Update cheatsheet.en.md

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

@@ -522,7 +522,7 @@ I recognize here the law of conservation of charge.
   <br>
   *$`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)=0`$*,   
 
-!!!!
+
 !!!! <details markdown=1>
 !!!! <summary>Reminders about the scalar triple product</summary>
 !!!! The scalar triple product of three vectors $`\vec{a}, \vec{b}, \vec{c}`$, denoted $`(\vec{a}, \vec{b}, \vec{c})`$
@@ -549,7 +549,7 @@ I recognize here the law of conservation of charge.
 !!!! </details>
 !!!!
 
-* $`\Longrightarrow`$ the **work of the Lorentz force** simplifies to:
+* $`\Longrightarrow`$ the **work of the Lorentz force** simplifies to :   
   <br>
   **$`d\mathcal{W}_{Lorentz} = q\,\overrightarrow{E}\cdot\overrightarrow{dl}`$**
 
@@ -578,7 +578,7 @@ I recognize here the law of conservation of charge.
   <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>
   $`d\mathcal{P}_{yielded} = \big(n\, q\big)\,\overrightarrow{E}\cdot\overrightarrow{v}\,d\tau = \rho\,\overrightarrow{E}\cdot\overrightarrow{v}\,d\tau`$   
 
@@ -590,7 +590,7 @@ I recognize here the law of conservation of charge.
 ##### 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}}`* :   
+  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>
@@ -604,7 +604,7 @@ I recognize here the law of conservation of charge.
   <br>
   **$`\large{\mathbf{d\mathcal{P}_{yielded} = \big(\overrightarrow{j}\cdot\overrightarrow{E}\big)\,d\tau}}`$**   
 
-* The *power yielded* by the electromagnetic field *in a volume $`\tau`* is called **$`\large{\text{Joule Effect}}`**,   
+* The *power yielded* by the electromagnetic field *in a volume $`\tau`* is called **$`\large{\text{Joule Effect}}`$**,   
   <br>
   **$`\large{\displaystyle\mathbf{\mathcal{P}_{yielded} = \iiint_{\tau}\big(\overrightarrow{j}\cdot\overrightarrow{E}\big)\,d\tau}}`$**   
 
@@ -620,7 +620,7 @@ I recognize here the law of conservation of charge.
 
 * An electromagnetic field $`\big(\overrightarrow{E},\,\overrightarrow{B}\big)`$ extending through space,
   the energy contained in the field is described by
-  a **volumetric energy density of the electromagnetic field $`\rho_{energy-EM}^{3D}`** defined at each point in space.
+  a **volumetric energy density of the electromagnetic field $`\rho_{energy-EM}^{3D}`$** defined at each point in space.
 
 <!-------------------
 @@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@
@@ -659,10 +659,7 @@ I recognize here the law of conservation of charge.
   \text{Recall that } \vec{u}\dfrac{\partial \vec{u}}{\partial t}=\dfrac{1}{2}\,\dfrac{\partial (\vec{u}\cdot\vec{u})}{\partial t}=\dfrac{1}{2}\,\dfrac{\partial u^2}{\partial t}}}`$   
   <br>
   $`\text{div}\,\big(\overrightarrow{E}\times\overrightarrow{B}\big)`$
-  $`\quad =-\,\mu_0\,\vec{j}\cdot\overrightarrow{E}\,-\,\dfrac{\mu_0\,\epsilon_0}{2}\,\dfrac{\partial E^2}{\partial t}
-  \,
-  -\,\dfrac{1}{2}\,\dfrac{\partial B^2}{\partial t}
-  `$$
+  $`\quad =-\,\mu_0\,\vec{j}\cdot\overrightarrow{E}\,-\,\dfrac{\mu_0\,\epsilon_0}{2}\,\dfrac{\partial E^2}{\partial t}\,-\,\dfrac{1}{2}\,\dfrac{\partial B^2}{\partial t}`$   
   <br>
   $`\color{blue}{\scriptsize{\text{Recognizing the Joule effect term }\vec{j}\cdot\vec{E}=\dfrac{d\mathcal{P}_{yielded}}{d\tau}}}`$   
   $`\color{blue}{\scriptsize{\text{encourages dividing each term of the equation by }\mu_0 }}`$   
@@ -872,7 +869,7 @@ 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 *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 $`c`$*.
@@ -880,7 +877,7 @@ I recognize here the law of conservation of charge.
 * The **Poynting vector** translates this fact and allows the *calculation of the energy* of an electromagnetic wave
   incident *on any surface per second*.
 
-* The **Poynting vector**, defined at each point in space, is *defined by* the relation:
+* The **Poynting vector**, defined at each point in space, is *defined by* the relation :   
   <br>
   *$`\large{\mathbf{d\mathcal{P}=\overrightarrow{\Pi}\cdot\overrightarrow{dS}}}`$*   
   <br>
@@ -889,17 +886,17 @@ I recognize here the law of conservation of charge.
 
 ![Poynting Vector](poynting-vector-1_L1200.jpg)
 
-* Its **expression** in terms of the fields $`\overrightarrow{E}`$ and $`\overrightarrow{B}`$ is:
+* Its **expression** in terms of the fields $`\overrightarrow{E}`$ and $`\overrightarrow{B}`$ is :   
   <br>
   **$`\large{\mathbf{\overrightarrow{\Pi}=\dfrac{\overrightarrow{E}\times\overrightarrow{B}}{\mu_0}}}`$**
 
-* *SI Unit*: **$`\mathbf{W\,m^{-2}}`$**
+* *SI Unit* : **$`\mathbf{W\,m^{-2}}`$**
 
 ! *Note 1*:
 !
 ! 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
+! $`\rho_{charge}^{3D}`$ (SI: $`C\,m^{-3}`$) at a velocity
 ! $`\overrightarrow{v_d}`$ (SI: $`m\,s^{-1}`$) :   
 ! * allows defining a volume electric current density vector
 ! $`\overrightarrow{j}_{current}^{3D}=\rho_{charge}^{3D}\,v_d`$
@@ -957,23 +954,23 @@ I recognize here the law of conservation of charge.
 
 #### How to Calculate the Power Crossing a Surface of Any Area and Orientation?
 
-* The **power** is simply calculated by the expression:
+* The **power** is simply calculated by the expression :   
   <br>
   **$`\displaystyle\mathcal{P}=\iint_S \overrightarrow{\Pi}\cdot\overrightarrow{dS}`$**
 
-* Consider a **monochromatic electromagnetic wave** with a time period $`\mathbf{T_{wave}}`$.
-  * **$`\mathbf{T_{wave}}`** is the *time period of $`\overrightarrow{E}`$*, the electric field of the wave.
+* Consider a **monochromatic electromagnetic wave** with a *time period $`\mathbf{T_{wave}}`$*.
+  * **$`\mathbf{T_{wave}}`$** is the *time period of $`\overrightarrow{E}`$*, the electric field of the wave.
   * 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}}`$*.
+    the *period of energy variations* of the wave is **$`\mathbf{T_{energy}}`$** *$`\mathbf{=\dfrac{T_{wave}}{2}}`$*.
 
 * Every **sensor** is characterized by a **response time $`\mathbf{\Delta t_{response}}`$** that quantifies its *speed*.   
   <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
+  * 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 $`\Delta t_{response}`$ :   
     <br>
     **$`\displaystyle\large{\mathbf{<\mathcal{P}(t)>\;=\iint_S <\overrightarrow{\Pi}(t)>\cdot\overrightarrow{dS}}}`$**