Planck’s Law Calculator
Calculate blackbody spectral radiance, inverse temperature, peak wavelength, and total exitance from temperature, wavelength, frequency, radiance, and emissivity.
Calculator is for informational purposes only. Terms and Conditions
Choose what to solve for
Select the blackbody radiation calculation and output units.
Enter the known values
All calculations use SI base units internally before converting the final answer.
Blackbody Curve Visual
The curve shows relative spectral radiance versus wavelength, with the peak and selected wavelength marked.
Solution
Live result, quick checks, warnings, and full solution steps.
Quick checks
- Check—
Show solution steps See the equation, substitutions, constants, and assumptions
- Enter values to see the full solution steps and checks.
Source, Standards, and Assumptions
Calculation basis, constants, assumptions, and limitations.
Source/standard information updates after a valid calculation.
- Assumptions will appear after a valid calculation.
On this page
Calculator Guide
How to Use the Planck’s Law Calculator
The Planck’s Law Calculator above estimates blackbody spectral radiance from temperature and wavelength or frequency. It can also help check peak wavelength using Wien’s Law, total blackbody exitance using the Stefan-Boltzmann Law, and inverse temperature from wavelength-based radiance.
Use it when you need a quick blackbody radiation calculation for physics, optics, remote sensing, thermal imaging, heat transfer, or astronomy. The most common workflow is entering temperature \(T\), wavelength \(\lambda\), and emissivity \(\varepsilon\) to calculate wavelength-based spectral radiance \(B_{\lambda}\).
Quick Answer
Planck’s Law calculates the spectral radiance emitted by an ideal blackbody at a specific temperature and wavelength or frequency. In wavelength form, the result \(B_{\lambda}\) tells you how much radiant power is emitted per area, per solid angle, and per wavelength interval.
When not to rely on the simplified calculator alone
Do not use a simple blackbody result as a final calibrated radiometry result when emissivity varies with wavelength, atmospheric absorption matters, the source is not close to thermal equilibrium, or detector response must be modeled. For instrument work, use measured calibration data and a full radiometric model.
Inputs and Outputs Used by the Calculator
The calculator uses the selected solve mode to decide which inputs are required. Most users enter temperature, wavelength, and emissivity to calculate \(B_{\lambda}\), but frequency, inverse temperature, peak wavelength, and total exitance checks are also useful.
| Type | Value | What It Means | Common Unit |
|---|---|---|---|
| Input | Temperature, \(T\) | Absolute blackbody temperature. The formula is evaluated in Kelvin. | K, °C, °F |
| Input | Wavelength, \(\lambda\) | Wavelength where spectral radiance is calculated. | m, µm, nm |
| Input | Frequency, \(\nu\) | Frequency where frequency-based spectral radiance is calculated. | Hz, GHz, THz |
| Input | Emissivity, \(\varepsilon\) | Graybody multiplier. Use 1 for an ideal blackbody. | dimensionless |
| Input | Spectral Radiance, \(B_{\lambda}\) | Known wavelength-based radiance used when solving for temperature. | W/(m²·sr·m), W/(m²·sr·µm) |
| Output | \(B_{\lambda}\) | Spectral radiance per wavelength interval. | W/(m²·sr·m), W/(m²·sr·µm) |
| Output | \(B_{\nu}\) | Spectral radiance per frequency interval. | W/(m²·sr·Hz) |
| Output | \(\lambda_{max}\) | Peak wavelength from Wien’s displacement law. | m, µm, nm |
| Output | \(M\) | Total radiant exitance integrated over all wavelengths. | W/m², kW/m² |
Planck’s Law Formula
Planck’s Law can be written in wavelength form or frequency form. The calculator uses the wavelength form for \(B_{\lambda}\) calculations and the frequency form for \(B_{\nu}\) calculations.
Wavelength Form
This form returns spectral radiance per wavelength interval. It is the most common form when users enter wavelength in micrometers or nanometers.
Frequency Form
This form returns spectral radiance per frequency interval. \(B_{\nu}\) and \(B_{\lambda}\) are related, but they are not numerically interchangeable by simply replacing \(\nu\) with \(c/\lambda\).
Peak Wavelength from Wien’s Law
Wien’s Law gives the wavelength where the blackbody curve reaches its wavelength-based peak.
Total Exitance from Stefan-Boltzmann Law
This gives total emitted radiant power per unit surface area over all wavelengths.
Important unit insight
A result in W/(m²·sr·m) is not the same numeric value as the same result in W/(m²·sr·µm). Because \(1\,\mu m=10^{-6}\,m\), per-meter and per-micrometer spectral radiance values differ by a factor of \(10^6\).
