Analysis of Characteristics of the Electric Field Induced by an Angularly Rotating and Oscillating Magnetic Object

Abstract

A mathematical model for an electric field induced by an angularly oscillating magnetic dipole was proposed with magnetic vector potential to analyze the characteristics of the electric field induced by a rotating and angularly oscillating magnetic object. This mathematical model was constructed for the electric field induced by a magnetic object oscillating at a certain angle. On this basis, the phase relationship among the three components of the induced electric field was analyzed (defining the right-hand Cartesian coordinate system). Evidently, a phase difference of π/2 always existed between the horizontal components of the electric field induced by a magnetic dipole rotating around the z-axis. The phase difference between the vertical and transverse components in the x–z plane was also π/2. A phase difference of π was observed in the y–z plane. The above theoretical analysis was verified through simulation and experiment. The results showed that the frequency of the induced electric field was related to the angular velocity and angle of rotation. The amplitude was associated with the magnetic moment and the angular velocity and angle of oscillation. The maximum amplitude did not exceed the amplitude of the electric field induced by a magnetic object angularly oscillating at the same velocity. With regard to the amplitude and phase relationship, the three components of the induced electric field measured in the experiment were consistent with the results of the theoretical analysis.

1. Introduction

With the increasing attention paid by countries around the world to marine resources, how to further develop and explore marine resources has become a research hotspot. At the same time, with the rapid development of various underwater equipment, such as UUVs (unmanned underwater vehicles), AUVs (autonomous underwater vehicles), underwater buoy systems, underwater biomimetic robots, and underwater unpowered/powered gliders, underwater target detection methods are gradually improving, and the focus of marine science and technology research is gradually shifting to underwater target detection. As a traditional underwater detection method, acoustic detection is widely used in various types of underwater target detection due to its advantages, such as slow underwater attenuation, long detection distance, and the ability to detect small targets. However, acoustic detection also has problems, such as being greatly affected by ocean background noise and being easily exposed by active acoustic detection. With the rapid development of seismic and noise reduction technology in various countries around the world, underwater acoustic detection technology is further limited, appearing inadequate in complex marine environments and adversarial fields. In recent years, countries around the world have vigorously developed underwater non-acoustic detection technology, studying the electric field, magnetic field, and gravity field of underwater targets.

The underwater electric field is an important physical field for ships. According to the signal frequency band, the electric field is mainly divided into a steady-state electric field (mainly including a corrosion-related static electric field) and an alternating electric field (mainly including a shaft frequency and power frequency electric field). Among them, the shaft frequency electric field signal generated by the rotation of the propeller is often used for long-distance detection of underwater targets due to its strong ability to resist natural interference, obvious near-field characteristics, and line spectrum characteristics of the power spectrum. The static electric field signal generated by corrosion and anti-corrosion currents is easy to track and locate due to its obvious near-field characteristics and high signal-to-noise ratio. Therefore, it is often used for end positioning of positioning systems, mine electric field fuses, and underwater array detection [1,2,3,4,5]. In recent years, attention has been widely paid to electric field detection. Many achievements have been made in electric field detection, in such aspects as theoretical modeling, detection and identification, and tracking and localization of underwater objects with an electric field. The electric field of an underwater object has been widely used in the methods used to detect objects [6,7,8,9,10,11,12].

Unlike a corrosion-related underwater electric field, an induced electric field is generated by the variation of the magnetic flux in the adjacent space during the motion, rolling, or swaying of a magnetic object. Vessels, underwater vehicles, and their corresponding rotating accessories, such as the main axis and propeller, contain ferromagnetic metals with residual magnetism in most cases. A large magnetic field is also caused if these metals are in a geomagnetic field for a long time and the metals become magnetized. In underwater target detection, electric field detection is gradually combined with underwater moving platforms, and the research focus on moving platforms has gradually shifted from instability platforms, such as buoys and submersibles, to platforms with good underwater mobility and a larger detection range, such as unmanned boats and UUVs. As for a detection platform, the electric field induced by the motion of a magnetic object exists as a background disturbance in the underwater object detection of the platform. The induced electric field should, therefore, be inhibited. 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 Gauss’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 Coulomb’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 rotation of ship propellers. Using Faraday’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 π/2 rad and π 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 angularly 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. Based on this, a mathematical model of the induced electromagnetic field generated by the angular oscillation of magnetic objects at a certain angle has been established, and further theoretical analysis was conducted on the phase characteristics of the three components of the induced electric field generated by the angular oscillation of the magnet. Finally, the correctness of the theoretical model and analysis results was verified through experiments. This manuscript has important research significance for in-depth study of the mechanism and characteristics of ship electric field generation, detection of ship electric field signals, platform adaptation of electric field sensors, and research on electric field fuses for underwater weapons.

