Update cheatsheet.en.md

12.temporary_ins/90.electromagnetism-in-vacuum/10.maxwell-equations/20.overview/cheatsheet.en.md

@@ -555,10 +555,10 @@ I recognize here the law of conservation of charge.
 
 ! *Note:*
 !
-! The *magnetic force $`\overrightarrow{F}_{magn.}=q\,\overrightarrow{v}\times\overrightarrow{B}`*,
+! The *magnetic force $`\overrightarrow{F}_{magn.}=q\,\overrightarrow{v}\times\overrightarrow{B}`$*,
 ! by nature perpendicular to the velocity vector $`\overrightarrow{v}`$ and thus to the elementary displacement
 ! vector $`\overrightarrow{dl}=\overrightarrow{v}\,dt`$ at every point on the particle's trajectory
-! with charge $`q`*, *does no work*:
+! with charge $`q`$*, *does no work*:
 !
 ! $`\mathbf{d\mathcal{W}_{magn} = \overrightarrow{F}_{magn}\cdot \overrightarrow{dl}=0}`$
 !
@@ -568,29 +568,29 @@ I recognize here the law of conservation of charge.
 
 * The **elementary power yielded by the field** to this particle is written as:
   <br>
-  **$`\mathbf{\mathcal{P}_{yielded} = \dfrac{d\mathcal{W}_{Lorentz}}{dt} = q\,\overrightarrow{E}\cdot\overrightarrow{v}}`**
+  **$`\mathbf{\mathcal{P}_{yielded} = \dfrac{d\mathcal{W}_{Lorentz}}{dt} = q\,\overrightarrow{E}\cdot\overrightarrow{v}}`$**
 
 ##### Power yielded in a material with a single type of charge carrier
 
-* If the **material medium** contains *$`n`* identical charge carriers of charge $`q`* per unit volume*,
+* If the **material medium** contains *$`n`$ identical charge carriers of charge $`q`$ per unit volume*,
   then an elementary volume $`d\tau`$ contains $`n\,\tau`$ charge carriers
-  and the **elementary power yielded** by the electromagnetic field is written as:
+  and the **elementary power yielded** by the electromagnetic field is written as :   
   <br>
   **$`d\mathcal{P}_{yielded} = n\,\big( q\,\overrightarrow{E}\cdot\overrightarrow{v}\big)\,d\tau`$**   
 
-* Expressed *with the volume charge density $`\rho=n\,q`*:
+* Expressed *with the volume charge density $`\rho=n\,q`* :   
   <br>
   $`d\mathcal{P}_{yielded} = \big(n\, q\big)\,\overrightarrow{E}\cdot\overrightarrow{v}\,d\tau = \rho\,\overrightarrow{E}\cdot\overrightarrow{v}\,d\tau`$   
 
-* Expressed *with the volume current density vector $`\overrightarrow{j}=\rho\,\overrightarrow{v}`*, noting that
-  $`\overrightarrow{E}\cdot\overrightarrow{v}=\overrightarrow{v}\cdot\overrightarrow{E}`$:
+* Expressed *with the volume current density vector $`\overrightarrow{j}=\rho\,\overrightarrow{v}`$*, noting that
+  $`\overrightarrow{E}\cdot\overrightarrow{v}=\overrightarrow{v}\cdot\overrightarrow{E}`$ :   
   <br>
   **$`d\mathcal{P}_{yielded} = \big(\overrightarrow{j}\cdot\overrightarrow{E}\big)\,d\tau`$**   
 
 ##### Power yielded in a material with multiple types of charge carriers
 
-* When a material contains **multiple types of charge carriers $`q_i`**
-  in *concentrations $`n_i`* and with *drift velocities $`\overrightarrow{v_{d\,i}}`*:
+* When a material contains **multiple types of charge carriers $`q_i`$**   
+  in *concentrations $`n_i`* and with *drift velocities $`\overrightarrow{v_{d\,i}}`* :   
   <br>
   $`\displaystyle d\mathcal{P}_{yielded} = \sum_{i=1}^p \big(n_i\,q_i\,\overrightarrow{E}\cdot\overrightarrow{v_i}\big)\,d\tau`$   
   <br>
@@ -600,7 +600,7 @@ I recognize here the law of conservation of charge.
   <br>
   *$`d\mathcal{P}_{yielded} = \overrightarrow{j}_{total}\cdot\overrightarrow{E}\,d\tau`$*   
 
