# Characteristics of Electric Field Induced by Oscillating Metal Underwater Vehicle
## 2 Mathematical Model of Induced Electric Field for Metal Motion  
For moving electric field detection platforms such as metal navigation bodies, electric field sensors are fixed on the detection platform. According to the principle of electric field measurement, electric field sensors are arranged in pairs as shown in Figure 1, and orthogonal pairs are used to measure the potential difference between two points. The expression is as follows:  
\[E_{I}=\lim_{\Delta I\to 0}\frac{U_{1}-U_{2}}{\Delta I} \tag{1}\]  
In Equation (1), \(\Delta I\) is the distance between electrode pairs, also known as electrode distance; \(U_{1}\) represents the potential measured by sensor 1 in each pair of electric field sensors, and similarly, \(U_{2}\) represents the potential measured by sensor 2 in each pair of electric field sensors; \(E_{I}\) is the electric field between electrode pairs. According to Equation (1), for electric field measurement, the smaller the electrode distance, the more accurate it is. However, for target detection, the larger the electrode distance, the higher the system sensitivity, and the farther the detection distancecan be achieved. When the motion platform moves, the measurement system electric field sensor and its wire will cut the geomagnetic field and produce induced electromotive force; if the movement speed is constant, the induced electromotive force is a stable static potential difference, which will only affect the measurement of the static electric field or potential and will not have a significant impact on the detection of low-frequency electric fields. The interference electromotive force generated by unstable motion speed or periodic shaking is an alternating signal, which will have an impact on the detection of low-frequency electric fields. The following will be derived separately. Assume that the measurement systems 1-1 and 1-2 as shown in Figure 1 are oriented along the \(x\)-axis, 2-1 and 2-2 along the \(y\)-axis, and 3-1 and 3-2 along the \(z\)-axis. \(\mathbf{E}=\mathbf{E}(E_{x},E_{y},E_{z})\) is the induced electric field, where \(E_{x},E_{y},E_{z}\) represents the components along the three directions; the velocity of the probe platform is \(\mathbf{v}=(v_{x},v_{y},v_{z})\), where \(v_{x}\), \(v_{y}\), and \(v_{z}\) are the velocity components along the three directions, respectively, which are obtained by [PERSON]'s law of electromagnetic induction:  
\[\mathbf{E}=\mathbf{E}(E_{x},E_{y},E_{z})=\mathbf{v}\times\mathbf{B}_{E} \tag{2}\]  
In the formula, \(\mathbf{B}_{E}=(B_{x},B_{y},B_{z})\) is the geomagnetic field, which is substituted into the solution to obtain  
\[E_{x}=v_{y}B_{z}-v_{z}B_{y}=-F\big{[}v_{y}\sin(I)+v_{z}\cos(I)\sin(\gamma)\big{]} \tag{3}\]  
\[E_{y}=v_{z}B_{x}-v_{x}B_{z}=F\big{[}v_{z}\cos(I)\cos(\gamma)+v_{x}\sin(I)] \tag{4}\]  
\[E_{z}=v_{x}B_{y}-v_{y}B_{x}=F\big{[}v_{x}\cos(I)\sin(\gamma)-v_{y}\cos(I)\cos( \gamma)\big{]} \tag{5}\]  
In the above equation, \(F\) is the modulus of the geomagnetic field, \(I\) is the magnetic inclination angle, and \(\gamma\) is the magnetic declination angle (the dip angle \(I\) denotes the angle between \(\mathbf{B}_{E}\) and a plane tangent to Earth; the angle \(\gamma\) is the deviation between the magnetic north and the \(x\)-axis). Equations (3)-(5) are the electric field obtained from the potential difference of the output of the three-axis measuring system, and it can be seen from the expressions that the electric field is only related to the platform speed and the three components of the geomagnetic field, and it is a static signal that does not change with changes in time. When the motion detection platform shakes at a certain angle, the measurement system will also shake accordingly. Assume that the angular velocity of the x-y plane shaking, i.e., the rate of change in direction angle, is \(\omega_{d}\), the angular velocity of the x-z plane shaking, i.e., the rate of change in pitch angle, is \(\omega_{p}\), and the angular velocity of the y-z plane shaking, i.e., the rate of change in roll angle, is \(\omega_{r}\).