= Solution
Each electromagnetic <normal mode> is a <quantum harmonic oscillator> with $E_n=(n+1/2)h\nu$. For $x=h\nu/(k_BT)$ its <canonical partition function> and occupation <probabilities> are
$$
Z_\nu=\frac{e^{-x/2}}{1-e^{-x}},\qquad
P_n=(1-e^{-x})e^{-nx},\qquad
\langle n\rangle=\frac1{e^x-1}.
$$
The <geometric distribution> is normalized and gives thermal mode <energy> $h\nu/(e^x-1)$. The zero-point term $h\nu/2$ is not part of the thermally emitted black-body <intensity>. The equilibrium <photon> <chemical potential> is zero because <photon number> is not conserved.
Counting two transverse <polarizations> gives $8\pi\nu^2/c^3$ modes per volume per <frequency> interval. Thus the spectral <energy density> is $u_\nu=8\pi h\nu^3/[c^3(e^x-1)]$. In an isotropic radiation field, <spectral radiance> is $B_\nu=cu_\nu/(4\pi)$. Transforming the spectral measure, $B_\lambda=B_\nu|d\nu/d\lambda|$, gives
$$
\boxed{B_\lambda(T)=\frac{2hc^2}{\lambda^5}\frac1{e^{hc/(\lambda k_BT)}-1}.}
$$
Here <intensity> <means> <radiance> per <wavelength> and solid angle; the hemispheric spectral exitance of a black surface is $M_\lambda=\pi B_\lambda$. The extra Jacobian is essential: the <wavelength> peak is not found by simply converting the <frequency> peak using $\lambda=c/\nu$.
At short <wavelength> the exponential suppresses radiation. At long <wavelength> the Rayleigh-Jeans limit is $B_\lambda\simeq2ck_BT/\lambda^4$; using classical equipartition at every <frequency> instead would produce the <ultraviolet catastrophe>. The <wavelength> maximum solves $5(1-e^{-x})=x$, with nonzero root $x=4.9651$, yielding <Wien's displacement law>. For the specified <temperature>,
$$
\boxed{\lambda_{\max}=\frac{2.898\times10^{-3}\,\mathrm{m\,K}}{2.7\,\mathrm K}
\simeq1.07\,\mathrm{mm}.}
$$
Integrating the hemispheric flux gives the Stefan-Boltzmann law, $M=\sigma T^4\simeq3.01\times10^{-6}\,\mathrm{W\,m^{-2}}$. The plot uses the specified ideal <temperature>, rather than substituting a later measured value. The physical black-body description of the <cosmic microwave background> is tested by comparing the sky spectrum with an absolute reference blackbody and separating instrumental and Galactic contributions.<The Cosmic Microwave Background Spectrum from the Full COBE FIRAS Data Set by Dale J. Fixsen and others>[abstract; FIRAS comparison with a precise blackbody and limits on spectral residuals]{ref}
\Image[/past-exam-of-the-physics-a-course-of-the-university-of-cambridge/2014/paper-1-planck.png]
{title=Original ideal spectral radiance per wavelength at 2.7 K, with the wavelength peak near 1.07 mm}
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