An intuitive guide to Maxwell's equations (2020) | Hacker News Reader

An intuitive guide to Maxwell's equations (2020)

photonlines.substack.com

553 pointsgballan133 comments

I would have killed for content like this back when I was getting my Physics degree. The diagrams are so beautiful and go straight to the heart of the key vector calculus concepts needed for E&M.

I remember struggling through Jackson[1] as a rite of passage, but there's no reason future generations should have to suffer as we did. This is what the web was meant to be.

[1]: https://en.wikipedia.org/wiki/Classical_Electrodynamics_(boo...

It really is a shame that in the 20th Century, the "best" math and science books were judged not for their educational power, but for how difficult and impressive they were to fellow professionals. It seems as though the professors were afraid that they'd lose their lecturer jobs if the books were too educational on their own.

Subject matter experts are not experts in pedagogy. Because pedagogy is a seperate subject entirely. And teaching is not about getting people to say aha. Thats just performance or entertainment. Seen everywhere these days thanks to the Attention Econnomy. You can gets ahas out of people playing great music. But dont equate that with getting people to play great music. Cuz that requires getting people to do lot of mundane mindless work for long long periods of time.

> And teaching is not about getting people to say aha. Thats just performance or entertainment.

Thank you. Reading the article will not in fact give you an easier time at the Jackson Problem sets.

I think many people who think this would have helped them back in the day have simply forgotten what the actual hard part of the degree was.

Are you sure the hard part is the most important part?

Yes. There's a difference between thinking you understand something and having to prove it via problems.

Often that's how I discover I didn't really understand something at all.

What does "aha" mean to you? To me it means understanding. And isn't that the point of teaching?

I can explain memory pointers to a layperson in terms of numbered boxes and yellow notes. They're still long long way to go from that even to reversing a single linked list successfully.

A few years ago, I tried teaching for a couple years. Something that struck me was that to teach at the elementary or high school level you need specific degrees, but to teach at a university you don't. There is this thinking that because you have a PhD you can teach, which is very far from the truth. Being a good communicator is a skill in itself.

> Something that struck me was that to teach at the elementary or high school level you need specific degrees, but to teach at a university you don't

So, the thing about elementary and high school is that everyone goes to it, but only people who are good at studying go to university.

Given that the students are highly selected in the latter, you can get away with much worse instruction.

> only people who are good at studying go to university.

I think this is arse over elbow; the purpose of an undergraduate degree course is to teach you to study and do research. The "research" done by undergraduates isn't novel research; the student repeats "research" that has been done by generations of students before them. I.e., it's practice.

For this reason, writing undergraduate essays felt to me like being an impostor; you try to write in the manner of a researcher, knowing that you're faking it.

Where I live you study pedagogy and practice it while doing your PhD. If you suck at it you can still pass, but at least they take a shot at teaching it to you. When applying for positions your record on teaching might make it harder to get to those where you're expected to do it regularly.

The usual nepotism, corruption and fraud in academia will of course allow some bad teachers to advance anyway.

That sounds good, wish more places did like that.

I don’t think this is true at all… Jackson is a good book not because it’s an easy introduction to EM but because it exposes you to more complex problems than would typically be looked at in undergraduate courses. There’s clearly a place for advanced texts for this reason.

Thanks a lot man - I'm really happy to have this kind of feedback. The reason I wrote this is because I found most of the modern explanations lacking in intuition behind the equations - along with also not explaining what the actual equations meant. If you found this useful please share and subscribe - I'm also trying to provide intuitive guides to other concepts (Schrodinger's equation, Black Holes, Quantum Mechanics, other complex topics) and eventually I'm hoping to write books on some of these topics which present math and physics in a much more clear and intuitive manner. Math shouldn't be hard to grasp. At the very bottom level it's very simple but presenting it in a clear and intuitive manner I will admit is very hard. Also full credit to a lot of the material as well goes to Grant Sanderson (3Blue1Brown) and most of the diagrams there were generated using Vexlio which I also highly promote: https://vexlio.com/

This is a really good article. A minor nit - it'll read easier without the exclamation points after every sentence.

You explain concepts really well. Wish I had professors like you in college.

Also check out the YouTube videos of eigenchris, especially his series on tensor calculus and relativity. Probably the clearest explanations I’ve seen on these subjects.

Thanks @photon_lines! In your temperature diagram, you mention that every point will take the average of the neighboring points. However, the equation is not a constraint on the temperature but on the "change of the slope (or gradient) of the temperature". The bigger the slope (in space), the faster (in time) the temperature changes at that point!

