Maxwell's Equations from the Ambient Electrical Energy Axioms

Maxwell's equations form the foundation of classical electrodynamics, optics, and electrical engineering. They describe every electromagnetic phenomenon with extraordinary precision. But all these equations continue to remain as empirical postulates. Gauss's law was inferred from Coulomb's measurements, Faraday's law from his induction experiments, and the Ampère-Maxwell law from a combination of experiment and theoretical consistency. Lenz's law has remained an observed direction with no mechanism. The displacement current was Maxwell's theoretical addition to close a mathematical inconsistency. Most derivations of these laws contain some circularity and do not proceed from generic primitive axioms. This paper derives all the Maxwell equations, Lenz's law, the wave equation, and the speed of light c = 1/ √ (µ_0 ε0) from the single axiom: the Ambient Electrical Energy E_0 fills all space, charges and currents create&nbsp;<span>smooth deformations of E_0, and minimizes its total deformation energy. The&nbsp;</span><span>dynamic deformation is described by a scalar potential V and a vector potential A.&nbsp;</span><span>The total energy functional is written entirely in terms of these potentials before&nbsp;</span><span>the notion of field is introduced.&nbsp;</span><span>The minimization of energy deformation with respect to V gives Gauss’s law&nbsp;</span><span>and the minimization with respect to A gives the Amp`ere-Maxwell&nbsp;law, with the&nbsp;</span><span>displacement current arising naturally as the kinetic energy of the time-varying&nbsp;</span><span>translational deformation rather than as a theoretical addition. The electric field E&nbsp;</span><span>and the magnetic field B are then defined as the combinations that the minimization&nbsp;</span><span>produced. Faraday’s law and the absence of magnetic monopoles follow immediately&nbsp;</span><span>as mathematical identities of these definitions.&nbsp;</span><span>The wave equation follows from combining Faraday’s law and the Amp`ere-Maxwell&nbsp;</span><span>law, and c = 1/√(μ_0ε_0) is E_0’s natural propagation speed, set by the ratio of its&nbsp;</span><span>translational stiffness ε_0 to its rotational inertia μ_0.</span>