{"id":35739,"date":"2021-09-30T13:00:13","date_gmt":"2021-09-30T13:00:13","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=35739"},"modified":"2025-12-29T03:30:33","modified_gmt":"2025-12-29T03:30:33","slug":"le-flux-electrique-et-la-loi-de-gauss","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/le-flux-electrique-et-la-loi-de-gauss\/","title":{"rendered":"Le Flux \u00c9lectrique et la Loi de Gauss"},"content":{"rendered":"<style>\np, ul, ol{\ntext-align: justify;\n}\nh1{\ntext-align:center;\ntext-transform: uppercase;\n}\nh2{\ntext-align:center;\ntext-transform: uppercase;\nfont-size:24pt;\n}\nh3 { \n    text-align: center;\n    text-transform: uppercase;\n    font-size: 24px !important;\n}\n<\/style>\n<h1>Le Flux \u00c9lectrique et la Loi de Gauss<\/h1>\n<p>\nEn \u00e9lectrostatique, calculer le champ \u00e9lectrique \u00ab \u00e0 partir de z\u00e9ro \u00bb peut devenir tr\u00e8s co\u00fbteux lorsque la g\u00e9om\u00e9trie de la distribution de charge n\u2019est pas triviale. Le flux \u00e9lectrique et la loi de Gauss offrent une voie plus intelligente : au lieu de se battre avec des int\u00e9grales interminables, on choisit une surface ferm\u00e9e appropri\u00e9e et on exploite la sym\u00e9trie du syst\u00e8me afin d\u2019obtenir des r\u00e9sultats clairs et v\u00e9rifiables. En pratique, cela se traduit par moins d\u2019\u00e9tapes, moins d\u2019erreurs et un meilleur contr\u00f4le conceptuel de ce que l\u2019on fait. Si vous souhaitez passer de \u00ab je connais la recette \u00bb \u00e0 \u00ab je comprends la m\u00e9thode \u00bb, vous verrez ici comment Gauss transforme des probl\u00e8mes qui semblent lourds en solutions directes, et dans quels cas son utilisation est r\u00e9ellement pertinente.\n<\/p>\n<p style=\"text-align:center;\"><b>Objectifs d\u2019Apprentissage<\/b><\/p>\n<ol>\n<li><strong>Expliquer<\/strong> le fonctionnement de la loi de Gauss pour le champ \u00e9lectrique.<\/li>\n<li><strong>Utiliser<\/strong> la loi de Gauss pour calculer des champs \u00e9lectriques en exploitant les sym\u00e9tries des coordonn\u00e9es cart\u00e9siennes, cylindriques et sph\u00e9riques.<\/li>\n<li><strong>Relier<\/strong> les formes int\u00e9grale et diff\u00e9rentielle au moyen du th\u00e9or\u00e8me de la divergence, en identifiant ce que repr\u00e9sente chaque terme.<\/li>\n<li><strong>Comparer<\/strong> l\u2019approche de Gauss avec le calcul direct par l\u2019int\u00e9grale de Coulomb, en expliquant quand elle r\u00e9duit la complexit\u00e9 et quand elle ne fournit pas de solution ferm\u00e9e.<\/li>\n<\/ol>\n<p style=\"text-align:center;\"><b><u>INDEX DES CONTENUS<\/u>:<\/b><br \/>\n<a href=\"#1\">R\u00e9solution de l\u2019\u00e9lectrostatique<\/a><br \/>\n<a href=\"#2\">Lignes de champ \u00e9lectrique<\/a><br \/>\n<a href=\"#3\">Remarque sur la densit\u00e9 des lignes de champ et leur repr\u00e9sentation<\/a><br \/>\n<a href=\"#4\">Flux du Champ \u00c9lectrique<\/a><br \/>\n<a href=\"#5\">Loi de Gauss<\/a><br \/>\n<a href=\"#6\">Probl\u00e8mes \u00e0 sym\u00e9trie sph\u00e9rique<\/a><br \/>\n<a href=\"#7\">Autres Sym\u00e9tries<\/a><br \/>\n<a href=\"#8\">Probl\u00e8mes \u00e0 sym\u00e9trie cylindrique et planaire<\/a>\n<\/p>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/b96XremJiKQ\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><br \/>\n<a name=\"1\"><\/a><\/p>\n<h2>R\u00e9solution de l\u2019\u00e9lectrostatique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=128s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">D\u2019apr\u00e8s ce que nous avons examin\u00e9 jusqu\u2019\u00e0 pr\u00e9sent, nous constatons que<\/span><\/strong><\/a> il suffit de conna\u00eetre la forme de l\u2019\u00e9l\u00e9ment de champ \u00e9lectrique et sa distribution dans l\u2019espace pour d\u00e9terminer le champ \u00e9lectrique total. Si nous avons une distribution