2. Modeling

2.1. A Model for an Electric Field Induced by a Magnetic Object Angularly Rotating at a Constant Velocity

Magnetic field modeling can be implemented for magnetic vessels, propellers, or other objects in various ways. Normally, a variety of models can be constructed, including a magnetic dipole model, uniformly magnetized rotating ellipsoid model, magnetic dipole array model, or magnetic dipole model plus uniform magnetized rotating ellipsoid model. Different models have different accuracies, but they are developed similarly for the electric field induced by a magnetic object angularly rotating at a constant velocity. For a simple process of modeling, a magnetic dipole rotating at a constant velocity was taken as an example to develop a mathematical model for the electric field induced by a magnetic object angularly rotating at a constant velocity.

As shown in Figure 1, the magnetic target is equivalent to a magnetic dipole with a magnetic moment of $\mathit{m}=m_x\mathit{i}+m_y\mathit{j}+m_z\mathit{k}$, located in the right-handed Cartesian coordinate system of original point O, and rotating uniformly around the z-axis at $\omega$ rad/s angular velocity. Here, $\mathit{i}$, $\mathit{j}$, and $\mathit{k}$ are the unit vectors in the x, y, and z directions of the Cartesian coordinate system, and $m_x$, $m_y$, and $m_z$ are the components of the magnetic dipole on the x, y, and z axes, respectively.

If the magnetic moment of the magnetic dipole at the time $t=0$ is $\mathit{m}=\left(m_x, m_y, m_z\right)$ (revised), the vector of the magnetic moment at the time t can be expressed as:

$${\mathit{m}}^{\prime }=(m_x\cos (\omega t)-m_y\sin (\omega t),m_x\sin (\omega t)+m_y\cos (\omega t),m_z)$$

(1)

When the magnetic dipole is at any point $p\left(x, y, z\right)$ in space at the time t (points in space are measured in meters), its magnetic vector potential is expressed as:

$$\mathit{A}=\frac{{\mu }_0{\mathit{m}}^{\prime }\times \mathit{r}}{4\pi r^3}$$

(2)

where $\mathit{r}=\left(x,y,z\right)$, and ${\mu }_0$ is the vacuum magnetic permeability.

By substituting Equation (1) into Equation (2), we can obtain the three components of the magnetic vector potential at the point p, as follows:

$${\mathit{A}}_x=\frac{{\mu }_0}{4\pi r^3}\left\{z[m_x\sin (\omega t)+m_y\cos (\omega t)]-y{m}_z\right\}$$

(3)

$${\mathit{A}}_y=\frac{{\mu }_0}{4\pi r^3}\left\{x{m}_z-z[m_x\cos (\omega t)-m_y\sin (\omega t)]\right\}$$

(4)

$$\begin{array}{ll}{\mathit{A}}_z& =\frac{{\mu }_{0}y}{4\pi r^3}[m_x\cos (\omega t)-m_y\sin (\omega t)]\\ & -\frac{{\mu }_{0}x}{4\pi r^3}[m_x\sin (\omega t)+m_y\cos (\omega t)]\end{array}$$

(5)

Based on its expression, the magnetic vector potential satisfies $\nabla ·\mathit{A}=0$.

For the formula $\mathit{E}=-\nabla U-\partial \mathit{A}/\partial t$, $U$ stands for the static potential of the field point in the space. The magnetic dipole is in uniform rotational motion at the origin O with angular velocity around the z-axis, so that $\mathrm{\Delta }U=0$. The three-axis components of the electric field induced by the magnetic dipole are obtained as follows:

$${\mathit{E}}_x=-\frac{{\mu }_0\omega z}{4\pi r^3}[m_x\cos (\omega t)-m_y\sin (\omega t)]$$

(6)

$${\mathit{E}}_y=-\frac{{\mu }_0\omega z}{4\pi r^3}[m_x\sin (\omega t)+m_y\cos (\omega t)]$$

(7)

$$\begin{array}{ll}{\mathit{E}}_z& =\frac{{\mu }_0\omega y}{4\pi r^3}[m_x\sin (\omega t)+m_y\cos (\omega t)]\\ & +\frac{{\mu }_0\omega x}{4\pi r^3}[m_x\cos (\omega t)-m_y\sin (\omega t)]\end{array}$$

(8)

where ${\mathit{E}}_x$, ${\mathit{E}}_y$, and ${\mathit{E}}_z$_z represent the longitudinal, transverse, and vertical components of the electric field, respectively.

