## 3 Surface Charge Distribution Modeling  
Surface charge distribution modeling of the SSEP begins with spherical scattering electrical field analysis. The model of the electrical field is simplified, as shown in Figure 2. The shaft is not shown in Figure 2 because of its negligible effect, due to its delicately designed multi-coaxial-layer active shielding and grounding structure.  
The electrical field analysis problem here can be taken as an electrostatic boundary value problem with a Dirichlet boundary condition, in which both the surface being measured and the far-field boundary are grounded, and the electrical potential of the probing ball is known. The electrical potential distribution \(U\) in the field can be expressed with [PERSON]'s equation, as shown in Equation (1) below.  
\[\
abla^{2}U=0, \tag{1}\]  
[PERSON]'s equation is expressed in a spherical coordinate as shown in Equation (2) in order to facilitate the calculation of the surface charge distribution on the probing ball.  
\[\
abla^{2}U=\frac{1}{r^{2}}\frac{\partial}{\partial r}\bigg{(}r^{2}\frac{ \partial U}{\partial r}\bigg{)}+\frac{1}{r^{2}\sin\theta}\frac{\partial}{ \partial\theta}\bigg{(}\sin\theta\frac{\partial}{\partial\theta}\bigg{)}+ \frac{1}{r^{2}\sin^{2}\theta}\frac{\partial^{2}U}{\partial\phi^{2}}=0, \tag{2}\]  
To solve [PERSON]'s equation, a 3D seven-point finite difference scheme is proposed as shown in Equation (3), and the spatial relationship of the seven points is shown in Figure 3. This finite difference  
Figure 1: Working principle of spherical scattering electrical field probe (SSEP).  
Figure 2: Model for analysis.  
scheme is derived from the spherical coordinates with the origin located at the center of the probing ball. Thus, the electrical potential distribution \(U\) can be obtained with proper gridding and iteration.  
\[U_{0}i^{+1}=\left\{\begin{array}{c}\frac{r^{2}\sin^{2}\theta_{h _{3}}h_{4}h_{5}h_{6}(r_{+}h_{2}-h_{1})+\sin^{2}\theta_{h_{1}}h_{2}h_{3}h_{4}h_{ 2}\left[+\left(h_{6}-h_{5}\right)\cot\theta\right]+2h_{1}h_{2}h_{3}h_{6}}{ \cdot\left[\begin{array}{c}\frac{2(r+h_{2})}{r_{h_{1}}(h_{1}+h_{2})}U_{1}^{ \,i}+\frac{2(r-h_{1})}{r_{h_{2}}(h_{1}+h_{2})}U_{2}^{\,i}+\frac{2}{r^{2}\sin^{2 }\theta_{h_{3}}(h_{3}+h_{4})}U_{3}^{\,i}\\ +\frac{2}{r^{2}\sin^{2}\theta h_{4}(h_{3}+h_{4})}U_{4}^{\,i}+\frac{2+h_{6}\cot \theta}{r^{2}h_{5}(h_{5}+h_{6})}U_{5}^{\,i}+\frac{2-h_{6}\cot\theta}{r^{2}h_{6 }(h_{5}+h_{6})}U_{6}^{\,i}\\ \frac{h_{4}}{h_{3}+h_{4}}U_{3}^{\,i}+\frac{h_{3}}{h_{3}+h_{4}}U_{4}^{\,i} \end{array}\right]_{\theta=0\text{ or }\theta=\pi} \tag{3}\]  
where \(I=1,\,2,\,3,\,\dots\).  
Considering the fact that the spherical scattering electrical field is extremely non-uniform, the grids in this method are generated non-uniformly from the spherical coordinates. As shown in Figure 4, fine grids are generated near the probing point while coarse grids are kept away from it. In addition, quick convergence is achieved by combining the over-relaxation iterative method and the non-uniform gridding in this study.  
Electric field intensity distribution \(\overset{\rightarrow}{E}\) can be obtained using Equation (4):  
\[\overset{\rightarrow}{E}=-\
abla U \tag{4}\]  
According to [PERSON]'s law, charge density distribution \(\sigma\) on the surface of the probing ball can be expressed as Equation (5) since the probing ball is a conductive sphere,  
\[\sigma=\varepsilon E, \tag{5}\]  
where \(\varepsilon\) is the dielectric constant.  
The distribution \(\sigma\) of the spherical surface charge on the probing ball can thus be obtained.  
Figure 4: Schematic diagram of nonuniform gridding in spherical coordinates.  
Figure 3: The spatial relationship of 7 points.