Update textbook.fr.md

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

@@ -33,12 +33,12 @@ important to find some of the results later in this chapter. The
 boundary conditions can be obtained from the Maxwell equations. The
 first two equations
 
-@@@@@@@@@@@@ $`\quad (equ. 3.4)`$
+équation 3.4
 
 will give information on the normal components of $`\overrightarrow{D}`$ and $`\overrightarrow{B}`$,
 while the third and fourth
 
-@@@@@@@@@@@@ $`\quad (equ. 3.5)`$
+équation 3.5
 
 will give information on the tangential components of $`\overrightarrow{E}`$ and $`\overrightarrow{H}`$.
 
@@ -64,7 +64,7 @@ figure [3.1] which extends from one side to the other on
 the separation surface. This box has a base surface A and an
 infinitesimally small thickness $`\delta`$. We get:
 
-@@@@@@@@@@@@@@@@@@@@@@@@@ $`\quad (equ. 3.6)`$
+équation 3.6
 
 If we now let $`\delta\longrightarrow 0`$ symmetrically with respect to the separation
 surface such that the cylinder gets "pressed" onto the surface:
@@ -82,20 +82,20 @@ surface such that the cylinder gets "pressed" onto the surface:
 
 We obtain:
 
-@@@@@@@@ $`\quad (equ. 3.7)`$
+équation 3.7
 
 Now, considering that $`d\overrightarrow{a_2}=-d\overrightarrow{a_1}`$ we can write:
 
-@@@@@@@@@@@@@ $`\quad (equ. 3.8)`$
+équation 3.8
 
 Finally, as $`S_1=S'`$ and $`d\overrightarrow{a_2}=\overrightarrow{n}_{2\rightarrow 1}\,da_1`$ we can
 write:
 
-@@@@@@@@@ $`\quad (equ. 3.9)`$
+équation 3.9
 
 or
 
-@@@@@@@@@@@ $`\quad (equ. 3.10)`$
+équation 3.10
 
 The normal component of the vector $`\overrightarrow{D}`$ is in general discontinuous.
 It is continuos only if there are no conduction charges at the
@@ -107,11 +107,11 @@ The situation is identical for the vector $`\overrightarrow{B}`$, the only diffe
 being that the right hand side of the equation is always 0. We
 conclude that:
 
-@@@@@@@@@ $`\quad (equ. 3.11)`$
+équation 3.11
 
 or
 
-@@@@@@@@@ $`\quad (equ. 3.12)`$
+équation 3.12
 
 The normal component of $`\overrightarrow{B}`$ is always conserved.
 
@@ -130,22 +130,22 @@ figure [3.2.]. We chose to integrate the line integral
 following the right-hand sense relative to the surface normal
 $`\overrightarrow{n_a}`$. By letting $`\delta\rightarrow 0`$, we get
 
-@@@@@@@@@@@ $`\quad (equ. 3.13)`$
+équation 3.13
 
 as the line integral along the sides goes to zero and the flux of the
 induction field $`\overrightarrow{B}`$, which is a finite quantity, approaches 0.
 Considering that $`\overrightarrow{CD}-\overrightarrow{AB}=d\vec{l}`$, we get:
 
-@@@@@@@@@@ $`\quad (equ. 3.14)`$
+équation 3.14
 
 or again
 
-@@@@@@@@@@@@ $`\quad (equ. 3.15)`$
+équation 3.15
 
 i.e. the tangential components of the electric field are always
 conserved at the interface. This condition can also be written:
 
-@@@@@@@@@@ $`\quad (equ. 3.16)`$
+équation 3.16
 
 __$`\overrightarrow{H}`$ vector__
 
@@ -153,7 +153,7 @@ Following the same reasoning as for the $`\overrightarrow{E}`$ vector we write f
 left hand side when $`\delta\rightarrow 0`$ :
 
 
-@@@@@@@@@@@@@ $`\quad (equ. 3.17)`$
+équation 3.17
 
 The right hand side needs more attention. The flux of the vector $`\overrightarrow{D}`$
 approaches 0. However the flux of the vector $`\overrightarrow{J}`$ over an infinitesimal surface
@@ -165,31 +165,31 @@ not $`[A/m*^2]`$ as $`\overrightarrow{J}`$) can. This is typically the case of a
 perfect conductor where a finite current can flow through a
 infinitesimally small area. We get:
 
-@@@@@@@@ $`\quad (equ. 3.18)`$
+équation 3.18
 
 or again
 
-@@@@@@@@ $`\quad (equ. 3.19)`$
+équation 3.19
 
 i.e. the tangential components of the magnetic field $`\overrightarrow{H}`$ is
 discontinuous unless no surface currents exist. As for the electric
 field, this condition can also be written:
 
