  We present a detailed experimental study on the electrostatic interaction
between a quantum dot and the metallic tip of a scanning force microscope. Our
method allowed us to quantitatively map the tip-induced potential and to
determine the spatial dependence of the tip's lever arm with high resolution.
We find that two parts of the tip-induced potential can be distinguished, one
that depends on the voltage applied to the tip and one that is independent of
this voltage. The first part is due to the metallic tip while we interpret the
second part as the effect of a charged dielectric particle on the tip. In the
measurements of the lever arm we find fine structure that depends on which
quantum state we study. The results are discussed in view of scanning gate
experiments where the tip is used as a movable gate to study nanostructures.
