Intuitive Guide to Maxwell’s Equations | Hacker News
Intuitive Guide to Maxwell’s Equations | Hacker News
These kinds of visual, well thought out explanations of topics which are often taught with terse, obscure and uninviting methods are a gift.
On a tangent: I remember asking my Calculus 101 professor what the “intuitive meaning” of divergence and curl was, outside of the formal math and equations. He was shocked that one could ask to sully these perfect mathematical concepts with dirty intuitive reductions. A guide like this, or 3-Blue-1-Brown videos would have made my day, then.
Wikipedia is probably the most obvious modern exemplar of this. The editors of most mathematics articles clearly are much fonder of playing with the equations editor than they are of actually explaining things.
A good description I read of Wikipedia maths articles was that they’re like the blackboard of a graduate-level class after everyone has left. All the information is there, but if you’re not already familiar with the subject there’s no way you’ll grasp anything.
lol agreed. Every Wikipedia article about math is:
Me: I learned nothing from that.
(Yes, which is partly my fault, but it’s really not helpful for an intro to any math topic unless you already have mastered 99% of the topic and just want to resolve the last few things. The encyclopedia model really goes against what you need for the general population here. Arbital tried to do a more helpful model here but is defunct; Khan Academy is a generally better for this but requires more investment.)
The problem is that math is inherently more abstract than any other topic. It’s so abstract that the vast majority of math terms have no simple, intuitive translation into everyday English. Everything is built upon a tower of abstractions that have developed over centuries.
In this community, we’ve witnessed the difficultly first-hand with the much-maligned monad tutorials. A mathematician would have no trouble understanding monads —- they’re just a matter of reading the definition and the laws. That’s what you do every day when you’re studying math in university. Read some definitions and some theorems, play with things a bit, try to prove stuff, then move on to the next topic.
I wish there was an “intuition” wiki where you can learn the intuition first and then a link at the bottom takes you back to the Wikipedia page where you can stare at the math until you really get it.
Probably true!
That said, Wikipedia’s mathy pages have proved repeatedly useful to me as a reference, e.g., whenever I remember a mathematical concept only vaguely or intuitively and just need to find a detailed formalization to implement it in code. My browser is always open, so Wikipedia is often the “reference of least resistance.”
I’m at least ambivalent about a general purpose encyclopedia having whole categories of articles that are mostly only accessible to specialists. In an ideal world, there would probably be a companion “Mathepedia” or something along those lines with a different target audience. But, of course, Wikipedia doesn’t really have the concept of a specific reader persona like a conventionally edited reference does.
I’m ambivalent too, i.e., I agree with your original comment but also, personally, frequently find Wikipedia’s “general purpose encyclopedia for specialists” useful.
FWIW, there’s Simple Wikipedia (https://simple.wikipedia.org), but it is, quite frankly, terrible for learning about anything of a mathematical nature beyond high-school algebra.
Perhaps Wikipedia should add a “Learn About This” section to math, physics, engineering, and hard-science pages?
Well, it’s not Wikipedia’s fault if people producing intuitive content are not contributing to it. It’s not like editors go out of their way to remove nice explanations.
There are many anecdotal reports of tyrannical Wikipedia editors doing exactly this. They claim dominion over a subset of Wikipedia articles and then manicure them exactly to their own personal tastes. I’m sure the situation you describe happens often in various corners of the website.
I nearly based my Master’s dissertation on a topic I discovered on a wikipedia math page when I was looking for something to cover. Turns out that subsection was maintained by the ‘inventor’ of topic and essentially served as a vanity page. The topic itself had no recognition in the community and if I had forged ahead on it, I would have failed my dissertation pretty hard.
Oh, I got rid of it years ago. Thanks though. Eventually a big-name mathematician (who some years later won a Fields medal) waded into the discussion and sided with me. I used my real name in that discussion so I won’t give more detail than that 🙂
Well, I’ve personally contributed to small parts, and never encountered the problem. I don’t doubt there are jerks on Wikipedia, but I would also bet that neither the author of this nice github repo or the praised youtuber mentioned in the comments even tried to commit anything into it.
