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z8_OG0w3loS^j&B(d8SU;6rs6qm;Ty#XxrY-xhKE;xZ-0Hv%R8H@$!rrKG#*V)r?AS zmelTAxp(Vg-Mt%@-r1bB;`6Fys@BrezNk%0i;Azv_{%Waxh-SHeHUj|0bUu8!`AD4 zr|4;XoxA7pxA$eWW#x&Rwu$8Wy8PUG@y4vL^-)znk4;NGeq{ThI(VYIX(Ust`up<K zi)#ZP-ab9=-_7lQ|9`m~t)JO*{ed~Gz+m2-QS$57tzXHDXNvp0*|9zSXY?b@KW(sN z$MB5%qQQR4Y9<B_B`Jmmg++{b8A<LQ3<@3UybKNk7a15>1V<U8fiT!oLW9OH^YX&R WlnWCw-!m{UFnGH9xvX<aXaWGP_3c3b literal 0 HcmV?d00001 diff --git a/rapport/rapport.md b/rapport/rapport.md index 992fea1..edaad85 100644 --- a/rapport/rapport.md +++ b/rapport/rapport.md @@ -4,13 +4,13 @@ Gawen Ackermann et Florian Burgener ## Introduction -Le but de ce travail est de créer une simulation permettant de visualiser le phénomène physique du champ électrique entre plusieurs particules chargées. Plus précisément, ce travail permet d'observer le comportement des lignes de champ par rapport à des particules chargées positivement ou négativement placé arbitrairement dans un univers discrétisé. Ci-dessous, vous pouvez observer l'application de la théorie du champ électrique entre deux charges, pour la partie de gauche les deux paritcules sont chargées positivement et ont la même charge et sur la partie de droite les particules sont également de même charge, cependant la première est chargée négativement tandis que le deuxième est chargée positivement. +Le but de ce travail est de créer une simulation permettant de visualiser le phénomène physique du champ électrique entre plusieurs particules chargées. Plus précisément, ce travail permet d'observer le comportement des lignes de champ par rapport à des particules chargées positivement ou négativement, placé arbitrairement dans un univers discrétisé. Ci-dessous, vous pouvez observer l'application de la théorie du champ électrique entre deux charges, pour la partie de gauche les deux particules sont chargées positivement et ont la même charge et sur la partie de droite les particules sont également de même charge, cependant la première est chargée négativement tandis que le deuxième est chargée positivement. Nous devrions arriver à un résultat similaire avec notre simulation.  <center>Figure 1 - Résultat que l'on devrait obtenir avec cette simulation (si même configuration)</center> -Ce rapport se décompose en trois parties, la première partie est une explication de la théorie physique du champ électrique ainsi que des formules que nous avons utilisé pour créer cette simulation, la deuxième partie est une description détaillée de la façon dont nous avons appliqué la théorie, la dernière partie est une synthèse de ce travail ainsi qu'une réflexion sur les différentes possibilités d'améliorations que nous pourrions apporter à celui-ci. +Ce rapport se décompose en trois parties, la première partie est une explication de la théorie physique du champ électrique ainsi que des formules que nous avons utilisé pour créer cette simulation, la deuxième partie est une description détaillée de la façon dont nous avons appliqué la théorie ainsi qu'une argumentation sur la réussite de l'objectif, la dernière partie est une synthèse de ce travail ainsi qu'une réflexion sur les différentes possibilités d'améliorations que nous pourrions apporter à celui-ci. ## Théorie @@ -18,7 +18,7 @@ Ce rapport se décompose en trois parties, la première partie est une explicati Un champ électrique est un champ qui entoure des particules chargées électriquement, il exerce une force sur toutes les autres particules chargées à intérieur de celui-ci ce qui a pour effet d'attirer ou de repousser les particules. La formule du champ électrique découle d'une transformation de la loi de Coulomb. -Loi de Coulomb : +**Loi de Coulomb :** $$ F = \frac{1}{4\pi\epsilon_0} \cdot \frac{\lvert q_0q_1 \rvert}{r^2} = K\frac{\lvert q_0q_1 \rvert}{r^2} $$ @@ -26,17 +26,17 @@ où K est la constante de Coulomb $K = \frac{1}{4\pi\epsilon_0}$, $\epsilon_0$ Cette loi représente la force exercée entre deux particules chargées. La variable $r$ est la distance qui sépare les deux particules chargées. -La direction de la force appliquée sur la particule 0 est le vecteur unitaire allant de la particule 0 à la particule 1, si et seulement si les deux particules ont un signe différent (par exemple, la particule 0 est chargée positivement et la particule 1 est chargée négativement). Pour la particule 1, il faut changer la particule 0 par la particule 1 et la particule 1 par la particule 0 (pour la particule 1, la direction de la force est le vecteur unitaire allant de la particule 1 à la particule 0). +La direction