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In summary, the key physical assumption needed to derive the Lorentz force law is that the field is locally phase invariant.

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https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

For those advanced enough, you define the EM tensor Fuv from A, show it is antisymmetric, has six independent components that can be grouped as the 3 component vector E and the three-component vector B. We note ∂u∂vFuv = 0, meaning δvFuv defined as the 4 current Ju can be interpreted as a continuity equation of something we will call charge. Up to now, it has been nothing but math. This is the key physical assumption needed. You have Maxwell's Equations. However, the Lagrangian formalism is needed to derive the Lorentz force law.

The above is quite mathematical. A more physical approach is the following:

https://iopscience.iop.org/article/10.1088/0143-0807/36/6/065036

I do not expect your audience to be advanced enough to understand any of the above. But it could be mentioned justifications exist once more advanced material is understood and handed out as supplementary reading for when they are more advanced. It is of value to see how it all fits together when students are learning the more advanced material. Often it is not part of standard treatments of the tensor formulation of SR, which IMHO is a pity.

For those really keen Lenny Susskind has written an extremely good book, IMHO understandable even by advanced high school students, bringing this all together, including the needed math:

https://www.amazon.com.au/dp/0141985011/

Just a few thoughts some of your more interested students may benefit from.

Thanks

Bill

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bhobba said:For those advanced enough, you define the EM tensor Fuv from A, show it is antisymmetric, has six independent components that can be grouped as the 3 component vector E and the three-component vector B. We note ∂u∂vFuv = 0, meaning δvFuv defined as the 4 current Ju can be interpreted as a continuity equation of something we will call charge. Up to now, it has been nothing but math. This is the key physical assumption needed. You have Maxwell's Equations. However, the Lagrangian formalism is needed to derive the Lorentz force law.

It seems you didn't state the key physical assumpiton needed, or I'm not getting it.

EDIT: I think you are saying that if you want conservation of the charge, then the Ampere-Maxwell law and the Gauss Law (laws that don't arise from the gauge invariance of the field tensor) do the trick. But that doesn't seem to be a neccesary condition.

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andresB said:It seems you didn't state the key physical assumpiton needed, or I'm not getting it.

First I gave a link to an argument from QM requiring local not just global phase invariance justifying the existence, at least theoretically, of a 4 vector with gauge invariance. The rest is just math. You define the EM tensor, Fuv, which is seen to be gauge invariant by construction (the subtraction in its definition cancels changes in gauge). Hence it does not depend on the choice of gauge - as physically measurable quantities would not. You notice it is antisymmetric, meaning on has 3 E components and 3 B components. Then you notice ∂u∂vFuv = 0, meaning δvFuv defined as the 4 current Ju obeys ∂uJu = 0 ie is the equation of conservation or continuity of a physical quantity - if Ju was an actual physical quantity. At this point, it is nothing but math. But what if we take seriously that Ju could be the current of something that is conserved. Making that assumption is a key physical idea that can be experimentally tested. The quantity is called charge. And indeed experiment does show it exists and is conserved. E and B are then seen to be fields called the electrical and magnetic fields respectfully and obey Maxwell's equations. Only experiments can validate the truth or otherwise of the physical assumptions made, and the resultant equations. It is just an interesting argument showing that one can reasonably justify the equations. That's all. It is an extension of the method Maxwell used to add the term he used to complete the EM equations. Its purpose is to give students some inkling that these ideas have theoretical underpinnings independent of physical experimentation.

It's like when Feynman and Murray hit on a key physical idea about the weak interaction. But there was some experimental evidence at the time that contradicted it. But as Feynman said - everything just fit theoretically. They both had faith that their idea was correct. So they said - let's wait a while and see what the experimentalists come up with. Sure enough, they were right. This is something rather strange about physics students should know about.

Thanks

Bill

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The free space Maxwell's equations don't follow from what you are saying. Something is lacking (like a superposition principle) because Faraday's law and the magnetic Gauss law (Bianchi identities for the field tensor) + conservation of charge don't imply the other two free-space Maxwell's equations, as evidenced by the existence of many non-linear electrodynamics.

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andresB said:The free space Maxwell's equations don't follow from what you are saying.

I am not quite sure of the exact context of your comment. They do follow from δvFuv = Ju as is well known and found in many standard texts and writeups on the internet e.g.:

https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html

Now is my 'derivation' airtight in that every assumption is fleshed out? Almost certainly not, although I have not carefully gone through it to find all the premises. In many textbooks, you will read, it is possible to derive Maxwell's equations from Coulombs Law and Relativity eg

http://cse.secs.oakland.edu/haskell/Special Relativity and Maxwells Equations.pdf

I heard a comment made in Jackson was he thought such derivations were silly. I don't have Jackson (I chose Schwinger as my advanced EM textbook), but I have no reason to doubt it. So I went through the derivation carefully and did manage to find unstated assumptions. But even in basic electrostatics, unstated assumptions are made. Suppose we have a charge generating an electric field. We put another charge near it - it will experience a force from Coulombs Law. We remove the charge generating the field and put another near it. Again it will create a force from Coulombs law. So far it is simple elementary school science. Now let's put the first charge back. The unstated assumption made is the presence of the other charge does not change the force it creates, and the forces are additive. It is so intuitive and obvious nobody states it. This is the type of thing you pick up by doing problems. I think Feynman discusses this sort of thing in his lectures and is one of the things separating math from physics.

