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I actually have happened to delved quite a bit to "really" understand the Maxwell's equations. I've bought original treatise, books with its commentary and plai
by sytelus 7y ago
I actually have happened to delved quite a bit to "really" understand the Maxwell's equations. I've bought original treatise, books with its commentary and plain old "for idiots" sort of books. His original treatise is super dense and unapproachable. Right now we can wear Maxwell's equations on t-shirt but their original form were forbidding. Even with modern form you "really" need to get concepts of differential geometry if you want to just play beyond abstract. There are tons of hand wavy explanations of div and curl out there but almost all can be broken with crafting clever question like “ok, so what you think curl of that would look like?”. I don't think even today I can claim I really get these concepts.
In any case, some of my biggest takeaways were these:
1. There is no such thing as "proof" of Maxwell's equations. Just like Einstein's field equations, Newton's laws and many other things in Physics, Maxwell equations are also simply laid out as lets assume these. Vast majority of “greatness” in Physics is simply assuming something without needing to fully understand it and then cross your fingers to see if some good predictions comes out of it.
2. The major achievement of Maxwell's equations is that you can predict velocity of light using other physical constants that have seemingly nothing to do with light. A consequence that we only later realized was that this was literally a constant and not relative to who is measuring it! This is easily one of the most non-obvious achievement in Physics.
- Certhas 7y agoYou should try to understand differential forms, if you have those then Maxwells equations in vaccuum really aren't strange any more. But Differential forms aren't that strange really. They are the mathematical objects that allow you to integrate along surfaces and curves. Of course their theory hadn't been developed when Maxwell wrote them. And Maxwell was very much concerned with EM in matter, which mixes the properties of the EM fields and materials, and that can thus be expected to get a bit messy. But I don't see what you are striving for when you say "really" understand them. I think this is a psychological category, rather than a hard criterion. Can you apply the formalism to calculate consequences? That's the main issue. Maybe you can have a better or worse intuition about the consequences, but that is often mainly due to practice. You can't expect to correctly intuit all possible consequences of a system as rich as EM. > Young man, in mathematics you don't understand things. You just get used to them. -- John von Neumann
- Koshkin 7y agoI don't think the equations of electrodynamics themselves are difficult to understand and accept at an intuitive level, as one can easily describe them in plain words. https://en.wikipedia.org/wiki/Maxwell's_equations#Conceptual_descriptions https://en.wikipedia.org/wiki/Maxwell's_equations#Conceptual...
- senderista 7y agoAnd a bit more deeply than just differential forms and the exterior derivative: connections on vector bundles and their curvature. Deeper still: connections on a principal bundle and the induced connections on associated bundles.
- tobinfricke 7y agoCould you recommend a good (accessible) reference on differential forms?
- 6gvONxR4sf7o 7y agoI'm late to the party and not the person you asked, but I've been trying to work through https://smile.amazon.com/gp/product/0817683038 https://smile.amazon.com/gp/product/0817683038 It's good so far.
- jjdredd 7y agoDifferential forms aren't necessary for this though.
- fluffything 7y ago> There is no such thing as "proof" of Maxwell's equations. Is there proof of any "physics"? Physics is the science that studies "physical models", which are just that, "models". Whether a model is useful (or not), or how accurate a model is, is determined by using the model to make a prediction, and then doing an experiment to check how bad the prediction is. Measuring a prediction is hard, and typically requires repeating the same experiment many many times, which at best produces a probability density distribution of the prediction the model should produce. This process is called "model validation", but it does not prove the model correct. For example, we can validate Newton's laws to predict the weight of many objects on earth relatively accurately. But this does not mean that Newton's laws are correct, that they would produce correct outcomes if you were moving at the speed of light, etc. When people talk about "proving physical models correct", I honestly have no idea what is it that they want, or how do they expect that this is done. If God was real and would answer to us, I guess we could ask God if a particular model is correct. But that's the only way I can think of to deliver one of these "proofs".
