diff --git a/03_charge_electrique_champs_electrique.md b/03_charge_electrique_champs_electrique.md
index 805aa98d7190a41b49a3ae213086b5d0f85069a0..ad16c275e826697f3ecabdfd5329b017302eccf1 100644
--- a/03_charge_electrique_champs_electrique.md
+++ b/03_charge_electrique_champs_electrique.md
@@ -303,11 +303,7 @@ Exercice (Symétrie) #
Soient deux forces $Q_1=10\mu\mathrm{C}$ et $Q_2=100\mu\mathrm{C}$ séparées par une distance $r$. Quelle charge ressent la plus grande force?
----
-
----
-
-Correction (Symétrie) #
+Solution (Symétrie) #
La force ressentie par les deux forces est la même. En effet,
$$
@@ -324,7 +320,7 @@ par deux comme des vecteurs. Ainsi, si nous avons un système
de trois charges, $Q_1$, $Q_2$, et $Q_3$, la charge $Q_1$
ressent la somme de la force $Q_2\rightarrow Q_1$ et $Q_3\rightarrow Q_1$.
-## Quelques exercices
+## Deux exercices
Les exercices suivants sont très similaires à ce que nous avons fait
avec les forces au chapitre précédent. L'unique différence, c'est que les forces
@@ -334,7 +330,7 @@ sont obtenues avec la loi de Coulomb.
Exercice (Sur une ligne) #
-Soient trois particules chargées, $Q_1$, $Q_2$, et $Q_3$ alignées de gauche à droite.
+Soient trois particules chargées, $Q_1$, $Q_2$, et $Q_3$ alignées de gauche à droite (voir @fig:charges_aligned).
Calculer la force exercée sur $Q_3$ (la particule la plus à droite) si les charges sont
données par
$$
@@ -346,8 +342,35 @@ r_{12}=0.3\ \mathrm{m},\quad r_{23}=0.2\ \mathrm{m}.
$$
Ensuite calculer les forces agissant sur $Q_1$ et $Q_2$.
+{#fig:charges_aligned width=100%}
+
+
+Solution (Sur une ligne) #
+
+Par le principe de superposition, on a que la force résultante, $\vec F$ que ressent $Q_3$ est donnée par
+$$
+\vec F=\vec F_{2\rightarrow 3}+\vec F_{1\rightarrow 3}.
+$$
+La loi de Coulomb nous donne immédiatement $F_{2\rightarrow 3}$ et $F_{1\rightarrow 3}$
+\begin{align}
+F_{2\rightarrow 3}&=k\frac{Q_2Q_3}{r_23^2}\cong 1.2\mathrm{N},\\
+F_{1\rightarrow 3}&=k\frac{Q_1Q_3}{r_13^2}\cong 2.7\mathrm{N}.
+\end{align}
+La charge $Q_3$ étant négative la force de $Q_1$ est répulsive, alors que celle de $Q_2$ est attractive.
+On a donc finalement
+$$
+F=-F_{2\rightarrow 3}+F_{1\rightarrow 3}=-1.5\mathrm{N},
+$$
+et la force pointe vers la gauche.
+
+Les force sur $Q_1$ et $Q_2$ sont à faire pour vous comme des grand·e·s.
+
---
+Cet exercice nous montre une propriété importante de la
+force électrique. La charge $Q_2$ ne masque pas du tout
+la charge $Q_1$. En effet, la force que $Q_1$ applique sur $Q_3$ est la même que si $Q_2$ n'était pas là.
+
---
Exercice (Avec des vrais vecteurs) #
@@ -357,6 +380,61 @@ sur $Q_3$ dûes aux charges $Q_1$ et $Q_2$.
Ajouter une charge $Q_4=-50\mu\mathrm{C}$ de telle façon à ce que la force sur $Q_3$ soit nulle. Quelle doit être sa position?
+{#fig:charges width=80%}
+
+Solution (Avec des vrais vecteurs) #
+
+Cet exercice se résout de façon très similaire à l'exercice précédent, simplement que le problème est bi-dimensionnel. La force résultante sur $Q_3$ est donnée
+par
+$$
+\vec F=\vec F_{2\rightarrow 3}+\vec F_{1\rightarrow 3}.
+$$
+Les normes de $\vec F_{2\rightarrow 3}$ et $\vec F{1\rightarrow 3}$ se calculent avec la loi de Coulomb
+\begin{align}
+F_{2\rightarrow 3}&=k\frac{Q_2Q_3}{r_23^2}\cong 325\mathrm{N},\\
+F_{1\rightarrow 3}&=k\frac{Q_1Q_3}{r_13^2}\cong 140\mathrm{N}.
+\end{align}
+Avec vos talents de trigonométrie, on a ensuite que les vecteurs sont donnés par
+\begin{align}
+\vec F_{1\rightarrow 3} &= F_{1\rightarrow 3}\vectwo{\cos 30}{-\sin 30}=\vectwo{120}{-70}\mathrm{N},\\
+\vec F_{2\rightarrow 3} &= F_{2\rightarrow 3}\vectwo{\cos 90}{\sin 90}=\vectwo{0}{325}\mathrm{N}.
