Weber Electrodynamics

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Re: Weber Electrodynamics

Unread postby Researcher720 » Fri Jul 06, 2018 10:46 am

For Weber electrodynamics, we don't need to specify a specific gauge. I understand that in Maxwell's electrodynamics, in order to have Gauss's law mathematically correct when the scalar and vector potential functions are explicitly specified, the Coulomb gauge is used. No need to do this in Weber electrodynamics. Gauge freedom applies to Weber's electrodynamics. No need to specify a gauge.
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Re: Weber Electrodynamics

Unread postby Researcher720 » Fri Jul 06, 2018 11:02 am

After talking to other researcher's, I think I understand their point of view on Maxwell much better. They effectively say:
1) Maxwell's equations are postulated to hold true. So you can not mathematically verify them. They are stated to be true as the starting postulate.
2) Then, writing the fields E and B in terms of scalar and vector potential functions, phi and A, is postulated to be true (without specifying what the scalar and vector potential functions are) and so you can't mathematically verify Maxwell's equations of the fields in terms of the potential functions.
3) But once you explicitly write the scalar and vector potential functions in terms of the charge density and current density functions, then Maxwell's equations are not mathematically valid, in general. But some of Maxwell's equations may be valid in specific gauges. But not all of Maxwell's equations, for a specific fixed gauge, are mathematically valid at the same time. For example, Gauss's law (electric) is valid in the Coulomb gauge, but that doesn't mean any of the other Maxwell's equations are valid in the Coulomb gauge. And Gauss's Law is not necessarily mathematically valid in other gauges.

It seems a bit convoluted and confusing to me.... On the other hand, for Weber's electrodynamics, we don't have to deal with such issues. Weber is mathematically valid right from the start, assuming only that the energy function for the energy between two charged bodies, is valid. Everything else in Weber's electrodynamics follows from that single function. Seems like a much simpler approach.
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Re: Weber Electrodynamics

Unread postby celeste » Sat Jul 07, 2018 6:38 pm

Researcher720 wrote:After talking to other researcher's, I think I understand their point of view on Maxwell much better. They effectively say:
1) Maxwell's equations are postulated to hold true. So you can not mathematically verify them. They are stated to be true as the starting postulate.
2) Then, writing the fields E and B in terms of scalar and vector potential functions, phi and A, is postulated to be true (without specifying what the scalar and vector potential functions are) and so you can't mathematically verify Maxwell's equations of the fields in terms of the potential functions.
3) But once you explicitly write the scalar and vector potential functions in terms of the charge density and current density functions, then Maxwell's equations are not mathematically valid, in general. But some of Maxwell's equations may be valid in specific gauges. But not all of Maxwell's equations, for a specific fixed gauge, are mathematically valid at the same time. For example, Gauss's law (electric) is valid in the Coulomb gauge, but that doesn't mean any of the other Maxwell's equations are valid in the Coulomb gauge. And Gauss's Law is not necessarily mathematically valid in other gauges.

It seems a bit convoluted and confusing to me.... On the other hand, for Weber's electrodynamics, we don't have to deal with such issues. Weber is mathematically valid right from the start, assuming only that the energy function for the energy between two charged bodies, is valid. Everything else in Weber's electrodynamics follows from that single function. Seems like a much simpler approach.


Thank you for this. Can you be a little more specific on the first line of point 3)?
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Re: Weber Electrodynamics

Unread postby jacmac » Sat Jul 07, 2018 8:06 pm

Higgsy said:
I suppose you are wondering why your post has got zero traction when you claim to have discovered some fundamental things about electromagnetic theory, which ought to interest the EU crowd. Well, the reason is that EU supporters on this forum fail to understand that physics is a mathematically based science and not a single one of them understands the first thing about what your post and your website are claiming. Not a single one of them is capable of solving undergraduate problems in electromagnetism.


Confession : I only got as far as one year of calculus in University.
I do believe however that I still know how to think.
On these forums I am interested in Ideas, not necessarily
the mathematical basis or explanation or proof of the ideas.

Reading this thread I really don't know what you all are talking about.
Is there an underlying idea here.
(Thank you celeste for speaking in English)

Does either Maxwell or Weber or the Sagnac formula say anything about
why the electron and the proton are attracted to each other until they get in close enough to start dancing
around in a fog , or orbit, or probability equation,or whatever.

Reading all the above does not instill in me any angst about not going further in mathematics.