Not that they should for some reason, but too often editing wikipedia is presented/thought of as something reserved to a happy few with deep social consequences, when it’s in fact the simplest thing in the world.
The problem is more due to a general culture that prefer rigour over clarity. Mathematics by definition is made of layers upon layers of intuitive conclusions. Then why do universities prescribe books with long dry derivations over clear explanations? Rigour is important to ensure correctness – and that should never be compromised. But that isn’t the reason why humans learn mathematics. People learn it because it extends their imagination. Clarity and intuition are important for that and those who can’t see it becomes disillusioned.
Rigour is better left to computers these days. They are better at following rules. We can have better proofs with software like metamath. Surprisingly, you can even gain deep insights by writing automated proofs (compared to manual proofs). Ideally, practitioners should consider simple and clear explanations as their primary goal and equations and proofs as necessary supplements. Think of this as literate programming for mathematics.
I think it is wikipedia’s fault, but that it’s not far from how it should be. Wikipedia is a secondary source, designed have be a reference taken from primary sources, rather than be a primary source teaching information. It’s a weird blurry distinction, but seems to work well. It’d be great to have a pedagogy section that either explicitly linked out to pedagogical primary sources, or a section explicitly designed to teach.
This, yes. To this day of the several dozen that I visited, I’m yet to find a math related article that serves the purpose of explaining the concepts laid in it.
The thing is that writing an intuitive explanation is hard, while writing down the actual definition is easy. Also, one person’s intuition is frequently another person’s gibberish (hence the “monads are like a burrito” phenomenon).
True generally in Physics as well. What’s the point of “understanding” Quantum Mechanics intuitively when you can do computations blindly without it!
This statement is misguided. A lot of physical phenomena doesn’t lend itself to intuitive understanding. Quantum Mechanics is a great example of this. You cannot expect subatomic particles to behave the same way that macroscopic objects do (and macroscopic behavior is what humans find intuitive because we experience it on a day to day basis).
Its the same misguided thinking that leads to the popular PopSci explanation of electron behavior being both wave and particle like. Why should we try to categorize the behavior a subatomic particle (an entity so distant from our experiential reality) as being analogous to one of two macroscopic entities?
I totally get the desire for intuitive understanding, and it should be encouraged, but sometimes you just have to put intuition aside and come to conclusions with pure mathematical reasoning.
Isn’t this just a misunderstanding of what we mean by intuitive? It’s possible to have an intuitive understanding of a thing that’s not “intuitive” to humans.
Exactly. Just imagine the quantum world as a bunch of liquids floating around carrying their associated particles, and you have an intuitive grasp of (the debroglie bohm interpretation of) quantum physics.
I’d even be more ok with that, if mathematicians used more rigor in their notation. Math developed in a time where it was tedious to write this out by hand and a lot was hidden in abbreviations and assumptions. One thing that programming gets very right is that it is way more explicit (although obviously not perfectly) the behavior behind any notation.
It’s one thing that Structure and Interpretation of Classical Mechanics (which I haven’t read) got really right in concept. It’s a shame, that approach hasn’t been adopted more widely.
Take for example page 7 of the pdf with the heat equation. Even the description of the Laplacian is wrong. The value isn’t the average of the points surrounding it. But if this simple function were written in python as a “update” function over a multi-dim array of temperature values it’d be clear exactly what’s happening.
Another example of simple inconsistency in notation: superscript, does that mean squared or not? When writing two terms adjacent is it multiplication or an operator being applied? Where is the behavior of the operator defined that a student and “read the code”?
English is an extremely verbose and imprecise language. Mathematicians replaced a verbose imprecise language with a terse imprecise language. It’s about time for rigorous fields to take the last step and introduce precise concepts using a precisely defined language.
> Even the description of the Laplacian is wrong. The value isn’t the average of the points surrounding it.
The description on page 7 is split into two parts. The first part describes the value of the Laplacian correctly:
> It tells us how the temperature value at the point compares to the average value of its neighboring points.