volumique, alors<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\int_V d\\vec{E}(\\vec{r})= \\int_V \\frac{\\rho(\\vec{r}^\\prime)}{4\\pi\\epsilon_0}\\frac{\\vec{r}-\\vec{r}^\\prime}{\\|\\vec{r}-\\vec{r}^\\prime\\|^3}dV<\/span>\n<p>o\u00f9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rho(\\vec{r}^\\prime)<\/span><\/span> est la densit\u00e9 volumique de charge. Dans le cas d\u2019une densit\u00e9 surfacique ou lin\u00e9ique de charge, nous remplacerons <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> par <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span> ou <span class=\"katex-eq\" data-katex-display=\"false\">\\lambda<\/span>, respectivement. \u00c0 partir de ce point, ce qui d\u00e9termine si nous pouvons ou non trouver le champ \u00e9lectrique est notre capacit\u00e9 \u00e0 r\u00e9soudre ou non l\u2019int\u00e9grale.<\/p>\n<p>Bien que la formulation du probl\u00e8me soit g\u00e9n\u00e9ralement directe, nous d\u00e9couvrirons t\u00f4t ou tard qu\u2019il n\u2019est pas toujours facile de l\u2019\u00e9valuer. En effet, une grande partie de l\u2019\u00e9tude de l\u2019\u00e9lectrostatique consiste \u00e0 d\u00e9velopper des strat\u00e9gies permettant d\u2019\u00e9viter le calcul d\u2019int\u00e9grales inutilement compliqu\u00e9es. Bon nombre de ces simplifications proviennent de l\u2019analyse vectorielle, en particulier de l\u2019utilisation de la divergence.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Lignes de champ \u00e9lectrique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=235s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Avant d\u2019introduire l\u2019analyse vectorielle<\/span><\/strong><\/a> dans notre \u00e9tude de l\u2019\u00e9lectrostatique, nous pr\u00e9senterons quelques id\u00e9es qui contribueront \u00e0 rendre le sujet un peu plus intuitif. Il s\u2019agit des <strong>lignes de champ \u00e9lectrique<\/strong>.<\/p>\n<p>Commen\u00e7ons par le cas le plus simple : le champ \u00e9lectrique d\u2019une charge ponctuelle situ\u00e9e \u00e0 l\u2019origine du syst\u00e8me de coordonn\u00e9es. Celui-ci est de la forme<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\vec{E}(\\vec{r}) = \\frac{1}{4\\pi\\epsilon_0}\\frac{q}{\\|\\vec{r}\\|^2}\\hat{r}<\/span>\n<p>Cela nous permet de repr\u00e9senter le champ \u00e9lectrique dans l\u2019espace comme un ensemble de \u00ab fl\u00e8ches \u00bb dont la direction et la longueur d\u00e9crivent la direction et l\u2019intensit\u00e9 du champ \u00e9lectrique en chaque point.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEhlkOkwoYFrKYwfidmawFFb60AlBumv8u2irJnN87xYJnTfY7h2U1HL3Hzfh7kQZHhyctM7r70IfEXZkh0faUjZrGr3H_h6xrjeED-GqV39v_t2OzPD0zD9sjlm9t_twIEuaJcte9qhEGIKH3Bzn7a_AZ1rCh53DFunFLiXm09JalAMYQNAjKjv3oOosA\" width=\"400\" height=\"300\" alt=\"Champ \u00e9lectrique d\u2019une charge ponctuelle sous forme de vecteurs\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEhlkOkwoYFrKYwfidmawFFb60AlBumv8u2irJnN87xYJnTfY7h2U1HL3Hzfh7kQZHhyctM7r70IfEXZkh0faUjZrGr3H_h6xrjeED-GqV39v_t2OzPD0zD9sjlm9t_twIEuaJcte9qhEGIKH3Bzn7a_AZ1rCh53DFunFLiXm09JalAMYQNAjKjv3oOosA\" width=\"400\" height=\"300\" alt=\"Champ \u00e9lectrique d\u2019une charge ponctuelle sous forme de vecteurs\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>\u00c9tant donn\u00e9 que l\u2019intensit\u00e9 du champ \u00e9lectrique d\u00e9cro\u00eet avec le carr\u00e9 de la distance \u00e0 l\u2019origine, les vecteurs deviennent de plus en plus petits \u00e0 mesure que l\u2019on s\u2019en \u00e9loigne. De plus, ils pointent radialement depuis la charge vers l\u2019ext\u00e9rieur.<\/p>\n<p>Cette repr\u00e9sentation est utile, mais il en existe une autre encore plus informative : \u00ab relier le continuum de fl\u00e8ches \u00bb afin de former un champ de lignes. De cette mani\u00e8re, ce ne sera plus la longueur des fl\u00e8ches qui indiquera l\u2019intensit\u00e9 du champ \u00e9lectrique, mais la \u00ab densit\u00e9 des lignes de champ \u00bb dans le diagramme.