[Paper Review] Can we derive the Lorentz force from Maxwell's equations?

Valery P. Dmitriyev|ArXiv.org|Jun 10, 2002

Quantum and Classical Electrodynamics3 citations

TL;DR

This paper investigates whether the Lorentz force law can be derived solely from Maxwell’s equations in the Coulomb gauge. It shows that while Maxwell’s equations alone are insufficient, assuming the electric force is known and the magnetic force is perpendicular to velocity allows derivation of the Lorentz force via energy-momentum balance and Lagrangian formalism, revealing the need for additional physical postulates beyond Maxwell’s equations.

ABSTRACT

The Lorentz force can be obtained from Maxwell's equations in the Coulomb gauge provided that we assume that the electric portion of the force acted on a charge is known, and the magnetic component is perpendicular to the velocity of motion of the charged particle.

Motivation & Objective

• To determine whether the Lorentz force can be rigorously derived from Maxwell’s equations alone.
• To identify the minimal set of additional assumptions required to extract the force law from Maxwell’s equations.
• To clarify the role of the Lorentz force in classical electrodynamics by analyzing its consistency with energy and momentum conservation.
• To demonstrate that the Lorentz force emerges from a Lagrangian derived via energy balance and field-particle interaction terms.
• To investigate the necessity of postulating the magnetic force's velocity-perpendicular nature as a non-derivable physical input.

Proposed method

• Derives two energy-like integrals from Maxwell’s equations in the Coulomb gauge, using vector calculus and integration by parts.
• Applies the continuity equation and current density to model point charges via Dirac delta functions and indicator functions.
• Performs spatial integration over point charges to extract cross-term interactions between two charges.
• Uses the time derivative of the total energy expression to derive a key relation (equation 27) linking field potentials, velocities, and forces.
• Assumes the form of the force law with a velocity-perpendicular magnetic term and verifies consistency with energy conservation.
• Constructs a Lagrangian from the derived energy expression and derives the full Lorentz force law via variational principles.

Experimental results

Research questions

• RQ1 Can the Lorentz force be derived purely from Maxwell’s equations without additional assumptions?
• RQ2 What minimal set of physical postulates is required to derive the Lorentz force from Maxwell’s equations?
• RQ3 Why is the magnetic force term necessarily perpendicular to the particle’s velocity in the Lorentz force law?
• RQ4 How does energy conservation constrain the form of the electromagnetic force on a moving charge?
• RQ5 What is the role of the vector potential and scalar potential in determining the force via field-particle interaction?

Key findings

• The Lorentz force cannot be derived from Maxwell’s equations alone; additional postulates are required.
• Assuming the electric force is known and the magnetic force is perpendicular to velocity allows derivation of the Lorentz force.
• The derived energy balance relation (equation 27) links the time derivative of charge motion to field interactions and forces.
• The interaction Lagrangian (equation 31) yields the correct Lorentz force law, confirming consistency with classical electrodynamics.
• The magnetic force term arises naturally as proportional to the cross product of velocity and the curl of the vector potential.
• Self-interaction terms (ε₀, w₀) are shown to be negligible in the final force expression, confirming the validity of the derived force law.