Thanks

Bill.

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bhobba said:I am not quite sure of the exact context of your comment. They do follow from δvFuv = Ju as is well known and found in many standard texts and writeups on the internet e.g.:

https://quantummechanics.ucsd.edu/ph130a/130_notes/node451.html

What In trying to say is the following. We assume the existence of the electric field tensor as a gauge invariance quantity and we get magnetic Gauss-Faradays law

##\partial_{\rho}F^{\mu\nu}+\partial_{\mu}F^{\nu\rho}+\partial_{\nu}F^{\rho\mu}=0##. Then we say that there is something called the charge/current, and that it is conserved, and we get ##\partial_{\mu}J^{\mu}=0##.

It doesn't follow from those two equations that the field and the charge are related by the Gauss-Ampere-Maxwell laws ##\partial_{\mu}F^{\mu\nu}=J^{\nu}##. Something else is required to justify the last equation.

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andresB said:It doesn't follow from those two equations that the field and the charge are related by the Gauss-Ampere-Maxwell laws ##\partial_{\mu}F^{\mu\nu}=J^{\nu}##. Something else is required to justify the last equation.

I am somewhat perplexed. I have been under the impression for many years Maxwell's equations cover all of EM (except the Lorentz force law - you need the Hamiltonian or Lagrangian formalism as well to get that). In tensor form, Maxwell's Equations are δuFuv = Jv. I read many years ago a book that derived all the equations used in EM from Maxwell's equations, as logically you should be able to do. The Gauss-Faraday law would be no different to any of those equations such as Gauss's Law which in turn implies Coulomb's Law (along with the Lorentz force law of course - but as mentioned that requires the use of the Hamiltonian or Lagrangian formalism).

Thanks

Bill

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Indeed, as you can see, any gauge-invariant antisymmetric dielectric 4-tensor ##D^{\mu\nu}=D^{\mu\nu}(F)## that obeys the dielectric-like form of the Gauss-Ampere-Maxwell equation ##\partial_{\mu}D^{\mu\nu}=J^{v}## leads to conservation of charge. And several such dielectric-like corrections to free space Maxwell equations have been studied since at least Born-Infeld.

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bhobba said:

https://quantummechanics.ucsd.edu/ph130a/130_notes/node296.html

For those advanced enough, you define the EM tensor Fuv from A, show it is antisymmetric, has six independent components that can be grouped as the 3 component vector E and the three-component vector B. We note ∂u∂vFuv = 0, meaning δvFuv defined as the 4 current Ju can be interpreted as a continuity equation of something we will call charge. Up to now, it has been nothing but math. This is the key physical assumption needed. You have Maxwell's Equations. However, the Lagrangian formalism is needed to derive the Lorentz force law.

The above is quite mathematical. A more physical approach is the following:

https://iopscience.iop.org/article/10.1088/0143-0807/36/6/065036

I do not expect your audience to be advanced enough to understand any of the above. But it could be mentioned justifications exist once more advanced material is understood and handed out as supplementary reading for when they are more advanced. It is of value to see how it all fits together when students are learning the more advanced material. Often it is not part of standard treatments of the tensor formulation of SR, which IMHO is a pity.

For those really keen Lenny Susskind has written an extremely good book, IMHO understandable even by advanced high school students, bringing this all together, including the needed math:

https://www.amazon.com.au/dp/0141985011/

Just a few thoughts some of your more interested students may benefit from.

Thanks

Bill

Thanks Bill. I do remember going through this treatment in grad school. And I was impressed as it showed how the E and the B fields are equivalent, and that they only thing that affects whether it is an E or a B field was if you were translating with respect to a rest frame. I will take a look.

Thanks,

Chris KQ6UP

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kq6up said:I will take a look.

It's only for more advanced and/or motivated students. You don't actually have to do much; just refer them to Susskind's book.

Thanks

Bill

Maxwell's equations are a set of four fundamental equations in electromagnetism that describe the relationship between electric and magnetic fields, and their sources (charges and currents). They were first published by James Clerk Maxwell in 1865.

Maxwell's equations are important because they provide a complete and concise description of the behavior of electric and magnetic fields, and their interactions with matter. They have been essential in the development of modern technology, including telecommunications, electronics, and power generation.

The four equations in Maxwell's equations are Gauss's law, Gauss's law for magnetism, Faraday's law, and Ampere's law. Together, they form a complete and consistent set of equations that describe the behavior of electric and magnetic fields.

Maxwell developed his equations by combining the laws of electricity and magnetism that were already known at the time, along with his own insights and mathematical techniques. He also made use of experimental evidence and observations to refine and validate his equations.

Maxwell's equations are considered one of the most important and successful theories in modern physics. They have been extensively tested and verified through experiments, and have been used to make predictions about electromagnetic phenomena that have been confirmed by further observations. They also form the basis for more advanced theories, such as quantum electrodynamics.

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