| Scenario | Temperature | Point | Mode | Notes |
|---|---|---|---|---|
| Solar-like surface | 5800 K | 500 nm | Bλ per nm | Radiance near the visible peak. |
| Heated metal | 1200 °C | 2.0 µm | Bλ per µm | Strong infrared emission for furnaces. |
| Room-temperature object | 300 K | 30 THz | Bν per THz | Thermal radiation in far infrared. |
This calculator uses Planck's law for blackbody spectral radiance. Choose a wavelength form or a frequency form. Emissivity scales the ideal result for real surfaces.
- Wavelength form: Bλ(λ,T) = (2hc²/λ⁵) / (exp(hc/(λkT)) − 1)
- Frequency form: Bν(ν,T) = (2hν³/c²) / (exp(hν/(kT)) − 1)
- Wien peak: λmax = b/T, with b ≈ 2.897771955×10⁻³ m·K
- Total exitance: M = εσT⁴, and L = M/π for hemispherical radiance.
- Band integration: numerical trapezoids for ∫Bλdλ or ∫Bνdν.
- Select wavelength or frequency mode.
- Enter the temperature and choose its unit.
- Set emissivity to match your surface (0–1).
- Provide the wavelength or frequency point, then choose the output scale.
- Optional: enable band integration and enter a range.
- Press Calculate. Results appear above the form.
- Use Download CSV or Download PDF to export.
1) Why spectral radiance matters
Spectral radiance describes how much power an ideal emitter sends in a specific direction per surface area and per wavelength or frequency interval. It is the core quantity behind thermal cameras, lamp calibration, remote sensing, and radiometric design, because it links temperature to measurable optical power.
2) Planck curve shape and scaling
The Planck curve rises steeply at short wavelengths, peaks, then decays gradually toward longer wavelengths. The curve grows rapidly with temperature: doubling temperature increases total emitted power by a factor of 16 because total exitance follows M = εσT⁴. This calculator reports both spectral radiance and totals.
3) Typical temperatures and peak wavelengths
Wien’s displacement law predicts λmax = b/T. For a solar-like surface near 5800 K, the peak wavelength is about 500 nm, close to visible light. At 300 K, the peak shifts to roughly 9.7 µm, in the thermal infrared. This is why room-temperature objects glow in IR, not visible.
4) Wavelength vs frequency forms
Bλ and Bν describe the same physics but use different “bin widths” (dλ vs dν). The numerical value at the same point will differ because converting between variables introduces a Jacobian factor. Use wavelength form for optics and filter bands; use frequency form for spectroscopy and RF/THz contexts.
5) Emissivity and real materials
Real surfaces emit less than an ideal blackbody. Emissivity (0–1) scales the radiance and the totals. Polished metals can have low emissivity, while painted or oxidized surfaces can be high. If emissivity changes with wavelength, treat the result as an approximation using an effective value.
6) Band integration for sensors and filters
Instruments often respond over a finite range, such as 8–14 µm for thermal imaging or 400–700 nm for visible photometry. The band option integrates radiance numerically over your selected limits, then reports the hemispherical exitance by multiplying the integrated radiance by π.
7) Units and practical interpretation
Radiance units are direction-sensitive. The calculator displays W·sr⁻¹·m⁻² per wavelength or frequency unit. Scaling “per nm” or “per THz” helps match common datasheets. For engineering comparisons, use the totals: radiance (M/π) and exitance (M).
8) Accuracy tips and numerical stability
Extremely small wavelengths or huge frequencies can produce large exponent terms. This tool uses safe handling for exp(x) − 1 to avoid overflow and improve low-x accuracy. For band results, increase steps for narrower bands or higher temperatures. Always keep inputs positive and in realistic ranges.
1) What is the difference between radiance and exitance?
Radiance is directional power per area per solid angle. Exitance is the total power leaving a surface into a hemisphere. For a diffuse emitter, exitance and radiance relate by M = πL.
2) Why do Bλ and Bν peaks occur at different places?
Peaks depend on whether you plot per wavelength or per frequency. Converting variables changes the density per interval, so the maximum shifts. Both forms describe the same spectrum, but with different binning.
3) What temperature corresponds to a 10 µm peak?
Using Wien’s law, T ≈ b/λ. With b ≈ 2.898×10⁻³ m·K and λ = 10 µm, temperature is about 290 K, close to room temperature.
4) Can I use this for graybody surfaces?
Yes. Set emissivity between 0 and 1 to scale results. This matches a graybody assumption where emissivity is constant with wavelength. For strongly wavelength-dependent emissivity, use band averages or separate runs.
5) Why does the band result multiply by π?
The integral computes radiance over a band. For a Lambertian emitter, integrating over a hemisphere yields exitance equal to π times radiance. The calculator reports both band radiance and band exitance.
6) How many integration steps should I use?
For broad bands, 500–1000 steps is usually fine. For narrow filters, steep peaks, or very hot sources, increase steps for smoother convergence. If results change noticeably with steps, keep increasing until stable.
7) Why is my radiance near zero at short wavelengths?
At low temperatures, the exponential term becomes very large at short wavelengths, making the Planck factor tiny. That is expected: cool objects emit mostly in infrared. Try longer wavelengths or raise the temperature.