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Concept
Liénard–Wiechert potential
Summary
The 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. 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. 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.
retarded potential
The retarded time is defined, in the context of distributions of charges and currents, as
where is the observation point, and is the observed point subject to the variations of source charges and currents.
For a moving point charge whose given trajectory is ,
is no more fixed, but becomes a function of the retarded time itself. In other words, following the trajectory
of yields the implicit equation
which provides the retarded time as a function of the current time (and of the given trajectory):
The Liénard–Wiechert potentials (scalar potential field) and (vector potential field) are, for a source point charge at position traveling with velocity :
and
where:
is the velocity of the source expressed as a fraction of the speed of light;
is the distance from the source;
is the unit vector pointing in the direction from the source and,
The symbol means that the quantities inside the parenthesis should be evaluated at the retarded time .
This can also be written in a covariant way, where the electromagnetic four-potential at is:
where and is the position of the source and is its four velocity.
We can calculate the electric and magnetic fields directly from the potentials using the definitions:
and
The calculation is nontrivial and requires a number of steps. The electric and magnetic fields are (in non-covariant form):
and
where , and (the Lorentz factor).
Official source
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