Liénard–Wiechert potential

Stemming directly from Maxwell's equations, these describe the complete, relativistically correct, time-varying electromagnetic field for a point charge in arbitrary motion, but are not corrected for quantum mechanical effects.

These expressions were developed in part by Alfred-Marie Liénard in 1898[1] and independently by Emil Wiechert in 1900.

The first term is link to the static field of the charge when it moves at a constant velocity.

This term is connected with the "static" part of the electromagnetic field of the charge.

On the illustration, we observe an event that happened in the center of the sphere and that propagated at the speed of light.

Note that outside the propagation sphere, the electric field has its initial state (no connection with the event we observe).

vector potentials satisfy the nonhomogeneous electromagnetic wave equation where the sources are expressed with the charge and current densities

Generally, the retarded solutions for the scalar and vector potentials (SI units) are

satisfy the homogeneous wave equation with no sources and boundary conditions.

These integrals are difficult to evaluate in their present form, so we will rewrite them by replacing

The study of classical electrodynamics was instrumental in Albert Einstein's development of the theory of relativity.

Analysis of the motion and propagation of electromagnetic waves led to the special relativity description of space and time.

The Liénard–Wiechert formulation is an important launchpad into a deeper analysis of relativistic moving particles.

Multiplying electric parameters of both problems by arbitrary real constants produces a coherent interaction of light with matter which generalizes Einstein's theory[7] which is now considered as founding theory of lasers: it is not necessary to study a large set of identical molecules to get coherent amplification in the mode obtained by arbitrary multiplications of advanced and retarded fields.

It is important to take into account the zero point field discovered by Planck.

Moreover, introducing the fluctuations of the zero point field produces Willis E. Lamb's correction of levels of H atom.

It introduces quantization of normal modes of the electromagnetic field in assumed perfect optical resonators.

The force on a particle at a given location r and time t depends in a complicated way on the position of the source particles at an earlier time tr due to the finite speed, c, at which electromagnetic information travels.

Only electromagnetic wave effects depend fully on the retarded time.

The first term describes near field effects from the charge, and its direction in space is updated with a term that corrects for any constant-velocity motion of the charge on its distant static field, so that the distant static field appears at distance from the charge, with no aberration of light or light-time correction.

This term, which corrects for time-retardation delays in the direction of the static field, is required by Lorentz invariance.

A charge moving with a constant velocity must appear to a distant observer in exactly the same way as a static charge appears to a moving observer, and in the latter case, the direction of the static field must change instantaneously, with no time-delay.

Thus, static fields (the first term) point exactly at the true instantaneous (non-retarded) position of the charged object if its velocity has not changed over the retarded time delay.

The second term, however, which contains information about the acceleration and other unique behavior of the charge that cannot be removed by changing the Lorentz frame (inertial reference frame of the observer), is fully dependent for direction on the time-retarded position of the source.

Thus, electromagnetic radiation (described by the second term) always appears to come from the direction of the position of the emitting charge at the retarded time.

For example, if the source charge has existed for an unlimited amount of time, during which it has always travelled at a speed not exceeding

Intuitively, as the source charge moves back in time, the cross section of its light cone at present time expands faster than it can recede, so eventually it must reach the point

This is not necessarily true if the source charge's speed is allowed to be arbitrarily close to

In this case the cross section of the light cone at present time approaches the point

The intuitive interpretation is that one can only ever "see" the point source at one location/time at once unless it travels at least at the speed of light to another location.

Electromagnetic field (arbitrary unit) of a positive point charge moving at constant speed. When , the electromagnetic field reduces to electrostatic field (in blue). Due to its insignificance at large distance, this field is ignored in high energy physics when computing electromagnetic radiation power. However, it is used for wakefield modeling. The Coulomb field is also known as pancake field when . [ 5 ] [ 6 ]
Electromagnetic field (arbitrary unit) of a positive accelerated point charge (far from the source). Acceleration is parallel to velocity. When the particle decelerates, this leads to braking radiations which can be strong when particles collide.
Electromagnetic field (arbitrary unit) of a positive accelerated point charge (far from the source). Acceleration is perpendicular to velocity. It is the synchrotron radiation .
Electrostatic field of a positive charge moving at a constant velocity. See Note.