Proof of Gauss' Law for Electrodynamics?

[Michael.7]

I’ve been searching for a proof, using the equation for the electric field due to a moving point charge – given, for example, on page 438 of the Third Edition of David Griffith’s Introduction to Electrodynamics (equation 10.65) – that Gauss’s law holds for a moving point charge. There is no such proof in Griffith’s textbook, or in the Third edition of Jackson’s Classical Electrodynamics, and the proof of Gauss’s law in the case of electrostatics that Jackson gives in the case of electrostatics on pages 27-8 would not seem to generalize to electrodynamics, since the varying retarded times and locations of the point charge giving rise to the electric field at different parts of the surrounding surface would seem to entail that one cannot add up the solid angles and arrive at the value 4π. The other textbook that I have is the Third Edition of Foundations of Electromagnetic Theory by John Reitz et al, and I have not been able to find any proof of Gauss’s law in the case of electrodynamics. (The last reference to Gauss’s law given in the index refers to page 336, and the Liénard-Wiechert equations for retarded potentials, followed by the general equations for the field due to a moving point charge, are not introduced until pages 470ff.)

In short, does anyone know of a textbook where there is a proof, using the equations for the field due to a moving point charge, that Gauss’s law holds for a moving point charge? Thanks!

 

[vanhees71]

There is no proof for Gauss's Law. It's one of the most fundamental equations of contemporary physics, namely one of Maxwell's equations, which cannot be "derived". If you have a moving charge, the proper solution (neglecting radiation reactions on the charge by its own electromagnetic field, which is a very difficult problem) for the electromagnetic field is given by the Lienard-Wiechert retarded potentials (or the equivalent Jefimenko equations for the electromagnetic field itself, which have the advantage of being gauge invariant). It's clear that they must solve all Maxwell's equations, including Gauss's Law, because that's what they are supposed to solve ;-).

 

[stedwards]

There is one sort of proof, if the axioms of electromagnetism are constructed differently than maxwell's 4 equations as vanhees said.

If we are to posit that the lagrangian (density) of electromagnetism is extremal, that the charge and current are invariant with respect to the vector potentials, and the lagrangian has the usual form, then we get all of maxwell's equations as a result.

These are equivalent structures. One could posit maxwell's equations and derive the lagrangian.

 

[vanhees71]

Well, Maxwell's equations are equivalent to the action principle, and of course the modern way to motivate Maxwell's equations is to write down a Lagrangian for a massless spin-1 field, but of course one must know that the electromagnetic field is described by such a field beforehand. From this it follows necessary from some group-representation theory of the Poincare group (Wigner 1940; see Weinberg Quantum Theory of Fields, vol. 1) that it must be described as a gauge field, and from the U(1) gauge symmetry nearly everything follows :-).

 

[tech99]

Well, Maxwell's equations are equivalent to the action principle, and of course the modern way to motivate Maxwell's equations is to write down a Lagrangian for a massless spin-1 field, but of course one must know that the electromagnetic field is described by such a field beforehand. From this it follows necessary from some group-representation theory of the Poincare group (Wigner 1940; see Weinberg Quantum Theory of Fields, vol. 1) that it must be described as a gauge field, and from the U(1) gauge symmetry nearly everything follows :-).

Apologies for the following. The interpretation of Gauss's Law we were taught at school said that 4xpi lines of force originate from a unit charge (cgs system of units). If the charge moves, the lines may become distorted (hence radiation of energy), but their number is unaltered - they cannot detach from the charge.

Well, Maxwell's equations are equivalent to the action principle, and of course the modern way to motivate Maxwell's equations is to write down a Lagrangian for a massless spin-1 field, but of course one must know that the electromagnetic field is described by such a field beforehand. From this it follows necessary from some group-representation theory of the Poincare group (Wigner 1940; see Weinberg Quantum Theory of Fields, vol. 1) that it must be described as a gauge field, and from the U(1) gauge symmetry nearly everything follows :-).

It looks to me that if we try to ascertain the magnitude of the moving charge at a great distance, then it is not possible, because we might be measuring a radiation field, and it could originate from a small charge accelerating greatly or a large charge accelerating a little. On the other hand, closer than a certain distance, where the field is essentially radial, we know that 4 pi lines of force originate from a unit charge (cgs system of units), and when the charge moves we might expect the field lines to become distorted but not change in number.

 

[tech99]

Apologies for the following. The interpretation of Gauss's Law we were taught at school said that 4xpi lines of force originate from a unit charge (cgs system of units). If the charge moves, the lines may become distorted (hence radiation of energy), but their number is unaltered - they cannot detach from the charge.

