APPENDIX A.
Derivation of the Relativistic Angular Momentum of the
Spinning Electron.
In the main text it was stated that for electrostatic reasons the electron
was to be considered as a spinning spherical shell. Consequently it is
necessary to determine the spin angular momentum of such a configuration and
this is the subject of this Appendix. Consider Fig. A.1
Fig.A.1 - A Spinning Electron Spherical Shell.
From Fig.A.1, the rest mass of the elemental is
The velocity of the elemental is
The energy mass of the elemental is therefore
|
Dm = |
dm0 Ge2 sinq dq df
|
|
æ
è
|
1 - |
w¢2sp Ge2 sin2q
c2
|
ö
ø
|
1/2
|
|
|
| (A.3) |
So that the relativistic angular momentum of the elemental is
|
DMsp* = |
dm0 w¢sp Ge4 sin3q dq df
|
|
æ
è
|
1 - |
w¢2sp Ge2 sin2q
c2
|
ö
ø
|
1/2
|
|
|
| (A.4) |
Taking a second order relativistic approximation
|
DMsp* = dm0 w¢sp Ge4 sin3q |
æ
è
|
1 + |
w¢2sp Ge2 sin2q
2c2
|
ö
ø
|
dq df |
| (A.5) |
Integrating over the surface of the sphere
|
Msp* = dm0 w¢sp Ge4 |
ó
õ
|
p
q = 0
|
|
ó
õ
|
2p
f = 0
|
|
æ
è
|
sin3q + |
w¢2sp Ge2 sin5q
2c2
|
ö
ø
|
dq df |
| (A.6) |
First with respect to f gives
|
Msp* = 2pdm0 w¢sp Ge4 |
ó
õ
|
p
q = 0
|
|
æ
è
|
sin3q + |
w¢2sp Ge2 sin5q
2c2
|
ö
ø
|
dq |
| (A.7) |
then with respect to q to yield after minor reduction
|
Msp* = 2pdm0 w¢sp Ge4 |
æ
è
|
4
3
|
+ |
8
15
|
|
w¢2sp Ge2
c2
|
+¼ |
ö
ø
|
|
| (A.8) |
Taking an approximate binomial contraction this gives
|
Msp* = |
2/3 m0 w¢sp Ge2
|
|
æ
è
|
1 - 4/5 |
w¢2sp Ge2
c2
|
ö
ø
|
1/2
|
|
|
| (A.9) |
When wspGe ® c this approximation gives about
a 10% error. It is however, adequate for the purpose required in this
paper.
P3 Version 1.0.2
Ó
P.G.Bass, April 2008
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