  The effect of an external bias voltage and fluctuating electromagnetic fields
on both the fundamental frequency and damping of cantilever vibrations is
considered. An external voltage induces surface charges causing
cantilever-sample electrostatic attraction. A similar effect arises from
charged defects in dielectrics that cause spatial fluctuations of electrostatic
fields. The cantilever motion results in charge displacements giving rise to
Joule losses and damping. It is shown that the dissipation increases with
decreasing conductivity and thickness of the substrate, a result that is
potentially useful for sample diagnostics. Fluctuating electromagnetic fields
between the two surfaces also induce attractive (Casimir) forces. It is shown
that the shift in the cantilever fundamental frequency due to the Casimir force
is close to the shift observed in recent experiments of Stipe et al. Both the
electrostatic and Casimir forces have a strong effect on the cantilever
eigenfrequencies, and both effects depend on the geometry of the cantilever
tip. We consider cylindrical, spherical, and ellipsoidal tips moving parallel
to a flat sample surface. The dependence of the cantilever effective mass and
vibrational frequencies on the geometry of the tip is studied both numerically
and analytically.
