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Why is Maxwell's theory so hard to understand? (2007) [pdf]

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Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#191

Post by s1artibartfast »

steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I wish most explanations wouldn't skip over the fact that field lines arent real, and just a tool to graphically depict what is going on. Statements like the following gets the causality entirely backwards.

>the strength of an electric field depends on the number of electric field lines.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#192

Post by toth »

adrian_b wrote: post #151
toth wrote: post #105

Earlier quoted context omitted.

> The originally published equations were "20 or so" because one equation was written for each scalar component. > Rewriting the equations in vector form reduces the number to the modern number. And if you use the differential form or 4d tensor notation they get reduced to 1 equation. Of course, for a lot of practical problems this is not very useful and it's better to work with the 3d vector form. > The variant with…

Even the vacuum version is incomplete without adding an equation for force or energy, because no meaning can be assigned to the electromagnetic field or potential otherwise than by its relationship with the force or energy. Even today, there exists no consensus about which is the correct expression for the electromagnetic force. Most people are happy to use approximate expressions that are known to be valid only in r…

I agree that to fully specify electromagnetism you also need to include how the fields affect charged matter. So EM = Maxwell's equations + Lorentz force equation (not sure why you say there is no consensus about what this is, that is new to me).

This is just a matter of taste, but OTOH I would not include descriptions of how some materials respond to the fields in the continuous limit as part of a definition of EM.

It is true that for most terrestrial applications you do need those to do anything useful with EM. But if you want to study plasmas you need to add Navier-Stokes to EM, doesn't mean hydrodynamics is part of EM. To study charged black holes you need EM + GR, but it still makes sense to treat them as mostly separate theories.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#193

Post by timeagain »

s1artibartfast wrote: post #191
steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I wish most explanations wouldn't skip over the fact that field lines arent real, and just a tool to graphically depict what is going on. Statements like the following gets the causality entirely backwards. >the strength of an electric field depends on the number of electric field lines.

It’s like saying that rain falls where there are blue regions on the weather map :)

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#194

Post by rolph »

s1artibartfast wrote: post #191
steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I wish most explanations wouldn't skip over the fact that field lines arent real, and just a tool to graphically depict what is going on. Statements like the following gets the causality entirely backwards. >the strength of an electric field depends on the number of electric field lines.

we can fix that.

the number of electric field lines, depends on the strength of an electric field. ,

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#195

Post by zinclozenge »

mitthrowaway2 wrote: post #14
bsder wrote: post #3

Maxwell's theory is not hard to understand--once you have the proper tools. The problem is that because of trying to cram a degree into 4 years, you wind up having a class on electromagnetics without any understanding of vector fields . Electrical engineering is particularly bad about this. You never get exposed to the Hamiltonian formulations of classical mechanics, and you never get exposed to vector analysis. Cons…

I think the essay is not about Maxwell's theory being hard for college students to understand, but rather, for other 19th century physicists to understand. And that's mainly because Maxwell himself didn't do a very good job of communicating his theory at the time, so it took other talented physicists to rework, explain, and popularize his ideas.

Which is interesting considering Freeman Dyson was the guy that made the connection between Schwinger's and Feynman's QED.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#196

Post by seanhunter »

Just today I was watching a cool video by Angela Collier[1] about how Faraday's experimental work really laid the groundwork for Maxwell by proving that light polarity could be affected by an electromagnetic field.

[1] https://www.youtube.com/watch?v=Fbi-_8zOuR8

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#197

Post by Tainnor »

mrguyorama wrote: post #182
lupire wrote: post #137

Earlier quoted context omitted.

Continuity just hides the ball. You say you can't comprehend how something can move from 1 to 2 discretely. But the paradoxical notion of infinite continuous change has been known since antiquity. It's faith either way. Discrete doesn't mean state changes are wholly globally arbitrary. Imagine a graph with nodes and edges, a state machine as computer sciences call it. I think it's easy to agree that the universe coul…

Zeno's """Paradox""" was nonsense even in it's own time. Easier now that we understand Newton's laws of motion but his contemporaries were able to sufficiently dispute his idea even without them.

Not nonsense. The argument goes that if time and space are both discrete, then to move from A to B in finite time means that you have to perform infinitely many actions in finite time.

Zeno didn't believe that the latter was possible. But he wasn't stupid, he obviously knew that motion was happening all the time in real life. His paradox really only makes sense in the context of Eleatic philosophy which assumes that reality is an illusion because change is fundamentally impossible (how can something come from nothing?).

If you want to reframe it in more modern terms, Zeno's paradox shows a contradiction in axioms. If you want to get rid of the contradiction, you have to change some of the axioms.

In real analysis, loosely speaking, we remove the axiom that an infinite process cannot result in a finite outcome - this way we are allowed to sum (some) infinite series, for example. But we don't "know" if reality behaves that way.

The atomists found a different solution: they argued that reality was fundamentally discrete. This way, Zeno's paradox also doesn't arise.

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#198

Post by seanhunter »

hiAndrewQuinn wrote: post #43

My proudest moment in high school was getting a 5/5 on the calculus based AP Physics C exams at 15 with no calculus and only rudimentary algebra knowledge at the time. That experience permanently colored my thinking, and made me much more open to practicing thorough visual imagination as a way to solve problems. I found that practice useful all the way through my EE degree's vector fields courses a decade later. I th…

> If you have examples of things like this in other areas like mathematical finance, I'd love to hear about them

There are tons of examples in mathematical finance but the obvious one is the Black/Scholes [1] paper where one of the key assumptions they make (which they know to be untrue but helpful) is that you can replicate a portfolio in continuous time. This allows them to use a constructed portfolio of a risk-free intrest-bearing instrument and the underlying to replicate the price of an option, and the process is a Brownian motion. Everyone knows that actual trading (and thus price processes) are discrete in real markets, but continuous time is much easier to model. Much later on people like Heston and Matytsyn(?sp) came up with stochastic vol models with jumps to replicate discrete price discontinuities, but they're a lot harder to work with in many ways.

[1] https://www.cs.princeton.edu/courses/archive/fall09/cos323/p...

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#199

Post by duped »

ThePhysicist wrote: post #93

Maxwell didn't have the nice differential geometric notations that we use today, which allow us to write his equations in a very concise and easy to understand form. His original paper is way more convoluted, so at the time it must have been really difficult to understand for everyone except the subject matter experts. And he was of course building on the work of Faraday, Ampere and others. But like with other theori…

Faraday didn't even know trigonometry, allegedly (he never studied mathematics). It's interesting that his student (Maxwell) who did have the mathematical background would extend his theories and figure out the math to explain it all

Re: Why is Maxwell's theory so hard to understand? (2007) [pdf]

#200

Post by psuedobrain »

s1artibartfast wrote: post #191
steamer25 wrote: post #168

Prior to computer-generated 3D animation, I can imagine it was very difficult to float and spin vector-arrows in mid-air with enough accuracy to show what goes on without having to resort to reams of explanatory paragraphs. Eugene Khutoryansky is something of a lesser-known 3b1b that's more focused on physics than math. I found his animations very helpful for building intuition around Maxwell's equations: https://www…

I wish most explanations wouldn't skip over the fact that field lines arent real, and just a tool to graphically depict what is going on. Statements like the following gets the causality entirely backwards. >the strength of an electric field depends on the number of electric field lines.

I think the question of whether field lines are real is more of a philosophical (of physics) question so it usually falls outside the scope of introductory material on E&M. However, some texts like Purcell and Morin do kinda take a stance on whether fields are real: "since it works, it doesn’t make any difference."
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