= Solution
A black body absorbs all incident radiation; a small opening into an equilibrated opaque cavity approximates it because internal reflections strongly suppress escape without absorption. Equilibrium radiation has zero <photon chemical potential>. Counting electromagnetic modes and assigning <mean> occupation $(e^{h\nu/(k_BT)}-1)^{-1}$ gives the spectral <energy> density
$$
\boxed{u_\nu(T)=\frac{8\pi h\nu^3}{c^3}\frac1{e^{h\nu/(k_BT)}-1}}.
$$
At low <frequency> this becomes the <Rayleigh-Jeans law>, $u_\nu\simeq8\pi k_BT\nu^2/c^3$; using that limit at all <frequencies> gives the <ultraviolet catastrophe>. At high <frequency> the exponential suppresses the spectrum. Quantized mode <energy> resolves the classical divergence.
Integration gives $u=a_{\rm rad}T^4$. An isotropic radiation field sends outward flux $cu/4$ through a small aperture, so
$$
\boxed{M=\sigma T^4,\qquad\sigma=\frac{2\pi^5k_B^4}{15h^3c^2}}.
$$
Differentiating the <wavelength> spectrum gives $\lambda_{\max}T\simeq2.90\times10^{-3}\,\mathrm{m\,K}$; the frequency-spectrum maximum is a different transformed peak, not simply $c/\lambda_{\max}$. A real surface has <emissivity> less than one and possibly dependent on <wavelength>. Cavity <temperature> uniformity, aperture geometry, calibrated detector response and background subtraction are necessary when testing the spectrum or inferring <temperature> from it.
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