We show that by virtue of the Rontgen interaction, a vortex of order n in a
Bose-Einstein condensate (BEC), in which the constituent atoms are charact
erized by an electric dipole, gives rise to a magnetic monopole distributio
n, which correctly preserves magnetic charge neutrality. If, on the other h
and, the BEC atoms are characterized by a magnetic dipole, we show that the
Aharonov-Casher interaction term leads to the prediction of an electric ch
arge distribution associated with the vortex state, which too, exhibits ele
ctric charge neutrality. Our theory displays an exact symmetry between the
electric and magnetic aspects of the problem and can thus be presented in a
unified fashion. Illustrations are given for finite-size atomic gas BECs,
superfluid helium, and spin-polarized hydrogen BECs in the n = 1 vortex sta
te.