'The bigger the slope (in space), the faster (in time) the temperature changes at that point!' - Sorry but I'm not really reading you here. If the points around an 'atom' a symmetrically and equally far away when it comes to the point in question but are opposite in magnitude (i.e. imagine having a point with temperature 12 degrees Celsius which is surrounded by a neighboring points which have temperatures of 8 degrees and 16 degrees (so the delta is +4 and -4) then the temperature here will stay the same. The slope of the temperature field has nothing to do with this - unless maybe you're alluding to the slope of something else? I think I should have maybe explained this equation in terms of 'concavity' instead of using the methodology which I used - you can get a good grasp of this in this link: https://www.youtube.com/watch?v=b-LKPtGMdss

Thanks for taking the time to respond and analyze my comment! - The bigger the slope (in space), the faster (in time) the temperature changes at that point -

I have to confess that I got it wrong, indeed: the right side of the equation is a Laplacian. But, rather than describing an average in temperature, it describes the divergence of the temperature field.

I wanted to command you on your excellent work! I am curious how easy is to use Vexlio, is a steep curve? And any favorite books you want to share, I'm going on holidays soon, and haven't planned much for reading yet.

Thank you!! Vexlio has no learning curve - it's literally so easy to use that I haven't had to read ANYTHING in order to get accustomed to doing what I need to do. I simply open the program and the UI is so intuitive that literally you will simply have no issues figuring out what you need to do to accomplish what you want to accomplish. When it comes to books: what are you interested in? Math / physics books or more general stuff? My favorite book of all time is 'Crime and Punishment' - it literally shows you how Dostoevsky thinks and puts you inside of his mind - not many books can do this.

I will give Vexlio a try! I don't have a book category per se, but I read freakonomics last year, and was very impressed by it, if you have similar recommendation, would appreciate it! I read Crime and Punishment but long time ago in high school, don't remember much of it though.

Sure - like I said, it's hard for me to say what my favorite is without naming a category, but since we're on the topic of Electromagnetism I'll stick to physics and recommend 1) Introduction to Elementary Particles by Griffiths (best into to standard model in particle physics) and 2) QED by Richard Feynman. QED btw is very well done and literally shows you the 'algorithm' behind standard particle physics and does so in a clear and unambiguous manner - really an amazing work by Feynman.

I agree with the parent comment that the article was quite good and useful, although I do have a nit to pick with the section on unification of the electric and magnetic fields. I think needs to look at an additional scenario.

That section looks at three scenarios:

1. An electrically neutral straight wire with an electron current and a test charge near the wire moving in parallel to it at the same velocity as the electrons in the electron current, observed from an observer stationary with respect to the positive charges in the wire analyzed without taking into account relativity.

The analysis shows that there is no electrostatic force on the test charge because the wire is electrically neutral, but there is a magnetic force because the test charge is moving in the magnetic field caused by the electron current.

(Nit within a nit: the drawing for this shows the positive and negative charges in the wire separated with the positive charges quite a bit closer to the test charge. That would result in an electric field from the wire that would attract the test charge. Maybe insert a short note saying that the positive and negative charges in the wire are actually mixed together so that their electric fields cancel outside the wire?)

2. Same as #1 except the observer is stationary with respect to the test charge.

The observer now sees no electron current in the wire, but does see a current from the positive charges. But the magnetic field from that positive current should not exert a force on the test charge because magnetic fields only affect moving charges and the test charge is not moving in the observer's frame.

3. The Lorentz contraction is introduced, and #2 is re-analyzed taking that into account. That Lorentz contraction applied to the positive current manifests to the observer as an increased density of positive charges. There wire now appears to the observer to no longer be electrically neutral. It has a net positive charge and the resulting electric fields attracts the electron to the wire.

What's missing is circling back and looking at scenario #1 again but including the Lorentz contraction. In scenario #1 the observer sees the negative charges moving, so should see increased negative charge density due to the Lorentz contraction, and the wire should appear to them to have a net negative charge, which would try to repel the test charge.

#1 with Lorentz included then is a fight between the magnetic attraction and the electrostatic repulsion.

Assuming objective reality and so requiring the test charge to actually feel the same force no matter who is observing we can infer that if the electrostatic force toward the wire in #3 is F then the magnetic force toward the wire in #1 must be 2F, which when opposed by the -F electrostatic force from the Lorentz contraction of the negative charges in the wire gives a net force toward the wire of F.

Thank you for the feedback. I'll review my notes and see if I can clarify this section - my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field. If I made any errors I give you my sincere apologies. Btw if you want to make edits to my work directly - you can find it as it's fully open source: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-E...

> my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field.

This isn't quite right, there are field configurations where the magnetic field doesn't vanish in any reference frame. This is actually the typical case: consider, for instance, two point charges moving relative to one another.

The right takeaway from SR isn't that the magnetic field is fake and the electric field is real, it's that both magnetic and electric fields are frame-dependent and it's the electromagnetic field tensor that's the real physical object.

You're 100% correct. The explanations I was reading gave me the impression that we can get rid of the magnetic field - but this only works in special cases. Right now I'm reviewing my notes and realizing that yup - they're both needed and will need to modify my explanation there or simply get rid of it entirely and keep things simple. Once again - thank you for the correction - you're a life saver this is definitely something which I overlooked and I wish I had more time to explain properly!

This is really excellent. I particularly like the outline of div and curl, the dot product and the cross product, and the connections drawn between the differential an integral forms. Thanks.

Fantastic work.

