PHSX 446
Homework 22:
Solutions
Spring 2015
Blackbody Radiation
1. Using the energy derived in class for a photon gas, derive the pressure. Express the
answer in terms of the temperature, the Stefan-Boltzmann constant, and other numbers
and constants.
As we found in class:
E = aT 4 V.
Photons have zero chemical potential, so the First Law gives
dE = T dS − pdV.
The entropy is evaluated to be (see eq. 21.19):
4
S = aT 3 V.
3
The First Law gives
4
4aT V dT + aT dV = T 4aV T dT + aT 3 dV
3
3
4
2
− pdV.
Solve for p to obtain
1
u
p = aT 4 = ,
3
3
where u is the energy density.
2. Plot the energy distribution of blackbody radiation versus versus x = hc/(kT λ) where
λ is the wavelength of the radiation. You may use any convenient scale for the energy
distribution, since a multiplicative factor does not change the shape of the curve. Plot
also the derivative of the energy spectrum in x versus x. The peak of the blackbody curve
is given by the zero of this function. Blow up your graph as need to obtain the position
of the peak in x to three significant figures. Obtain Wien’s Law. Do not use a “solve”
command or the like to do this, rather, obtain the solution graphically.
The energy distribution is given by
d 3
.
eβ − 1
To get Wien’w Law, we need the wavelength distribution. In terms of x:
d 3
dλ x5
−→
.
eβ − 1
ex − 1
The maximum in this function will gives the Wien peak.
Below is a plot of the function, its derivative, and a blowup of the where the derivative
passes through zero.
The blowup shows that the peak is at x0 = 4.9650. The peak wavelength is given by
λp T =
hc
= 0.290 cm K,
kx0
which is Wien’s Law.
3. Derive eq. 21.10 of the text by doing the angle average described.
The energy flux in the z direction is the angle-average photon speed into one hemisphere
hvic, times the energy density u, times 1/2 (since the flux is going into one hemisphere):
Jz =
u
hviv.
2
The project of a photon’s velocity vector in the z direction is z cos θ, were θ is the angle
between the z axis and the photon’s velocity vector. The angle average of the speed into
one hemisphere is the integral of the speed over solid angle divided by the solid angle 2π
of one hemisphere:
Z 2π
Z π/2
c
c
hvi =
dφ
dθ sin θ cos θ = .
2π 0
2
0
The theta integral is from zero to π/2 because the photons are moving into one hemisphere.
We get
c
Jz = u = σT 4 .
4
Since the z-axis is arbitrary, this is the energy flux in any direction for a photon gas.