# Method for Measuring Surface Charge on Insulating Materials Based on the Vibrating Capacitor Method
## 4 Conclusions  
This research successfully addressed the challenge of measuring surface charge density in insulating materials using Comsol simulation methods. By analyzing parameters such as charged area size, probe height, and position, and utilizing the spatial distribution of potential measurement values of surface point charges, an integral relationship closely associated with the distribution of surface charge density was established. Validation against numerical simulation results from four common charge distribution scenarios confirmed the feasibility of using this integral relationship calculation method to replace complex electrostatic field simulation processes for determining the relationship between surface charge density and measured voltage. Additionally, this method provides a reliable approach for measuring surface charge density in insulating materials. These findings contribute to a deeper understanding of contact electrification phenomena and offer practical applications in fields such as material science, electronics, and energy storage.  
The vibrating capacitor method, as a non-contact and high-precision technique for measuring electric fields, exhibits promising prospects in various domains including electronic devices and material analysis. By accurately exploring the actual surface charge distribution characteristics of materials, a better understanding of material properties and device design can be achieved, thereby providing new avenues for technological advancement. Althoughsignificant progress has been made in this research, there are still limitations. Due to constraints in experimental conditions, it is currently challenging to accurately determine the charge density on a material surface. Future studies will explore the utilization of KPFM experimental equipment for microscopic-scale experiments to refine the theoretical framework. Additionally, further optimization of the vibrating capacitor method's model and methodology is essential. Practical shielding structures will be integrated to enhance model realism and accuracy. Validation through comparison with alternative methods will be pursued. Furthermore, an in-depth investigation into the effects of various parameters on electric field distribution is warranted to broaden application areas. Refinement of the integral formula is necessary to precisely establish the relationship between surface charge density and measured potential, facilitating seamless conversion. By implementing the vibrating capacitor method in practical scenarios, viable solutions to real-world problems can be offered.  
Conceptualization, J.F. and X.X.; methodology, J.F. and X.X.; software, J.F. and X.X.; validation, J.F. and X.X.; formal analysis, J.F. and X.X.; investigation, J.F. and X.X. resources, J.F. and X.X.; data curation, J.F. and X.X.; writing--original draft preparation, J.F. and X.X.; writing--review and editing, J.F. and X.X.; visualization, J.F. and X.X.; supervision, J.F. and X.X.; project administration, J.F. and X.X.; funding acquisition, J.F. and X.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (No. 52375166).  
Not applicable.  
The data presented in this study are available in this article.  
The authors declare no conflicts of interest.