10 ms·

Intuitive Guide to Maxwell's Equations

385 points by photon_lines 6y ago

contravariant 6y ago
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).
- layoutIfNeeded 6y ago
  How do you define curl in 4d?
  - JadeNB 6y ago
    Div, grad, and curl are a manifestation of de Rham cohomology that make use of a lot of lucky coincidences. See https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/De_Rham_cohomology for the formal definition (in which curl becomes the exterior derivative from 1-forms to 2-forms) and https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_of_cohomology.pdf https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_... for a nice exposé.
  - contravariant 6y ago
    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.
chronolitus 6y ago
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.
- ghaff 6y ago
  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.
  - cs702 6y ago
    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."
    - ghaff 6y ago
      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.
      - cs702 6y ago
        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 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?
pgt 6y ago
Link to 3Blue1Brown video on Divergence and Curl: https://youtu.be/rB83DpBJQsE https://youtu.be/rB83DpBJQsE
antman 6y ago
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.
- pa7x1 6y ago
  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.
  - OscarCunningham 6y ago
    You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.
    - danharaj 6y ago
      escape your hodge duals fam
      - OscarCunningham 6y ago
        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).
    - tobinfricke 6y ago
      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".
Daub 6y ago
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.
- alanbernstein 6y ago
  Or, you're standing on his shoulder, you might say.
- reikonomusha 6y ago
  Small nit, it’s “Clerk” not “Clark”.
  - Daub 6y ago
    Well spotted. Thanks.
tarsinge 6y ago
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).
- l33tman 6y 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.
king07828 6y ago
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.
- photon_lines 6y ago
  Thanks for the feedback and I agree! I'm going to try to create guides for relativity and quantum mechanics. After I finish those, I'll keep expanding into general math topics, so hopefully those will help as well. Also an FYI - I made a similar guide to linear algebra which you can also find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-...
aimor 6y ago
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?
SAI_Peregrinus 6y ago
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 https://www.av8n.com/physics/maxwell-ga.htm
- photon_lines 6y ago
  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!
  - SAI_Peregrinus 6y ago
    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.
Aardwolf 6y ago
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
- discordance 6y ago
  If I’m getting you right, take a look into ‘dimensional analysis’
- augustt 6y ago
  Usually easier to move around the units to match the standard equations, e.g. voltage is rate of change of magnetic flux.
  - deleted 6y ago
    [deleted]
- Rayhem 6y ago
  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.
Koshkin 6y ago
According to the epigraph, God said something about D and H, but those were carefully avoided in the text. What's up with that.
- c1ccccc1 6y ago
  They have to do with the behaviour of the electromagnetic field inside of different materials: https://en.wikipedia.org/wiki/Electric_displacement_field https://en.wikipedia.org/wiki/Electric_displacement_field , https://en.wikipedia.org/wiki/Magnetic_field#The_H-field https://en.wikipedia.org/wiki/Magnetic_field#The_H-field To get Maxwell's equations in a vacuum, just replace D with εₒE and replace H with B/μₒ.
- photon_lines 6y ago
  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!
signaru 6y ago
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.
muktabh 6y ago
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.
Jailbird 6y ago
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'
- photon_lines 6y ago
  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!
peter_d_sherman 6y ago
>"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!"
mikorym 6y ago
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.

385 points by photon_lines 6y ago

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).

layoutIfNeeded 6y ago
How do you define curl in 4d?
- JadeNB 6y ago
  Div, grad, and curl are a manifestation of de Rham cohomology that make use of a lot of lucky coincidences. See https://en.wikipedia.org/wiki/De_Rham_cohomology https://en.wikipedia.org/wiki/De_Rham_cohomology for the formal definition (in which curl becomes the exterior derivative from 1-forms to 2-forms) and https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_of_cohomology.pdf https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_... for a nice exposé.
- contravariant 6y ago
  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.

chronolitus 6y ago

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.

ghaff 6y ago
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.
- cs702 6y ago
  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."
  - ghaff 6y ago
    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.
    - cs702 6y ago
      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 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?

pgt 6y ago

Link to 3Blue1Brown video on Divergence and Curl: https://youtu.be/rB83DpBJQsE https://youtu.be/rB83DpBJQsE

antman 6y ago

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.

pa7x1 6y ago
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.
- OscarCunningham 6y ago
  You can even write them in terms of the four-vector A rather than F (related by F = dA) to reduce them to ddA = J.
  - danharaj 6y ago
    escape your hodge duals fam
    - OscarCunningham 6y ago
      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).
  - tobinfricke 6y ago
    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".

Daub 6y ago

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.

alanbernstein 6y ago
Or, you're standing on his shoulder, you might say.
reikonomusha 6y ago
Small nit, it’s “Clerk” not “Clark”.
- Daub 6y ago
  Well spotted. Thanks.

tarsinge 6y ago

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).

l33tman 6y 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.

king07828 6y ago

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.

photon_lines 6y ago
Thanks for the feedback and I agree! I'm going to try to create guides for relativity and quantum mechanics. After I finish those, I'll keep expanding into general math topics, so hopefully those will help as well. Also an FYI - I made a similar guide to linear algebra which you can also find here: https://github.com/photonlines/Intuitive-Overview-of-Linear-Algebra-Fundamentals https://github.com/photonlines/Intuitive-Overview-of-Linear-...

aimor 6y ago

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?

SAI_Peregrinus 6y ago

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 https://www.av8n.com/physics/maxwell-ga.htm

photon_lines 6y ago
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!
- SAI_Peregrinus 6y ago
  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.

Aardwolf 6y ago

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

discordance 6y ago
If I’m getting you right, take a look into ‘dimensional analysis’
augustt 6y ago
Usually easier to move around the units to match the standard equations, e.g. voltage is rate of change of magnetic flux.
- deleted 6y ago
  [deleted]
Rayhem 6y ago
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.

Koshkin 6y ago

According to the epigraph, God said something about D and H, but those were carefully avoided in the text. What's up with that.

c1ccccc1 6y ago
They have to do with the behaviour of the electromagnetic field inside of different materials: https://en.wikipedia.org/wiki/Electric_displacement_field https://en.wikipedia.org/wiki/Electric_displacement_field , https://en.wikipedia.org/wiki/Magnetic_field#The_H-field https://en.wikipedia.org/wiki/Magnetic_field#The_H-field To get Maxwell's equations in a vacuum, just replace D with εₒE and replace H with B/μₒ.
photon_lines 6y ago
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!

signaru 6y ago

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.

muktabh 6y ago

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.

Jailbird 6y ago

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'

photon_lines 6y ago
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!

peter_d_sherman 6y ago

>"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!"

mikorym 6y ago

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.