# Proof of Gauss' Law for Electrodynamics?

• Michael.7
In summary: In short, the charge cannot be measured at a great distance, but it can be measured closer than a certain distance.
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!

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 ;-).

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.

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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 :-).

vanhees71 said:
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.
vanhees71 said:
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 said:
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.

vanhees71 said:
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.

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Michael.7 said:
[...]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?

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stedwards said:
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

vanhees71 said:
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 ## (*,*;*,*)##.)

## 1. What is Gauss' Law for Electrodynamics?

Gauss' Law for Electrodynamics is a fundamental law in electromagnetism that relates the electric field at a point to the distribution of electric charges in space. It states that the electric flux through any closed surface is equal to the total enclosed charge divided by the permittivity of free space.

## 2. How is Gauss' Law for Electrodynamics derived?

Gauss' Law for Electrodynamics can be derived from the more general Gauss' Law, which relates the flux of any vector field through a closed surface to the divergence of that field within the surface. By applying this principle to the electric field, we can derive Gauss' Law for Electrodynamics.

## 3. What is the significance of Gauss' Law for Electrodynamics?

Gauss' Law for Electrodynamics is significant because it provides a mathematical framework for understanding the behavior of electric fields and charges. It allows us to calculate the electric field at any point in space and understand how it is affected by the distribution of charges.

## 4. Can Gauss' Law for Electrodynamics be applied to any situation?

Yes, Gauss' Law for Electrodynamics is a fundamental law that applies to any situation involving electric fields and charges. However, in some cases, it may be more convenient to use other laws, such as Coulomb's Law, to solve specific problems.

## 5. Are there any limitations to Gauss' Law for Electrodynamics?

One limitation of Gauss' Law for Electrodynamics is that it assumes a static electric field. It does not account for time-varying fields or situations where charges are in motion. In those cases, we need to use more advanced laws, such as Maxwell's equations, to fully describe the behavior of the electric field.

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