## 3 Experiment and Analysis of Carbon Particle Adhesion Under Different Electric Field Strengths
The force situation of the particles during the movement under experimental conditions is shown in Figure 10.  
Figure 10: Schematic diagram of particle forces under different motion states. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Force Type** & **Variable Symbol** & **Source of Force** \\ \hline Electric field force & \(F_{\text{E}}\) & Particle charge and electric field \\ Dielectrophoresis force & \(F_{\text{Dep}}\) & Particle polarization and electric field gradient \\ Thermophoresis force & \(F_{Th}\) & Temperature gradient of fluid \\ Gravity and buoyancy & \(F_{G}\) & Inherent properties of particles \\ Drag force & \(F_{D}\) & Relative motion between particles and fluids \\ Additional quality force & \(F_{Aq}\) & Relative acceleration between particles and fluids \\ Basset force & \(F_{B}\) & Relative acceleration between particles and fluids \\ Magnus force & \(F_{M}\) & Particle rotation and relative motion \\ Saffman force & \(F_{S}\) & Velocity gradient in fluid \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The force acting on particles in fluids. Assuming a particle carries a positive charge under the action of an electric field, the particle will move downward under the combined action of the electric field and gravity, and will experience an oil drag force opposite to its direction of motion, as shown in Figure 10a. When a particle collides with the grounded epoxy resin board, its movement speed decreases to 0, and it is no longer subjected to oil drag force. Instead, the particle is subjected to adhesion force directed towards the surface of the epoxy resin, as well as vertical, upward elastic force that acts as a balance. At this point, the particle adheres to the surface of the epoxy resin, as shown in Figure 10b. As the contact time between the particle and the surface of the epoxy resin increases, the positive charge carried by the particle will gradually be offset by the negative charge due to the conductive charging effect, until the polarity reversal occurs and the particle becomes negatively charged as a whole. At this time, the electric field force acting on the particle becomes vertically upward, but still adheres to the surface of the epoxy resin, as shown in Figure 10c. As the negative charge carried by the particle increases, the electric field force it experiences gradually increases. If the electric field force increases enough to overcome the effects of gravity and adhesion force, the particle adhering to the surface of the epoxy resin will be pulled apart, return to the transformer oil, and move towards the other electrode until collision occurs again, as shown in Figure 10d. The gravity and buoyancy acting on particles are both in the vertical direction and can usually be considered together. Carbon particles are considered as spherical particles, and the combined force of gravity and buoyancy \(F_{G}\) can be calculated by Equation (4). \[F_{G}=\frac{4}{3}\pi R^{3}(\rho_{p}-\rho_{oil})g \tag{4}\]  
Among them, \(R\) is the particle radius, \(\rho_{p}\) and \(\rho_{oil}\) are the densities of particles and transformer oil, respectively, and \(g\) is the acceleration due to gravity. Due to the higher density of carbon particles compared to transformer oil, the direction of \(F_{G}\) is vertically downward. The electric field force \(F_{E}\) experienced by particles after being charged is shown in Equation (5). and its direction is determined by the direction of the electric field and the polarity of the charge carried by the particles. \[F_{E}=\lambda_{E}QE \tag{5}\]  
Among them, \(\lambda_{E}\) is the electric field force correction coefficient introduced considering the mirror charge caused by the electrode. When the particle comes into contact with or approaches the electrode, \(\lambda_{E}=0.832\). When the electric field force causes the particles to move away from the electrode surface and the distance is greater than five times the particle radius, \(\lambda_{E}=1\).