## 6 Conclusions  
This analysis confirms that neglecting the dependence on the electric field of the dielectric constant \(\varepsilon_{\parallel}\) in the analysis of the capacitance curves at high voltage may be dangerous and lead to wrong values of the extrapolation length [27]. This effect is clearly at the origin of the breakdown of the analysis of the capacitance curves with the linear law given in Equation (6) which gives values of \(L\) strongly dependent on the voltage range used to fit the data when \(\varepsilon_{\parallel}\) is taken as a constant. I notice that the behavior observed in my analysis, namely a value \(L\) that systematically decreases when the fits are performed at large voltages, is the same as the one reported by [PERSON] et al. [5] by using the YvS or the RV techniques. This result suggests that the problems faced by these authors are due to the variation with the electric field of the dielectric constant \(\varepsilon_{\parallel}\) rather than to anchoring inhomogeneities in the samples. This effect should be particularly important in LC such as 8 CB which is composed of molecules with a strong electric dipole moment. To measure the extrapolation length, I thus propose a new procedure that is more general than the one proposed by [PERSON] et al. [27] and perhaps easier to implement. This procedure consists of solving directly the full Equations (3) and (4) governing the problem, which is much more rigorous than using the approximate Equation (5) as these authors do. Indeed, this equation assumes that the anchoring angle \(\theta\) is small, typically less than 0.2 rad, which is a strong limitation at large field (I also mention here that Formula (2) in [27] is wrong and must be read \(B=\sqrt{\frac{(1-y_{p})(1+xy_{p})}{1+\gamma y_{p}}}\)). Another advantage of my method is that it simultaneously gives the values of the two dielectric constants and of the elastic constants \(K_{1}\) and \(K_{3}\). My method can also be used to test the shape of the anchoring potential, as it does not assume that the anchoring potential is of the Rapini-Papoular form. This method is also easy to implement because the Equations (3) and (4) are easily solved numerically, for instance with Mathematica. Last but not least, this method can also be generalized to the case of samples with a pretilt angle, as the ones obtained by using a rubbed polymer. In that case, it is sufficient to replace \(\theta\) by \(\theta-\theta_{a}\) in the l.h.s. of Equation (5) where \(\theta_{a}\) is the pretilt angle. Note also that the measurement of \(V_{c}\) must be performed with a thick _parallel_ planar sample (the reason for this is explained in [24]). This is important because badly measuring \(V_{c}\) changes the slope of the experimental curves \(\overline{C}(\overline{V})\) and thus the value of \(\kappa\). By contrast, the measurements of the extrapolation length must be performed with an _antiparallel_ sample in order for Equations (3) and (4) (or (9)) to apply. This method could also be applied to LC with a negative dielectric anisotropy to measure the polar anchoring energy under homeotropic anchoring. Figure 13: Comparison between the best Rapini–Papoular potential (blue dashed curve) and the best modified Rapini–Papoular potential (red solid curve). The two potentials are almost indistinguishable below \(\theta=0.4\) rad, meaning that in our experiment, the Rapini–Papoular potential can be used to fit the data. In the future, it would be interesting to perform experiments at larger field to better test the validity of the Rapini-Papoular form of the anchoring potential. It would also be interesting to compare the values of \(L\) and \(A\) obtained this way with those obtained, for instance, by measuring the critical voltage \(V_{\ell}^{\prime}\) in a wedge sample [47] and by using more sophisticated methods such as the one based on spectroscopic ellipsometry [48,49] or the all-optical compensated method of [PERSON] et al. [14]. This research received no external funding. Not applicable.