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Stefan-Boltzmann Law: How Bodies Radiate Energy Through Their Temperature

Important Formula

Formula: Stefan-Boltzmann Law

$$\class{green}{P} ~=~ \sigma \, A \, {\class{red}{T}}^4$$ $$\class{green}{P} ~=~ \sigma \, A \, {\class{red}{T}}^4$$ $$A ~=~ \frac{ \class{green}{P} }{ \sigma \, \class{red}{T}^4 }$$ $$\class{red}{T} ~=~ \sqrt[4]{ \frac{ \class{green}{P} }{ \sigma \, A } }$$ $$\sigma ~=~ \frac{ \class{green}{P} }{ A \, {\class{red}{T}}^4 }$$

Rearrange formula

What do the formula symbols mean?

Radiant power

$$ \class{green}{P} $$ Unit $$ \mathrm{W} = \frac{\mathrm J}{\mathrm s} $$

Radiant power [J/s] is the energy emitted by a black body per time.

Surface area

$$ A $$ Unit $$ \mathrm{m}^2 $$

Surface area of a black body (for example of the sun).

Temperature

$$ \class{red}{T} $$ Unit $$ \mathrm{K} $$

Surface temperature of the black body.

Stefan-Boltzmann constant

$$ \sigma $$

The Stefan-Boltzmann constant has the value $ \sigma = 5.67 \cdot 10^{-8} \, \frac{\mathrm J}{\mathrm{m}^2 \, \mathrm{K}^4 \, \mathrm{s}} $.

Each body, which has a temperature (thus all), radiates energy.

How much energy does our sun radiate and how much of it reaches us on earth?
How much energy does the Earth radiate into space?
What is the surface temperature of our sun?
What energy does your body radiate?

Such and similar questions can be answered by the Stefan Boltzmann law. This law describes the relation between the temperature $ \class{red}{T} $ of a body and its radiant power $ \class{green}{P} $. Also the surface $ A $ of the body influences the radiated power:

$$ \begin{align} \class{green}{P} ~=~ \sigma \, A \, {\class{red}{T}}^4 \end{align} $$

Emitted Radiant Power of a Black Body

Here $ \sigma $ is the Stefan Boltzmann constant with the value $ \sigma = 5.67 \cdot 10^{-8} \, \frac{\mathrm J}{\mathrm{m}^2 \, \mathrm{K}^4 \, \mathrm{s}} $. This is a constant that has an exact value that depends only on other physical constants:

$$ \begin{align} \sigma ~&=~ \frac{ 2 \pi^5 \, {k_{\text B}}^4 }{15 h^3 \, c^2} \\\\
~&=~ 5.67 \cdot 10^{-8} \, \frac{\mathrm J}{\mathrm{m}^2 \, \mathrm{K}^4 \, \mathrm{s}} \end{align} $$

What is the radiant power $ \class{green}{P} $? It indicates how much energy per second the surface of the body radiates. The unit of radiant power is W [watts] or J/s [joules per second].

Furthermore, the Stefan-Boltzmann law applies only to so-called black bodies or bodies that are almost a black body. However, the name black has nothing to do with the color of the body. For example, the sun is an approximately black body, although, as you know, it is not black. A black body absorbs all radiation that hits it. It does not reflect any radiation and does not let any radiation pass.

From the Stefan-Boltzmann law you can see how strongly the radiated energy depends on the temperature. Doubling the surface temperature of the sun would result in 16 times as much solar energy reaching the earth.

Example: Temperature of the sun

From the energy of the sun arriving at us, we can infer the total emitted energy of the sun. The sun emits a power of $ 3.8 \cdot 10^{26} \, \mathrm{W} $ (energy per second). The radius of the sun is $ 6.96 \cdot 10^8 \, \mathrm{m} $.

The sun is a sphere (yes, it is a sphere...). Therefore, we can use the formula for the surface area of a sphere to determine the surface area $ A $ of the sun: $ A = 4\pi \, r^2 $. Rearrange the Stefan-Boltzmann law 1 with respect to the temperature:

$$ \begin{align} \class{red}{T} ~&=~ \sqrt[4]{ \frac{ \class{green}{P} }{ \sigma \, 4 \pi \, r^2 } } \\\\

~&=~ \sqrt[4]{ \frac{ \class{green}{ 3.8 \cdot 10^{26} \, \mathrm{W} } }{ 5.67 \cdot 10^{-8} \, \frac{\mathrm J}{\mathrm{m}^2 \, \mathrm{K}^4 \, \mathrm{s}} ~\cdot~ 4 \pi ~\cdot~ (6.96 \cdot 10^8 \, \mathrm{m})^2 } } \\\\

~&=~ 5760 \, \mathrm{K}

\end{align} $$

The surface temperature of the sun is 5760 Kelvin or 5486 degrees Celsius.