-* By simply setting *$`\overrightarrow{j}_{total}=\overrightarrow{j}`*:
+* By simply setting *$`\overrightarrow{j}_{total}=\overrightarrow{j}`* :   
   <br>
   **$`\large{\mathbf{d\mathcal{P}_{yielded} = \big(\overrightarrow{j}\cdot\overrightarrow{E}\big)\,d\tau}}`$**   
 
@@ -627,7 +627,7 @@ I recognize here the law of conservation of charge.
 ##### Is the expression of the volumetric energy density of the electromagnetic field contained in Maxwell's equations?
 --------------------->
 
-* Start with the mathematical identity:
+* Start with the mathematical identity :   
   <br>
   $`\mathbf{\text{div}\,\big(\overrightarrow{U}\times\overrightarrow{V}\big)=
   \overrightarrow{V}\cdot\big(\overrightarrow{\text{curl}}\,\overrightarrow{U}\big)\,-\,\overrightarrow{U}\cdot\big(\overrightarrow{\text{curl}}\,\overrightarrow{V}\big)}`$   
@@ -677,7 +677,7 @@ I recognize here the law of conservation of charge.
   \,-\,\dfrac{\epsilon_0}{2}\,\dfrac{\partial E^2}{\partial t}
   \,
   -\,\dfrac{1}{2\,\mu_0}\,\dfrac{\partial B^2}{\partial t}
-  `$$
+  `$
   <br>
   $`\color{blue}{\scriptsize{\text{which you can rewrite:}}}`$   
   <br>
@@ -689,7 +689,7 @@ I recognize here the law of conservation of charge.
   \right)
   }`$**
 
-* Thus appears the **volumetric energy density of the electromagnetic field** with *SI unit: $`J\,m^{-3}`*:
+* Thus appears the **volumetric energy density of the electromagnetic field** with *SI unit: $`J\,m^{-3}`$* :   
   <br>
   **$`\large{\mathbf{\rho_{energy-EM}^{3D}=\dfrac{\epsilon_0\,E^2}{2}+\dfrac{B^2}{2 \mu_0}}}`$**   
 
@@ -697,7 +697,7 @@ I recognize here the law of conservation of charge.
   * an *electric component* **$`\;\rho_{elec}^{3D}=\dfrac{\epsilon_0\,E^2}{2}`$**
   * a *magnetic component* **$`\;\rho_{mag}^{3D}=\dfrac{B^2}{2 \mu_0}`$**.
 
-* The electromagnetic energy $`\mathcal{E}_{EM}`$ contained **in a volume $`\tau`** is expressed as:
+* The electromagnetic energy $`\mathcal{E}_{EM}`$ contained **in a volume $`\tau`$** is expressed as :   
   <br>
   **$`\displaystyle\large{\mathbf{\mathcal{E}_{EM}=\iiint_{\tau} \left(\dfrac{\epsilon_0\,E^2}{2}+\dfrac{B^2}{2 \mu_0}\right) d\tau}}`$**
 
@@ -711,29 +711,29 @@ I recognize here the law of conservation of charge.
 
