Why normalize the spectrum on the graph?

B_λ changes by many orders of magnitude with temperature. Normalizing to the peak in the displayed λ range keeps the curve on screen while preserving the shape and the marked Wien peak.

Why is ∫πB_λ dλ on the page not exactly equal to σT⁴?

σT⁴ equals the integral of πB_λ over all wavelengths from zero to infinity. The simulator integrates only from 50 nm to 3500 nm, so the ratio is below 100% until almost all power lies in that band.

Is the colored bar what a blackbody “looks like”?

It is a rough perceptual mix: each visible wavelength is weighted by B_λ at your temperature. Human color vision and display gamuts are more complex, but the trend (cooler → redder, hotter → whiter/bluer) is right.

Black Body: Planck Spectrum

B_λ(λ,T); Wien λ_max ∝ 1/T; Stefan–Boltzmann M = σT⁴ and numeric ∫πB_λ dλ.

Live graphs

About this model

This page plots Planck’s spectral radiance B_λ(T, λ) for an ideal black body. Wien’s displacement law locates the wavelength of peak emission (λ_max T ≈ 2.898×10⁻³ m·K). Stefan–Boltzmann’s law states that the total hemispherical exitance is M = σT⁴ with σ ≈ 5.67×10⁻⁸ W·m⁻²·K⁻⁴. The chart normalizes B_λ to unit peak on a fixed wavelength window so the shape is easy to compare across temperatures; a vertical segment marks the theoretical λ_max. A trapezoidal integral of πB_λ over the same window is shown next to σT⁴ so you can see how much of the total power lies inside the visible band’s context window.

Who it's for: Introductory thermal radiation, astrophysics (stellar colors), and engineering heat transfer students.

Key terms

Planck law
Wien displacement
Stefan–Boltzmann law
spectral radiance
black body

How it works

Spectral radiance of a black body: Planck’s law, Wien’s displacement (λ_max ∝ 1/T), and Stefan–Boltzmann total emitted power ∝ T⁴.

Key equations

B_λ = (2hc²/λ⁵) / (e^{hc/(λkT)} − 1)

λ_max T ≈ 2.898×10⁻³ m·K · M = σT⁴

Frequently asked questions

Why normalize the spectrum on the graph?: B_λ changes by many orders of magnitude with temperature. Normalizing to the peak in the displayed λ range keeps the curve on screen while preserving the shape and the marked Wien peak.
Why is ∫πB_λ dλ on the page not exactly equal to σT⁴?: σT⁴ equals the integral of πB_λ over all wavelengths from zero to infinity. The simulator integrates only from 50 nm to 3500 nm, so the ratio is below 100% until almost all power lies in that band.
Is the colored bar what a blackbody “looks like”?: It is a rough perceptual mix: each visible wavelength is weighted by B_λ at your temperature. Human color vision and display gamuts are more complex, but the trend (cooler → redder, hotter → whiter/bluer) is right.

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