Jack
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Re: Weber Electrodynamics

Unread postby Researcher720 » Sun Jul 08, 2018 4:34 am

Hi Jack, There is an idea in physics that using mathematics can help discover new physical things and help explore new ideas about the physical Universe. And that if the mathematics is contradictory then there is something wrong with the mathematical description of the physics indicating that we don't understand what is going on. The idea is that once the physical can be comprehended by the logical mind, then it can be written down in mathematics so that other people can comprehend it. But if the math leads to a logical/mathematical contradiction, then there is something wrong with the understanding of the physical Universe. But, don't forget, doing math is not necessarily doing physics. Need experimentation, for sure. Experimentation and math work together to make up "physics"....

So, I think it is very important to have a logical, verifiable mathematical description of electrodynamics. Maxwell's mathematical description doesn't appear to be adequate. Weber seems to be better, but is not the last word for electrodynamics. I am hoping others will start looking at Weber electrodynamics
http://www.weberelectrodynamics.com/
and help extend it.
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Re: Weber Electrodynamics

Unread postby Researcher720 » Sun Jul 08, 2018 4:56 am

Hi celeste, For (3), consider Gauss's Law (does anyone know how to enter math on this forum?)
div E = (1/epsilon_0)rho
This is one of Maxwell's equations (SI units). This is assumed to be true for electrodynamics, but note it was derived only for electrostatics (for example, in Chapter 1 of Jackson on electrostatics). When E is specified in terms of the scalar and vector potential functions
E = -grad Phi - @ A/@t
(here "@" is the partial derivative symbol) Gauss's law above is still assumed to be true.
But once you write the scalar potential function, Phi, in terms of the charge density function, and the vector potential function, A, in terms of the current density function, then it is easy to show that
div E = (1/epsilon_0) rho - @(div A)/@t
which is not Gauss's law because of the @(div A)/@t part. So, in general, once the potential functions are written in terms of the charge and current density functions, Maxwell's equations are not mathematically valid, in general.
However, if the special condition (Coulomb gauge condition) that div A = 0 is imposed, then Gauss's law remains valid. But 1) this fixes the gauge, removing the "gauge freedom" of Maxwell's electrodynamics, and 2) there is no guarantee that any of the other Maxwell's equations remain valid. Indeed, the Ampere's Law is not valid....
I've written a lot about this on my Weber electrodynamics website at
http://www.weberelectrodynamics.com/
In particular, for this topic
http://www.weberelectrodynamics.com/Max ... xwell.html
Weber electrodynamics doesn't have this problem.
Last edited by Researcher720 on Sun Jul 08, 2018 4:58 am, edited 1 time in total.
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Re: Weber Electrodynamics

Unread postby Higgsy » Mon Jul 09, 2018 3:48 pm

Researcher720 wrote:For Weber electrodynamics, we don't need to specify a specific gauge.
I know next to nothing about Weber electrodynamics, but from some superficial research it seems that it is discounted because, inter alia, it does not lead to the prediction of electromagnetic waves and the correct speed of light. If that is true, that would indeed be a devastating reason to reject it. Would you care to comment?
I understand that in Maxwell's electrodynamics, in order to have Gauss's law mathematically correct when the scalar and vector potential functions are explicitly specified, the Coulomb gauge is used. No need to do this in Weber electrodynamics. Gauge freedom applies to Weber's electrodynamics. No need to specify a gauge.
Gauge freedom falls mathematically and necessarily out of the definitions of the vector magnetic potential and the electric scalar potential. By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory.

Note that in QFT the magnetic vector potential has physical meaning in that it can be used to predict experimentally verifiable physical effects such as the Aharonov-Bohm effect.

Fixing the gauge is equivalent to setting certain constraints on the physical problem, which in turn allows you to recover effects which are not immediately apparent (for example in the A-B effect the charged particles traverse regions of zero magnetic and electrical field and yet are affected). So the Coulomb gauge is defined by setting the divergence of the vector potential to zero, and it is trivial to recover Gauss's law. The Lorenz gauge is relevant to problems where the effects of changing charge distribution and magnetic flux propagate at the speed of light, and it is explicitly Lorentz invariant.

As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years.

