# On Anisothermal Electromagnetic Elastic Deformations in Flight in Fair Weather and Lightning Storms
## 3 Electromagnetic and Thermoelastic Effects in a Slab
### Elastic Effects: Displacement, Strain and Stress  
The magnetic force  
\[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}\mathbf{J}\wedge\mathbf{H},\] (51a) associated with the electric current ( 35d ) and magnetic field ( 37a ), is given by \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\] (51b) and thus lies in the \[x\] -direction normal to the slab and is given by \[\mathbf{F}_{\mathrm{m}}=\chi\frac{\sqrt{\varepsilon\mu}}{c}E_{0}^{2}\exp\left( -\frac{2x}{l}\right)\mathbf{e}_{\mathrm{x}}=\frac{\chi}{c_{\mathrm{em}}}E_{0} ^{2}\exp\left(-\frac{2x}{l}\right)\mathbf{e}_{\mathrm{x}},\] (51c) where the electromagnetic wave speed ( 35a ) appears. Taking the displacement vector in the \[x\] -direction, \[\mathbf{u}(x)=u(x)\mathbf{e}_{\mathrm{x}},\] (52a) and the only non-zero component of the strain is \[e_{\mathrm{xx}}=\frac{\mathrm{d}u}{\mathrm{d}x}, \tag{52b}\]and the corresponding stress (21d) is  
\[\tau_{\rm xx}(x)=-\alpha[T(x)-T_{\rm ref}]+\
u\frac{\mathrm{d}u}{\mathrm{d}x}, \tag{52c}\]  
where \(T_{\rm ref}\) is a reference temperature at which the thermal stress is zero, which involves the effective elastic modulus  
\[\
u=\frac{K}{1+\sigma}\frac{1-\sigma}{1-2\sigma}. \tag{53}\]  
For steady conditions, there is no inertia force in (17), and, since there are no electric charges (39d), there is no electric force in (17), which simplifies to the balance of the magnetic force (51c) and stress (52c)  
\[-F_{\rm m}=\frac{\mathrm{d}\tau_{\rm xx}}{\mathrm{d}x},\] (54a) leading to the differential equation for the displacement \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]0 \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]1 (54b) which is solved next, together with boundary conditions. The differential equation for the displacement (54b) can be rewritten as  
\[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]2 (55a) involving the constants \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]3 \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]4 (55b) where \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]5, so \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]6, and thus the last term in square brackets can be omitted. The integration of ( 55a ) leads to the following: \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]7  
Assuming zero displacement at the inner boundary,  
\[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]8 (57a) eliminates one constant: \[\mathbf{F}_{\mathrm{m}}=\frac{\mu}{c}J_{0}H_{0}\exp\left(-\frac{2x}{l}\right) \mathbf{e}_{\mathrm{y}}\times\mathbf{e}_{\mathrm{z}},\]9 (57b) The displacement ( 57b ) corresponds to the strain \[x\]0and stress at the outer boundary balancing the atmospheric pressure  
\[x\]1  
taking (47b) as the reference temperature for the thermal stresses. This determines the constant \(C_{3}\),  
\[x\]2  
involving  
\[x\]3  
Substitution of (58d) in (57b) specifies the displacement  
\[x\]4  
the strain  
\[x\]5  
and the stress  
\[x\]6  
using (60), (58d), (55b) and (49). The terms on the right-hand side of the stress are due to (i) the external pressure (58b); (ii) the temperature difference between the local temperature and the reference temperature; (iii) the magnetic force (51c) in a form that vanishes at the inner boundary.