  A quite straightforward approximation for the electrostatic interaction
between two perfectly conducting surfaces suggests itself when the distance
between them is much smaller than the characteristic lengths associated to
their shapes. Indeed, in the so called "proximity force approximation" the
electrostatic force is evaluated by first dividing each surface into a set of
small flat patches, and then adding up the forces due two opposite pairs, the
contribution of which are approximated as due to pairs of parallel planes. This
approximation has been widely and successfully applied to different contexts,
ranging from nuclear physics to Casimir effect calculations. We present here an
improvement on this approximation, based on a derivative expansion for the
electrostatic energy contained between the surfaces. The results obtained could
be useful to discuss the geometric dependence of the electrostatic force, and
also as a convenient benchmark for numerical analyses of the tip-sample
electrostatic interaction in atomic force microscopes.