The second part describes the long-run behavior of the differential equation
> The temperature value that this point takes is the average temperature of the points surrounding it
> The second part describes the long-run behavior of the differential equation
> The temperature value that this point takes is the average temperature of the points surrounding it
And this is not correct. Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren’t describing and evolution if its inaccurate for all points except a final state.
On top of that, even during the evolution the temperature isn’t what’s taking on the “average” of surrounding points, it the change in temperature wrt time that’s being change by the average of points. And yet again, it’s not an average of the surrounding temperatures is and average of the surrounding differences in temperature.
And this interaction we’ve had highlights exactly my point. English is an impressive language, and Mathematics is full of hand wavy explanations that come with simplicity context that isn’t explicity and precisely defined. Mathematicians are used to it, and of course it’s learnable like anything. My point is it’s not necessary, it evolved in a different time with different constraints. Its a similar example of “the medium is the message” – math evolved when the only way to write was by hand, and duplication of definitions was manually expensive. We don’t need these handwavy shortcuts now – we can make it easier for students to learn by being precise and providing definitions for them to see exactly what’s happening – not expect them to learn from trying to recall every imagined context. And the fall back of “I had to learn it this way, so they should too”, is a horrible excuse.
> Excluding a final state when all temperature values are identical, at no point is the temperature value equal to the average of the surrounding values. You aren’t describing and evolution if its inaccurate for all points except a final state.
The average of the surrounding points is the attractor temperature for the system. It is an asymptote which the temperature of the point is moving towards. It’s like saying an oscillator (such as a spring) wants to be at neutral, even though it never comes to rest at neutral.
I’m not engaging with your larger point, I’m just quibbling with you saying page 7 is wrong. I think that “takes” in the english description says that the temperature approaches the value over time, whereas you interpreted “takes” as referring to the temperature at every point in time.
> I’m not engaging with your larger point, I’m just quibbling with you saying page 7 is wrong. I think that “takes” in the english description says that the temperature approaches the value over time, whereas you interpreted “takes” as referring to the temperature at every point in time.
I hear you, and I think we’ve probably approaching the end of the productive part of our conversation.
I do want to mention that the quote above, and interpretation of “takes” is exactly my larger point though. These definitions are all sloppy and prone to interpretation. Precise definitions would eliminate the need for all of this.
And since I can’t help myself, thinking about this a little more, even your interpretation above is either faulty, or has to change the definition of “surrounding”. If I have a point of average temperature, a doughnut or sphere of warmer points a small distance immediately around it and then the majority of all other points around that being colder, then the asymptote is actually towards the average of the of outer colder points, not the surrounding warmer points.
What’s the point of “understanding” QM intuitively if it brings confusion rather than understanding?
QM can be understood intuitively, the problem is we are still teaching the bullshit ‘wave particle duality’ when we basically know that fields are what is real.
There’s also no such thing as ‘observation’. Fields interact by a precise and well-tested mathematical function. That function does have the effect of mutating the state, but why should there be such a thing as pure observation? There’s nothing magic about conscious observation.
You literally can’t understand something if you are confused by it. Your question makes no sense.
Perhaps the comment was intended to convey the very different potential for intuitive “understanding” between visualisable classical theories and non-visualisable quantum theory.
At a fundamental conceptual level it does not map to anything in our normal experience – attempting to force analogies etc can end up confusing things further.
I agree with your original observation btw, but it’s important to recognise the limits we may be bound by in explanation and comparison.
There is a difference between understanding and “understanding” (intuitively, no less!).
The book Geometrical Vectors does exactly that. In fact, it considers the gradient vector to be a different kind of vector than a normal distance vector. If I have a vector between two points, and I compress space so that the points move closer together, then the distance vector gets smaller. However, the gradient vector gets bigger. The gradient is basically a density, it’s units are units-of-whatever-your-taking-the-gradient-of / meters.
And, or course, the vector you get from a cross product only has 3 degrees of freedom because we live in a 3D world. The cross product of two 2D vectors is a scalar, the cross product of two 4D vectors needs 6 numbers to describe it. Even in 3D, if you reflect the original vectors in a mirror, the cross product now points the opposite direction, i.e. it depends on the handednes of the coordinate system.
