
Intuitive Guide to Maxwell's Equations - photon_lines
https://github.com/photonlines/Intuitive-Guide-to-Maxwells-Equations
======
chronolitus
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
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.

~~~
C4stor
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.

~~~
uoaei
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.

~~~
soVeryTired
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.

~~~
QuesnayJr
If this is a math topic, you should bring it to the attention of the
WikiProject Mathematics talk page:
[https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mat...](https://en.wikipedia.org/wiki/Wikipedia_talk:WikiProject_Mathematics).
They might do something about it.

~~~
soVeryTired
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 :)

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pgt
Link to 3Blue1Brown video on Divergence and Curl:
[https://youtu.be/rB83DpBJQsE](https://youtu.be/rB83DpBJQsE)

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signaru
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.

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Daub
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.

~~~
reikonomusha
Small nit, it’s “Clerk” not “Clark”.

~~~
Daub
Well spotted. Thanks.

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antman
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
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: _d_ F=J dF=0

Geometric algebra produces two equations too, by the way, not one.

~~~
OscarCunningham
You can even write them in terms of the four-vector A rather than F (related
by F = dA) to reduce them to _d_ dA = J.

~~~
tobinfricke
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".

~~~
jabl
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.

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peter_d_sherman
>"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!_ "

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mikorym
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.

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contravariant
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
How do you define curl in 4d?

~~~
JadeNB
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_...](https://web.ma.utexas.edu/users/a.debray/lecture_notes/idea_of_cohomology.pdf)
for a nice exposé.

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tarsinge
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
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.

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king07828
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
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-...](https://github.com/photonlines/Intuitive-Overview-of-Linear-
Algebra-Fundamentals)

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muktabh
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.

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Jailbird
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
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!

------
Aardwolf
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

~~~
augustt
Usually easier to move around the units to match the standard equations, e.g.
voltage is rate of change of magnetic flux.

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SAI_Peregrinus
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
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
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.

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aimor
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?

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Koshkin
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
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 _ε_ ₒE and replace
H with B/ _μ_ ₒ.

