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The electric field of a uniformly moving charge is cylindrically symmetric around an axis parallel to its velocity vector. It varies inversely to the square of the distance.

The electric field of a linear electric quadrupole moment is cylindrically symmetric around the axis parallel to the separation vector between the two electric dipoles which make it up. It varies inversely to the fourth power of the distance.

For this and other reasons, I have wondered if a uniformly moving charge in effect produces an electric quadrupole moment density at each point in surrounding space whose density varies inversely to the distance, such that the total quadrupole moment contained in a sphere centered on the charge varies proportionally to the square of that sphere's radius.

For instance, consider the following quote from the article "Experimental verification of quadrupole model of the electric field of a rotating magnet"

Magnetic field of an axially symmetric magnetic field source is stationary in the lab frame when it rotates about the axis parallel to its magnetic moment. The electric field produced by such a rotation will be potential.

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Current theoretical models of the potential electric field of a rotating magnet contradict Eq. (1) [10–12]. They all give the quadrupole electric field, when there are the density of electric charge of one sign on the magnet poles and one of the opposite sign — at the equator. Herewith, the total electric charge of an isolated rotating magnet is always equal to zero [12]. This fact gives a possibility to make another experiment on verification of the quadrupole model of the electric field of a rotating magnet. The electric field outside a conducting shield must be zero for the quadrupole if we enclose the magnet in the shield and bring the magnet into rotation. We made the first such an experiment in 2001 [13]. The experiment showed that the electric field of the rotating insulated magnet, enclosed in an insulated conductive shield, was non-zero, and its magnitude was about 1/3 of the field without the shield. It was found that the electric field is not detected when the shield is connected to earth. The experiment was a qualitative, and experimental error was not evaluated. It is of interest to repeat the experiment [13] and to carry out measurements to verify the quadrupole model.

Since the magnet descibed as according to

Magnetic field of an axially symmetric magnetic field source is stationary in the lab frame when it rotates about the axis parallel to its magnetic moment. The electric field produced by such a rotation will be potential.

could be modeled as the superposition of moving charges, and since such a magnet in motion could possess an electric quadrupole moment, it seems reasonable that the electric quadrupole moment density of the magnet in motion could be in part contributed by the linear combination of the induced electric quadrupole moment densities produced by each individual moving charge used to model the magnet. If that is the case, then that might help explain why the experimenters behind the above quoted article observed that the rotating magnet produced an electric field that was able to penetrate a conductive shield, as the electric quadrupole moment densities that these moving charges produced outside the shield should be able to produce electric fields which the shield is unable to screen against.

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The basic story here is that multipole moments don't have well-defined transformation properties. For example, the field of a boosted dipole is time-dependent, so it becomes ambiguous to assign it multipole moments. For this reason, there's an ambiguity in how to define the transformation properties of the dipole moment. This is the basic issue behind Mansuripur's paradox about the Lorentz force law.

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