That sounds like the beginnings of intuition for exterior algebra. While the gradient is a vector (i.e. it lives in a vector space), it’s better thought of as being different to a standard ‘displacement’ vector.
Look at a topographical map of a landscape and note the contour lines. As you zoom into the contour lines (if they’re detailed enough), they’ll start to look more and more like parallel lines, densely spaced for a steep slope, or sparsely spaced for a mild slope. These parallel lines are the gradient.
A gradient and a vector ‘fit together’ to give a real number. The more parallel lines the vector pierces, the bigger the number [0]. So a gradient ‘eats’ a vector, spitting out a real number, and vice-versa. Just like a row vector ‘eats’ a column vector and spits out a real number.
I’m saying ‘gradient’ here, but really what I mean is ‘one-form’. Language deliberately imprecise for all y’all mathematicians out there.
I’m not personally knowledgeable enough to suggest how this would behave in geometric algebra, I’m just smart enough to think that this would behave much cleaner in that framework.
Would really like to have that free time to delve and be able to suggest the alternative formulation myself.
I love explanations of concepts that appeal to people in new and different ways, and I thank anyone clever, patient, and gifted enough to produce them. The world is a better place when more people understand are not afraid of more things.
But I think it’s important to remember that mathematics is not some evil thing! There are people who train carefully so that the terse mathematical explanations are the intuition. There’s nothing wrong with not being one of those people, but knocking the mathematical explanation is denying them their intuition just as their explanation denies you yours.
In a word, it takes all kinds, and I think we should all be as patient with others’ ways of learning and understanding as we hope that they will be with ours.
Most mathematicians consider the equations for div and curl to be the “dirty equations” while the “perfect mathematical concept” is the exterior derivative of a 1-form.
If the objective is to generalize, then differentiation is simply the adjoint of the boundary operation, which is a homomorphism; extending the notions of closed and exact forms to general differential forms along with that the differential operator is nilpotent allows setting up de Rham cohomology, which also has analogs by multiple authors.
I guess there’s also an aesthetic component to limit the extent of generalization
This was one nice thing about studying physics, most of the math we learned was motivated by some application. I think my first exposure to div and curl was using the analogy of fluids flowing in the plane.
Intuitively, divergence is how much fluid is being created (or destroyed) at a point, curl is how much a waterwheel would spin if placed at a point.
just a quibble, if your Calculus 101 covers div and curl, you are going to an intense STEM school and really could be expected to understand the math.
Understanding the math and understanding the importance of the mathematical concept are two different things.
Personally, div and curl didn’t quite click for me until I took fluid dynamics in my final year. I could do the homework but didn’t really get why they were useful until it made sense why “div=0 always for incompressible fluids”.
ok, but my point was that div and curl were taught in 102, not 101 (which you easily could have placed out of). therefore, the reasoning goes, if they were in 101, that must be a hyper 101.
source: went to intense STEM school
A chapter in Shen and Kong’s Applied Electromagnetism helped me a lot back in college. It is graphically explaining the integrals and vector differential equations, something I didn’t see in other textbooks. My intuition was then subsequently reinforced by coding a lot of FDTD (Finite Difference Time Domain) for research. The FDTD algorithm is so simple, that I wish physics teachers covered it in class.
As a side note there was a historical debate, and the shape of these equations is the end result of the chosen system.
In geometric algebra which did not prevail, these are one equation.
In modern relativisitic formulations using tensors they are one equation also, derivable from the action of the electromagnetic field tensor. This formulation also shows that electric and magnetic fields are the same, simply rotated under a relativisitic transform, which is the modern view.
Start here for the ideas
https://en.wikipedia.org/wiki/Electromagnetic_tensor
Geometric algebra is a bit awkward notationally. Physicist prefer to use an alternative notation based on exterior calculus that does provide a compact representation of Maxwell equations: dF=J
dF=0
Geometric algebra produces two equations too, by the way, not one.
