Planck’s Law (Blackbody) Calculator

Planck’s Law: The Complete Guide to Blackbody Spectral Radiance

Planck’s Law describes how an ideal blackbody emits electromagnetic radiation at a given temperature. It tells you the spectral radiance—how much power is emitted per unit area, per unit solid angle, per unit wavelength (or frequency). If you just used the calculator above, this section explains the equations behind the numbers, how to interpret the output, and how to apply the law in astronomy, thermal imaging, and materials science.

The same law can be expressed per frequency instead of per wavelength:

These two forms are consistent but not numerically equal at corresponding \(\lambda\) and \(\nu=c/\lambda\) because they use different spectral variables. Your choice depends on whether your sensor or dataset is wavelength-based (nm/µm) or frequency-based (Hz/THz).

Variables, Units, and Physical Constants in Planck’s Law

The calculator computes in SI units internally and converts to popular display units like W·sr\(^{-1}\)·m\(^{-2}\)·nm\(^{-1}\) or W·sr\(^{-1}\)·m\(^{-2}\)·µm\(^{-1}\). To switch spectral domains, remember \(\mathrm{d}\nu = -c/\lambda^{2}\,\mathrm{d}\lambda\), which is why \(B_{\lambda}\) and \(B_{\nu}\) look different.

Related Blackbody Laws: Wien’s Displacement and Stefan–Boltzmann

Wien’s displacement law gives the peak wavelength of the Planck curve for a temperature \(T\):

This is useful to estimate which part of the spectrum (UV, visible, infrared) dominates the emission. For example, the Sun’s photosphere around \(T\approx 5778\,\text{K}\) peaks near \(\lambda_{\max}\approx 500\,\text{nm}\) (green).

Stefan–Boltzmann law integrates Planck’s Law over all wavelengths and directions to yield the total emitted power per unit area:

While Planck’s Law is spectral (per wavelength/frequency), Stefan–Boltzmann is broadband (total). Use them together: Planck for shape, Stefan–Boltzmann for area under the curve.

Worked Examples Using Planck’s Law

Example 1 — Spectral Radiance at a Visible Wavelength

Compute \(B_{\lambda}\) at \(\lambda=550\,\text{nm}\) for a blackbody at \(T=5778\,\text{K}\) (approximate solar photosphere). Convert \(\lambda\) to meters: \(550\,\text{nm}=5.5\times 10^{-7}\,\text{m}\).

First compute the dimensionless exponent: \(\displaystyle x=\frac{h c}{\lambda k T}\). With \(h=6.62607015\times10^{-34}\,\text{J·s}\), \(c=2.99792458\times10^{8}\,\text{m·s}^{-1}\), \(k=1.380649\times10^{-23}\,\text{J·K}^{-1}\), \(\lambda=5.5\times10^{-7}\,\text{m}\), and \(T=5778\,\text{K}\), we get a finite \(x\) that produces a moderate exponential term. After evaluating the numerator \(2hc^{2}\) and the \(\lambda^{-5}\) factor, then dividing by \(e^{x}-1\), you obtain a spectral radiance on the order of \(10^{13}\)–\(10^{14}\ \text{W·sr}^{-1}\text{·m}^{-3}\). If you prefer per-nanometer units, multiply by \(10^{-9}\) to get W·sr\(^{-1}\)·m\(^{-2}\)·nm\(^{-1}\).

What does it mean physically? This value tells you how bright the blackbody appears around 550 nm, per steradian, per square meter of surface, per nanometer of bandwidth.

Example 2 — Frequency Form at 100 THz

Now compute \(B_{\nu}\) at \(\nu=100\,\text{THz} = 1\times10^{14}\,\text{Hz}\) for \(T=300\,\text{K}\) (room temperature, mid-IR).

The exponent \(\frac{h\nu}{kT}\) is modest in the thermal IR at room temperature, so the denominator \(e^{h\nu/(kT)}-1\) does not explode. Plug the constants and simplify to get a radiance on the order of \(10^{-13}\)–\(10^{-12}\ \text{W·sr}^{-1}\text{·m}^{-2}\text{·Hz}^{-1}\). If you need an equivalent wavelength, use \(\lambda=c/\nu\) (here, \(\lambda\approx 3\,\mu\text{m}\)).

Example 3 — Peak Wavelength with Wien’s Law

For \(T=900\,\text{K}\) (heated metal), the peak wavelength is:

This explains why very hot objects glow red, then white: as temperature rises, \(\lambda_{\max}\) shifts to shorter wavelengths, bringing more emission into the visible band.

How to Use Planck’s Law in Practice

• Astronomy: Fit a blackbody curve to stellar spectra to estimate effective temperature; convert between \(B_{\lambda}\) and \(B_{\nu}\) to match instrument response.
• Thermal imaging: Model IR camera signals by integrating \(B_{\lambda}\) over sensor bands and applying optics throughput and detector responsivity.
• Material characterization: For real surfaces with emissivity \(\varepsilon(\lambda,T)\le1\), scale: \(L_{\lambda}=\varepsilon(\lambda,T)\,B_{\lambda}\).
• Lighting & LEDs: Compare blackbody loci (CCT) to measured spectra to assess color temperature and deviations.
• Remote sensing: Retrieve land-surface temperature by inverting radiance with atmospheric corrections and known emissivity.

