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.
Regarding "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 ...
I'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.
He's trying to get the point across that "X don't really know on a fundamental level why X" for any group of people X and observation Y, if by "on a fundamental level" you mean "is not based on something assumed to be true". He's not uncomfortable with it, anyone with a scientific or engineering background knows that (or at least should know that). That doesn't stop predictions obtained from well-tested models from being useful.
Well, maybe that's what he's trying to get across, but it seems largely irrelevant to the interviewer's question. He could just have given a physics textbook level explanation of how magnets work. I'm not convinced that there's some kind of profound misunderstanding evident in the interviewer's question that needs addressing in long and rambling terms.
There is actually a subtle point on why the question of "physicists don't really know on a fundamental level why magnets repel each other" is nonsense ultimately which doesn't stop physics from producing useful or interesting in some other way models.
Those who think that a textbook explanation could have been the answer are missing the point: it is not about magnets—it is about what it means to know anything in terms of anything else.
There is no need for useful models to form a nested hierarchy converging to a single "reality" i.e., it is not necessary for a model to be more fundamental than another model even if they relate to what we observe as the same phenomenon.
What physics textbooks say about magnetic attraction is very far from being "trivially" true. It's a wealth of utterly non-obvious information resulting from centuries of scientific research.
Also, "cucumbers" is not an answer to the question you posed.
So you are telling me, you know the answer to my own question better then its author :)
I'm curious why do you think magnets repel? What terms would you use to describe it? How these terms are defined? What terms in turn are used in these definitions? How these terms are defined in turn? etc.
I love this interview it’s typical of Feynman; what do you mean by how magnet works is a fair question but it’s also a multi form one. We don’t really know we just know up to a certain level he had a lecture about how Mayans predicted astronomical events with beans, were the bean the answer to why?. I’ve learned a lot from this interview on how to look at science. Why is indeed a profound concept
"Interviewer: If you get hold of two magnets, and you push them, you can feel this pushing between them. Turn them around the other way, and they slam together. <q1> Now, what is it, the feeling between those two magnets? </q1>
Feynman: What do you mean, "What's the feeling between the two magnets?"
Interviewer: <q2> There's something there, isn't there? </q2> The sensation is that there's something there when you push these two magnets together.
Feynman: Listen to my question. What is the meaning when you say that there's a feeling? Of course you feel it. Now what do you want to know?
Interviewer: What I want to know is <q3>what's going on between these two bits of metal </q3>?
Feynman: They repel each other.
Interviewer: <q4> What does that mean, or why are they doing that, or how are they doing that? </q4> I think that's a perfectly reasonable question.
Feynman: Of course, it's an excellent question. But the problem, you see, when you ask why something happens, how does a person answer why something happens? For example, Aunt Minnie is in the hospital. Why? Because she went out, slipped on the ice, and broke her hip. That satisfies people. It satisfies, but it wouldn't satisfy someone who came from another planet and who knew nothing about why when you break your hip do you go to the hospital. How do you get to the hospital when the hip is broken? Well, because her husband, seeing that her hip was broken, called the hospital up and sent somebody to get her. All that is understood by people. And when you explain a why, you have to be in some framework that you allow something to be true. Otherwise, you're perpetually asking why."
I think Feynman properly responded to the questions asked -- people do thing that they have to "find a meaning" and "why" and talk about "the feeling."
Feynman properly answers there "of course you feel it!"
Follow very carefully his whole response (I link the transcript) -- it's deeply thought through and applicable to much more than just "feeling -- meaning -- why -- magnets." It's about the "why questions" and "meaning" questions in general, from the view of physics.
>But "how" was not a starting question of the interviewer
As your transcript shows, the interviewer asks "why are they doing that, or how are they doing that?".
That seems like it would have been a good point to respond with an explanation of why magnets repel each other.
I don't buy all this stuff about 'how' vs. 'why' questions anyway. Lots of 'why' questions are perfectly sensible scientific questions. E.g., 'Why don't magnets stick to aluminum?'
Exactly, and that "why" is answered thoroughly, see the transcript.
The part of "how" is also there:
"If you're somebody who doesn't know anything at all about it, all I can say is the magnetic force makes them repel, and that you're feeling that force.
