The exact solutions for fully three‐dimensional electrostatic quantities due to a free charge in a nanoscale tip/base junction, such as the scanning‐tunneling microscope, are given for the problem as modeled in the prolate‐spheroidal coordinate system. These exact solutions consist of summation series of integrals of conical functions, and methods for calculating these exact solutions with high accuracy are described in detail in this paper. Calculated results for the surface‐charge distribution on the tip and the base due to a free electron in a nanoscale tip/base junction are also presented. This analysis has important implications for the ultimate objective of quantifying electron tunneling, from first principles, in nonplanar geometries such as sharp vacuum field emitters. ©1995 American Institute of Physics.