It looks to me that if we try to ascertain the magnitude of the moving charge at a great distance, then it is not possible, because we might be measuring a radiation field, and it could originate from a small charge accelerating greatly or a large charge accelerating a little. On the other hand, closer than a certain distance, where the field is essentially radial, we know that 4 pi lines of force originate from a unit charge (cgs system of units), and when the charge moves we might expect the field lines to become distorted but not change in number.

On further thought, I suppose that only the radial field is of interest in Gauss's Law, so that the radiation field can be disregarded.

 

[stedwards]

Well, Maxwell's equations are equivalent to the action principle, and of course the modern way to motivate Maxwell's equations is to write down a Lagrangian for a massless spin-1 field, but of course one must know that the electromagnetic field is described by such a field beforehand. From this it follows necessary from some group-representation theory of the Poincare group (Wigner 1940; see Weinberg Quantum Theory of Fields, vol. 1) that it must be described as a gauge field, and from the U(1) gauge symmetry nearly everything follows :-).

Of course, its undoubtably true that experimental data alone wouldn't have motivate a Lagrangian solution without knowing ahead of time. There is, however a perfectly classical lagrangian without consideration of nonclassical gauges, though there are an uncountable number of gauges that emerge in classical electromagnetism. The gauge invariance that emerges for the usual Lagrangian, is simply $A'=A+\phi$ such that $F_{\mu \nu}=\partial _\mu A_\nu- \partial_\nu A_\mu=\partial _\mu A'_\nu- \partial_\nu A'_\mu$. (The Christoffel connection vanishes.)

On a third approach, in a somewhat equation free way, one could simply postulate a vector field $A\subset R^3$ and associate components of various derivatives with $B, E, \phi, \rho, ...$ scaled by $c$ and the impedance of space, where the associations and scaling are also applied to obtain and interpret maxwell's equations.

Where in Weinberg? I should really learn more qed. I don't understand where U(1) comes from.

 

[stedwards]

[...]In short, does anyone know of a textbook where there is a proof, using the equations for the field due to a moving point charge, that Gauss’s law holds for a moving point charge? Thanks!

Not explicitly. But isn't all that is required is the additional consideration of charge continuity?

 

[vanhees71]

Of course, its undoubtably true that experimental data alone wouldn't have motivate a Lagrangian solution without knowing ahead of time. There is, however a perfectly classical lagrangian without consideration of nonclassical gauges, though there are an uncountable number of gauges that emerge in classical electromagnetism. The gauge invariance that emerges for the usual Lagrangian, is simply $A'=A+\phi$ such that $F_{\mu \nu}=\partial _\mu A_\nu- \partial_\nu A_\mu=\partial _\mu A'_\nu- \partial_\nu A'_\mu$. (The Christoffel connection vanishes.)

On a third approach, in a somewhat equation free way, one could simply postulate a vector field $A\subset R^3$ and associate components of various derivatives with $B, E, \phi, \rho, ...$ scaled by $c$ and the impedance of space, where the associations and scaling are also applied to obtain and interpret maxwell's equations.

Where in Weinberg? I should really learn more qed. I don't understand where U(1) comes from.

The main didactical problem is to motivate gauge invariance as a guiding principle in building the Lagrangian for electromagnetism. First you need that the electromagnetic field is a massless vector field. This you can infer from observations. It took quite a while in the history of science to come to Maxwell's equations (BTW it's 150 year anniversary of the Maxwell equations this year, which started the whole development of modern physics)! Then everything follows from the representation theory of the Poincare group, because from there you get the necessity of a gauge theory from symmetry considerations, which are the most fundamental laws of physics (or better said the most fundamental way to think about the fundamental laws).

In Weinberg you find this in the first chapters of vol. 1. You may also have a look at Appendix B of my lecture notes on QFT

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

 

[stedwards]

The main didactical problem is to motivate gauge invariance as a guiding principle in building the Lagrangian for electromagnetism. First you need that the electromagnetic field is a massless vector field. This you can infer from observations. It took quite a while in the history of science to come to Maxwell's equations (BTW it's 150 year anniversary of the Maxwell equations this year, which started the whole development of modern physics)! Then everything follows from the representation theory of the Poincare group, because from there you get the necessity of a gauge theory from symmetry considerations, which are the most fundamental laws of physics (or better said the most fundamental way to think about the fundamental laws).

In Weinberg you find this in the first chapters of vol. 1. You may also have a look at Appendix B of my lecture notes on QFT

http://fias.uni-frankfurt.de/~hees/publ/lect.pdf

I'll take a look at your lecture notes, though the your delimiter notation is unfamiliar to me. (Such as the brackets in $[ ket ]$, and $(*,*;*,*)$.)