Commit eee8b904 authored by Claude Meny's avatar Claude Meny

Update cheatsheet.en.md

parent e4fe462f
Pipeline #21620 canceled with stage
...@@ -456,7 +456,7 @@ I recognize here the law of conservation of charge. ...@@ -456,7 +456,7 @@ I recognize here the law of conservation of charge.
$`\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens^{3D}}{\partial t}=0`$ $`\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens^{3D}}{\partial t}=0`$
---------------------> --------------------->
* We can *integrate this local equality* over any volume $`\tau`$: * We can *integrate this local equality* over any volume $`\tau`$ :
<br> <br>
$`\displaystyle\iiint_{\tau} \Big(\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens}{\partial t}\big)\,d\tau=0`$ $`\displaystyle\iiint_{\tau} \Big(\text{div}\,\overrightarrow{j} +\dfrac{\partial \dens}{\partial t}\big)\,d\tau=0`$
<br> <br>
...@@ -469,16 +469,16 @@ I recognize here the law of conservation of charge. ...@@ -469,16 +469,16 @@ I recognize here the law of conservation of charge.
<br> <br>
*Applied to the first term* of the equality, we obtain : *Applied to the first term* of the equality, we obtain :
<br> <br>
**$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} + \iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$** . **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} + \iiint_{\tau}\dfrac{\partial \dens}{\partial t}\,d\tau=0`$**.
* Noting again that *space and time are independent in classical physics*, the order of derivation or integration with respect to a spatial variable and a time variable does not matter: * Noting again that *space and time are independent in classical physics*, the order of derivation or integration with respect to a spatial variable and a time variable does not matter :
<br> <br>
**$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{\partial}{\partial t} \left(\iiint_{\tau}\dens \,d\tau\right)=0`** . **$`\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{\partial}{\partial t} \left(\iiint_{\tau}\dens \,d\tau\right)=0`$**.
* Noting that *$`\displaystyle\iiint_{\tau}\dens^{3D} \,d\tau`$ is the total charge $`Q_{int}`* * Noting that *$`\displaystyle\iiint_{\tau}\dens^{3D} \,d\tau`$ is the total charge $`Q_{int}`$*
contained in the volume $`\tau`$, we obtain the **integral expression of the conservation law** of charge: contained in the volume $`\tau`$, we obtain the **integral expression of the conservation law** of charge :
<br> <br>
**$`\mathbf{\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{dQ_{int}}{dt}=0}`** . **$`\mathbf{\displaystyle\oiint_S \overrightarrow{j}\cdot \overrightarrow{dS} +\dfrac{dQ_{int}}{dt}=0}`$**.
<br> <br>
...@@ -493,15 +493,15 @@ I recognize here the law of conservation of charge. ...@@ -493,15 +493,15 @@ I recognize here the law of conservation of charge.
* The **sensitivity** of a particle **to electromagnetic interaction** is quantified * The **sensitivity** of a particle **to electromagnetic interaction** is quantified
by the parameter called *electric charge* of the particle. by the parameter called *electric charge* of the particle.
* The force that describes the *action of an electromagnetic field $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`* * The force that describes the *action of an electromagnetic field $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`$*
on a particle with charge $`q`$ is the **Lorentz force**, with the expression: on a particle with charge $`q`$ is the **Lorentz force**, with the expression :
<br> <br>
**$`\overrightarrow{F}_{Lorentz}=q\Big(\overrightarrow{E}+\overrightarrow{v}\times\overrightarrow{B}\Big)`$** **$`\overrightarrow{F}_{Lorentz}=q\Big(\overrightarrow{E}+\overrightarrow{v}\times\overrightarrow{B}\Big)`$**
<br> <br>
&nbsp;&nbsp;where $`\overrightarrow{v}`$ is the velocity vector of the particle in the inertial frame of the observer. &nbsp;&nbsp;where $`\overrightarrow{v}`$ is the velocity vector of the particle in the inertial frame of the observer.
* *During an elementary displacement $`\overrightarrow{dl}`* of the particle in the electromagnetic field * *During an elementary displacement $`\overrightarrow{dl}`$* of the particle in the electromagnetic field
$`\big(\overrightarrow{E}, \overrightarrow{B}\big)`*, the **work of the Lorentz force** is written as: $`\big(\overrightarrow{E}, \overrightarrow{B}\big)`$*, the **work of the Lorentz force** is written as :
<br> <br>
**$`d\mathcal{W}_{Lorentz}=\overrightarrow{F}_{Lorentz}\cdot\overrightarrow{dl}`**, **$`d\mathcal{W}_{Lorentz}=\overrightarrow{F}_{Lorentz}\cdot\overrightarrow{dl}`**,
<br> <br>
...@@ -517,8 +517,8 @@ I recognize here the law of conservation of charge. ...@@ -517,8 +517,8 @@ I recognize here the law of conservation of charge.
<br> <br>
where $`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)`$ is the scalar triple product of the sequence of the three vectors. where $`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)`$ is the scalar triple product of the sequence of the three vectors.
* Since the *vectors $`\overrightarrow{v}`$ and $`\overrightarrow{dl}=\overrightarrow{v}\,dt`* are *collinear*, the scalar triple product * Since the *vectors $`\overrightarrow{v}`$ and $`\overrightarrow{dl}=\overrightarrow{v}\,dt`$* are *collinear*, the scalar triple product
is zero: is zero :
<br> <br>
*$`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)=0`$*, *$`\Big(\overrightarrow{v},\overrightarrow{B},\overrightarrow{dl}\Big)=0`$*,
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