Deserves to be widely used to teach Maxwell's equations.

THANK YOU.

Speaking of 3Blue1Brown - would you be interested in presenting this same material in a video? 3Blue1Brown has regular showcases for this type of material. I will read your article when I get a moment but after a hard day's of screen time I sometimes like to kick back to a video.

The basic ideas, the pictures, the diagrams, etc, found here, typically show up in enough books if you look for them that I don't feel that this was the main limiting factor in my physics education. The difficulty of Jackson (which doesn't show up until grad school for most students) is in the problem sets, not the ideas behind the equations (which most students have a had at least two courses in already).

I don't believe that having a more 'intuitive' idea of the equations really helps all that much, as the intuition needed for solving the problems isn't really physical, but mathematical. Which integrals are solvable, which order of integration will make this tractable, do I need to use properties of Bessel functions here, etc.

We can argue whether getting good at this sort of thing is actually useful for physicists, but I wouldn't know. Very few of us ended up becoming researchers in the field.

Jackson's the standard here in Spain (undergrad), after working through a course based off of Griffith's.

I am confused.

1. While the posted guide is excellently written, it's not particularly novel. I was taught EM in a very similar fashion. Diagrams similar to those in the guide were drawn on the board by my professors.

2. Jackson is a graduate EM text. It is mathematically difficult, because when you read it, you should have been familiar with EM and all this conceptual underpinning for at least 3-4 years. The goal of Jackson is to solve the equations for scenarios that undergrads would find challenging. What did you study in your undergrad?

Re #2: Jackson was the standard text for undergrads like me doing a Mathematics degree. It was a late second year or early third (final) year text if I recall rightly. This was 1992, so I'm still amazed to read that its still a commonly used text.

Fwiw, other standard texts used in Durham (UK) back then were Spivak on Calculus, Goldstein on mechanics, and for the mathematical physics kids, landau and lifschitz on mechanics and electromagnetism, and (an absolute doorstop) Misner, Wheeler and Thorne on Gravitation (relativity).

From the first preface (1962) of Jackson

> Typically, the undergrad program in electricity and magnetism involves two or perhaps three semesters beyond elementary physics.... As a general rule, a two-semester course in electromagnetic theory is given to beginning graduate students. It is for such a course that my book is designed.

So, your professors did you injustice by using an inappropriate book. Spivak, Goldstein and MWT are undergraduate books and appropriate. Landau and Lifschitz is great and accessible to smart undergraduates, but I don't see why you would use it for mathematical physics. Sure, Landau emphasized methods a lot, but there are better books for it.

As someone studying Physics (Bachelor) in Germany Jackson is what my electrodynamics professor recommended. My professor greatly shortened the chapter maxwell in matter and opted to give an intro into quantum electrodynamics instead.

At my uni it's a fourth semester course with theoretical mechanics (second semester) and quantum mechanics (third semester) preceeding it.

> So, your professors did you injustice by using an inappropriate book.

Not necessarily: undergraduate and pre-undergraduate education differs a lot between the UK and the US.

MTW ("the telephone book") is definitely not an undergrad textbook (although you might be able to cobble together an undergrad course out of bits and pieces of it). It is very heavy on intuition and visualization, though, which is why I like it (e.g. the "egg carton" visualization of differential forms).

From the preface of MTW:

> It supplies two tracks through the subject. The first track ... is suitable for a one-semester course at the junior or senior level or in graduate school.

As you say, it picks out bits and pieces that an undergrad can understand.

Today, there are better GR books, so use those.

There are certainly many better GR books (going back at least as far as Wald), but there's still nothing quite like the whimsy of MTW.

'What did you study in your undergrad?' - Computer Science. I study applied math and physics in my spare time - I'm currently teaching myself quantum field theory and other topics. For the most part - they're incomprehensible to an average person which is why I'm so passionate about doing what I'm doing - all of this stuff is extremely simple underneath but we humans find ways to make it complicated. Why not untangle that complexity and simply explain things in a clear and intuitive manner? Also - your comments on your undergraduate ease of grasping Maxwell's equations usually don't apply to everyone. Many professors don't sketch out what they mean and many books don't go through the fundamentals that students need in order to grasp what they mean. This guide is supposed to give someone a good background on 1) what they need to understand in order to grasp the equations and 2) what the equations actually mean in clear human language. Hopefully this helps - I also haven't had a chance to read Jackson but he's been mentioned so many times that right now I'll make a note to actually read the book and see how well he explains the concepts and see if I can maybe find other ways of making things simpler.

I am happy that you are studying math+physics as a hobby. And I will say once again, that you distilled the conceptual parts of EM in the guide very well.

I will also agree with you that many professors don't teach well. I was a physics prof for a few years, and it is difficult to distill stuff well. Not everyone has the skill, passion and the job incentives to do it well. I was lucky enough to be graced with profs who did.

