# Interesting Derivation of Maxwell's Equations

• I
Mentor

## Main Question or Discussion Point

I really love seeing derivations of the EFE's, Maxwell's equations, Schrodinger equation etc.

I have seen a number of derivations of Maxwell's Equations but this is the shortest, most illuminating and best I have come across - it basically just uses covarience - and as it says - a little bit more:
http://iopscience.iop.org/article/10.1088/0143-0807/36/6/065036/pdf

Brings up a number of interesting points such as there are a number of different but really equivalent forms of EM, how Magnetic monopoles fits in etc. Hope everyone finds it as interesting as I did.

Oh - does anyone really feel like doing the full detail of the proof in the appendix

Thanks
Bill

Last edited:
Ibix and Dale

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Dale
Mentor
I really love seeing derivations
I like test theories. It seems like this could be used as a test theory for a few things, although it wasn’t presented as such

bhobba
stevendaryl
Staff Emeritus
I really love seeing derivations of the EFE's, Maxwell's equations, Schrodinger equation etc.

I have seen a number of derivations of Maxwell's Equations but this is the shortest, most illuminating and best I have come across - it basically just uses covarience - and as it says - a little bit more:
http://iopscience.iop.org/article/10.1088/0143-0807/36/6/065036/pdf

Brings up a number of interesting points such as there are a number of different but really equivalent forms of EM, how Magnetic monopoles fits in etc. Hope everyone finds it as interesting as I did.

Oh - does anyone really feel like doing the full detail of the proof in the appendix

Thanks
Bill
What's the reason (equation 2.2) for assuming that $\frac{dp^\alpha}{d\tau}$ is linear in the 4-velocity?

stevendaryl
Staff Emeritus
What's the reason (equation 2.2) for assuming that $\frac{dp^\alpha}{d\tau}$ is linear in the 4-velocity?
They compute the tensor $\frac{\partial p_\alpha}{\partial x^\beta}$. That only makes sense if $p_\alpha$ is a covector field, defined for all spacetime. But why should $p_\alpha$ be uniquely defined anywhere except along the particle's path?

Demystifier
Mentor
They compute the tensor $\frac{\partial p_\alpha}{\partial x^\beta}$. That only makes sense if $p_\alpha$ is a covector field, defined for all spacetime. But why should $p_\alpha$ be uniquely defined anywhere except along the particle's path?
I think you figured out your first question. The unstated assumption, even though they handwavy sort of justify it, not very well IMHO, is the Fαβ they define is physically interpreted as a field.

Thanks
Bill

vanhees71
Gold Member
2019 Award
Well, the proof in appendix A is (almost) given in any good textbook on electromagnetism extending it by the introduction of magnetic charges ;-)). It's a bit lengthy for a forum post since you need to explain carefully gauge invariance and the possibility to choose the Lorenz gauge for the two vector potentials.

The paper is of course flawed, because there are in principle many more possible local force laws, and indeed ##p_{\mu}## are co-vector components in the extended Hamilton formalism but no covector-field components. You need to make the additional assumption that the force field is a massless vector field to get almost naturally electromagnetism.

Of course, no fundamental law can be mathematically derived but has to be found empirically and then brought into the most convenient mathematical form. I don't know a single example for a fundamental law derived by pure mathematics. All have been found by observation in nature. That's, in my opinion, the reason why amazing attempts to find fundamental physical laws by pure thought like SUSY or even String Theory have failed to get some useful ones yet.

Dale, bhobba and dextercioby
Mentor
The paper is of course flawed,
It is, in fact all of them are, a point made by Jackson in his textbook I believe. The word someone mentioned he used was 'silly'. Even the one I post of the deviation from Coulomb's law, and stated in some textbooks as possible, is flawed. After hearing about Jackson's view I carefully went through that derivation and some unstated assumptions are made - but it took me a while to spot. Someone may enjoy spotting them as well:

Its simply an amusing heuristic justification. I kind of like those things though - it amuses me you can, via math alone 'guess' at quite a few laws - of course Dirac was a master at it. And be wrong many times as well.

BTW never did get a copy of Jackson - I was about to click on Amazon to purchase it but for some reason the one by Schwinger grabbed me:
https://www.amazon.com/dp/0738200565/?tag=pfamazon01-20

I of course have looked at Jackson in a library - but from what I saw I think Schwinger the better book. Just my view. Anyway both are nice books but for some reason Jackson is the more famous - don't really know why.

That's, in my opinion, the reason why amazing attempts to find fundamental physical laws by pure thought like SUSY or even String Theory have failed to get some useful ones yet.
Yes. Maybe too enamoured by Dirac who pulled it off a couple of times. While he was Feynman's hero, I am pretty sure Feynman was in your camp as well. Actually it's my view as well - but I do get a kick out of trying hence why I like this sort of stuff. Yes I know - I am a hypocrite - but hey I am retired - its whatever amuses me .

