We experimentally demonstrate that one-dimensional Bose-Einstein condensates brought close to microfabricated wires on an atom chip are a very sensitive sensor for magnetic and electric fields reaching a sensitivity to potential variations of ∼10−14eV at 3μm spatial resolution. We measure a two-dimensional magnetic field map 10μm above a 100-μm-wide wire and show how the transverse current-density component inside the wire can be reconstructed. The relation between the field sensitivity and the spatial resolution is discussed and further improvements utilizing Feshbach-resonances are outlined.