Maxwell's Equations
Maxwell's equations are the four coupled partial differential equations that, together with the Lorentz force law, constitute classical electromagnetic theory: Gauss's law for the electric field, the absence of magnetic monopoles, Faraday's law of induction, and the Ampère–Maxwell law with its displacement-current term.
The standard account
James Clerk Maxwell built the theory in stages — "On Physical Lines of Force" (1861–62), which developed it from a mechanical model of molecular vortices in an aether; "A Dynamical Theory of the Electromagnetic Field" (1865); and the Treatise on Electricity and Magnetism (1873). The decisive step was the displacement current, a term added to Ampère's circuital law which makes the system consistent with conservation of charge and, crucially, yields a wave equation. The waves travel at 1⁄√(ε₀μ₀), a quantity computable from purely electrostatic and magnetostatic measurements, and Maxwell found it equal within experimental error to the measured speed of light — the identification of light as an electromagnetic wave.
Two historical facts matter for what follows. First, the four equations universally taught today are not the ones Maxwell wrote. The Treatise presented a larger system — commonly counted as twenty equations in twenty variables, using quaternion notation — and it was Oliver Heaviside who in the mid-1880s recast them into the compact four-equation vector form, discarding the scalar and vector potentials as primary quantities. Second, the equations govern the fields only; they do not by themselves say what force a field exerts on a charge. That requires the separately postulated Lorentz force.
The equations are Lorentz-invariant as they stand, which is why they required no amendment when special relativity appeared in 1905; historically the influence ran the other way, with electromagnetism serving as the model relativity was built to match. Their predictions are among the best-tested in physics, and quantum electrodynamics reproduces them in the classical limit.
On this wiki
The literature collected here does not generally dispute that the standard equations work. It disputes their foundation, their completeness, and in several cases their physical interpretation. The main lines of objection are set out in detail at Electromagnetism; what follows identifies where each is argued.
The Heaviside reduction. A recurring historical claim is that the compression of Maxwell's system into four vector equations discarded physically meaningful structure. Tom Bearden argues this in Maxwell's Lost Unified Field Theory (1988); Eric R Laithwaite treats the episode historically in Oliver Heaviside - Establishment Shaker. Roland H Dishington makes a related but narrower point in Correcting the Texts: Fundamental Errors in Electromagnetics (2012) — that the scalar and vector potentials are the fundamental physical entities and E and B merely their force-related representations, and that treating the fields as primary produces wrong energy densities and confusion about energy flow. Whether physical content was actually lost is disputed among the critics themselves; what they agree on is that "Maxwell's equations" is a misnomer for Heaviside's.
Displacement current. The most radical objection here comes from Ivor Catt, who argues across his own Displacement Current (1978), Maxwell's Equations Revisited (1980) and The Death of Electric Current (1982) that displacement current is a fiction introduced to patch a theory, that electric current as ordinarily conceived does not exist, and that what propagates is a transverse electromagnetic step in the space between conductors. The Catt Question puts the challenge in its sharpest form. David Tombe approaches the same term from the opposite direction in his own Displacement Current (2008) and Ampère’s Circuital Law and Displacement Current, arguing that Maxwell conceived it as a real displacement in a sea of molecular vortices and that the modern justification by charge conservation misrepresents both its origin and its effect. Harry Hamlin Ricker traces the history in The History of Displacement Current, and Lefteris A Kaliambos examines its consequences in Impact of Maxwell's Equation of Displacement Current on Electromagnetic Laws and Comparison of the Maxwellian Waves With our Model of Dipolic Particles.
Modified and generalised systems. Thomas E Phipps proposed replacing the partial time derivative with a total convective derivative, yielding equations invariant under the Hertzian rather than the Lorentz group. Jaroslav G Klyushin has developed a generalised system from which, he argues, the Lorentz, Ampère, Whittaker, Weber and Spencer force expressions all follow as special cases — see A Generalized Formula for the Lorentz Force Density and Maxwell Equations and Wave Solution of Generalized Maxwell Equations and Quantum Mechanics – Part I. Domina Eberle Spencer and colleagues worked on the Gaussian form of the equations (The Gaussian Form of the Maxwell Equations, 1994), and Uma Y Shama on New Maxwell Equations (1995). Charles William Lucas sets the electrodynamics of extended, elastic particles against the standard system in Electrodynamics of Real Particles vs. Maxwell's Equations, Relativity Theory and Quantum Mechanics.
Derivation from a medium. Since Maxwell's own route was through an aether model, several contributors here attempt to recover the equations from a mechanical medium rather than postulate them — for example Deriving Maxwell's Equations from a Postulated Two-Component Solid Aether. See Aether and Category:Aether.
The force law is separate. Georg Galeczki argues in What Does the Lorentz Force Have to do with Maxwell's Equations? (1998) that the Lorentz force has nothing mathematically or physically to do with the field equations, and that electrodynamics can be built from a force law between moving charges without postulating field equations at all. This is the entry point to the Weber and Ampère traditions documented at Coulomb's Law and Lorentz Force.
Open questions
Genuine unsettled questions surround the classical equations even in mainstream physics: the divergent self-energy of a point charge, the pathological runaway and pre-acceleration solutions of the Abraham–Lorentz radiation-reaction equation, and the fact that no magnetic monopole has ever been found although nothing in the theory forbids one. These are acknowledged difficulties, not manufactured ones, and several of the papers above take them as their starting point.