 ##### Wave Equation
 
-* For a *vector field $`\overrightarrow{U}(\overrightarrow{r},t)`*, the **d'Alembert wave equation** is written as:
+* For a *vector field $`\overrightarrow{U}(\overrightarrow{r},t)`*, the **d'Alembert wave equation** is written as :   
   <br>
   **$`\Delta \overrightarrow{U} - \dfrac{1}{v^2} \; \dfrac{\partial^2 \;\overrightarrow{U}}{\partial t^2}=0`$**   
 
-* The expression of the *vector Laplacian operator $`\Delta`* in terms of the $`\text{grad}`$, $`\text{div}`$, and $`\text{curl}`$ operators is:
+* The expression of the *vector Laplacian operator $`\Delta`* in terms of the $`\text{grad}`$, $`\text{div}`$, and $`\text{curl}`$ operators is :   
   <br>
   *$`\Delta = \overrightarrow{\text{grad}} \left(\text{div}\right) - \overrightarrow{\text{curl}} \left(\overrightarrow{\text{curl}}\right)`$*   
 
-* The **idea** is to *calculate for each of the fields $`\overrightarrow{E}`$ and $`\overrightarrow{B}`*
+* The **idea** is to *calculate for each of the fields $`\overrightarrow{E}`$ and $`\overrightarrow{B}`$*
   the expression of *its Laplacian*, to see if it can be identified with the wave equation.   
 
 ##### Study of the $`\overrightarrow{E}`$ Component of the Electromagnetic Field
 
-* To **establish the expression $`\Delta \overrightarrow{E}`**, I calculate
+* To **establish the expression $`\Delta \overrightarrow{E}`$**, I calculate
   $`\overrightarrow{\text{curl}}\left(\overrightarrow{\text{curl}} \overrightarrow{E}\right)`$ and then
-  $`\overrightarrow{\text{grad}} \left(\text{div} \overrightarrow{E}\right)`* from Maxwell's equations.
+  $`\overrightarrow{\text{grad}} \left(\text{div} \overrightarrow{E}\right)`$* from Maxwell's equations.
 
 * $`\overrightarrow{\text{curl}} \left(\overrightarrow{\text{curl}} \overrightarrow{E}\right) =
   \overrightarrow{\text{curl}} \left(-\dfrac{\partial \overrightarrow{B}}{\partial t}\right)`$   
   <br>
   In classical physics, space and time are decoupled. The spatial coordinates
   and the time coordinate are independent. The order of differentiation or integration between
-  spatial coordinates and the time coordinate does not matter, so:
+  spatial coordinates and the time coordinate does not matter, so :   
   <br>
   $`\overrightarrow{\text{curl}} \left(\overrightarrow{\text{curl}} \overrightarrow{E}\right) =
   -\dfrac{\partial}{\partial t} \left(\overrightarrow{\text{curl}} \overrightarrow{B}\right)`$   
@@ -751,12 +751,12 @@ I recognize here the law of conservation of charge.
 * Reconstructing
   $`\Delta \overrightarrow{E} = \overrightarrow{\text{grad}} \left(\text{div} \overrightarrow{E}\right) -
   \overrightarrow{\text{curl}} \left(\overrightarrow{\text{curl}} \overrightarrow{E}\right)`$
-  gives:
+  gives :   
   <br>
   $`\Delta \overrightarrow{E} = \overrightarrow{\text{grad}} \left(\dfrac{\rho}{\epsilon_0}\right) +
   \mu_0 \dfrac{\partial \overrightarrow{j}}{\partial t} + \mu_0 \epsilon_0 \dfrac{\partial^2 \overrightarrow{E}}{\partial t^2}`$   
   <br>
-  which, by identifying with the first term of the wave equation, gives:
+  which, by identifying with the first term of the wave equation, gives :   
 
   **$`\mathbf{\Delta \overrightarrow{E} - \mu_0 \epsilon_0 \dfrac{\partial^2 \overrightarrow{E}}{\partial t^2} =
   \dfrac{1}{\epsilon_0} \overrightarrow{\text{grad}} \left(\rho\right) + \mu_0 \dfrac{\partial \overrightarrow{j}}{\partial t}}`$**   
@@ -766,7 +766,7 @@ I recognize here the law of conservation of charge.
 ##### Study of the $`\overrightarrow{B}`$ Component of the Electromagnetic Field
 
 * A *similar study* (proposed as a self-test in the advanced section) would lead me
-  to the propagation equation for the magnetic field $`\overrightarrow{B}`$:
+  to the propagation equation for the magnetic field $`\overrightarrow{B}`$ :   
   <br>
   **$`\mathbf{\Delta \overrightarrow{B} - \epsilon_0 \mu_0 \dfrac{\partial^2 \overrightarrow{B}}{\partial t^2} =
   -\mu_0 \overrightarrow{\text{curl}} \overrightarrow{j}}`$**   
@@ -775,7 +775,7 @@ I recognize here the law of conservation of charge.
 
