On the Origin of Maxwell's Displacement Current, a "sea" of magnetic vortices capacitor; 旋涡发生体产生的旋涡作用力交替作用在探头两侧，使探头内的差动电容发生差值变化

On the Origin of Maxwell's Displacement Current (self.AskHistorians)

submitted 5 months ago by dargscisyhp

Maxwell's equation governing light and all other electromagnetic phenomena would not have been possible without the modification of Ampere's law to include the displacement current. However, I'm a bit curious about the historical origin of the displacement current term. While I know the term is necessary for the continuity equation, in class it was presented as though Maxwell simply added the term for mathematical beauty: it made the form of the equations governing the electric and magnetic fields symmetric. And from there, he was able to derive the electromagnetic wave and all the other phenomena associated with classical electrodynamics.
But physicists are not historians, and I cannot believe that there was no more compelling reason for the addition of the displacement current term. Historians, what was the real reason Maxwell added the displacement current term to Ampere's law? Or am I off-base, and the term really was added simply for symmetry reasons?

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[–]LeftoverNoodles 2 points3 points4 points 5 months ago* (2 children)

I am going to make the assumption, that my mathematical beauty your professor means Simple and Symmetric. Simple in this context stemming from the fewest axioms or "Laws" and Symmetric, meaning unchanging under rotation or accelerating frames of reference. I am also probably going to over simplify some of the math.
From his paper is was fairly clear*** that Maxwell was attempting to integrate the theory of Magnetism and the Theory of Electricity into a single mathematical framework i.e. the Theory of Electromagnetism.

I propose now to examine magnetic phenomena from a mechanical point of view, and to determine what tensions in, or motions of, a medium are capable of producing the mechanical phenomena observed.

At the time the two phenomena where obviously connected, but there was not mathematical way to describe both simultaneously. The displacement current he postulated as a way of connecting Electricity and Magnetism was based around the concept of a "sea" of magnetic vortices that moved energy from one side of a capacitor to another. Maxwell was correct in how he described the effects, but incorrect in identifying the underlying mechanism.
What is critical for context is that Maxwell was doing his work before the invention of vector calculous. This made his work neither simple or beautiful, but rather a pages upon pages of rigorous calculation performed in a fixed set of coordinates. Without vectors, rotating the coordinate axis becomes almost more complex than the underlying mathematics, and the concept of symmetry is almost meaningless. Nor did Maxwell reduce the number of existing Law's, he in fact extended them.
It wasn't until a couple of decades later, when the same equations were reformulated in vector notation that they beauty of what Maxwell did became apparent. When written in vector form Maxwell's equations were both highly symmetric (unchanging under rotation and frame of reference) and simplified. The Symmetry techniques that Maxwell discovered as part of his unification when on to underlie the totality of modern physics, but the technique was the bi-product of a different goal.
In short, Maxwell invented a phenomena, which he described with the displacement current, so we could combine electricity and magnetism. His phenomena was wrong, but his math wasn't. It wasn't until they the E&M equations were reformulated in vector calculous, that they became beautiful. It's concept what very similar to Einstein's Cosmological Constant, in that's its a correct concept implemented for the wrong reasons.
*** It occurs to me that this is not actually clear. Buts its mentioned in both Griffeths text and on the Wiki page.

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[–]dargscisyhp[S] 0 points1 point2 points 5 months ago* (1 child)

What I mean by symmetric is that the form of the equations for both the electric and magnetic fields is the same; not symmetry in the sense of Noether. Explicitly, before the addition of the displacement current the curl of the electric field was proportional to the time derivative of the electric field, whereas the curl of the magnetic field was not dependent on the time-derivative of the electric field. With the addition of the displacement current the curl of both fields depend on the time derivative of the other.
It is not particularly clear to me how one would approach the study of electromagnetism without vector calculus short of doing everything component by component. However, if we were to do it this way the asymmetry between the laws governing the electric field and the magnetic field should still be readily apparent. While I understand that there was an empirically established connection between electricity and magnetism, there was no experimental evidence at the time of any such thing as a displacement current from what I've been told. As such, Maxwell's laws as stated before the addition of the displacement current already embodied the connection between electricity and magnetism as far as it had been empirically verified.
All that said, what you're telling me is that the origin of the displacement current comes from physical considerations rather than mathematical ones. Namely, it was postulated "around the concept of a "sea" of magnetic vortices that moved energy from one side of a capacitor to another." I am curious as to why he was investigating this in the first place, and how the addition of the displacement current was able to solve his problem. Can you provide a link to this, please (or a page number in Griffiths or Jackson would suffice).
It seems plausible to me that one might "symmetrize" the pre-Maxwell equations for the electric and magnetic fields and then see what happens just for funsies. We would obviously arrive at the laws of electrodynamics by doing so. However, it seemed bizarre to me that this is how arguably the biggest advance in 19th century physics came about. I would love to read the actual argument that drove Maxwell to incorporate the displacement current into the electromagnetic field equations.
Thanks for your reply!
Edit: Do you know if any of Maxwell's contemporaries were working on the same thing? If the motivation behind all this was to explain a physical phenomena, surely others were trying to explain the same thing. From what you've told me, Maxwell's approach seems to be fundamentally different than Einstein's i.e. Einstein primarily worked deductively where he built his theories off of some rather simple axioms, whereas Maxwell worked inductively and tried to explain a particular phenomenon by his work and then generalized. While Maxwell's work was obviously monumental, this take away a little bit from his work in my eyes. Then again, I'm a bit biased. I find it particularly beautiful to see physics advance motivated primarily by mathematical considerations.

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[–]LeftoverNoodles 0 points1 point2 points 5 months ago* (0 children)

without vector calculus short of doing everything component by component.

Bingo. Its a lot more raw calculation.
You should be able to find his paper online if you want to read exactly what he published. My Griffith's is in storage, but its in the preface. Again my interpretation of the events is to solve the weakness of the contemporary Ampere's law in describing currents that very overtime. There was no mention of the grander notion of Symmetry, but you are correct in that this was putting the cap stone on E&M becoming Symmetric / fully integrated.
The general timeline of events are:

• 1861 was the displacement current
• 1862 was the speed of propagation of an electric field (same as light)
• 1865 - The paper on EM waves, and how they were close to the speed of light.
• 1870's - Vector Calculous by Gibbs (Edit).
• 1889 - Physical Optics (Edit)

I think Faraday also speculated that light was electrical in nature, so the idea was not wholly Maxwell's. From Maxwell's arguments and hypotheses he was obviously building up from the existing knowledge base, trying to model and understand the phenomina of his day. It wasn't later until the EM Wave connection and the linkage with light, and how that all gets mixed up with the orbits of electrons and Quantum Mechanics, that the Symmetry and to some degree simplicity kicks into high gear.
Its very easy for the modern physicist looking back to see how Symmetrical and well designed everything looks, but back when Maxwell was working they just didn't have the mathematical tools or the understanding of sub-atomic world to pull everything together. Einstein 50 years later, had the benefit of seeing how Electrodynamics came together, and the power of vector calculous and differential geometry when we was working on general relativity.

Then again, I'm a bit biased. I find it particularly beautiful to see physics advance motivated primarily by mathematical considerations.

My QM professor would tell me this type of thinking is Philosophy. And that here we follow the scientific method.