Abstract
We reintroduce an alternative expression for the Lagrangian density that governs the interaction of a charged particle with external electromagnetic fields, proposed by Livens about one century ago. This Lagrangian is written in terms of the local superposition of the particle fields with the applied electromagnetic fields, not in terms of the particle charge and of the electromagnetic potentials as is usual. Here, we show that the total Lagrangian for a set of charged particles assumes a simple elegant form with the alternative formulation, giving an aesthetic support for it. We also show that the alternative Lagrangian is equivalent to the traditional one in their domain of validity and that it provides an interesting description of the Aharonov-Bohm effect.
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References
Y. Aharonov, D. Bohm, Phys. Rev. 115, 485 (1959)
R.G. Chambers, Phys. Rev. Lett. 5, 3 (1960)
A. Tonomura, et al., Phys. Rev. Lett. 56, 792 (1986)
M. Peshkin, A. Tonomura. Lecture Notes in Physics, vol. 340 (Springer, New York, 1989)
G. Matteucci, G. Pozzi, Phys. Rev. Lett. 54, 2469 (1985)
R.A. Webb, S. Washburn, C.P. Umbach, R.B. Laibowitz, Phys. Rev. Lett. 54, 2696 (1985)
A. van Oudenaarden, M.H. Devoret, Y.V. Nazarov, J.E. Mooij, Nature. 391, 768 (1998)
A. Bachtold, et al., Nature. 397, 673 (1999)
H. Peng, et al., Nature Mater. 9, 225 (2010)
G.H. Livens, Philos. Mag. 32, 195 (1916)
G.H. Livens. The Theory of Electricity, chap. VI, 2nd edn. (Cambridge University Press, Cambridge, 1926)
J.D. Jackson. Classical Electrodynamics, 3rd edn. (Wiley, New York, 1999)
L.D. Landau, E.M. Lifshitz. The Classical Theory of Fields, 4th edn. (Butterworth-Heinemann, Oxford, 2000)
F. Rohrlich. Classical Charged Particles, 3rd edn. (World Scientific Publishing, Singapore, 2007)
B. Liebowitz, Nuovo Cimento. 38, 932 (1965)
T.H. Boyer, Phys. Rev. D. 8, 1679 (1973)
T.H. Boyer, Found. Phys. 32, 41 (2002)
A. Caprez, B. Barwick, H. Batelaan, Phys. Rev. Lett. 99, 210401 (2007)
T.H. Boyer, Found. Phys. 38, 498 (2008)
M. Peshkin, Phys. Rep. 80, 375 (1981)
L. Vaidman, Phys. Rev. A. 86, 040101 (2012)
R.P. Feynman, R.B. Leighton, M. Sands. Lectures on Physics, chap. 28, vol. II (Addison-Wesley, Reading, 1977)
Acknowledgments
I would like to acknowledge Kirk McDonald for very useful discussions and for calling my attention to the works of Livens [10, 11]. I would also like to acknowledge Ricardo Schor for useful discussions and Júlia Parreira for useful comments on the manuscript. This work was supported by the Brazilian agencies CNPq and CAPES.
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Saldanha, P.L. Alternative Expression for the Electromagnetic Lagrangian. Braz J Phys 46, 316–320 (2016). https://doi.org/10.1007/s13538-016-0417-4
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DOI: https://doi.org/10.1007/s13538-016-0417-4