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So some profound questions remain:- Can we take the low-frequency limit of the radiation field and recover the Coulomb field?
The short answers are (a) no, (b) no, and (c) yes.>The only way to explain the two behaviors is to realize that there are two different contributions to the field, one of which is > dominant at short distances and long times, while the other is dominant at long distances and high frequencies.I think, question 2. (that is, (b) for consistency with labeled answers) here is ill-posed. One cannot wiggle the Coulomb field. We can only wiggle the source producing field, in the given case - the point charge. So, question 2./(b) should be formulated as:"(b) Can we wiggle a point charge and thereby produce radiation field?" And my answer to this would be (b) Yes. The generated radiation field would be a special case of what we call "Dipole radiation".