  Quantum fluctuations of the electromagnetic field in the medium surrounding
two discharged macroscopic polarizable bodies induce a force between the two
bodies, the so called Casimir force. In the last two decades many experiments
have accurately measured this force, and significant efforts are made to
harness it in the actuation of micro and nano machines. The inherent many body
character of the Casimir force makes its computation very difficult in
non-planar geometries, like the standard experimental sphere-plate
configuration. Here we derive an approximate semi-analytic formula for the
sphere-plate Casimir force, which is both easy to compute numerically and very
accurate at all distances. By a comparison with the fully converged exact
scattering formula, we show that the error made by the approximate formula is
indeed much smaller than the uncertainty of present and foreseeable Casimir
experiments.
