Blackbody Radiation Calculator - Planck Spectrum Peak & Exitance
The blackbody radiation calculator turns any temperature into the Planck spectrum peak wavelength, total exitance, and spectral radiance at any wavelength.
Blackbody Radiation Calculator
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What Is Blackbody Radiation Calculator?
A blackbody radiation calculator turns one absolute temperature into the full Planck spectrum: the Wien peak wavelength, the Stefan-Boltzmann radiant exitance, and the spectral radiance at any wavelength you pick. It is built for thermal physics, astrophysics, and optics coursework where the same closed-form formulas are needed repeatedly.
- • Sun-like spectra: Plug in 5778 K to read the peak wavelength near 502 nm and the exitance around 6.3 × 10⁷ W/m² for stellar-physics checks.
- • Tungsten filaments: Estimate the 2500 K peak near 1.16 μm and the total exitance of an incandescent bulb for an electronics or lighting problem.
- • Thermal cameras: Read the 293 K peak around 9.9 μm and the exitance around 419 W/m² to calibrate long-wave-infrared sensors.
- • Cosmic microwave background: Check the 2.725 K peak near 1.06 mm against observational cosmology data; the exitance is 3.1 × 10⁻⁶ W/m².
The page assumes a Lambertian surface; if your source is a grey body, drop the emissivity below 1 and the exitance scales linearly while the peak wavelength and spectral radiance shape stay the same.
For the inverse problem of taking a temperature and reading out the thermal emission from a horizon, the Black Hole Temperature Calculator pairs the same NIST constants with the same Wien law.
How Blackbody Radiation Calculator Works
The calculator reads temperature, wavelength, emissivity, and area, converts everything to SI base units, then evaluates the Planck law, Wien displacement law, and Stefan-Boltzmann law with the NIST CODATA 2018 constants.
- T: Absolute temperature of the black body in kelvin (K). Converted from °C or °F when needed.
- lambda: Wavelength at which the Planck spectral radiance is evaluated, converted to meters.
- epsilon: Emissivity between 0 and 1. A perfect blackbody is 1; a polished mirror is 0.
- A: Surface area in m² used for total emitted power.
- h, c, k_B, sigma, b: Planck constant, speed of light, Boltzmann constant, Stefan-Boltzmann constant, and Wien displacement constant from NIST CODATA 2018.
The Planck law sets the spectral radiance per unit wavelength B_lambda, the Wien displacement law gives the peak wavelength, and the Stefan-Boltzmann law is the integral of the Planck spectrum shown separately as M and P = M × A. All three formulas share the same NIST constant set, so the calculator reports a consistent picture once the temperature is set.
Sun's surface (5778 K)
T = 5778 K, wavelength = 500 nm, emissivity = 1, area = 1 m²
lambda_max = 2.898e-3 / 5778 = 5.015e-7 m = 501.5 nm; M = 5.670e-8 × 5778⁴ = 6.32e7 W/m²; B_lambda(5778 K, 500 nm) ≈ 2.64e4 W/(m²·sr·nm)
Peak wavelength = 501.5 nm, exitance = 6.32 × 10⁷ W/m², spectral radiance at 500 nm = 2.64 × 10⁴ W·m⁻²·sr⁻¹·nm⁻¹
The peak sits in the green visible band; a 1 m² patch at this temperature would radiate 63 MW.
Room-temperature radiator (293.15 K)
T = 293.15 K (20 °C), wavelength = 9880 nm, emissivity = 1, area = 1 m²
lambda_max = 2.898e-3 / 293.15 = 9.885e-6 m = 9885 nm; M = 5.670e-8 × 293.15⁴ ≈ 418.8 W/m²
Peak wavelength = 9885 nm, exitance = 418.8 W/m²
Twenty degrees Celsius peaks in the long-wave infrared at about 9.9 μm.
According to NIST CODATA - Stefan-Boltzmann constant, the Stefan-Boltzmann constant sigma = 2 pi^5 k_B^4 / (15 h^3 c^2) evaluates to 5.670374419e-8 W m^-2 K^-4, the value used here for the radiant exitance.
When the question shifts from a continuous thermal spectrum to a discrete atomic transition, the Bohr Model Calculator uses the same Planck constant and speed of light to read the photon wavelength and energy between two quantum levels.
