{"id":25650,"date":"2022-07-12T00:00:56","date_gmt":"2022-07-12T00:00:56","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25650"},"modified":"2024-05-21T09:42:23","modified_gmt":"2024-05-21T09:42:23","slug":"la-vitesse-de-la-lumiere-et-les-ondes-electromagnetiques-dans-lespace-vide","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/la-vitesse-de-la-lumiere-et-les-ondes-electromagnetiques-dans-lespace-vide\/","title":{"rendered":"La Vitesse de la Lumi\u00e8re et les Ondes \u00c9lectromagn\u00e9tiques dans l&#8217;Espace Vide"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>La Vitesse de la Lumi\u00e8re et les Ondes \u00c9lectromagn\u00e9tiques dans l&#8217;Espace Vide<\/h1>\n<p class=\"eq\"><em><strong>R\u00e9sum\u00e9 :<\/strong><br \/>\nDans ce cours, nous examinerons comment, \u00e0 partir du comportement des ondes \u00e9lectromagn\u00e9tiques dans le vide, on obtient la solution des \u00e9quations de Maxwell de l&#8217;\u00e9lectromagn\u00e9tisme dans l&#8217;espace vide. En cons\u00e9quence, il s&#8217;av\u00e8re que la vitesse de propagation des ondes \u00e9lectromagn\u00e9tiques dans le vide est une constante qui ne d\u00e9pend d&#8217;aucun r\u00e9f\u00e9rentiel inertiel.<\/br><\/em><\/p>\n<p><strong>OBJECTIFS D&#8217;APPRENTISSAGE<\/strong><br \/>\n\u00c0 la fin de ce cours, l&#8217;\u00e9tudiant sera capable de :<\/center><\/p>\n<ol>\n<li><strong>D\u00e9montrer<\/strong> la connexion entre les \u00e9quations de Maxwell dans le vide et la propagation des ondes \u00e9lectromagn\u00e9tiques.<\/li>\n<li><strong>Argumenter<\/strong> pourquoi la vitesse de la lumi\u00e8re est constante dans le vide et comment cela contredit les transformations de Galil\u00e9e.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Les \u00e9quations de Maxwell dans le vide<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>La propagation des ondes \u00e9lectromagn\u00e9tiques<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>La vitesse de la lumi\u00e8re est une constante universelle<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>Conclusions<\/strong><\/a>\n<\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/FsHuY-aQGdA\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center>\n<\/div>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Les \u00e9quations de Maxwell dans le vide<\/h2>\n<p style=\"text-align:justify;\">L&#8217;\u00e9lectromagn\u00e9tisme de l&#8217;espace vide pr\u00e9sente certaines propri\u00e9t\u00e9s qui m\u00e9ritent d&#8217;\u00eatre mentionn\u00e9es. Il s&#8217;av\u00e8re que les \u00e9quations de Maxwell d\u00e9crivant les champs \u00e9lectriques et magn\u00e9tiques prennent la forme suivante dans le vide<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rlr}\n\n\\vec{\\nabla} \\cdot \\vec{E}  &amp;= 0 &amp; [1]\\\\ \\vec{\\nabla} \\cdot \\vec{B}  &amp;= 0 &amp; [2]\\\\ \\vec{\\nabla} \\times \\vec{E} &amp;\\displaystyle = -\\frac{\\partial \\vec{B}}{\\partial t} &amp; [3]\\\\ \\vec{\\nabla} \\times \\vec{B} &amp;\\displaystyle = \\mu_0\\epsilon_0 \\frac{\\partial \\vec{E}}{\\partial t} &amp; [4] \\end{array}\n\n<\/span>\n<p style=\"text-align:justify;\">\u00c0 partir de cela, on peut confirmer que toute perturbation dans les champs \u00e9lectriques et magn\u00e9tiques se propage comme une onde dans l&#8217;espace vide. Comment le savons-nous ? Parce qu&#8217;en analysant ces expressions, on obtient une \u00e9quation d&#8217;onde pour les deux champs.