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.
