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 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 The integral in (4) can be calculated by using the following substitution of (3): where Substituting Eq. (5) into Eq. (4) and performing integration, one obtains 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 11 F el $_{nor}$( a, ϕ ) = − 2 πϵ$_{0}$ ( V − V$_{0}$ ) 2 [ R 2 a 1 √ 1 − β 2 + c$_{0}$ + c$_{1}$ a R + c$_{2}$ a $^{2}$(2 + β $^{2}$) 2 R 2 + c$_{3}$ a $^{3}$(2 + 3 β $^{2}$) 2 R 3 + c$_{4}$ a $^{4}$(8 + 24 β 2 + 3 β $^{4}$) 8 R 4 + c$_{5}$ a $^{5}$(8 + 40 β 2 + 15 β $^{4}$) 8 R 5 + c$_{6}$ a $^{6}$(16 + 120 β 2 + 90 β 4 + 5 β $^{6}$) 16 R 6 ] . (7) β ≡ β ( a, ϕ ) = 1 a ( A 2 1 + A 2 − 2 A$_{1}$A$_{2}$ cos ϕ ) 1 / $^{2}$, tan α = ( A$_{2}$ cos ϕ − A$_{1}$ ) / ( A$_{1}$ sin ϕ ) . (6) z$_{s}$ − z$_{p}$ = a [1 + β cos(2 πx/ Λ − α )] , (5) F el $_{nor}$( a, ϕ ) = − 2 πϵ$_{0}$ ( V − V$_{0}$ ) 2 1 Λ ∫ Λ 0 dx Φ( z$_{s}$ − z$_{p}$ ) . (4) z$_{s}$ − z$_{p}$ = a + A$_{2}$ sin(2 πx/ Λ + ϕ ) − A$_{1}$ sin(2 πx/ Λ) . (3)