Update textbook.fr.md

12.temporary_ins/96.electromagnetism-in-media/20.reflexion-refraction-at-interfaces/20.metallic-waveguides/10.main/textbook.fr.md

@@ -276,52 +276,26 @@ Let's consider a $`TE_{0,n}`$ mode as described by equations
 $`k\,\cos\theta=k_y=n\pi/b`$ and $`k\,\sin\theta=k_z`$. We get
 
 $`\overrightarrow{E}_{\perp}=`$$`E_0\,\sin\Big(\dfrac{n\pi}{b}\,y\Big)\,\sin(k_z\,z-\omega\,t})\overrightarrow{e_x}`$
+$`\quad(eq. 4.11)`$ 
 
 and
 
    
+$`\overrightarrow{B}_{\perp}=`$
+$`\left(\begin{array}{l}
+0\\
+\dfrac{E_0}{c}\dfrac{k_z}{k}\sin\big(\frac{n\pi}{b}\,y\big)\sin\,(k_z\,z -\omega\,t)\\
+\dfrac{E_0}{c}\dfrac{n\pi}{b\,k}\cos\big(\frac{n\pi}{b}\,y\big)\sin\,(k_z\,z -\omega\,t)
+\end{array}\right)`$$`\quad(eq. 4.12)`$  
 
-\cdot \sin\big(\underbrace{k\,\sin\theta}_{\large{wavevector}} z-\omega t\big)\overrightarrow{e_z}`$$`\quad(eq.1)`$    
-
-
-
-
-and
-
-⊥ *b z* *x*
->
-0
->
-$`\overrightarrow{B}`$~⊥~ =  *E*0 *kz* sin ( *nπ y*) sin (*k~z~z* − *ωt*)
->
-(4.12)
-
- *E*0 *nπ* cos ( *nπ y*) sin (*k~z~z* − *ωt*)
-
-1 *E*^2^ *k~z~*
-
-2 *[nπ]{.underline}*
+The time-averaged Poynting vector becomes:
 
-(**P**) = 2 *cµ*0
 
-sin
->
-*k*
->
-*y \_e~z~* (4.13)
->
-*b*
->
 The power transmitted by the guide (units \[*W* \]) can be found by
 integrating the previous results over the cross-section of the
 waveguide
 
-= { *a* { *b*
 
-= 1 *E*^2^ *k~z~*
->
-(4.14)
->
 i.e. the transmitted power is proportional to the cross-sectional area
 of the waveguide. The practical limit of transmittable power is set by
 the dielectric breakdown of the dielectric filling the waveguide. In