# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 3 Electromagnetic and Thermoelastic Effects in a Slab
### Thermal Effects: Temperature and Heat Flux  
The Joule effect of heat production by dissipation of electric currents in a resistive medium [8; 13] is specified by the heat source  
\[Q=\mathbf{J}\cdot\mathbf{E}=\chi|\mathbf{E}|^{2}=\frac{|\mathbf{j}|^{2}}{\chi}, \tag{41}\]  
leading, in the case of the skin effect (35d), to the depth dependence  
\[Q(x)=Q_{0}\exp\biggl{(}-\frac{2x}{l}\biggr{)},\] (42a) with surface value \[Q_{0}=E_{0}J_{0}=\chi(E_{0})^{2}=\frac{(J_{0})^{2}}{\chi}. \tag{42b}\]  
The steady heat conduction balances the heat sources \(Q\) in a domain \(D\) of volume \(\mathrm{d}V\) against the heat flux \(\mathbf{G}\) through the boundary \(\partial D\) with area element \(\mathrm{d}\mathbf{A}\):  
\[\int_{D}Q\,\mathrm{d}V=\int_{\partial D}\mathbf{G}\cdot\mathrm{d}\mathbf{A}= \int_{D}(\mathbf{\
abla}\cdot\mathbf{G})\,\mathrm{d}V,\] (43a) Use of the divergence theorem leads to the steady heat equation with sources [15] \[\mathbf{\
abla}\cdot\mathbf{G}=Q.\] (43b) Using the Fourier heat flux ( 22b ), and neglecting thermoelectric coupling, \[\mathbf{G}=-k\mathbf{\
abla}T\] (44) leads [15] to the steady heat equation with sources \[\
abla^{2}T=-\frac{Q}{k},\] (45) which specifies the temperature and reduces to ( 29c ) in the absence of heat sources.  
Thus, in the interior of the slab, the temperature satisfies  
\[\frac{\mathrm{d}^{2}T}{\mathrm{d}x^{2}}=-\frac{Q(x)}{k}=-\frac{Q_{0}}{k}\exp \biggl{(}-\frac{2x}{l}\biggr{)}=-\frac{\chi}{k}(E_{0})^{2}\exp\biggl{(}-\frac {2x}{l}\biggr{)}, \tag{46}\]  
together with boundary conditions on the two sides. The integration of (46) specifies the temperature profile  
\[T(x)=-T_{*}\exp\biggl{(}-\frac{2x}{l}\biggr{)}+C_{1}x+C_{2}, \tag{47a}\]where (i) the constant \(T_{*}\) has the dimensions of temperature:  
\[T_{*}=\frac{Q_{0}l^{2}}{4k}=\frac{\chi}{4k}(E_{0}l)^{2}=\frac{\varepsilon}{4\mu k \chi}(E_{0}c)^{2}; \tag{47b}\]  
(ii) the temperature at the outer boundary specifies the constant \(C_{1}\):  
\[T_{0}\equiv T(0)=C_{2}-T_{*}; \tag{48a}\]  
(iii) the temperature at the inner boundary specifies the constant \(C_{2}\):  
\[T_{1}\equiv T(L)=-T_{*}\exp\left(-\frac{2L}{l}\right)+C_{1}L+T_{0}+T_{*}. \tag{48b}\]  
Solving (48a) and (48b) for \(C_{1}\) and \(C_{2}\), and substitution in (47a), specifies the temperature profile  
\[T(x)=T_{0}+\frac{T_{1}-T_{0}}{L}x+T_{*}\left(1-\frac{x}{L}\right)-T_{*}\left[ \exp\left(-\frac{2x}{l}\right)-\frac{x}{L}\exp\left(-\frac{2L}{l}\right)\right] \tag{49}\]  
where, on the right-hand side of (49), (i) the first two terms correspond to the boundary temperatures (48a) and (48b) without sources; (ii) the last two terms are due (47b) to the Joule heating--Equations (41), (42a) and (42b). The heat flux (44) is related by  
\[\mathbf{G}=-k\frac{\mathrm{d}T}{\mathrm{d}x}\mathbf{e}_{\mathrm{x}},\] (50a) to the gradient of the temperature ( 49 ): \[\mathbf{G}(x)=\mathbf{e}_{\mathrm{x}}k\bigg{\{}\frac{T_{*}+T_{0}-T_{1}}{L}-T_ {*}\left[\frac{2}{l}\exp\left(-\frac{2x}{l}\right)+\frac{1}{L}\exp\left(- \frac{2L}{l}\right)\right]\bigg{\}}. \tag{50b}\]  
The temperature gradient causes thermal stresses that, together with electromagnetic forces, balance the elastic stresses, thus specifying the displacement, strain and stress (Section 3.3).