Feynman's proof utilizes a geometrical and fundamental integration argument. I like it, except this bit:
What makes me unconfortable somehow is that in (c) we are counting in some of the charge we counted at (b). It seems to me that it is this extra counting which makes the potential to be larger than expected, and I am uncomfortable with it. To see why, consider the following situation with discrete charges:
Here, the yellow line represents the light cone (the observer being, of course at its apex), and the blue dots, the places where the observer "sees" each of the constituent charges. However, it is clear that if the charge cloud was small enough, or if we were far enough, the potential would be just the potential for a point charge of charge equal to the total charge of the cloud, as no charge is "overcounted" (something which is also due to the cloud's speed being less than c).
This is in disagreement with what Feynman derives and with the Liénard-Wiechert's potential of a moving charge. So why is my argument wrong? Is the continuity of the cloud, somehow crucial for the proof? If so why?
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