The Coulomb interaction of a dielectric probe tip with a uniform field exis
ting above a semi-infinite, homogeneous dielectric substrate is studied. Th
e induced polarization surface charge density and the field distribution at
the bounding surface of the dielectric medium with the geometry of half of
a two sheeted hyperboloid of revolution located above the dielectric half
space interfaced with a uniform surface charge density is calculated. The f
orce density on the hyperboloidal probe medium is calculated as a function
of the probe tip shape. The calculation is based on solving Laplace's equat
ion and employing a newly derived integral expansion for the vanishing diel
ectric limit of the potential. The involved numerical simulations comprise
the evaluation of infinite double integrals involving conical functions. (C
) 2001 American Institute of Physics.