content_image/362825.jpg content_image/362826.jpg
Here, the parameter β ≡ β ( a, ϕ ) is defined in Eq. (6) and the coefficients Y$_{i}$ ≡ Y$_{i}$ ( β $^{2}$) are given by 

<img src='content_image/362825.jpg'>

By fitting the force data in Eq. (14) to Eq. (19), the quantities k$_{ben}$ and z$_{0}$ were found. 

This was repeated for four different electrostatic force measurements and the average values obtained are 

<img src='content_image/362826.jpg'>

Using an independent quad-cell AFM measurement of the torsional movement of such a cantilever, the torsional spring constant was found to be K$_{tor}$ = 0 . 28 ± 0 . 5 N/m. The value of the bending deflection coefficient m$_{ben}$ = 209 ± 3 nm/unit S was obtained in the same way as in normal Casimir force measurements [42, 49] from the change in position of the contact point for application of different voltages to the plate during the normal electrostatic force measurement. Then the value of the bending spring constant K$_{ben}$ = (6 . 1 ± 0 . 3) × 10 − $^{3}$N/m was found from Eqs. (11) and (21). By comparing the two spring constants, we find that K$_{tor}$/K$_{ben}$ ≈ 46 which means that our system is really much more sensitive to the lateral rather than to the normal Casimir force. 

In recent literature [50] there is a discussion that in several measurements of the Casimir force the residual potential difference V$_{0}$ depends on separation, whereas in some other it is constant. Because of this, we have performed an additional control measurement of V$_{0}$ at short separation a = 127 . 3 nm using the parabolic dependence of F el lat in Eq. (19) on V . For this purpose five different voltages close to the residual potential difference were applied to the grating and the cantilever deflection was measured as a function of V leading to V$_{0}$ = − 39 . 4 mV. This is consistent with the result (10) obtained at large separation a = 1 µ m and confirms that in our experiment the residual potential difference is separation independent. 

The same calibration procedures, as described above for the first set of measurements, were repeated for a second set with a larger corrugation amplitude of 25.5 nm on the sphere. 

Here we obtained a residual potential difference ˜ V$_{0}$ = − 28 . 35 ± 1 . 15 mV and the following 

