# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 2 Fundamental Equations of Dissipative Electromagnetic Thermoelasticity
### Isotropic Media and Steady Fields
Similarly, the isotropic conditions (19b) apply to the thermoelectric diffusion relations (14a) and (15), leading to the electric current  
\[\theta_{ij}=0. \tag{19c}\]1 (22a) and heat flux \[\theta_{ij}=0. \tag{19c}\]2  
The fundamental equations for isotropic matter may be obtained equivalently by replacing the isotropic constitutive (21a)-(21d) and diffusion (22a)-(22b) relations in the equations of (i) electricity (13a) and (21c):  
\[\theta_{ij}=0. \tag{19c}\]3  
(ii) magnetism (14b), (21b) and (22a):  
\[\theta_{ij}=0. \tag{19c}\]4  
(iii) energy (16) and (22b):  
\[\theta_{ij}=0. \tag{19c}\]5(iv) momentum (17), (21b) and (21c):  
\[\theta_{ij}=0. \tag{19c}\]6 \[\theta_{ij}=0. \tag{19c}\]7  
where the divergence of the stress tensor (21d) is used in vector form,  
\[\theta_{ij}=0. \tag{19c}\]8 \[\theta_{ij}=0. \tag{19c}\]9 \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]0  
The same equations of electricity (13a)\(\rightarrow\)(23), magnetism (14b)\(\rightarrow\)(24), energy (16)\(\rightarrow\)(25) and momentum (18)\(\rightarrow\)(26) follow from anisotropic\(\rightarrow\)isotropic matter using the isotropic constitutive relations--(19a), (19b), (19c), (20a) and (20d)--and diffusion relations (20a). Thus, the electric \(\mathbf{E}\) and magnetic \(\mathbf{H}\) fields, displacement vector \(\mathbf{u}\) and temperature \(T\) satisfy the fundamental equations of electricity (23), magnetism (24), energy (25) and momentum (26) (i) for an isotropic elastic solid subject to electromagnetic forces; (ii) with coupled heat flux and electric current; (iii) assuming constant constitutive and diffusion coefficients; (iv) in unsteady conditions. In steady conditions \(\partial/(\partial t)=0\), (i) the electric field decouples (23) with zero curl:  
\[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]1 (28a) (28b) (ii) the divergence of the electric field shows it is generated by electric charges: \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]2 (28c) (iii) the magnetic induction satisfies the following: \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]3 (29a) (iv) the equation of magnetism ( 24 ) also decouples considering Ohmic electric currents and no thermoelectric coupling \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]4 : \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]5 (29b) (v) the energy equation ( 25 ) considering only thermal conduction and no thermoelectric coupling \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]6 simplifies to [PERSON]'s equation \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]7 (29c) for temperature in the absence of heat sources; (vi) the momentum equation ( 17 ) with external electric charges \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]8 and currents \[\big{\{}\varepsilon_{ij},\mu_{ij},\alpha_{ij},k_{ij},\chi_{ij},\eta_{ij}\big{\}} =\delta_{ij}\big{\{}\varepsilon,\mu,\alpha,k,\chi,\eta\big{\}},\]9 and elastic stresses ( 27 ) simplifies to \[\varepsilon\]0  
These equations are solved next in the one-dimensional case of a parallel-sided slab (Section 3).