Web Simulation 

  

 

 

 

Blackbody Radiation & Stefan-Boltzmann Law 

This tutorial visualizes blackbody radiation: an ideal emitter whose spectrum depends only on temperature. You can see how total power radiated grows as P ∝ T4 (Stefan-Boltzmann), how the peak wavelength shifts with λmax ∝ 1/T (Wien's law), and how the object's apparent color follows the Planckian locus from dull red to blue-white.

 

Sections

Mathematical foundation

1. Stefan-Boltzmann law

Total power radiated grows with the fourth power of temperature:

P = σ A T4      σ ≈ 5.67×10−8 W/(m²·K4)

Doubling the temperature T increases power by a factor of 24 = 16.

2. Planck's law

Spectral radiance (power per unit area, solid angle, and wavelength) sets the shape of the spectrum:

L(λ,T) = (2hc²/λ⁵) / (ehc/(λkT) − 1)

The graph plots this curve; the area under it is related to total power.

3. Wien's displacement law

The peak wavelength shifts inversely with temperature:

λmax = b / T      b ≈ 2.898×10−3 m·K

Hotter objects peak at shorter (bluer) wavelengths. The dashed line on the graph shows λmax.

Worked example — the Sun: at T ≈ 5800 K, Wien gives λmax = 2.898×10−3/5800 ≈ 500 nm — green-ish, in the middle of the visible band, which is why sunlight looks white. A 300 K human body peaks near 9.7 µm, deep in the infrared.

4. Color (Planckian locus)

The color of the "emitter" circle approximates the chromaticity of a blackbody at that temperature: red → orange → yellow → white → blue-white as T increases.

Simulation

The interactive simulator is below. Use the controls to explore the concepts described above.

Stefan-Boltzmann Lab

3000 K

Total radiance P = σ T4:

Peak wavelength λmax:

Doubling T → 16× power.

Tutorial: hue visible at all T

Emitter (Planckian color)

Spectral radiance vs wavelength. Shaded band: visible (380–700 nm). Dashed line: Wien λmax. Overlay: auto-X (max X where Y < 5% of peak), comp (compare curves), Lin/Log (X-axis scale).

 

Usage

Drag the Temperature slider from 300 K to 10,000 K:

  • Emitter circle: Color and glow follow the Planckian locus (red hot → white → blue-white). Glow intensity scales with T4. Toggle the eye icon for Real vs Tutorial mode.
  • Graph: Spectral radiance vs wavelength. Default X range is 100 nm–3000 nm. Use auto-X to set the max wavelength to where the curve drops below 5% of peak (linear Y); X tick labels adjust automatically. Lin/Log switches the X-axis scale. Enable Fixed Y-scale to compare power at different T.
  • λmax (dashed line): Wien's law—peak shifts to shorter wavelength as T increases.
  • Visible band: Light tint shows 380–700 nm; you can see which part of the spectrum enters the visible range as T rises.
  • comp: Overlay button to plot multiple curves at T, 0.9T, 0.8T, 0.7T, 0.6T.
  • Show Classical Prediction: When checked, a red dashed line shows the Rayleigh-Jeans (classical) curve. It matches Planck in the infrared but shoots toward infinity at short wavelengths—the Ultraviolet Catastrophe that led to quantum mechanics.

Lab ideas

1. Stefan-Boltzmann: Note the "Total radiance" value. Double the temperature (e.g. 3000 K → 6000 K) and confirm the power increases by about 16×.

2. Wien's law: At 3000 K the peak is in the infrared (~966 nm). At 6000 K it moves to ~483 nm (visible). At 10,000 K it is in the UV (~290 nm).

3. Color: Compare 800 K (dim red), 3000 K (orange-white), 6000 K (sun-like white), 10,000 K (blue-white).

Try this: tick Show Classical Prediction. The Rayleigh–Jeans curve hugs Planck’s in the infrared but diverges to infinity at short wavelengths — the Ultraviolet Catastrophe. Planck’s quantization of energy was the fix, and the birth of quantum mechanics.

Limitations

  • Ideal blackbody: a perfect emitter/absorber with emissivity ε = 1. Real surfaces have ε < 1 and wavelength-dependent emissivity, so they radiate less and with a modified spectrum.
  • Thermal equilibrium only: the spectrum assumes a single well-defined temperature. Non-thermal sources (lasers, fluorescent lamps, LEDs) do not follow Planck’s law.
  • Approximate color: the emitter color approximates the Planckian locus mapped to sRGB; it is illustrative, not a calibrated chromaticity.
  • Display scaling: radiance axes and glow are rescaled for visibility, so on-screen heights are relative, not absolute W/m².