Coulomb's Law
Coulomb's law states that the electrostatic force between two point charges acts along the line joining them, is proportional to the product of the charges, and falls off as the inverse square of their separation: F = k q₁q₂⁄r². It is the foundation of electrostatics and, for most of the researchers catalogued on this wiki, the one piece of classical electrodynamics nobody wants to change.
The standard account
Charles-Augustin de Coulomb published the law in 1785, having measured the force with a torsion balance of his own design. Henry Cavendish had established the inverse-square dependence more precisely a decade earlier, by a null method — showing that no charge resides on the inner of two connected concentric spheres, which holds only if the exponent is exactly 2 — but did not publish, and the result became known only when Maxwell edited his papers a century later.
The null method remains the basis of the modern tests, and they are extraordinarily tight: modern experiments constrain any departure of the exponent from 2 to less than about one part in 10¹⁵. The same experiments are read as an upper bound on the photon rest mass, since a massive photon would give a Yukawa-type exponential cutoff rather than a pure inverse square.
In the standard formulation Coulomb's law is a special case, not an axiom: it is what Gauss's law — the first of Maxwell's equations — gives for a static point charge, and it is the low-velocity limit of the full electrodynamic interaction. The law as written applies to charges at rest; once charges move, the standard theory says that the electric force is supplemented by a magnetic one, and the whole is expressed by the Lorentz force law.
The inverse-square form has a structural consequence worth noting: it makes the electrostatic self-energy of a genuine point charge infinite. Classical electrodynamics has never resolved this, and it is renormalised away rather than answered in quantum electrodynamics.
On this wiki
Coulomb's law occupies a distinctive position in the critical literature gathered here. Where relativity, the Lorentz force and even Maxwell's equations are contested from many directions (see Electromagnetism), Coulomb's law is far more often treated as bedrock — and several researchers argue that it is not merely correct but sufficient, that the rest of electrodynamics is Coulomb's law applied properly to charges in motion.
Coulomb's law as the whole of electromagnetism. Jan Olof Jonson has pursued this position for two decades. In Turning Back to Coulomb's Law as a Basis for Electromagnetism (2008) he argues that the Liénard–Wiechert potentials from which the standard fields are derived were fallaciously obtained, and that if the potentials fail the theory built on them must go; in their place he proposes to recover magnetic forces, electromagnetic induction and the wave–particle behaviour of light from Coulomb's 1785 law alone. He extends the argument to induction in The Use of Finite Differences on Electric Currents Gives Credit to Coulomb's Law as Causing Electromagnetic Forces, thereby Explaining Electromagnetic Induction (2013), and has applied it to the Ampère bridge experiments, which he reads as refuting the Lorentz force while confirming Coulomb's law.
Generalisation to moving charges: the Gauss–Weber line. A second and older tradition generalises Coulomb's law directly rather than replacing it with fields. Wilhelm Weber's force law of 1846 is a Coulomb law modified at higher orders by terms depending on the relative velocity and relative acceleration of the two charges, acting directly between them with no field as intermediary; it reduces to Coulomb's law in the static limit. Thomas E Phipps derived a modernised version from first principles in Derivation of a Modernized Weber Force Law (1992), invoking a limiting relative particle speed, with Weber's original recovered as the low-speed case. Andre K T Assis is the tradition's principal modern advocate — see Relational Mechanics and Modern Experiments Related to Weber's Electrodynamics — and Paul Wesley extended it to fields, waves and radiation.
Jaroslav G Klyushin gives the historical framing explicitly in On Electrodynamic Forces (2005): electrodynamics began when Gauss and Weber generalised Coulomb's law to moving charges, that programme was eclipsed rather than refuted by the Maxwell field approach, and it has not been exhausted. His own generalised formula is intended to contain the Lorentz, Ampère, Whittaker, Weber and Spencer expressions as special cases. Domina Eberle Spencer and colleagues pursued the Gauss–Weber route experimentally and theoretically — see The Force Between Current Elements (1994).
Coulomb's law and particle structure. Charles William Lucas derives a single universal force law for finite-size elastic charged particles, arguing in The Universal Electrodynamic Force (2011) that the point-charge idealisation behind the textbook Coulomb law is precisely what fails in scattering experiments. James Keele reworks the electron–proton case in Force and Rotation (2010), obtaining the Coulomb force in a form he argues differs from the customary presentation. Athanassios A Nassikas treats the Coulomb interaction within a unified scheme in Minimum Contradictions Everything & Ether and Forces' Unification.
Further material is indexed under Category:Electrodynamics and Category:Electromagnetism.
Open questions
Two difficulties with the law are genuine and acknowledged in mainstream physics rather than manufactured here. The first is the infinite self-energy of a point charge, mentioned above, which has no classical resolution. The second is that Coulomb's law is strictly a statement about static charges; extending it to moving ones is not a matter of substitution, and exactly how it should be extended is the historical fork — Weber's direct action-at-a-distance on one side, the Maxwell–Lorentz field formulation on the other — that much of the literature above is still arguing about.