So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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Re: Weber Electrodynamics

Unread postby Higgsy » Mon Jul 09, 2018 4:08 pm

Aardwolf wrote:
Higgsy wrote:
There's nothing wrong with using maths to describe a given phenomena as you may understand it, the problem is when you start putting the cart before the horse and believing the formulas somehow prove your theory to be the one and only arbiter of all things truthful. Unfortunately all you do is chase the rabbit down the hole.
Who said anything about using the "formulas" to prove the theory? The expressions, such as Maxwell's equations, are a mathematical description of reality. Whether or not they are an accurate description depends on experiment. But you can't understand the description, and therefore the physics, without the maths. The classical theory of electromagnetism is Maxwell's. It's mathematical and relies on vector calculus of fields. Without the maths, you don't understand how electricity and magnetism behave. In the real world.
I’m not sure whether you are deliberately missing the point or if it genuinely escapes you. Yes, of course a mathematical description of reality can support and/or even predict experimental outcome but that’s not the point. The point is the theory CAN STILL BE WRONG. Hence 1,400 year errors like Ptomely. Arguing over Maxwell and Weber is like arguing over Shakespeare’s grammar. Both meaningless and not worthy of contemplation, which is why no-one is answering, but you just took it as an opportunity to disparage the members here.

You are beating a strawman. No-one thinks that the maths proves anything on its own. I have already said that in a previous post (emboldened above), which you have chosen to ignore. Of course a theory can be wrong - all you have to do to show it is wrong is to show that it does not comport with reality. If your theory is that the force due to gravity is given by: F=Gm1m2/r3, then that's a perfectly valid mathematical expression that does not match observation, so it's phyically wrong. So you keep beating up on a point that I am not making. The point that I am making, is that you cannot attempt a description of reality in physics without using maths (whether that description is right or wrong). Can you point to any theories in physics which do not require maths to describe them quantitatively and accurately? If Maxwell describes elctromagnetism accurately and Weber does not then don't you think that discussing which to accept is actually the crux of the matter rather than some irrelevant point as you would have it.

One thing that I do know is that those people who argue against the importance of maths in physics are universally mathematically illiterate.
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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Re: Weber Electrodynamics

Unread postby Higgsy » Mon Jul 09, 2018 4:41 pm

neilwilkes wrote:Firstly, anybody who genuinely believes that physics is all done "with maths" is part of the problem.
All physics descriptions of reality or Nature are mathematical. Every one.

Getting back to Maxwell - there are many more than 4 equations.
There are four equations in Heaviside's modern vector notation. These are entirely equivalent to Maxwell's 20 equations.
Maxwell's original work has been edited & altered, both by Heaviside and even more by Lorentz who literally discarded the entire class of Maxwellian systems that are in disequilibrium. Lorentz revised the Maxwell-Heaviside equations to make them amenable to separation of variables and closed analytical solutions. These are not Maxwell's equations, nor are they the truncation of Maxwell's theory by Heaviside et al
Can you show this mathematically here, or provide a link to a peer reviewed paper in a respectable journal? In any case, in current physics, Maxwell's theory is Lorentz invariant (which it must be to be an accurate description of observed reality), and it is a subset of QED in the limit.
Additionally, Maxwell's electrodynamics is a material fluid flow theory, and it assumes a material ether
No - you are stretching the analogy too far. Because the theory has similarities with fluid dynamics does not mean that there is a material aether. In fact that idea was discarded in 1905, and there is nothing in Maxwell's theory that demands such a thing.
Anything that fluid systems can do, electrodynamics systems can do (at least in theory) because their mathematical models are the same form. So when one cites known examples of fluid driven systems where the energy to run the system is freely furnished by the active environment, analogous electrodynamic systems in active environments - and in disequilibrium exchange with that environment - must also exist in nature.
Examples? With supporting empirical data?
Indeed, particle physics requires it and proves it
Can you show why that must be so, using the actual theory rather than hand-waving?
These are the systems arbitrarily discarded by Lorentz' symmetrical regauging in every university.
Show us.

Oh I see. You somehow think that Maxwell's original 20 expressions contain some wisdom that has been lost. That's a common error. You'll need to demonstrate that claim mathematically - the information that is contained in the original formulation that is not in the modern vector notation, and that that information accurately reflects some aspect of reality that can be demonstrated empirically. Otherwise you're just hand-waving.
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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Re: Weber Electrodynamics

Unread postby Higgsy » Mon Jul 09, 2018 5:00 pm

jacmac wrote:Confession : I only got as far as one year of calculus in University.
I do believe however that I still know how to think.
On these forums I am interested in Ideas, not necessarily
the mathematical basis or explanation or proof of the ideas.
Then you are telling pretty stories, not doing physics.

Reading this thread I really don't know what you all are talking about.
Is there an underlying idea here.
(Thank you celeste for speaking in English)
Yes there are some quite elementary and foundational ideas in physics being discussed. Sorry you are left out.

