## 1 Introduction  
Nowadays, parts adopting miniature internal structures with high aspect ratios, such as deep, small holes and grooves [1, 2], can transmit working medium, energy, and information over relatively long distances, and are of great significance in achieving high integration and low energy consumption in the aviation, aerospace, and automotive industries. The machining precision of these structures often reaches the sub-micron level; moreover, their aspect ratio is up to several tens and even several hundreds [3, 4, 5], posing a great challenge to conventional measurement methods in terms of measurability. Various probing methods have been investigated to measure this particular kind of structure, such as miniaturized ball probes with flexible hinges [6, 7, 8], fiber optical probes [9, 10, 11, 12, 13],as well as other creative probing methods based on the principle of the tunneling effect [14], optical interference [15; 16], pneumatics [17], capacitance [18], atomic force microscope [19], and the piezoelectric effect [20]. However, not having the three features of point probing capability, non-contact probing, and 3D isotropic sensing at the same time makes it difficult to accurately measure high aspect ratio structures with these conventional methods. As part of our efforts to find a possible solution including all of these required capabilities, a spherical scattering electrical field probe (SSEP) was proposed, based on the detection of charge signal on the probing ball, and nanometer resolution displacement sensing, non-contact probing and 3D isotropy sensing capability were achieved simultaneously [1; 2]. The only unclear part of the SSEP is point probing capability, on which only qualitative analysis was performed for the time being.  
The point probing capability of the SSEP is closely related to surface charge distribution on the probing ball. To analyze the point probing characteristics of the SSEP, a quantitative investigation of the spherical scattering electrical field needs to be conducted to illustrate the law of surface charge distribution. The difficulties are: (1) Conventional theoretical analysis methods [21; 22] cannot complete the modeling task with complicated boundary conditions, such as the possibility that the surface being probed could be a plane, spherical, cylindrical, or free geometrical shape; (2) On the other hand, the spherical scattering electrical field is also difficult to model with conventional numerical methods due to the problem of balancing modeling accuracy and computational load. This is because the diameter of the probing ball is on a hundreds of microns to millimeter level, while the probing gap is on a micro and sub-micrometer level, and this creates a multi-scale problem, posing a challenge to gridding and computational accuracy; (3) Experimental methods [23; 24; 25; 26; 27] are not applicable here due to the lack of micro-/nano probes with the ability to detect the spherical scattering electrical field in the micro probing gap without introducing violent disturbance into the field. Therefore, an appropriate method of modeling and quantitative analysis of spherical scattering electrical field is the key issue to be solved for the analysis of the surface charge distribution characteristics as well as point probing characteristics of the SSEP.  
In this paper, this problem is solved using a proposed 3D seven-point finite difference scheme. In the accordingly built spherical coordinates, the spherical scattering electrical field is modeled by 3D gridding and finite difference computation, solved through iteration calculation, and validated by finite element method (FEM) numerical simulation under special boundary conditions. In this way, the surface charge distribution characteristics and the resulting approximate point probing characteristics of the SSEP are quantificationally analyzed and represented.