Blackbody Peak Frequency (Wien)
Calculate the peak frequency of blackbody radiation with Wien law (f_max ≈ 5.879×10¹⁰·T) from the temperature in kelvin.
f_máx (Hz)
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Black-body peak frequency: f_max = 2.82·k·T/h
Take Planck's 1900 distribution and ask where a black body pours out the most spectral power per unit frequency. The answer is f_max = α·k·T/h. Here α ≈ 2.821 comes from solving a transcendental equation, k is Boltzmann's constant, and h is Planck's constant. Work it through and you get f_max ≈ 5.879·10¹⁰ Hz/K · T. The cosmic microwave background (CMB) sits at T = 2.725 K and peaks near 160 GHz, in the microwave band. That is precisely where the COBE, WMAP and Planck satellites took their measurements. The Sun, at 5778 K, peaks around 340 THz in the near-infrared.
Applications
Cosmology leans on it heavily: the Planck mission mapped the CMB's 160 GHz peak to pin down cosmological parameters. It also shows up in Hawking radiation from black holes, where temperature falls as mass rises, in reading stellar surface temperatures off spectroscopy, and in designing millimeter-wave detectors for radio astronomy.
FAQ
Why is the constant different from Wien's law in wavelength? The Planck distribution takes a different shape depending on whether you express it per unit wavelength or per unit frequency. So the two peaks land at separate points of the spectrum, and they do not satisfy f_max = c/λ_max.
Where does the factor 2.821 come from? It's the numerical root of the equation (3 − x)·eˣ = 3, the point that maximizes the Planck frequency distribution.
Does the formula work for any opaque object? Roughly, yes. Most thermal sources behave like gray bodies (ε ≈ constant), so the peak position comes out about right. What changes is total intensity, which scales with emissivity.
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