Maxwell's equations give a field-centric view of electricity and magnetism. They describe how electromagnetic fields arise, evolve, and propagate in space and time. By themselves, however, they do not specify how charged matter responds to a given field. That role is played by classical electrodynamics.
At a fundamental level, the motion of a charged particle influences the electromagnetic field, and the modified field in turn influences the particle’s motion. This feedback loop is fully captured only when the field equations and particle dynamics are treated together. In many practical settings, however, we consider charged particles whose influence on the field is negligible compared to that of the dominant sources. In this regime, the particle may be treated as a test charge moving in a prescribed electromagnetic field.
The central object of this section is the Lorentz force law, which governs the dynamics of charged matter in electromagnetic fields. This law plays a role in electrodynamics analogous to that of Newton’s second law in classical mechanics. When coupled with Maxwell’s equations and charge conservation, it yields a self-consistent classical theory of electromagnetic interactions.
Conservation of Charge
Conservation of charge is a fundamental principle stating that electric charge cannot be created or destroyed, only transported from one region of space to another. Throughout this section, we will primarily use differential (local) formulations, which describe conservation laws at each point in space and time.
Here, \(\rho\) denotes the charge density and \(\mathbf{J}\) the current density. This equation states that any local change in charge density must be accompanied by a corresponding flux of charge. In other words, charge can accumulate or deplete at a point only if there is a net flow of charge into or out of that region.
Lorentz Force Law
The motion of charged matter in electromagnetic fields is governed by the Lorentz force law. A particle of charge \(q\) moving with velocity \(\mathbf{v}\) in electric and magnetic fields \(\mathbf{E}\) and \(\mathbf{B}\) experiences a force
The electric field term produces a force parallel to the field and can perform work on the particle, changing its kinetic energy. The magnetic field term produces a force perpendicular to the particle’s velocity and therefore alters the direction of motion without changing the particle’s speed.
When combined with Newton’s second law, \(\mathbf{F} = m\mathbf{a}\), the Lorentz force law defines a system of differential equations governing the trajectory of a charged particle. In this sense, the Lorentz force plays a role in electrodynamics analogous to that of Newton’s second law in classical mechanics.
In many applications, the electromagnetic fields are treated as prescribed inputs generated by external sources, and the influence of the particle’s motion on the fields is neglected. This test-charge approximation is valid when the charge and associated currents are sufficiently small compared to the dominant field sources.
The rate of work done by the electromagnetic field on a charged particle is given by
which shows explicitly that magnetic forces do no work.
Together, conservation of charge and the Lorentz force law provide the fundamental link between electromagnetic fields and the motion of matter. These principles form the basis for analyzing the dynamics of charged particles, the behavior of currents in conductors, and the operation of electromechanical systems. In more complex settings, where many charges interact or where the motion of matter significantly alters the electromagnetic field, these ideas must be coupled self-consistently with Maxwell’s equations. Such regimes lead naturally to continuum descriptions, collective effects, and fully coupled field–matter models.