Discovering that the Maxwell equations can be written in this succinct form is kind of mind-blowing, but I’m wondering whether it provides any additional insight? It seems like one needs to do a significant “unpacking” to actually use the equation or gain insight from it. I would love to hear an explanation of electrodynamics starting with “star ddA=J”.
Yes, I kinda agree. The underlying physics is, by definition, the same, and what you gain by reducing the number of equations you lose by having more complicated “objects” and needing more advanced maths to handle them (e.g. the electromagnetic field tensor, external calculus and whatnot vs. just vector fields and basic vector calculus).
To get an intuition of the physics, I think the traditional 4 equation form is actually more useful, as you can construct toy examples and study the equations one at a time in isolation.
Where the more advanced formulations are useful, and actually are used, is for stuff like relativistic physics where 4-vectors, curved spacetime etc. are needed and not just a gimmick.
But for more down-to-earth applications of electrodynamics like antennas, field propagation in various forms of matter etc., the classical version is fine.
You get new insights, that F is a curvature 2-form. Written in this way it’s explicit that EM is also a geometric theory. This observation opens the door to Yang-Mills theories which are behind all the Standard Model.
Ha that’s funny. I’d been copying my notation from the parent post, and I assumed ‘d‘ was just their notation. I didn’t realise they meant to say
*d
(apparently HN doesn’t let you escape asterisks).
Yes, but that’s the whole point: math is incredibly information dense. I’m trying to avoid that. Math shouldn’t be hard – some concepts are incredibly easy to grasp, but the way that they’re taught makes me shudder. It’s like seeing code with one letter variable names -written by someone who was scared of losing their job. If you have a way of coming up with an intuitive way of conveying the concepts using geometric algebra, let me know, and I’ll be happy to include it in the guide though!
James Clark Maxwell is one of the Demi-gods of Colour science. He produced the first Colour photograph, and was the first to quantify Colour. Every time you define an RGB value, he is sitting on your shoulder.
>”The Greeks were the originators of this conception. They imagined that ‘things’ were built out of smaller things, like atoms and molecules. When the atomic theory came about, they expected the atoms themselves to have some sort of mass, shape and size, and to be a microcosm of more things. Let’s take sand as an example. To the careless eye, sand seems like a fluid, since quantities of it appear to freely merge and split, but on closer inspection, it’s just a bunch of tiny objects which can be described as individual `things’ interacting with each other.
The world of quantum mechanics and quantum field theory introduce a different conception of what things are though. It turns out that elementary particles can’t be thought of as individual ‘things’ which have a volume.
In fact, if atoms did have a volume, physics wouldn’t work.
We would end up with “surfaces” of electrons behaving in an impossible manner and spinning faster than the speed of light.
Well then, you say, what if atoms aren’t objects with a volume, but points in space? It turns out that this notion isn’t easily prone to interpretation either! No one actually knows or understands what a ‘point mass’ is! It also has a bizarre implication: if indeed we did have volume-less point masses, we obtain something with an infinite density!
In theory, the entire universe could be squeezed to a single point!“
I didn’t have 1st year physics, so this was very informative for me.
Maxwell 1 + 2: Field per surface area. The asymmetry between magnetic fields and electrical fields is that magnetic ones have a total field of zero; that is, al magnetic lines loop back on themselves.
Maxwell 3: The principle behind power generating turbines.
Maxwell 4: The principle behind electrical motors.
To be honest, I didn’t even know you can condense it into four equations. I was only exposed to the history of electromagnetism, not the latest expression thereof.
This is awesome. What a concept, identifying the variables in the equations and providing diagrams to show the geometric meaning of the equations. Really wish wikipedia (or some sort of companion website) would similarly present an explanation of the practical meaning of variables and operators in the heavy math pages so that people with an understanding of algebra could understand the formulaic descriptions in addition to the qualitative descriptions after landing on a random page with heavy math.
For a truly intuitive understanding I think you will have to understand relativistic differential geometry. Once you have that you just have 1 (4-dimensional) vector potential, the Laplacian of which is equal to the current.
This single remaining equation corresponds to the Maxwell equations for the electric field, the equations for the magnetic field just correspond to the fact that the ‘curl’ of this vector potential has 0 divergence (which is just a basic fact of geometry, and is also why magnetic monopoles are unlikely).
