Lorenz gauge condition

From Wikipedia, the free encyclopedia

The Lorenz gauge (or Lorenz gauge condition) was published by the Danish physicist Ludvig Lorenz. The Lorenz gauge is often erroneously spelled as 'Lorentz gauge', many people believing that Dutch physicist Hendrik Lorentz was the first to state the condition presumably because the condition is Lorentz invariant; it doesn't change under the transformations due to H. Lorentz. In fact, it was the Danish physicist, L. Lorenz, who first published this condition.

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' work was not received well by James Clerk Maxwell (primarily because of his own labors over the electric and magnetic fields). Lorenz' 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' 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, 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.