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.<p>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.<p>[1]: <a href="https://en.wikipedia.org/wiki/Classical_Electrodynamics_(book)" rel="nofollow">https://en.wikipedia.org/wiki/Classical_Electrodynamics_(boo...</a>
One of my favorite parts of my education was going through E&M to arrive at the beauty of Maxwell's Equations.<p>I later found out that you can squeeze <i>even more</i> beauty out of them by boiling them down even further using differential geometry.<p><a href="http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handouts/maxwell.pdf" rel="nofollow">http://virtualmath1.stanford.edu/~conrad/diffgeomPage/handou...</a>
“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.”<p>There is a lot of interesting discussion on whether fields are real, and the dialogue goes back centuries: <a href="https://youtu.be/j2oSyAfPzWg?si=BHRv8lodGhqZBtbl" rel="nofollow">https://youtu.be/j2oSyAfPzWg?si=BHRv8lodGhqZBtbl</a>
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.
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.<p>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.
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.
> 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 [...]<p>Maxwell's original equations connected light and electricity. Maxwell's original 20 equations had 20 unknowns, using 'quaternion-based notation', which no one understood.<p>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.<p>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.<p>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: <a href="https://en.wikipedia.org/wiki/Gravitoelectromagnetism" rel="nofollow">https://en.wikipedia.org/wiki/Gravitoelectromagnetism</a>
More on relativistic equivalence between electric and magnetic field:<p><a href="https://physics.stackexchange.com/questions/489291/how-did-einstein-know-the-speed-of-light-was-constant/489614#489614" rel="nofollow">https://physics.stackexchange.com/questions/489291/how-did-e...</a>
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?
Nobody has explained Maxwell's equations better than Parth G. The video is a bit old, but very intuitive!<p><a href="https://www.youtube.com/watch?v=0jW74lrpeM0" rel="nofollow">https://www.youtube.com/watch?v=0jW74lrpeM0</a>
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.<p>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.<p>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:<p><a href="https://www.clerkmaxwellfoundation.org/DysonFreemanArticle.pdf" rel="nofollow">https://www.clerkmaxwellfoundation.org/DysonFreemanArticle.p...</a><p>In the penultimate paragraph, he writes <i>"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.</i>" 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 <i>inverse</i> 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!<p>True. But we do suspect the existence of massless points, and surely have many pointless masses.
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.
I want to mention this video here: <a href="https://www.youtube.com/watch?v=9Tm2c6NJH4Y" rel="nofollow">https://www.youtube.com/watch?v=9Tm2c6NJH4Y</a> I think it's a good intro to Maxwell and afterwards one could read this blog post..
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.
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”.
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
>"The [Maxwell's] equations in their original format amounted to <i>20 equations</i>, not the 4 elegant ones we have familiarity with today."<p>>"What exactly are fields?<p>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<p><i>stresses</i><p>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<p><i>stresses</i><p>and mechanical forces as being a consequence of such fields, and not vice-versa."<p>Question:<p>Is <i>stress</i> -- the <i>cause</i> or the <i>effect</i> -- of <i>electromagnetic fields</i>?<p>?
>basic intuition of having a mathematical function spread out throughout space and time<p>Citing Wiktionary definition of intuition:<p>> Immediate cognition without the use of conscious rational processes.
> A perceptive insight gained by the use of this faculty.<p>So, that might be a great exposure of the topic, but this won’t be an intuitive one.<p>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.<p>At least that page confesses half-heartedly that it’s title is actually a clickbait lie.<p>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.<p>All that said, thanks for the link and the publication, that’s an interesting reading.