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The Black Body

A blackbody is the simplest source: it absorbs and reemits radiation with 100% efficiency. The frequency content of blackbody radiation is given by the Planck Function:

${\displaystyle B_{\nu }={\frac {h\nu }{\lambda ^{2}}}{2 \over (e^{\frac {h\nu }{kT}}-1)}}$

${\displaystyle B_{\nu }={\frac {2h\nu ^{3}}{c^{2}(e^{\frac {h\nu }{kT}}-1)}}\neq B_{\lambda }}$

Derivation

The # density of photons having frequency between ${\displaystyle \nu }$ and ${\displaystyle \nu +d\nu }$ has to equal the # density of phase-space cells in that region, multiplied by the occupation # per cell. Thus:

${\displaystyle n_{\nu }d\nu ={\frac {4\pi \nu ^{2}d\nu }{c^{3}}}{\frac {2}{e^{\frac {h\nu }{kT}}-1}}}$

However,

${\displaystyle h\nu {\frac {n_{\nu }c}{4\pi }}=I_{\nu }=B_{\nu }}$

so we have it. In the limit that ${\displaystyle h\nu \gg kT}$:

${\displaystyle B_{\nu }\approx {\frac {2h\nu ^{3}}{c^{2}}}e^{-{\frac {h\nu }{kT}}}}$

If ${\displaystyle h\nu \ll kT}$:

${\displaystyle B_{\nu }\approx {\frac {2kT}{\lambda ^{2}}}}$

Note that this tail peaks at ${\displaystyle \sim {\tfrac {3kT}{h}}}$. Also, ${\displaystyle \nu B_{\nu }=\lambda B_{\lambda }}$