You say, "That's very strange, because I don't feel kind of force like that in other circumstances." When you turn them the other way, they attract. There's a very analogous force, electrical force, which is the same kind of a question, that's also very weird. But you're not at all disturbed by the fact that when you put your hand on a chair, it pushes you back. But we found out by looking at it that that's the same force, as a matter of fact (an electrical force, not magnetic exactly, in that case). But it's the same electric repulsions that are involved in keeping your finger away from the chair because it's electrical forces in minor and microscopic details. There's other forces involved, connected to electrical forces. It turns out that the magnetic and electrical force with which I wish to explain this repulsion in the first place is what ultimately is the deeper thing that we have to start with to explain many other things that everybody would just accept. You know you can't put your hand through the chair; that's taken for granted. But that you can't put your hand through the chair, when looked at more closely, why, involves the same repulsive forces that appear in magnets. The situation you then have to explain is why, in magnets, it goes over a bigger distance than ordinarily. There it has to do with the fact that in iron all the electrons are spinning in the same direction, they all get lined up, and they magnify the effect of the force 'til it's large enough, at a distance, that you can feel it. But it's a force which is present all the time and very common and is a basic force of almost - I mean, I could go a little further back if I went more technical - but on an early level I've just got to tell you that's going to be one of the things you'll just have to take as an element of the world: the existence of magnetic repulsion, or electrical attraction, magnetic attraction.
I can't explain that attraction in terms of anything else that's familiar to you. For example, if we said the magnets attract like if rubber bands, I would be cheating you. Because they're not connected by rubber bands. I'd soon be in trouble. And secondly, if you were curious enough, you'd ask me why rubber bands tend to pull back together again, and I would end up explaining that in terms of electrical forces, which are the very things that I'm trying to use the rubber bands to explain. So I have cheated very badly, you see. So I am not going to be able to give you an answer to why magnets attract each other except to tell you that they do. And to tell you that that's one of the elements in the world - there are electrical forces, magnetic forces, gravitational forces, and others, and those are some of the parts. If you were a student, I could go further. I could tell you that the magnetic forces are related to the electrical forces very intimately, that the relationship between the gravity forces and electrical forces remains unknown, and so on. But I really can't do a good job, any job, of explaining magnetic force in terms of something else you're more familiar with, because I don't understand it in terms of anything else that you're more familiar with."
Yeah, he answers the question. I just think people are giving Feynman too much credit here. He seems to have been in a grumpy gramps mood that day and taken some time to actually get round to answering the question.
No -- he used the question to demonstrate the basic premises of physics -- that the "whys" can never end, as long as somebody is not "satisfied" with the answer, and that to even understand "how" needs some precondition to be useful in any way to the one who asked, and that otherwise it's just "cheating" or practically giving somebody false sense that he'll know something because the analogies popularly used are just wrong.
Like he said, the bigger marvel is that, that the same electromagnetic forces are what keeps us from falling through the floor. Or what keeps the apple hanging on the tree.
Or, only specific to the human uses, how the movement of water is transformed to supply remotely the electrical machines with the power.
But that's what nobody asks, because they don't "feel" it unusual. The magnets are just a small manifestation of the same forces that "feels" unusual to the people.
It's his answers to "philosophers" who earn the points asking "whys" which, from his point of view, are too wrong to ask, having a false context.
Everyone understands that you can keep asking "why?". Four year old kids understand this.
What the interviewer obviously wanted was an explanation of a particular physical phenomenon targeted at the level of someone without any background in physics. Everyone, Feynman included, has been in that position.
> What the interviewer obviously wanted was an explanation of a particular physical phenomenon targeted at the level of someone without any background in physics.
How do you know that? I would claim that it's what you expected and even if you received that (as quoted before!) you double down on showing the dissatisfaction in what preceded that explanation, namely, Feynman explaining that the "satisfaction" impression of every answer depends on the already existing knowledge of the person who asks.
But the answer was completely honest: there aren't any intuitions about electromagnetic fields present in someone "without any background in physics" which would allow the decent (non-cheating) answer.