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgebGCEuU6Akvc8M-p8m0FjK-a1AGtq4H7A2fAuy1r6M08uAnWQRYJfHIRtcRAvGt3CQZHCI7EwCHv3os55aZpef1KDTHFDiS2Sf8nvyXH_ctiildMSeSK-suC7al5kbGmFReywKsEJh1GVsHDtTqRShAyiJZFQER2fKav4wOcn9z8q7zjmjk3T07UHTQ\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgebGCEuU6Akvc8M-p8m0FjK-a1AGtq4H7A2fAuy1r6M08uAnWQRYJfHIRtcRAvGt3CQZHCI7EwCHv3os55aZpef1KDTHFDiS2Sf8nvyXH_ctiildMSeSK-suC7al5kbGmFReywKsEJh1GVsHDtTqRShAyiJZFQER2fKav4wOcn9z8q7zjmjk3T07UHTQ\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<a name=\"3\"><\/a><\/p>\n<h3>Remarque sur la densit\u00e9 des lignes de champ et leur repr\u00e9sentation<\/h3>\n<p>Avant de poursuivre, il convient de noter un d\u00e9tail concernant le diagramme des lignes de champ \u00e9lectrique. Ce type de repr\u00e9sentation n\u2019est pas totalement fid\u00e8le lorsqu\u2019il est trac\u00e9 dans un plan (2D). Dans un dessin 2D, si l\u2019on consid\u00e8re un cercle de rayon <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span>, le nombre total de lignes se r\u00e9partit sur le p\u00e9rim\u00e8tre de la circonf\u00e9rence, de sorte que la densit\u00e9 lin\u00e9aire est<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n}{2\\pi r}<\/span>\n<p>Celle-ci d\u00e9cro\u00eet en fonction de <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> et non de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r^2<\/span><\/span>, comme on s\u2019attendrait \u00e0 ce que le fasse l\u2019intensit\u00e9 du champ \u00e9lectrique. Cependant, si l\u2019on interpr\u00e8te le mod\u00e8le en trois dimensions (comme un h\u00e9risson), alors le nombre total de lignes serait r\u00e9parti sur la surface d\u2019une sph\u00e8re<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n}{4\\pi r^2}<\/span>\n<p>et cela d\u00e9cro\u00eet bien en fonction de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">r^2<\/span><\/span>. Autrement dit, bien que la repr\u00e9sentation des lignes de champ soit habituellement r\u00e9alis\u00e9e en deux dimensions, ce que l\u2019on cherche en r\u00e9alit\u00e9 \u00e0 synth\u00e9tiser est une situation en trois dimensions. Nous n\u2019avons simplement pas de papier en trois dimensions pour la dessiner : nous repr\u00e9sentons en 2D ce que nous souhaitons communiquer en 3D.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Flux de Champ \u00c9lectrique<\/h2>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=665s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Lorsque nous nous interrogeons sur le nombre<\/span><\/strong><\/a> de lignes de champ \u00e9lectrique qui traversent une surface donn\u00e9e, la r\u00e9ponse est fournie par le flux du champ \u00e9lectrique \u00e0 travers cette surface. Ainsi, on d\u00e9finit le flux \u00e9lectrique d\u2019un champ <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{E}<\/span><\/span> \u00e0 travers une surface <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> comme<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\Phi_{\\vec{E},S} =\\displaystyle \\int_S \\vec{E}\\cdot d\\vec{S}<\/span>\n<p>Nous ne devons pas nous laisser tromper par la notion intuitive de \u00ab nombre de lignes de champ \u00e9lectrique qui traversent une surface \u00bb. Rappelons que ce nombre de lignes (ou la densit\u00e9 de lignes) constitue une mani\u00e8re de repr\u00e9senter l\u2019intensit\u00e9 du champ \u00e9lectrique. Par cons\u00e9quent, le flux \u00e9lectrique que nous calculons est une grandeur scalaire associ\u00e9e \u00e0 l\u2019intensit\u00e9 du champ \u00e9lectrique qui traverse la surface <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>.