 ##### Propagation of an Electromagnetic Wave in Matter
 
-* The study starts from Maxwell's equations and the two equations:
+* The study starts from Maxwell's equations and the two equations :   
   <br>
   $`\Delta \overrightarrow{E} - \mu_0 \epsilon_0 \dfrac{\partial^2 \overrightarrow{E}}{\partial t^2} =
   \dfrac{1}{\epsilon_0} \overrightarrow{\text{grad}} \left(\rho\right) + \mu_0 \dfrac{\partial \overrightarrow{j}}{\partial t}`$   
@@ -794,13 +794,13 @@ I recognize here the law of conservation of charge.
   *$`\rho_{\text{vacuum}}=0 \quad \text{and} \quad \overrightarrow{j}_{\text{vacuum}}=\overrightarrow{0}`$*.   
 
 * Therefore, the propagation of the electromagnetic wave in vacuum is expressed in the form
-  of a system of **two d'Alembert equations**:
+  of a system of **two d'Alembert equations** :   
   <br>
   **$`\large{\boldsymbol{\mathbf{\Delta \overrightarrow{E} - \mu_0 \epsilon_0 \dfrac{\partial^2 \overrightarrow{E}}{\partial t^2} = \overrightarrow{0}}}}`$**
   <br>
   **$`\large{\boldsymbol{\mathbf{\Delta \overrightarrow{B} - \mu_0 \epsilon_0 \dfrac{\partial^2 \overrightarrow{B}}{\partial t^2} = \overrightarrow{0}}}}`$**
 
-!!!! *Attention*:
+!!!! *Attention* :                   
 !!!!
 !!!! *Maxwell's equations imply the propagation of the electromagnetic field*.
 !!!!
@@ -825,7 +825,7 @@ I recognize here the law of conservation of charge.
   <br>
   *$`\large{\mathscr{v}=\dfrac{1}{\sqrt{\epsilon_0 \mu_0}}}`$*   
 
-* The *speed of light in vacuum*, denoted *$`\mathbf{c}`$*, is a **fundamental constant** of the universe, and its exact value is:
+* The *speed of light in vacuum*, denoted *$`\mathbf{c}`$*, is a **fundamental constant** of the universe, and its exact value is :   
   <br>
   *$`\large{c=299,792,458 \, \text{m} \, \text{s}^{-1} \approx 3 \times 10^8 \, \text{m} \, \text{s}^{-1}}`$*   
 
@@ -872,9 +872,9 @@ I recognize here the law of conservation of charge.
 #### What Is the Poynting Vector?
 
 * The **electromagnetic wave contains energy**
-  * with *at each point in space* a **volumetric energy density $`\rho_{energy-EM}^{3D}`**,
-    - with an *electric component $`\rho_{energy-EM}^{3D}=\dfrac{\epsilon_0\,E^2}{2}`*,
-    - with a *magnetic component $`\rho_{energy-EM}^{3D}=\dfrac{B^2}{2 \mu_0}`*.
+  * with *at each point in space* a **volumetric energy density $`\rho_{energy-EM}^{3D}`$$**,
+    - with an *electric component $`\rho_{energy-EM}^{3D}=\dfrac{\epsilon_0\,E^2}{2}`$*,
+    - with a *magnetic component $`\rho_{energy-EM}^{3D}=\dfrac{B^2}{2 \mu_0}`$*.
   * which **moves in vacuum** *at speed $`c`$*.
 