Does either Maxwell or Weber or the Sagnac formula say anything about
why the electron and the proton are attracted to each other until they get in close enough to start dancing
around in a fog , or orbit, or probability equation,or whatever.
What!?

No, for that you need the (non-relativistic) Schroedinger equation which expresses the wave function in terms of the Bohr radius, generalised Laguerre polynomials and the relevant quantum numbers.

Reading all the above does not instill in me any angst about not going further in mathematics.
That's because you are not a physicist.
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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Re: Weber Electrodynamics

Unread postby Researcher720 » Mon Jul 09, 2018 5:25 pm

Hi Higgsy, Thanks for your comments. You wrote: Weber "... does not lead to the prediction of electromagnetic waves and the correct speed of light." Yes it does. The "old" or original interpretation of Weber did not include waves. It was an instantaneous action at a distance theory. But when I wrote Weber in terms of "fields", it looks like Maxwell's theory (in symbols). And when I introduced time retardation (following J. P. Wesley's method for introducing time retardation) Weber's theory gives waves with speed c. These are new results of my work, so I don't expect others to know about it all.

You wrote: "By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory." Yes. And that, now, also holds for Weber's electrodynamics. So then why must we specify the Coulomb gauge (in Maxwell) for Gauss's law to be valid? If the gauge choice doesn't change the E field, then Gauss's law should remain valid regardless of the gauge specified. But Gauss's law, in Maxwell, is only valid in the Coulomb gauge.

And Ampere's Law doesn't hold mathematically. Take the mathematical definition of the vector potential function, A, and put it in left-hand side of Ampere's law
(\nabla x B = \nabla x (\nabla x A))
and you can't get the right hand side of Ampere's law. Is there some other gauge that makes this work, because it doesn't work in the Coulomb gauge. So, picking a gauge may make some of Maxwell's 4 equations mathematically correct (left-hand side equals right-hand side) but doesn't make all 4 Maxwell equations correct.

My approach, which is very simple, makes all 4 "corrected" Maxwell's equations correct for the current density and charge density functions.

You wrote: "Note that in QFT the magnetic vector potential has physical meaning in that it can be used to predict experimentally verifiable physical effects such as the Aharonov-Bohm effect." Thanks for reminding me. I meant to look at the Aharonov-Bohm effect in terms of Weber. Note that Weber's force law includes the Lorentz force law and other terms. Maybe these other terms give the Aharonov-Bohm effect.... I'll look at it.

You wrote: "As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years." Yes, but those same tests also verifies Weber's electrodynamics. Note that Lorentz force law is contained in Weber's force law (as dominate terms). For experiments based physics, note that Ampere's force equation (not Ampere's law of Maxwell equations) was developed **from experimentation** by Ampere. (See Assis's book Ampere's electrodynamics, Graneau's book Ampere-Neumann Electrodynamics of Metals, Graneau's book Newtonian Electrodynamics, Wesley's book Selected Topics In Scientific Physics, and papers in the literature...) Why then isn't Ampere's force law in mainstream physics textbooks? It was developed from experiments... Oh, and Ampere's force law is also in Weber's force law. See http://www.weberelectrodynamics.com/Web ... Force.html
The typical experiment showing Lorentz force law alone is lacking are the so-called "exploding wire" experiments. This requires a longitudinal force along the current carrying wire to rip apart the wire. But the Lorentz force equation doesn't have a longitudinal force. Ampere's force equation (and hence Weber's force equation) does. There are other experiments that indicate the Lorentz force is not enough to describe all electrodynamics effects.

You wrote: "So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?" I am not working in any particular gauge. I consider Maxwell's equations and potentials and Lorentz force and continuity equation as a single mathematical **system** of equations. Meaning that they **should** all be mathematically consistent with one another. They aren't. Just try (as I know you have) Gauss's law with the E field written in terms of grad Phi and @A/@t . It doesn't hold. Why should it? It was derived only for the electrostatic case (E = -grad Phi) and not for the electrodynamics case. If you want to use the Coulomb gauge, o.k. Then try to get Ampere's law to hold in the same Coulomb gauge. It doesn't work. (It all works with Weber's electrodynamics.)
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Re: Weber Electrodynamics

Unread postby jacmac » Mon Jul 09, 2018 7:25 pm

Higgsy:
I know next to nothing about Weber electrodynamics, .

