= Solution
The constants are the <Planck constant> $h$, photon <frequency> $\nu$, <speed of light> $c$, <Boltzmann constant> $k_B$ and absolute <temperature> $T$. The supplied $U_\nu$ is a <spectral energy density>, with units $\mathrm{J\,m^{-3}\,Hz^{-1}}$. It describes equilibrium radiation per volume and frequency interval, rather than emitted <power> itself. For isotropic cavity radiation the corresponding surface <spectral radiant exitance> is $M_\nu=cU_\nu/4$.
At low <frequency> the curve rises as $\nu^2$; it has one positive maximum and an exponentially decaying high-frequency tail. The left panel below shades the area $u(T)=\int_0^\infty U_\nu\,d\nu$ under one curve. Put $x=h\nu/(k_BT)$; then
$$
U_\nu=\frac{8\pi(k_BT)^3}{h^2c^3}\frac{x^3}{e^x-1},\qquad d\nu=\frac{k_BT}{h}\,dx.
$$
The remaining dimensionless <integral> is independent of <temperature>, so
$$
\boxed{u(T)\propto T^4,\qquad M(T)=\frac c4u(T)=\sigma T^4.}
$$
Thus doubling <temperature> doubles the characteristic frequency scale, raises the spectrum's height by eight and increases its area by sixteen. Total emitted <energy> also depends on emitting <area> and elapsed time.
\Image[/past-exam-of-the-short-options-paper-of-the-physics-prelims-of-the-university-of-oxford/2019-quantum-sketches.png]
{title=Planck spectral energy density, Compton scattering geometry and square-well state amplitudes}
Back to article page
