## 4 Physical Phenomena and Ruling Equations
### Governing Equations  
Founded on [PERSON]'s local behavior, the differential form of the general EMFs four equations [44] is given by:  
\[\mathbf{\
abla}\times\mathbf{E}=-\partial_{t}\,\mathbf{B}\ \text{([PERSON]--Faraday)} \tag{1}\]  
\[\mathbf{\
abla}\times\mathbf{H}=\sigma\,\mathbf{E}+\partial_{t}\,\mathbf{D}\ \text{(Maxwell--Ampere)} \tag{2}\]  
\[\mathbf{\
abla}\cdot\mathbf{D}=\rho_{\text{e}}\ \text{(Maxwell--Gauss)} \tag{3}\]  
\[\mathbf{\
abla}\cdot\mathbf{B}=0\ \text{(Maxwell--Thomson)} \tag{4}\]  
The HT equation in its differential form is given by:  
\[\text{c}\ \rho\ \partial T/\partial t=\mathbf{\
abla}\cdot\text{(k}\ \mathbf{\
abla}\ T) \tag{5}\]  
Note that, if we assume a constant value of the materials' parameters, the typical time constant of electric/EM phenomena lasts about 3-6 periods of the wave, while thermal issues usually need a few minutes. It means that, for pulse work, sometimes, we can skip thermal analysis. In the case of EMF exposure of living tissues or devices, the EMF harmonic fields, the BH, and the heat source P\({}_{\text{d}}\) equations are given as follows:  
\[\mathbf{\
abla}\times\mathbf{H}=\mathbf{J} \tag{6}\]  
\[\mathbf{J}=\mathbf{J}_{e}+\sigma\,\mathbf{E}+\text{j}\ \omega\ \mathbf{D} \tag{7}\]  
\[\mathbf{E}=-\mathbf{\
abla}\ \text{V}-\text{j}\ \omega\ \mathbf{A} \tag{8}\]  
\[\mathbf{B}=\mathbf{\
abla}\times\mathbf{A}\ \text{with}\ \
abla\cdot\mathbf{A}=0\ \text{(gauge)} \tag{9}\]  
\[\text{c}\ \rho\ \partial T/\partial t=\mathbf{\
abla}\cdot\text{(k}\ \mathbf{\
abla}\ T)+\text{P}_{\text{d}}+\text{P}_{\text{t}}+\text{c}_{\text{f}} \ \rho_{\text{f}}\ \text{P}_{\text{f}}\ \text{(T}_{\text{f}}-T) \tag{10}\]  
\[\mathbf{\
abla}\times\mathbf{H}=\sigma\,\mathbf{E}+\partial_{t}\,\mathbf{D}\ \text{(Maxwell--Ampere)} \tag{2}\]0  
In (1)-(11), \(\mathbf{H}\) and \(\mathbf{E}\) are the magnetic and electric field vectors in A/m and V/m. \(\mathbf{B}\) and \(\mathbf{D}\) are the magnetic and electric induction vectors in T and C/m\({}^{2}\), and \(\mathbf{A}\) and \(V\) are the magnetic vector potential and electric scalar potential in Wb/m and volt. \(\mathbf{J}\) and \(\mathbf{J_{e}}\) are the total and source current densities vectors in A/m\({}^{2}\). The electric conductivity is denoted by \(\sigma\) in S/m, the volume density of electric charges by \(\rho_{\text{e}}\) in C/m\({}^{3}\), and the angular frequency by \(\omega=2\pi\)f, with the frequency f in Hz. The character \(\mathbf{\
abla}\) is a partial derivative vector operator. The character \(\partial_{t}\) is the partial time derivative operator. The behavior's magnetic and electric laws, respectively, given by \(\mathbf{B/H}\) and \(\mathbf{D/E}\), are signified by the permeability \(\mu\) in H/m and the permittivity \(\varepsilon\) in F/m. The symbol \(\varepsilon^{\prime\prime}\) denotes the imaginary part of the complex permittivity (j \(\omega\ \mathbf{D}=\text{j}\ \omega\ (\varepsilon^{\prime}-\text{j}\ \varepsilon^{\prime\prime})\)\(\mathbf{E}\)), and \(\rho\) represents the material density in kg/m\({}^{3}\). E represents the electric field strength (absolute peak value) in V/m; c represents the substance-specific heat (at constant pressure) in J/(kg \({}^{\circ}\)C), with k the thermal conductivity in W/(m \({}^{\circ}\)C) and T the temperature in \({}^{\circ}\)C. The dissipated power volume density in W/m\({}^{3}\) given by Equation (11) links to the principal dielectric EMF loss and will be used in the coupling of EMF and BH equations. The BH Equation (10), related to living tissues, involves a self-tissue heat source P\({}_{\text{t}}\), convective heat transfer via irrigating fluid, and an external heat source related to the EMF exposure P\({}_{\text{d}}\), all in W/m\({}^{3}\). T\({}_{\text{f}}\) and \(T\) are, respectively, the fluid temperature and the local temperature of tissue in \({}^{\circ}\)C, and c\({}_{\text{f}}\), \(\rho_{\text{f}}\), and p\({}_{\text{f}}\) are, respectively, fluid, specific heat in J/(kg \({}^{\circ}\)C), density in kg/m\({}^{3}\), and perfusion rate in 1/s.