I guess I will take that on face value.
Enjoy your argument.
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Re: Weber Electrodynamics

Unread postby Aardwolf » Tue Jul 10, 2018 7:24 am

Higgsy wrote:One thing that I do know is that those people who argue against the importance of maths in physics are universally mathematically illiterate.
Yet again you are attacking members without any reason or provocation. Normally that's the resort of someone who has either lost the argument or doesn't have confidence in their own position. Either way, you can stick to your meaningless arguments about how may angels are dancing on the point of a pin and I'll follow that famously mathematically illiterate Einstein's advice and wallow in experience and knowledge;

Pure logical thinking cannot yield us any knowledge of the empirical world: all knowledge of reality starts from experience and ends in it. (Albert Einstein, 1954)
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Re: Weber Electrodynamics

Unread postby Higgsy » Tue Jul 10, 2018 5:26 pm

Aardwolf wrote:
Higgsy wrote:One thing that I do know is that those people who argue against the importance of maths in physics are universally mathematically illiterate.
Yet again you are attacking members without any reason or provocation. Normally that's the resort of someone who has either lost the argument or doesn't have confidence in their own position. Either way, you can stick to your meaningless arguments about how may angels are dancing on the point of a pin and I'll follow that famously mathematically illiterate Einstein's advice and wallow in experience and knowledge;

Pure logical thinking cannot yield us any knowledge of the empirical world: all knowledge of reality starts from experience and ends in it. (Albert Einstein, 1954)

And yet again you avoid the material point. You make a speciality of setting up and knocking down strawmen don't you? Who ever claimed that experimentation and observation are not the beginning and the confirmation of knowledge of the physical world? Not me, not any physicist I know (especially not me, as I am an experimentalist). But the point, which you will do anything to avoid, is that in physics, having done your experiments and carried out your observations, then that knowledge of the physical world that you have acquired is always, always described mathematically. And if you don't understand the maths, then you can't understand the description. I ask you again: Can you point to any reasonably accurate and universal theories or descriptions of the behaviour of the physical world that do not require mathematics to describe them quantitatively and accurately? Just one?

Anyone who doesn't follow vector algebra cannot possibly understand the most elementary undergraduate level classical electromagnetism. It's impossible to express the knowledge of electromagnetism that has been gained by experiment without it. And yet any number of you guys, lacking ability in vector algebra, claim to know better than professional physicists. That's silly and deluded.
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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Re: Weber Electrodynamics

Unread postby Higgsy » Tue Jul 10, 2018 6:40 pm

Researcher720 wrote:Hi Higgsy, Thanks for your comments. You wrote: Weber "... does not lead to the prediction of electromagnetic waves and the correct speed of light." Yes it does. The "old" or original interpretation of Weber did not include waves. It was an instantaneous action at a distance theory. But when I wrote Weber in terms of "fields", it looks like Maxwell's theory (in symbols). And when I introduced time retardation (following J. P. Wesley's method for introducing time retardation) Weber's theory gives waves with speed c. These are new results of my work, so I don't expect others to know about it all.

OK I take you word for it, as I have not studied Weber electrodynamics as I said before. However, everything I read about the reason Maxwell's theory superseded Weber's in the 1870s states that Weber's was shown not to predict the speed of light correctly, and moreover to violate energy conservation. Maybe you have corrected those defects, but you'll excuse me if I remain skeptical that such fundamental flaws can be corrected. Furthermore, I see no motivation to spend the time learning and studying an alternative theory of electromagnetism, when Maxwell's has been shown to be accurate after more than 150 years of experiment, and also, I remind you, falls out of QED, the most accurate physical theory ever, in the classical limit.

You wrote: "By definition the magnetic field remain unchanged under transformations of the vector potential by addition of the gradient of a scalar function, and the electric field remains unchanged under transformation of the scalar potential by the addition of the time derivative of a scalar function. So the classical theory of electromagnetism has this gauge freedom and is a gauge theory." Yes. And that, now, also holds for Weber's electrodynamics. So then why must we specify the Coulomb gauge (in Maxwell) for Gauss's law to be valid? If the gauge choice doesn't change the E field, then Gauss's law should remain valid regardless of the gauge specified. But Gauss's law, in Maxwell, is only valid in the Coulomb gauge.
The original form of Gauss's law was derived to express the form of the electric field and the static charges that cause it, and so is valid in the Coulomb gauge. But you haven't shown that a dynamic version of Gauss's law, the electric field resulting from electric charges in motion, doesn't follow correctly from the potentials, in other gauges. As far as I can see there is no inconsistency in the Maxwell description using the potential definitions of the fields.

