The set of equations known today as Maxwell's equations along with a few constitutive equations lie at the heart of classical electromagnetism. A common misconception held by many is that Maxwell's equations are essential, and that classical electromagnetic theory is settled science and is no longer an active field of investigations. We will review the four Maxwell's equations and related equations, their supporting experimental evidence, the field concept, and the Lorentz and Ritz force laws. We will give a brief outline of two approaches to classical electromagnetism which bypass Maxwell's equations, the propagated potential approach and the direct action approach which bypasses even the field concept. We will also give a few indications of some current research being done in comparing the fitness of the Lorentz and Ritz force laws, and point out some area of research yet to be explored concerning the direct action approach to classical electromagnetism using the Ritz force formula.
Tran, Max, "Evidence for Maxwell's equations, fields, force laws and alternative theories of classical electrodynamics" (2018). CUNY Academic Works.