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Thermodynamics

Stefan-Boltzmann Law of Radiation

A surface emits thermal radiation according to its absolute temperature, area, and emissivity.

P_{emit} = σ ϵ A T^{4}

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Source & review

Difficulty

Intermediate

Read time

5 min

Prerequisites

Basic algebra, consistent units, and the stated model assumptions

Review statusNeeds review

Source

OpenStax, College Physics 2e, §14.7 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example. The textbook section reference was corrected.

Definition

Understanding the core concept

A surface emits thermal radiation according to its absolute temperature, area, and emissivity. This expression is emitted power, not net heat loss after absorbed environmental radiation is included.

Variables & Units

Understanding each component

Symbol	Meaning	Units
$P_{emit}$	Power (radiated)	$W$
$σ$	Stefan-Boltzmann Constant	$W m^{- 2} K^{- 4}$
$ϵ$	Emissivity	$1$
$A$	Area	$m^{2}$
$T$	Temperature	$K$

Real-World Applications

Where this formula is used in practice

Thermal emission

Estimate gross emitted thermal-radiation power.

Temperature comparisons

Compare emission at different absolute temperatures.

Worked Example

Step-by-step calculation with real numbers

Problem

Find the emitted power of a 1 m² ideal black surface at 300 K using the listed Stefan–Boltzmann constant.

Given

σ = 5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}

ϵ = 1

A = 1 m^{2}

T = 300 K

Solution

P_{emit} = σ ϵ A T^{4}

P_{emit} = (5.67 \times 1 0^{- 8} W m^{- 2} K^{- 4}) \cdot (1) \cdot (1 m^{2}) \cdot (300 K)^{4}

P_{emit} \approx 459.3 W

Final Answer

P_{emit} \approx 459.3 W

Formula Guide

Info

DifficultyIntermediate

Read Time5 min

PrerequisitesBasic algebra, consistent units, and the stated model assumptions

Tools

Source & review

StatusNeeds review

Last reviewedPending

Source

OpenStax, College Physics 2e, §14.7 (Paul Peter Urone and Roger Hinrichs)

OpenFormulaGraph equation reference; FormuLab adaptation details and license are provided below.

Adapted from OpenFormulaGraph, under CC BY 4.0.

FormuLab adapted wording, notation, applicability conditions and topic placement, and expanded the numerical check into a worked example. The textbook section reference was corrected.

Assumptions

• Use absolute temperature in kelvin and an appropriate thermal emissivity between zero and one.
• For net exchange with large uniform surroundings, an additional surroundings-temperature term is required.

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Stefan-Boltzmann Law of Radiation

Thermal emission

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