The relativistic connection between harmonic bidimensional electrostatic and magnetostatic fields


  • Eugenio Ley Koo Instituto de Fisica, UNAM
  • Diego Mora de la Fuente Instituto de Física, UNAM
  • Joshua Cornejo Gómez Instituto de Física, UNAM



bidimensional, harmonic, electrostatic, magnetostatic, force fields, sources, potentials, Lorentz transformations


The antecedent of this contribution is [1], which constructed the harmonic bidimensional expansions for the electrostatic and magnetostatic potentials, produced by a straight line with uniform charge and uniform current distributions, respectively, in Cartesian, and cylindrical circular, elliptic and parabolic coordinates. For the successive geometries, the sources are confined in the respective cylinders containing the source line, plus induced sources in two grounded flat, elliptical and parabolic plates; the potentials are continuous at the source cylinder and vanish at the grounded plates. In the electrostatic case, the electric intensity field is evaluated as the negative of the gradient of the potential; in the magnetostatic case, the magnetic induction field is the rotational of the axial potential. Both potential and force fields are bidimensional, and the equipotential surfaces and force fields are orthogonal. The normal components of the electric field at the source cylinder show a discontinuity, which according to Gauss’s law is a measure of the surface charge distribution; in contrast, the tangential components are continuous due to the conservative character of the electrostatic force. The normal components of the induction field are continuous due to its solenoidad character; its tangential components show a discontinuity which by Ampere’s law is a measure of the linear current intensity. Figures 1-4 illustrate the equipotentials on the left and electric field lines on the right; and the magnetic field lines on the left and the equipotentials on the right, exhibiting also their respective orthogonalities. The differences between the electric and magnetic multipoles are recognized, but we can still ask if there is a connection between them. The answer is given here in terms of the Lorentz transformations of the four-vector potentials and sources, and of the antisymmetric force field four-tensor.


A. Gongora y M.A. Ortiz E. Ley Koo. In: rev. Mex Fìs 37 (1991), pp. 795–785.

E.M. Purcell. Electricity and Magnetism, Berkeley Physics Course 2. Second Edition. McGraw-Hill, 1985.

L. Eyges. The Classical Electromagnetic Field. New York: Dover, 1972.

F.J. Milford J.R. Reitz and R.W. Christy. Foundations of Electromagnetic Theory. Reading: AddisonWesley, 1979.

G. Arfken. Mathematical Methods for Physicists. Second Edition. New York: Academic Press, 1970. 11

R. Feynman. The Feynman Lectures on Physics. Mainly Electromagnetism and Matter. Second Edition. UK: Addison-Wesley, 1975.

A. Einstein. “On the Electrodynamics of Moving Bodies”. In: Annalen der Physik 17 (1905), pp. 891– 921.

H. Minkowski. “Raum und Zeit”. In: Jahresbericht der Deutschen Mathematiker-Vereinigung 18 (1909), pp. 75–88.

B.F. Schutz. A First Course in General Relativity. Second Edition. UK: Cambridge University Press, 2009.




How to Cite

E. Ley Koo, D. Mora de la Fuente, and J. Cornejo Gómez, “The relativistic connection between harmonic bidimensional electrostatic and magnetostatic fields”, Rev. Mex. Fis. E, vol. 20, no. 2 Jul-Dec, pp. 020207 1–, Jun. 2023.