The relativistic connection between harmonic bidimensional electrostatic and magnetostatic fields

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.