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The approximate expression for the electric force in the case when the surfaces of a sphere and a plate are covered with sinusoidal corrugations can be obtained using the PFA, as applied to electric forces. For this purpose we consider the separation along the z -axis between any two points z$_{s}$ on the corrugated sphere and z$_{p}$ on the corrugated plate 

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Here a , as defined above, is the closest separation between a perfectly shaped sphere and a plate, and ϕ ≡ 2 πx$_{0}$/ Λ is the phase shift between the corrugations of both bodies. Then, according to the PFA, the normal electric force between corrugated a sphere and a plate is obtained from Eq. (1) by averaging over a period as 

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The integral in (4) can be calculated by using the following substitution of (3): 

where 

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Substituting Eq. (5) into Eq. (4) and performing integration, one obtains 

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It is well known that for corrugated surfaces the use of the PFA may not lead to pre- cise expressions for the Casimir force [23–25]. For a static electric force, however, the PFA should work much better because in this case the diffraction effects are absent. To verify the validity of Eq. (7), we applied a 3-dimensional finite element analysis (FEA) using Com- sol Multiphysics [46] to numerically solve the Poisson equation with appropriate boundary 