The expressions in Equations (6)–(8) are the same as those in [16], but this method is simpler to implement than the loop integration proposed in [16].

The phase characteristics of the induced electric field were further analyzed. With Equations (6)–(8), we obtain:

$${\mathit{E}}_x=-\sqrt{{m_x}^2+{m_y}^2}\frac{{\mu }_0\omega z}{4\pi r^3}\sin (\omega t+{\phi }_x)$$

(9)

$${\mathit{E}}_y=-\sqrt{{m_x}^2+{m_y}^2}\frac{{\mu }_0\omega z}{4\pi r^3}\sin (\omega t+{\phi }_y)$$

(10)

$${\mathit{E}}_z=-\sqrt{(x^2+y^2)({m_x}^2+{m_y}^2)}\frac{{\mu }_0\omega }{4\pi r^3}\sin (\omega t+{\phi }_z)$$

(11)

where ${\phi }_x$, ${\phi }_y$, and ${\phi }_z$ represent the initial phases for the three components of the electric field, respectively. They are determined by Equations (6)–(8), as follows:

$${\phi }_x=\arctan (-\frac{m_x}{m_y})$$

(12)

$${\phi }_y=\arctan (\frac{m_y}{m_x})$$

(13)

$${\phi }_z=\pi +\arctan (\frac{y{m}_y+x{m}_x}{y{m}_x-x{m}_y})$$

(14)

With Equations (12)–(14), we obtain:

$${\phi }_y={\phi }_x+\frac{\pi }{2}$$

(15)

$${\phi }_z=\left\{\begin{array}{c}{\phi }_y+\frac{\pi }{2}\begin{array}{cc}& y=0,x\ne 0\end{array}\\ {\phi }_y+\pi \begin{array}{cc}& x=0,y\ne 0\end{array}\end{array}\right.$$

(16)

With Equations (15) and (16), it is concluded that a phase relationship exists among the three components of the induced electric field at the angular rotation of the magnetic object, as follows:

(1)

The phase difference between the components ${\mathit{E}}_y$ and ${\mathit{E}}_x$ is always π/2.

(2)

When y= 0,x≠ 0 (x-z plane), the phase difference between the components ${\mathit{E}}_z$ and ${\mathit{E}}_y$ is π/2; when x= 0, y ≠ 0 (y-z plane), the phase difference between the components E_z and E_y is π.

2.2. The Model for the Electric Field Induced by an Angularly Oscillating Magnetic Object

A mathematical model for the electric field induced by an angularly oscillating magnetic object can be developed using the model for the electric field induced by a magnetic object angularly rotating at a constant velocity. The difference between the two models of this type must be the range of motion. A magnetic object angularly rotating at a constant velocity moves at the angle θ, ranging from 0 to 2π, but it can oscillate back and forth within a certain angle range. By limiting the rotation angle of the magnet, the uniform rotation-induced electric field model can be transformed into an angularly oscillating magnet-induced electric field model.

Assuming that the magnet oscillates uniformly around the z-axis at an angular velocity of $\omega$ rad/s, with an oscillated angle of 0~θ_m, the sway period can be considered as $t_{\mathrm{oT}}={\theta }_m/\omega$ (the subscript O stands for oscillate), indicating that the target swaying process is composed of countless $t_{\mathrm{oT}}$ (s). As analyzed in the previous section, the induced electric field expression obtained during the process of swaying from 0 to θ_m is the same as Equations (6)–(8). When the oscillating angle changes from θ_m to 0, let the initial time $t_{\mathrm{o}0}=0$, oscillating initial angle θ = θm, and the expression for the magnetic moment of $t_{\mathrm{o}}$ at any continuous time during this oscillating stage is:

$$\left\{\begin{array}{l}{\mathit{m}}_x^{\prime }=m_x\cos ({\theta }_m-\omega t_o)-m_y\sin ({\theta }_m-\omega t_o)\\ {\mathit{m}}_y^{\prime }=m_x\sin ({\theta }_m-\omega t_o)+m_y\cos ({\theta }_m-\omega t_o)\\ {\mathit{m}}_z^{\prime }=m_z\end{array}\right.$$

(17)

