Update textbook.fr.md

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

@@ -275,27 +275,33 @@ Let's consider a $`TE_{0,n}`$ mode as described by equations
 "-2" from the amplitudes and use the fact that 
 $`k\,\cos\theta=k_y=n\pi/b`$ and $`k\,\sin\theta=k_z`$. We get
 
-$`\overrightarrow{E}_{\perp}=`$$`E_0\,\sin\Big(\dfrac{n\pi}{b}\,y\Big)\,\sin(k_z\,z-\omega\,t})\overrightarrow{e_x}`$
+$`\overrightarrow{E}_{\perp}=`$$`E_0\,\sin\Big(\dfrac{n\pi}{b}\,y\Big)\,\sin(k_z\,z-\omega\,t)\overrightarrow{e_x}`$
 $`\quad(eq. 4.11)`$ 
 
 and
 
    
-$`\overrightarrow{B}_{\perp}=`$
-$`\left(\begin{array}{l}
+$`\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)`$  
+\end{array}\right)`$
+$`\quad(eq. 4.12)`$  
 
 The time-averaged Poynting vector becomes:
 
+$`\langle\overrightarrow{P}\rangle=
+\dfrac{1}{2}\,\dfrac{E_0^{\;2}}{c\mu_0}\,\dfrac{k_z}{z}\,\sin^2\Big(\dfrac{n\pi}{b}\,y\Big)\,\overrightarrow{e_z}`$
+$`\quad(eq. 4.13)`$  
 
 The power transmitted by the guide (units \[*W* \]) can be found by
 integrating the previous results over the cross-section of the
 waveguide
 
 
+
+
 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