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Although the electromagnetic field of a given distribution of charge and current is unambiguously defined by the Maxwell equations, there is no unique inverse correspondence between this field and point-like sources. We prove this statement for electric and magnetic dipoles and discuss two examples when the same field at some point or in some region is generated by either a dipole or an electric charge that are varying in different ways. The electric and magnetic fields of the charge depend on its speed and the distance between the charge and the observation point. It is shown that it is possible to find a law of motion for the charge in which the variation in the distance to the observer is compensated for by variation in the velocity of the charge. One particular example of the motion of a charge is discussed, wherein the electric and magnetic fields at a specific point of observation remain constant. The issues discussed could be of interest to both scientists and scholars.