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>La propagation des ondes \u00e9lectromagn\u00e9tiques<\/h2>\n<p style=\"text-align:justify;\">\u00c0 partir de [4] et [5], il est \u00e9tabli que le champ \u00e9lectrique satisfait la relation suivante :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{llr}\n\n\\vec{\\nabla} \\times (\\vec{\\nabla} \\times \\vec{E}) &amp;= \\displaystyle \\vec{\\nabla} \\times \\left( -\\frac{\\partial \\vec{B}}{\\partial t} \\right) &amp;\\\\\n\n                                      &amp;=\\displaystyle -\\frac{\\partial}{\\partial t}\\left(\\vec{\\nabla} \\times \\vec{B}\\right) = -\\frac{\\partial}{\\partial t} \\left(\\mu_0\\epsilon_0 \\frac{\\partial \\vec{E}}{\\partial t} \\right) = -\\mu_0\\epsilon_0  \\frac{\\partial^2 \\vec{E}}{\\partial t^2}&amp; [6]\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">Ensuite, comme tout champ vectoriel satisfait la relation :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\vec{\\nabla} \\times \\left(\\vec{\\nabla} \\times \\vec{A} \\right) = \\vec{\\nabla}(\\vec{\\nabla} \\cdot \\vec{A}) - \\nabla^2 \\vec{A},\\;\\;\\;[7]<\/span>\n<p style=\"text-align:justify;\">\u00c0 partir de [2, 6] et [7], on peut \u00e9crire :<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n&amp;\\displaystyle \\vec{\\nabla}(\\underbrace{\\vec{\\nabla} \\cdot \\vec{E}}_{=0}) - \\nabla^2 \\vec{E} = \\vec{\\nabla} \\times \\left(\\vec{\\nabla} \\times \\vec{E} \\right) = -\\mu_0\\epsilon_0  \\frac{\\partial^2 \\vec{E}}{\\partial t^2} &amp; \\\\\n\n\\equiv &amp;\\displaystyle \\color{blue}{\\nabla^2 \\vec{E} =  \\mu_0\\epsilon_0  \\frac{\\partial^2 \\vec{E}}{\\partial t^2}}  &amp; [8]\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">Ce qui est marqu\u00e9 en bleu est justement une \u00e9quation de propagation des ondes pour le champ \u00e9lectrique.<\/p>\n<p style=\"text-align:justify;\">De mani\u00e8re compl\u00e8tement analogue, cela se produit pour le champ magn\u00e9tique<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{ll}\n\n\\vec{\\nabla} \\times (\\vec{\\nabla} \\times \\vec{B}) &amp;= \\displaystyle \\vec{\\nabla} \\times \\left(\\mu_0\\epsilon_0 \\frac{\\partial \\vec{E}}{\\partial t} \\right)\\\\\n\n                                      &amp;=\\displaystyle \\mu_0\\epsilon_0 \\frac{\\partial}{\\partial t}\\left(\\vec{\\nabla} \\times \\vec{E}\\right) = \\mu_0\\epsilon_0 \\frac{\\partial}{\\partial t} \\left(- \\frac{\\partial \\vec{B}}{\\partial t} \\right) = -\\mu_0\\epsilon_0 \\displaystyle  \\frac{\\partial^2 \\vec{B}}{\\partial t^2}\n\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">et ensuite<\/p>\n<p id=\"eq\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rll}\n\n&amp; \\displaystyle \\vec{\\nabla}(\\underbrace{\\vec{\\nabla} \\cdot \\vec{B}}_{=0}) - \\nabla^2 \\vec{B} = \\vec{\\nabla} \\times \\left(\\vec{\\nabla} \\times \\vec{B} \\right) = -\\mu_0\\epsilon_0  \\frac{\\partial^2 \\vec{B}}{\\partial t^2} &amp;\\\\\n\n\\equiv &amp;\\color{blue}{\\nabla^2 \\vec{B} = \\displaystyle \\mu_0\\epsilon_0  \\frac{\\partial^2 \\vec{B}}{\\partial t^2}}&amp; [9]\n\\end{array}<\/span>\n<p style=\"text-align:justify;\">C&#8217;est \u00e0 partir de cela que l&#8217;on dit que les champs \u00e9lectromagn\u00e9tiques dans le vide ont de nombreux modes possibles d&#8217;\u00eatre, et une famille de ces modes prend la forme d&#8217;une onde \u00e9lectromagn\u00e9tique qui se propage dans l&#8217;espace et le temps.