Key Concepts Explained
Four ideas hold the calculator together: the Planck law, the Wien displacement law, the Stefan-Boltzmann law, and the role of emissivity on a real surface.
Planck's law
B_lambda(T, lambda) = 2 h c^2 / (lambda^5 (exp(h c / (lambda k_B T)) - 1)) gives the spectral radiance per unit wavelength of a perfect black body. It reduces to the Wien form at short wavelengths and the Rayleigh-Jeans form at long wavelengths.
Wien displacement law
lambda_max = b / T places the peak of the Planck spectrum, with b = 2.898e-3 m K. Doubling the temperature halves the peak wavelength, which is why hot stars look blue and cool stars look red.
Stefan-Boltzmann law
M = epsilon sigma T^4 is the integral of the Planck spectrum over all wavelengths and angles. The T^4 dependence makes total radiated power extremely sensitive to temperature.
Emissivity
Real surfaces absorb and emit less than a perfect black body, captured by emissivity epsilon between 0 and 1. Polished metals have epsilon below 0.1, while soot and matte paint are closer to 1.
These four concepts share the NIST constants, so the same h, c, and k_B appear in the Planck law, the Wien displacement constant, and the Stefan-Boltzmann constant, which is why the calculator reads consistently across outputs. Color temperature is the Wien law applied the other way: an LED or photographic light is described by the temperature of a black body that would peak at the same wavelength, even when the actual source is not a black body. According to NIST CODATA - Wien displacement constant, b = 2.897771955e-3 m K is the fixed SI value that sets the peak wavelength of every Planck spectrum this calculator reports.
To read the Planck spectrum as a thermal population of photons, the Boltzmann Factor Calculator uses the same Boltzmann constant to turn k_B T into a probability factor exp(-E / k_B T) at any chosen energy.
How to Use This Calculator
Pick a temperature and a wavelength, set the emissivity and area if your surface is not a perfect black body, then read the six outputs from the same panel.
- 1 Enter the temperature: Type the absolute temperature of the source. Defaults to 5778 K, the effective surface temperature of the Sun. Switch to Celsius or Fahrenheit in the unit menu when the source value is in those units.
- 2 Enter the wavelength: Set the wavelength at which the Planck spectral radiance is evaluated. Defaults to 500 nm (green), close to the Sun's peak.
- 3 Set the emissivity: Leave at 1 for a perfect black body. Drop toward 0 for highly reflective surfaces; the radiant exitance and total power scale linearly with emissivity.
- 4 Set the surface area: Pick the surface area of the emitter in square meters, square centimeters, or square feet. Total emitted power uses this input.
- 5 Read the Wien peak: Headline output is the peak wavelength in nm where the Planck curve is maximum for that temperature.
- 6 Check exitance and power: Radiant exitance in W/m² and total emitted power in W are computed from M = epsilon sigma T^4 and P = M × A.
For a thermal-camera calibration target at 20 °C, switch to °C and enter 20; the peak lands around 9885 nm in the long-wave infrared, and the exitance comes out near 419 W/m².
If the emitter loses heat by conduction as well as by radiation, the Heat Transfer Conduction Calculator takes the same surface geometry and adds the conductive heat flux through the wall.
Benefits of Using This Calculator
The blackbody radiation calculator packages the Planck law, the Wien displacement law, and the Stefan-Boltzmann law behind one temperature input so coursework and lab checks share the same numbers.
- • Six outputs from one temperature: Peak wavelength, peak frequency, peak wavenumber, radiant exitance, total power, and spectral radiance at any wavelength all come from the same input.
- • NIST CODATA 2018 constants: Planck constant, speed of light, Boltzmann constant, Stefan-Boltzmann constant, and Wien displacement constant match published values to four or more significant figures.
- • Unit-aware inputs: Accept temperature in K, °C, or °F; wavelength in nm, μm, or m; area in m², cm², or ft². Everything is converted to SI internally so the formulas stay clean.
- • Emissivity slider: Adjust emissivity between 0 and 1 to scale the exitance for grey-body surfaces without changing peak wavelength or spectral shape.
- • Worked examples built in: Defaults are set to the Sun's 5778 K surface; worked examples cover the Sun and a 293 K room-temperature radiator.
If the question shifts from a continuous blackbody spectrum to a discrete atomic transition or a horizon temperature, the same NIST constants drive the neighboring calculators in this category.
