aus Wikisource, der freien Quellensammlung
und
p
2
=
i
′
B
(
−
σ
sin
2
ϕ
+
cos
ϕ
1
−
σ
2
sin
2
ϕ
)
{\displaystyle p_{2}={\frac {i'}{B}}\left(-\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\right)}
(8)
oder
p
2
=
i
0
−
σ
sin
2
ϕ
+
cos
ϕ
1
−
σ
2
sin
2
ϕ
B
+
1
−
σ
2
sin
2
ϕ
=
{\displaystyle p_{2}=i_{0}{\frac {-\sigma \sin ^{2}\phi +\cos \phi {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}_{+}{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}=}
=
i
0
cos
ϕ
−
σ
1
−
σ
2
sin
2
ϕ
B
(
1
−
σ
2
)
1
−
σ
2
sin
2
ϕ
.
{\displaystyle =i_{0}{\frac {\cos \phi -\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}{{\mathfrak {B}}(1-\sigma ^{2}){\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}.}
(9)
Setzen wir dies in (2) ein, so wird:
L
=
2
π
c
D
i
0
⋅
∫
0
π
/
2
sin
ϕ
d
ϕ
B
(
1
−
σ
2
)
1
−
σ
2
sin
2
ϕ
{\displaystyle L=2\pi cDi_{0}\cdot \int _{0}^{\pi /2}{\frac {\sin \phi \ d\phi }{{\mathfrak {B}}(1-\sigma ^{2}){\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}}}
[
cos
ϕ
(
1
B
−
−
1
B
+
)
+
σ
1
−
σ
2
sin
2
ϕ
(
1
B
−
+
1
B
+
)
]
=
{\displaystyle \left[\cos \ \phi \left({\frac {1}{{\mathfrak {B}}_{-}}}-{\frac {1}{{\mathfrak {B}}_{+}}}\right)+\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\left({\frac {1}{{\mathfrak {B}}_{-}}}+{\frac {1}{{\mathfrak {B}}_{+}}}\right)\right]=}
=
4
π
c
D
i
0
σ
B
2
(
1
−
σ
2
)
2
∫
0
π
/
2
sin
ϕ
d
ϕ
1
+
cos
2
ϕ
−
σ
2
sin
2
ϕ
1
−
σ
2
sin
2
ϕ
.
{\displaystyle ={\frac {4\pi cDi_{0}\sigma }{{\mathfrak {B}}^{2}(1-\sigma ^{2})^{2}}}\int _{0}^{\pi /2}\sin \phi \ d\phi {\frac {1+\cos ^{2}\phi -\sigma ^{2}\sin ^{2}\phi }{\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}}.}
Setzen wir noch den Energieinhalt des ruhenden Hohlraumes
R
{\displaystyle R}
4
π
i
0
D
B
=
E
0
,
{\displaystyle {\frac {4\pi i_{0}D}{\mathfrak {B}}}=E_{0},}
so wird nach Durchführung der Integration
L
=
E
0
c
2
B
2
1
(
1
−
σ
2
)
2
[
1
2
+
1
2
σ
2
−
(
1
−
σ
2
)
2
4
σ
3
log
1
+
σ
1
−
σ
]
.
{\displaystyle L=E_{0}{\frac {c^{2}}{{\mathfrak {B}}^{2}}}{\frac {1}{(1-\sigma ^{2})^{2}}}\left[{\frac {1}{2}}+{\frac {1}{2\sigma ^{2}}}-{\frac {(1-\sigma ^{2})^{2}}{4\sigma ^{3}}}\log {\frac {1+\sigma }{1-\sigma }}\right].}
Vernachlässigen wir hierin Größen von der Ordnung
σ
3
{\displaystyle \sigma ^{3}}
L
=
E
0
c
2
B
2
⋅
4
3
.
{\displaystyle L=E_{0}{\frac {c^{2}}{B^{2}}}\cdot {\frac {4}{3}}.}
Dieser Ausdruck hat nun die Form und die Dimension einer kinetischen Energie. Man kann also sagen, daß die kinetische Energie unseres Systems scheinbar um
L
{\displaystyle L}
![{\displaystyle \left[\cos \ \phi \left({\frac {1}{{\mathfrak {B}}_{-}}}-{\frac {1}{{\mathfrak {B}}_{+}}}\right)+\sigma {\sqrt {1-\sigma ^{2}\sin ^{2}\phi }}\left({\frac {1}{{\mathfrak {B}}_{-}}}+{\frac {1}{{\mathfrak {B}}_{+}}}\right)\right]=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2efcc4a057507c4fa263a92d32bdaa8d9a86a5bb)
![{\displaystyle L=E_{0}{\frac {c^{2}}{{\mathfrak {B}}^{2}}}{\frac {1}{(1-\sigma ^{2})^{2}}}\left[{\frac {1}{2}}+{\frac {1}{2\sigma ^{2}}}-{\frac {(1-\sigma ^{2})^{2}}{4\sigma ^{3}}}\log {\frac {1+\sigma }{1-\sigma }}\right].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a7dbb536e9d251a4b5d5d64f805b32c459b71cd4)
