# 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-z Plane  
Similarly, the measurement system wobbles in the \(x\)-\(z\) plane, again with the 1-1 sensor at the origin of the coordinates, at time \(t\), the velocity at point \(l\) between electrodes 1-1 and 1-2 is \(\mathbf{v}_{1l}=\omega_{p}l\), and its three components are represented as:  
\[v_{1 xl}=-\omega_{p}l\sin(\omega_{p}t) \tag{27}\]  
\[v_{1 yl}=0 \tag{28}\]  
\[v_{1 zl}=\omega_{p}l\cos(\omega_{p}t) \tag{29}\]  
The three components of the electric field at \(l\) for electrodes 1-1 and 1-2 can be obtained as  
\[E_{1 xl}=v_{1 yl}B_{z}-v_{1 zl}B_{y}=-B_{y}\omega_{p}l\cos(\omega_{p}t) \tag{30}\]  
\[E_{1 yl}=v_{1 zl}B_{x}-v_{1 xl}B_{z}=\omega_{p}l[B_{x}\cos(\omega_{p}t)+B_{z}\sin (\omega_{p}t)] \tag{31}\]  
\[E_{1 zl}=v_{1 xl}B_{y}-v_{1 yl}B_{x}=-B_{y}\omega_{p}l\sin(\omega_{p}t)] \tag{32}\]  
Thus, the induced electric field on the electrode pair 1-1 and 1-2 is  
\[\begin{array}{c}E_{1p}=\int_{0}^{\Delta l}{[E_{1 xl}\cos(\omega_{p}t)+E_{1 zl }\sin(\omega_{p}t)]dl}/{\Delta l}\\ =\ \int_{0}^{\Delta l}{-B_{y}\omega_{p}ldl}/{\Delta l}=-\frac{1}{2}B_{y}\omega_ {p}\Delta l\end{array} \tag{33}\]  
Similarly, the linear velocity used at any point \(l\) between 2-1 and 2-2 in the \(y\)-axis direction is \(\mathbf{v}_{2l}=\frac{1}{2}\omega_{p}\Delta l\). The three components of the electric field at \(l\) on electrodes 2-1 and 2-2 can be obtained as follows:  
\[E_{2 xl}=v_{2 yl}B_{z}-v_{2 zl}B_{y}=-\frac{1}{2}B_{y}\omega_{p}\Delta l\cos( \omega_{p}t) \tag{34}\]  
\[E_{2 yl}=v_{2 zl}B_{x}-v_{2 xl}B_{z}=\frac{1}{2}\omega_{p}\Delta l[B_{x}\cos( \omega_{p}t)+B_{z}\sin(\omega_{p}t)] \tag{35}\]  
\[E_{2 zl}=v_{2 xl}B_{y}-v_{2 yl}B_{x}=-\frac{1}{2}B_{y}\omega_{p}\Delta l\sin( \omega_{p}t)] \tag{36}\]  
Therefore, when the measurement system shakes in the x-z plane, the induced electric field on the 2-1 and 2-2 electrode pairs is as follows:  
\[\begin{array}{c}E_{2p}=\int_{0}^{\Delta l}{E_{2 yt}dl}/{\Delta l}\\ =\ \int_{0}^{\Delta l}{\frac{1}{2}\omega_{p}\Delta l[B_{x}\cos(\omega_{p}t)+B_{ z}\sin(\omega_{p}t)]dl}/{\Delta l}\\ =\ \frac{1}{2}\omega_{p}\Delta l[B_{x}\cos(\omega_{p}t)+B_{z}\sin(\omega_{p}t)] \end{array} \tag{37}\]  
The electrode pairs 3-1 and 3-2 are on the \(z\)-axis, and the linear velocity near the connection point between the 3-2 end and the 1-1 and 1-2 electrodes is \(\mathbf{v}_{3l}=\omega_{p}\sqrt{l^{2}+0.25\Delta l^{2}}\).  
The three components of the electric field at 1 for electrodes 3-1 and 3-2 can be obtained as follows:  
\[E_{3 xl}=v_{3 yl}B_{z}-v_{3 zt}B_{y}=-B_{y}\omega_{p}\sqrt{l^{2}+0.25\Delta l^{2 }}\cos(\omega_{p}t) \tag{38}\]  
\[E_{3 yl}=v_{3 zt}B_{x}-v_{3 xl}B_{z}=\omega_{p}\sqrt{l^{2}+0.25\Delta l^{2}}[B_{ x}\cos(\omega_{p}t)+B_{z}\sin(\omega_{p}t)] \tag{39}\]  
\[E_{3 zl}=v_{3 xl}B_{y}-v_{3 yt}B_{x}=-B_{y}\omega_{p}\sqrt{l^{2}+0.25\Delta l^{2 }}\sin(\omega_{p}t)] \tag{40}\]  
Therefore, the induced electric field on the 3-1 and 3-2 electrode pairs when the measurement system is shaking in the x-z plane is as follows:  
\[\begin{array}{c}E_{3p}=\int_{-\Delta l/2}^{\Delta l/2}{[-E_{3 xl}\sin(\omega _{d}t)+E_{3 zl}\cos(\omega_{d}t)]dl}/{\Delta l}\\ =\ \int_{0}^{\Delta l}{\frac{1}{2}\omega_{p}\Delta l}dl/{\Delta l}=0\end{array} \tag{41}\]