Factors That Affect Your Results
Three input factors change every output, with emissivity as a fourth caveat. The limits below describe where the closed-form model is reliable and where it stops being exact.
Temperature T
Peak wavelength scales as 1/T, radiant exitance as T^4, and spectral radiance depends on T through the Planck law.
Wavelength lambda
The Planck law B_lambda rises steeply toward the peak and falls off on the Wien tail. Sampling too far from the peak gives a small spectral radiance.
Emissivity epsilon
Reduces the radiant exitance and total emitted power by the factor epsilon without shifting the peak wavelength.
Numerical regime
Above about 1 K the Planck law is highly accurate; near the Planck temperature quantum gravity is expected to modify the spectrum.
- • The model assumes a Lambertian black body; directional or anisotropic surfaces need angle-dependent emissivity data.
- • Spectral radiance below about 1e-30 W m^-2 sr^-1 m^-1 is reported as zero because the closed-form Planck law underflows in IEEE-754 double precision.
- • The calculator does not include grey-body transmission factors or atmospheric absorption, so results represent the emitter in vacuum.
Planck's law was derived for an idealized cavity radiator, so the calculator describes an ideal black body. Real laboratory sources are calibrated to match it within their emissivity factor, and the emissivity input lets the calculator reflect that calibration. The same equations describe the cosmic microwave background, which has a blackbody spectrum at 2.725 K measured by the FIRAS instrument on the COBE satellite.
According to Hyperphysics - Blackbody Radiation, the Planck radiation law gives the spectral radiance of a black body as B_lambda(T) = 2 h c^2 / (lambda^5 (exp(h c / (lambda k_B T)) - 1)), the formula used to evaluate spectral radiance in this calculator.
When the wavelength and temperature are tied to a wave picture rather than a thermal spectrum, the Harmonic Wave Equation Calculator uses the same speed of light and frequency relation to read the period and wavelength of any harmonic wave.
Frequently Asked Questions
Q: What does a blackbody radiation calculator compute?
A: A blackbody radiation calculator reads one absolute temperature and returns the Planck spectrum peak wavelength from Wien's law, the peak frequency and wavenumber, the Stefan-Boltzmann radiant exitance, the total emitted power for the chosen area, and the spectral radiance at any wavelength you pick. Defaults are set to the Sun's surface temperature and visible green.
Q: How is the peak wavelength of blackbody radiation found?
A: The peak wavelength follows from Wien's displacement law lambda_max = b / T with b = 2.898e-3 m K. At T = 5778 K the peak sits at about 501.5 nm in the visible green; at T = 2.725 K the peak is near 1.06 mm in the microwave band; at T = 293.15 K the peak is near 9885 nm in the long-wave infrared.
Q: What is the Stefan-Boltzmann law used for in this calculator?
A: The Stefan-Boltzmann law M = epsilon sigma T^4 gives the radiant exitance, which is the total power emitted per unit area across all wavelengths. With sigma = 5.670e-8 W m^-2 K^-4, a surface at 5778 K exits about 6.32e7 W/m^2; the same surface at 293 K exits about 419 W/m^2, illustrating the strong T^4 dependence.
Q: What is the difference between spectral radiance and total emitted power?
A: Spectral radiance B_lambda is the power per unit area, per unit solid angle, per unit wavelength at a chosen wavelength; it changes with the wavelength slider. Total emitted power P is the integral of the Planck spectrum over all wavelengths and angles multiplied by the surface area, computed here from M = epsilon sigma T^4 times the input area.
Q: Does emissivity change a blackbody radiation result?
A: Emissivity scales the radiant exitance and the total emitted power linearly without shifting the peak wavelength or changing the shape of the Planck spectrum. A polished metal at epsilon = 0.05 emits 20 times less power than a soot-covered surface at epsilon = 1 with the same temperature and area.
Q: How does the Sun's surface temperature show up in the blackbody curve?
A: The Sun's effective surface temperature is about 5778 K. Plugging that into Wien's law gives lambda_max = 502 nm in the visible green, and the Stefan-Boltzmann law gives an exitance around 6.3e7 W/m^2. The Sun's actual spectrum follows this blackbody curve closely because the photosphere is close to a perfect black body with emissivity near 1.