Assumptions and Limitations of Planck’s Law

• Ideal blackbody only: Real objects emit less than a blackbody. Use emissivity: \(0\le\varepsilon(\lambda,T)\le1\).
• No atmospheric effects: Gases and aerosols absorb/emit; field measurements require atmospheric transmittance and path-radiance corrections.
• Angular dependence: Planck’s Law gives radiance; emissivity and reflectance can depend on view angle for real materials.
• Spectral variable matters: \(B_{\lambda}\) and \(B_{\nu}\) are not equal at \(\lambda=c/\nu\); convert carefully using \(|\mathrm{d}\nu/\mathrm{d}\lambda| = c/\lambda^{2}\).
• Instrument bandpass: Sensors integrate over finite spectral widths; compare band-integrated radiance, not a single-point value.
• Numerical stability: At very short \(\lambda\) or high \(\nu\), the exponential can overflow; calculators should use stable forms (e.g., \(\mathrm{expm1}\)).

Planck’s Law — Frequently Asked Questions

What is Planck’s Law in simple terms?

It’s a formula that predicts how much light an idealized hot object emits at each wavelength or frequency. The hotter the object, the more it emits overall and the more its peak shifts toward shorter wavelengths.

What’s the difference between \(B_{\lambda}\) and \(B_{\nu}\)?

They describe the same physics with different spectral variables. \(B_{\lambda}\) is per unit wavelength; \(B_{\nu}\) is per unit frequency. Because \(\mathrm{d}\nu\neq\mathrm{d}\lambda\), their numeric values differ at corresponding \(\lambda\) and \(\nu=c/\lambda\).

Which units should I use?

Use W·sr\(^{-1}\)·m\(^{-2}\)·nm\(^{-1}\) for visible/near-IR work, W·sr\(^{-1}\)·m\(^{-2}\)·µm\(^{-1}\) for thermal IR, and W·sr\(^{-1}\)·m\(^{-2}\)·Hz\(^{-1}\) for frequency-domain data. Internally, the equations need SI meters and hertz.

How does emissivity affect results?

Real surfaces rarely behave as perfect blackbodies. Multiply by emissivity: \(L_{\lambda}=\varepsilon(\lambda,T)\,B_{\lambda}\). Emissivity depends on material, surface roughness, wavelength, and temperature.

Why doesn’t the peak of \(B_{\lambda}\) match the peak of \(B_{\nu}\)?

Because the spectral axes are non-linearly related. The “peak” depends on which axis you plot against. Wien’s law gives the peak for the wavelength form; the frequency form peaks elsewhere.

Can Planck’s Law handle reflected light or transmitted light?

No. Planck’s Law models thermal emission from the object itself. Reflected solar radiation and atmospheric effects must be modeled separately and combined with emissivity and reflectance.

How do I convert between per-meter and per-nanometer radiance?

Multiply \(B_{\lambda}\) (per m) by \(10^{-9}\) to get per-nm, or by \(10^{-6}\) to get per-µm. The calculator above performs these conversions automatically.

Is Rayleigh–Jeans or Wien’s approximation good enough?

Rayleigh–Jeans \(\big(B_{\lambda}\approx 2ckT/\lambda^{4}\big)\) holds at long wavelengths/low frequencies; Wien \(\big(B_{\lambda}\approx (2hc^{2}/\lambda^{5})e^{-hc/(\lambda kT)}\big)\) holds at short wavelengths/high frequencies. For precise work, use the full Planck function.

Practical Tips for Accurate Calculations

• Always use kelvin. Celsius must be converted: \(T[\text{K}] = T[^\circ\text{C}] + 273.15\).
• Be consistent with spectral units. Convert nm ↔ µm ↔ m before evaluating the exponential.
• Use stable numerics. When \(\frac{hc}{\lambda kT}\) is very small, use \(\mathrm{expm1}(x)\) to avoid subtractive cancellation in \(e^{x}-1\).
• Compare band-integrated values. If your detector spans \([\,\lambda_1,\lambda_2\,]\), integrate \(B_{\lambda}\) over that interval and apply optics throughput and detector efficiency.
• Account for emissivity and environment. For real measurements, include \(\varepsilon(\lambda,T)\), background reflections, and atmospheric terms.

Bottom Line: Why Planck’s Law Matters

Planck’s Law is the foundation for interpreting thermal radiation. With it, you can predict where an object emits most strongly, convert between spectral domains, and estimate temperatures from radiance measurements. Paired with Wien’s displacement and Stefan–Boltzmann laws, it becomes a versatile toolkit for astronomy, remote sensing, and thermal engineering. Use the calculator above to explore how temperature shifts the spectrum and how radiance scales across wavelengths or frequencies—then apply emissivity and instrument models for real-world accuracy.