Electrostatic force density for a scanned probe above a charged surface

The Coulomb interaction of a dielectric probe tip with a uniform field existing above a semi-infinite, homogeneous dielectric substrate is studied. The 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 force density on the hyperboloidal probe medium is calculated as a function of the probe tip shape. The calculation is based on solving Laplace's equation and employing a newly derived integral expansion for the vanishing dielectric limit of the potential. The involved numerical simulations comprise the evaluation of infinite double integrals involving conical functions.