Liénard–Wiechert potential

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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 quadropoles in the same way as the Liénard-Wiechert potentials are related to a point charge were computed by Ribarič and Šušteršič in 1995.^[2]

Contents 1 Implications 2 Equations 2.1 Terms 2.2 Definition of Liénard-Wiechert potentials 2.3 Corresponding values of electric and magnetic fields 3 See also 4 References

[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

[edit] Terms

• r, the field point.
• t, time.
• s(t), the position of the point charge (which, of course, may vary in time).
• v(t), the velocity of point charge.
• T, the retarded time—implicitly determined by the equation . Loosely speaking, the retarded time takes into account the time it takes for electromagnetic information to propagate from the source (point charge) to the observer (at the field point).

• V, scalar potential field.
• A, vector potential field.
• (or its magnitude, as will be clear from context).

[edit] Definition of Liénard-Wiechert potentials

[edit] Corresponding values of electric and magnetic fields

[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

[edit] References

1. ^ http://verplant.org/history-geophysics/Wiechert.htm
2. ^ 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.