You use the outer derivative which generalizes the curl as well as a few similar constructs. Technically it’s slightly different as it returns a bivector (which is a bit like a plane spanned by 2 vectors), but in 3D both vectors and bivectors form a 3D space and you can freely convert between the two. The difference between the divergence and the curl is basically whether you switch between vectors and bivectors before or after you take the outer derivative.
Interesting, I always thought of fields as the interface to measure underlying particles interactions instead of the “base thing” itself. My reasoning was that since it was impossible to model effects from individual particles behavior, we were modeling the aggregate effect using fields, but underneath it was just “small objects” interactions.
Edit: reading the rest of the guide, I find it very enlightening and at the perfect level of complexity perfect for me (no formal maths since Uni 15 years ago).
At this level of abstraction and energy scale it’s modelled as a mathematical field, but things get more tricky “underneath” just as you suggest.
This is so well written. I remember being taught these in my first year engineering and I really never understood a lot intuitively. (I could solve numericals and clear the course but I did not get it then. Being a CS student, I never had to care about it ever since). I wonder how the world would have been different if all courses were taught this way.
OK since photon_lines is reading, do you want any comments?
For instance, right on page 4 of the (not numbered) PDF – I tripped over the word “discreet”, where ostensibly you meant “discrete”.
If not – s’ok. Just askin’
Indeed and thank you!! Yes – if you have any improvement suggestions, or anything which you’d like added, let me know and I’ll see what I can do!
I also see that I made a few spelling mistakes, and a few kind folks here have already submitted issues to let me know, so I’ll correct them soon! Thank you all for the feedback and for the help and suggestions! I really appreciate it!
I’d be interested in an intuitive explanation of the units of electromagnetic quantities.
E.g. the unit of magnetic flux is equivalent to volt times seconds, and inductance is volt times seconds per ampere. But I can’t find intuitive explanations for this
This is phenomenally interesting to me because that very question led me to understand a lot more about the nature of units themselves. In particular, EM practitioners often use “Gaussian units” that name the math much cleaner (factors of 4pi appear and then disappear, instead of carrying around ε0 everywhere), but they have the weird side effect of perverting SI units. For example, in Gaussian units the standard unit of charge, the statcoulomb, has dimensions involving length to the 1/2 power. I think that has rather interesting implications on the nature of length; it seemd to me that the dimensions we take for granted might not be as fundamental as we think.
Usually easier to move around the units to match the standard equations, e.g. voltage is rate of change of magnetic flux.
It’s also interesting once you understand what Maxwell’s equations mean to look at their geometric algebra formulation[1]. In particular it makes their use in special and general relativity somewhat more elegant, since the GA form explicitly includes a spacetime component. Of course that page isn’t an elementary introduction, it assumes familiarity with GA and the divergence & curl operators, as well as some concepts of special & general relativity.
https://www.av8n.com/physics/maxwell-ga.htm
Actually, at the end of the guide, I tried to include an explanation which states that the magnetic field is just a by-product of relativity, and that the equations really only describe one field. A comment on the other approach: the geometric formulation to me looks interesting, but it’s still very information dense and a bit un-intuitive! I’ll take a look when I get more time though and see if I can re-formulate the ending chapter using this notation. Thank you for the feedback!
That article I linked is definitely not written for beginners, but I do find its notation more intuitive. But that’s only because I’m used to working in the notation of geometric algebra, so using it for Maxwell’s equations makes sense.
Unrelated to the subject, but while viewing the pdf in the GitHub viewer using Firefox on Android: halfway through Firefox crashed and my phone’s wallpaper changed. Strange bug, and I can recreate it. Anyone else?
According to the epigraph, God said something about D and H, but those were carefully avoided in the text. What’s up with that.
Apologies – I didn’t even realize that the epigraph expressed some of the equations in terms of flux density! That’s what those extra terms mean! I did actually include an explanation of what flux is and how to intuitively derive it within the guide, and you can find it near the integral explanations. Hopefully that helps!
