In this work we measure the inversion charge density for short UTBB-FDSOI MOSFETs (down to 45 nm) using an accurate method for the extraction of the parasitic components which take into account the effect of the back bias. Based on these results, we propose a modification of the inversion charge density model that uses the simplified Lambert function in order to include the back bias effect. This model is then validated with the experimental data for different gate lengths (from 185 nm to 45 nm). It is shown that this inversion charge model is valid for the mobility extraction and gives very good assessment of the mobility if we know only the IV-characteristic, the threshold voltage (under back bias effect) and the inversion charge capacitance. Finally, this method is used to extract the mobility for short channel devices (down to 40 nm) and it is found that the short channel mobility suffers from degradation even for large back biases. For the shortest device, it is demonstrated that the main source of mobility degradation is caused by oxide charges and interface states located near the S/D extensions and then it is deduced that the mobility reduction for large back biases is related to neutral defects from Si-crystal close to the channel edge.