OK, so what is my point in the preceding derivations?
1) We assumed only the 1/r electrostatic potential, the Special theory of Relativity and a few very basic ideas from Quantum Mechanics. From this we showed that the electromagnetic forces follow from these three ideas. In particular I think these derivations show how a relatively simple force between stationary particles can be transformed by Special Relativity into a rather complex set of forces (Electromagnetism). I think this is an important fundamental concept. It shows a deep relationship between Electromagnetism, Relativity and Quantum Mechanics that is not taught in books on these subjects.
2) Note that in deriving how to use the 4-vector potential to calculate forces, we had to assume that the energy of interaction between the source and test charge was located at the source charge. This is in direct conflict with the conventional understanding that the energy is in the electromagnetic field in the space between and around the particles. I think this may be a very important result. In fact, it challenges some very basic ideas in physics, such as that the energy of the interaction propagates through space from the source to the test charge.
I recently read "The Maxwellians" by Bruce Hunt (see http://www.amazon.com/Maxwellians-Cornell-History-Science/dp/0801482348). This excellent book tells the history of electromagnetism, and the people who developed the theory in the 1880's and 1890's after Maxwell. I did not know until reading this book that Maxwell never wrote the equations that carry his name. It was Heaviside that took Maxwell's 20 field equations and distilled them down to 4 vector differential equations. Maxwell's original equations included a combination of potentials and fields apparently. Heaviside's reworking of them emphasized the fields and almost completely suppressed the potentials. Heaviside was also one of the promoters of the idea that the energy is carried in the electric and magnetic fields, like we have all been taught. As a result of the derivations just posted I am starting to question this dogma.
There is no doubt that solving electromagnetic problems using E and B fields (and D and H) is tremendously convenient. The power of this method is that through the use of some simple boundary conditions, one can almost completely ignore the presence of charges on metal and in dielectric materials. The alternative, to integrate the potentials from all these charges, requires that you first solve for the motions of all the charges acted on by whatever is driving the system (i.e. the radio transmitter, or light source, etc.). This presents a bit of a chicken and egg problem. How can you find the fields without knowing the motions of the charges? How can you find the motion of the charges without knowing the fields?
In the physical world this is not a problem because each charge is affected only by charges on its past light cone. And the charges on the past light cone are not affected by charges on their future light cone. (Actually, in a future post I will argue that this is not completely true.) In principle this problem can be solved, but it can be very difficult mathematically without using Maxwell's equations for figuring out the fields first. As a practical tool Maxwell's equations are brilliant.
But I wonder if they are the correct physical model of the world. In the preceding posts I've shown that the forces on charges can be calculated without referring to electric or magnetic fields, but only using potentials. What is the real, physical, thing, the potentials or the fields? This may be a pointless philosophical question if they both give the same results. But it appears to me that they might not. The potential approach requires the energy of interaction to be located on the source charge. The potential approach also subtly implies action at a distance.
In fact, I learned in Hunt's book that there was a great debate about this very subject in the early 1890's at a conference in England. It is sometimes called the "murder of phi". One argument against the potentials was that they imply action at a distance, which was strongly disliked. The result was that everyone agreed that it was the fields that are real and that the potentials are mathematical fictions that are useful in some problems, but are not real. This was Heaviside's position apparently. But this was all done before Relativity and Quantum Mechanics. Relativity is all done with potentials. Relativistic electrodynamics does use the electromagnetic field tensor, consisting of components of the E and B fields. But it is defined in terms of derivatives of the 4-potential.
Quantum Mechanics almost demands action at a distance. Wouldn't action at a distance solve the issues with quantum entanglement and EPR experiments? Of course that would violate Special Relativity unless the interaction is on the light cone, but of course it is! The Aharonov-Bohm effect (http://en.wikipedia.org/wiki/Aharonov-Bohm) strongly implies that the potentials are real. This effect demonstrates that there are observable changes to the phase of a charged particle's wave function due to the potential, even when there are no gradients in the potential to cause an "electric" or "magnetic" field.
If the potentials are real, and not just mathematical conveniences, then it seems to me that that also means that the energy of interaction is NOT located in the space between the charges, but rather at the charges themselves. That would be a significant development in our understanding. I probably need to do more to justify this claim, however, if I'm going to convince people that this is correct.
In future posts I will derive "electromagnetic" radiation from an oscillating electric dipole using potentials, and get the classical result. I will also calculate radiation resistance without using Poyntings Theorem, but only potential interactions between the radiating charges and charges absorbing the "radiation". I also plan to show that blackbody radiation can be derived using potentials and NOT using cavity modes. (I'm still working on that, but I think I see how to do it.) Finally, I have some posts on General Relativity that fit in with all this and give some insight into how the force of gravity is fundamentally different from electromagnetism.
I would very much like to discuss these ideas with anyone interested. Especially if you don't buy into my arguments!
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