## 2 Modeling  
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 \(\mathbf{m}=m_{x}\mathbf{i}+m_{y}\mathbf{j}+m_{z}\mathbf{k}\), located in the right-handed Cartesian coordinate system of original point \(\mathbf{O}\), and rotating uniformly around the z-axis at \(\omega\) rad/s angular velocity. Here, \(\mathbf{i}\), \(\mathbf{j}\), and \(\mathbf{k}\) 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 \(\mathbf{m}=\left(m_{x},\ m_{y},\ m_{z}\right)\) (revised), the vector of the magnetic moment at the time t can be expressed as:  
\[\mathbf{m}^{\prime}=\left(m_{x}\cos(\omega t)-m_{y}\sin(\omega t),m_{x}\sin(\omega t )+m_{y}\cos(\omega t),m_{z}\right) \tag{1}\]  
When the magnetic dipole is at any point \(p(x,\ y,\ z)\) in space at the time \(t\) (points in space are measured in meters), its magnetic vector potential is expressed as:  
\[\mathbf{A}=\frac{\mu_{0}\mathbf{m}^{\prime}\times\mathbf{r}}{4\pi r^{3}} \tag{2}\]  
where \(\mathbf{r}=(x,y,z)\), 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:  
\[\mathbf{A}_{x} =\frac{\mu_{0}}{4\pi r^{3}}\left\{z[m_{x}\sin(\omega t)+m_{y}\cos( \omega t)]-ym_{z}\right\} \tag{3}\] \[\mathbf{A}_{y} =\frac{\mu_{0}}{4\pi r^{3}}\left\{xm_{z}-z[m_{x}\cos(\omega t)-m_{ y}\sin(\omega t)]\right\}\] (4) \[ \begin{array}{cc}A_{z}&=\frac{\mu_{0}\omega}{4\pi r^{3}}[m_{x}\cos( \omega t)-m_{y}\sin(\omega t)]\\ &-\frac{\mu_{0}\omega}{4\pi r^{3}}[m_{x}\sin(\omega t)+m_{y}\cos(\omega t)] \end{array}  \tag{5}\]  
Based on its expression, the magnetic vector potential satisfies \(\
abla\cdot\mathbf{A}=0\). For the formula \(\mathbf{E}=-\
abla U-\partial\mathbf{A}/\partial t\), \(U\) 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 \(\Delta U=0\). The three-axis components of the electric field induced by the magnetic dipole are obtained as follows:  
\[\mathbf{E}_{x} =-\frac{\mu_{0}\omega z}{4\pi r^{3}}[m_{x}\cos(\omega t)-m_{y}\sin( \omega t)] \tag{6}\] \[\mathbf{E}_{y} =-\frac{\mu_{0}\omega z}{4\pi r^{3}}[m_{x}\sin(\omega t)+m_{y}\cos (\omega t)]\] (7) \[\mathbf{E}_{z} =\frac{\mu_{0}\omega y}{4\pi r^{3}}[m_{x}\sin(\omega t)+m_{y}\cos (\omega t)]\] (8) \[+\frac{\mu_{0}^{2}\omega}{4\pi r^{3}}[m_{x}\cos(\omega t)-m_{y} \sin(\omega t)]\]  
where \(\mathbf{E}_{x}\), \(\mathbf{E}_{y}\), and \(\mathbf{E}_{zz}\) 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.