In a dipole, two opposite charges nearly cancel out their effects; the remaining force results from the difference of distance between each of the dipole's charges and the remote sensing charge or dipole.
- One charge creates on a remote charge a force varying like d-2
- A dipole on a remote charge, like d-3
- A dipole on a remote dipole, like d-4
The distance reduces the force more and more quickly because the compensation itself between two charges gets more perfect as the distance increases, since the relative separation of the charges gets smaller as compared with the distance of the sensing charge.
Though, whether it's like d
-3 or d
-4 results from a series expansion of (d+h/2)
-2 and (d-h/2)
-2, as Corribus already wrote. You may observe that a difference between f(d+h/2) and f(d-h/2) is a derivative when h is small, and the derivatives of d
-2 are d
-3 and d
-4; but is that more or less
understanding than the plain series expansion?
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More interesting: electric forces are so strong that a piece of matter is electrically neutral. Technology achieves charges ridiculously small as compared with the amount of electrons and protons. So most often, electric forces result from dipoles at best, rarely from monopoles.
Even true ions are exceedingly rare; in chemistry they exist in solutions, where + and - ions are linked by polar -+ molecules that bridge them like +-+-+-+-, telling that no charge is separated. The solvent molecules have less than one charge at each end, but every solute ion has many solvent molecules around it. In a solid too, each + ion is surrounded by - ions and they all touch an other, so that telling where the electrons are isn't trivial, and in fact the difference is tiny. Only this makes the ionic state possible.
Magnetic dipoles differ from electric ones in that no magnetic monopole (single charge) has ever been observed. Electromagnetism makes a fundamental difference here. Though, magnetic dipoles are more conveniently described and computed as if they consisted of two magnetic charges like an electric dipole.
The static field of a dipole is described as a "near-field" because it drops quickly with the distance. Take a small DC current loop: it may conduct current to the left at one side and create an induction like d
-2, but the current flows back to the right at the other side, compensating in part the left current, and the difference drops as d
-3.
Though, if the current changes quickly enough, the induction by the left and right currents arrive to the observer with different delays, and this is an additional cause for them not to compensate fully. This one needs a distance difference comparable with the wavelength, so an antenna must have such a size, and the difference provided by the propagation time doesn't decrease further with distance, so this "far-field" varies like d
-2.