de la force appliquée sur la particule 0 est le vecteur unitaire allant de la particule 0 à la particule 1, si et seulement si les deux particules ont un signe différent (par exemple, la particule 0 est chargée positivement et la particule 1 est chargée négativement). Pour obtenir la direction de la force appliquée sur la particule 1, il faut changer la particule 0 par la particule 1 et la particule 1 par la particule 0 (pour la particule 1, la direction de la force est le vecteur unitaire allant de la particule 1 à la particule 0). Dans le même principe, si les deux particules ont le même signe (par exemple, les deux particules sont chargées positivement) la force va dans la direction opposée, pour la force appliquée sur la particule 0, la direction de la force est l'opposé du vecteur unitaire allant de la particule 0 à la particule 1. Pour la particule 1, il faut changer la particule 0 par la particule 1 et la particule 1 par la particule 0 (de la même façon que précédemment). -Loi du champ électrique : +**Loi du champ électrique :** $$ E = \frac{F}{q_0} = K\frac{\lvert q_0q_1 \rvert}{r^2} \cdot \frac{1}{q_0} = K\frac{\lvert q_1 \rvert}{r^2} $$ Il est important de noter que $q_0$ (la charge d'essai) est toujours une charge positive ayant une charge beaucoup moins importante que $q_1$ afin qu'elle ne perturbe pas le champ électrique. Cette formule est le résultat de l'annulation de $q_0$ dans la loi de Coulomb. Le champ électrique n'étant dépendant que de la particule $q_1$, alors la direction de celui-ci à la position de $q_0$ va de la particule $q_0$ à $q_1$ si la particule $q_1$ est chargée négativement et va dans la direction opposée si la particule $q_1$ est chargé positivement. -Principe de superposition du champ électrique : +**Principe de superposition du champ électrique :** $$ E = \frac{F_1}{q_0} + \frac{F_2}{q_0} + \frac{F_3}{q_0} + \cdots \\ @@ -71,13 +71,13 @@ Pour passer à la suite, le point courant devient le nouveau point et on recalcu Dans cette partie, nous allons détailler comment nous sommes arrivées à créer notre simulation de lignes de champ et pourquoi nous pensons que notre résultat est bonne représentation du phénomène physique. Ci-dessous, vous pouvez observer le résultat de notre simulation avec deux particules. La première particule est chargée positivement et placée à la position x=0.25 et y=0.5, la deuxième particule est chargée négativement et placé à la position x=0.75 et y=0.5. Il est important de noter que notre univers est de taille [0, 1] x [0, 1], le point (0, 0) étant en haut à gauche et le point (1, 1) en bas à droite de l'écran. - +| Résultat souhaité | Résultat obtenu | +| :---------------------------------------: | :---------------------------------------: | +|  |  | -<center>Figure 2 - Premier résultat de la simulation</center> +En comparant l'image décrivant la théorie et le résultat de notre simulation, on observe que celle-ci ce comporte comme prédite. Cependant, nous allons créer deux configurations différentes supplémentaires afin de pouvoir être quasiment certain que notre simulation est représentative du phénomène physique et cohérente par rapport à la théorie. -Si vous comparer l'image ci-dessus et l'image initial que nous avons mis dans l'introduction, la simulation ce comporte comme prédite, cependant nous allons créer un cas plus complexe afin de pouvoir être plus confiant et pouvoir dire que notre simulation est correcte par rapport à la théorie. - -Afin de pouvoir valider notre simulation, nous avons utilisé ce site internet https://academo.org/demos/electric-field-line-simulator qui nous permet de créer des prédictions et de les comparer à notre simulation. Le site academo.org est un site gratuit qui met à disposition des resources sur de sujets comme les mathématiques, la physique, etc. Nous avons estimé qu'il était suffisament de confiance afin de pouvoir se baser dessus. +Afin de pouvoir valider notre simulation, nous avons utilisé ce site internet https://academo.org/demos/electric-field-line-simulator qui nous permet de créer des prédictions et de les comparer à notre simulation. Le site academo.org est un site gratuit qui met à disposition des resources sur de sujets comme les mathématiques, la physique, etc. Nous avons estimé qu'il était suffisament de confiance afin de pouvoir se baser dessus. Au dessus de chaque image est décris l'état de la particule de la façon suivante : (charge électrique, position). | Prédiction | Notre simulation | | :--------------------------: | :--------------------------: | -- GitLab