## 2 Models and Analyses
### Static Models
For the first stage, the total vertical force received by the particles is:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]0  
The force of the particle in the \(x\) direction is:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]1  
Then the particle velocity in the \(x\) direction \(u_{x}\) can be obtained according to [PERSON]'s second law:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]2  
Similarly, the speed in the \(y\) direction \(u_{y}\) can be obtained as:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]3  
According to the definition of displacement:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]4  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]5  
Assuming that the height of the surface of the workpiece is \(h_{0}\), when the particles reach the surface of the workpiece after time \(t_{0}\), the displacement of \(s_{x1}\) occurs, and the velocity in the vertical direction becomes 0 through elastic collision, and the velocity in the horizontal direction is \(u_{x0}\). Then, to discuss the limit traction distance of the particles, the friction force of the particle must be greater than the magnetic traction force, the particle starts to make variable deceleration motions, until the horizontal direction \(u_{x}\) decreases to 0, the displacement is \(s_{x2}\). Marking the horizontal coordinate as \(x_{2}\), and the particle is in a critical state. If the maximum static friction force is greater than the current magnetic traction force, the particles will remain stationary at a speed of 0, otherwise the particles will start to accelerate again until they reach the defect. The limit state of the second stage, can be obtained according to the force balance equation of [PERSON]'s third law when the particle is at rest:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]6  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]7By reducing the volume term on both sides of the equation, we can get:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]8  
Under the condition of fixed defect size and magnetic charge, \(\mathrm{H}(x_{2})\
abla\mathrm{H}(x_{2})\) can be considered as a function on \(x\), and can be simply written as \(H^{\prime}(x)\), and we get:  
\[H(x)=\frac{2.65\mu_{0}H_{w}}{4\pi^{2}}\Bigg{[}\arctan\frac{dc(x+w_{0})}{(x+w_{ 0})^{2}}-\arctan\frac{dc(x-w_{0})}{(x-w_{0})^{2}}\Bigg{]} \tag{2}\]9  
That is, as long as the static friction coefficient \(K_{1}\) is determined, the value of \(x_{2}\) can be obtained. The adsorption limit distance of small particles is \(L_{lim}=x_{2}+s_{x1}+s_{x2}\).