[HN Gopher] An intuitive guide to Maxwell's equations (2020)
___________________________________________________________________
An intuitive guide to Maxwell's equations (2020)
Author : gballan
Score : 490 points
Date : 2024-06-02 20:54 UTC (1 days ago)
(HTM) web link (photonlines.substack.com)
(TXT) w3m dump (photonlines.substack.com)
| olooney wrote:
| 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...
| lupire wrote:
| 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.
| jc6 wrote:
| 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.
| archgoon wrote:
| > 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.
| lupire wrote:
| Are you sure the hard part is the most important part?
| kaashif wrote:
| 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.
| esafak wrote:
| What does "aha" mean to you? To me it means understanding.
| And isn't that the point of teaching?
| varjag wrote:
| 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.
| pezezin wrote:
| 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.
| vkou wrote:
| > 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.
| denton-scratch wrote:
| > 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.
| cess11 wrote:
| 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.
| pezezin wrote:
| That sounds good, wish more places did like that.
| physicsguy wrote:
| 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.
| photon_lines wrote:
| 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/
| WalterBright wrote:
| This is a really good article. A minor nit - it'll read
| easier without the exclamation points after every sentence.
| wayoverthecloud wrote:
| You explain concepts really well. Wish I had professors like
| you in college.
| cgh wrote:
| 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.
| robertbarbe wrote:
| 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!
| photon_lines wrote:
| '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
| robertbarbe wrote:
| 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.
| 3abiton wrote:
| 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.
| photon_lines wrote:
| 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.
| tzs wrote:
| 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.
| photon_lines wrote:
| 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...
| nyssos wrote:
| > 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.
| kayo_20211030 wrote:
| 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.
| cs702 wrote:
| Fantastic work.
|
| Deserves to be widely used to teach Maxwell's equations.
|
| THANK YOU.
| archgoon wrote:
| 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.
| angra_mainyu wrote:
| Jackson's the standard here in Spain (undergrad), after
| working through a course based off of Griffith's.
| abdullahkhalids wrote:
| 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?
| kjellsbells wrote:
| 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).
| abdullahkhalids wrote:
| 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.
| macbr wrote:
| 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.
| denotational wrote:
| > 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.
| senderista wrote:
| 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).
| abdullahkhalids wrote:
| 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.
| senderista wrote:
| 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.
| photon_lines wrote:
| '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.
| abdullahkhalids wrote:
| 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.
| wglb wrote:
| 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.
| richk449 wrote:
| What was an acceptable answer to "derive maxwells equations"?
| hxriv wrote:
| "Be Maxwell" - op, probably.
| wglb wrote:
| Show the steps such as faraday's law and other things that
| led up to it.
| lupire wrote:
| 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.
| sanderjd wrote:
| 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...
| TheRealDunkirk wrote:
| 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.
| senderista wrote:
| What did you dislike? I went through all of HRW as an
| undergrad (25+ years ago) and recall generally liking the
| presentation.
| fraserharris wrote:
| David J Griffiths' Introduction to Electrodynamics was a
| fantastic undergraduate-level E&M textbook
| https://en.wikipedia.org/wiki/Introduction_to_Electrodynamic...
| xeonmc wrote:
| More on relativistic equivalence between electric and magnetic
| field:
|
| https://physics.stackexchange.com/questions/489291/how-did-e...
| humanfromearth9 wrote:
| 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.
| RachelF wrote:
| A great article. Good to see they give Heavyside the credit he
| deserves for what everyone now calls "Maxwell's equations".
| sesm wrote:
| Is 'curl E' a standard notation nowadays? When I was at uni we
| used 'rot E'.
| jonlong wrote:
| Wiki says that curl is standard in North America, while rot is
| common in "the rest of the world, particularly in 20th century
| scientific literature". As a North American I can confirm that
| I was always taught curl and only saw rot in older books.
|
| That said [?]x is what I've seen most commonly overall.
| dboreham wrote:
| Scotland we used curl. I never heard of rot. Although Maxwell
| was Scottish, the vector analysis notation was invented
| later.
| xanderlewis wrote:
| In England we also seem to use curl. I've similarly never
| heard of rot.
| defrost wrote:
| Ditto 1980's era Australian physics and engineering
| courses.
|
| Throw in a right hand thumbs up for "direction" of curl
| (fingers indicate rotation, orthogonal thumb direction is
| orientation) and other results about paths having to have
| a zero rotation between places with opposing rotation,
| etc.
| 082349872349872 wrote:
| -- Hey, what's up?
|
| -- Forward cross left!
| xigoi wrote:
| I'm from Czechia and we mostly use rot, sometimes [?]x.
| proee wrote:
| I took an EM 300 level class and our professor made a speech at
| the beginning of the course that he would build on the
| fundamentals of electromagnetics and introduce us to Maxwells
| equations in the end, with the goal being to provide us with a
| foundation to truly understand them. However, our class failed
| rather miserably in that we bombed the tests and clearly did not
| master the fundamentals. Three quarters of the way through the
| course I had the gall to ask if we were going to get to maxwells
| equations. He glared at me with disgust and said "No".
| lupire wrote:
| Such a shame and a waste of money that your professor was so
| terrible. What other job allows employees who show such disdain
| and contempt for the paying customer?
| vkou wrote:
| If the happiness of the paying customer were the top priority
| of a university class, everyone in one should get a gold
| sticker and an A.
|
| Alas, that sort of thing only happens in fairy tales and at
| Harvard.
