The question of how can the twist ring work gets a clear guide from the (non) effect of E fields on a photon. I worried about whether 1/r^2 – 1/r^3 could be right given the symmetry between the E field and B field equations–but it is the only possible solution, I realized. If it is only due to B field curvature, then the solution is unstable, it will collapse to nothing on perturbation. And the E field having no effect on a photon clearly points the way–there cannot be attraction of E field components, only attraction/repulsion of twists, that is, rotating field components. And indeed, setting the ring twist induces two effects: an E field, which will diminish as 1/r^2, and a dragged twist of E field components about the ring curve at any point on the ring. This dragged twist is the current loop that will generate the far field 1/r^3 magnetic component.

So, I began the math. I looked at the various Maxwell’s equation field forms, and because of the ring cylindrical symmetry, chose the cylindrical form of Maxwell’s. I derived the expanded form from the grad cross F equations to grad^2 cross F = 1/c^2 partial^2(F)/partial(t)^2 (dang… wordpress needs mathematical notation), and now have a big ugly mess whose initial conditions are defined by the twists at radius re, omega as a function t/(re c) and dont care everywhere else. The idea is to prove that this gives a solution with 1/r^2 – 1/r^3 form and does not dissipate.

But–there’s no way Mathematica is going to do this (I couldn’t even get mathematica to correctly derive grad^2 F, but that might just be because it’s harder to manipulate than I know how to manage). I searched for some assumptions that would allow me to simplify the equations, but there is only the fact that F(phi) is zero on the ring radius. I can’t assume it is zero elsewhere because E field components are only normal to the ring circumference at the radial point–adjacent E field components are not normal and will contribute to the F(phi) term.

So–right now I only see that I can do an iterative solution, and perhaps that will show me some symmetries of the solution that will allow me to simplify enough to get an analytic solution. But, finding a dynamic Maxwell’s 3D solver that works in cylindrical components is unlikely at best (there are commercial tools, as I mentioned previously, but they are static). And writing such a tool is going to be a major project. I’m first going to trawl the net to see if there’s something I can work from…

Agemoz

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