What is the Stefan-Boltzmann law?

The Stefan-Boltzmann law gives the total power radiated by a body of surface area A at absolute temperature T: P = ε·σ·A·T⁴, where σ = 5.67×10⁻⁸ W/(m²·K⁴) is the Stefan-Boltzmann constant and ε is the emissivity (0–1, with 1 = perfect black body).

Why does a body glow red, then orange, then white as it heats up?

As temperature rises, the peak wavelength of emitted radiation shifts to shorter wavelengths (Wien's displacement law: λ_peak ≈ 2898 μm·K / T). At about 1000 K an iron rod glows dim red. By 3000 K (light bulb filament) it glows yellow-white. The Sun at 5800 K appears white. At 10000 K and above, blue-white.

Emissivity ε is a dimensionless property of a surface (0–1) that compares its radiation to a perfect black body. Polished metal has very low emissivity (0.03–0.1), human skin and oxidised surfaces have ε ≈ 0.95. A perfect black body radiator has ε = 1. Lower ε means less power radiated for the same temperature.

Stefan-Boltzmann Radiation Simulator

P = εσAT⁴ — Glowing body, T⁴ law • Simulate • Explore • Practice • Quiz

Mode

Units

📖 User Guide

P 566 W

T 1500 K

λ_peak 1.93 μm

ε 0.90

Presets

Temperature T K

Emissivity ε

Surface Area A cm²

Ambient T_amb K

P (emitted)

566 W

P_net

566 W

Flux M

5.66e4 W/m²

λ_peak

1.93 μm

Color region

Red glow

T (Celsius)

1227 °C

T (Fahrenheit)

2240 °F

Power vs ambient

566×

📖 Learning panels

Σ Live equations — values substituted

⚖ Temperature scale — P at common T

💡 What-if coach — insights

User Guide — Stefan-Boltzmann Simulator

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 λ_peak 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, λ_peak, 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 T_amb) is P_net = εσA(T⁴ − T_amb⁴). 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

Stefan-Boltzmann radiation simulator showing a glowing spherical body with temperature controls, emissivity slider, and a visible-spectrum colour bar indicating which part of the electromagnetic spectrum the body is radiating in, plus a live readout of the total power P equals epsilon sigma A T to the fourth — Default state. Drag the temperature slider and watch the spectral peak shift toward shorter wavelengths while the total radiated power climbs as T⁴.

Stefan-Boltzmann simulator at hot iron rod preset showing 1500 K orange-glowing sphere with peak wavelength in the near infrared and power output 2584 watts — Hot iron at 1500 K. Visibly orange-red glow; most power still in IR.

Stefan-Boltzmann simulator at 300 K room temperature showing dark sphere with peak wavelength in far infrared and dramatically lower power output — Cool body at 300 K. Same area, fraction of the power, all far-IR.

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: λ_peak = 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.

The T⁴ Scaling — Why a 2× Temperature Means 16× Power

The single most important thing about Stefan-Boltzmann is the fourth-power dependence. Double the absolute temperature, the radiated power increases by 2⁴ = 16. Triple it, 81×. This is dramatic and unintuitive.

A practical example: a kitchen oven at 200 °C (473 K) radiates 2840 W/m² from its walls. The same oven at 250 °C (523 K) radiates 4250 W/m² — 50% more power for a 10% temperature rise. This is why oven-cleaning self-cleaning cycles work: heating to 500 °C (773 K) jumps radiated power to 20,200 W/m², which is enough to incinerate grease deposits but obviously requires a sealed cavity.

How the Sun’s Power Reaches Earth — The Full Calculation

This is the calculation that started the field. Sun surface temperature 5778 K. Solar radius 6.96×10⁸ m. Emissivity essentially 1 (close to a perfect blackbody).

Step	Working	Result
Solar surface flux	P/A = σT⁴ = 5.67×10⁻⁸ × 5778⁴	6.32×10⁷ W/m²
Solar surface area	4πR² = 4π(6.96×10⁸)²	6.09×10¹⁸ m²
Total solar luminosity	P = flux × area	3.85×10²⁶ W
Spread over Earth’s orbit (4πd², d = 1.5×10¹¹ m)	P / (4πd²)	1361 W/m²

1361 W/m² is the “solar constant.” Measured by satellites at the top of Earth’s atmosphere; matches the calculation to within 0.1 %. That number is the budget every solar panel, climate model, and crop-growth model on Earth ultimately works from.

Emissivity — Why a Polished Surface Stays Cool

The ε factor in the equation matters as much as T⁴. Polished metals have ε below 0.1; anodised aluminium around 0.8; oxidised iron 0.85; the human body 0.97. A perfect mirror would have ε = 0 and not radiate at all (it would also reflect everything that hit it).

This is why thermos flasks use silvered inner walls (low emissivity blocks radiation heat transfer), why spacecraft are wrapped in gold foil (reflects sunlight, has low IR emission), and why thermal cameras can detect surface treatments at a glance: a polished area looks “cold” even when it isn’t.

References

Incropera — Fundamentals of Heat and Mass Transfer, 7th ed., Chapter 12 (Radiation).
Stefan, J. (1879) — Über die Beziehung zwischen der Wärmestrahlung und der Temperatur. Sitzungsberichte der Akademie. The original derivation.
NIST — Reference values for emissivity of common materials.

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.

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