## 1 Introduction
With regard to electric field detection, the induced electric field generated by a detected object because of the motion of the magnetic object is a signal source. Therefore, its characteristics must be extracted. For both inhibitions of disturbances and extraction of characteristics, it is necessary to determine the generation mechanism, mathematical model, and signal characteristics of the electric field. At present, there are relatively few research aspects on the induced electric field generated by the angular oscillation of magnetic objects. The authors of [13] took the magnetic dipole as the research object and adopted the relativistic electromagnetic transformation method to derive the calculation formula of the induced electric field around the magnetic dipole moving at uniform speed in an arbitrary direction. Based on this, the spatial distribution characteristics and characteristics of the induced electric field, which were generated by the uniform motion of magnetic dipoles along the magnetic moment direction, were analyzed, and the correctness of the theoretical method was verified through experimental verification and analysis. The authors of [14], according to the theory of relativistic electromagnetic transformations, derived the formula for the calculation of the three components of the induced electric field generated by a single magnetic dipole, moving at a uniform speed along any direction, and obtained the spatial distribution of the induced electric field in the inertial system through the transformation of the coordinates, solving the defects via [PERSON]'s law (can only be calculated by the induced electric field of the magnetic dipole moving along the direction of the polar moment), providing a basis for fitting the distribution of the magnetic field of the magnetic object by a magnetic dipole, and then investigating the related induced electric field. The authors of [15] aimed to study the induced electric field generated by the magnetic hull of a moving ship. Based on basic electromagnetic field theories, such as [PERSON]'s law and Biot-Savart's law, a mathematical model for the induced electric field generated by the motion of a magnetic ship was established. Through simulation and experimental verification, an important conclusion was drawn: the induced electric field generated by the motion of a magnetic dipole has obvious three-axis passing characteristics, rather than two directions obtained from relativistic electromagnetic transformation. The authors of [16] studied the induced electric field generated by the magnetic hull of a moving ship. electric field generated by the rotation of ship propellers. Using [PERSON]'s law of electromagnetic induction and the loop integration method, a mathematical model of the induced electric field of a uniformly rotating magnetic dipole was derived. Through experimental verification, it was found that the intensity of the induced electric field generated by the rotation of the propeller is proportional to the rotation speed and magnetic moment. The frequency of this electric field is equal to the rotational frequency of the magnet, and the phase difference of the three-axis components is \(\pi/2\) rad and \(\pi\) rad, respectively. This conclusion improves the generation mechanism of the ship axial frequency electric field but lacks theoretical analysis of the phase characteristics of the induced electric field generated by the self-shaking of magnetic targets, and the phase relationship conclusions obtained from simulation analysis are not comprehensive. In order to improve the theoretical model of the underwater-target-induced electric field, this manuscript analyzes the characteristics of the induced electric field generated by the angular oscillating magnetic objects themselves. The vector magnetic potential method is used to derive the mathematical model of the induced electric field generated by the rotation of magnetic dipoles. Compared with the loop integration method, the derivation process is more concise.