I am glad that you have the passion for this. I will say this though, that once you become a formal teacher (school/university), then it becomes clear to your that your responsibility is not complete until your students have the skills to use the concepts that you are teaching them. Skill here means being able to model actual physical systems and get both the behavior and numbers out. When teaching a course, you have limited contact time with students and students have limited total time to spend on the course. You have to balance teaching conceptual understanding and modelling skills in that time. That balance is extremely difficult to attain, the reasons for which will easily fill a small book.

You can go all in on concepts, and what happens is that within a few months students have completely blanked out on everything, because you need the mathematical framework and have solved difficult problems for things to stick in your brain long term. And conversely teaching only maths is terrible because no one knows and what and why.

We used the John Kraus book on Electromagnetics for the dynamic fields course. This was preceded by a course on static fields. That course’s final had the shortest test statement I had ever encountered: “Derive Maxwell’s Equations”. I found the Kraus book satisfactory.

What was an acceptable answer to “derive maxwells equations”?

Show the steps such as faraday’s law and other things that led up to it.

Maxwell's equations are statements of experimental results in differential form, plus Maxwell's guess that there are no magnetic monopoles and that there is a displacement current to carry current through a capacitor, providing continuity.

You don't really derive them (unless the professor had in mind, "Produce them from the integral forms"). A better question would be, "Derive the EM wave equation in free space from Maxwell's equation and determine the phase velocity."

> Produce them from the integral forms

Pretty much what he was looking for. Keep in mind that this was at the end of a course covering pretty much all of static fields. And this course was a precursor and prerequisite for the following course which was about the application and implications of Maxwell's equations.

The latter was taught by my favorite professor, who seemed to have a radar to know when the class was not following. Without request, he would erase what he had written on the board and restart.

The lab was fun. Three quarters of the way through the first 2-hour lab, he said "By the way, anyone who gets the right answer loses points on this exercise. The point is to teach you how hard it is to come to the right answer."

He told a story of his work during WW II. His favorite thing then was to build a little $50 piece of equipment to render million-dollar radar sets useless. Clearly his task was to help improve the radar set. "The odds are stacked in favor of the jammer." Who obviously cheer for the inverse of the distance to the fourth power.

People often make comments like this, forgetting that want they are marvelling at was actually in the book they read or class they took the first, and then actual different is that they forgot, or they've had more time to stew on the material so it feels more familiar the second time through. Hence the adage that the best book on the subject is whatever book you read second. It seems so much more intuitive the second time through.

Yeah reading this a couple decades out from my undergrad physics classes, my thought was "I remember learning all of this very painstakingly over multiple years and multiple different classes".

But also, I'm not sure I would have grokked much in this article without having taken those classes already, with the benefit of lectures and graded homework and group study sessions and TAs answering questions and all that...

At least it wasn't Halliday and Resnick. It's been 35 years since my BSME at Purdue, and I can still remember their names. God I hated those textbooks. If someone tells me that this Jackson book was worse, I won't believe it.

What did you dislike? I went through all of HRW as an undergrad (25+ years ago) and recall generally liking the presentation.

David J Griffiths' Introduction to Electrodynamics was a fantastic undergraduate-level E&M textbook https://en.wikipedia.org/wiki/Introduction_to_Electrodynamic...

One of my favorite parts of my education was going through E&M to arrive at the beauty of Maxwell's Equations.

I later found out that you can squeeze even more beauty out of them by boiling them down even further using differential geometry.

http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handou...

I prefer the spacetime algebra version.

https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

“Maxwell’s theory only becomes simple and elegant once we start to think of the fields (mathematical functions) as being primary and the electromagnetic stresses and mechanical forces as being a consequence of such fields, and not vice-versa.”

There is a lot of interesting discussion on whether fields are real, and the dialogue goes back centuries: https://youtu.be/j2oSyAfPzWg?si=BHRv8lodGhqZBtbl

I would have assumed Rupert Sheldrake would be completely taboo to most on HN, so I'm glad to see your comment voted highly! His telepathy "pseudo" scientific work is what he often leads into with any discussions of fields, and this was my favorite talk he gave: https://www.youtube.com/watch?v=GS1Drlyr37Q

This is incredibly well explained. Everything is simple, yet it is packed with so much details that memorising this and understanding this cannot be done without effort and focus. This whole stuff is fascinating when explained in such a way that it makes sense. I fought with this during my 2nd year of engineering studies, but did certainly not understand half of it at the time. With that explanation, I would have enjoyed studying the subject so much more. I guess I was not smart enough to understand my textbook and all the consequences of the formulas, so that I was unable to be fascinated by the subject.

A great article. Good to see they give Heavyside the credit he deserves for what everyone now calls "Maxwell's equations".

I love articles like this, but they all make the same mistake of starting with vector algebra instead of geometric algebra. In 3D space, vector algebra works, but it falls flat on its face in both 2D and 4D scenarios. It's intuitive until it is completely broken.

I would love to see the same style of article, but using bivectors and the like where appropriate, such that the whole thing generalises neatly to 4D space-time, not just 3D space.

