Physics · Ch 10 — Thermal Properties of Matter

Stefan-Boltzmann Law and Boltzmann's Correction

10.14

While Wien's displacement law (Section 10.13) describes where, in the spectrum, a black body's emission

is concentrated, it says nothing about how much total energy the black body is radiating altogether.

That question is answered by Stefan's law.

Stefan's law, established experimentally by the Austrian physicist Josef Stefan and shortly afterward

derived from theoretical thermodynamic principles by Ludwig Boltzmann (so that it is very commonly called

the Stefan-Boltzmann law), states that the total radiant energy emitted, per unit time, per unit

surface area, by a black body is directly proportional to the fourth power of its absolute

temperature:

E = \sigma T^4

where $E$ is the energy radiated per second per unit area (with SI unit $\text{W m}^{-2}$ ), $T$ is the

absolute temperature in kelvin, and $\sigma$ is a universal constant, the Stefan-Boltzmann constant,

$\sigma \approx 5.67\times10^{-8}\ \text{W m}^{-2}\text{K}^{-4}$ .

For a black body with total surface area $A$ , the total power radiated (energy radiated per unit

time, over the whole surface) is therefore

P = \sigma A T^4

and for a real surface, which is not a perfect black body, this is scaled down using its emissivity $e$

(introduced in Section 10.11, and equal to $1$ only for an ideal black body):

P = e\sigma A T^4

Because the law involves the fourth power of the absolute temperature, radiated power is extremely

sensitive to temperature: doubling a body's absolute temperature increases the power it radiates by a

factor of $2^4 = 16$ , and even a comparatively modest fractional rise in $T$ produces a much larger

fractional rise in radiated power.

Stefan's law, exactly as written above, gives the total power a body radiates outward. In any real,

practical situation, however, the body doing the radiating is never actually sitting in surroundings at

absolute zero -- its surroundings are always at some finite temperature $T_0 > 0\ \text{K}$ , and are

therefore themselves continuously radiating energy, some of which the body under study absorbs back from

them. Boltzmann's correction accounts for this two-way exchange by subtracting the power the body

absorbs from the surroundings from the power it radiates outward, giving the net rate at which the

body actually loses radiant energy:

P_{\text{net}} = e\sigma A\left(T^4 - T_0^4\right)

This corrected, net form is the one genuinely needed whenever the aim is to work out how fast a hot body …