Electromagnetic angular momentum of an orbiting charge. (English) Zbl 1476.78003

Trompetter, W. J.

Found. Phys. 51, No. 5, Paper No. 102, 22 p. (2021).

Summary: The electric field of an orbiting charge or electron observed in the rotating frame takes on a circular trajectory with a maximum radius of \(R=\frac{c}{\omega '} \). The resultant extended electromagnetic structure is used to derive the spin-orbit energy of the orbiting electron. A surprising result of the derived expression is that the orbital velocity has a specific value \(( \beta_0 = \frac{1}{136.96} )\) in close agreement \((99.94\%)\) with the experimentally determined value for the fine structure constant \(( \hbar =\frac{ke^2}{c}\frac{1}{\alpha_{\text{exp}}} =\frac{ke^2}{c}\frac{1}{\beta_0} \rightarrow \alpha_{\text{exp}}=\beta_0)\). Furthermore, the derived spin-orbit expression does not include a g-factor (i.e \(g=1)\) which means that the Larmor and Thomas precessions are equal and opposite, resulting in zero net precession in the lab frame. These results suggest that the quantised fine structure constant and by extension Planck’s constant, are a natural extension of classical electromagnetism and conservation of energy in a rotating frame.

MSC:

78A35

Motion of charged particles

Keywords:

electromagnetic mass; angular momentum; spin-orbit interaction; relativity in rotating frames; fine structure constant; Planck’s constant

References:

[1]	Jackson, JD, Classical Electrodynamics (1999), New York: Wiley, New York · Zbl 0920.00012
[2]	Fisher, GP, The Electric Dipole Moment of a Moving Magnetic Dipole, Am. J. Phys., 39, 12, 1528-1533 (1971)
[3]	Holstein, BR, Gyroscope precession and general relativity, Am. J. Phys., 69, 12, 1248-1256 (2001)
[4]	Munoz, G., Spin-orbit interaction and the Thomas precession: a comment on the lab frame point of view, Am. J. Phys., 69, 5, 554-556 (2001)
[5]	Thomas, LH, The motion of the spinning electron, Nature (London), 117, 514 (1926)
[6]	Thomas, LH, Kinematics of an electron with an axis, Philos. Mag., 3, 1-22 (1927) · JFM 53.0875.07
[7]	Muller, RA, Thomas precession: where is the torque?, Am. J. Phys., 60, 4, 313-317 (1992)
[8]	Spavieri, G.; Mansuripur, M., Origin of the spin-orbit interaction, Phys. Scr., 90, 085501 (2015)
[9]	Newburgh, RG, thomas precession and extended structures, Lett. Nuovo Cimento, 5, 5, 387-388 (1972)
[10]	Cavalleri, G., Thomas precession of macroscopic objects, Lett. Nuovo Cimento Ser. 2, 7, 15, 575-576 (1973)
[11]	Abraham, M., Prinzipien der Dynamik des Elektrons, Ann. Phys., 10, 105-179 (1903) · JFM 34.0915.02
[12]	Lorentz, H.A.: The Theory of Electrons (1909)
[13]	Lorentz, H.A.: Dover reprint, New York (1952)
[14]	Schott, GA, Electromagnetic Radiation (1912), Cambridge: Cambridge U.P., Cambridge · JFM 43.0994.01
[15]	Feynman, R.P., Leighton, R.P., Sands, M.: The Feynman Lectures on Physics, Vol. 2. Addison-Wesley, Section 28-1 (1964)
[16]	Rohrlich, F., Self-energy and stability of the classical electron, Am. J. Phys., 28, 7, 639-643 (1960) · Zbl 0094.41303
[17]	Rohrlich, F., Classical Charged Particles (1965), Reading: Addison-Wesley, Reading · Zbl 0131.43602
[18]	Rohrlich, F., Electromagnetic momentum, energy, and mass, Am. J. Phys., 38, 11, 1310-1316 (1970)
[19]	Rohrlich, F.: Classical Charged Particles, 3rd ed. Singapore: World Scientific, ISBN 978-981-270-004-9 (2007) · Zbl 1187.78001
[20]	Moylan, P., An elementary account of the factor 4/3 in the electromagnetic mass, Am. J. Phys., 63, 9, 818-820 (1995)
[21]	Teukolsky, SA, The explanation of the Trouton-Noble experiment revisited, Am. J. Phys., 64, 9, 1104-1109 (1996)
[22]	Griffiths, DJ, Resource letter EM-1: electromagnetic momentum, Am. J. Phys., 80, 1, 7-18 (2012)
[23]	Hobson, A., There are no particles, there are only fields, Am. J. Phys., 81, 3, 211-223 (2013)
[24]	Jennison, RC, Ray paths in a rotating system, Nature, 199, 739-741 (1963) · Zbl 0113.21002
[25]	Jennison, RC; Ashworth, DG, Surveying in rotating systems II. Triangulation radius, J. Phys. A: Math. Gen., 9, 1257-1260 (1976)
[26]	Ashworth, DG; Jennison, RC, Surveying in rotating systems, J. Phys. A: Math. Gen., 9, 35-43 (1976)
[27]	Ashworth, DG; Davies, PA, Geodesics in rotating systems, Int. J. Theor. Phys., 16, 11, 845-861 (1977)
[28]	Ashworth, DG; Davies, PA, Transformations between inertial and rotating frames of reference, J. Phys. A: Math. Gen., 12, 9, 1425-1440 (1979)
[29]	Grøn, Ø., Relativistic description of a rotating disk with angular acceleration, Found. Phys., 9, 5-6, 353-369 (1979)
[30]	Grøn, Ø.; Rizzi, G.; Ruggiero, ML, Space geometry in rotating reference frames: a historical appraisal, Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames (2004), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1075.83001
[31]	Rizzi, G.; Ruggiero, ML, Relativity in Rotating Frames: Relativistic Physics in Rotating Reference Frames (2004), Dordrecht: Kluwer Academic Publishers, Dordrecht · Zbl 1050.83005
[32]	Ehrenfest, P., Gleichförmige Rotation starrer Körper und Relativitätstheorie, Phys. Z., 10, 918 (1909) · JFM 40.0931.01
[33]	Einstein, A., Zum Ehrenfestschen Paradoxon, Phys. Z., 12, 509-510 (1911) · JFM 42.0727.02
[34]	Einstein, A.; Infield, L., The Evolution of Physics, 239-242 (1938), Cambridge: Cambridge Univ. Press, Cambridge
[35]	Franklin, P., The meaning of rotation in the special theory of relativity, Proc. Natl. Acad. Sci., 8, 265-268 (1922) · JFM 48.1324.01
[36]	Berenda, CW, The problem of the rotating disk, Phys. Rev., 62, 280-290 (1942) · Zbl 0060.44012
[37]	Møller, C., The Theory of Relativity, 222-226 (1972), Oxford: Clarendon Press, Oxford
[38]	Einstein, A.: On the Electrodynamics of Moving Bodies (1905), revised and translated in The Principle of Relativity. Dover, pp. 35-65 (1923)
[39]	Resnick, R., Introduction to Special Relativity, 62-63 (1968), New York: Wiley, New York
[40]	Uhlenbeck, GE; Goudsmit, S., Naturwissenschaften, 47, 953 (1925)
[41]	Uhlenbeck, GE; Goudsmit, S., Nature, 117, 264 (1926)
[42]	Tiesinga, E.; Mohr, PJ; Newell, DB; Taylor, BN, CODATA recommended values of the fundamental physical constants: 2018, Rev. Mod. Phys., 93, 2, 84 (2021)
[43]	Panofsky, WKH; Phillips, M., Classical Electricity and Magnetism (2005), Boston: Addison-Wesley, Boston · Zbl 0067.19205
[44]	Strandberg, MWP, Special relativity completed: the source of some 2s in the magnitude of physical phenomena, Am. J. Phys., 54, 4, 321-331 (1986)
[45]	Gladman, B.; Quinn, DD; Nicholson, P.; Rand, R., Synchronous locking of tidally evolving satellites, Icarus, 122, 1, 166-192 (1996)
[46]	Peale, SJ; Burns, JA, Rotation histories of the natural satellites, Planetary Satellites, 87-112 (1977), Tucson: University of Arizona Press, Tucson
[47]	Janssen, M.; Mecklenburg, M.; Hendricks, VF, From classical to relativistic mechanics: electromagnetic models of the electron, Interactions: Mathematics, Physics and Philosophy, 65-134 (2007), Dordrecht: Springer, Dordrecht
[48]	Trouton, FT, Effect on charged condenser of motion through the ether, Sci. Proc. R. Dublin Soc., 7, 379-384 (1902)
[49]	Trouton, FT; Noble, HR, The mechanical forces acting on a charged condenser moving through space, Philos. Trans. R. Soc. Lond. Ser. A, 202, 165-181 (1904) · JFM 35.0875.02
[50]	Butler, JW, On the Trouton-Noble experiment, Am. J. Phys., 36, 936-941 (1968)
[51]	Rindler, W.; Denur, J., A simple relativistic paradox about electrostatic energy, Am. J. Phys., 56, 9, 795 (1988)
[52]	Herrmann, F., The unexpect path of the energy in a moving capacitor, Am. J. Phys., 61, 2, 119-121 (1993)
[53]	Butler, JW, A proposed electromagnetic momentum-energy 4-vector for charged bodies, Am. J. Phys., 37, 1258-1272 (1969)