Equation (17) is substituted into Equation (2) to obtain the following expressions of the magnetic vector potential when the magnetic object angularly oscillates from θ_m to 0:

$${\mathit{A}}_x^{\prime }=\frac{{\mu }_0}{4\pi r^3}\left\{z[m_x\sin ({\theta }_m-\omega t_o)+m_y\cos ({\theta }_m-\omega t_o)]-y{m}_z\right\}$$

(18)

$${\mathit{A}}_y^{\prime }=\frac{{\mu }_0}{4\pi r^3}\left\{x{m}_z-z[m_x\cos ({\theta }_m-\omega t_o)-m_y\sin ({\theta }_m-\omega t_o)]\right\}$$

(19)

$$\begin{array}{ll}{\mathit{A}}_z^{\prime }& =\frac{{\mu }_{0}y}{4\pi r^3}[m_x\cos ({\theta }_m-\omega t_o)-m_y\sin ({\theta }_m-\omega t_o)]\\ & -\frac{{\mu }_{0}x}{4\pi r^3}[m_x\sin ({\theta }_m-\omega t_o)+m_y\cos ({\theta }_m-\omega t_o)]\end{array}$$

(20)

Based on $\mathit{E}=-\nabla U-\partial \mathit{A}/\partial t$, the electric field induced by the angularly oscillating magnetic object can be determined by:

$${\mathit{E}}_x^{\prime }=-\frac{{\mu }_0\omega z}{4\pi r^3}[m_y\sin ({\theta }_m-\omega t_o)-m_x\cos ({\theta }_m-\omega t_o)]$$

(21)

$${\mathit{E}}_y^{\prime }=\frac{{\mu }_0\omega z}{4\pi r^3}[m_x\sin ({\theta }_m-\omega t_o)+m_y\cos ({\theta }_m-\omega t_o)]$$

(22)

$$\begin{array}{l}{\mathit{E}}_z^{\prime }=-\frac{{\mu }_0\omega y}{4\pi r^3}[m_x\sin ({\theta }_m-\omega t_o)+m_y\cos ({\theta }_m-\omega t_o)]\\ \begin{array}{cc}& \end{array}-\frac{{\mu }_0\omega x}{4\pi r^3}[m_x\cos ({\theta }_m-\omega t_o)-m_y\sin ({\theta }_m-\omega t_o)]\end{array}$$

(23)

3. Simulation Analysis of the Characteristics of the Electric Field Induced by an Angularly Oscillating Magnetic Object

To verify the theoretical analysis, a rectangular magnet with 1.7 cm × 1.7 cm × 8 cm was selected as an example simulation object. The magnetic field intensity at different positions was measured by a three-axis optical pump magnetometer, and the three-axis magnetic moment components were calculated using the least squares method to obtain 0.51 A·m², 6.70 A·m², and 0.66 A·m². Assuming that the magnet rotates uniformly in the direction of 0.66 A·m² magnetic moment with an angular velocity of $\omega =2\pi$ rad/s, the magnetic permeability of seawater is obtained as μ₀ = 4π × 10⁻⁷ H/m. The three components of the induced electric field at the observation points $p_1(1,0,1)$ (x-zplane) and $p_2(0,1,1)$ (y-zplane) were obtained through simulation calculations, as shown in Figure 2 and Figure 3.

Figure 2 and Figure 3 are visual simulation validations of Formulas (9)–(16).

As shown in Figure 2, the phase difference between ${\mathit{E}}_y$ and ${\mathit{E}}_x$ is π/2, and the phase difference between ${\mathit{E}}_z$ and ${\mathit{E}}_y$ is π/2. As revealed in Figure 3, the phase difference between the components ${\mathit{E}}_y$ and ${\mathit{E}}_x$ is π/2, and the phase difference between the components ${\mathit{E}}_z$ and ${\mathit{E}}_y$ is π.

Therefore, it can be proven that the theoretical analysis in Section 2.1 is completely correct, and the phase difference between the ${\mathit{E}}_x$ component and the ${\mathit{E}}_y$ component is always $\pi /2$. In the x–z plane, the phase difference between the ${\mathit{E}}_z$ component and the ${\mathit{E}}_y$ component is $\pi /2$, and in the y–z plane, the phase difference between the ${\mathit{E}}_z$ component and the ${\mathit{E}}_y$ component is $\pi$.

Through the above analysis, and compared with the loop integration method, it can be concluded that the phase relationship between the three components of the induced electric field generated during the rotation of the magnet verifies the theoretical correctness of the formula derivation.

