The radiation emitted by a blackbody at temperature $T$ has a frequency distribution given by the Planck spectrum:
$$
\epsilon_{T}(f)=\frac{2 \pi h}{c^{2}}\left(\frac{f^{3}}{e^{h f / k_{\mathrm{B}} T}-1}\right)
$$
where $\epsilon_{T}(f)$ is the energy density of the radiation per unit increment of frequency, $v$ (for example, in watts per square meter per hertz), $h=6.626 \cdot 10^{-34} \mathrm{~J} \mathrm{~s}$ is Planck's constant, $k_{\mathrm{B}}=1.38 \cdot 10^{-23} \mathrm{~m}^{2} \mathrm{~kg} \mathrm{~s}^{-2} \mathrm{~K}^{-1}$ is the Boltzmann constant,
and $c$ is the speed of light in vacuum. (We'll derive this distribution in Chapter 36 as a consequence of the quantum hypothesis of light, but here it can reveal something about radiation. Remarkably, the most accurately and precisely measured example of this energy distribution in nature is the cosmic microwave background radiation.) This distribution goes to zero in the limits $f \rightarrow 0$ and $f \rightarrow \infty$ with a single peak in between those limits. As the temperature is increased, the energy density at each frequency value increases, and the peak shifts to a higher frequency value.
a) Find the frequency corresponding to the peak of the Planck spectrum, as a function of temperature.
b) Evaluate the peak frequency at temperature $T=6.00 \cdot 10^{3} \mathrm{~K}$, approximately the temperature of the photosphere (surface) of the Sun.
c) Evaluate the peak frequency at temperature $T=2.735 \mathrm{~K}$, the temperature of the cosmic background microwave radiation.
d) Evaluate the peak frequency at temperature $T=300 . \mathrm{K}$, which is approximately the surface temperature of Earth.