Furthermore, you claim that the fact that classical electromagnetism is a gauge theory and has gauge freedom (which you agree with) now holds for Weber electrodynamics. Don't you see that that directly contradicts your earlier claim that there is no need to fix a gauge when working with the potential definition of fields in Weber? Either the potential definitions of the fields have that degree of freedom or they don't (and the fact that they do is a direct and necessary consequence of their definition). I would remind you again that the potentials are not just a convenient mathematical ploy, but that they have physical significance. So how are the potentials defined in Weber?

And Ampere's Law doesn't hold mathematically. Take the mathematical definition of the vector potential function, A, and put it in left-hand side of Ampere's law
(\nabla x B = \nabla x (\nabla x A))
and you can't get the right hand side of Ampere's law. Is there some other gauge that makes this work, because it doesn't work in the Coulomb gauge. So, picking a gauge may make some of Maxwell's 4 equations mathematically correct (left-hand side equals right-hand side) but doesn't make all 4 Maxwell equations correct.
I haven't worked through your claim here in detail, but Maxwell adds the displacement current to the original definition of Ampere's law, so you have to work with retarded time. Gauge freedom means that you pick a gauge to match the physical problem.
My approach, which is very simple, makes all 4 "corrected" Maxwell's equations correct for the current density and charge density functions.
Fine - can you show that those equations are empirically accurate where Maxwell's theory is not or vice versa?

You wrote: "As far as I am concerned the problem isn't whether there is gauge freedom in the theory or not, but whether the theory can be shown to be an accurate representation of experimental results, which test Maxwell's theory has passed with flying colours for more than 100 years." Yes, but those same tests also verifies Weber's electrodynamics. Note that Lorentz force law is contained in Weber's force law (as dominate terms). For experiments based physics, note that Ampere's force equation (not Ampere's law of Maxwell equations) was developed **from experimentation** by Ampere. (See Assis's book Ampere's electrodynamics, Graneau's book Ampere-Neumann Electrodynamics of Metals, Graneau's book Newtonian Electrodynamics, Wesley's book Selected Topics In Scientific Physics, and papers in the literature...) Why then isn't Ampere's force law in mainstream physics textbooks? It was developed from experiments... Oh, and Ampere's force law is also in Weber's force law. See http://www.weberelectrodynamics.com/Web ... Force.html
I'm sorry - you mean the law that describes the force between two current carrying wires? Not in textbooks?
The typical experiment showing Lorentz force law alone is lacking are the so-called "exploding wire" experiments. This requires a longitudinal force along the current carrying wire to rip apart the wire. But the Lorentz force equation doesn't have a longitudinal force. Ampere's force equation (and hence Weber's force equation) does. There are other experiments that indicate the Lorentz force is not enough to describe all electrodynamics effects.
Lorentz force law applies to force on a charged particle moving in a magnetic field. The Ampere forvce law refers to the force between two current carrying wires, and of course there can be "longitudinal" forces depending on whether the wires are parallel or not.

You wrote: "So I ask again, in your attempted derivation of Gauss's law using Maxwell's theory, which gauge are you working in?" I am not working in any particular gauge. I consider Maxwell's equations and potentials and Lorentz force and continuity equation as a single mathematical **system** of equations. Meaning that they **should** all be mathematically consistent with one another. They aren't. Just try (as I know you have) Gauss's law with the E field written in terms of grad Phi and @A/@t . It doesn't hold. Why should it? It was derived only for the electrostatic case (E = -grad Phi) and not for the electrodynamics case. If you want to use the Coulomb gauge, o.k. Then try to get Ampere's law to hold in the same Coulomb gauge. It doesn't work. (It all works with Weber's electrodynamics.)
You have to fix a gauge, because gauge freedom is an inevitable consequence of the potential definitions of the fields. And in this post you say that Weber electrodynamics has gauge freedom, so the same thing must apply.

Furthermore you haven't convinced me that there is any inconsistency. In the Coulomb gauge, Gauss's law can be recovered trivially. And in other gauges - well, if you are working in the Lorenz gauge then I think you have a mathematical error in your derivation since you have to take care of retarded time.

Have you looked at the relativistic formulations of Maxwell in tensor calculus which also fall exactly out of QED in the classical limit?
"Every single ion is going to start cooling off instantly as far as I know…If you're mixing kinetic energy in there somehow, you'll need to explain exactly how you're defining 'temperature'" - Mozina
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