Can someone refer me to an article or two that discusses this conversion, that is, electric field strength measured at a specified distance to RF watts input to the dipole and ground plane antenna?
Sure. One could derive the formula from scratch, of course, by assuming the RF in question is radiated from a fictional antenna that distributes power equally in all directions at once, up, down, or sideways (an "isotropic" antenna); then take into account the impedance of free space to translate field intensity into power, integrating that over a sphere having a radius equal to the measurement distance. Fortunately, it boils down to a relatively simple formula. The result is what's termed Effective Isotropic Radiated Power, or EIRP.
In apps.fcc.gov/eas/comments/GetPublishedDocument.html?id=204&tn=255011 we see the essential relationship:
eirp = (E×d)2/30
where eirp is in watts, E is field intensity in V/m and d is the measurement distance in meters. Now, our legal limit of 15,848 μV/m equals .015848 V/m, and the measurement distance is 30 meters, so we plug those into the equation and evaluate it as follows:
eirp = (.015848×30)2/30 = (0.47544)2/30 = 0.22604/30 = 0.007535 W
That's 7.535 mW EIRP, meaning 15,848 μV/m is the voltage you'd get at 30 m from an imaginary isotropic antenna if you put 7.535 mW into it. That's a little less than twice 4.8 mW, not orders of magnitude off, fortunately. But why even that much difference? Because you don't have an isotropic antenna.
Every real-world antenna radiates more power in some directions than it does others, which means that all real antennas (in free space conditions, at least) have gain over an isotropic antenna. An ideal dipole radiates nothing along its axis, but that leaves more signal to radiate perpendicularly than would be radiated by the fictional isotropic antenna, so a dipole is rated for an antenna gain of 2.15 dBi (per the FCC's OET; others give it as 2.19 dBi). That means you'd need to cut the power into a dipole by 2.15 dB, or a factor 1.64, to produce the same field strength in its maximum lobe as an isotropic antenna.
Now, 7.535 mW/1.64 = 4.59 mW, pretty close to our number. Why that little discrepancy? It depends on how tightly you round the numbers that comprise the EIRP formula, and the textbook value you use for dipole gain. It's entirely possible for the number to turn out anywhere from just above 4.5 to nearly 4.8 mW, depending on your references.
John
(P.S. - At the risk of complicating things further, there's also the case of a monopole antenna. A ground-mounted vertical over perfect ground only has half a sphere within which to radiate, compared to a full spherical region around a dipole, so you only have to input half the power [2.3 mW] for the same signal strength as the dipole. Hence, the theoretical gain of a monopole is 3 dB greater than that of a dipole, or 5.15 to 5.19 dB [depending on your source] above that of an isotropic antenna. Why a monopole complicates things is multi-fold, though. One big reason at these frequencies is that the real world ground losses mean you won't achieve the full antenna gain from a ground mounted monopole, but it's hard to predict by how much without actual measurements. In the case of a vertical over an elevated ground plane, such an antenna only simulates the monopole case near-in because the radiated wave begins spreading into the lower hemisphere as soon as it leaves the edge of the plane, which lowers the apparent gain depending on just where you measure it and how far off the ground the base is. Meanwhile, a ground plane with downward angled radials looks more like a vertically oriented dipole right from the start, and can be treated as such...if the transmission line is sufficiently decoupled from the antenna; whereas one whose radials stick out horizontally for a ways and then droop is harder to analyze.)