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The three dimensional potential of a charged particle tunneling between a flat metal surface and a spherical metal tip is calculated within the framework of the hydrodynamic description of metallic electrons. It is demonstrated that the inclusion of coupling of surface modes in the two electrodes, even for separations as small as 10 times the screening length in either of them, contributes less than 5% of the total potential of a point charge. Hence the potential is obtained as a superposition of contributions from a planar surface and a charge neutral, conducting sphere (and can include a simple classical term for a sphere at a fixed potential). This should enable accurate determination of three dimensional tunnel currents in scanning tunneling microscope geometry.

Interaction Potential of a Charged Particle in a Plane?Sphere Geometry