  There is a recurring question in solar physics about whether or not electric
currents are neutralized in active regions (ARs). This question was recently
revisited using three-dimensional (3D) magnetohydrodynamic (MHD) numerical
simulations of magnetic flux emergence into the solar atmosphere. Such
simulations showed that flux emergence can generate a substantial net current
in ARs. Another source of AR currents are photospheric horizontal flows. Our
aim is to determine the conditions for the occurrence of net vs. neutralized
currents with this second mechanism. Using 3D MHD simulations, we
systematically impose line-tied, quasi-static, photospheric twisting and
shearing motions to a bipolar potential magnetic field. We find that such
flows: (1) produce both {\it direct} and {\it return} currents, (2) induce very
weak compression currents - not observed in 2.5D - in the ambient field present
in the close vicinity of the current-carrying field, and (3) can generate
force-free magnetic fields with a net current. We demonstrate that neutralized
currents are in general produced only in the absence of magnetic shear at the
photospheric polarity inversion line - a special condition rarely observed. We
conclude that, as magnetic flux emergence, photospheric flows can build up net
currents in the solar atmosphere, in agreement with recent observations. These
results thus provide support for eruption models based on pre-eruption magnetic
fields possessing a net coronal current.
