Derive an expression for the casimir force between a plane and a sphere of radius r, separated by some distance a, in the limit a r. to do this make use of the proximity theorem. wherein you treat the two surfaces as nearly flat and assume the force from each region is given by the casimir pressure, p, between two planes (2.182), or in other words: df p (d(x)) da you should expand the separation between the surfaces, d(x), to second order. (these approximations actually produce extremely accurate results for the casimir force because the distance dependence is so steep.) also derive the casimir force for two spheres of radius r, again separated by a