86
Mr. E. Cunningham
[Feb. 11,
Now pass to the consideration of the equilibrium radiation within a cavity in a body. The pressure is normal to the surface and the vector
σ
4
π
[
E
H
]
{\displaystyle {\frac {\sigma }{4\pi }}[EH]}
vanishes at all points.
Thus
P
X
=
L
P
,
P
Y
=
M
P
,
P
Z
=
N
P
.
{\displaystyle P_{X}=LP,\ P_{Y}=MP,\ P_{Z}=NP.}
Hence
p
x
′
d
s
=
L
P
d
S
=
P
l
d
s
,
p
y
′
d
s
=
1
β
M
P
d
S
=
P
m
d
s
,
p
z
′
d
s
=
1
β
N
P
d
S
=
P
n
d
s
,
{\displaystyle {\begin{array}{lrl}p'_{x}ds=&LPdS&=Plds,\\\\p'_{y}ds=&{\frac {1}{\beta }}MPdS&=Pmds,\\\\p'_{z}ds=&{\frac {1}{\beta }}NPdS&=Pnds,\end{array}}}
Thus the pressure in the moving cavity is in the direction (l, m, n ), and is equal to P .
Let the expression for the energy of the field be now similarly treated.
1
8
π
(
e
2
+
h
2
)
=
1
8
π
{
E
X
2
+
H
X
2
+
β
2
(
1
+
v
2
c
2
)
(
E
Y
2
+
H
Y
2
+
E
Z
2
+
H
z
2
)
+
4
β
2
v
c
(
E
Y
H
z
−
E
z
H
Y
)
}
.
{\displaystyle {\frac {1}{8\pi }}\left(e^{2}+h^{2}\right)={\frac {1}{8\pi }}\left\{E{}_{X}^{2}+H_{X}^{2}+\beta ^{2}\left(1+{\frac {v^{2}}{c^{2}}}\right)\left(E_{Y}^{2}+H_{Y}^{2}+E_{Z}^{2}+H_{z}^{2}\right)+{\frac {4\beta ^{2}v}{c}}\left(E_{Y}H_{z}-E_{z}H_{Y}\right)\right\}.}
In the cavity at rest when the radiation is in equilibrium, the mean values of
(
E
X
2
+
H
X
2
)
,
(
E
Y
2
+
H
Y
2
)
,
(
E
Z
2
+
H
Z
2
)
{\displaystyle \left(E_{X}^{2}+H_{X}^{2}\right),\ \left(E_{Y}^{2}+H_{Y}^{2}\right),\ \left(E_{Z}^{2}+H_{Z}^{2}\right)}
are each equal to
1
3
(
E
2
+
H
2
)
{\displaystyle {\frac {1}{3}}\left(E^{2}+H^{2}\right)}
, and that of
(
E
Y
H
Z
−
E
Z
H
Y
)
{\displaystyle \left(E_{Y}H_{Z}-E_{Z}H_{Y}\right)}
is zero, the average being taken over any interval of time very small, but large compared with the periods of the constituent radiation.
Hence, denoting mean values by a stroke,
1
8
π
(
e
2
+
h
2
)
¯
=
1
8
π
(
E
2
+
H
2
)
¯
{
1
3
+
2
(
c
2
+
v
2
)
3
(
c
2
−
v
2
)
}
,
{\displaystyle {\frac {1}{8\pi }}{\overline {\left(e^{2}+h^{2}\right)}}={\frac {1}{8\pi }}{\overline {\left(E^{2}+H^{2}\right)}}\left\{{\frac {1}{3}}+{\frac {2\left(c^{2}+v^{2}\right)}{3\left(c^{2}-v^{2}\right)}}\right\},}
or
ϵ
¯
=
3
c
2
+
v
2
3
(
c
2
−
v
2
)
E
¯
.
{\displaystyle {\overline {\epsilon }}={\frac {3c^{2}+v^{2}}{3\left(c^{2}-v^{2}\right)}}{\overline {E}}.}
In identical manner, Planck's equation
g
¯
=
4
v
3
(
c
2
−
v
2
)
E
¯
{\displaystyle {\overline {g}}={\frac {4v}{3\left(c^{2}-v^{2}\right)}}{\overline {E}}}
is obtained.
Finally,
p
¯
=
P
¯
=
1
3
E
¯
=
c
2
−
v
2
3
c
2
+
v
2
ϵ
¯
,
{\displaystyle {\overline {p}}={\overline {P}}={\frac {1}{3}}{\overline {E}}={\frac {c^{2}-v^{2}}{3c^{2}+v^{2}}}{\overline {\epsilon }},}
g
¯
=
4
v
3
c
2
+
v
2
ϵ
¯
.
{\displaystyle {\overline {g}}={\frac {4v}{3c^{2}+v^{2}}}{\overline {\epsilon }}.}