The scalar potential and electric field distributions in the gap region of a probe-substrate system are calculated. The probe, modeled as a dielectric medium with the geometry of a one sheeted hyperboloid of revolution, is located above a charged substrate surface which is modeled as a dielectric half space interfaced with a uniform surface charge density. The potential and field distributions are then calculated as functions of the dielectric constant of the medium filling the space between the tip and the surface, and as functions of the hyperboloidal shape parameter. Comparisons are made with the case of a dielectric spheroidal body. The analytical results attained can be used to study other related quantities such as energy density or Coulomb interaction in the neighborhood of the nanometer sized apex region without resorting to numerical methods. This investigation allows for tip shape related field variation in various dielectric media to be studied. Application of this approach to modeling probe tip-sample (probe tip-substrate) interaction in scanning probe microscopy, or to modeling dielectric breakdown processes, are examples of the potential use of the method.