11 ms·
An intuitive guide to Maxwell's equations (2020)
- olooney 2y agoI 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_(book) https://en.wikipedia.org/wiki/Classical_Electrodynamics_(boo...
- lupire 2y agoIt 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 2y agoSubject 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 2y ago> 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.
- esafak 2y agoWhat does "aha" mean to you? To me it means understanding. And isn't that the point of teaching?
- varjag 2y agoI 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 2y agoA 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 2y ago> 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 2y ago> 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 2y agoWhere 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 2y agoThat sounds good, wish more places did like that.
- physicsguy 2y agoI 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 2y agoThanks 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/ https://vexlio.com/
- WalterBright 2y agoThis is a really good article. A minor nit - it'll read easier without the exclamation points after every sentence.
- wayoverthecloud 2y agoYou explain concepts really well. Wish I had professors like you in college.
- cgh 2y agoAlso 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.
- deleted 2y ago[deleted]
- robertbarbe 2y agoThanks @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!
- archgoon 2y agoThe 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 2y agoJackson's the standard here in Spain (undergrad), after working through a course based off of Griffith's.
- abdullahkhalids 2y agoI 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 2y agoRe #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 2y agoFrom 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 2y agoAs 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.
- wglb 2y agoWe 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 2y agoWhat was an acceptable answer to “derive maxwells equations”?
- hxriv 2y ago[flagged]
- wglb 2y agoShow the steps such as faraday’s law and other things that led up to it.
- wrycoder 2y agoMaxwell'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."
- wglb 2y ago> 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.
- lupire 2y agoPeople 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 2y agoYeah 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 2y agoAt 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 2y agoWhat did you dislike? I went through all of HRW as an undergrad (25+ years ago) and recall generally liking the presentation.
- fraserharris 2y agoDavid J Griffiths' Introduction to Electrodynamics was a fantastic undergraduate-level E&M textbook https://en.wikipedia.org/wiki/Introduction_to_Electrodynamics https://en.wikipedia.org/wiki/Introduction_to_Electrodynamic...
- xeonmc 2y agoMore on relativistic equivalence between electric and magnetic field: https://physics.stackexchange.com/questions/489291/how-did-einstein-know-the-speed-of-light-was-constant/489614#489614 https://physics.stackexchange.com/questions/489291/how-did-e...
- humanfromearth9 2y agoThis 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 2y agoA great article. Good to see they give Heavyside the credit he deserves for what everyone now calls "Maxwell's equations".
- sesm 2y agoIs 'curl E' a standard notation nowadays? When I was at uni we used 'rot E'.
- jonlong 2y agoWiki 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 ∇× is what I've seen most commonly overall.
- dboreham 2y agoScotland we used curl. I never heard of rot. Although Maxwell was Scottish, the vector analysis notation was invented later.
- xanderlewis 2y agoIn England we also seem to use curl. I’ve similarly never heard of rot.
- defrost 2y agoDitto 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 2y ago— Hey, what's up? — Forward cross left!
- xigoi 2y agoI’m from Czechia and we mostly use rot, sometimes ∇×.
- proee 2y agoI 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 2y agoSuch 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 2y agoIf 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 2y agoIn 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 2y agoYou 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 2y agoIt may also be helpful to note that Maxwell's equations are like burritos, in a sence.
- jiggawatts 2y agoI 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 2y agoI 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 2y agoInteresting paper to skim that seems to look into it. https://vixra.org/pdf/1206.0021v1.pdf https://vixra.org/pdf/1206.0021v1.pdf
- ptarjan 2y agoThe 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.
- samantha-wiki 2y agoOne 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/handouts/maxwell.pdf http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handou...
- itishappy 2y agoI prefer the spacetime algebra version. https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Spacetime_algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
- sn41 2y agoJust 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 2y agoMaxwell'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 2y agoSussman and Wisdom do it themselves (briefly) in chapter 10 of Functional Differential Geometry.
- nathan_compton 2y agoThey also have a book on field theory, which is just about E&M basically.
- omnicognate 2y agoAre you sure? I don't see such a book at https://mitpress.mit.edu/author/gerald-jay-sussman-2078/ 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 2y agoYou're right: I'm thinking of the theoretical minimum books by Leonard Susskind and Art Friedman.
- Davidzheng 2y agoBy 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 2y agohttps://news.ycombinator.com/item?id=23700295 https://news.ycombinator.com/item?id=23700295 (Link from 2020 w/ 93 comments)
- senderista 2y agoEvery 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 2y agoDo 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 2y agoJust search for videos, stuff like this: https://www.youtube.com/watch?v=eEwZeY51mT0 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 https://www.youtube.com/watch?v=c0MR-vWiUPU
- the__alchemist 2y agoNote: 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 2y agoWikipedia explains the basics https://en.m.wikipedia.org/wiki/Del https://en.m.wikipedia.org/wiki/Del and the related articles on gradient (slope), divergence (flow across a boundary), and curl (circulation)
- taxicabjesus 2y ago> 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 https://en.wikipedia.org/wiki/Gravitoelectromagnetism
- bfuller 2y ago> 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 2y agoSome 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 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 https://news.ycombinator.com/item?id=39071316 Another comment: https://news.ycombinator.com/item?id=25238488 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?
- WalterBright 2y ago> 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 2y agoFantastic job on the 2D plots stretched out over time in 3D. That's really hard to pull off. How were they made?
- photon_lines 2y agoThose 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=PLZHQObOWTQDNPOjrT6KVlfJuKtYTftqH6&index=2 https://www.youtube.com/watch?v=ly4S0oi3Yz8&list=PLZHQObOWTQ...
- alok-g 2y agoI 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.
- MockObject 2y agoYou might enjoy this book. https://www.amazon.com/Principles-Electrodynamics-Dover-Books-Physics/dp/0486654931 https://www.amazon.com/Principles-Electrodynamics-Dover-Book...
- alok-g 2y agoLooks awesome! Ordering right away. Thanks a lot for the recommendation. :-)
- julianeon 2y agoGreat link and great article: I'm slowly working through it. A treat.
- psychoslave 2y ago>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 2y ago'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.pdf 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.
- sriram_malhar 2y agoI'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 2y agoI want to mention this video here: https://www.youtube.com/watch?v=9Tm2c6NJH4Y 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 2y ago“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 https://youtu.be/j2oSyAfPzWg?si=BHRv8lodGhqZBtbl
- waskosky 2y agoI 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 https://www.youtube.com/watch?v=GS1Drlyr37Q
- fraukeadeline 2y ago[dead]
- petre 2y agoThank 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 2y agoThe 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 2y agoThe "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 2y agoNobody 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 https://www.youtube.com/watch?v=0jW74lrpeM0
- deleted 2y ago[deleted]
- peter_d_sherman 2y ago>"The [Maxwell's] equations in their original format amounted to 20 equations, not the 4 elegant ones we have familiarity with today." >"What exactly are fields? Well, fields can be thought of as a function acting throughout space and time. The predominant thinking at the time tried to account for such fields through mechanical structures composed of ‘wheels’ and ‘vortices’ which carried the mechanical stresses that the these fields propagated. Of course, such thinking made it difficult to grasp the beauty and meaning of the equations. 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." Question: Is stress -- the cause or the effect -- of electromagnetic fields? ?