| sien wrote:
| In the past record store employees.
|
| Also, to this day, some clothing store employees if you don't
| fit what they want as their 'look'.
|
| But it is rare.
|
| Some professors regarded their courses as being for weeding
| out people who would not become academics.
| golergka wrote:
| You could argue that in a sense the customer of a teaching
| institution is not a student, but a future employer who uses
| the diploma or grades as valuable information source.
| avodonosov wrote:
| It may also be helpful to note that Maxwell's equations are like
| burritos, in a sence.
| jiggawatts wrote:
| 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.
| sbrorson wrote:
| 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?
| BoiledCabbage wrote:
| Interesting paper to skim that seems to look into it.
|
| https://vixra.org/pdf/1206.0021v1.pdf
| ptarjan wrote:
| 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.
| nathan_compton wrote:
| Field theorists pretty much already have abandoned the vector
| calculus version of the equations, though.
| photon_lines wrote:
| 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!!
| evanb wrote:
| 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.
| jordibc wrote:
| 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 :)
| cygx wrote:
| _even more in geometric algebra: dF = J_
|
| You can do that using differential forms as well - using
| the co-differential d, we can write a single equation (d +
| 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...
| jordibc wrote:
| 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.
| samantha-wiki wrote:
| 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...
| itishappy wrote:
| I prefer the spacetime algebra version.
|
| https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
| sn41 wrote:
| 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?
| almostgotcaught wrote:
| 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.
| omnicognate wrote:
| Sussman and Wisdom do it themselves (briefly) in chapter 10 of
| Functional Differential Geometry.
| nathan_compton wrote:
| They also have a book on field theory, which is just about
| E&M basically.
| omnicognate wrote:
| 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.
| nathan_compton wrote:
| You're right: I'm thinking of the theoretical minimum
| books by Leonard Susskind and Art Friedman.
| gtsnexp wrote:
| https://fab.cba.mit.edu/classes/862.22/index.html
|
| Ch. 6?
| Davidzheng wrote:
| 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.
| senthil_rajasek wrote:
| https://news.ycombinator.com/item?id=23700295
|
| (Link from 2020 w/ 93 comments)
| senderista wrote:
| 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.
| martyvis wrote:
| 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)
| Jensson wrote:
| 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
| the__alchemist wrote:
| 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.
| lupire wrote:
| 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)
| lupire wrote:
| Have you ever seen a book that doesn't? It's in every book I've
| seen.
| bhaney wrote:
| 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.
| sampo wrote:
| > Have you ever seen a book that doesn't?
|
| The brown Rudin.
| taxicabjesus wrote:
| > 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
| bfuller wrote:
| > 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 :)
| taxicabjesus wrote:
| 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?
| nathan_compton wrote:
| The universe clearly isn't completely random. I'm not sure
| anyone believes that.
| seanhunter wrote:
| 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
| aap_ wrote:
| 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 = -r
|
| [?]B - dE/dt = J
| teleforce wrote:
| 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...
| nathan_compton wrote:
| 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.
| teleforce wrote:
| 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.
| nathan_compton wrote:
| 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.
| WalterBright wrote:
| > 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.
| klysm wrote:
| Fantastic job on the 2D plots stretched out over time in 3D.
| That's really hard to pull off. How were they made?
| photon_lines wrote:
| 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...
| alok-g wrote:
| 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.
| julianeon wrote:
| Great link and great article: I'm slowly working through it. A
| treat.
| psychoslave wrote:
| >basic intuition of having a mathematical function spread out
| throughout space and time
|
| Citing Wiktionary definition of intuition:
|
| > Immediate cognition without the use of conscious rational
| processes. > A perceptive insight gained by the use of this
| faculty.
|
| So, that might be a great exposure of the topic, but this won't
| be an intuitive one.
|
| It's a bit disappointing when a document promise that it's going
| to teach something thanks to some (presumably mostly) universal
| intuition, and then actually require the reader to be comfortable
| with some abstract notions to begin with.
|
| At least that page confesses half-heartedly that it's title is
| actually a clickbait lie.
|
| There is nothing wrong with asking readership some prior
| knowledge. But what can we expect when we are pretending we ask
| individuals to follow their curiosity and just come with their
| intuition and attention? That smells like a receipt for
| disappointment or possibly even leading people to lose confidence
| in what they can get out of good will, curiosity, attention and
| intuition.
|
| All that said, thanks for the link and the publication, that's an
| interesting reading.
| mannykannot wrote:
| '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
| Schrodinger 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.
| sriram_malhar wrote:
| 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.
| apples_oranges wrote:
| I want to mention this video here:
| https://www.youtube.com/watch?v=9Tm2c6NJH4Y I think it's a good
| intro to Maxwell and afterwards one could read this blog post..
| mensetmanusman wrote:
| "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
| petre wrote:
| Thank you. These were nicely explained by my electrical
| engineering professors, albeit with coarser diagrams. Your
| article refreshed my memory and reminded me why I've grown to
| like vector calculus and math put to good use in engineering.
| Lovely diagrams.
| openrisk wrote:
| The relativistic version of the Maxwell equations simplifies them
| to a ridiculous degree but the price to pay is yet another layer
| of mathematical abstraction and fewer opportunities for intuitive
| visualization
| kordlessagain wrote:
| 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.
| arunc wrote:
| 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
___________________________________________________________________
(page generated 2024-06-03 23:02 UTC)