I will probably get downvoted for pointing this out, but the reality is that the geometric algebra approach to E&M, while interesting for its own reasons, will not replace the formalism based on Gibbs's vector calculus. One reason is simply that vector calculus is pretty intuitive and easy to learn. The major reason, however, is that the vector calculus approach is totally entrenched in the worlds of engineering and physics. After 100 years, nobody actually practicing those disciplines will make the notation change just so they can replace the 4 Maxwell's equations with one geometric algebra equation.

Also, Gibbs's vector calculus is used in fluid dynamics and other engineering disciplines, and as far as I know, nobody it touting the advantages of geometric algebra to folks working in fluid dynamics. I can be pretty sure that some HN reader will show me I am wrong about this by pointing out one lonely researcher who has found a way to express the Navier-Stokes equations using the geometric product ... but so what? ... My main point is that traditional vector calculus is a language everybody knows how to speak, geometric algebra is just another way to say the same things, so why would anybody change?

Interesting paper to skim that seems to look into it.

https://vixra.org/pdf/1206.0021v1.pdf

The metric system seems like a similar analog to geometric algebra vs vector calculus. You are saying the same thing but the language you are using is much more internally consistent.

Adoption has been bumpy given the US resistance but I think in the long run it (or something even more consistent) will win out. Similarly I think geometric algebra will be adopted. Maybe not in our lifetimes but eventually.

Field theorists pretty much already have abandoned the vector calculus version of the equations, though.

I actually took a look at doing this, but most human minds aren't tuned to 4D space-time, so if you have ideas on presenting this sort of thing to most people let me know and I'll be more than happy to modify my approach!!

Here's a critical take on geometric algebra: https://alexkritchevsky.com/2024/02/28/geometric-algebra.htm...

tl;dr: GA's geometric product is a mixed-grade differential form, which is quite weird. Why not just think in terms of differential forms? Maxwell's equations are so sweetly summarized as dF=0 and d*F = J.

Just to give a brief answer to those reasonable criticisms:

The mixed-grade already exists in complex numbers (it is very useful there, and even more so in geometric algebra).

Differential forms are included in geometric algebra (the exterior/outer products are isomorphic). Turns out, combining that product with the inner product gives you an invertible product (as Clifford found out). That by itself already is a huge advantage.

Finally, Maxwell's equations are sweetly summarized in differential forms, but even more in geometric algebra: dF = J . Not only it is just one equation instead of two, but in addition the "d" (or "nabla") is directly invertible thanks to the geometric product (which differential forms lack and then have to use more indirect methods, including the Hodge dual).

By the way, I'm very partial to geometric algebra, but wouldn't say it is an "error" not to use it! Maybe just a big missed opportunity :)

even more in geometric algebra: dF = J

You can do that using differential forms as well - using the co-differential δ, we can write a single equation (δ + d)F = J. However, from the perspective of Yang-Mills theory, that's a rather questionable approach as we're stitching together the Bianchi identity and the Yang-Mills equation for no particular reason...

Cool, I didn't know that. Still, the main point of the geometric algebra version is that it's not a "stitching" exercise, but a natural operation in the algebra -- and even better, an invertible one.

Every vector calculus instructor should teach their students the intuitive (by which I mean visual/physical) meaning of grad, div, and curl (and the intuition behind results like Stokes's and Gauss's theorems). Even engineering students uninterested in proofs should be able to grok the intuition.

Do you have other good resources for this? (I'm sure I understood this better back doing my degree but it was 40 years ago)

Just search for videos, stuff like this:

https://www.youtube.com/watch?v=eEwZeY51mT0

You can make similar kinds of videos for all 3 of them. That video shows a divergence free field since number of particles aren't changing, I easily see that since I know the intuitive explanation for divergence, it is useful to have intuition for those things.

Gradient is just the equivalent of slope but for higher than 1 dimension.

Edit: Or no, that field has divergence, I'm dumb I didn't watch the start, many particles accumulate at a few points, that is due to divergence. Divergence is essentially areas that attracts or repels particles in that simulation.

Found the divergence video, in case it is hard to understand what I said above: https://www.youtube.com/watch?v=c0MR-vWiUPU

Note: This series (Multivariate calc on Kahn Academy) is done by Grant Sanderson of 3Bl1Brown. It's outstanding, and goes over the concepts related to how the article here describes the fields.

Wikipedia explains the basics

https://en.m.wikipedia.org/wiki/Del and the related articles on gradient (slope), divergence (flow across a boundary), and curl (circulation)

Have you ever seen a book that doesn't? It's in every book I've seen.

The textbook my Emag professor wrote himself made sure to avoid anything intuitive or visual, and was just a dense tome of text and equations with nothing else. He had a lot of trouble getting it published, but made sure to teach from it for decades. If you asked nicely, he'd give you a copy of the errata that he never fixed in the book. That class was essentially "vector calc for EEs" so it was my introduction to all these concepts, and I never intuitively understood them until much later.

University education in a nutshell.

We need to burn it down, better sooner than later.

> Have you ever seen a book that doesn't?

The brown Rudin.

