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Stefan–Boltzmann law

Stefan–Boltzmann law is a physics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Stefan–Boltzmann law rather than just read about it. In short: The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

Stefan–Boltzmann law — main illustration
Stefan–Boltzmann law — illustration

Key takeaways

  • Stefan–Boltzmann law belongs to physics; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Stefan–Boltzmann law to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Stefan–Boltzmann law from memory before moving on to harder problems.

Reference excerpt

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically. For an ideal absorber/emitter or black body, the Stefan–Boltzmann law states that the total energy radiated per unit surface area per unit time (also known as the radiant exitance) is directly proportional to the fourth power of the black body's temperature, T {\displaystyle T} :

M ∘ = σ T 4 . {\displaystyle M^{\circ }=\sigma \,T^{4}.}

The constant of proportionality, σ {\displaystyle \sigma } , is called the Stefan–Boltzmann constant. It has the value

In the general case, the Stefan–Boltzmann law for radiant exitance takes the form:

M = ε M ∘ = ε σ T 4 , {\displaystyle M=\varepsilon \,M^{\circ }=\varepsilon \,\sigma \,T^{4},}

where ε {\displaystyle \varepsilon } is the emissivity of the surface emitting the radiation. The emissivity is generally between zero and one. An emissivity of one corresponds to a black body.

Detailed explanation The radiant exitance (previously called radiant emittance), M {\displaystyle M} , has dimensions of energy flux (energy per unit time per unit area), and the SI units of measure are joules per second per square metre (J⋅s−1⋅m−2), or equivalently, watts per square metre (W⋅m−2). The SI unit for absolute temperature, T, is the kelvin (K). To find the total power, P {\displaystyle P} , radiated from an object, multiply the radiant exitance by the object's surface area, A {\displaystyle A} :

P = A ⋅ M = A ε σ T 4 . {\displaystyle P=A\cdot M=A\,\varepsilon \,\sigma \,T^{4}.}

Matter that does not absorb all incident radiation emits less total energy than a black body. Emissions are reduced by a factor ε {\displaystyle \varepsilon } , where the emissivity, ε {\displaystyle \varepsilon } , is a material property which, for most matter, satisfies 0 ≤ ε ≤ 1 {\displaystyle 0\leq \varepsilon \leq 1} . Emissivity can in general depend on wavelength, direction, and polarization. However, the emissivity which appears in the non-directional form of the Stefan–Boltzmann law is the hemispherical total emissivity, which reflects emissions as totaled over all wavelengths, directions, and polarizations. The form of the Stefan–Boltzmann law that includes emissivity is applicable to all matter, provided that matter is in a state of local thermodynamic equilibrium (LTE) so that its temperature is well-defined. (This is a trivial conclusion, since the emissivity, ε {\displaystyle \varepsilon } , is defined to be the quantity that makes this equation valid. What is non-trivial is the proposition that ε ≤ 1 {\displaystyle \varepsilon \leq 1} , which is a consequence of Kirchhoff's law of thermal radiation.) A so-called grey body is a body for which the spectral emissivity is independent of wavelength, so that the total emissivity, ε {\displaystyle \varepsilon } , is a constant. In the more general (and realistic) case, the spectral emissivity depends on wavelength. The total emissivity, as applicable to the Stefan–Boltzmann law, may be calculated as a weighted average of the spectral emissivity, with the blackbody emission spectrum serving as the weighting function. It follows that if the spectral emissivity depends on wavelength then the total emissivity depends on the temperature, i.e., ε = ε ( T ) {\displaystyle \varepsilon =\varepsilon (T)} . However, if the dependence on wavelength is small, then the dependence on temperature will be small as well. Wavelength- and subwavelength-scale particles, metamaterials, and other nanostructures are not subject to ray-optical limits and may be designed to have an emissivity greater than 1. In national and international standards documents, the symbol M {\displaystyle M} is recommended to denote radiant exitance; a superscript circle (°) indicates a term relative to a black body. (A subscript "e" is added when it is important to distinguish the energetic (radiometric) quantity radiant exitance, M e {\displaystyle M_{\mathrm {e} }} , from the analogous human vision (photometric) quantity, luminous exitance, denoted M v {\displaystyle M_{\mathrm {v} }} .) In common usage, the symbol used for radiant exitance (often called radiant emittance) varies among different texts and in different fields. The Stefan–Boltzmann law may be expressed as a formula for radiance as a function of temperature. Radiance is measured in watts per square metre per steradian (W⋅m−2⋅sr−1). The Stefan–Boltzmann law for the radiance of a black body is:

… excerpt ends here. Continue reading the full article.

Illustrations

Stefan–Boltzmann law: Total emitted energy, j ≡ M ∘ {\displaystyle j\equiv M^{\circ }} , of a black body as a function of its temperature, T {\displaystyle T} . The upper (black) curve depicts the Stefan–Boltzmann law, M ∘ = σ T 4 {\displaystyle M^{\circ }=\sigma \,T^{4}} . The lower (blue) curve is total energy according to the Wien approximation, M W ∘ = M ∘ / ζ ( 4 ) ≈ 0.924 σ T 4 {\displaystyle M_{W}^{\circ }=M^{\circ }/\zeta (4)\approx 0.924\,\sigma T^{4}\!\,}
Total emitted energy, j ≡ M ∘ {\displaystyle j\equiv M^{\circ }} , of a black body as a function of its temperature, T {\displaystyle T} . The upper (black) curve depicts the Stefan–Boltzmann law, M ∘ = σ T 4 {\displaystyle M^{\circ }=\sigma \,T^{4}} . The lower (blue) curve is total energy according to the Wien approximation, M W ∘ = M ∘ / ζ ( 4 ) ≈ 0.924 σ T 4 {\displaystyle M_{W}^{\circ }=M^{\circ }/\zeta (4)\approx 0.924\,\sigma T^{4}\!\,}
Stefan–Boltzmann law: Log–log graphs of peak emission wavelength and radiant exitance vs. black-body temperature. Red arrows show that 5780 K black bodies have 501 nm peak and 63.3 MW/m2 radiant exitance.
Log–log graphs of peak emission wavelength and radiant exitance vs. black-body temperature. Red arrows show that 5780 K black bodies have 501 nm peak and 63.3 MW/m2 radiant exitance.
Stefan–Boltzmann law: Deriving the Stefan–Boltzmann Law using Planck's law.
Deriving the Stefan–Boltzmann Law using Planck's law.

Worked examples

Example 1 — a first encounter with Stefan–Boltzmann law

Start with the simplest possible case. Write down what Stefan–Boltzmann law claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In physics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Stefan–Boltzmann law before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Stefan–Boltzmann law ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Stefan–Boltzmann law

In research
Stefan–Boltzmann law appears in physics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Stefan–Boltzmann law in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Stefan–Boltzmann law is common in secondary-school and first-year university syllabi. It links to neighbouring topics Heat transfer, Laws of thermodynamics, Ludwig Boltzmann, so understanding it makes those chapters shorter.
In everyday life
Look for Stefan–Boltzmann law outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Stefan–Boltzmann law in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Stefan–Boltzmann law means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Stefan–Boltzmann law out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Stefan–Boltzmann law in simple terms?

The Stefan–Boltzmann law, also known as Stefan's law, describes the intensity of the thermal radiation emitted by matter in terms of that matter's temperature. It is named for Josef Stefan, who empirically derived the relationship, and Ludwig Boltzmann who derived the law theoretically.

Why does Stefan–Boltzmann law matter?

Because it connects several physics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Stefan–Boltzmann law?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Stefan–Boltzmann law.

Tags

  • Heat transfer
  • Laws of thermodynamics
  • Ludwig Boltzmann
  • Power laws

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