2.1 Blackbody radiation and photon gas 29
and the number of reflected photons to N
ref
. It is clear that the total number of
reflected and absorbed photons is equal to the number of photons incident on the
wall, i.e.,
N
inc
= N
abs
+ N
ref
. (2.43)
In thermodynamical equilibrium the number of photons absorbed and emitted
by the wall must be the same, i.e., N
abs
= N
em
. At the same time the internal
energy of a wall and the energy of radiation do not change with time.
Let us write the momentum transferred to the wall during the time t:
p = (N
abs
+ 2 N
ref
+ N
em
)
h
-
ω
c
= 2 N
inc
h
-
ω
c
=
1
3
n
ω
h
-
ωL
2
t. (2.44)
Knowing the momentum transferred by the photons with frequency ω, we can
find the pressure, P
ω
(T ), experienced by the wall:
P
ω
(T ) =
F
L
2
=
1
L
2
p
t
=
1
3
n
ω
h
-
ω =
u
ω
(T )
3
. (2.45)
The total pressure of the photon gas, P(T ), is the sum of individual pressures,
P
ω
(T ), of photons of all frequencies:
P(T ) =
∞
0
P
ω
(T )dω =
1
3
∞
0
u
ω
(T )dω. (2.46)
By substituting into the integrand in Eq. (2.46) Planck’s expression for the spectral
density (2.27), we obtain
P(T ) =
u(T )
3
=
k
4
B
T
4
3π
2
c
3
h
-
3
I =
4σ
3c
T
4
, (2.47)
where we introduced u(T ) =
∞
0
u
ω
(T )dω, the volume energy density of photon
gas, the integral I ,
I =
∞
0
γ
3
dγ
e
γ
− 1
=
π
4
15
, (2.48)
and the Stefan–Boltzmann constant σ :
σ =
π
2
k
4
B
60c
2
h
-
3
= 5.67 × 10
−8
Wm
−2
K
−4
. (2.49)
It follows from Eq. (2.47) that the photon-gas pressure does not depend on
the volume filled by the photon gas. In contrast to an ideal gas of molecules,
whose pressure linearly depends on the temperature, the pressure of a photon
gas is proportional to T
4
, which is observed in many physical phenomena. Thus,
the gravitational squeezing of stars because of their enormous mass (compared
with the mass of the Earth) is prevented by the pressure of the photon gas.
As a result of the thermonuclear reactions taking place inside of the stars their
temperature is T ∼ 10
8
K. At these temperatures the pressure of the photon gas
is about P ∼ 10
16
Pa (in the center of the Earth P ∼ 10
11
Pa). After having