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added stuff. must still add kirchoff law

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05_courant_electrique.md
+ 5
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......@@ -391,10 +391,10 @@ plt.title("Puissance instantanée et moyenne du courant alternatif.")
plt.legend()
```
La puissance moyenne, $\bar P$, est facilement calculée (on va le faire le calcul ici, mais on le voit bien
La puissance moyenne, $\bar{P}$, est facilement calculée (on va le faire le calcul ici, mais on le voit bien
sur l'illustration de @fig:puissance_alternatif) et est donnée par
$$
\bar P=\frac{1}{2}I_0^2R=\frac{1}{2}\frac{V_0^2}{R}.
\bar{P}=\frac{1}{2}I_0^2R=\frac{1}{2}\frac{V_0^2}{R}.
$$
---
......@@ -453,7 +453,7 @@ De l'équation
\end{equation*}
on obtient
\begin{equation*}
I_\mathrm{rms}=\frac{\bar P}{V_\mathrm{rms}}=\frac{1000}{230}=4.35\A.
I_\mathrm{rms}=\frac{\bar{P}}{V_\mathrm{rms}}=\frac{1000}{230}=4.35\A.
\end{equation*}
Ainsi le courant de pic est donné par
\begin{equation*}
......@@ -463,9 +463,9 @@ La résistance est donc donnée par
\begin{equation*}
R=\frac{V_\mathrm{rms}}{I_\mathrm{rms}}=\frac{V_0}{I_0}=\frac{325}{6.15}=52.8\Omega.
\end{equation*}
Aux USA, la tension étant de $120\V$, on obtient pour la puissance $\bar P$ disponible
Aux USA, la tension étant de $120\V$, on obtient pour la puissance $\bar{P}$ disponible
\begin{equation*}
\bar P=\frac{V_\mathrm{rms}^2}{R}=\frac{120^2}{52.8}=272\W.
\bar{P}=\frac{V_\mathrm{rms}^2}{R}=\frac{120^2}{52.8}=272\W.
\end{equation*}
On voit qu'on risque d'avoir un problème pour faire fonctionner notre sèche cheveux à plein régime. A l'inverse un sèche-cheveux américain va très probablement griller si on le branche en Europe.
......
06_cicruits_electriques.md
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......@@ -180,8 +180,74 @@ Deux résistances de $100\Omega$ sont connectées en parallèle ou en série sur
---
On peut faire évidemment beaucoup plus compliqué. Et on va le faire immédiatement.
---
Exemple (Plein de résistances) #
Soit le circuit de la @fig:complex_res. Si la tension $V=9\V$.
1. Quel est le courant sort de la source de tension? $
2. Quel est le courant dans la résistance de $6\Omega$?
![Un certain nombre de résitances...](figs/complex_res.svg){#fig:complex_res width=50%}
---
La méthodologie globale pour résoudre ce genre d'exercice, est de déterminer la résistance équivalente du circuit en le décomposant en sous-circuits série/parallèle puis en utilisant la loi d'Ohm.
---
Solution (Plein de résistances) #
1. On commence par calculer la résistance équivalent des résistances $8\Omega$ et $4\Omega$. Ces résistances sont en parallèle. On a donc
$$
\frac{1}{R_\mathrm{eq1}}=\frac{1}{8}+\frac{1}{4}=\frac{3}{8}\frac{1}{\Omega}.
$$
On a donc que $R_\mathrm{eq1}=2.7\Omega$ (voir @fig:complex_res_1)
![Un certain nombre de résitances, moins 1...](figs/complex_res_1.svg){#fig:complex_res_1 width=50%}
Puis vient le temps de calculer la résistance équivalente des résistances de $6\Omega$ et $2.7\Omega$. Ces résistances sont en série, donc il suffit de les sommer et on a $R_\mathrm{eq2}=6+2.7=8.7\Omega$ (voir @fig:complex_res_2)
![Un certain nombre de résitances, moins 2...](figs/complex_res_2.svg){#fig:complex_res_2 width=50%}
On a à présent les résistances de $10\Omega$ et $8.7\Omega$ qui sont en parallèle et on calcule
$$
\frac{1}{R_\mathrm{eq3}}=\frac{1}{10}+\frac{1}{8.7}=0.21\frac{1}{\Omega},
$$
et il vient $R_\mathrm{eq3}=4.8\Omega$ (voir @fig:complex_res_3).
![Un certain nombre de résitances, moins 3...](figs/complex_res_3.svg){#fig:complex_res_3 width=50%}
Finalement, on a trois résistances en série qu'on peut simplement sommer et il vient
$$
R_\mathrm{eq}=0.5+5+4.8=10.3\Omega.
$$
On peut à présent calculer le courant dans le circuit avec la loi d'Ohm
$$
V=R_\mathrm{eq}I\Leftrightarrow I=\frac{9}{10.3}=0.87\A.
$$
2. Le courant dans la résistance de $6\Omega$ doit être la même que dans la branche correspondant à $\R_\mathrm{eq2}$. On doit encore calculer la tension $V_2$ aux bornes du sous-circuit du haut. On a que
$$
V=V_1+V_2+_3\Leftrightarrow V_2=V-V_1-V_3=V-(0.5+5)\cdot 0.87=4.3\V.
$$
Et finalement que
$$
I=\frac{4.3}{8.7}=0.48\A.
$$
---
Pour des circuits beaucoup plus compliqués que ceux vus ici, il faut noter qu'il y a un ensemble
de règles qu'on peut appliquer pour déterminer le courant dans un circuit: les **lois de Kirchoff**. On ne va pas les lister ou les étudier ici.
de règles qu'on peut appliquer pour déterminer le courant dans un circuit: les **lois de Kirchoff**.
## Les circuits contenant des capacités en parallèle ou en série
......@@ -366,4 +432,31 @@ plt.xlabel('temps [s]')
plt.ylabel('tension [V]')
plt.title("Tension du condensateur")
plt.legend()
```
\ No newline at end of file
```
---
Exemple (Décharge du condensateur) #
Si un condensateur de $C=10\mu \mathrm{F}$ est connectée à une résistance de $R=100\Omega$ et possède une tension de $V_0$. Quand le condensateur se décharge, combien de temps faut-il pour que sa tension tombe à $10\%$ de sa valeur originale?
---
---
Solution (Décharge du condensateur) #
Comme la tension d'un condensateur se diminue comme
\begin{equation*}
V_C=V_0e^{-t/(RC)},
\end{equation*}
et qu'on veut connaître le temps nécessaire pour que $V_C=0.1V_0$, on peut écrire l'équation
\begin{align*}
0.1V_0&=V_0e^{-t/(RC)},\\
0.1&=e^{-t/(RC)},\\
0.1&=e^{-t/(10^{-5}\cdot 100)},\\
\ln{0.1}&=-10^3t,\\
t&=0.0023\s.
\end{align*}
---
figs/complex_res.svg 0 → 100644
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