-@@@@@@@@ $`\quad (equ. 3.20)`$
+équation 3.20
 
 !! *Summary*   
 !! 
 !! *vector form*   
-!! * @@@@@
-!! * @@@@@
-!! * @@@@@
-!! * @@@@@
+!! * x
+!! * x
+!! * x
+!! * x
 !!
 !! *component form*
-!! * @@@@@
-!! * @@@@@
-!! * @@@@@
-!! * @@@@@
+!! * x
+!! * x
+!! * x
+!! * x
 
 
 ! *Remarks*   
@@ -207,7 +207,7 @@ field, this condition can also be written:
 ! of $`\overrightarrow{E}`$ and $`\overrightarrow{B}`$ alone using the constitutive relations
 ! $`\overrightarrow{D}=\epsilon\,\overrightarrow{E}`$ and $`\overrightarrow{B}=\mu\,\overrightarrow{H}`$.
 
-&&&&&&&&&&&&&&&&&&&&&&&&&
+équation 
 
 <br>
 
@@ -233,7 +233,7 @@ conductor. For the perfect conductor, as *σ* , its penetration depth
 transmission can occur, the wave is totally reflected. The incident
 and reflected waves are:
 
-@@@@@@@@ $`\quad (equ. 3.21)`$
+équation 3.21
 
 ![](electromag-in-media-reflexion-transmission-fig-33a.jpg)   
 ![](electromag-in-media-reflexion-transmission-fig-33b.jpg)     
@@ -243,25 +243,25 @@ _different times._
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.22)`$
+équation 3.22
 
 Using the boundary conditions at the interface (*z* = 0) we have:
 
-@@@@@@@@ $`\quad (equ. 3.23)`$
+équation 3.23
 
 as the boundary conditions are valid in any point of the surface and
 at any time. We cab therefore write the reflected wave as:
 
-@@@@@@@@ $`\quad (equ. 3.24)`$
+équation 3.24
 
 In medium 1 we have a superposition of the incident and reflected
 waves. The total wave is given by:
 
-@@@@@@@@ $`\quad (equ. 3.26)`$
+équation 3.26
 
 or in real notation
 
-@@@@@@@@@@@
+équation 
 
 This wave represents a **stationary wave**1: a wave the oscillates in
 place. The nodes and antinodes of the electric field and magnetic
@@ -276,23 +276,23 @@ net energy. This can be readily understood as as much energy is
 carried by the incident wave from right to left as the reflected one
 in the opposite direction. Calculating the Poynting vector we have:
 
-@@@@@@@@ $`\quad (equ. 3.27)`$
+équation 3.27
 
 which at any point *z* changes direction periodically. Its
 time-averaged values is indeed *\<* $`\overrightarrow{S}`$*~tot~ \>~t~*= 0.
 
 Due to the discontinuity of the magnetic field, a surface current
-density must
+density must be present at the interface. Using the fourth boundary condition we get:
 
 ![](electromag-in-media-reflexion-transmission-fig-34a.jpg)   
 ![](electromag-in-media-reflexion-transmission-fig-34b.jpg)     
 _Figure 3.4: Left: The incident, reflected and resulting wave at a_
 _particular time. Right: The superposition of several resulting waves_
-_at different times. See the video of the simulation on the_
-_[moodle_page.](https://moodle.insa-toulouse.fr/course/view.php?id=621)_
-_be present at the interface. Using the fourth boundary condition we get:_
+_at different times._
 
-@@@@@@@@@
+![](standing_wave_partial_reflection_SWR=2_v2.gif)
+
+équation 
 
 ##### Case 2: 2 perfect dielectrics
 
@@ -301,7 +301,7 @@ will consider that the materials are non-magnetic (*µ* = *µ*~0~) and
 that no charges nor currents exist at the interface (*σ~s~* = 0*,*
 $`\overrightarrow{j}`$*~s~* = 0). The incident ware is given by
 
-@@@@@@@@ $`\quad (equ. 3.28)`$
+équation 3.28
 
 ![](electromag-in-media-reflexion-transmission-fig-35a.jpg)   
 ![](electromag-in-media-reflexion-transmission-fig-35b.jpg)     
@@ -313,35 +313,35 @@ _$`\overrightarrow{E}`$*~tot~* is no more 0._
 where **k**~1~ and *n*~1~ are the wavevecvtor and refractive index of
 medium 1. The reflected and transmitted waves are:
 
