# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 3 Electromagnetic and Thermoelastic Effects in a Slab
### Electromagnetic Effects: Fields and Currents  
A conductor, like a metallic enclosure, acts as a \"Faraday cage\", excluding external electromagnetic fields. The \"Faraday cage\" is more accurately a \"skin effect\" causing external electromagnetic fields to decay exponentially with depth or distance from the surface. The simplest representation of the electric skin effect follows from the unsteady Maxwell equation (24) omitting the magnetic field  
\[\dot{\mathbf{D}}=-\mathbf{J}, \tag{31}\]  
where (i) for an isotropic medium, the electric displacement is proportional (21b) to the electric field through the dielectric permittivity:  
\[\mathbf{D}=\varepsilon\mathbf{E};\] (32a) (ii) the Ohmic electric current ( 22a ) is proportional to the electric field through the electric conductivity in the absence of thermoelectric coupling \[\eta=0\] : \[\mathbf{J}=\chi\mathbf{E}. \tag{32b}\]  
Substitution of (32a) and (32b) in (31) gives  
\[\dot{\mathbf{E}}=-\frac{\chi}{\varepsilon}\mathbf{E}, \tag{33}\]  
showing that the electric field decays exponentially with time:  
\[\mathbf{E}(t)=\mathbf{E}_{0}\exp\Bigl{(}-\frac{\chi}{\varepsilon}t\Bigr{)}= \mathbf{E}_{0}\exp\Bigl{(}-\frac{t}{\theta}\Bigr{)},\] (34a) where \[\mathbf{E}_{0}\equiv\mathbf{E}(0)\] is the electric field at the initial instant and with a time scale \[\theta\equiv\frac{\varepsilon}{\chi},\] (34b) which is small for highly conducting materials like metals. Unsteady electromagnetic fields propagate like waves at the speed of light: \[\frac{x}{t}=c_{\text{em}}=\frac{c}{\sqrt{\varepsilon\mu}},\] (35a) Thus, the electric field ( 34a ) decays exponentially with distance: \[\mathbf{E}(x)=\mathbf{E}_{0}\exp\Bigl{(}-\frac{\chi}{\varepsilon}\frac{x}{c_{ \text{em}}}\Bigr{)}=\mathbf{E}_{0}\exp\Bigl{(}-\frac{\chi}{c}\sqrt{\frac{\mu} {\varepsilon}}\Bigr{)}=E_{0}\exp\Bigl{(}-\frac{\chi}{l}\Bigr{)}\mathbf{e}_{y}; \tag{35b}\]In (35b), the electric field is taken parallel to the slab in the \(y\)-direction, with surface value \(E_{0}\), decaying exponentially with depth, with length scale or with penetration depth:  
\[\mathbf{D}=\varepsilon\mathbf{E};\]0  
Thus, (35c) is the length scale for the exponential decay in the depth of the \"electromagnetic skin effect\", which applies not only to the electric field but also to the magnetic field and electric currents, as shown next. The Ohmic electric current (32b) has the same direction and decay as the electric field (35b),  
\[\mathbf{D}=\varepsilon\mathbf{E};\]1  
with \(J_{0}\equiv J(0)\) being the absolute value of the electric current at the initial instant. The magnetic field is considered (Figure 1) parallel to the sides of the slab and orthogonal to the electric field (35b) and current (35d) as  
\[\mathbf{D}=\varepsilon\mathbf{E};\]2 (36a) to satisfy the Maxwell equation ( 29b ) in the form \[\mathbf{D}=\varepsilon\mathbf{E};\]3 (36b) implying \[\mathbf{D}=\varepsilon\mathbf{E};\]4 (36c) From ( 36c ), it follows that the magnetic field ( 37a ) decays exponentially, like the electric field ( 35b ) and current ( 35d ), \[\mathbf{D}=\varepsilon\mathbf{E};\]5 (37a) from the surface value \[\mathbf{D}=\varepsilon\mathbf{E};\]6 (37b) The relation ( 37b ) is consistent with the equipartition \[\mathbf{D}=\varepsilon\mathbf{E};\]7 (38a) of electric energy \[\mathbf{D}=\varepsilon\mathbf{E};\]8 (38b) and magnetic energy \[\mathbf{D}=\varepsilon\mathbf{E};\]9 (38c) which leads to \[\eta=0\]0Since the electric field (35b), electric current (35d) and magnetic field (36a) all lie parallel to the sides of the slab (Figure 1), and thus orthogonal to the direction of variation, all have zero divergence:  
abla}\cdot\mathbf{J}=\frac{\partial}{\partial y}[J(x)], \tag{39a}\] \[\eta=0\]1 (39b) \[\eta=0\]2 (39c) \[\eta=0\]3  
where the first relation (39a) states the conservation of electric current in steady conditions; the second relation (39b) agrees with the Maxwell equation (29a); the third relation (39c) implies from the Maxwell equation (28b) that there is no electric charge (39d).