Lorentz Force
The Lorentz force is the force exerted on a charged particle by electric and magnetic fields, F = q(E + v × B). Together with Maxwell's equations, which govern the fields themselves, it constitutes classical electromagnetic theory: the equations say what the fields do, and the Lorentz force says what the fields do to charges.
The standard account
The expression is older than its name. Hermann Grassmann published the equivalent force between current elements in 1845; Maxwell gave the form in the Treatise (1873); Oliver Heaviside wrote it for a moving point charge in 1889; and Hendrik Lorentz made it the foundation of his electron theory in the 1890s, which is how it acquired his name.
Its content is straightforward and heavily used. The electric term acts along E regardless of the particle's motion. The magnetic term acts perpendicular to both the velocity and B, does no work, and bends a charged particle moving in a uniform magnetic field into a circle of radius mv⁄qB — the basis of the cyclotron, the mass spectrometer, magnetic focusing in electron optics, and the operation of every electric motor.
Two features are important for what follows. First, the Lorentz force is not derivable from Maxwell's equations; it is an independent postulate of the theory, added alongside them. Second, the Grassmann–Lorentz form is not the only candidate. The original force law of André-Marie Ampère, published between 1820 and 1825, contains a component acting along the direction of current flow, which Grassmann's does not. For a complete closed circuit the two laws give identical results, so ordinary laboratory and engineering practice cannot distinguish them. They differ only for isolated current elements — that is, where a circuit is broken, deformed, or destroyed.
On this wiki
The Lorentz force is one of the most heavily contested items in the literature collected here, and the objections are of several distinct kinds. The broader picture is set out at Electromagnetism.
It is a separate postulate, and the theory does not need it. Georg Galeczki argues in What Does the Lorentz Force Have to do with Maxwell's Equations? (1998) that the force has nothing, mathematically or physically, to do with Maxwell's field equations — that properly written it is a phenomenological expression describing motion in external fields originating from decoupled systems, and that electrodynamics can be built instead from a force law directly between moving charges. He extends the point to relativity in What Does the Lorentz Force Have to do with Special Relativity?.
It violates Newton's third law. The Lorentz force between two moving charges is in general not equal and opposite; one charge can act on another without a matching reaction. Jaroslav G Klyushin presses this in On Electrodynamic Forces (2005), calling the formula asymmetric and non-universal, and offers a generalised expression intended to contain the Lorentz, Ampère, Whittaker, Weber and Spencer forms as special cases — see A Generalized Formula for the Lorentz Force Density and Maxwell Equations and A Field Generalization for the Lorentz Force Formula. Standard electrodynamics answers this objection by assigning momentum to the field itself, so that momentum is conserved overall; whether that answer is physical or bookkeeping is precisely what is disputed here.
Ampère's original law is the correct one. The largest body of work here concerns the longitudinal force. Peter Graneau and Neal Graneau argued in Ampere Electrodynamics (1993) and Only Ampere Forces Explain Railgun Recoil and Wire Explosions (1994) that wires carrying very large currents fragment into short segments under longitudinal tension, and that railgun recoil is transmitted through the conductors in a way the Lorentz force does not predict. Paul Wesley supplied theoretical support; Thomas E Phipps reported positive experimental results and argued in Ampere Tension and Newton's Laws (1993) that the whole issue is about Newton's third law; Panos Pappas conducted the much-discussed Ampère bridge experiments; Stefan Marinov contributed Experimentum Crucis for Magnetic Interaction; and James Keele, Jorge A Guala-Valverde, Rémi Saumont and Harold Aspden each added measurements or analyses. Domina Eberle Spencer and colleagues compared the candidate laws directly in The Force Between Current Elements (1994), noting that the Grassmann expression "often called the Lorentz force" is one of several consistent with the experimental record for closed circuits. Mainstream electrodynamics regards these results as artefacts of thermal, mechanical or contact effects rather than of a new force; the dispute is empirical and has never been fully closed.
It is unnecessary because Coulomb's law suffices. Jan Olof Jonson argues that no separate magnetic force exists — that magnetism is what Coulomb's law looks like when the analysis of moving charges is done correctly — and reads the Ampère bridge results as refuting the Lorentz force outright. See Turning Back to Coulomb's Law as a Basis for Electromagnetism (2008).
It follows from Galilean, not Lorentz, invariance. Charles William Lucas, deriving a universal force law for finite-size elastic charged particles in The Universal Electrodynamic Force (2011) and Derivation of the Classical Universal Electrodynamic Force, argues that the Lorentz force can be obtained assuming Galilean invariance and that the relativistic derivation is therefore not required.
Further material is indexed under Category:Electrodynamics and Category:Electromagnetism.
Open questions
Two matters here are genuinely unsettled rather than merely disputed. The first is the status of the Ampère longitudinal force: the experiments are difficult, the effects are small, and the interpretation of exploding-wire and railgun results remains contested in the primary literature. The second is more theoretical — the correct treatment of radiation reaction, the force a charge exerts on itself, where the standard Abraham–Lorentz equation has pathological runaway and pre-acceleration solutions that no classical formulation has removed.