+\end{align}
+Et on a finalement
+$$
+\vec F=\vectwo{120+0}{325-70}=\vectwo{120}{255}\mathrm{N}.
+$$
+On peut en déduire la norme $||\vec F||$ à l'aide du théorème de Pythagore
+$$
+||\vec F||=\sqrt{F_x^2+F_y^2}\cong 280\mathrm{N}.
+$$
+L'orientation, $\theta$, de $\vec F$ se calcule également avec un peu de trigonométrie
+$$
+\tan\theta=\frac{F_y}{F_x}=\frac{255}{120}=2.13 \Leftrightarrow \theta=65^\circ.
+$$
+Afin de déterminer l'endroit où il faut placer $Q_4$, il faut que la somme de $\vec F$ avec $\vec F_{4\rightarrow 3}$ soit nulle. On doit donc avoir que la norme de ces deux forces doivent être égales et que leur direction doivent être opposées. Cela nous donne deux conditions
+\begin{align}
+F=F_{4\rightarrow 3},\\
+\theta_4=360-\theta.
+\end{align}
+Nous connaissons déjà la direction de la force et donc dans quelle direction se trouvera $Q_4$ (la direction pointée par l'angle $\theta_4$). La première condition nous permet finalement de calculer la distance entre $Q_3$ et $Q_4$, $r_{34}$
+\begin{align}
+F&=k\frac{Q_3Q_4}{r_{34}^2}=\frac{9\cdot 10^9\cdot 50\cdot 10^{-6}\cdot 65\cdot 10^{6-}}{r_{34}^2},\nonumber\\
+r_{34}^2&=k\frac{Q_3Q_4}{r_{34}^2}=\frac{29.2}{280},\nonumber\\
+r_{34}&=0.32\mathrm{m}.
+\end{align}
+
+---
+
+Encore une fois on utilise le principe de superposition
+pour déterminer l'emplacement de la nouvelle charge comme si les charges $Q_1$ et $Q_2$ n'existaient pas.
+
+---
+
+Exercice (Flotter) #
+
+Soient deux charges, $Q$ et $-Q$ qui sont à une distance $r$ l'une de l'autre. Pouvez-vous trouver l'endroit où
+on peut placer une charge $Q$ entre les deux pour qu'elle ne ressente aucune force? Même question mais pour les deux charges initiales ayant une charge positive.
+
---
## Le champs électrique
@@ -375,8 +453,35 @@ $$
E=k\frac{Qq}{r^2}\frac{1}{q}=k\frac{Q}{r^2}=\frac{1}{4\pi \epsilon_0}\frac{Q}{r^2}.
$$
On voit donc que $E$ ne dépend que de $Q$ et plus de $q$.
+Le champs électrique est une quantité ne dépendant que de la charge qui l'émet. Il décroît comme le carré de la distance
+entre la charge et le point de la mesure.
Si à présent on inverse le problème et qu'on nous donne un champs électrique $\vec E$, et qu'on aimerait connaître la force dûe à ce champs électrique sur une charge $q$ quelconque. On aurait
$$
\vec F=\vec E q.
$$
+
+---
+
+Exercice (Photocopieuse) #
+
+Une photocopieuse fonctionne en arrangeant des charges
+(suivant la forme de ce qu'on veut copier)
+sur la surface d'un rouleau et ensuite en disposant
+gentiment l'encre sèche, des particules chargées négativement (le toner), sur le rouleau pour qu'elles s'y attachent. Ainsi, le toner s'attache en suivant
+la forme des particules chargées et est ensuite chauffé
+lorsqu'il se trouve au contact du papier pour y être transféré. En supposant que chaque particule de toner a un excès de 20 électrons et que leur masse est de $9\cdot 10^{-16}\mathrm{kg}$. Quelle doit être le champs électrique du rouleau pour que les particules restent attachées au rouleau, s'il faut au moins une force de deux fois leur poids pour qu'elles ne s'en détachent pas?
+
+Solution (Photocopieuse) #
+
+La charge de chaque particule de toner est de $q=20e$. La force électrostatique qui s'applique sur chacune d'entre elles est de $F=qE$. De plus, on nous dit que pour que le toner ne se détache, la force électrostatique doit être au moins égale à deux fois le poids de chaque particule. Il vient donc
+$$
+q\cdot E=2\cdot m\cdot g.
+$$
+On résout pour $E$ et on trouve
+$$
+E=\frac{2mg}{q}=\frac{2\cdot 9\cdot 10^{-16}\cdot 9.8}{20\cdot 1.6\cdot 10^{-19}}=5.5\cdot 10^3\mathrm{N}/\mathrm{C}.
+$$
+
+---
+
diff --git a/figs/charges.svg b/figs/charges.svg
new file mode 100644
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