Planck’s Law Calculator
Compute blackbody spectral radiance at a wavelength, or solve for temperature from measured radiance using Planck’s distribution.
Calculation Steps
Practical Guide
Planck’s Law Calculator: Blackbody Spectral Radiance Made Practical
Use this guide to get trustworthy results from the Planck’s Law Calculator. We’ll walk through the equation, unit handling, assumptions (blackbody vs graybody), and how to interpret spectral radiance across different wavelengths. You’ll also see worked examples matching real physics and engineering use cases like thermal imaging and solar spectra.
Quick Start
- 1 Pick what you want to solve for: spectral radiance \(B_{\lambda}\) or temperature \(T\). The calculator hides the row you’re solving for.
- 2 Enter the wavelength \(\lambda\) and choose units (nm, µm, mm, cm, or m). Most thermal problems use µm.
- 3 If solving for radiance, enter temperature and its unit (K, °C, or °F). The calculator converts internally to Kelvin.
- 4 If solving for temperature, enter the measured radiance \(B_{\lambda,obs}\) and pick whether it’s in SI per-meter units (W·m⁻³·sr⁻¹) or per-µm (W·m⁻²·sr⁻¹·µm⁻¹). The tool converts to SI.
- 5 Check the equation banner to confirm you’re using the wavelength form of Planck’s Law (not the frequency form). This matters for units and interpretation.
- 6 Review Quick Stats for a sanity check: peak wavelength from Wien’s law and radiance in SI.
- 7 Use “Show Steps” to verify substitutions, especially if you’re matching a textbook or lab report.
Tip: Always think in spectral terms. \(B_{\lambda}\) is radiance per unit wavelength, not total emitted power. To compare to band-limited sensors, you may need to integrate over a wavelength interval.
Common mistake: Mixing wavelength units (e.g., entering 10 but selecting nm instead of µm) can shift the result by orders of magnitude.
Choosing Your Method
Planck’s Law comes in a few equivalent forms. The calculator uses the wavelength form because it matches the way most engineering instruments (IR cameras, spectrometers) are specified. Pick the approach that matches your data and goal.
Method A — Wavelength Form (what this calculator uses)
Best for spectra expressed versus wavelength, \(\lambda\), which is the most common plotting convention in labs and engineering.
- Direct match to radiance per wavelength data.
- Easy to pair with Wien’s displacement law.
- Most IR and optical sensors are banded in µm or nm.
- Units can feel odd (per meter). You must convert per-µm properly.
- Not interchangeable with frequency-form numbers without conversion.
Method B — Frequency Form (for \(B_{\nu}\))
Use when your spectrum is in terms of frequency \(\nu\) or when doing RF/astronomy work.
- Natural for frequency-domain measurements.
- Often simpler for some integrals over \(\nu\).
- Different functional shape; peaks shift compared to wavelength plots.
- Values are not numerically equal to \(B_{\lambda}\) at the same “color.”
Method C — Brightness Temperature (Inversion)
If you measure radiance at a wavelength and want a temperature estimate, invert the wavelength form.
- Directly used in thermal imaging and pyrometry.
- Gives a “brightness temperature” at that wavelength.
- Assumes blackbody emissivity \( \varepsilon = 1\). Real surfaces can read low.
- Very sensitive to measurement noise in the far tails of the spectrum.
What Moves the Number the Most
Radiance rises rapidly with temperature. In the IR, doubling \(T\) can increase \(B_{\lambda}\) by several orders of magnitude because of the exponential term.
Two effects compete: the \(\lambda^{-5}\) factor pushes radiance up at short wavelengths, while the exponential term suppresses emission when \(hc/(\lambda k_BT)\) is large. This creates a peak that shifts with \(T\).
Planck’s law is extremely unit-sensitive. A wavelength input off by \(10^3\) moves you to a totally different spectral region. The calculator converts to meters internally to avoid hidden scaling errors.
The calculator assumes \(\varepsilon=1\). Real materials emit: \[ B_{\lambda,\text{real}}=\varepsilon(\lambda)B_{\lambda} \] If emissivity is \(0.7\), your measured radiance is 30% lower and the inverted temperature will be underestimated.
Your instrument usually captures a wavelength band. The correct comparison is: \[ L_{\text{band}}=\int_{\lambda_1}^{\lambda_2} B_{\lambda}\,d\lambda \] A single-wavelength radiance is a point sample, not total flux.
At very small \(\lambda\) or low \(T\), the exponential dominates and radiance becomes tiny. Small measurement errors blow up when solving for temperature. Use mid-spectrum wavelengths when possible.