I actually think his explanation is right on. Physicists are practical. They make model to quantitatively describe their experiments. Understanding fundamental truth is kind of a by-product.
They generally follows a reductionist approach, so a simpler model that can explain a wider set of experiments is a better model. But there's no guarantee that a simpler explanation is the "truth". All we can say is this approach "makes sense".
> Obviously, explanations can be more or less fundamental.
That's not obvious, in fact, it's not even true. Feynman's point is that people bring context to a question. People will accept different things as 'fundamental'; different assumptions and axioms. Can you sometimes explain things using fewer axioms? Sometimes; but is it more fundamental? Is it better to build a theory on fewer axioms, even if they can't be directly validated? Would you prefer a theory based on 4 axioms, none of which can be directly validated, over one with 6, but you can measure all six directly? What if the four can't be re-derived from the 6, but all observable statements can? What if either set can be re-derived from the other?
Feynman isn't saying that "physicists don't really know on a fundamental level why magnets repel each other". He's alluding to the idea that that question isn't well defined. He's encouraging the questioner to figure out what sort of answer would satisfy them, and why.
I think it's obviously true that some explanations are more fundamental than others. We have, e.g. a more fundamental explanation of tides than anyone did in 500AD. That is not to deny that it is difficult to make precise what exactly it is that makes one explanation more 'fundamental' than another. As always, individual cases are clear; the general principle is elusive.
>He's alluding to the idea that that question isn't well defined.
It's well-defined enough to answer. Hundreds of millions of school children learn a perfectly sensible answer to the question every year. (Would this answer satisfy someone with a PhD in physics? Obviously not. But that's not the point.)
It's in any case bizarre to insist that a layperson ask a question that's well-defined according to the standards of a particular field.
The odd thing is, we physicists actually have a pretty solid understanding of why magnets repel each other. It's a fun story, one that I try to include every time I teach the subject, and I'm pretty sure that Feynman knew it! So it's curious to me to see Feynman basically dodging this question. (His point that "I can't explain it in terms of anything else you're already familiar with" is entirely valid! But I kinda feel like that's what questions are for: learning new things, that we may not already be familiar with.)
With that in mind, I probably would have answered in a different way, though the questioner might get bored and regret asking if I didn't find a way to be unusually efficient about it: this answer relies on a lot of knowledge that seems at best tangentially related. :) I'd try to give some rough description of the way that moving currents generate magnetic fields (especially the field generated by a current loop), and the way that moving currents feel a force due to magnetic fields (especially the force on a current loop). That's enough to argue that current loops will interact with each other in just the same ways that magnets do. And then I could tell at least a sketch of the story of how regions of aligned spins in magnetic materials act on average like current loops. (But that's quite a long story to answer a simple question, so again, I can sympathize with Feynman for saying, "There isn't a straightforward answer that a non-expert would understand.")
Didn’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”
That 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.
I'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.
You 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
I 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.
And 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.
Similarly, 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.
In 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.
One of the things that really struck me reading The Structure of Scientific Revolutions was the observation in it that you could almost always distinguish scientific fields from other by whether they taught students using the original works of those who made important discoveries or if they re-wrote them into easier to understand textbooks. If you can't separate the truth of what someone said from the way they said it you might be doing something useful but you don't have the tools to be making actual progress.
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.
> Maxwell's equations are difficult to view as a whole since they are 6 or 7 separate things that should each be understood individually
How so? You can't understand electricity and magnetism separately because they both affect each other; the clearest triumph of Maxwell's equations is that they describe electromagnetic waves, but you need the complete system of equations to do that.
I guess this description is subjective in terms of wholes or parts, but to me the each of the equations in isolation makes sense too. They all handle separate parts of electricity and of magnetism. For me, personally, I find it difficult to visualise them all at the same time. I have to look at each one carefully and then relate that to the whole which would be what we now call electromagnetism.
If 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)
Maybe 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.
Sure. 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.
IIRC, 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.
Heaviside 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
I'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.
I 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.
Agreed 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.)
Well, 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. ]
Most 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.
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.
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.
That 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.
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.
One 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.
> 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.
Derivation 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.
> 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".
> 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.")
> 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.
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.