<\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h3>Loi de Gauss<\/h3>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=758s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">\u00c9tant donn\u00e9 que l\u2019intensit\u00e9 du champ \u00e9lectrique<\/span><\/strong><\/a> est proportionnelle \u00e0 la charge \u00e9lectrique, nous devrions pouvoir exprimer le flux \u00e9lectrique \u00e0 travers une surface qui renferme une certaine charge comme une quantit\u00e9 proportionnelle \u00e0 la charge enferm\u00e9e. En effet, il n\u2019est pas difficile de montrer que c\u2019est bien le cas. Consid\u00e9rons la figure suivante :<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgGcCL8WVnhwXmxDkhsW5W31AyJiEsJDsVZZDNm1kQ-MREYYaaBvYb7CBkGSCkfPgiNbDGFP-R4LHr_9pH6ijy0Ji7m1VgzO2pjJwjFDOqAd61VGMJfb4CDfmGyn9uacon7VcpXlB9cd7ZltDUEc3fhDQ86PuKqQb7kN-JuNgGxInlRKiyY91nU2zHfIg\" width=\"500\" height=\"400\" alt=\"Flux \u00e9lectrique \u00e0 travers une surface ferm\u00e9e\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEgGcCL8WVnhwXmxDkhsW5W31AyJiEsJDsVZZDNm1kQ-MREYYaaBvYb7CBkGSCkfPgiNbDGFP-R4LHr_9pH6ijy0Ji7m1VgzO2pjJwjFDOqAd61VGMJfb4CDfmGyn9uacon7VcpXlB9cd7ZltDUEc3fhDQ86PuKqQb7kN-JuNgGxInlRKiyY91nU2zHfIg\" width=\"500\" height=\"400\" alt=\"Flux \u00e9lectrique \u00e0 travers une surface ferm\u00e9e\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>\u00c0 partir de cela, on obtient :<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\oint_S \\vec{E}\\cdot d\\vec{S} &amp;= \\displaystyle \\oint_S \\left(\\frac{1}{4\\pi\\epsilon_0} \\frac{q_{enc}}{\\|\\vec{r}\\|^2}\\hat{r} \\right)\\cdot d\\vec{S} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{q_{enc}}{4\\pi\\epsilon_0} \\oint_S \\frac{\\hat{r}}{\\|\\vec{r}\\|^2}\\cdot d\\vec{S} \\\\ \\\\\n\n&amp; = \\displaystyle \\frac{q_{enc}}{4\\pi\\epsilon_0} \\underbrace{\\oint_S d{\\Omega}}_{= 4\\pi} = \\frac{q_{enc}}{\\epsilon_0}\n\n\\end{array}\n\n<\/span>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEjLQhqvLJZMuQHfXHj_WYfbajP9PYwVdNgs4eVflg_jQAJSFu5czNfBgMBTWOWXCE5Tx3-DYwrs8eNpOuJoflvQYbUwpl3BG4BaZxJdnJirqRPsbZM00TfnzyGQvuAimfenB3GUYnEJdZDh2xiXWX5ftu0bN-UYH3G4rydnrnBqEpKDNnNXgdpi5EP81w\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/blogger.googleusercontent.com\/img\/a\/AVvXsEjLQhqvLJZMuQHfXHj_WYfbajP9PYwVdNgs4eVflg_jQAJSFu5czNfBgMBTWOWXCE5Tx3-DYwrs8eNpOuJoflvQYbUwpl3BG4BaZxJdnJirqRPsbZM00TfnzyGQvuAimfenB3GUYnEJdZDh2xiXWX5ftu0bN-UYH3G4rydnrnBqEpKDNnNXgdpi5EP81w\" width=\"400\" height=\"300\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p>En synth\u00e8se, nous obtenons :<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle\\color{blue}{\\oint_S \\vec{E}\\cdot d\\vec{S} = \\frac{q_{enc}}{\\epsilon_0}}<\/span>\n<p>Il s\u2019agit de la <strong>Loi de Gauss pour le champ \u00e9lectrique sous sa forme int\u00e9grale<\/strong>, qui montre une relation de proportionnalit\u00e9 entre le flux \u00e9lectrique \u00e0 travers une surface ferm\u00e9e et la charge enferm\u00e9e. Notons que je l\u2019ai pr\u00e9sent\u00e9e sous sa \u00ab forme int\u00e9grale \u00bb afin de souligner qu\u2019il existe \u00e9galement une forme diff\u00e9rentielle, laquelle s\u2019obtient en utilisant le th\u00e9or\u00e8me de la divergence de Gauss dans le cadre de l\u2019analyse vectorielle.<\/p>\n<div style=\"background-color: #c0ffc0; padding: 20px;\">\n<h4>Th\u00e9or\u00e8me de la divergence de Gauss<\/h4>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=1007s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Si <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{F}<\/span><\/span> est un champ vectoriel diff\u00e9rentiable<\/span><\/strong><\/a> et si <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span> est une surface ferm\u00e9e qui enferme un volume <span class=\"katex-eq\" data-katex-display=\"false\">V<\/span>, alors on a<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\oint_S\\vec{F}\\cdot