Lorenz gauge condition

From Wikipedia, the free encyclopedia

In electromagnetism, the Lorenz gauge or Lorenz gauge condition is a particular choice of gauge for the electromagnetic field which was first proposed by the Danish physicist Ludvig Lorenz. It is a Lorentz invariant gauge condition. It is frequently called the "Lorentz gauge" because of confusion with Hendrik Lorentz, after whom Lorentz invariance is named.

1 Description
2 History
3 See also
4 External articles, references, and further reading

[edit] Description

In electromagnetism, the Lorenz gauge condition is a general method of calculation of time-dependent electromagnetic fields in which retarded potentials are introduced. ^[1] The condition is a gauge fixing in which,

$\partial_{a}A^a = A^a{}_{,a} = 0 \!$

where $A a$ is the four-potential, the comma denotes a partial differentiation and the repeated index indicates that the Einstein summation convention is being used. This gauge has the advantage of being Lorentz invariant. It still leaves some residual gauge degrees of freedom, but they propagate freely at the speed of light, so they are insignificant.

In ordinary vector notation and SI units, the condition is:

$\nabla\cdot{\mathbf A} + \frac{1}{c^2}\frac{\partial\phi}{\partial t}=0.$

where A is the magnetic vector potential and φ is the electric potential; see also Gauge fixing.

In Gaussian units the condition is:

$\nabla\cdot{\mathbf A} + \frac{1}{c}\frac{\partial\phi}{\partial t}=0.$

[edit] History

When originally published, Lorenz's work was not received well by James Clerk Maxwell. Maxwell had eliminated the Coulomb electrostatic force from his derivation of the electromagnetic wave equation since he was working in what would nowadays be termed the Coulomb gauge. The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying electric field. Lorenz's work was the first symmetrizing shortening of Maxwell's equations after Maxwell himself published his 1865 paper. In 1888, retarded potentials came into general use after Heinrich Rudolf Hertz's experiments on electromagnetic waves. In 1895, a further boost to the theory of retarded potentials came after J. J. Thomson's interpretation of data for electrons (after which investigation into electrical phenomena changed from time-dependent electric charge and electric current distributions over to moving point charges). ^[2]

[edit] See also

[edit] External articles, references, and further reading

General

Eric W. Weisstein, "Lorenz Gauge".
^ Kirk T. McDonald, "The Relation Between Expressions for Time-Dependent Electromagnetic Fields Given by Jefimenko and by Panofsky and Phillips". Dec. 5, 1996
- ^ Ibid.

Further reading

L. Lorenz, "On the Identity of the Vibrations of Light with Electrical Currents" Philos. Mag. 34, 287-301, 1867.
J. van Bladel, "Lorenz or Lorentz?". IEEE Antennas Prop. Mag. 33, p. 69, 1991.
R. Becker, "Electromagnetic Fields and Interactions", chap. DIII. Dover Publications, New York, 1982.
A. O'Rahilly, "Electromagnetics", chap. VI. Longmans, Green and Co, New York, 1938.

History

R. Nevels, C.-S. Shin, "Lorenz, Lorentz, and the gauge", IEEE Antennas Prop. Mag. 43, 3, pp. 70-1, 2001.
E. Whittaker, "A History of the Theories of Aether and Electricity", Vols. 1-2. New York: Dover, p. 268, 1989.