Analysis of the motion and propagation of electromagnetic waves led to the special relativity description of space and time. Schwarzschild and Fokker considered the advanced field of a system of moving charges, and the retarded field of a system of charges having the same geometry and opposite charges. To compute energy, it is necessary to use the absolute fields which includes the zero point field; otherwise, an error appears, for instance in photon counting. It is important to take into account the zero point field discovered by Planck M.
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Feynman frames it nicely, so I should print it and put it on the kitchen door, so I can look at it everyday. Hmm… Stupid question, perhaps, but why is there no wave equation above? Here we have everything. How do we find it? That solution for a small blob of charge, i. However, his equations did not get much attention, apparently, because a German physicist, Emil Johann Wiechert, worked on the same thing and found the very same equations just two years later.
These are the equations: Now, you may wonder why I am mentioning them, and you may also wonder how we get those integrals above, i. Frankly, I would like to give you the same answer as above, i. So why did I write this post? Well… I am not sure. I guess I just wanted to sum things up for myself, so I can print it all out and put it on the kitchen door indeed.
Is it E or B or what? I hope you enjoyed it. Note that it talks about the electric field only, as the magnetic field is so tiny and, in any case, if we have E then we can find B. So the formula is: The geometry of the situation is depicted below. We have some charge q that, we assume, is moving through space, and so it creates some field E at point P.
Well… It points to where the charge was at the time just a little while ago, i. It might be anywhere. But look at the terms in the equation. And so we have the second term, which sort of compensates for that.
It suggests that we should calculate the delayed Coulomb field but add a correction to it, which is its rate of change times the time delay that we use. Nature seems to be attempting to guess what the field at the present time is going to be, by taking the rate of change and multiplying by the time that is delayed. It does, but in a fairly complicated way.
Click here for PDF version updated I. This laid the basis for a kinematic consideration of the motion of charged particles and the resulting retarded potentials and fields. Their solution, which is the basis of the corresponding derivations in all textbooks about electrodynamics e. Jackson, Griffiths or Feynman , involves a spatial integration over the retarded density distribution i.
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Potenziale di Liénard-Wiechert
Liénard–Wiechert potential | Wikipedia audio article