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...
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/
I agree with the parent comment that the article was quite good and useful, although I do have a nit to pick with the section on unification of the electric and magnetic fields. I think needs to look at an additional scenario.
That section looks at three scenarios:
1. An electrically neutral straight wire with an electron current and a test charge near the wire moving in parallel to it at the same velocity as the electrons in the electron current, observed from an observer stationary with respect to the positive charges in the wire analyzed without taking into account relativity.
The analysis shows that there is no electrostatic force on the test charge because the wire is electrically neutral, but there is a magnetic force because the test charge is moving in the magnetic field caused by the electron current.
(Nit within a nit: the drawing for this shows the positive and negative charges in the wire separated with the positive charges quite a bit closer to the test charge. That would result in an electric field from the wire that would attract the test charge. Maybe insert a short note saying that the positive and negative charges in the wire are actually mixed together so that their electric fields cancel outside the wire?)
2. Same as #1 except the observer is stationary with respect to the test charge.
The observer now sees no electron current in the wire, but does see a current from the positive charges. But the magnetic field from that positive current should not exert a force on the test charge because magnetic fields only affect moving charges and the test charge is not moving in the observer's frame.
3. The Lorentz contraction is introduced, and #2 is re-analyzed taking that into account. That Lorentz contraction applied to the positive current manifests to the observer as an increased density of positive charges. There wire now appears to the observer to no longer be electrically neutral. It has a net positive charge and the resulting electric fields attracts the electron to the wire.
What's missing is circling back and looking at scenario #1 again but including the Lorentz contraction. In scenario #1 the observer sees the negative charges moving, so should see increased negative charge density due to the Lorentz contraction, and the wire should appear to them to have a net negative charge, which would try to repel the test charge.
#1 with Lorentz included then is a fight between the magnetic attraction and the electrostatic repulsion.
Assuming objective reality and so requiring the test charge to actually feel the same force no matter who is observing we can infer that if the electrostatic force toward the wire in #3 is F then the magnetic force toward the wire in #1 must be 2F, which when opposed by the -F electrostatic force from the Lorentz contraction of the negative charges in the wire gives a net force toward the wire of F.
Thank you for the feedback. I'll review my notes and see if I can clarify this section - my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field. If I made any errors I give you my sincere apologies. Btw if you want to make edits to my work directly - you can find it as it's fully open source: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-E...
Thanks @photon_lines! In your temperature diagram, you mention that every point will take the average of the neighboring points. However, the equation is not a constraint on the temperature but on the "change of the slope (or gradient) of the temperature". The bigger the slope (in space), the faster (in time) the temperature changes at that point!
'The bigger the slope (in space), the faster (in time) the temperature changes at that point!' - Sorry but I'm not really reading you here. If the points around an 'atom' a symmetrically and equally far away when it comes to the point in question but are opposite in magnitude (i.e. imagine having a point with temperature 12 degrees Celsius which is surrounded by a neighboring points which have temperatures of 8 degrees and 16 degrees (so the delta is +4 and -4) then the temperature here will stay the same. The slope of the temperature field has nothing to do with this - unless maybe you're alluding to the slope of something else? I think I should have maybe explained this equation in terms of 'concavity' instead of using the methodology which I used - you can get a good grasp of this in this link: https://www.youtube.com/watch?v=b-LKPtGMdss
I wanted to command you on your excellent work! I am curious how easy is to use Vexlio, is a steep curve? And any favorite books you want to share, I'm going on holidays soon, and haven't planned much for reading yet.
Thank you!! Vexlio has no learning curve - it's literally so easy to use that I haven't had to read ANYTHING in order to get accustomed to doing what I need to do. I simply open the program and the UI is so intuitive that literally you will simply have no issues figuring out what you need to do to accomplish what you want to accomplish. When it comes to books: what are you interested in? Math / physics books or more general stuff? My favorite book of all time is 'Crime and Punishment' - it literally shows you how Dostoevsky thinks and puts you inside of his mind - not many books can do this.
You explain concepts really well. Wish I had professors like you in college.
This is really excellent. I particularly like the outline of div and curl, the dot product and the cross product, and the connections drawn between the differential an integral forms. Thanks.
Fantastic work.
Deserves to be widely used to teach Maxwell's equations.
THANK YOU.
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.
This is a really good article. A minor nit - it'll read easier without the exclamation points after every sentence.
