368
THE RADIATION PROBLEM
Let
v
denote
the velocity of the mirror
at
time t.
Owing
to
the
frictional force
mentioned
above,
this
velocity
decreases
by
PvT
m
in the
small time
interval
T,
where
m
denotes the
mass
of
the mirror
and
P
the
retarding force
corresponding to
unit
velocity of the
mirror.
Further,
we
denote
by
A
the
velocity
change
of
the mirror
during
T
corresponding to
the
irregular
fluctuations of the radiation
pressure.
The
velocity of the
mirror
at
time t
+ T
is
PT
v
-
m
v + A .
[43]
For the
condition that
on
the
average
v
shall
remain
unchanged
during
T, we
obtain
v
-
PT
v + A
=
HP
or,
if
we
omit
relatively infinitesimal
quantities and
take into
account
that
the
average
value of
vk
obviously
vanishes:
[44]
A2
=
ML
^
tn
.
In this
equation v2
can
be
replaced using
the
equation
[45]
Imp
1
RT
2
N
,
[46]
[47]
which
can
be
derived
from
the entropy-probability
equation.
Before
giving
the
value
of
the friction
constant P,
we
specialize the
problem
under
considera-
tion
by
assuming
that the mirror
completely
reflects the radiation
of
a
certain
frequency range (between
v
and
v +
dv)
and
is
completely
transpar-
ent
to
radiation
of
other frequencies.
By
a
calculation omitted
here
for the
sake of brevity,
one
obtains
from
a
purely electrodynamic
investigation the
following
equation, which is
valid for
any
arbitrary
radiation
distribution:
P
=
2
c
-j'£
dvf
,