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/
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.
I wanted to command you on your excellent work! I am curious how easy is to use Vexlio, is a steep curve? And any favorite books you want to share, I'm going on holidays soon, and haven't planned much for reading yet.
Thank you!! Vexlio has no learning curve - it's literally so easy to use that I haven't had to read ANYTHING in order to get accustomed to doing what I need to do. I simply open the program and the UI is so intuitive that literally you will simply have no issues figuring out what you need to do to accomplish what you want to accomplish. When it comes to books: what are you interested in? Math / physics books or more general stuff? My favorite book of all time is 'Crime and Punishment' - it literally shows you how Dostoevsky thinks and puts you inside of his mind - not many books can do this.
I will give Vexlio a try! I don't have a book category per se, but I read freakonomics last year, and was very impressed by it, if you have similar recommendation, would appreciate it! I read Crime and Punishment but long time ago in high school, don't remember much of it though.
Sure - like I said, it's hard for me to say what my favorite is without naming a category, but since we're on the topic of Electromagnetism I'll stick to physics and recommend 1) Introduction to Elementary Particles by Griffiths (best into to standard model in particle physics) and 2) QED by Richard Feynman. QED btw is very well done and literally shows you the 'algorithm' behind standard particle physics and does so in a clear and unambiguous manner - really an amazing work by Feynman.
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.
Thanks @photon_lines! In your temperature diagram, you mention that every point will take the average of the neighboring points. However, the equation is not a constraint on the temperature but on the "change of the slope (or gradient) of the temperature". The bigger the slope (in space), the faster (in time) the temperature changes at that point!
'The bigger the slope (in space), the faster (in time) the temperature changes at that point!' - Sorry but I'm not really reading you here. If the points around an 'atom' a symmetrically and equally far away when it comes to the point in question but are opposite in magnitude (i.e. imagine having a point with temperature 12 degrees Celsius which is surrounded by a neighboring points which have temperatures of 8 degrees and 16 degrees (so the delta is +4 and -4) then the temperature here will stay the same. The slope of the temperature field has nothing to do with this - unless maybe you're alluding to the slope of something else? I think I should have maybe explained this equation in terms of 'concavity' instead of using the methodology which I used - you can get a good grasp of this in this link: https://www.youtube.com/watch?v=b-LKPtGMdss
Thanks for taking the time to respond and analyze my comment! - The bigger the slope (in space), the faster (in time) the temperature changes at that point -
I have to confess that I got it wrong, indeed: the right side of the equation is a Laplacian. But, rather than describing an average in temperature, it describes the divergence of the temperature field.
Speaking of 3Blue1Brown - would you be interested in presenting this same material in a video? 3Blue1Brown has regular showcases for this type of material. I will read your article when I get a moment but after a hard day's of screen time I sometimes like to kick back to a video.
I agree with the parent comment that the article was quite good and useful, although I do have a nit to pick with the section on unification of the electric and magnetic fields. I think needs to look at an additional scenario.
That section looks at three scenarios:
1. An electrically neutral straight wire with an electron current and a test charge near the wire moving in parallel to it at the same velocity as the electrons in the electron current, observed from an observer stationary with respect to the positive charges in the wire analyzed without taking into account relativity.
The analysis shows that there is no electrostatic force on the test charge because the wire is electrically neutral, but there is a magnetic force because the test charge is moving in the magnetic field caused by the electron current.
(Nit within a nit: the drawing for this shows the positive and negative charges in the wire separated with the positive charges quite a bit closer to the test charge. That would result in an electric field from the wire that would attract the test charge. Maybe insert a short note saying that the positive and negative charges in the wire are actually mixed together so that their electric fields cancel outside the wire?)
2. Same as #1 except the observer is stationary with respect to the test charge.
The observer now sees no electron current in the wire, but does see a current from the positive charges. But the magnetic field from that positive current should not exert a force on the test charge because magnetic fields only affect moving charges and the test charge is not moving in the observer's frame.
3. The Lorentz contraction is introduced, and #2 is re-analyzed taking that into account. That Lorentz contraction applied to the positive current manifests to the observer as an increased density of positive charges. There wire now appears to the observer to no longer be electrically neutral. It has a net positive charge and the resulting electric fields attracts the electron to the wire.
What's missing is circling back and looking at scenario #1 again but including the Lorentz contraction. In scenario #1 the observer sees the negative charges moving, so should see increased negative charge density due to the Lorentz contraction, and the wire should appear to them to have a net negative charge, which would try to repel the test charge.
#1 with Lorentz included then is a fight between the magnetic attraction and the electrostatic repulsion.
Assuming objective reality and so requiring the test charge to actually feel the same force no matter who is observing we can infer that if the electrostatic force toward the wire in #3 is F then the magnetic force toward the wire in #1 must be 2F, which when opposed by the -F electrostatic force from the Lorentz contraction of the negative charges in the wire gives a net force toward the wire of F.
Thank you for the feedback. I'll review my notes and see if I can clarify this section - my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field. If I made any errors I give you my sincere apologies. Btw if you want to make edits to my work directly - you can find it as it's fully open source: https://github.com/photonlines/Intuitive-Guide-to-Maxwells-E...
> my key point there was simply to show that the magnetic field isn't really necessary - I wanted to show that it's all part of relativistic contractions made by the electric field.
This isn't quite right, there are field configurations where the magnetic field doesn't vanish in any reference frame. This is actually the typical case: consider, for instance, two point charges moving relative to one another.
The right takeaway from SR isn't that the magnetic field is fake and the electric field is real, it's that both magnetic and electric fields are frame-dependent and it's the electromagnetic field tensor that's the real physical object.
You're 100% correct. The explanations I was reading gave me the impression that we can get rid of the magnetic field - but this only works in special cases. Right now I'm reviewing my notes and realizing that yup - they're both needed and will need to modify my explanation there or simply get rid of it entirely and keep things simple. Once again - thank you for the correction - you're a life saver this is definitely something which I overlooked and I wish I had more time to explain properly!