What the Variables Mean
Every Planck’s Law input must be interpreted with the correct unit and spectral interval. A temperature mistake or wavelength-unit mistake can change the result by orders of magnitude.
| Symbol | Meaning | How to Enter or Use It |
|---|---|---|
| \(B_{\lambda}\) | Spectral radiance per wavelength interval. | Use W/(m²·sr·m) internally or W/(m²·sr·µm) for practical optics output. |
| \(B_{\nu}\) | Spectral radiance per frequency interval. | Use W/(m²·sr·Hz). |
| \(T\) | Absolute temperature. | Enter K directly or let the calculator convert °C or °F to K. |
| \(\lambda\) | Wavelength. | Enter as m, mm, µm, nm, or Å. Must be positive. |
| \(\nu\) | Frequency. | Enter as Hz, GHz, or THz. Must be positive. |
| \(\varepsilon\) | Emissivity. | Use 1 for an ideal blackbody. Use 0 to 1 for a simplified graybody estimate. |
| \(h\) | Planck constant. | \(6.62607015\times10^{-34}\,J\cdot s\) |
| \(c\) | Speed of light in vacuum. | \(299792458\,m/s\) |
| \(k_B\) | Boltzmann constant. | \(1.380649\times10^{-23}\,J/K\) |
How to Use the Calculator
Start by choosing the solve mode that matches what you know. For most blackbody problems, use spectral radiance from wavelength.
Select the solve mode
Choose whether you want \(B_{\lambda}\), \(B_{\nu}\), inverse temperature, peak wavelength, or total blackbody exitance.
Enter temperature correctly
Planck’s Law requires absolute temperature. If you use °C or °F, the calculator converts the value to Kelvin internally.
Choose wavelength or frequency units
Use µm for infrared and optical work, nm for visible or ultraviolet work, and Hz or THz for frequency-based calculations.
Review the graph and quick checks
Compare the selected wavelength with the peak wavelength. If the selected wavelength is far from the peak, the radiance may be very small.
How to Interpret the Result
A Planck’s Law result is a spectral density, not the total emitted power. It tells you how much radiance occurs at a specific wavelength or frequency interval.
| Result Pattern | What It May Mean | What to Check Next |
|---|---|---|
| Selected wavelength near \(\lambda_{max}\) | The radiance is near the wavelength-based peak for that temperature. | Check whether the peak falls in UV, visible, infrared, or microwave range. |
| Very small radiance | The wavelength may be far from the blackbody peak or the temperature may be too low. | Check the exponent term \(hc/(\lambda k_B T)\). |
| Very large radiance | The wavelength is near the peak at high temperature or output units are per micrometer. | Verify whether output is W/(m²·sr·m) or W/(m²·sr·µm). |
| Peak in visible light | The source is likely thousands of Kelvin, such as a solar-like or incandescent source. | Compare \(T\) with known source temperatures. |
| Peak in long-wave infrared | The source is closer to room temperature or human-body temperature. | Use µm units and compare with thermal imaging bands. |
What to do with the result
Use the result to compare spectral output at different temperatures, estimate sensor-band behavior, or check whether a wavelength is near the blackbody peak. Do not treat one spectral radiance value as total radiated power unless you integrate over wavelength and solid angle.
What changes the result most?
Temperature and wavelength dominate the result. Temperature affects both the height and location of the blackbody curve, while wavelength appears in a \(\lambda^5\) term and in the exponential term. Small wavelength-unit mistakes can therefore create huge result errors.
Quick sanity check
For a source near \(5778\,K\), Wien’s Law gives \(\lambda_{max}\approx0.502\,\mu m\), which is in the visible range. For a source near \(300\,K\), the peak should be near \(9.7\,\mu m\), in the long-wave infrared range. If your result points to a wildly different region, check the temperature and wavelength units.
Input Quality Checklist
Before relying on the calculator result, verify the input values and units. Most Planck’s Law errors come from unit mismatches rather than the formula itself.
Temperature
Confirm the value is physically above absolute zero and converted to Kelvin before calculation.
Wavelength Scale
Check whether your wavelength is in meters, micrometers, nanometers, or angstroms.
Radiance Interval
Do not mix \(B_{\lambda}\) in W/(m²·sr·µm) with \(B_{\nu}\) in W/(m²·sr·Hz).
Emissivity
Use \(\varepsilon=1\) for an ideal blackbody and a realistic value below 1 for graybody estimates.
Worked Example: Solar Spectral Radiance at 0.5 µm
This example uses a solar-like blackbody temperature and a visible wavelength. It matches the most common use case for checking how Planck’s Law predicts the blackbody spectrum.
Use the Wavelength Form
Substitute Values
Final Answer
Result Check
This is reasonable because Wien’s Law gives \(\lambda_{max}\approx0.502\,\mu m\) for \(5778\,K\), which is very close to the selected \(0.5\,\mu m\) wavelength.