> Virtually every force we experience in everyday life (with the exception of gravity) is electromagnetic in origin. [...] It wasn’t until the arrival of Oliver Heaviside, who reformulated and simplified the equations [...]

Maxwell's original equations connected light and electricity. Maxwell's original 20 equations had 20 unknowns, using 'quaternion-based notation', which no one understood.

Heaviside restated Maxwell's 20 equations into 4 equations using vector calculus. The restatements helped with simplification, but I believe it wasn't without cost.

There's a lot that's still unexplained in our modern world, especially with regards to individual humans' experiences. I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about.

There ought to be a link between electromagnetism and gravity, we just haven't figured it out yet. This wikipedia article was cited by Bing CoPilot in response to my query. It's above my pay grade, maybe one of you can translate it for me: https://en.wikipedia.org/wiki/Gravitoelectromagnetism

> I got a window on these as a taxi driver, where I was sent people who helped me figure out things I'd been wondering about.

I'm curious now, would you indulge me? If its woo woo we can just pretend no one is reading :)

Some people are of the philosophical bent that our world is entirely random. But this doesn't commonly match our experience. For example, I often asked people how they met their significant other. Sometimes it was nothing special, some couples had quite a story.

I had the sense that I got certain passengers for more than just transportation. Some people were having a rotten day, and I was able to cheer them up. One lady had some time to kill before her bus' departure time, so we went to the 24 hour diner, ordered our own pies and compared notes. When we got to the bus station she said it was the best birthday she'd had in quite a long time.

This was a semi-recent comment about the matching algorithm: https://news.ycombinator.com/item?id=34402081

The most important thing I learned in my taxi was about substance abuse. This HN poll didn't get any upvotes, but it references some of the diaries I never finished: https://news.ycombinator.com/item?id=39071316

Another comment: https://news.ycombinator.com/item?id=25238488

If you're so inclined, I'm curious if you've experience is also that our universe is more than random?

The universe clearly isn't completely random. I'm not sure anyone believes that.

I am the founder of randonautica, so in short yes I have found that our universe is more than random. Thanks for the answer, sometimes I wish a little more intuition was allowed here rather than only analytics.

Here is a set of lecture slides on the changing form of Maxwell's equations including the component form (which was apparently Maxwell's very first version), the quaternion form which came second and then Heaviside's version[1]

Fun fact about Heaviside (that noone asked for) - he's also the guy who invented the "cover up" method of doing partial fraction decomposition quickly.[2]

[1] https://www.thp.uni-koeln.de/gravitation/mitarbeiter/hehl/Ma...

[2] https://math.mit.edu/~jorloff/suppnotes/suppnotes03/h.pdf

I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful. He certainly got rid of the quaternions but that seems like a minor difference because Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.

The major difference I found was that Maxwell was expressing things in terms of the scalar and vector potential (which is what you have to do in QED) whereas Heaviside got rid of that and just had an electric and magnetic field instead. I found that you need 7 of Maxwell's equations to derive the 4 Heaviside(?) equations.

If you actually wanted to embrace quaternions you could write the famous 4 equations as just two (using natural units):

∇E + dB/dt = -ρ

∇B - dE/dt = J

> Maxwell was also not really taking advantage of them much and always split them up into scalar and vector part.

Did Maxwell actually use quaternions? If I recall correctly, at least in A Treatise on Electricity and Magnetism, quaternions were not actually used. Instead, he did most things in Cartesian coordinates, and all equations were applied to a vector's x, y, z components tediously. But many sources claimed Maxwell used quaternions, including quotes from Lord Kelvin. My reading on this part of history is limited, so my guess is that he did use them in personal research or in later papers. On the other hand, some other physicists of the same era used quaternions extensively, including applying them to Maxwell's electromagnetism, that is a sure fact...

Coincidentally, A Treatise on Electricity and Magnetism was written as an overview all electromagnetic phenomena as a whole, so it paid very little special attention to the generation and transmission of electromagnetic waves. Combining that with its difficult math, the book would puzzle physicists for another decade before they see the light from the book, and made it a rather curious period of history in electromagnetism.

> I've been trying to find these 4 equations in Heaviside's writing but so far have not been successful.

In 1885, Heaviside published Electromagnetic Induction and Its Propagation in The Electrician, and formulated what he called the "Duplex Form" of Maxwell's equation. This was a long series of papers published in several months, and later republished in Electric Papers, Volume I. Basically, following his physical intuition, he felt that electric and magnetic fields should be symmetric and generate each other, and that should be directly highlighted in equations.

The logic of the paper went like the following.

First, he started with a definition of electric current [1]:

    C = kE
    D = cE / 4π
    Γ = C + D

in which, E denotes electric force, C denotes conduction current, k denotes specific conductivity constant, D denotes displacement current, and c denotes dielectric constant. Finally, Γ denotes true electric current, which is the sum of the conduction and displacement terms.

