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Contributions of Abraham, Lorentz and Poincaré to Classical Theory of Electrons

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Abstract

Max Abraham, Hendrik Lorentz and Henri Poincaré developed theories of electrons in the years following the discovery of electrons in cathode rays and in a component of the radiation from radioactive elements. The subjects covered in their theories include electrodynamics of moving particles, the origins of mass and implications for other fundamental forces. Highlights include electromagnetic mass, the stability of electrons, and experimental evidence for the dependence of relativistic mass on velocity.

Attention is given to distinguishing the contributions of the three physicists.

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Figure 9.1
Figure 9.2
Figure 9.3
Figure 9.4

Notes

  1. 1.

    For more recent context for this problem, the reader may consult (Feynman, Leighton, and Sands 1963) Volume II, Chapter 28, Electromagnetic Mass.

  2. 2.

    And for this problem, the reader may consult (Jackson, Classical Electrodynamics 1999) Chapter 16, Radiation Damping, Classical Models of Charged Particles (roughly sections 16.1–16.4).

  3. 3.

    For possible comparison with Einstein see §10 of (Einstein, Zur Elektrodynamik bewegter Körper 1905).

  4. 4.

    In Equation 9.2 W is the total energy, not the Lagrangian, so this reasoning would not apply there.

  5. 5.

    Here and elsewhere, the first page number refers to the original publication and the second page number (following the semicolon) refers to the page number in Part I of this book.

  6. 6.

    Equation 19a in (Abraham , Dynamik des Electrons, 1902a, p. 33) has the same relation between moving and resting charge density as equation 7 in (Lorentz, Electromagnetic phenomena in a system moving with any velocity smaller than that of, 1904, p. 813; p. 263).

  7. 7.

    This point is discussed further in a following section on Poincaré’s contribution and starts on page 184.

  8. 8.

    And the typo is worth pointing out because it goes unremarked and repeated by (Miller, 1973) in equation 33d.

  9. 9.

    If you’re keeping count and wondering what the second divergence is, it is that charged electrons are not stable if there are only electrical forces. This is discussed starting page 189 below. The additional forces required for stability are now called the Poincaré stresses. Poincaré brings the need for this additional force to Lorentz’s attention in a third letter in the same timeframe (Kox 2008, p. 179, letter 128; p. 40). There is no extant reply from Lorentz to Poincaré to any of these three letters.

  10. 10.

    I have done the corresponding analysis with the data from (Kaufmann , Die electromagnetishe Masse des Elektrons, 1902a), but the standard deviation is much larger so it didn’t seem worth presenting the results.

  11. 11.

    For possible comparison with Einstein concerning shape see §4 of (Einstein, Zur Elektrodynamik bewegter Körper 1905).

  12. 12.

    For possible comparison with Einstein see §9 of (Einstein, Zur Elektrodynamik bewegter Körper 1905).

  13. 13.

    For possible comparison with Einstein see §5 of (Einstein, Zur Elektrodynamik bewegter Körper 1905).

  14. 14.

    Because the continuity equation is unchanged by Lorentz transformation, it indicates that (, J) is a four-vector. For more, see (Jackson, Classical Electrodynamics, 1999, pp. 554–555, eqn. 11.129).

  15. 15.

    For possible comparison with Einstein see §6 and §9 of (Einstein, Zur Elektrodynamik bewegter Körper, 1905).

  16. 16.

    For possible comparison with Einstein see last pages of §3 of (Einstein, Zur Elektrodynamik bewegter Körper, 1905) for an argument that φ(v) = 1 based on a translation and its inverse.

  17. 17.

    There is limited further discussion of this work in the following chapter, Chapter 10.

  18. 18.

    In fact, the discussion of real and ideal electrons reads as if it were written in the context of Lorentz’s corresponding states; it is a notable contrast to the discussion of the volume of a moving sphere and the derivation of the transformation of charge density and the equation of continuity for charge.

  19. 19.

    This is not correct and introduces an incorrect factor of order unity. The core of the problem is an incorrect application of the Lorentz transformation; an approach similar to that taken by Poincaré with the transformation of the charge density in §1 and discussed above is needed. That is to say a manifestly covariant four-vector approach is needed. Poincaré’s equation for the momentum is the same as, for example, equation 2–13 in (Rohrlich 2007, p. 15). But as Rohrlich indicates, it is not correct. Rohrlich on page 17 provides a brief history of the multiple times this problem and the correction were found and provides a detailed discussion of the resolution in section 6–3, pages 129–134.

  20. 20.

    Interestingly, Poincaré, while noting that this formula applies to a spherical electron, does not follow the reference to Searle mentioned by Abraham at the top of the following page and provided a few pages earlier which does have the corresponding potential for an ellipsoid (specifically equation 23 (Searle 1897, p. 340)); he manages to make do.

  21. 21.

    In the Introduction, (Poincaré, Sur la dynamique de l’électron 1906, p. 130; p. 46) referred to it as an external pressure. Positive external pressure and negative internal pressure largely amounts to the same thing. Since his argumentation is based on the Lagrangian, he does not have a specific mechanism to distinguish between an external and internal origin of the pressure.

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Popp, B.D. (2020). Contributions of Abraham, Lorentz and Poincaré to Classical Theory of Electrons. In: Henri Poincaré: Electrons to Special Relativity. Springer, Cham. https://doi.org/10.1007/978-3-030-48039-4_9

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