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varga said:Ok, but all I need is full expression of all the terms in any equation and calculator to see what result it gives with whatever arbitrary parameters, or I can use some actual experimental measurements and hence confirm the numbers and validity or accuracy of the equation. I do not need to know even any math to do that, just one example to see what is what. I can not solve this example myself because my argument is that it can not be done and that at best it will give the same result as what I get via Lorentz force equation.
So let's start there .. what do you get for the Lorentz force equation?
Thank you, but that would prove my point as that is not one of the four equations that we call "Maxwell's equations".
Ummm ... no. Maxwell's equations are *general* ... this is the solution for a specific case, which is directly relevant to your question. The E & B fields are *derived from* the charge density and current density *using* Maxwell's equations. The Lienard-Weichert potentials are the *inputs* to the Maxwell's equations to allow you to calculate the fields.
What error are we talking about anyway? The one due to limited speed of propagation of gravity and em field potentials? If so, then what that has to do with this example where there is only one charge? Even with two or more, how large would that error be with closely spaced charges if this effect is not even concern for interplanetary distances where gravity field of the Sun is supposed to "lag" eight *minutes* and yet we can not measure it? I again object to the involvement of special relativity, I'm talking about the velocity of ~700m/s, but nevertheless I'm very interested to compare actual results and see how large this error correction really is.
No, read it again .. there are two corrections, one for the relativistic effects to the Coulomb field, and the other for *emission of radiation* ... you know, that phenomenon that you didn't think existed for accelerated charges? Note how the Maxwell's equations derive the existence of such a phenomenon (radiating EM waves) from the basic, simple inputs, in a manner that is consistent with experiment. That is a nice example of why the rest of us do not find them "insufficient" or "superfluous".