-@@@@@@@@ $`\quad (equ. 3.29)`$
+équation 3.29
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.30)`$
+équation 3.30
 
 We need now two boundary conditions to determine the reflected and
 trans- mitted waves. Using the tangential boundary conditions for
 $`\overrightarrow{E}`$ and $`\overrightarrow{H}`$ we have:
 
-@@@@@@@@ $`\quad (equ. 31)`$
+équation 3.31
 
 Solving for *E*^0^ and *E*^0^ we obtain:
 
-@@@@@@@@@
+équation 
 
 We can now define the
 
-@@@@@@@@@@@@@
+équation 
 
 *   Reflection coefficient *r* = *^E^*0 = [*n* −*n*]{.underline}
 
 and
 
-@@@@@@@@@@@@
+équation 
 
 *   the Transmission coefficient *t* = *^E^*0 = [2*n *]{.underline} .
 
-@@@@@@@@@
+équation 
 
 <br>
 
@@ -398,7 +398,7 @@ _Figure 3.7: General case of reflection and refraction_
 
 monochromatic wave are:
 
-@@@@@@@@@@@@@@
+équation 
 
 The boundary conditions are applied to the interface (*z* = 0) and
 must be valid for **all points on the interface** and **for all
@@ -407,29 +407,27 @@ times**.
 In *z* = 0, for the tangential component of the electric field we
 have:
 
-@@@@@@@@@@@@
+équation 
 
-@@@@@@@@
-
-@@@@@@@@ $`\quad (equ. 3.35)`$
+équation 3.35
 
 where **r***~s~* is a vector on the interface.
 
 i.  For this condition to be valid *t* we must have *ω* = *ω*^t^ =
     *ω*^tt^. As a consequence we have
 
-@@@@@@@@ $`\quad (equ. 3.36)`$
+équation 3.36
 
-@@@@@@@@ $`\quad (equ. 3.37)`$
+équation 3.37
 
 ii. For this condition to be valid []{#_bookmark65 .anchor}∀**r***~s~*
     we must have:
 
-@@@@@@@@ $`\quad (equ. 3.38)`$
+équation 3.38
 
 Taking the first two terms we can write
 
-@@@@@@@@ $`\quad (equ. 3.39)`$
+équation 3.39
 
 As **r***~s~* belongs to the interface, (**k***~i~* **k***~r~*) is
 normal to it. This means that the vectors **k***~i~*, **k***~r~* and
@@ -438,18 +436,18 @@ the normal belong to the same plane, i.e. the plane of incidence.
 !! *First law of reflexion*   
 !! ....
 
-iii. Equation [3.38](#_bookmark65) can be recast as
+iii. Equation [3.38] can be recast as
 
-@@@@@@@@ $`\quad (equ. 3.40)`$
+équation 3.40
 
 These relations can be satisfied only if all the *x* and *y*
 components are the same. Taking the *x* components we can write
 
-a.  @@@@@@@@@
+a. équation 
 
-b.  @@@@@@@@
+b. équation
 
-c.  @@@@@@@
+c. equation
 
 From the first two relations we get *θ~i~* = *θ~r~* as \|*k~i~*\| =
 \|*k~r~*\|.
@@ -474,7 +472,7 @@ sin *θ~t~* or
 From the previous section we can write the incident, reflected and
 transmitted waves as:
 
-@@@@@@@@@@
+equation
 
 where the indices "1" and "2" indicate the first and second medium
 physical quantities. We will now separate the discussion for a TE and
@@ -496,16 +494,16 @@ From figure [3.8](#_bookmark67) we can write using the two tangential
 boundary conditions and the relations $`\overrightarrow{B}`$ = *µH*, \|$`\overrightarrow{B}`$\| =
 ^[\|$`\overrightarrow{E}`$\|]{.underline}^ *n*:
 
-@@@@@@@@
+equation
 
 As before, we define the reflection and transmission coefficient for a
 TE ( or
 
-@@@@@@@@ $`\quad (equ. 3.42)`$
+equation 3.42
 
 which by solving the previous equations can be evaluated to:
 
-@@@@@@@@ $`\quad (equ. 3.43)`$
+equation 3.43
 
 ! *Remarks*   
 ! ....
@@ -517,12 +515,12 @@ transverse. From figure [3.8](#_bookmark67) we can write using the two
 tangential boundary conditions and the relations $`\overrightarrow{B}`$ = *µH*,
 \|$`\overrightarrow{B}`$\| = ^[\|$`\overrightarrow{E}`$\|]{.underline}^ *n*:
 
