Two Perspectives On James Clerk Maxwell And His Equations

We are unabashed fans of [The History Guy’s] YouTube channel, although his history videos aren’t always about technology, and even when they are, they don’t always dig into the depths that we’d like to see. That’s understandable since the channel is a general interest channel. However, for this piece on James Clerk Maxwell, he brought in [Arvin Ash] to handle the science side. While [The History Guy] talked about Maxwell’s life and contributions, [Arvin] has a complementary video covering the math behind the equations. You can see both videos below.

Of course, if you’ve done electronics for long, you probably know at least something about Maxwell’s equations. They unified electricity and magnetism and Einstein credited them with spurring one of his most famous theories.

Deriving Maxwell’s equations is a math nightmare, but [Arvin] doesn’t do that. He uses some amazing graphics to explain how the equations relate electricity and magnetism. A great deal of our modern world — especially related to any sort of radio technology — builds on these four concise equations.

One thing we didn’t realize is how wide-ranging Maxwell’s interest were. He contributed to astronomy by explaining Saturn’s rings, derived statistical laws about gasses, and worked on color vision, including creating the first light-fast color photograph. He also contributed to thermodynamics, control theory, and optics. Those were the days!

We really enjoyed the way the two videos support each other. Understanding the math is a big deal to us, but understanding the man’s life and the context he lived in is pretty interesting, too. We hope we see more such collaborations.

Not that we don’t do our part to try to tell technology history with a bit more depth than a typical history book. Do you remember [Rufus Turner]? We do. We’ve even had a few debates over who really invented radio. If you want more of that, you can always browse our history tag.

25 thoughts on “Two Perspectives On James Clerk Maxwell And His Equations

  1. As I understand it Maxwells equations as well as being intrinsically relativistic, is readily quantised and the symmetry he “shoehorned” in is a good example of gauge symmetry.

    Wonderful serendipity?

  2. I’ve always meant to spend some time trying to understand how to apply Maxwell’s equations (and possibly learn how to use them to simulated 3D space of maybe 1000x1000x1000 voxels for an iterative non-intuitive approach for antenna design), but I have to admit that the maths in those four concise equations has always scare the hell out of me. Is there less concise version with hopefully hell of a lots more equations that is far more approachable, to the mathematically illiterate (I did study three dimensional partial differential complex equations a long long long long time ago, but that maths hurt my head back then as well).

    1. “Is there less concise version with hopefully hell of a lots more equations that is far more approachable,”

      Uh… no? Electromagnetism at this level is the bane of physics students in graduate school: the seminal text (Jackson’s Classical Electrodynamics) is usually considered a weed-out class in grad school.

      Solving Maxwell’s equations in a simulated 3D space is Not Easy. The typical way to do it is with finite-difference time domain (FDTD) simulations, which can be very expensive (both computationally and cost-wise).

      There is an open-source free FDTD simulation called MEEP, but without access to grid-scale computing you’re likely to struggle to use it for iterative design due to memory/computing constraints.

      To scare you even more, Maxwell’s equations aren’t even really 4 equations: relativistically, they’re just one. They’re just saying that the wave (d’Alembert) operator of the electromagnetic field is defined by the 4-dimensional electromagnetic current. Or, in far fewer words: the electromagnetic field is wave-generated by charges and currents. That’s it.

      1. Yes, the reason the F-117 and related projects are so angular is BECAUSE it’s a problem modelling Maxwell’s equations in 3D. They had to do the bulk of it in 2D… and this is people with access to the supercomputers of the time, well into the 80s, probably most of the 90s. Don’t think it’s been long since they could do it without dimming lightbulbs in the whole rest of the state.

        1. “Yes, the reason the F-117 and related projects are so angular is BECAUSE it’s a problem modelling Maxwell’s equations in 3D.”

          The reason the F-117 and related projects are so angular is BECAUSE IT WORKS.

          1. Yes, but the flat facets and sharp joints compromise flight characteristics considerably. Later stealth projects aren’t nearly as angular because there was sufficient computing power by then to allow the much more desirable non-faceted stealthy surfaces and cross-sections.

          2. We all can create stealth equipment. If the stealth bomber or DDG Navy destroyer were made of mirrors you would see very little of yourself. This contributes to the Grand majority of the reduced radar signature. In fact the Aeigis DDG destroyer uses no radar absorbent paint(believe me I have painted the USS Benfold DDG-65 many times), and when seen from another ship or aircraft looks like a small ship or nothing at all depending on the angle. The ships hull is more traditional and gives away some of the cross section.

    2. Get a copy of the paperback “Div Grad Curl and All That” and a used copy of David Griffiths ‘Introduction to Electrodyanamics. On YouTube check 3Blue1Brown’s videos.

    3. As Jesse Jenkins alluded to, Oliver Heaviside already simplified Maxwell’s equations. His simplifications are commonly referred to as “Maxwell’s equations.” From the Wikipedia article on Heaviside: “In 1884 he recast Maxwell’s mathematical analysis from its original cumbersome form (they had already been recast as quaternions) to its modern vector terminology, thereby reducing twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns we now know as Maxwell’s equations. “

  3. He neglected monopole current. If only we had a loop of some material that would support monopole current, we could have a transformer that would work on DC! Was he friends with Tesla or something?!

  4. Haven’t looked at the videos yet but read an interesting book a while back called The Maxwellians. My recollection is Faraday independently captured the data. Maxwell developed ~16 equations and a team of others (Heaviside, Hertz, etc) distilled it to the 4 equations. A bit more of teamwork?

  5. Interesting how after Thomas Young in 1827 rejected the theory that light is made of material particles moving through empty space in favor of the alternative theory of a medium of material particles that collide off each other in a transverse wave motion, way back in 1864 Maxwell adopted and piled on his more complex theory on top of that- viewing light not as just a single transverse wave in a material particle medium, but as a dual transverse wave. see Maxwell’s originally published graphic at Google books: I am fascinated by the alternative theory that lost- the view that light is more like a tennis ball moving through empty space rebirthed very briefly by Descartes in his Le Monde, and amplified by Newton in the 17th and 18th centuries, it is amazing to me to think that maybe way back on the timeline a terrible mistake was made resulting in a nearly 200 year dim age in the human understanding of light. I think it is healthier to keep an open mind and entertain the main alternative like we do a two-party system in government.

    1. You’re really making a distinction where there isn’t one. Think about the surface of a drum: a drum can only admit certain standing wave patterns on it. If you try to establish some other pattern, it won’t work. So the drum’s excitations are quantized: they can only be established in certain fixed modes.

      Light, like everything else that carries energy/momentum, is the same thing. Just imagine the 4D electromagnetic field as the surface of the drum, and light as the quantized excitation of it.

  6. Thanks for posting this! Dare I look forward to a similar piece on Heavyside?

    But here’s a question: [Why] [do] [you] [insist] [on] [putting] [names] [in] [brackets] [?]

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