Here's something fun to think about. Consider the contrast:
a) In electromagnetic radiation, the E-field of a point source
dies off like 1/r. The power and the energy density scale like
E squared, so they scale like 1/r^2, which is consistent with the
idea that energy is conserved as the wave spreads out.
b) In electrostatics, the E-field of a point source dies off
like 1/r^2. The energy density falls off like 1/r^4, but it isn’t
/transporting/ any energy, so conservation doesn’t have anything
to say about it.
The question arises, can we obtain a consistent view of these two
facts? This is not going to be easy, because starting with the
1/r^2 Coulomb field of a point charge, I don’t see any way to
explain the 1/r radiation field. By way of contrast, if there is
some sort of cancellation, I can arrange something that falls off
/faster/ than 1/r^2 – such as a dipole field that falls off like
1/r^3 – but I cannot cook up anything that falls of slower than
1/r^2. I’ve seen a number of books that claim to explain things
this way, but it never made any sense to me.
So some profound questions remain:
a) Can we take the low-frequency limit of the radiation field
and recover the Coulomb field?
b) Can we wiggle the Coulomb field and get the radiation field?
c) Or are they related in some other way?
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.
To make a long story short: Starting from the four-vector-ish
potential of a point charge, the Coulomb field results from taking
the spatial derivative of the potential, whereas the radiation
field results from taking a couple of time derivatives. For
the next level of detail on this, see: https://www.av8n.com/physics/lienard-wiechert.htm#sec-fields
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Tangential remark: When I refer to the four-vector-ish potential,
that "-ish" is a reminder that it's not really a four-vector,
although you can "almost" treat it as one. A lot of authors who
ought to know better are sloppy about this. For details, including
ways of making the problems go away, see: https://www.av8n.com/physics/lienard-wiechert.htm#sec-four-vector-ish-potential
Even more tangential remark: You may be in the habit of
converting my https URLs into http URLs. That should no
longer be necessary. I'm now using a security certificate
that should be trusted by your browser. If you encounter
any problems with this, please let me know off-list.