-@@@@@@@@
+equation 
 
 Solving the system we obtain the reflection and transmission
 coefficients for a TM (/I or *p*) wave as:
 
-@@@@@@@@ $`\quad (equ. 3.44)`$
+equation 3.44
 
 
 ##### Brewster's angle
@@ -534,9 +532,9 @@ the relative value of the two media refrac- tive indices *n*~1~ and
 *n*~2~. By using Snell-Descartes law *n*~1~ sin(*θ*~1~) = *n*~2~
 sin(*θ*~2~) we have that
 
-a.  If @@@@@@@
+a.  If equation 
 
-b.  If @@@@@@@@
+b.  If equation
 
 
 ##### Case a)
@@ -545,7 +543,7 @@ In this case, we consider the range 0 *θ*~1~ *π/*2 for the incident
 angle *θ*~1~. Correspondingly, the refraction angle *θ*~2~ will vary
 in the range 0 *θ*~2~ *θ*2*max*. Using Snell we have:
 
-@@@@@@@@@@@
+equation
 
 ![](electromag-in-media-reflexion-transmission-fig-39a.jpg)    
 ![](electromag-in-media-reflexion-transmission-fig-39b.jpg)    
@@ -561,7 +559,7 @@ _Figure 3.10: Plot of the reflection_
 _coefficients (*r*~⊥~ and *r*I/) and the correspond- ing reflectivities_
 _(*R*~⊥~ and *R*I/) for case a (top) and case b (bottom)._
 
-@@@@@@@@@
+equation
 
 This signifies that for the particular angle *θ~B~* (the Brewster's
 angle) *r*I/(*θ~B~*) = 0, as shown in figure [3.10.](#_bookmark68) By
@@ -600,11 +598,11 @@ will still hold.
 
 The condition *r~p~* = 0 inserted in the Fresnel equation gives
 
-@@@@@@@@ $`\quad (equ. 3.45)`$
+equation 3.45
 
 while from Snell's law we have
  
-@@@@@@@@ $`\quad (equ. 3.46)`$
+equation 3.46
 
 ![]()    
 _Figure 3.12: Photographs of a window taken_
@@ -617,7 +615,7 @@ _[https://en.wikipedia.org/wiki/Brewster's_angle](https://en.wikipedia.org/wiki/
 By multiplying the left and the right terms of the previous equations
 together we get
 
-@@@@@@@@ $`\quad (equ. 3.47)`$
+equation 3.47
 
 which gives as only possible solution *θ~B~* + *θ*~2~ = *π/*2 ( i.e.
 the transmitted and reflected beams are perpendicular, see figure
@@ -630,44 +628,44 @@ always smaller than the limit angle for total reflection.
 Let's consider the oblique incidence shown in figure
 [3.8.](#_bookmark67) The transmitted field is given by
 
-@@@@@@@@ $`\quad (equ. 3.48)`$
+equation 3.48
 
 Using the boundary conditions we have previously found that
 
-@@@@@@@@ $`\quad (equ. 3.49)`$
+equation 3.49
 
 Using this relation we can write
 
-@@@@@@@@ $`\quad (equ. 3.50)`$
+equation 3.50
 
 When the incidence occurs at the limit angle *θ*~1~ = *θ*1*lim*, Snell
 law gives *n*~2~ = *n*~1~ sin *θ*1*lim*. If now we increase the
 incidence angle, we have *θ*~1~ *\θ*1*lim* and thus *n*~2~ *\< n*~1~
 sin *θ*~1~ or *n*^2^ − *n*^2^ sin2 *θ*~1~ *\<* 0. Plugging this into
-equation [3.49,](#_bookmark71) we
+equation [3.49,] we
 
 obtain:
 
-@@@@@@@@ $`\quad (equ. 3.51)`$
+equation 3.51
 
 i.e. (*k*~2~)*~z~* is purely imaginary and the transmitted wave
 becomes:
 
-@@@@@@@@ $`\quad (equ. 3.52)`$
+equation 3.52
 
 The wave has a propagating character in the *x* direction and an
 evanescent character in the *z* direction. Let's consider the case of
 a TE wave. The Fresnel relation gives
  
-@@@@@@@@ $`\quad (equ. 3.53)`$
+equation 3.53
 
 As (*k*~2~)*~z~* is purely imaginary we have, as expected,
 
-@@@@@@@@
+equation
 
 but also
 
-@@@@@@@@ $`\quad (equ. 3.55)`$
+equation 3.55
 
 i.e. the reflected wave is totally reflected with a phase shift.
 
