## 5 Conclusions  
This article adopted the magnetic dipole model of the magnet and derived the mathematical model of the induced electric field generated by the uniform rotation of the magnet using the vector magnetic potential method. Compared with the loop integration method, the derivation process is more concise. Based on this, a mathematical model of the induced electric field generated by the magnet angularly oscillating at a certain angle was further derived and established. Through theoretical analysis, the phase relationship between the three components of the induced electric field generated by the angularly oscillating  
Figure 7: Comparison of measured and theoretical values of (**a**) \(E_{x}\), (**b**) \(E_{y}\), and (**c**) \(E_{z}\). (**a**) \(E_{x}\) component of an induced electric field, (**b**) \(E_{y}\) component of an induced electric field, and (**c**) \(E_{z}\) component of an induced electric field.  
magnet was obtained: there was always an \(\pi/2\) phase difference between the horizontal components, a phase difference of \(\pi/2\) between the vertical and horizontal components in the x-z plane, and a phase difference of \(\pi\) in the y-z plane. Model simulation was conducted to verify the correctness of the mathematical model, and it was found that the amplitude of the induced electric field generated by the angularly oscillating magnet was not greater than that generated by the magnet rotation. The conclusion is that as the angularly oscillating angle decreased, the amplitude of the induced electric field generated by the angularly oscillating magnet also decreased for the \(E_{y}\) component of the induced electric field. Finally, the correctness of the theoretical model and analysis was verified in the experimental results.  
With the deepening of research on underwater target detection by combining electric field detection systems with underwater motion platforms, the induced electric field generated by the angular oscillation of magnetic motion platforms themselves has become an undeniable background interference in the entire detection system. Therefore, a detailed analysis of the induced electric field characteristics generated by the angular oscillation or rotation of magnetic bodies is the key to solving the problem of electric field detection in underwater motion platforms. The model and conclusions obtained in this article, especially the phase relationship between the three components of the induced electric field, have theoretical guidance significance for identifying, extracting, or eliminating the induced electric field generated by magnet rotation.  
Data curation, T.X.; formal analysis, J.Z. and D.X.; investigation, D.X.; methodology, J.Z., D.X., T.X., and Q.J.; writing--original draft, J.Z.; writing--review and editing, T.X. and Q.J. All authors have read and agreed to the published version of the manuscript.  
This project was supported by the Young Scientists Fund of the National Natural Science Foundation of China (No. 51509252).  
Not applicable.  
The data presented in this study are available upon request from the corresponding author. The data are not publicly available due to the special significance of the materials used in the experimental process (such as magnetic materials used for rotation), which involve privacy in some research fields, it is not convenient to fully disclose their data.  
The authors declare no conflicts of interest.