LienardWiechert

The Liénard–Wiechert potentials are a set of mathematical expressions in classical electromagnetism that describe the electromagnetic potential generated by a moving point charge. These potentials are crucial for understanding how the electric and magnetic fields propagate from a source charge and are fundamental to describing radiation from accelerating charges.

These potentials, developed by Alfred-Marie Liénard and Emil Wiechert independently in the early 20th century, take into account the finite speed of light. Unlike the static potentials (Coulomb potential for electric field and the Biot-Savart law for magnetic field), the Liénard–Wiechert potentials are time-dependent and reflect the fact that changes in the charge's position and velocity do not instantaneously affect the fields at distant points.

The scalar potential, denoted by $\phi$, and the vector potential, denoted by $\mathbf{A}$, are given by specific formulas involving the charge's position, velocity, and acceleration at a "retarded time." The retarded time is the time in the past when the charge was at a position such that its influence, traveling at the speed of light, is just reaching the observation point at the current time.

From these potentials, one can derive the electric and magnetic fields at any point in space and time. The Liénard–Wiechert potentials are essential for deriving the formula for electromagnetic radiation emitted by an accelerating charge, a cornerstone of classical electrodynamics.