Liénard–Wiechert potential

Liénard–Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential in the Lorenz gauge. Built directly from Maxwell's equations, these potentials describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum-mechanical effects. Electromagnetic radiation in the form of waves can be obtained from these potentials. These expressions were developed in part by Alfred-Marie Liénard in 1898 and independently by Emil Wiechert in 1900.

The Liénard–Wiechert potentials ${\displaystyle \varphi }$ (scalar potential field) and ${\displaystyle \mathbf {A} }$ (vector potential field) are for a source point charge ${\displaystyle q}$ at position ${\displaystyle \mathbf {r} _{s}}$ traveling with velocity ${\displaystyle \mathbf {v} _{s}}$:

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