Next, a definition of magnetic current [2]:

    B  = µH
    G  = Ḃ / 4π = µḢ / 4π
    G' = gH + µḢ / 4π

H denotes magnetic force, B denotes magnetic induction, µ denotes permeability, G denotes magnetic current, Ḃ and Ḣ are derivatives of B and H (Newton's notation). Hypothetically, suppose that magnetic monopoles exist (Heaviside did so), G' would denote the "true magnetic current", with an extra conduction term gH, where g is a constant similar to k.

Then, he introduced the concepts of divergence and curl, and their physical significance [3]. After more discussion and derivation, he finally wrote [4]:

    curl (H - h)  = 4πΓ = 4πkE + cĖ
    -curl (e - E) = 4πG = 4πgH + µḢ

in which, e and H denote impressed electric and magnetic forces to take static fields into account. Finally, since magnetic monopoles don't exist, he made g = 0, but kept this term in the equations for symmetry and elegance. [0]

This is the core of Heaviside's Duplex Form of Maxwell's equations. one can clearly see the co-evolution of electric and magnetic fields, and is the precursor of the modern Maxwell's equations as we know today in its vector calculus formulation. As far as I know, his treatment of "physical" vectors as first-class objects is his original invention (independently invented by Gibbs as well), although the concepts themselves came from quaternions.

This is not a complete summary, as he continued his analysis in a series of publications.

A good book on this part of history is Oliver Heaviside: the life, work, and times of an electrical genius of the Victorian age, by Paul J. Nahin.

[0] So the claim "Maxwell's equations need modifications if magnetic monopole has been discovered" is historically inaccurate, it should rather be, "be restored to Heaviside's original form."

[1] Electric Papers, Volume I, Page 429, https://archive.org/details/electricalpapers01heavuoft/page/...

[2] Page 441: https://archive.org/details/electricalpapers01heavuoft/page/...

[3] Page 443: https://archive.org/details/electricalpapers01heavuoft/page/...

[4] Page 449: https://archive.org/details/electricalpapers01heavuoft/page/...

Also, in page 452 [5], Heaviside wrote:

    div B = 0

Finally in page 475 [6]:

    div D = ρ

So yes, essentially all 4 Maxwell's equations were here.

[5] https://archive.org/details/electricalpapers01heavuoft/page/...

[6] https://archive.org/details/electricalpapers01heavuoft/page/...

Didn't see the reply this late, hopefully you will see mine.

He did indeed use quaternions but it's not easy to find: https://archive.org/details/atreatiseonelec02maxwgoog/page/2...

Unfortunately these equations are not quite without mistakes (I remember a missing dot for a time derivative) compared to the component form. They're correct in the wikipedia article: https://en.wikipedia.org/wiki/History_of_Maxwell's_equations...

If you replace S.∇ with ∇· and V.∇ with ∇× you essentially get the vector calculus version of the equations.

Thank you for extracting the core ideas out of this lengthy text. But I'm still wondering where this very concise present-day formulation with just 4 equations was first written down, even if you can somehow find them scattered around in the book. I found something about Hertz but didn't try to follow up on it, i think he may have only considered a vacuum.

I remember reading a great answer [1] from Stack Exchange, that claims:

> the 1873 treatise used a pre-Heavisde form of vector calculus cannnibalized from Hamilton's quaternions ... only sparingly, to present the equations in capsule summary form.

Thanks for the reply. From your link, I now understand what does "vector calculus cannnibalized from Hamilton's quaternions ... only sparingly" means.

[1] https://hsm.stackexchange.com/a/15618

Quaternion was crucial and instrumental tool in Maxwell discovery and the formulation of the electromagnetics (EM) equations. When Terence Tao was asked how come nobody has proof of the Riemann hypothesis, arguably the hardest of the Math problems, and according to him this is because there is no appropriate tools available at the moment to proof it. I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis.

Unlike other waves for example sound waves, EM has a unique polarization property. In order to completely and correctly model EM based phenomena quaternion based formulation and representation is necessary. One of the reasons that almost all existing wireless modulation are not utilizing polarization is due to most of the microwave and wireless engineers are not familiar with quaternion. Ironically their biased attitude is not unlike early mathematicians and scientists that were very much opposed to complex number, and it turn out that almost all of the modern wireless modulation for example OFDM are utilizing complex number.

For the derivation of the Maxwell’s equations using geometric algebra involving quaternion please check these articles and they can be summarized the into one elegant equation [1][2].

[1] Maxwell’s eight equations as one quaternion equation:

https://pubs.aip.org/aapt/ajp/article/46/4/430/1050887/Maxwe...

[2] A derivation of the quaternion Maxwell’s equations using geometric algebra:

https://peeterjoot.com/2018/03/05/a-derivation-of-the-quater...

Lay people seem to have this weird obsession with Quaternions and love to suggest that somehow theoretical physicists are missing something because they don't use them. But physicists are almost disgustingly familiar with SU(2) which is isomorphic to the quaternions and easier to work with and understand (quite obviously, in my opinion). It is hard to imagine, from my point of view, that a mere isomorphism stands between physicists and progress, especially given that physicists have long generalized _beyond_ SU(2) and the quaternions in their understanding of fundamental fields. Formulating an SU(3) gauge theory in terms of quaternions would at least be difficult and almost certainly be goofy, if not impossible.