- Koshkin 7y ago> Physics is the science that studies "physical models" This is a completely wrong thing to say. Physics studies (some aspects of) Nature by creating models and testing their predictions. (You as a student might well be studying "physical models", but it would be a funny thing to say that for example "biology is a science where the teacher is yelling at students while trying to attract their attention to some nasty-looking posters.")
- tlholaday 7y ago> Is there proof of any "physics"? Arguably there is `disproof`.
- Smoosh 7y agoI'm not sure that God understands and can explain physics.
- nabla9 7y agoSimilarly, if you look at the Newton's first papers of differential calculus, they are very hard to understand and he explains them very confusing way. Whoever discovers these things must do so without concepts and formulations the people who came later made to simplify and understand them. The amount refinement that happens between invention and teaching the concepts to undergraduates is huge.
- avip 7y agoTry looking at his Chemistry notebooks though...
- orbifold 7y agoIn the case of differential calculus Leibniz (a German) came up with it at roughly the same time and his notation and approach is what actually stuck. In the case of Maxwell's equations it was Heaviside who came up with the vector formulation as four equations and it took Grassman, Cartan and Hodge to arrive at the modern two equation formulation in terms of differential forms.
- azepoi 7y agoSpeaking of Leibniz, this paper on the relation between the logarithm and the chainette is interesting: https://www.maa.org/sites/default/files/pdf/awards/college.math.j.47.2.95.pd https://www.maa.org/sites/default/files/pdf/awards/college.m... (via https://fermatslibrary.com/s/how-to-find-the-logarithm-of-any-number-using-nothing-but-a-piece-of-string https://fermatslibrary.com/s/how-to-find-the-logarithm-of-an... )
- mikorym 7y agoI would add that, unlike Newton: 3. Maxwell's equations are difficult to view as a whole since they are 6 or 7 separate things that should each be understood individually. This is true of Newton's 3 laws too, but they are easy enough to bring together that you can teach them to high school students. While it is true that Newton's laws are about different things too, I find it personally more coherent to call them "Newton's Laws". Maxwell's equations are to me more like "Maxwell's List of Equations". Maxwell also served a similar role that Euclid did: He did a lot of curation of contemporary results. Edit: In terms of point 2. I think as you say that Einstein was in fact not the first to make that assumption. Maxwell already knew about it, and his contemporaries did, but Einstein was successful in taking the assumption (about light being constant) further.
- selimthegrim 7y agoIf someone taught you that way they failed as a teacher. I am sorry to say that but they are a whole (with the exception of displacement current which needs Faraday tensor (which you can get relativistic invariants from too) to understand)
- mikorym 7y agoMaybe one can approach this from the other side. Knowing electromagnetism today, what is the most complete and simple framework for modelling it? The set of equations from Maxwell's book are to me a heterogeneous presentation. If there is a category of electromagnetic objects or perhaps some other pure mathematics framework that synthesises everything together, then I would call that thing the whole.
- selimthegrim 7y agoSure. You can start from relativity and show that for example the antisymmetric Faraday tensor (P in the link [1] below) can get you a lot of phenomena (not all the way, accelerating charges are hairy, even for Feynman, see the link [2]) To note, the reference to finding the relativistic invariant from the tensor in the first link goes back to the first edition of Landau and Lifshitz. The problem was removed from later editions because only a masochist wants to find invariants of 4x4 matrices by hand. [1] https://www.mathpages.com/home/kmath647/kmath647.htm https://www.mathpages.com/home/kmath647/kmath647.htm [2] https://www.mathpages.com/home/kmath528/kmath528.htm https://www.mathpages.com/home/kmath528/kmath528.htm
- pvg 7y agoand many other things in Physics, Maxwell equations are also simply laid out as lets assume these. I'm not sure I understand what you mean by this. These are all experimentally verifiable.
- TheOtherHobbes 7y agoThat's the point - they're concise descriptions of observable behaviour in terms of a specific set of mathematical metaphors. They have no independent mathematical derivation from some underlying set of first principles.
- pvg 7y agoThat sounds like a definition of 'just assumed' by which most things are just assumed. That would be broad to the point of uninterestingness and I wonder if that's what the poster really meant.
