One of the oldest unresolved problems in physics is the mechanism of charge exchange between contacting surfaces when at least one of them is insulating. We describe a new technique, using force microscopy, for studying this problem with greater lateral resolution than has been previously possible. The force microscope is shown to have 0.2 \ensuremath{\mu}m lateral resolution and the sensitivity to detect 3 electronic charges. In contact-charging experiments between the microscope tip and polymethyl methacrylate, the charged region was much larger than the expected contact area and bipolar charge exchange was observed.