 * The **Poynting vector** translates this fact and allows the *calculation of the energy* of an electromagnetic wave
@@ -900,7 +900,7 @@ I recognize here the law of conservation of charge.
 ! The displacement of a charge (SI unit: $`C`$) contained in an elementary volume
 ! $`d\tau`$ (SI: $`m^3`$) with volume charge density
 ! $`\rho_{charge}^{3D}`$ (SI: $`C\,m^{-3`$) at a velocity
-! $`\overrightarrow{v_d}`$ (SI: $`m\,s^{-1}`$):
+! $`\overrightarrow{v_d}`$ (SI: $`m\,s^{-1}`$) :   
 ! * allows defining a volume electric current density vector
 ! $`\overrightarrow{j}_{current}^{3D}=\rho_{charge}^{3D}\,v_d`$
 ! (SI: $`A\,m^{-2}`$),
@@ -966,15 +966,15 @@ I recognize here the law of conservation of charge.
   * Since electrical energy is proportional to $`E^2`$, <br>
     the *period of energy variations* of the wave is **$`\mathbf{T_{energy}}`** *$`\mathbf{=\dfrac{T_{wave}}{2}}`$*.
 
-* Every **sensor** is characterized by a **response time $`\mathbf{\Delta t_{response}}`** that quantifies its *speed*.
+* Every **sensor** is characterized by a **response time $`\mathbf{\Delta t_{response}}`$** that quantifies its *speed*.   
   <br>
   Consider a sensor sensitive to electromagnetic energy:
-  * If *$`\mathbf{\Delta t_{response}\ll T_{energy}}`*, then the sensor is sensitive to the *instantaneous power*:
+  * If *$`\mathbf{\Delta t_{response}\ll T_{energy}}`*, then the sensor is sensitive to the *instantaneous power* :   
     <br>
     *$`\displaystyle\large{\mathbf{\mathcal{P}(t)=\iint_S \overrightarrow{\Pi}(t)\cdot\overrightarrow{dS}}}`$*   
     <br>
   * If **$`\mathbf{\Delta t_{response}\gg T_{energy}}`**, then the sensor cannot follow the temporal variations of
-    the instantaneous power and only measures the **average value of the power** estimated over $`\Delta t_{response}`$:
+    the instantaneous power and only measures the **average value of the power** estimated over $`\Delta t_{response}`$ :   
     <br>
     **$`\displaystyle\large{\mathbf{<\mathcal{P}(t)>\;=\iint_S <\overrightarrow{\Pi}(t)>\cdot\overrightarrow{dS}}}`$**   
 
@@ -983,7 +983,7 @@ I recognize here the law of conservation of charge.
 !!! The *visible domain* corresponds to:
 !!! * a *wavelength* in vacuum on the order of 500 nanometers: *$`\mathbf{\lambda = 5\cdot 10^{-7}\,m}`$* <br>
 !!!   <br>
-!!!   This corresponds to a time period of the electric field $`T_{wave}`$ of: <br>
+!!!   This corresponds to a time period of the electric field $`T_{wave}`$ of : <br>
 !!!   $`T_{wave}=\dfrac{\lambda}{c}=\dfrac{5\cdot 10^{-7}}{3\cdot 10^{8}} = 1.7\times 10^{-15}\,s`$, <br>
 !!!   or <br>
 !!!   $`T_{energy}=8.5\times 10^{-16}\,s`$ <br>
@@ -1012,7 +1012,7 @@ For transverse and classical resources, parallel links
   <br>
   $`\Longrightarrow`$ A *whole new world of "lights"* is revealed, called the **electromagnetic spectrum**.   
 
-![Electromagnetic Spectrum](astro-electromagnetic-spectrum-N4_1_fr_L1200.jpg)
+![Electromagnetic Spectrum](astro-electromagnetic-spectrum-N4_1_en_L1200.jpg)
 
 * In particular, **knowledge of the universe** resulted *before Maxwell* from the sole observation of the *visible domain*,