An alternative to Maxwell's equation based on the conservation of acceleration
Résumé
Equivalence between the discrete laws of electromagnetism and mechanics is revealed by the derivation of a non-linear equation intertwining electrical and magnetic effects. The alternative physical model to Maxwell’s is based on the conservation of the acceleration of electric current, the only variable in the law of discrete electromagnetism. The modeling of this law is based solely on the essential elements of electrostatics and magnetostatics, where Ohm’s and Ampère’s laws are reformulated in a dynamic framework. The Maxwell-Thomson law is abandoned and the conservation of magnetic flux is ignored, this emancipation paves the way for the potential existence of magnetic monopoles. The Maxwell-Gauss equation expressing the conservation of electric flux is also set aside, as it is implicitly satisfied in the discrete formulation. The discrete point of view consists in observing that direct and induced currents flow on the same conductor without interacting in the steady state, and in an intertwined manner in the variable state. The sum of these two contributions forms the material derivative of the electric current density, the current acceleration. This quantity can be decomposed into a curl-free and a divergence-free components, a Helmholtz-Hodge decomposition to extract both direct and induced contributions. The law of discrete electromagnetism presents complete symmetry in the exchange of electric and magnetic fields. Inertial terms of the current acceleration contained in its material derivative are none other than the non-linear Lorentz forces. The new formulation is equivalent to the law of motion of discrete mechanics.
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