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Liénard-Wiechert potentials describe the classical electromagnetic effect of a moving electric point charge in terms of a vector potential and a scalar potential. 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^[1] and continued into the early 1900s.

The Liénard-Wiechert potentials can be generalized according to gauge theory.

The explicit expressions for potentials related to moving dipoles and quadrupoles in the same way as the Liénard-Wiechert potentials are related to a point charge were computed by Ribarič and Šušteršič in 1995.^[2]

[edit] Implications

The study of classical electrodynamics was instrumental in 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 more complex analysis of relativistic moving particles.

The Liénard–Wiechert description is accurate for a large, independent moving particle, but breaks down at the quantum level.

Quantum mechanics sets important constraints on the ability of a particle to emit radiation. The classical formulation, as laboriously described by these equations, expressly violates experimentally observed phenomena. For example, an electron around an atom does not emit radiation in the pattern predicted by these classical equations. Instead, it is governed by quantized principles regarding its energy state. In the later decades of the twentieth century, quantum electrodynamics helped bring together the radiative behavior with the quantum constraints.

[edit] Equations

The force on a particle at a given location $r$ and time $t$ depends on the position of the source particles at an earlier time $t r$ due to the finite speed, c, at which electromagnetic information travels. For example, a particle on Earth 'sees' a charged particle on the Moon as it was 1.5 seconds ago and a charged particle on the Sun as it was 500 seconds ago. This earlier time in which an event happens such that a particle at location $r$ 'sees' this event at a later time $t$ is called the retarded time, $t r$ . The retarded time varies with position; for example the retarded time at the Moon is 1.5 seconds before the current time and the retarded time on the Sun is 500 s before the current time. The retarded time can be calculated as:

$t_r=t-\frac{\mathcal{R}}{c},$

where $\scriptstyle\mathcal{R}$ = |r-s_r| and s_r is the location of the particle at the retarded time.

[edit] Definition of Liénard-Wiechert potentials

The Liénard-Wiechert potentials $V$ and $A$ , where $V$ is the scalar potential field and $A$ is the vector potential field, forms a potential representation of the fields of a moving point charge having a charge $q$ and velocity $v r$ (at the retarded time $t r$ ) such that:

$V(\mathbf{r}, t) = \frac{1}{4\pi\epsilon_0} \left(\frac{q c}{\mathbf{\mathcal{R}}c - \mathbf{\mathcal{R}} \cdot \mathbf{v}(T)}\right)$

and

$\mathbf{A}(\mathbf{r}, t) = \left(\frac{\mathbf{v}(T)}{c^2}\right) V(\mathbf{r}, t).$

[edit] Corresponding values of electric and magnetic fields

$\mathbf{E} = - \nabla V - \dfrac {\partial \mathbf{A}} { \partial t }$

$\mathbf{B} = \nabla \times \mathbf{A}$

[edit] See also

Maxwell's equations which govern classical electromagnetism
Classical electromagnetism for the larger theory surrounding this analysis
Special relativity, which was a direct consequence of these analyses
Rydberg formula for quantum description of the EM radiation due to atomic orbital electrons
Jefimenko's equations
Larmor formula
Abraham-Lorentz force
Inhomogeneous electromagnetic wave equation
Wheeler-Feynman absorber theory also known as the Wheeler-Feynman time-symmetric theory

[edit] References

^ http://verplant.org/history-geophysics/Wiechert.htm
^ Ribarič, M., and L. Šušteršič, Expansion in terms of time-dependent, moving charges and currents, SIAM J. Appl. Math. 55, 593-624.

Griffiths, David. Introduction to Electrodynamics. Prentice Hall, 1999. ISBN 0-13-805326-X.

[0] ttp://verplant.org/history-geophysics/Wiechert.htm

[1] Ribarič, M., and L. Šušteršič, Expansion in terms of time-dependent, moving charges and currents, SIAM J. Appl. Math. 55, 593-624.

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