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Maxwell published his Treatise on Electricity and Magnetism in 1873, 150 years ago; before the discovery of quantized charges, special relativity, quantum field theory etc. How has the mathematical and physical modelling of classical electrodynamics today changed since then?

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  • $\begingroup$ Well, the whole computers thing let’s one model very complex geometry and fast temporal features… $\endgroup$
    – Jon Custer
    Commented Apr 27, 2022 at 2:12

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We will take the cutoff to be 1873, the publication date of the first edition of Maxwell's treatise. I'm not an expert in electrodynamics by any stretch, but here are some of the things that come to mind.

Mathematically, the most significant difference between Maxwell and 20th century (and beyond) electrodynamics is the use of vector notation. Maxwell worked with vectors component-wise, and this sort of trend continued into the early parts of the 20th century, despite the development of vector notation by Heaviside and Gibbs in the late 19th century. Aside from the change in notation, much of the work was related to partial differential equations, which were fairly well developed by Maxwell's time. Most of our analytical "tricks" in electrostatics, like spherical harmonics, Green's functions, method of images, etc were already developed and thoroughly studied, but the conception of real analysis was certainly lagging behind (e.g. Dirac delta and distribution theory). Mathematical techniques used to approximate the solution to partial differential equations, like variational techniques, were certainly possible (the calculus of variations is old) but I personally haven't seen much in this vain in older literature. It's also usually dependent on integral equations, or functional analysis, two fields which saw most of their development in the 20th century. Even still, most textbooks going into the 20th century were based on Maxwell's coverage, so e.g. Jackson's Classical Electrodynamics or Smythe's Static and Dynamic Electricity are, in their first few chapters, mostly restatements and summaries of this work, but with the benefit of many decades of refinement.

Certain problems were awaiting more satisfactory solutions, for example the electromagnetic field due to a moving point charge, but some of the groundwork was done and already gave hints towards a relativistic theory. The use of electric and magnetic scalar and vector potentials was justifiably seen as essential to the theory of electromagnetism, but the concept of gauge freedom was a point of contention. Problems in radiation were still new and underdeveloped, but the wave behavior was embedded in the theory and emphasized by Maxwell, despite a lack of direct experimental evidence.

The most significant changes in the theory are more related to the interpretation. An obvious example is the aether, which was essential to Maxwell's conception of the electromagnetic field. This is probably the most significant change, but also the most immaterial to the mathematical side. The aether theory was put to rest at the turn of the 20th century, in light of growing evidence that the aether had essentially no role to play in the behavior of the electromagnetic field, and could be neglected without much reformulating. Other somewhat metaphysical theories and perspectives as well as conceptual tools (e.g. tubes of force) would grow and wane in popularity, before mostly disappearing from literature.

Another significant change was the introduction of the Poynting vector (1884), a term in the power balance of electromagnetic waves which is consistent with Maxwell's theory. The interpretation of this "power flow" term was debated going into the 20th century, but its importance was accepted and it now plays a central role in the study of electromagnetic field energy.

The limits of classical electrodynamics have been refined, as our understanding of the structure of matter grows. Things which were in Maxwell's time simply hypotheses were found to be true (atomic nature of matter, as well as electric charge). The application of electrodynamics to the dynamics of charged particles has taken a central role in electrodynamic theory. The study of radiation, and the study of electromagnetic waves, have both been developed and applied to things like optics, microwaves, and RF communication. Examples being the theory of antennas, or the theory of waveguides.

Depending on your scope of "electrodynamics", the rise of computing has been hugely impactful in industry, and for theoretical study. But I would say this is more application-related, as it does not develop the theory itself.

Extension of the theory to other states of matter (notably plasma) might be getting close to the quantum nature of matter, and this isn't something I have experience with. But it's certainly been a large area of study for decades, and could be a significant development since Maxwell's time.

There have also been important experimental results, as you can imagine. Trying to keep to experiments which apply classically, we have Hertz's verification of electromagnetic waves was significant for obvious reasons; Michelson-Morely was another nail in the aether's coffin; the Stewart-Tolman experiment demonstrated that the charge carriers in various conductors were negatively charged, in an experiment originally proposed by Maxwell; verifications of Coulomb's law by Plimpton, then later by Bartlett, and others.

All in all, while the PDEs at the heart of classical electrodynamics have not changed much (in substance), and many of our problem solving strategies are carried over from Maxwell's time, there was still a huge amount of work to be done in applications and in setting the limits. Much of the theoretical work was closely tied to the emerging quantum theory and the theory of special relativity, the latter introducing the more modern tensor notation to electromagnetism, and providing a closer connection between the electric and magnetic fields. Still, Maxwell's most outdated ideas relate to the structure of matter and space, not so much to the mathematical theory, which were just more immature.

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