  The electromagnetic Casimir interaction between two spheres is studied within
the scattering approach using the plane-wave basis. It is demonstrated that the
proximity force approximation (PFA) corresponds to the specular-reflection
limit of Mie scattering. Using the leading-order semiclassical WKB
approximation for the direct reflection term in the Debye expansion for the
scattering amplitudes, we prove that PFA provides the correct leading-order
divergence for arbitrary materials and temperatures in the sphere-sphere and
the plane-sphere geometry. Our derivation implies that only a small section
around the points of closest approach between the interacting spherical
surfaces contributes in the PFA regime. The corresponding characteristic length
scale is estimated from the width of the Gaussian integrand obtained within the
saddle-point approximation. At low temperatures, the area relevant for the
thermal corrections is much larger than the area contributing to the
zero-temperature result.