Worked Examples
Example 1 — Thermal IR Radiance from a Room-Temperature Blackbody
- Given: \(\lambda = 10\,\mu\text{m}\), \(T = 300\,\text{K}\)
- Goal: Compute \(B_{\lambda}\)
- Constants: \(h=6.626\times10^{-34}\,\text{J·s}\), \(c=2.998\times10^8\,\text{m/s}\), \(k_B=1.381\times10^{-23}\,\text{J/K}\)
This result is a point radiance at 10 µm. If you were comparing to a thermal camera with an 8–14 µm band, you would integrate across that interval rather than use a single value.
Example 2 — Visible Radiance from a Solar-Temperature Blackbody
- Given: \(\lambda = 500\,\text{nm}\), \(T=5800\,\text{K}\) (approx solar surface)
- Goal: Compute \(B_{\lambda}\)
The magnitude difference versus Example 1 is expected: higher temperature and shorter wavelength place the spectrum near its peak. The calculator’s Quick Stats should show a Wien peak near \(\lambda_{max}\approx 0.5\,\mu\text{m}\), aligning with sunlight.
Example 3 — Solve for Brightness Temperature from Measured IR Radiance
- Given: \(\lambda=10\,\mu\text{m}\), \(B_{\lambda,obs}=1.0\times10^{7}\,\text{W·m}^{-3}\text{·sr}^{-1}\)
- Goal: Estimate \(T\)
This “brightness temperature” matches the ballpark of Example 1. If the surface emissivity were 0.9, the true physical temperature would be slightly higher than this estimate.
Common Layouts & Variations
Planck’s law appears in several practical configurations. The calculator covers the core blackbody wavelength form, so you can map your situation onto that model.
| Scenario / Configuration | Typical Inputs | What You Compare Against | Notes / Pros & Cons |
|---|---|---|---|
| Thermal imaging (8–14 µm cameras) | \(\lambda\) mid-IR band, approximate \(T\) | Band-integrated radiance | Use Planck point values for intuition; real calibration needs band integration and emissivity correction. |
| Single-color pyrometry | One wavelength, measured \(B_{\lambda}\) | Brightness temperature \(T_b\) | Fast temperature estimate; emissivity uncertainty dominates error. |
| Solar/visible spectra | \(\lambda\) in nm, \(T\sim 5800\,K\) | \(B_{\lambda}\) curve shape | Peak in visible; good for confirming Wien shift and overall spectral distribution. |
| Furnace or molten metal radiation | Near-IR/visible \(\lambda\), high \(T\) | Radiance or inferred \(T\) | Choose \(\lambda\) near peak for sensitivity; avoid spectral tails. |
| Graybody approximation | \(\varepsilon\), \(\lambda\), \(T\) | \(\varepsilon B_{\lambda}\) | The calculator assumes \(\varepsilon=1\); multiply outputs by emissivity to estimate real radiance. |
| Frequency-domain work | \(\nu\), \(T\) | \(B_{\nu}\) | Use frequency form; do not plug \(\nu\) into wavelength form. |
- Match your plotted axis to the correct form (\(\lambda\) vs \(\nu\)).
- For per-µm data, convert using \(1\,\mu\text{m}=10^{-6}\,\text{m}\).
- Expect Wien peak shifts: hotter → peak at shorter wavelengths.
- Use mid-band wavelengths for best temperature inversion stability.
- Apply emissivity if the surface is not a blackbody.
- Sensor readings are band-limited unless explicitly monochromatic.
Specs, Logistics & Sanity Checks
Engineers usually encounter Planck’s law through instruments. Here’s what to verify before you trust a number.
Instrument Bandwidth
Most detectors integrate over a finite band. If your camera reports radiance in W·m⁻²·sr⁻¹, it’s typically already integrated. When comparing to this calculator, integrate: \[ L_{\text{band}}=\int_{\lambda_1}^{\lambda_2}B_{\lambda}\,d\lambda \] or use a representative wavelength near the band center for quick checks.
Emissivity & Surface Finish
Polished metals, ceramics, painted surfaces—each has its own \(\varepsilon(\lambda)\). If you invert temperature from radiance without emissivity, you get brightness temperature: \[ T_b < T_{\text{true}} \quad \text{if } \varepsilon<1 \] Use tabulated emissivities or measured values when accuracy matters.
Atmospheric Effects
IR radiation can be absorbed by water vapor or CO₂. For long paths, measured \(B_{\lambda,obs}\) may be less than source emission. Short distances in clear air are usually safe; otherwise consider transmissivity.
Sanity check: Wien’s law provides a quick reality test: \[ \lambda_{max}T \approx 2.898\times10^{-3}\,\text{m·K} \] If your chosen \(\lambda\) is far from the peak, radiance will be tiny and inversion noisy.
Don’t over-interpret a point value: \(B_{\lambda}\) is not total power. Total hemispherical exitance requires integrating in wavelength and solid angle (or using Stefan–Boltzmann).