d\\vec{S} = \\int_V (\\vec{\\nabla}\\cdot \\vec{F})dV<\/span>\n<\/div>\n<p>&nbsp;<\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=b96XremJiKQ&amp;t=1055s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">En appliquant le th\u00e9or\u00e8me de la divergence<\/span><\/strong><\/a> au flux du champ \u00e9lectrique \u00e0 travers la surface ferm\u00e9e <span class=\"katex-eq\" data-katex-display=\"false\">S<\/span>, on obtient<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\oint_S\\vec{E}\\cdot d\\vec{S} = \\int_V (\\vec{\\nabla}\\cdot\\vec{E})dV = \\frac{q_{enc}}{\\epsilon_0}<\/span>\n<p>Par ailleurs, on a \u00e9galement<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{q_{enc}}{\\epsilon_0} = \\int_V \\frac{\\rho}{\\epsilon_0} dV<\/span>\n<p>\u00c0 partir de ces deux derni\u00e8res \u00e9quations, on obtient finalement<\/p>\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\color{blue}{\\vec{\\nabla}\\cdot\\vec{E} = \\frac{\\rho}{\\epsilon_0}}<\/span>\n<p>Il s\u2019agit de la <strong>Loi de Gauss pour le champ \u00e9lectrique sous sa forme diff\u00e9rentielle.<\/strong><\/p>\n<p>Nous pouvons d\u00e9sormais utiliser la loi de Gauss afin de mieux exploiter les sym\u00e9tries g\u00e9om\u00e9triques de certains probl\u00e8mes et de simplifier, dans une large mesure, le calcul des int\u00e9grales qui conduisent au champ \u00e9lectrique.<\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h2>Probl\u00e8mes \u00e0 sym\u00e9trie sph\u00e9rique<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/04itEuVNDN4\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<ol>\n<li>D\u00e9terminer le champ \u00e9lectrique \u00e0 une distance <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span> du centre d\u2019une surface sph\u00e9rique de rayon <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span> qui poss\u00e8de une densit\u00e9 de charge uniforme <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span>. Analyser les deux cas : lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z\\lt R<\/span><\/span>, et lorsque <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">z\\geq R<\/span><\/span>.<\/li>\n<li>Effectuer la m\u00eame analyse que dans l\u2019exercice pr\u00e9c\u00e9dent, mais en consid\u00e9rant cette fois une sph\u00e8re pleine et uniform\u00e9ment charg\u00e9e avec une densit\u00e9 volumique <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span>. R\u00e9aliser ensuite un graphique de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{E}\\|<\/span><\/span> en fonction de <span class=\"katex-eq\" data-katex-display=\"false\">z<\/span>.<\/li>\n<li>Supposons que le champ \u00e9lectrique, \u00e0 une distance <span class=\"katex-eq\" data-katex-display=\"false\">r<\/span> de l\u2019origine du syst\u00e8me de coordonn\u00e9es, soit <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{E}=kr^2\\hat{r}<\/span><\/span>, avec <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> constante. Trouver la densit\u00e9 de charge <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> associ\u00e9e \u00e0 ce champ.<\/li>\n<\/ol>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Autres Sym\u00e9tries<\/h3>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/6eMaax9orAo\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><\/center><\/p>\n<p><a href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=122s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">La loi de Gauss est toujours vraie<\/span><\/strong><\/a>, mais elle n\u2019est pas toujours utile. Dans les exemples pr\u00e9c\u00e9dents, si <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> n\u2019\u00e9tait pas uniforme, si nous ne disposions pas d\u2019une sym\u00e9trie sph\u00e9rique, ou si une autre forme \u00e9tait choisie pour la surface gaussienne, il resterait vrai que le flux \u00e9lectrique vaut <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">q_{enc}\/\\epsilon_0<\/span><\/span>, mais le champ \u00e9lectrique ne serait pas n\u00e9cessairement constant ni orient\u00e9 dans la m\u00eame direction que l\u2019\u00e9l\u00e9ment <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">d\\vec{S}<\/span><\/span> ; et sans ces conditions, nous ne pouvons pas extraire <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\|\\vec{E}\\|<\/span><\/span> de l\u2019int\u00e9grale.