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Physics 220
Homework #1
Spring 2015
Due Wednesday 4/8/15
1. Using Excel or Mathematica, plot the thermal spectrum (the Rayleigh-Jeans law with
Planck’s correction) for temperatures of 300K , 1500K , 3000K , and 5900K .
Makes all of the curves on the same graph. What features do your graphs show?
What happens to the maximum wavelength of emission from the blackbody as the
temperature of the object increases? Is this as expected? What does the area under
each curve represent? Now on a separate plot, plot the Rayleigh-Jeans law with
Planck’s correction and just the Rayleigh-Jeans law for a temperature of T = 5900K .
What do you notice?
2. We derived the Rayleigh-Jeans law with Planck’s correction in terms of the
wavelength of light. What does the Rayleigh-Jeans law with Planck’s correction look
like in terms of the frequency of light?
3. The rate at which energy from the Sun reaches a unit area on the Earth is called the
solar constant ( S ). The solar constant is measured to be S = 1350 mW2 on the earth at a
distance rE→S = 1.5 × 1011 m . What is the energy output of the Sun? If the Sun has an
average radius of RSun = 6.96 × 10 8 m , what is the approximate surface temperature of
the Sun?
4. Suppose that the Sun had no internal source of energy and was formed as a hot ball of
hydrogen at some very hot initial temperature ( Ti ). How long would it take to cool to
a final temperature T f = 6000K ? Use the Stefan-Boltzmann law and assume that the
total kinetic energy of the particles in the Sun is proportional to the temperature and is
3
given by E = NkT , where N is the number of particles in the Sun. Assume that the
2
entire mass of the sun ( M Sun = 2 × 10 30 kg ) is composed of protons. (Of course the
sun does have an internal source of energy. Proton fusion heats the sun and this
converts matter into kinetic energy so the lifetime of the Sun is clearly much larger
than the value you obtained.)
5. Assume that you are a blackbody radiator. How much energy do you radiate in one
hour if your temperature is T = 310K ? At what maximum wavelength do you
radiate?
6. A light bulb emits 40W of energy every second at a wavelength of 6 × 10 −7 m . What
is the total number of photons emitted every second and what is the energy of one of
these photons?
7. Estimate the temperature of a “white” hot light bulb filament. If the light bulb has a
radiated power of 100W , what is the surface area of the filament? How many
photons are radiated per second?
8. A thermal light source has a temperature of T = 6000K and a total radiated power of
dS
100W . What is the power per unit area per µ m (
) at the peak of the thermal
dλ
spectrum?
9. For thermal radiation, show that the relationship between the energy per volume per
du
dn
unit wavelength (
) and the number of photons per volume per wavelength (
)
dλ
dλ
du ⎛ hc ⎞ dn
=⎜ ⎟
is
.
dλ ⎝ λ ⎠ dλ