We have calculated the plasma potential and dust potential throughout the border region of semiinfinite dust clouds embedded in a plasma. A relevant ordering parameter is P=(Ndo/N0)a&mgr; (interior dust density/exterior plasma density)¿dust radius (in microns). For comparatively high dust densities (P>10-3) the cloud potential is high, screening out electrons, with low interior dust charges as a result. For decreasing P the cloud potential becomes less, with a corresponding increase in the dust charges. At P<10-5 the cloud potential is practically zero and the dust particles have their maximum charge which is identical to that of a single dust particle in a plasma. For diffuse dust border regions with density scale length H≫&lgr;D (Debye length), the deviations from charge neutrality are small. When H≲&lgr;D, large deviations from charge neutrality occur in the border region and also some penetration of electric fields inside the border region.