As for "I'm not a mathematician but I've got a strong feeling that quaternion will be one of the potent tools to proof Riemann hypothesis" I'd love to understand your intuition here, because I just don't see it.

Please read my comments properly, I'm saying that currently engineers are not familiar with quaternion because they have been exposed to vector calculus in their formal education not physicists. About thirty years ago in my EM class we have had a combined class of engineers and physicists, for some unknown reasons the lecturer was a microwave engineer. Thanks to physicists like David Hestenes that are more physicists now who are familiar with quaternion but for engineers it's still very much a minority thanks to Oliver Heaviside who really hated quaternions and popularized the inferior vector notations rather than superior quaternion versor notations.

As I've also mentioned in my comments the quaternion is necessary in order to fully describe polarization in EM, and there other comments in this post that upholds Heaviside vector can provide the exact representation of EM that is not correct. Heaviside vector representation is the simplication of the more comprehensive quaternion representation but do not mislead to say otherwise i.e the same thing.

For Riemann hypothesis, I just providing my intuition that whoever want to proof it need to have quaternion in their toolbox while Terence commented that whoever want to proof it needs a proper set of tools but he did not mention the exact tool just merely saying that current tools are inadequate. For me whatever the set of tools that will be used to proof Riemann hypothesis, one of them will be most probably quaternion.

Gravitoelectromagnetism doesn't actually have anything to do with electromagnetism except that certain formal features of the theory of general relativity correspond roughly to the mathematical structures we talk about in electromagnetism, albeit with the proviso that the symmetries underlying the two theories are different.

More on relativistic equivalence between electric and magnetic field:

https://physics.stackexchange.com/questions/489291/how-did-e...

Just curious: Sussman and Wisdom have written a book called "Structure and Interpretation of Classical Mechanics" following the classic SICP Scheme book. Has anyone attempted a similar approach for electromagnetics?

Maxwell's equations are a classical field theory (no quantization). That means Maxwell's equations are one of the theories of the body called classical mechanics. So if you wanted to, you could write down the Lagrangian (density) or Hamiltonian for various experimental configurations (eg charged particle in a field) and derive Maxwell's equations (there are a couple of papers like this). Nothing stopping you from using SICM's formalism either. Would it be a useful exercise? No clue.

Sussman and Wisdom do it themselves (briefly) in chapter 10 of Functional Differential Geometry.

They also have a book on field theory, which is just about E&M basically.

Are you sure? I don't see such a book at https://mitpress.mit.edu/author/gerald-jay-sussman-2078/. I'd be very interested to read it if there is one.

Functional Differential Geometry is about the maths required for field theories but focuses on relativity as the main example.

You're right: I'm thinking of the theoretical minimum books by Leonard Susskind and Art Friedman.

https://fab.cba.mit.edu/classes/862.22/index.html

Ch. 6?

I'm throwing money at the screen and nothing is happening!!

Please make a book of this and other associated topics. You write very well.

Nobody has explained Maxwell's equations better than Parth G. The video is a bit old, but very intuitive!

https://www.youtube.com/watch?v=0jW74lrpeM0

The "displacement current in the medium" that Maxwell originally included in his equations was directly tied to his concept of the "luminiferous ether" as the medium through which light and electromagnetic waves propagated.

The ether was never definitively proven not to exist; however, extensive experiments, including those in space, have consistently failed to detect its presence. Notably, frame-dragging effects observed in experiments such as Gravity Probe B support the predictions of general relativity without requiring an ether.

Very sad.

'Why is Maxwell’s Theory so hard to understand?' - an essay by Freeman Dyson on how Maxwell's theory brought a sea-change (or perhaps I should say paradigm shift) to physics:

https://www.clerkmaxwellfoundation.org/DysonFreemanArticle.p...

In the penultimate paragraph, he writes "For example, the Schrödinger wave-function is expressed in a unit which is the square root of an inverse cubic meter. This fact alone makes clear that the wave-function is an abstraction, for ever hidden from our view. Nobody will ever measure directly the square root of a cubic meter." This has me wondering if there is a reason he could not have ended with "Nobody will ever measure directly the square root of an inverse cubic meter", other than that the as-written version makes the point just as well.

By the way i think the modern formulation of Maxwell equations as four equations is an intuitive reformulation of the original formulation of maxwell i believe.

> No one actually knows or understands what a ‘point mass’ is!

True. But we do suspect the existence of massless points, and surely have many pointless masses.

Fantastic job on the 2D plots stretched out over time in 3D. That's really hard to pull off. How were they made?

Those visuals are directly pulled from Grand Sanderson (3Blue1Brown). I gave him full credit and linked the videos at the end of the article but in case anyone needed a reference you can view the visuals here: https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQ...

I would love to see the approach extended to explain the special relativistic aspects of electromagnetism that, as I understand, links electric and magnetic fields, capacitance and inductance, etc. like space and time coordinates. There seems very limited material available on the same on the Internet.