Blackbody Radiation Curve Diagram
The blackbody curve shows how spectral radiance is distributed across wavelength. As temperature increases, the curve becomes taller and shifts toward shorter wavelengths.
How to read the curve
The peak marker shows \(\lambda_{max}\), while the selected wavelength marker shows where the calculator evaluates \(B_{\lambda}\). If the selected marker is far from the peak, the result can be much lower than the maximum spectral radiance for that temperature.
Reference Values for Blackbody Radiation
The values below are useful sanity checks. They are approximate and assume ideal blackbody behavior.
| Object or Source | Approximate Temperature | Peak Wavelength | Main Region |
|---|---|---|---|
| Cosmic microwave background | 2.725 K | 1064 µm | Microwave |
| Earth surface | 288 K | 10.1 µm | Long-wave infrared |
| Human body | 310 K | 9.35 µm | Long-wave infrared |
| Incandescent filament | 2700 K | 1.07 µm | Near infrared |
| Sun-like surface | 5778 K | 0.502 µm | Visible light |
| Hot blue-white star | 10000 K | 0.290 µm | Ultraviolet |
Practical Ranges and Engineering Judgment
Planck’s Law is exact for an ideal blackbody, but real engineering use depends on emissivity, optical path, detector bandwidth, surface finish, and calibration.
Thermal Imaging
Room-temperature objects peak near \(10\,\mu m\), which is why many thermal cameras operate in long-wave infrared bands.
Solar and Visible Light
A source near \(5800\,K\) peaks near \(0.5\,\mu m\), so visible wavelengths are a natural check for solar-like radiation.
Graybody Surfaces
Emissivity below 1 reduces the radiance estimate, but real emissivity may vary with wavelength and viewing angle.
When a correct result is still incomplete
A single Planck’s Law value does not account for finite sensor bandwidth, atmospheric absorption, reflected radiation, optical transmission, or detector response. Use band integration and calibration data for final radiometric measurements.
Planck’s Law Units and Conversions
Unit conversion is the most common source of confusion. The wavelength form and frequency form use different spectral intervals, and per-meter values do not match per-micrometer values numerically.
| Quantity | Common Units | Conversion Reminder |
|---|---|---|
| Temperature | K, °C, °F | \(T_K=T_C+273.15\) |
| Wavelength | m, mm, µm, nm, Å | \(1\,\mu m=10^{-6}\,m\), \(1\,nm=10^{-9}\,m\) |
| Frequency | Hz, GHz, THz | \(1\,THz=10^{12}\,Hz\) |
| Wavelength radiance | W/(m²·sr·m), W/(m²·sr·µm) | \(1\,W/(m^2\cdot sr\cdot m)=10^{-6}\,W/(m^2\cdot sr\cdot \mu m)\) |
| Frequency radiance | W/(m²·sr·Hz) | Do not directly compare this number to \(B_{\lambda}\) without interval conversion. |
Planck’s Law vs. Wien’s Law vs. Stefan-Boltzmann Law
These three radiation laws answer different questions. Planck’s Law gives the full spectral distribution, Wien’s Law gives the peak wavelength, and Stefan-Boltzmann Law gives total emitted power per surface area.
| Method | What It Calculates | Best Use | Main Limitation |
|---|---|---|---|
| Planck’s Law | Spectral radiance at a specific wavelength or frequency. | Optics, sensors, spectral analysis, and blackbody curve checks. | One point on the spectrum unless integrated over a band. |
| Wien’s Law | Peak wavelength. | Quickly identifying whether emission peaks in UV, visible, IR, or microwave. | Does not give radiance magnitude. |
| Stefan-Boltzmann Law | Total radiant exitance over all wavelengths. | Heat transfer and total emission estimates. | Does not show spectral distribution. |
| Band integration | Radiance over a finite wavelength band. | Detector, camera, and remote-sensing applications. | Requires numerical integration and instrument bandpass data. |
Common Mistakes When Using Planck’s Law
Planck’s Law is sensitive to units and spectral intervals. A small input mistake can create a result that looks scientific but is physically misleading.
Common Mistakes
- Using Celsius directly in the formula instead of Kelvin.
- Entering nanometers while the calculator is set to micrometers.
- Comparing \(B_{\lambda}\) and \(B_{\nu}\) as if they are the same type of output.
- Forgetting the \(10^6\) factor between per meter and per micrometer units.
- Assuming emissivity is always constant across wavelength.
- Treating a single spectral radiance value as total emitted power.
Better Practice
- Convert every temperature to Kelvin before evaluating the formula.
- Use µm for most infrared work and nm for many visible-light calculations.
