# Method for Measuring Surface Charge on Insulating Materials Based on the Vibrating Capacitor Method
## 3 Results and Discussion
### Relationship between Measured Potential and Charge Density Distribution  
During the process of measuring surface potential using the vibrating capacitor method, the measured potential may be influenced by surface charges outside the measurement position. In order to establish the relationship between measured potential and surface charge density, simulations are first conducted to calculate the measured potentials at different positions under a point charge density distribution. Considering the influence of probe height on the surface charge of dielectric materials, the distance between the probe and the material surface is strictly maintained at 2 mm. Surface charges at different distances will result in different measurement values of potential. Therefore, in the three-dimensional model, a circular point charge area with a diameter of 1 mm is established at the center of the surface of the dielectric material. The probe is gradually moved horizontally from directly above the initial point charge area, with each movement being 2 mm. The results are shown in Figure 6.  
Figure 5: Relationship between measured potential and probe distance: (**a**) overall charged surface of the material; (**b**) locally charged. nation of these diverse scenarios, our objective is to evaluate the feasibility of employing integration calculations as an alternative to simulation methods. For the point charge region within the circular ring friction area, one of the 12 regions in Figure 1b is selected to set the surface charge density to \(\sigma_{ex}\), while the remaining regions are set to 0 C/m\({}^{2}\). The probe initially moves away from the charged region to other regions. The results are shown in Figure 7a: The x-coordinate of the probe section represents the position of the probe. As shown in Figure 1b, when the probe is directly above the charged region, the x-value is 1. When the probe is positioned above an adjacent region next to the charged region, the x-value is 2, and so on. Since only one region is charged, and the entire model exhibits symmetry, it is only necessary to measure half of the circular ring area; namely, probe section 1 to 7. It can be seen that as the probe moves away from the charged region, the measured potential rapidly decreases and gradually approaches 0 V. Moreover, the simulation results of the model are close to the results of the integration calculation. In tribological experiments, after repeated cyclic friction in circular regions, measurements of surface charge along the friction trajectory exhibit characteristics reminiscent of sinusoidal signals. When the charge distribution characteristic is similar to the friction experiment, surface charge densities with sinusoidal distribution characteristics are sequentially set in the 12 regions in Figure 1b. The probe measures each charged region in sequence. The results, as shown in Figure 7b, demonstrate that the simulation model results are very close to the numerical results obtained from the integrated calculations. For the case of charges randomly distributed on the surface of the dielectric material, 12 random numbers representing the surface charge density in each subregion of the circular ring area are generated. The corresponding measured potentials are solved using both simulation methods and formula integration calculations, as shown in Figure 7c. The gray line represents the randomly generated surface charge density values below the probe area, while the red and blue lines represent the simulated measured potential values obtained  
Figure 7: Relationship comparison between simulation models and integral calculation results: (**a**) region with point charge distribution along circular trajectory; (**b**) region with sine feature charge distribution along circular trajectory; (**c**) region with random feature charge distribution along circular trajectory; (**d**) region on the surface of a rectangle with random feature charge distribution. using the Comsol model and the calculated values obtained using the formula, respectively.