4 ms·

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 ha

by bsder 3y ago

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. Consequently, you are stuck with the Heaviside-Hertz pedagogy with silly things like "displacement current" and stupid, weird-ass integration contours (which don't work in motors, LOL)--and the attendant difficulty in understanding Maxwell's theory. However, if you have vector analysis and fields, then you can understand formulations like Carver Mead's "Collective Electrodynamics": https://www.amazon.com/Collective-Electrodynamics-Quantum-Foundations-Electromagnetism/dp/0262632608 https://www.amazon.com/Collective-Electrodynamics-Quantum-Fo... Suddenly, emag is a whole lot more straightforward to understand. It's not EASY as it's very math heavy, but it has a lot fewer weird things that are just "we say it works."

MichaelZuo 3y ago
I've always thought the Heaviside notation is a bit bizarre... is there any advantage to them at all?
- teleforce 3y ago
  Heaviside and his proponents avoided quaternion like a plague, and like a classic false messiah he somehow convinced people not to use it. If we want to easily and completely model the EM waves its entirety including polarization we need to embrace quaternion, there is no two ways about it. The intuitive understanding of EM can be only developed by using quaternion and me personally waiting for someone to write a Pozar's book version in quaternion approach.
mitthrowaway2 3y ago
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.
- nimish 3y ago
  Admittedly the tools to make the mathematics more compact and clearer weren't invented yet (except as quaternions). We did get stuck with Gibbs' vector calculus formulation as the canonical view unfortunately.
- bsder 3y ago
  People forget that Maxwell's Theory contradicts the prevailing belief of the time that waves ALWAYS require a medium for propagation--the so called "aether". That's a really large conceptual jump and physicists did not make that jump lightly or easily. Maxwell's paper was 1865. The Michelson–Morley experiment was 1887. Michelson himself couldn't make the shift. Quoting Wikipedia (https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper...): "The negative result led Michelson to the conclusion that there is no measurable aether drift.[1] However, he never accepted this on a personal level, and the negative result haunted him for the rest of his life (Source; The Mechanical Universe, episode 41[8])." In an attempt to preserve "aether", Lorentz contraction then enters the picture as an ad hoc explanation for the Michelson-Morley result. It turns out Lorentz contraction is correct, but not because of the existence of "aether" but because of the constancy of the speed of light--c (Einstein Special Relativity--1905). Once you finally give up on "aether" after 40 years of trying otherwise, you can finally just roll with the mathematical implications of Maxwell's equations.
  - mitthrowaway2 3y ago
    Maxwell saying that his theory "attributes electric action to tensions and pressures in an all-pervading medium... the medium being identical with that in which light is supposed to be propagated" suggests that Maxwell himself did not view his theory to be contradicting the existence of an aether (or was being coy about it). Which is especially interesting because Maxwell based his theory on Faraday's, and Faraday didn't believe in the aether.[1] [1] Michael Faraday's Thoughts on Ray Vibrations, 1846, cited by Maxwell in his paper. Faraday says: "The view which I am so bold to put forth considers, therefore, radiation as a kind of species of vibration in the lines of force which are known to connect particles and also masses of matter together. It endeavors to dismiss the aether, but not the vibration." (https://pwg.gsfc.nasa.gov/Education/wfarad1846.html https://pwg.gsfc.nasa.gov/Education/wfarad1846.html)
LeapingLennie 3y ago
Are there any textbooks you would recommend for learning vector analysis / vector fields before studying EM?
- bsder 3y ago
  I do not. None of the books I know of are very good because they are mostly targeted at Mathematics majors rather than physicists or engineers. Gerard ’t Hooft used to have a humongous list of textbooks for aspiring theoretical physicists. I'd sure look there for starters.
- sidlls 3y ago
  EM’s vector fields formulation is fairly straightforward: it’s all curls and divergence. Any 3rd (-ish) semester undergrad multivariate calculus course is likely to cover it in sufficient depth. “Mathematical Methods for Physicists” covers it in sufficient depth, for example, provided you already have a thorough understanding of the prerequisite material. Most undergrad physics degree curriculums should have E&M courses whose texts (e.g. Introduction to Electricity and Magnetism, Griffiths) cover enough of the details. If you want to pursue it further than an undergraduate level study, you’ll also want a good text on differential equations that has or is supplemented by material covering, e.g., spherical harmonics and Bessel functions (among other things). I wish I could remember what I used, but it was…more years ago than I care to say when I was an grad student.
- rramadass 3y ago
  I was recommended Nathan Ida's Engineering Electromagnetics as being comprehensive in that all the necessary Mathematics is introduced in place as needed. Lookup the reviews for this book on the web. Perhaps somebody who has read this book can comment in more detail.
deleted 3y ago
[deleted]
jiggawatts 3y ago
Vector fields are just as incorrect, and ought to be relegated to history books as a mere stepping stone on the way to understanding. Geometric algebra can reduce Maxwell’s equations into a single, hilariously terse equation: ∇F = J Ref: https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Spacetime_algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...
- guardingit 3y ago
  Thanks for the pointer on Geometric Algebra. This looks to be a promising path to understanding relativity/QM/EM, and goes some way to explaining my unease with cross products and imaginary numbers. Disclaimer: maths degree, so my unease was not a plain lack of understanding.
  - nyssos 3y ago
    > maths degree Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM. The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.
    - guardingit 3y ago
      Thanks for your comment. > They do not do a good job of tracking the physical structure of electromagnetism What do you mean by tracking the physical structure? By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > and in fact end up baking in a lot of assumptions about the setting that fail to generalize. Can you explain a bit more what you mean here please?
      - 3y ago

by bsder 3y ago

MichaelZuo 3y ago

I've always thought the Heaviside notation is a bit bizarre... is there any advantage to them at all?

