# Characteristics of Electric Field Induced by Oscillating Metal Underwater Vehicle
## 2 Mathematical Model of Induced Electric Field for Metal Motion
### The Measurement System Shakes in the x-y Plane  
The measurement system shakes in the x-y plane, assuming that at time \(t=0\), as shown in Figure 1, the three pairs of electric field sensors in the electric field measurement system are aligned with the coordinate axis direction, and 1-1 and 1-2 are on the \(x\)-axis; the state over time \(t\) is shown in Figure 2. The distance between electrodes 1-1 is located at the coordinate origin, and the linear velocity at the distance from the coordinate origin is \(\mathbf{v}_{1l}=\omega_{d}l\). At time \(t\), the three components of velocity at \(l\) when shaking in the x-y plane can be obtained as follows:  
\[v_{1 xl}=-\omega_{d}l\sin(\omega_{d}t) \tag{6}\]  
\[v_{1 yl}=\omega_{d}l\cos(\omega_{d}t) \tag{7}\]  
\[v_{1 zl}=0 \tag{8}\]  
Substituting Equations (6)-(8) into (2), the three components of the electric field at the 1-1 and 1-2 electrode pairs can be obtained as follows:  
\[E_{1 xl}=v_{1 yl}B_{z}-v_{1 zl}B_{y}=B_{z}\omega_{d}l\cos(\omega_{d}t) \tag{9}\]  
\[E_{1 yl}=v_{1 zl}B_{x}-v_{1 xl}B_{z}=B_{z}\omega_{d}l\sin(\omega_{d}t) \tag{10}\]  
\[E_{1 zl}=v_{1 xl}B_{y}-v_{1 yl}B_{x}=-\omega_{d}l[B_{y}\sin(\omega_{d}t)+B_{x} \cos(\omega_{d}t)] \tag{11}\]  
Combining Equation (1) shows that the electric field is approximated by the potential difference, which in turn is equal to the integral of the electric field \(U=\int Edl\), so that the induced electric field on the 1-1 and 1-2 electrode pairs when the measurement system is shaking in the \(x\)-\(y\) plane is  
\[ \begin{array}{l}E_{1d}=\int_{0}^{\Delta l}[E_{1 xl}\cos(\omega_{d}t)+E_{1 yl} \sin(\omega_{d}t)]dl/\Delta l\\ =\int_{0}^{\Delta l}B_{z}\omega_{d}ldl/\Delta l=\frac{1}{2}B_{z}\omega_{d} \Delta l\end{array}  \tag{12}\]  
Similarly, 2-1 and 2-2 on the \(y\)-axis and thestate at moment \(t\) are shown in Figure 3. Therefore, it can be inferred that the linear velocity near the junction point \(l\) between the 1-2 end and the 1-1 and 1-2 electrodes is \(\mathbf{v}_{2l}=\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}}\). \[v_{2 xl}=-\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}}\sin(\omega_{d}t) \tag{13}\]  
\[v_{2 yl}=\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}}\cos(\omega_{d}t) \tag{14}\]  
Figure 2: Schematic diagram of the shaking of the electrode pair 1-1 measurement system (this figure illustrates the dynamic response of the electrode pair 1-1 to shaking movements, highlighting the induced variations in the electric field measurement due to the electrode’s oscillation in the x-\(y\) plane. It shows the theoretical setup used to model how shaking intensity and direction can influence the electric potential difference detected by the system, crucial for understanding the electric field’s sensitivity to vehicular movement under simulated conditions). The three components of the electric field at \(l\) for electrodes 2-1 and 2-2 can be obtained as follows:  
\[E_{2 xl}=v_{2 yl}B_{z}-v_{2 xl}B_{y}=B_{z}\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}}\cos (\omega_{d}t) \tag{16}\]  
\[E_{2 yl}=v_{2 xl}B_{x}-v_{2 xl}B_{z}=B_{z}\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}} \sin(\omega_{d}t) \tag{17}\]  
\[E_{2 xl}=v_{2 xl}B_{y}-v_{2 yl}B_{x}=-\omega_{d}\sqrt{l^{2}+0.25\Delta l^{2}}[B_{ y}\sin(\omega_{d}t)+B_{x}\cos(\omega_{d}t)] \tag{18}\]  
The measurement system shakes in the \(x\)-\(y\) plane, and the induced electric field on the 2-1 and 2-2 electrode pairs is  
\[ \begin{array}{l}E_{2d}=\int_{-\Delta l/2}^{\Delta l/2}[-E_{2 xl}\sin(\omega_{d} t)+E_{2 yl}\cos(\omega_{d}t)]dl/\Delta l\\ =\ \int_{0}^{\Delta l}\partial dl/\Delta l=0\end{array}  \tag{19}\]  
Similarly, 3-1 and 3-2 are on the \(z\)-axis, and since the electrode pairs of 3-1 and 3-2 are perpendicular to the x-y plane, the linear velocity at any point between 3-1 and 3-2 is \(\mathbf{v}_{3l}=\frac{1}{2}\omega_{d}\Delta l\).