# Method for Measuring Surface Charge on Insulating Materials Based on the Vibrating Capacitor Method
## 1 Introduction  
The phenomenon of surface charging induced by mutual contact and friction between materials is referred to as contact electrification or triboelectrification [1; 2; 3; 4; 5]. This process involves the transfer of charges, leading to the accumulation of charges of different polarities and densities on the surface of objects, resulting in electrostatic effects. Although static electricity plays a positive role in various fields such as electrostatic dust removal [6; 7; 8; 9], electrostatic spraying [10; 11; 12; 13], electrostatic flocking [14; 15; 16; 17], and electrostatic copying [18; 19], the accumulation discharge of charges and electrostatic attraction also pose significant hazards. For example, electrostatic discharge may cause electromagnetic interference and circuit breakdown in electronic devices [20; 21], high-voltage electrostatic discharge poses a threat to personal safety [22], and electrostatic discharge in dusty environments may lead to explosions and fires [23; 24; 25]. Additionally, dust adhesion caused by electrostatic attraction can significantly reduce the quality of industrial products [26]. Therefore, precise measurement of the surface charge density of objects becomes crucial for better understanding and controlling electrostatic effects, achieving safer, more efficient, and reliable applications. Currently, various methods exist for measuring the surface charge density of materials. [PERSON] et al. [27] utilized electrostatic induction to derive the relationship between the contact separation distance and the surface charge density by measuring the voltage of the circuit during the contact separation process. Furthermore, by measuring the voltage of the same liquid metal after the contact separation with the tested material, the contact electrification charge density of different materials for the same material can be inferred. The Faraday cup method records the change in current when the tested material enters the Faraday cup through electrostatic induction to calculate the quantity of charge, thereby obtaining the charge density [28; 29]. The aforementioned methods provide the average surface charge density of materials and cannot reveal the distribution characteristics of charge density on the surface. Electrostatic field strength meters can measure electric field strength, which can then be used to deduce the charge distribution [30]. This method does not require direct contact with the material, thus avoiding interference with the tested material. However, the spatial resolution of electrostatic field strength meters is often limited and cannot accurately calculate the surface charge density of materials. In 1898, [PERSON] invented a technique for measuring the contact potential difference of metals, known as the Kelvin method [31]. In 1932, [PERSON] optimized this method by designing circuits to make it more sophisticated and accurate [32; 33; 34; 35]. The Kelvin method is also known as the vibrating capacitor method. In this method, the contact potential difference between two metals is generated due to the different work functions of the metals [36; 37; 38]. A capacitor is formed between the probe and the tested metal, and by applying vertical vibration to the probe perpendicular to the test surface, the contact potential difference induces a weak current in the measuring circuit. Applying a certain bias voltage to the probe ensures no current flows in the measuring circuit. At this point, the probe-to-ground voltage equals the material's contact potential difference. Later, the vibrating capacitor method was used to measure the surface potential of dielectric materials. As the probe size decreased, the spatial resolution of surface potential measurement improved. Subsequently, the Kelvin method was combined with Atomic Force Microscopy (AFM) technology, gradually developing into Kelvin Probe Force Microscopy (KPFM) technology, achieving atomic-level spatial resolution in surface potential measurement.