898×10⁻³ / T (give T in Kelvin, get λ in metres)
- ›Find the temperature: T = 2.898×10⁻³ / λ_max (measure the peak wavelength, compute T)
Example, identify a star's temperature from its colour: if peak emission is at 400 nm (blue-violet), then T = 2.898×10⁻³ / 400×10⁻⁹ ≈ 7245 K.
What is the Stefan-Boltzmann law and what does T⁴ mean in practice?
The Stefan-Boltzmann law gives total radiated power per unit area:
σ = 5.670 374 × 10⁻⁸ W m⁻² K⁻⁴
The T⁴ dependence means small temperature increases lead to large power increases:
- ›Double T (×2): power increases by 2⁴ = 16×
- ›Triple T (×3): power increases by 3⁴ = 81×
- ›Sun (5778 K) vs human body (310 K): power per m² ratio = (5778/310)⁴ ≈ 120 000×
- ›This also applies to cooling: an object radiates rapidly at high T, slows as it cools
Why does the Sun appear white/yellow even though its peak emission is in the green?
The peak wavelength is not the same as the perceived colour. Here is why the Sun looks white:
- ›Wien's law gives the peak, but the Sun emits at all visible wavelengths (400–700 nm), not just at 501 nm
- ›Blue + green + red light mixed together = white light (the human eye mixes them)
- ›Rayleigh scattering in Earth's atmosphere removes some blue (scatters it sideways, that's why the sky is blue), leaving the Sun appearing slightly yellowish from the surface
- ›Astronauts above the atmosphere confirm: the Sun is white
This is analogous to how incandescent lights look warm-yellow even though they peak in the near-infrared, the eye integrates over the whole visible portion of the spectrum.
What is emissivity and which value should I use for my material?
Emissivity tells you how efficiently a surface radiates compared to an ideal blackbody at the same temperature.
Common values:
- ›Human skin: 0.97–0.99 (behaves almost like a perfect blackbody in thermal IR)
- ›Matte black paint: 0.95–0.98
- ›Water / ice: 0.95–0.98
- ›Brick, concrete: 0.85–0.95
- ›Oxidised iron: 0.64–0.80
- ›Polished aluminium: 0.03–0.10 (very low, reflects rather than emits)
- ›Gold foil: 0.01–0.03 (used in spacecraft MLI insulation for this reason)
Important: emissivity is wavelength-dependent. The values above are broad averages in the thermal infrared (8–14 μm), relevant for room-temperature objects.
What is the cosmic microwave background (CMB) and why is it a blackbody?
The CMB is the afterglow of the hot early universe, preserved as the most perfect blackbody spectrum ever measured:
- ›380 000 years after the Big Bang: the universe cooled to ~3000 K, allowing hydrogen atoms to form
- ›Before that, the universe was an opaque plasma, a near-perfect blackbody cavity
- ›Once neutral, photons could travel freely, this "last scattering surface" is the CMB
- ›13.8 billion years of expansion red-shifted the temperature from ~3000 K to 2.725 K
- ›Peak wavelength: b / 2.725 ≈ 1.06 mm, in the microwave range
The COBE satellite (1992) measured the CMB spectrum and confirmed it matches a perfect blackbody to within 50 parts per million, one of the most precise agreements between theory and experiment in all of science.
How do infrared thermometers and thermal cameras use blackbody radiation?
Infrared devices exploit the fact that all objects above 0 K emit thermal radiation:
- ›IR thermometers detect power in a specific IR band, typically 8–14 μm
- ›They use Stefan-Boltzmann (M = εσT⁴) in reverse: measure M → compute T
- ›Emissivity setting matters, most IR thermometers default to ε = 0.95 (fine for skin, bad for polished metal)
- ›Thermal cameras array hundreds of thousands of microbolometer pixels, each measuring local IR power
- ›Applications: fever detection, building insulation surveys, electrical inspection, firefighting
Important limitation: if emissivity is unknown or wrongly set, the temperature reading will be wrong. A polished metal with ε = 0.05 will appear much colder than it really is because it reflects rather than emits.
Does the calculator save my inputs?
Yes, all inputs are automatically saved to your browser's localStorage:
- ›Temperature value and unit (K/°C/°F)
- ›Emissivity value
- ›Surface area (if entered)
- ›All data stays local, nothing is transmitted to any server
Click Reset All to clear all fields and delete the saved data.