The characteristics of the electric field induced by the magnetic dipole oscillating back and forth within the angle range of 0–θ_m were further analyzed in the simulation. The simulation calculation conditions remained unchanged, the angular velocity was still ω = 2π rad/s while oscillating, and the magnetic permeability of seawater also remained unchanged. The three components of the induced electric field with θ_m = π/6 and θ_m = π/3 at the observation point p₁(1, 0, 1) were obtained, as shown in Figure 4 and Figure 5, respectively.

As revealed in Figure 4 and Figure 5, under the same angularly oscillating velocity, the larger the oscillating angle, the lower the oscillating frequency. When the angle was θ_m = π/6, the oscillating frequency was 6 Hz. When the angle was θ_m = π/3, the oscillating frequency was 3 Hz. The relationship between the oscillating frequency with the angle and angular velocity of oscillating is theoretically expressed by:

$$f=\frac{\omega }{2{\theta }_m}$$

(24)

Comparing Figure 2 with Figure 4 and Figure 5, it can be seen that under the same angular velocity, the amplitude of the induced electric field generated by the angular oscillation of the magnet was not greater than that generated by the rotation of the magnet. For the ${\mathit{E}}_y$ component of the induced electric field, as the oscillation angle decreased, the amplitude of the induced electric field generated by the magnet oscillation also decreased.

4. Verification Experiment

To verify the correctness of the theoretical simulation, the rectangular magnetic dipole used in the simulation was used in an experiment to maintain the magnetic moment parameters unchanged. The three components of its magnetic moment were 0.51 A·m², 6.70 A·m², and 0.66 A·m². It was attached to a copper rod in the direction of the magnetic moment 0.66 A·m² component and placed in a seawater pool. With the copper rod as an axis, it was manually rotated on the seawater surface (no difference from underwater rotation because of the consistent magnetic conductivity between seawater and air). An electric field measurement system was fixed underwater. The seawater in the pool was prepared with sea salt to achieve a conductivity of approximately 2.8 S/m.

In the electric field detection system, considering the actual engineering application background, the sensor uses six Ag/AgCl electrodes (Ag/AgCl electrodes have the advantages of better stability, lower sub-noise, etc., as a reference electrode is widely used in the ship’s anti-corrosion system, and can be underwater for more than ten years of continuous work), constructing a three-component electric field measurement system. Here, two electrodes are in the same direction as a group of electrode pairs and each pair is perpendicular to each other. The distance between the measuring electrodes and the reference electrode in the x, y, and z directions in the measurement system is 10 cm, the water depth of the experimental pool is 85 cm, the reference electrode is 30 cm from the water surface, and the horizontal distance between the magnet and the reference electrode is 5 cm (the sensor needs to be immersed in salt water before the experiment to stabilize the polar deviation). The signals measured by the electric field acquisition system were amplified, filtered, and conditioned, and then collected and stored by the data acquisition card, with a sampling frequency of 20 Hz.

It was measured that the rotational velocity was approximately ω ≈ 5π rad/s since the magnetic dipole was manually rotated. The schematic diagram of the experiment is shown in Figure 6. The theoretical and measured values were compared, as shown in Figure 7.

Based on the analysis of experimental results in Figure 7, it was concluded that:

(1)

The measured and theoretical values of the electric field induced by a rotating magnetic dipole were basically equal in terms of scale and phase characteristics. The difference between them must be mainly attributed to the manual rotation in the experiment, causing the fluctuation of rotational velocity.

(2)

Based on the measured results, the phase difference between E_y and E_x was π/2, and the phase difference between E_z and E_y was π. These values are consistent with the theoretical analysis.

The above experimental results demonstrate the correctness of the mathematical models and theoretical analysis.

Due to the limitations of experimental conditions, the experiment on the induced electric field generated by magnet oscillation in Section 2.2 was limited. Due to the difficulty in maintaining and controlling the oscillation angles π/6 and π/3 during manual oscillation, mechanical oscillation equipment can generate significant mechanical noise and irremovable electromagnetic interference during the oscillation process. In addition, currently, in non-magnetic laboratories or experimental seawater pools, there is no device that can degauss existing oscillating electromagnetic equipment, so for the time being, the induced electromagnetic field generated by shaking can only be theoretically guided by computational simulation.

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 magnet was obtained: there was always an π/2 phase difference between the horizontal components, a phase difference of π/2 between the vertical and horizontal components in the x–z plane, and a phase difference of π 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.