Thanks
Bill

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vanhees71
Gold Member
2019 Award
Sure, it's nice to see, what one can "derive", because it sheds light on some aspects of the theory, you don't see in the standard treatment.

I fully agree with you about Schwinger's textbooks, which are the most underrated books ever. Indeed, his electrodynamics book is better than Jackson's, but the problem is Schwinger's books is that they are very advanced and do not use the standard arguments, but they are gems to get an alternative view and also very witty mathematical methods. Alone the treatment of the cylinder functions in his Classical Electrodynamics book is worth buying it. The same holds for the epilogue of the Quantum Mechanics textbook, where you get the functional derivative version of Feynman's path integrals and the Schwinger action principle. Another great achievement is the heat-kernel method in QFT, but I'm not aware whether Schwinger put this in a textbook. I learnt it from the textbook by Donoghue et al, Dynamics of the Standard Model.

If you want to get Jackson's book after all, consider whether you'd rather want the 2nd than the newest edition, because the 2nd edition is consistently written in the Gaussian system of units, while in the newest edition (I think the English version is the 3rd, the German one the 4th edition) he switches between SI and Gaussian units, which is pretty confusing.

bhobba
Mentor
Indeed, his electrodynamics book is better than Jackson's, but the problem is Schwinger's books is that they are very advanced and do not use the standard arguments, but they are gems to get an alternative view and also very witty mathematical methods. Alone the treatment of the cylinder functions in his Classical Electrodynamics book is worth buying it.
When people ask my opinion on what sequence to study I say the following:
Susskind: https://www.amazon.com/dp/B0747BK692/?tag=pfamazon01-20
The free Feynman Lectures volume 2 (of course overall you should study all three): http://www.feynmanlectures.caltech.edu/
Griffiths - Electrodynamics: https://www.amazon.com/dp/1108420419/?tag=pfamazon01-20
Zangwell - Modern Electrodynamics: https://www.amazon.com/dp/0521896975/?tag=pfamazon01-20

Then Schwinger. Really I don't think Jackson is the best choice after Griffiths - Zingwell explains - not necessarily proves mind you, important things such as stuff from complex analysis such as contour integration students may not know. I could have put Jackson in as well - but I like both Zingwell and Schwinger better. Zingwell for me is very clear and a good preparation for Schwinger. I know its supposed to be graduate level - but having seen the book upper undergraduate is fine.

Just a note to those reading this, complex analysis is really necessary to more advanced work - not a rigorous mathematical treatment I had to do undergrad - but an applied treatment is fine for physics eg:
https://www.amazon.com/dp/0071615695/?tag=pfamazon01-20

Thanks
Bill

vanhees71
Gold Member
2019 Award
I've not formed an opinion about Zangwill yet. My first reaction was, why writing another oldfashioned textbook, where there are so many excellent ones like Becker & Sauter and of course Jackson. I think the best book after Feynman and Griffiths for E&M is Landau+Lifshitz vol. II which is much more modern than many new textbooks although written already in the 60ies (I guess).

poseidon721 and bhobba
Mentor
I've not formed an opinion about Zangwill yet. My first reaction was, why writing another oldfashioned textbook, where there are so many excellent ones like Becker & Sauter and of course Jackson. I think the best book after Feynman and Griffiths for E&M is Landau+Lifshitz vol. II which is much more modern than many new textbooks although written already in the 60ies (I guess).
I have and love Landau as well. Nice choice - should have thought of it myself. Should really read volume 1 first though - it explains Lagrangian's and symmetry implications very well and you really need it for volume 2.

Thanks
Bill

atyy
Well, the proof in appendix A is (almost) given in any good textbook on electromagnetism extending it by the introduction of magnetic charges ;-)). It's a bit lengthy for a forum post since you need to explain carefully gauge invariance and the possibility to choose the Lorenz gauge for the two vector potentials.

The paper is of course flawed, because there are in principle many more possible local force laws, and indeed ##p_{\mu}## are co-vector components in the extended Hamilton formalism but no covector-field components. You need to make the additional assumption that the force field is a massless vector field to get almost naturally electromagnetism.

Of course, no fundamental law can be mathematically derived but has to be found empirically and then brought into the most convenient mathematical form. I don't know a single example for a fundamental law derived by pure mathematics. All have been found by observation in nature. That's, in my opinion, the reason why amazing attempts to find fundamental physical laws by pure thought like SUSY or even String Theory have failed to get some useful ones yet.
They are not attempt to find laws by pure thought. Rather they are meant to map out the space of possibilities that "pure thought" exposes for future theories. Pure thought exposes that our current quantum theory of gravity fails at high energies, so it is a legitimate question to ask: what completions are possible?

And yes, people knew those may not be the only possibilities. Polchinksy's string text book mentions asymptotic safety, for example.

vanhees71