teleforce 3y ago
Heaviside and his proponents avoided quaternion like a plague, and like a classic false messiah he somehow convinced people not to use it. If we want to easily and completely model the EM waves its entirety including polarization we need to embrace quaternion, there is no two ways about it. The intuitive understanding of EM can be only developed by using quaternion and me personally waiting for someone to write a Pozar's book version in quaternion approach.

mitthrowaway2 3y ago

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.

nimish 3y ago
Admittedly the tools to make the mathematics more compact and clearer weren't invented yet (except as quaternions). We did get stuck with Gibbs' vector calculus formulation as the canonical view unfortunately.
bsder 3y ago
People forget that Maxwell's Theory contradicts the prevailing belief of the time that waves ALWAYS require a medium for propagation--the so called "aether". That's a really large conceptual jump and physicists did not make that jump lightly or easily. Maxwell's paper was 1865. The Michelson–Morley experiment was 1887. Michelson himself couldn't make the shift. Quoting Wikipedia (https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_experiment https://en.wikipedia.org/wiki/Michelson%E2%80%93Morley_exper...): "The negative result led Michelson to the conclusion that there is no measurable aether drift.[1] However, he never accepted this on a personal level, and the negative result haunted him for the rest of his life (Source; The Mechanical Universe, episode 41[8])." In an attempt to preserve "aether", Lorentz contraction then enters the picture as an ad hoc explanation for the Michelson-Morley result. It turns out Lorentz contraction is correct, but not because of the existence of "aether" but because of the constancy of the speed of light--c (Einstein Special Relativity--1905). Once you finally give up on "aether" after 40 years of trying otherwise, you can finally just roll with the mathematical implications of Maxwell's equations.
- mitthrowaway2 3y ago
  Maxwell saying that his theory "attributes electric action to tensions and pressures in an all-pervading medium... the medium being identical with that in which light is supposed to be propagated" suggests that Maxwell himself did not view his theory to be contradicting the existence of an aether (or was being coy about it). Which is especially interesting because Maxwell based his theory on Faraday's, and Faraday didn't believe in the aether.[1] [1] Michael Faraday's Thoughts on Ray Vibrations, 1846, cited by Maxwell in his paper. Faraday says: "The view which I am so bold to put forth considers, therefore, radiation as a kind of species of vibration in the lines of force which are known to connect particles and also masses of matter together. It endeavors to dismiss the aether, but not the vibration." (https://pwg.gsfc.nasa.gov/Education/wfarad1846.html https://pwg.gsfc.nasa.gov/Education/wfarad1846.html)

LeapingLennie 3y ago

Are there any textbooks you would recommend for learning vector analysis / vector fields before studying EM?

bsder 3y ago
I do not. None of the books I know of are very good because they are mostly targeted at Mathematics majors rather than physicists or engineers. Gerard ’t Hooft used to have a humongous list of textbooks for aspiring theoretical physicists. I'd sure look there for starters.
sidlls 3y ago
EM’s vector fields formulation is fairly straightforward: it’s all curls and divergence. Any 3rd (-ish) semester undergrad multivariate calculus course is likely to cover it in sufficient depth. “Mathematical Methods for Physicists” covers it in sufficient depth, for example, provided you already have a thorough understanding of the prerequisite material. Most undergrad physics degree curriculums should have E&M courses whose texts (e.g. Introduction to Electricity and Magnetism, Griffiths) cover enough of the details. If you want to pursue it further than an undergraduate level study, you’ll also want a good text on differential equations that has or is supplemented by material covering, e.g., spherical harmonics and Bessel functions (among other things). I wish I could remember what I used, but it was…more years ago than I care to say when I was an grad student.
rramadass 3y ago
I was recommended Nathan Ida's Engineering Electromagnetics as being comprehensive in that all the necessary Mathematics is introduced in place as needed. Lookup the reviews for this book on the web. Perhaps somebody who has read this book can comment in more detail.

deleted 3y ago

[deleted]

jiggawatts 3y ago

Vector fields are just as incorrect, and ought to be relegated to history books as a mere stepping stone on the way to understanding. Geometric algebra can reduce Maxwell’s equations into a single, hilariously terse equation: ∇F = J Ref: https://en.wikipedia.org/wiki/Mathematical_descriptions_of_the_electromagnetic_field#Spacetime_algebra https://en.wikipedia.org/wiki/Mathematical_descriptions_of_t...

guardingit 3y ago
Thanks for the pointer on Geometric Algebra. This looks to be a promising path to understanding relativity/QM/EM, and goes some way to explaining my unease with cross products and imaginary numbers. Disclaimer: maths degree, so my unease was not a plain lack of understanding.
- nyssos 3y ago
  > maths degree Then you want differential forms for EM, differential geometry more broadly for GR, and a bit of functional analysis for QM. The hype around geometric algebras (Clifford algebras over R) just comes from the fact that it's not the plug'n'chug explicit numbers and coordinates approach, which is all most people ever see. They do not do a good job of tracking the physical structure of electromagnetism, and in fact end up baking in a lot of assumptions about the setting that fail to generalize.
  - guardingit 3y ago
    Thanks for your comment. > They do not do a good job of tracking the physical structure of electromagnetism What do you mean by tracking the physical structure? By this, do you mean the typical EM formulation of Maxwell's laws produces Gauss's law and Faraday's law (which are instructive) where as the Geometric Algebra formula produces ∇F = J (less instructive?). > and in fact end up baking in a lot of assumptions about the setting that fail to generalize. Can you explain a bit more what you mean here please?
    - 3y ago