<\/p>\n<p>La sym\u00e9trie est cruciale dans l\u2019application de la loi de Gauss \u00e0 la r\u00e9solution des probl\u00e8mes.<\/p>\n<p>Il existe de nombreux types de sym\u00e9tries que nous pouvons exploiter. Parmi toutes celles-ci, les trois suivantes sont les plus fr\u00e9quentes :<\/p>\n<ol>\n<li><strong>Sym\u00e9trie sph\u00e9rique :<\/strong> la surface gaussienne est une sph\u00e8re concentrique.<\/li>\n<li><strong>Sym\u00e9trie cylindrique :<\/strong> la surface gaussienne est un cylindre coaxial.<\/li>\n<li><strong>Sym\u00e9trie planaire :<\/strong> la surface gaussienne est une bo\u00eete rectangulaire.<\/li>\n<\/ol>\n<p><a name=\"8\"><\/a><\/p>\n<h3>Probl\u00e8mes \u00e0 sym\u00e9trie cylindrique et planaire<\/h3>\n<ol>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=630s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Consid\u00e9rez un c\u00e2ble cylindrique<\/span><\/strong><\/a> infiniment long, rectiligne, de rayon <span class=\"katex-eq\" data-katex-display=\"false\">R<\/span> et charg\u00e9 avec une densit\u00e9 de charge <span class=\"katex-eq\" data-katex-display=\"false\">\\rho<\/span> de la forme\n<p style=\"text-align: center;\" dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\rho(r) = \\left\\{\\begin{array}{lll}\n\nkr &amp; ; &amp; r\\lt R \\\\ \\\\\n\n0 &amp; ; &amp; R\\lt r \\\\ \\\\\n\n\\end{array}\\right.<\/span>\n<p>o\u00f9 <span class=\"katex-eq\" data-katex-display=\"false\">k<\/span> est une constante. Calculez le champ \u00e9lectrique \u00e0 l\u2019int\u00e9rieur du cylindre.<\/li>\n<li><a href=\"https:\/\/www.youtube.com\/watch?v=6eMaax9orAo&amp;t=1895s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Trouver le champ \u00e9lectrique produit<\/span><\/strong><\/a> par un plan infini muni d\u2019une densit\u00e9 uniforme de charge <span class=\"katex-eq\" data-katex-display=\"false\">\\sigma<\/span>.<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Le Flux \u00c9lectrique et la Loi de Gauss En \u00e9lectrostatique, calculer le champ \u00e9lectrique \u00ab \u00e0 partir de z\u00e9ro \u00bb peut devenir tr\u00e8s co\u00fbteux lorsque la g\u00e9om\u00e9trie de la distribution de charge n\u2019est pas triviale. Le flux \u00e9lectrique et la loi de Gauss offrent une voie plus intelligente : au lieu de se battre avec [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":35708,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":6,"footnotes":""},"categories":[722,647],"tags":[],"class_list":["post-35739","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electromagnetisme","category-physique"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Le Flux \u00c9lectrique et la Loi de Gauss - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Le flux \u00e9lectrique mesure la quantit\u00e9 de \u00ab sortie \u00bb du champ \u00e0 travers une surface ; la loi de Gauss le relie \u00e0 la charge enferm\u00e9e \u222eE\u00b7dA = Qenc\/\u03b50\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/le-flux-electrique-et-la-loi-de-gauss\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Le Flux \u00c9lectrique et la Loi de Gauss\" \/>\n<meta property=\"og:description\" content=\"Le flux \u00e9lectrique mesure la quantit\u00e9 de \u00ab sortie \u00bb du champ \u00e0 travers une surface ; 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