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,
\(k_{\mathrm{B}}=1.38 \cdot 10^{-23} \mathrm{JK}^{-1}\) 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 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 \(T=6.00 \cdot 10^{3} \mathrm{~K}\),
approximately the temperature of the photosphere (surface) of the Sun.
c) Evaluate the peak frequency at \(T=2.735 \mathrm{~K}\), the temperature of
the cosmic background microwave radiation.
d) Evaluate the peak frequency at \(T=300 . \mathrm{K},\) which is approximately
the surface temperature of Earth.
