Planck's Law Calculator (Black-Body Spectrum)
Spectral radiance of a black body at any wavelength and temperature, its peak and total emission, with the whole spectrum drawn against a second temperature.
Results
The dashed line marks the value you entered.
What this tool does
Max Planck's 1900 formula gives the light emitted by a perfectly black object from its temperature alone, and it is where quantum physics began: it only works if energy comes in packets of hν. The peak slides to shorter wavelengths as the object gets hotter (Wien's law) and the area under the curve grows with the fourth power of temperature (Stefan–Boltzmann). The Sun, at about 5772 K, peaks near 500 nm; a 3000 K filament peaks in the infrared, which is why an incandescent bulb turns most of its power into heat.
Formula
B_λ(T) = (2hc² ÷ λ⁵) × 1 ÷ (e^(hc ÷ λkT) − 1) · λ_peak = b ÷ T · M = σT⁴
Variables
| Symbol | Meaning | Unit |
|---|---|---|
t | Temperature | K |
wl | Wavelength | nm |
t2 | Comparison temperature | K |
B | Spectral radiance | kW/(sr·m²·nm) |
PK | Wavelength of the peak | nm |
BP | Spectral radiance at the peak | kW/(sr·m²·nm) |
RP | Compared with the peak | % |
M | Total power emitted per m² (σT⁴) | MW/m² |
B2 | Spectral radiance at the comparison temperature | kW/(sr·m²·nm) |
Worked example
- Temperature5772 K
- Wavelength500 nm
- Comparison temperature3000 K
- Spectral radiance26.2385 kW/(sr·m²·nm)
- Wavelength of the peak502.0 nm
- Spectral radiance at the peak26.2396 kW/(sr·m²·nm)
- Compared with the peak100.00 %
- Total power emitted per m² (σT⁴)62.9386 MW/m²
- Spectral radiance at the comparison temperature0.260268 kW/(sr·m²·nm)
Limitations
- The formula assumes ideal conditions: no friction losses, no air resistance and no efficiency losses unless you enter them.
- Mixing units is the most common source of error. Convert every input to the units shown next to each field before calculating.