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>La vitesse de la lumi\u00e8re est une constante universelle<\/h2>\n<p style=\"text-align:justify;\">En d&#8217;autres termes, les perturbations dans les champs \u00e9lectromagn\u00e9tiques se propagent toujours rapidement <span class=\"katex-eq\" data-katex-display=\"false\">c = 1\/\\sqrt{\\epsilon_0 \\mu_0}\\approx 3\\cdot 10^8[m\/s],<\/span> qui est la vitesse de la lumi\u00e8re dans le vide. Exp\u00e9rimentalement, on observe que cette vitesse est la m\u00eame pour tous les r\u00e9f\u00e9rentiels inertiels, ce qui ne correspond pas \u00e0 ce qui serait obtenu en appliquant les transformations de Galil\u00e9e, comme montr\u00e9 dans <a href=\"http:\/\/toposuranos.com\/material\/es\/las-transformaciones-de-galileo-y-sus-limitaciones\/\" rel=\"noopener\" target=\"_blank\">Les Transformations de Galil\u00e9e et leurs Limitations<\/a> ; car selon celles-ci, m\u00eame la structure m\u00eame de l&#8217;onde est alt\u00e9r\u00e9e lors du passage d&#8217;un r\u00e9f\u00e9rentiel inertiel \u00e0 un autre. Ces r\u00e9sultats sont la cl\u00e9 pour abandonner les transformations de Galil\u00e9e, faisant place aux transformations de Lorentz de la relativit\u00e9 restreinte car : une transformation de coordonn\u00e9es correctement formul\u00e9e doit pr\u00e9server les lois de la physique pour tous les observateurs inertiels.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\"><a name=\"4\"><\/a><\/p>\n<h2>Conclusions<\/h2>\n<p style=\"text-align:justify;\">\n        Cette \u00e9tude sur les ondes \u00e9lectromagn\u00e9tiques et la vitesse de la lumi\u00e8re dans le vide a r\u00e9v\u00e9l\u00e9 des aspects fondamentaux de la physique moderne. Les \u00e9quations de Maxwell dans le vide ne d\u00e9crivent pas seulement la propagation des champs \u00e9lectromagn\u00e9tiques sous forme d&#8217;ondes, mais mettent \u00e9galement en \u00e9vidence une constante universelle : la vitesse de la lumi\u00e8re. Cette d\u00e9couverte remet en question les notions classiques de la physique, telles que les Transformations de Galil\u00e9e, et souligne l&#8217;importance des Transformations de Lorentz dans la relativit\u00e9 restreinte. La constance de la vitesse de la lumi\u00e8re dans tous les cadres inertiels est un pilier fondamental dans notre compr\u00e9hension de l&#8217;univers, qui va au-del\u00e0 de l&#8217;intuition classique et ouvre la porte \u00e0 une exploration plus profonde des lois de la physique.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>La Vitesse de la Lumi\u00e8re et les Ondes \u00c9lectromagn\u00e9tiques dans l&#8217;Espace Vide R\u00e9sum\u00e9 : Dans ce cours, nous examinerons comment, \u00e0 partir du comportement des ondes \u00e9lectromagn\u00e9tiques dans le vide, on obtient la solution des \u00e9quations de Maxwell de l&#8217;\u00e9lectromagn\u00e9tisme dans l&#8217;espace vide. En cons\u00e9quence, il s&#8217;av\u00e8re que la vitesse de propagation des ondes \u00e9lectromagn\u00e9tiques [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":25601,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":10,"footnotes":""},"categories":[722,647,703],"tags":[],"class_list":["post-25650","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electromagnetisme","category-physique","category-relativite"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>La Vitesse de la Lumi\u00e8re et les Ondes \u00c9lectromagn\u00e9tiques dans l&#039;Espace Vide - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"D\u00e9couvrez comment les \u00e9quations de Maxwell r\u00e9v\u00e8lent la vitesse de la lumi\u00e8re comme une constante universelle. 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