- Keep \(B_{\lambda}\) and \(B_{\nu}\) separate unless you intentionally convert intervals.
- Check whether the output is per meter or per micrometer.
- Use measured emissivity when accuracy matters.
- Use Wien’s Law and Stefan-Boltzmann Law as sanity checks.
Troubleshooting Unexpected Results
If your result looks wrong, check the units, solve mode, and spectral interval first. Those issues explain most suspicious Planck’s Law results.
| Problem | Likely Cause | Fix |
|---|---|---|
| Radiance is nearly zero | The wavelength is far into the short-wavelength tail or the temperature is too low. | Check the exponent term and compare the selected wavelength to \(\lambda_{max}\). |
| Result differs by 1,000,000 | Output changed between per meter and per micrometer. | Check whether the unit is W/(m²·sr·m) or W/(m²·sr·µm). |
| Peak wavelength seems wrong | Temperature was entered in °C or °F but interpreted as K. | Convert to Kelvin and rerun the calculation. |
| \(B_{\lambda}\) and \(B_{\nu}\) do not match | They are different spectral densities. | Do not compare them directly without interval conversion. |
| Inverse temperature seems too high or low | Radiance unit, wavelength unit, or emissivity may be wrong. | Verify \(B_{\lambda}\) is wavelength-based and use a realistic emissivity value. |
Common edge case
At very short wavelengths and low temperatures, the exponential term can become extremely large. Mathematically, this drives spectral radiance close to zero. That does not always mean the calculator is broken; it may simply mean that wavelength is in a region where the body emits very little radiation.
Assumptions, Sources, and Limitations
This calculator is intended for educational analysis and preliminary engineering estimates. It uses blackbody radiation equations with an optional graybody emissivity multiplier.
Blackbody Assumption
The ideal formula assumes a perfect emitter. Real surfaces usually emit less radiation than an ideal blackbody.
Graybody Approximation
Emissivity is treated as constant, even though real emissivity may vary with wavelength, temperature, surface condition, and angle.
No Band Integration
A single spectral radiance point is not the same as radiance over a sensor band or total emitted power.
No Atmospheric or Detector Model
The calculator does not include atmospheric absorption, optical transmission, detector response, calibration, or reflected radiation.
Calculation basis
The formula uses standard Planck blackbody radiation relationships, Wien’s displacement law, and the Stefan-Boltzmann law with fixed physical constants. The constants used are consistent with SI fixed constants published by the NIST Reference on Constants, Units, and Uncertainty.
Glossary of Terms
These terms help explain what the calculator is doing and why the units matter.
Blackbody
An ideal object that absorbs all incoming radiation and emits the maximum possible thermal radiation for its temperature.
Spectral Radiance
Radiant power per projected area, per solid angle, and per wavelength or frequency interval.
Emissivity
A dimensionless factor that compares a real surface’s emission to an ideal blackbody.
Wavelength
The spatial period of electromagnetic radiation, commonly measured in micrometers or nanometers for optical and infrared work.
Frequency
The number of wave cycles per second, measured in hertz.
Peak Wavelength
The wavelength where the wavelength-based blackbody curve reaches its maximum.
Frequently Asked Questions
What does the Planck’s Law Calculator calculate?
The Planck’s Law Calculator calculates blackbody spectral radiance from temperature and wavelength or frequency. It can also help estimate temperature from wavelength-based radiance, peak wavelength from Wien’s Law, and total exitance from the Stefan-Boltzmann Law.
What units does Planck’s Law use?
The wavelength form commonly uses W/(m²·sr·m) or W/(m²·sr·µm). The frequency form commonly uses W/(m²·sr·Hz). Temperature must be converted to Kelvin before the formula is evaluated.
What is the difference between B lambda and B nu?
B lambda is spectral radiance per wavelength interval, while B nu is spectral radiance per frequency interval. They are related, but their numeric values and peak locations are not directly interchangeable because the interval size changes when converting between wavelength and frequency.
Why does my Planck’s Law result change by a factor of one million?
A common reason is switching between per meter and per micrometer spectral radiance units. Since one micrometer is \(10^{-6}\) meters, W/(m²·sr·m) and W/(m²·sr·µm) differ by a factor of one million.
Can Planck’s Law solve for temperature?
Yes, Planck’s Law can be rearranged to estimate blackbody temperature from wavelength-based spectral radiance and wavelength. The result depends strongly on emissivity and assumes the measured radiance behaves like ideal blackbody or graybody radiation.
Is emissivity always equal to 1?
No. Emissivity is 1 for an ideal blackbody. Real surfaces usually have emissivity below 1, and many materials have emissivity that changes with wavelength, temperature, surface finish, and viewing angle.