  We develop a spectral representation formalism to calculate the Casimir force
in the non-retarded limit, between a spherical particle and a substrate, both
with arbitrary local dielectric properties. This spectral formalism allows one
to do a systematic study of the force as a function of the geometrical
variables separately from the dielectric properties. We found that the force
does not follow a simple power-law as a function of the separation between the
sphere and substrate. As a consequence, the non-retarded Casimir force is
enhanced by several orders of magnitude as the sphere approaches the substrate,
while at large separations the dipolar term dominates the force.
