engleski

Roads to Maxwell's Equations

This paper introduces several new derivations of Maxwell's equations which vary considerably in content and perspective. They are designed to span as much ground as possible and explore the roads less traveled. The first one is a Dirac-like inference&#8213 ; it is based on enforcing the generalized Huygens principle by pulling a sort of a square root out of the free wave equation for a vector field. This field, which is later identified as the Faraday field, must be transversal for the procedure to work&#8213 ; this is justified both by experiments and symmetry arguments. The second derivation appears as a variation of the first one. The arguments are applied in a reverse order&#8213 ; transversality is the starting point and the wave equation is deployed in the end to fix the linear operator that governs the time evolution as demanded by the Huygens principle. The third one is a potential-based reasoning&#8213 ; it shifts the perspective since in classical electrodynamics introduction of the potentials seems to serve merely as a calculational convenience. Here, the D'Alembertian is not split right away&#8213 ; instead, the implications of applying a source (current) are considered. Demanding the transversality at this point kills the longitudinal component of the current&#8213 ; and spoils the locality which is the reason for having a field theoretic description in the first place. The rescue comes from the conservation of charge which permits us to trade a manifestly nonlocal term for an additional field component. Consequently, the D'Alembertian is split more naturally and a radiation gauge emerges. The fourth road is yet another potential-based approach&#8213 ; this time with emphasize on gauge invariance. Careful examination shows that the previous inference might have applied the transversality condition overzealously&#8213 ; even where experiments did not indicate it was necessary&#8213 ; at the location of the source. Hence, locality need not be spoiled entirely&#8213 ; only locality in time, but not in space, is to be sacrificed and no new degrees of freedom (fields) are required&#8213 ; so a temporal gauge comes out. In effect, this means removing the longitudinal part of the Laplacian (instead of the current). The fifth derivation has a more traditional taste&#8213 ; it focuses on mending Ampere's law through a series of approximations. They come from projecting out the longitudinal part of the current and repairing the damage it does to locality by adding further nonlocal terms containing time-derivatives. The result is recognized as a Born series expansion for the Green function of the D'Alembertian&#8213 ; temporal derivatives are treated pertubatively, which is the essence of quasistatic approximations. It proves beneficial to compare these approaches with the derivations of Maxwell, Feynman, Schwinger and Weinberg. Gilbert Strang wrote that &#8220 ; a revolution [in a textbook presentation] is very likely to end at 2&#61552 ; &#8221 ; &#8213 ; and this appears to apply very well here&#8213 ; the derivations we have presented start from dynamic considerations close to the modern elementary particle physics and gradually arrive very close to Maxwell's original ansatz. We are inclined to perceive this more as a blessing than a curse displaying the curvature of our minds.

Electromagnetic Theory; Maxwell's Equations

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