@@ -697,14 +695,14 @@ incident and reflected fields.
 Considering the situation depicted in figure [3.13,] the
 incident wavevector is given by:
 
-@@@@@@@@ $`\quad (equ. 3.56)`$
+equation 3.56
 
 Inserting this into the incident fields, and calculating the components of the
 $`\overrightarrow{E}`$ and $`\overrightarrow{B}`$ fields form the picture (or using 
 the plane wave rule $`\overrightarrow{B}_i=\dfrac{\overrightarrow{k}_i\times \overrightarrow{E}_i}{\omega}`$)
 , we obtain :
 
-@@@@@@@@@@@@ 
+equation
  
 2there is "-" sign in front of the fields as the chosen electric field
 drawn in figure [3.13](#_bookmark74) is antiparallel to the *x* axis.
@@ -716,42 +714,42 @@ _Figure 3.13 : Configuration des champs e.m pour le mode TE._
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.58)`$
+equation 3.58
 
 ##### Reflected fields
 
-@@@@@@@@@@@@
+equation
 
 The reflected electric field **[E]{.underline}***~r~* = *E*~0~ e*i*
 (*ky* cos *θ* + *kz* sin *θ* − *ωt*)**ˆe***~x~* is obtained by
 repeating the same procedure. Considering that the incident and
 reflected angles are the same, we have:
 
-@@@@@@@
+equation
 
 In addition, applying the boundary condition to the tangential
 component of the electric field (which is here the total electric
 field) we have *E~r~x* + *E~i~x* = 0. We finally obtain:
 
-@@@@@@@@ $`\quad (equ. 3.59)`$
+equation 3.59
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.60)`$
+equation 3.60
 
 ##### Total fields
 
-@@@@@@@@@@
+equation 3.61
 
 We calculate now the total fields **[E]{.underline}**
 
-@@@@@@@@@@
+equation 3.62
 
 existing in medium 1 for a TE wave:
 
-@@@@@@@@ $`\quad (equ. 3.61)`$
+equation 3.61
 
-@@@@@@@@ $`\quad (equ. 3.62)`$
+equation 3.62
 
 field is parallel to the interface.
 
@@ -780,19 +778,19 @@ _pattern with water waves._
 To this aim it is convenient to revert to the real notation for the
 fields:
 
-@@@@@@@@ $`\quad (equ. 3.63)`$
+equation 3.63
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.64)`$
+equation 3.64
 
 we obtain:
 
-@@@@@@@@ $`\quad (equ. 3.65)`$
+equation 3.65
 
 while its time average is:
 
-@@@@@@@@@@ $`\quad (equ. 3.66)`$
+equation 3.66
 
 Power is therfore carried along the positive *z* direction and is null
 on the nodal planes.
@@ -803,36 +801,36 @@ The discussion is similar to the TE case with the role of $`\overrightarrow{E}`$
 $`\overrightarrow{B}`$ exchanged. Referring to figure [3.13](#_bookmark74) we have
 again for the wavevectors:
 
-@@@@@@@ $`\quad (equ. 3.67)`$
+equation 3.67
 
 and
 
-@@@@@@@@ $`\quad (equ. 3.68)`$
+equation 3.68
 
 Using the boundary condition for the tangential component of the
 electric field *E~r~z* + *E~i~z* = 0 and the fact the field experience
 total reflection, $`\overrightarrow{E}`$*~i~* = $`\overrightarrow{E}`$*~r~* we obtain:
 
-@@@@@@@@@@@ $`\quad (equ. 3.69)`$
+equation 3.69
 
 and
 
-@@@@@@@@@@ $`\quad (equ. 3.70)`$ 
+equation 3.70
 
 Finally, the total fields existing in medium 1 for a TM ( or *p*) wave
 are given by:
 
-@@@@@@@@@@@@
+equation
 
 and
 
-@@@@@@@@@@@@
+equation
 
 The same remarks made for the total TE wave can repeated here for the
 total TM wave we the appropriate changes for the role of the $`\overrightarrow{E}`$ and
 $`\overrightarrow{B}`$ fields.
 
-@@@@@@@@@@@@@
+equation
 
 , i.e. the magnetic field is
 
@@ -842,9 +840,9 @@ parallel to the interface.
     axis (see figure [3.14).](#_bookmark75) The positions of the nodes
     and antinodes planes for the magentic fields are given by:
 
-@@@@@@@@@@
+equation
 
 Finally, the time-averaged Poytining vector is given by:
 
-@@@@@@@@
+equation