Formulação das equações de Maxwell: da forma integral à covariante

Abstract

Maxwell’s equations are part, together, of the fundamental basis of what we call classical electromagnetism, being partial differential equations capable of describing, directly or indirectly, through the application of appropriate mathematical theorems and algebraic manipulations, the totality of phenomena of an electromagnetic nature, with contributions to the electromagnetic phenomena, with contributions to classical optics, besides direct consequences in the field of technological applications. What we call Lorentz force is also in this context of fundamental bases of electromagnetic theory, which is a branch of physics active in various areas of theoretical, experimental and/or applied research. In this paper we show the integral, differential and covariant formulations for Maxwell’s equations. There are some mathematical representations of Maxwell’s equations that, in essence, describe the same physical phenomena, although particular formulation proves to be more appropriate, depending on the physical phenomena investigated, etc. The covariant formulation (in terms of covariant field tensors) of these equations is very useful in field theory and, in particular, in the study of special relativity.Integral and differential forms are commonly used in undergraduate physics curricula. In this paper, in addition the two most common formulations, we will show, through the mathematical deductions that Maxwell’s equations written in their most compact form, covariant, in terms of a quadratensor that clearly unifies the electric and magnetic fields into a single physical/mathematical entity, the electromagnetic field. With the covariant formulation, we aim to extend the training of a beginning student in an introductory electromagnetism course, inserting fundamental physical bases that, for sure, will find strong relevance for their academic life, as much for those who will go on to graduate studies in the areas mentioned above, as for any physics student at the undergraduate and/or graduate level, regardless of the specific area eventually chosen