= Planck's law
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{title2=$B_\nu(T),\ B_\lambda(T)$}
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= Planck law
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<Planck's law> gives the <spectral radiance> of an ideal <blackbody> in <thermal equilibrium> at absolute <temperature> $T>0$. It predicts how much <power> is carried in each small <frequency> or <wavelength> interval, per projected <area> and per <solid angle>. The whole curve is determined by $T$: no material-specific parameter enters this ideal <Planck function>.
Writing $\nu$ for <frequency> in hertz and $\lambda$ for <wavelength> in metres, the two forms of the <Planck function> are
$$
B_\nu(T)=\frac{2h\nu^3}{c^2}\frac1{e^{h\nu/(k_BT)}-1},
\qquad
B_\lambda(T)=\frac{2hc^2}{\lambda^5}\frac1{e^{hc/(\lambda k_BT)}-1}.
$$
Here $h$ is the <Planck constant>, $k_B$ the <Boltzmann constant>, and $c$ the <speed of light> in vacuum. Their exact values in the <International System of Units> are $h=6.62607015\times10^{-34}\,\mathrm{J\,s}$, $k_B=1.380649\times10^{-23}\,\mathrm{J\,K^{-1}}$, and $c=299792458\,\mathrm{m\,s^{-1}}$. https://www.nist.gov/si-redefinition/meet-constants[NIST, Meet the Constants]{ref}
The units matter: $B_\nu$ is measured in $\mathrm{W\,m^{-2}\,sr^{-1}\,Hz^{-1}}$, whereas $B_\lambda$ is measured in $\mathrm{W\,m^{-2}\,sr^{-1}\,m^{-1}}$. Both are <spectral radiance>, rather than <radiant exitance> or <energy density>. To report $B_\lambda$ per micrometre, multiply its per-metre value by $10^{-6}$; a micrometre-wide interval is $10^{-6}$ metres wide.
For a computed example, at $T=6000\,\mathrm K$ and $\lambda=0.500\,\mu\mathrm m$, the exponent is $hc/(\lambda k_BT)\approx4.79592$. Substitution into <Planck's law> gives $B_\lambda\approx3.17569\times10^{13}\,\mathrm{W\,m^{-2}\,sr^{-1}\,m^{-1}}$, or $3.17569\times10^7\,\mathrm{W\,m^{-2}\,sr^{-1}}$ per micrometre. At the same <wavelength>, the illustrative <temperatures> $3000\,\mathrm K$ and $4500\,\mathrm K$ give $2.60268\times10^5$ and $6.37793\times10^6$ in the latter units. These are ideal predictions for stipulated <temperatures>.
\Image[/planck-spectral-radiance.png]
{title=Ideal <Planck's law> spectra at three <temperatures>}
{description=The vertical axis is <spectral radiance> per micrometre, including the $10^{-6}$ conversion from per metre. Increasing <temperature> raises the curve at every fixed <wavelength> and moves its marked peak toward shorter <wavelengths>. The peaks are approximately $966$, $644$, and $483\,\mathrm{nm}$ for $3000$, $4500$, and $6000\,\mathrm K$, respectively; only $0.1$ to $3\,\mu\mathrm m$ of each spectrum is shown.}
The key <change of variables> is $\nu=c/\lambda$. Corresponding positive intervals carry the same <power>, so $B_\lambda\,d\lambda=B_\nu\,|d\nu|$ and
$$
B_\lambda(T)=B_\nu(T)\big|_{\nu=c/\lambda}\,\frac{c}{\lambda^2}.
$$
Equal-width <wavelength> intervals have unequal widths in <frequency>; the factor $c/\lambda^2$ compensates for this. Merely replacing the horizontal-axis labels of $B_\nu$ by $c/\nu$ does not produce $B_\lambda$. This explains the <frequency and wavelength peaks of Planck's law>.
A short derivation uses the <Boltzmann distribution> for a single electromagnetic mode with thermal <energies> $nh\nu$, $n=0,1,\ldots$. Put $q=e^{-h\nu/(k_BT)}$. The <geometric series> gives $Z=\sum_{n\geq0}q^n=(1-q)^{-1}$ and mean thermal <energy> $h\nu q/(1-q)$. The <electromagnetic mode density in a box>, including both independent <polarization of an electromagnetic wave> states of a <photon>, is $8\pi\nu^2/c^3$ per volume per unit <frequency>. Multiplication gives spectral <energy density> $u_\nu=8\pi h\nu^3/[c^3(e^{h\nu/(k_BT)}-1)]$; <isotropy> gives $B_\nu=cu_\nu/(4\pi)$. This counts thermal excitation <energy>, excluding <zero-point energy>. https://www.feynmanlectures.caltech.edu/I_41.html#Ch41-S3[Feynman Lectures, I.41.3, quantum oscillator and radiation]{ref}
<Planck's law> connects <quantum mechanics> to a measurable radiation curve. Its small-$h\nu/(k_BT)$ limit is the <Rayleigh-Jeans law>, $B_\nu\approx2k_BT\nu^2/c^2$; its large-$h\nu/(k_BT)$ limit is the <Wien approximation>, $B_\nu\approx(2h\nu^3/c^2)e^{-h\nu/(k_BT)}$. The suppression at high <frequency> makes the total <energy density> finite. Integrating the curve gives the <Stefan–Boltzmann law>, while locating its peak gives <Wien's displacement law>. Thus one formula predicts both the spectral shape and the total emission.
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