10
1 Basic Background Material
emitting photons at all frequencies, to radiation in an enclosure with perfectly
reflecting walls will essentially catalyse thermal equilibration between the radiation
and the walls of the enclosure. This mechanism thus allows arbitrary radiation to be
converted into isotropic radiation in thermodynamic equilibrium at temperature T .
If we designate the number of photons per unit volume having velocity components c x lying in the velocity range c x to c x + dc x by n(c x , T )dc x , then the
isotropy of space implies that the average number of photons lying within the
specified velocity range and moving to the right is equal to the average number
of photons n(−c x , T )dc x moving to the left. We note that photons, unlike classical
particles, all have fixed speed c, so that c x varies only because the direction of
motion changes. Moreover, as for classical particles, no special significance can be
attached to any of the three Cartesian axis directions. The photon number density,
n(T ) = N(T , V )/V , is then given by
n(T ) ≡
c
−c
n(c x , T )dc x = 2
c
0
n(c x , T )dc x .
Consider photons in a cylindrical container of length L, cross-sectional area A,
and volume V = AL (see Fig. 1.2) that are moving to the right with x-components
of velocity c x to c x + dc x and located at a distance c x t from the end wall. In a
time interval t L/c, the average number of photons within the volume element
Ac x t that will collide with the end wall is given by (Ac x x , T )dc x . The
x-component of photon momentum will be p x = pc x /c, so that the momentum
change arising from an elastic photon–wall collision will be 2p x c x /c, thereby giving
an average force 2p x c x /(cct) exerted on the wall during time t.
If we allow for a distribution of photon frequencies ν, then thermodynamically,
the average force exerted on the wall by collisions with photons having average
energy hν and x-component of velocity lying between c x and c x + dc x will be
given by
dF =
2hνc x
c 2 Ac x n(c x , T )dc x .
The pressure P associated with a photon gas is thus defined as
P =
2hν
c 2
c
0
n(c x , T )c
2
x dc x =
N(T , V )hν
V c 2
c 2
x ,
which, upon taking into account that c 2
x = c 2
y = c 2
z =
1
3 c 2 , gives
P =
1
3
hνN
V
,
(1.2.10)
for the equation of state for a photon gas. Upon identifying Nhν as the internal
energy, U(T , V ), of the photon gas, we see that the pressure and the internal energy
1 Basic Background Material
emitting photons at all frequencies, to radiation in an enclosure with perfectly
reflecting walls will essentially catalyse thermal equilibration between the radiation
and the walls of the enclosure. This mechanism thus allows arbitrary radiation to be
converted into isotropic radiation in thermodynamic equilibrium at temperature T .
If we designate the number of photons per unit volume having velocity components c x lying in the velocity range c x to c x + dc x by n(c x , T )dc x , then the
isotropy of space implies that the average number of photons lying within the
specified velocity range and moving to the right is equal to the average number
of photons n(−c x , T )dc x moving to the left. We note that photons, unlike classical
particles, all have fixed speed c, so that c x varies only because the direction of
motion changes. Moreover, as for classical particles, no special significance can be
attached to any of the three Cartesian axis directions. The photon number density,
n(T ) = N(T , V )/V , is then given by
n(T ) ≡
c
−c
n(c x , T )dc x = 2
c
0
n(c x , T )dc x .
Consider photons in a cylindrical container of length L, cross-sectional area A,
and volume V = AL (see Fig. 1.2) that are moving to the right with x-components
of velocity c x to c x + dc x and located at a distance c x t from the end wall. In a
time interval t L/c, the average number of photons within the volume element
Ac x t that will collide with the end wall is given by (Ac x x , T )dc x . The
x-component of photon momentum will be p x = pc x /c, so that the momentum
change arising from an elastic photon–wall collision will be 2p x c x /c, thereby giving
an average force 2p x c x /(cct) exerted on the wall during time t.
If we allow for a distribution of photon frequencies ν, then thermodynamically,
the average force exerted on the wall by collisions with photons having average
energy hν and x-component of velocity lying between c x and c x + dc x will be
given by
dF =
2hνc x
c 2 Ac x n(c x , T )dc x .
The pressure P associated with a photon gas is thus defined as
P =
2hν
c 2
c
0
n(c x , T )c
2
x dc x =
N(T , V )hν
V c 2
c 2
x ,
which, upon taking into account that c 2
x = c 2
y = c 2
z =
1
3 c 2 , gives
P =
1
3
hνN
V
,
(1.2.10)
for the equation of state for a photon gas. Upon identifying Nhν as the internal
energy, U(T , V ), of the photon gas, we see that the pressure and the internal energy