## 2 Basic Equation
### Dispersive Equation  
#### 2.2.1 Lorentz Force  
In a uniform and perfectly conductive elastic solid, the electromagnetic field is represented by [PERSON]'s equations,  
\[\
abla\times\mathbf{h}=\mathbf{J}+\varepsilon_{0}\dot{\mathbf{E}}, \tag{8}\] \[\
abla\times\mathbf{E}=-\mu_{0}\dot{\mathbf{h}},\] (9) \[\mathbf{E}=-\mu_{0}(\dot{\mathbf{u}}\times\mathbf{H}),\] (10) \[\
abla\cdot\mathbf{h}=0, \tag{11}\]  
where \(\
abla\) represents the Hamiltonian operator, with \(\mu_{0}\) and \(\varepsilon_{0}\) standing as the constants for magnetic permeability and electric field coefficient, respectively. Furthermore, \(\mathbf{H}\) denotes the external magnetic field strength, whereas \(\mathbf{h}\) and \(\mathbf{E}\) denote the internal magnetic field strength (induced magnetic field) and the electric field strength, respectively. \(\mathbf{J}\) is the displacement current density, and \(\mathbf{u}\) is the displacement vector.  
Based on the analysis of the strain gradient elasticity theory, the expressions for the tensor \(\tau\) and higher-order tensor \(\mu\) in the absence of a heat source are as follows:  
\[\tau=\mu(\
abla\mathbf{u}+\mathbf{u}\
abla)+\lambda\
abla\cdot\mathbf{u}\mathbf{\delta}, \tag{12}\]  
\[\mathbf{\mu}=c[\lambda(\
abla\
abla\cdot\mathbf{u})\mathbf{\delta}+\mu\
abla(\
abla\mathbf{u}+ \mathbf{u}\
abla)]. \tag{13}\]  
Expression of the Maxwell stress \(\mathbf{\eta}\):  
\[\mathbf{\eta}=\mu_{0}[\mathbf{H}\otimes\mathbf{h}+\mathbf{h}\otimes\mathbf{H}-(\mathbf{H}\cdot\mathbf{h}) \mathbf{\delta}]. \tag{14}\]  
Considering the plane strain problem \(y\geq 0\) in an electromagnetic half-space and assuming that the perturbation of the external magnetic field is \(\mathbf{H}=H_{0}\mathbf{e}_{z}\), \(\mathbf{e}_{z}\) is the unit vector along the \(z\)-axis and \(H_{0}\) is a constant.  
Substituting Equations (3) and (4) into Equation (7a) yields the equation of motion in terms of displacement:  
\[\Big{(}1-c\
abla^{2}\Big{)}[(\lambda+2\mu)\
abla\
abla\cdot\mathbf{u}-\mu\
abla \times\
abla\times\mathbf{u}]+\mathbf{F}=\rho\ddot{\mathbf{u}}-\frac{\rho d^{2}}{3}\
abla^ {2}\ddot{\mathbf{u}}, \tag{15}\]  
where \(\
abla^{2}\) denotes the Laplace operator. When the microstructural parameters \(c=d=0\), Equation (15) is reduced to the equation of motion in terms of classical elastic theory. The presence of the microstructural parameters \(c\) and \(d\) in the equation of motion suggests that they affect the propagation of waves in solids.  
\(\mathbf{F}\) is the Lorentz force, expressed as:  
\[\mathbf{F}=\mu_{0}(\mathbf{J}\times\mathbf{H}). \tag{16}\]  
By comparing the curl operator applied to both sides of Equation (10) with Equation (9), we obtain:  
\[\
abla\times\mathbf{E}=-\mu_{0}\
abla\times(\dot{\mathbf{u}}\times\mathbf{H})=-\mu_{0} \dot{\mathbf{h}}, \tag{17}\]  
simultaneously integrating both sides of Equation (17) to derive the expression for the induced magnetic field:  
\[\mathbf{h}=\
abla\times(\mathbf{u}\times\mathbf{H})=(-H_{0}\
abla\cdot\mathbf{u})\mathbf{e}_{z}. \tag{18}\]  
Substituting Equation (8) into Equation (16) obtains:  
\[\begin{split}\mathbf{F}=\mu_{0}(\mathbf{J}\times\mathbf{H})=\mu_{0}\Big{(} \
abla\times\mathbf{h}-\varepsilon_{0}\dot{\mathbf{E}}\Big{)}\times\mathbf{H}\\ =\mu_{0}H_{0}^{2}\
abla\
abla\cdot\mathbf{u}-\varepsilon_{0}\mu_{0}^ {2}H_{0}^{2}\ddot{\mathbf{u}}.\end{split} \tag{19}\]