- kryogen1c 7y ago>most things are just assumed. ah! but thats the problem - most assumed things are wrong. it is non-trivial to generate a set of assumptions that matches observations with the current technology and with all future technology, with no base from which to derive your new laws.
- pvg 7y agomost assumed things are wrong. That sounds suspiciously like an assumed thing. Compared to Maxwell's equations, at a minimum.
- ddrt 7y agoNo such proof... except for the T-shirt, right?
- tsbinz 7y agoRegarding "fully" understanding physics, I like to refer people to Feynman's answer to a question about why magnets repel each other[0]. "Assuming something" is one integral part of the scientific method. You formulate a question, you build a hypothesis from prior knowledge, you make predictions, you test them, you analyze your results which might lead you to change your hypothesis or not. "Greatness" is finding simple hypotheses (like "the laws of physics are the same as viewed from any inertial frame and the speed of light is the same for every observer") that predict/"explain" effects that were inconsistent with prior hypotheses. Of course you can never really prove a model of the world is correct - another model that predicts differences that are too small for us to measure might be the "truth", but, until we can measure it, there isn't much use in pondering ... [0] https://www.youtube.com/watch?v=36GT2zI8lVA https://www.youtube.com/watch?v=36GT2zI8lVA
- agumonkey 7y agoI'd love to have others high grade researcher pitch in about the art of good theory. Often it's more a matter of disregarding your own perceptions, and aim for the out-of-the-box stupid.
- foldr 7y agoI've never really been that impressed with Feynman's responses in that interview. He seems to be uncomfortable with just saying that physicists don't really know on a fundamental level why magnets repel each other. Obviously, explanations can be more or less fundamental.
- Swizec 7y agoDidn’t his answer boil down to “The math is the physics. That’s the explanation. That’s how it works. You can’t just make up an analogy because the math is the analogy”
- foldr 7y agoThat seems a bit extreme and not really consistent with the attitude he took in his public lectures and in Surely You're Joking. It amounts to giving up and saying that you can't explain basic physical phenomena to laypeople on any level.
- empiricus 7y agoDid you get to https://en.wikipedia.org/wiki/Electromagnetic_tensor https://en.wikipedia.org/wiki/Electromagnetic_tensor ?
- vhffm 7y agoYou may be interested to know that both Maxwell's and the Einstein Field Equations can be derived by minimizing an action, cf. https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_action https://en.wikipedia.org/wiki/Einstein%E2%80%93Hilbert_actio... https://en.wikipedia.org/wiki/Electromagnetic_tensor#Lagrangian_formulation_of_classical_electromagnetism https://en.wikipedia.org/wiki/Electromagnetic_tensor#Lagrang... Enjoy!
- evanb 7y agoWell, sure, but that only pushes the question back one step further: who told you which action to pick? Why pick R and not R^2, for GR, for example? [ Just to avoid someone spending too much time on an explanation IAAPhysicist; yes I understand effective field theory and the preference for Lagrangians built from relevant/marginal operators. ]
- sideshowb 7y agoCan't you get there by assuming electrical effects propagate at the speed of light as well? And magnetism falls out as a result?
- amelius 7y ago> There is no such thing as "proof" of Maxwell's equations. Yes. But still there is the problem of self-consistency which can be difficult especially if boundary conditions come into play. This is of course a purely mathematical affair.
- sn41 7y agoIIRC, the modern formulation of Maxwell's equations is due to Oliver Heaviside. He formulated the laws concisely using vector calculus. We probably need to look at Heaviside as well to get the full historical picture.
- JadeNB 7y agoHeaviside is well worth looking at anyway. I don't know much about him personally, aside from the fact that the unit step function is named after him (in a classical case of "let's reduce a profound body of work to one incidental concept to which we attach the name"; see also the Kronecker delta), but I do know that his work was deep and, like much work destined to be of importance in applied mathematics, initially offended mathematicians by its emphasis on practicalities rather than mathematical purity. https://en.wikipedia.org/wiki/Oliver_Heaviside https://en.wikipedia.org/wiki/Oliver_Heaviside
- sn41 7y agoI agree - there is a great book by Paul Nahin. I bought it since I wanted to understand how someone got the idea of using complex numbers in electrical engineering. But I have found the book to be very hard to skim. I really have to work along with it to understand the technicalities.