It really is a shame that in the 20th Century, the "best" math and science books were judged not for their educational power, but for how difficult and impressive they were to fellow professionals. It seems as though the professors were afraid that they'd lose their lecturer jobs if the books were too educational on their own.
Subject matter experts are not experts in pedagogy. Because pedagogy is a seperate subject entirely. And teaching is not about getting people to say aha. Thats just performance or entertainment. Seen everywhere these days thanks to the Attention Econnomy. You can gets ahas out of people playing great music. But dont equate that with getting people to play great music. Cuz that requires getting people to do lot of mundane mindless work for long long periods of time.
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.
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.
I think this is arse over elbow; the purpose of an undergraduate degree course is to teach you to study and do research. The "research" done by undergraduates isn't novel research; the student repeats "research" that has been done by generations of students before them. I.e., it's practice.
For this reason, writing undergraduate essays felt to me like being an impostor; you try to write in the manner of a researcher, knowing that you're faking it.
Where I live you study pedagogy and practice it while doing your PhD. If you suck at it you can still pass, but at least they take a shot at teaching it to you. When applying for positions your record on teaching might make it harder to get to those where you're expected to do it regularly.
The usual nepotism, corruption and fraud in academia will of course allow some bad teachers to advance anyway.
Thank you. Reading the article will not in fact give you an easier time at the Jackson Problem sets.
I think many people who think this would have helped them back in the day have simply forgotten what the actual hard part of the degree was.
Are you sure the hard part is the most important part?
Yes. There's a difference between thinking you understand something and having to prove it via problems.
Often that's how I discover I didn't really understand something at all.
What does "aha" mean to you? To me it means understanding. And isn't that the point of teaching?
I can explain memory pointers to a layperson in terms of numbered boxes and yellow notes. They're still long long way to go from that even to reversing a single linked list successfully.
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.
I am confused.
1. While the posted guide is excellently written, it's not particularly novel. I was taught EM in a very similar fashion. Diagrams similar to those in the guide were drawn on the board by my professors.
2. Jackson is a graduate EM text. It is mathematically difficult, because when you read it, you should have been familiar with EM and all this conceptual underpinning for at least 3-4 years. The goal of Jackson is to solve the equations for scenarios that undergrads would find challenging. What did you study in your undergrad?
Re #2: Jackson was the standard text for undergrads like me doing a Mathematics degree. It was a late second year or early third (final) year text if I recall rightly. This was 1992, so I'm still amazed to read that its still a commonly used text.
Fwiw, other standard texts used in Durham (UK) back then were Spivak on Calculus, Goldstein on mechanics, and for the mathematical physics kids, landau and lifschitz on mechanics and electromagnetism, and (an absolute doorstop) Misner, Wheeler and Thorne on Gravitation (relativity).
From the first preface (1962) of Jackson
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.
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).
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.
Not necessarily: undergraduate and pre-undergraduate education differs a lot between the UK and the US.
'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.
We used the John Kraus book on Electromagnetics for the dynamic fields course. This was preceded by a course on static fields. That course’s final had the shortest test statement I had ever encountered: “Derive Maxwell’s Equations”. I found the Kraus book satisfactory.
What was an acceptable answer to “derive maxwells equations”?
Show the steps such as faraday’s law and other things that led up to it.
“Be Maxwell” - op, probably.
The basic ideas, the pictures, the diagrams, etc, found here, typically show up in enough books if you look for them that I don't feel that this was the main limiting factor in my physics education. The difficulty of Jackson (which doesn't show up until grad school for most students) is in the problem sets, not the ideas behind the equations (which most students have a had at least two courses in already).
I don't believe that having a more 'intuitive' idea of the equations really helps all that much, as the intuition needed for solving the problems isn't really physical, but mathematical. Which integrals are solvable, which order of integration will make this tractable, do I need to use properties of Bessel functions here, etc.
We can argue whether getting good at this sort of thing is actually useful for physicists, but I wouldn't know. Very few of us ended up becoming researchers in the field.
Jackson's the standard here in Spain (undergrad), after working through a course based off of Griffith's.
At least it wasn't Halliday and Resnick. It's been 35 years since my BSME at Purdue, and I can still remember their names. God I hated those textbooks. If someone tells me that this Jackson book was worse, I won't believe it.
What did you dislike? I went through all of HRW as an undergrad (25+ years ago) and recall generally liking the presentation.
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...
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.