Stefan-Boltzmann Radiation Simulator
P = εσAT⁴ — Glowing body, T⁴ law • Simulate • Explore • Practice • Quiz
Σ Live equations — values substituted
⚖ Temperature scale — P at common T
💡 What-if coach — insights
1 Overview
The Stefan-Boltzmann Simulator visualises the famous T⁴ law: every object hotter than absolute zero radiates electromagnetic energy with power P = ε·σ·A·T⁴. Slide the temperature from 200 K to 6000 K and watch the body change colour from invisible (room T) through dim red (~1000 K), bright orange (~1500 K), yellow-white (~3000 K, light bulb filament), and finally to white-blue near the Sun’s 5800 K.
Built for high-school and technical-college physics students learning thermal radiation, with Practice and Quiz modes for self-testing.
2 Getting Started
The simulator opens at T = 1500 K (about hot iron), ε = 0.90, A = 100 cm². Power radiated is P ≈ 258 W — about half a household lightbulb. Try the Sun Surface preset (5800 K) and watch P jump to ~570 kW per the same area — almost a thousand times more, because of the T⁴ dependence.
3 Simulate Mode
The canvas shows a circular body glowing with a colour that matches its temperature (using a smoothed black-body colour mapping). Glow rays emanate from the body, with intensity proportional to P. The peak wavelength λpeak is computed via Wien’s law — in the red at ~1500 K, near-infrared at room T, ultraviolet for stellar temperatures.
Eight readout cards show emitted P, net P (after subtracting ambient back-radiation), flux M = P/A, λpeak, color region, T in °C and °F, and the ratio of emitted to ambient power. Six presets cover human body, hot iron, light-bulb filament, lava, the Sun, and red giants.
4 Explore Mode
Four categories: Basics (black body, temperature radiation, why heated objects glow), Formulas (Stefan-Boltzmann derivation, Wien’s law, emissivity), Applications (incandescent bulbs, infrared thermography, climate balance, stellar classification), Common Errors (Kelvin vs Celsius, T⁴ dependence, surface area vs volume).
5 Practice & Quiz
Practice generates problems — compute P at given T, ε, A; find required T for a target power; identify the dominant colour region. Quiz tests T⁴ intuition, emissivity, and Wien’s law.
6 Tips & Best Practices
- Always use absolute temperature in Kelvin! T(K) = T(°C) + 273.15. Plugging Celsius directly gives a wildly wrong answer.
- P scales with T4 — doubling T multiplies P by 16×. Tripling T multiplies P by 81×.
- Net radiation (when in surroundings at Tamb) is Pnet = εσA(T4 − Tamb4). At room temperature your body radiates and absorbs near-equal power.
- Polished metal has very low ε (0.03 for silver). Deliberately poor radiators are used to limit heat loss in vacuum flasks and satellites.
- Pair this with the Heat Transfer Modes simulator to compare conduction, convection, and radiation.
Understanding Stefan-Boltzmann Radiation
The Stefan-Boltzmann law is the master equation of thermal radiation: every body at absolute temperature T radiates total power P = ε·σ·A·T4, where σ = 5.67×10−8 W/(m²·K4) is the Stefan-Boltzmann constant, A is the surface area, and ε is emissivity (0–1, 1 = perfect black body).
Power Radiated at Different Temperatures
| Object | T (K) | T (°C) | Flux (W/m²) at ε = 1 | Color |
|---|---|---|---|---|
| Cosmic microwave background | 2.7 | −270 | 3.0×10−6 | Invisible |
| Frozen ice | 270 | −3 | 301 | Far IR |
| Human body | 310 | 37 | 523 | Far IR |
| Hot iron (dim red) | 1000 | 727 | 56 700 | Dim red |
| Lava | 1500 | 1227 | 287 000 | Bright orange |
| Light bulb filament | 3000 | 2727 | 4.59 MW/m² | Yellow-white |
| Sun surface | 5800 | 5527 | 64.2 MW/m² | White |
| Star (Sirius A) | 10 000 | 9727 | 567 MW/m² | Blue-white |
The T&sup4; Power Law
The fourth-power dependence is what makes radiation so dramatic. Doubling temperature gives 16× more power. The Sun at 5800 K radiates ~570 kW/m²; at 11 600 K it would radiate 9 MW/m². This is also why incandescent bulbs are so inefficient: only ~5% of the radiation falls in the visible band — the rest is wasted as infrared heat.
Wien’s Displacement Law
While total power follows T⁴, the peak wavelength of radiation shifts inversely with T: λpeak = 2898 μm·K / T. Cool objects radiate in the infrared (invisible). At ~700 K things start glowing dim red. At ~3000 K (incandescent bulbs) we get yellow-white. The Sun at 5800 K peaks in the green band (which is why our eyes evolved to see green most sensitively!), with broad spread giving white appearance.
Emissivity and Real Surfaces
A perfect black body (ε = 1) absorbs and emits all radiation. Real surfaces have lower emissivity. Polished aluminium has ε ≈ 0.04 — emits only 4% of black-body power. Anodised aluminium 0.77. Human skin 0.98. Soot 0.95. Low-emissivity ("low-E") windows and reflective paints exploit this to limit radiative heat exchange.
Engineering Applications
Incandescent bulbs use tungsten at 3000 K because higher T gives higher visible-light fraction. Infrared thermometers measure T from radiated power and assumed ε. Climate science balances incoming solar radiation against Earth’s outgoing infrared (P_in = P_out). Spacecraft thermal control uses high-ε radiator panels to dump heat into the cold of space, since vacuum eliminates conduction and convection.
Who Uses This Simulator?
This Stefan-Boltzmann simulator is used by high-school physics students learning thermal radiation, technical-college trainees in heat transfer, mechanical and aerospace engineers sizing thermal radiators, climate-science students modelling planetary energy balance, and astronomy students applying the formula to stars. Practice and Quiz modes ensure students master both the formula and the dramatic T⁴ intuition.
Explore Related Simulators
Explore our Heat Transfer Modes, Thermal Conductivity, Specific Heat Capacity, Phase Change & Latent Heat, and Heat Exchanger simulators.