- CamperBob2 7y agoAgreed as well. Nahin is comprehensive and authoritative, but a good biography of Heaviside for people who aren't already physicists or engineers has yet to be written (or at least yet to be discovered by me.)
- dboreham 7y agoI've been reading a book about Heaviside (Forgotten Genius). Something I had not appreciated was that he was an Engineer. I suppose I imagined a nerdy dude in his mother's basement trying to reformulate Maxwell's work. But in reality he was a guy who spent his working life trying to do things like find the location of breaks in subsea cables; figure out how to transmit higher data rates through long cables, and so on. It was in trying to solve these engineering problems that he ended up becoming interested in the mathematics of transmission line propagation and hence Maxwell's work.
- neutronicus 7y agoOne more point you may want to internalize: If you integrate the Faraday / Ampere equations, the divergence of the Magnetic field will never change, and the divergence of the Electric field ("charge") will obey a continuity equation with the current. So the two Gauss Laws don't really have any Dynamic content, they're just requirements on the initial conditions of the Fields.
- organicfigs 7y agoOne of my best memories in college was going through wedge product, differential forms and connections, and its application in the derivation of Maxwell's equations. When going through it, I always assumed someone came in and wrote those generalized forms (for students of differential forms: no pun intended ;) of his equations 50-100 years later. It's crazy to think that's how he approached E&M from a preliminary perspective.
- lmm 7y ago> When going through it, I always assumed someone came in and wrote those generalized forms (for students of differential forms: no pun intended ;) of his equations 50-100 years later. That's true though? Someone else here said the modern formulation is due to Heaviside; Maxwell's original version had dozens of equations because we didn't have all that vector calculus notation at the time.
- fpoling 7y agoDerivation of Maxwell equations from the corresponding Lagrange form does not require to know differential geometry. But some understanding of that is useful to realize that it is the Maxwell equations that sort-of follows from the constness of the speed of light which in turn reflects the underlying geometry of the model of space-time. And the speed of light is constant only so far as that model matches the reality.
- steve76 7y ago> “ok, so what you think curl of that would look like?”. When making a milk shake, open the blender's lid and look down so you see a whirlpool. Take a small pinwheel, and dip the pinwheel's face flat and level on the surface. Move it around, and notice it spins faster in the center than on the side. Write down the pinwheel's speed at all the points. https://mathinsight.org/curl_idea https://mathinsight.org/curl_idea ... ... ... I think I got that right. I also replace derivatives with the little triangle, and integral with sum to clear things up. The calculus part of it makes the resolution aspect of the concept, going from scanlines to 4K, irrelevant. With vectors and multiple dimensions, the normal way is to think of small components building up to a scalar value. X and Y components creating a hypotenuse. Another way I think of it is I have a sensor, and get one scalar reading. Can that one reading keep spitting out values? What resolution do I need to simulate that reading? Can I just use x and y and get a hypotenuse? Or does it go to infinity?
- coliveira 7y agoMost things in physics are not really provable at a mathematical level, and mathematical certainty is fundamentally not the goal whatsoever. The idea behind physics is to work as a detective, and use experimentation to establish if some mathematical model holds true or not. It is the experimentation that shows "proof" of a physical statement. Even if the mathematics behind some model shows a supposed proof of something, it has no meaning unless that "proved math" is tested against experiments.
- scythe 7y ago>The major achievement of Maxwell's equations is that you can predict velocity of light This is true, but the other major achievement of Maxwell's equations was his formulation of the displacement current: https://en.wikipedia.org/wiki/Displacement_current https://en.wikipedia.org/wiki/Displacement_current which underlies the wave equation and which "symmetrizes" the loop laws for electric and magnetic fields. This was the "missing link" between the previously known laws of electromagnetism (Ampere's law, Gauss's law, and Faraday's law) and the full theory given in Maxwell's equations.