= Black-body radiation
{title2=$u=a_{\rm rad}T^4,\quad p=u/3$}
{wiki}
= Blackbody radiation
{synonym}
<Black-body radiation> is the <electromagnetic radiation> in <thermal equilibrium> with an ideal <blackbody>. It can be described as a <photon gas>: a cavity whose walls are all at the same <temperature> contains radiation with the universal <Planck's law> spectrum. A small opening in such a cavity approximates a <blackbody> emitter because incoming radiation undergoes repeated absorption inside. The radiation's equilibrium spectrum depends on <temperature>, rather than the wall material. https://openstax.org/books/university-physics-volume-3/pages/6-1-blackbody-radiation[OpenStax, University Physics 3, §6.1, blackbody cavity and emission laws]{ref}
There are three useful quantities to keep distinct: directional <spectral radiance> $B_\nu$ or $B_\lambda$, total <energy density> $u$ in the cavity, and outward <radiant exitance> $j^*$ from its opening. <Planck's law> supplies the first; <integrals> over <frequency> and directions supply the others. For <isotropy>, every direction has the same $B_\nu$, so the full $4\pi$ <solid angle> gives
$$
u(T)=\frac{4\pi}{c}\int_0^\infty B_\nu(T)\,d\nu
=a_{\rm rad}T^4,\qquad
a_{\rm rad}=\frac{8\pi^5k_B^4}{15h^3c^3}
=\frac{\pi^2k_B^4}{15\hbar^3c^3}.
$$
Here $\hbar=h/(2\pi)$ is the reduced <Planck constant>, and the <radiation constant> is $a_{\rm rad}\approx7.565733250\times10^{-16}\,\mathrm{J\,m^{-3}\,K^{-4}}$. The explicit <Boltzmann constant> makes this an SI formula; adopting <natural units> with $k_B=1$ recovers $u=\pi^2T^4/(15\hbar^3c^3)$.
For a short <integral> outline, substitute $x=h\nu/(k_BT)$ and use
$$
\int_0^\infty\frac{x^3}{e^x-1}\,dx
=\sum_{n=1}^\infty\int_0^\infty x^3e^{-nx}\,dx
=6\sum_{n=1}^\infty\frac1{n^4}=\frac{\pi^4}{15}.
$$
The expansion is a <geometric series> with nonnegative terms; repeated <integration by parts> evaluates each <integral>, and $\zeta(4)=\pi^4/90$ is a value of the <Riemann zeta function>. This gives the coefficient above.
The <photon gas> also exerts <radiation pressure>, $p=u/3$. The factor $1/3$ comes from the <angular average> of the squared directional component of <photon> <momentum>, $\langle\cos^2\theta\rangle=1/3$. The <thermodynamic derivation of blackbody radiation pressure> gives the same result. In contrast, outgoing <power> involves one projected-direction factor over an outward hemisphere, yielding <radiant exitance>
$$
j^*=\pi\int_0^\infty B_\nu(T)\,d\nu
=\frac{cu}{4}=\sigma T^4,
\qquad \sigma=\frac{2\pi^5k_B^4}{15h^3c^2}.
$$
This is the <Stefan–Boltzmann law> with unit <emissivity>; $\sigma$ is the <Stefan-Boltzmann constant>. The factor $\pi$ follows from <Lambert's cosine law>, rather than the $4\pi$ full-sphere factor used for <energy density>.
For example, an ideal $6000\,\mathrm K$ cavity has $u\approx0.980519\,\mathrm{J\,m^{-3}}$, <radiation pressure> $p\approx0.326840\,\mathrm{Pa}$, and outward <radiant exitance> $j^*\approx7.34881\times10^7\,\mathrm{W\,m^{-2}}$. A stipulated $3000\,\mathrm K$ cavity has one-sixteenth each value because all three scale as $T^4$. These describe stored <energy>, mechanical <pressure>, and outgoing <power> per <area>, respectively. In a closed equilibrium cavity, equal opposing radiation balances the directional <power> flow even though $u$ and $p$ are nonzero.
The usefulness of <black-body radiation> is this common reference: <Planck's law>, <Wien's displacement law>, and the <Stefan–Boltzmann law> consistently connect spectral measurements, <temperature>, and total emission. The three illustrative curves in <Planck's law> are ideal equilibrium spectra; applying them to a material requires considering its <emissivity> and whether <thermal equilibrium> is a suitable approximation.
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