# Forces on charges, currents, and electromagnetic waves

Authored explanation; editorial review pending.

Electric charge is the intrinsic property of matter that produces and responds to electromagnetic fields. A charge q at rest in an electric field E experiences an electrostatic force F = q E. When moving with velocity v through a magnetic field B, the charge experiences an additional perpendicular magnetic force F = q (v x B). Together, these form the Lorentz force.

Moving a charge through an electric potential difference Delta V transfers potential energy Delta W = q Delta V. For a fundamental electron charge e = 1.602e-19 C accelerated across a potential of 1 V, the energy gained is defined as 1 electron-volt (1 eV = 1.602e-19 J).

In microscopic matter, an electron bound to an equilibrium position by a restoring force acts as a harmonic oscillator or resonator. When disturbed by incoming electromagnetic waves, it absorbs and reradiates energy at its natural resonant frequency nu_0. Einstein's 1905 light-quanta paper opened with Planck's model of such resonators in thermal equilibrium with radiant energy.

Maxwell's electrodynamics expresses four physical laws in words: electric charges act as sources of electric flux; magnetic field lines are closed loops with no isolated magnetic charges; a time-varying magnetic field induces a circulating electric field (Faraday induction); and electric currents alongside time-varying electric fields generate circulating magnetic fields (Maxwell-Ampere law). Combined, these equations govern self-propagating electromagnetic waves traveling at speed c in vacuum.

Einstein pointed out in 1905 that classical electrodynamics treated the relative motion of a magnet and a conductor with an artificial asymmetry: moving the magnet created an electric field in space that drove current, whereas moving the conductor created no electric field but rather a magnetic Lorentz force on electrons. Yet the physical current was identical in both descriptions.

## Worked example

$$
\mathbf{F} = q\left(\mathbf{E} + \mathbf{v} \times \mathbf{B}\right), \quad \Delta W = q \Delta V
$$

Force equals charge times the sum of electric field and the cross product of velocity with magnetic field, and work equals charge times potential difference.

Accelerating an electron of charge e = 1.602e-19 Coulombs across an electric potential difference of Delta V = 1.0 Volt gives kinetic energy Delta W = (1.602e-19 C)(1.0 V) = 1.602e-19 Joules = 1 eV. An electron bound with effective spring constant k_s and mass m oscillates at natural frequency nu_0 = (1 / 2 pi) sqrt(k_s / m).

Charge is the property that makes electric forces; fields mediate force between separated charges without action-at-a-distance.