  We present results on the determination of the differential Casimir force
between an Au-coated sapphire sphere and the top and bottom of Au-coated deep
silicon trenches performed by means of the micromechanical torsional oscillator
in the range of separations from 0.2 to 8 $\upmu$m. The random and systematic
errors in the measured force signal are determined at the 95\% confidence level
and combined into the total experimental error. The role of surface roughness
and edge effects is investigated and shown to be negligibly small. The
distribution of patch potentials is characterized by Kelvin probe microscopy,
yielding an estimate of the typical size of patches, the respective r.m.s.
voltage and their impact on the measured force. A comparison between the
experimental results and theory is performed with no fitting parameters. For
this purpose, the Casimir force in the sphere-plate geometry is computed
independently on the basis of first principles of quantum electrodynamics using
the scattering theory and the gradient expansion. In doing so, the
frequency-dependent dielectric permittivity of Au is found from the optical
data extrapolated to zero frequency by means of the plasma and Drude models. It
is shown that the measurement results exclude the Drude model extrapolation
over the region of separations from 0.2 to 4.8~$\upmu$m, whereas the
alternative extrapolation by means of the plasma model is experimentally
consistent over the entire measurement range. A discussion of the obtained
results is provided.
