## 3 Experiment and Analysis of Carbon Particle Adhesion Under Different Electric Field Strengths
E is the electric field strength, Q is the charge of the particles, which conforms to the normal distribution law, and the calculation formula for the average charge \(\overline{Q}\) of the particles is shown in Equation (6) [23]. \[\overline{Q}=\frac{6244\pi^{3}\varepsilon_{0}\varepsilon_{m}}{3}R^{2}E^{0.4218} \tag{6}\]  
where \(\varepsilon_{0}\) and \(\varepsilon_{m}\) are the vacuum dielectric constant and the relative dielectric constant of the transformer oil, respectively. According to Equation (6), it can be seen that the particle charge is positively correlated with the electric field strength. [PERSON] and other scholars proposed the famous JKR particle contact model in 1971 based on the classical theory of particle elastic deformation [24], which has become a commonly used method for studying particle adhesion phenomena [12]. According to the JKR model, the adhesion force \(F_{n}\) exerted on carbon particles adhering to the surface of epoxy resin satisfies the relationship shown in Equation (7). \[F_{n}=N+3\pi\omega^{*}R+\sqrt{\left(3\pi\omega^{*}R\right)^{2}+6\pi\omega^{*}RN} \tag{7}\]  
Among them, \(N\) is the external load of the particles, which includes electric field force, gravity, and buoyancy in the experiment, and \(\omega^{*}\) is the adhesion work; its value is related to the material properties of particles and solid surfaces, measured in J/m\({}^{2}\)[11]. According to Equation (7), it can be seen that \(\left(3\pi\omega^{*}R\right)^{2}+6\pi\omega^{*}RN\geq 0\) must be satisfied, within which the particles meet the adhesion conditions. Specifically, when \(\left(3\pi\omega^{*}R\right)^{2}+6\pi\omega^{*}RN=0\), the particles are in a critical state of adhesion and detachment, and the corresponding critical adhesion conditions are shown in Equation (8). \[N_{cs}=F_{E}+F_{G}=-\frac{3\pi\omega^{*}R}{2} \tag{8}\]  
In the formula, \(N_{cs}\) is the critical external load of the particle. Its physical meaning is that when the external load is negative, that is, tends to pull apart the particle from the solid surface, the particle cannot be immediately pulled apart due to the adhesion force. Only when the particle charge increases and thus lead to an increase in the electric field force until the external load exceeds the critical load \(N_{cs}\) when the particles are pulled apart. At this time, the charge is called the critical adhesion charge \(Q_{cs}\) of the particle in the static transformer oil, as shown in Equation (9). \[Q_{cs}=\frac{9\pi\omega^{*}R+8\pi R^{3}\big{(}\rho_{p}-\rho_{oil}\big{)}g}{6 \lambda_{E}E} \tag{9}\]  
When \(Q>Q_{cs}\), the particles adhering to the solid surface are pulled apart by the electric field force. Therefore, the adhesion probability of a single particle in a single collision process is the cumulative distribution probability of particle charge in \(Q<Q_{cs}\). According to Equation (9), when the electric field strength \(E\) increases, \(Q_{cs}\) decreases and the particle charge is more likely to exceed \(Q_{cs}\). Therefore, the adhesion probability of particles in a single collision process decreases with the increase in electric field strength. However, the adhesion degree of carbon particles on the surface of epoxy resin is not only related to the adhesion probability of individual particles in a single collision process, but also to the collision frequency \(f\) between particles and solid surfaces, as shown in Equation (10). \[f=\frac{nv_{p}}{d} \tag{10}\]  
where \(v_{p}\) is the vertical velocity of particles in the transformer oil, \(n\) is the total number of particles in the transformer oil between the electrode plates, and \(d\) is the spacing between epoxy resin boards. Neglecting the acceleration process of adhesive particles in oil after being pulled apart by electric field force, the charge of particles in oil remains basically unchanged during their movement, and their electric field force and drag force will reach an equilibrium state, as shown in Equation (11).