{"id":34810,"date":"2022-07-12T13:00:33","date_gmt":"2022-07-12T13:00:33","guid":{"rendered":"https:\/\/toposuranos.com\/material\/?p=34810"},"modified":"2025-09-21T02:34:05","modified_gmt":"2025-09-21T02:34:05","slug":"celeritas-lucis-et-undae-electromagneticae","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/la\/celeritas-lucis-et-undae-electromagneticae\/","title":{"rendered":"Celeritas Lucis et Undae Electromagneticae"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>Celeritas Lucis et Undae Electromagneticae in Spatio Vacuo<\/h1>\n<p class=\"eq\"><em><strong>Summarium:<\/strong><br \/>\nIn hac lectione recognoscemus quomodo, ex comportatione undarum electromagneticarum in vacuo, obtinetur ex resolutione aequationum Maxwellianarum electromagnetismi in spatio vacuo. Quamobrem habetur celeritatem propagationis undarum electromagneticarum in vacuo constantem esse quae nullo systemate inertiali refertur.<\/br><\/em><\/p>\n<p><strong>PROPOSITA DISCENDI<\/strong><br \/>\nPost hanc lectionem discipulus poterit:<\/center><\/p>\n<ol>\n<li><strong>Demonstrare<\/strong> nexum inter aequationes Maxwellianas in vacuo et propagationem undarum electromagneticarum.<\/li>\n<li><strong>Rationem reddere<\/strong> cur celeritas lucis in vacuo constans sit et quomodo hoc transformationibus Galilaeanis adversetur.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Aequationes Maxwellianae in vacuo<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>Propagatio undarum electromagneticarum<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>Celeritas lucis est constans universalis<\/strong><\/a><br \/>\n<a href=\"#4\"><strong>Conclusiones<\/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>Aequationes Maxwellianae in vacuo<\/h2>\n<p style=\"text-align:justify;\">Electromagnetismus spatii vacui quasdam proprietates habet quae commemorandae sunt. Fit ut aequationes Maxwellianae quae campos electricos et magneticos describunt, in vacuo hanc formam accipiant:<\/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;\">Ex his confirmari potest perturbationem quamlibet in campis electricis et magneticis propagari velut undam in spatio vacuo. Quomodo hoc scimus? Quia ex analysi harum expressionum aequatio undae pro ambobus campis obtinetur.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>Propagatio undarum electromagneticarum<\/h2>\n<p style=\"text-align:justify;\">Ex [4] et [5] habetur campum electricum sequentem relationem explere:<\/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;\">Deinde, sicut omne campus vectorialis relationem explere solet:<\/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;\">Ex [2, 6] et [7] scribi potest:<\/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;\">Hoc quod caeruleo notatum apparet est ipsa aequatio propagationis undarum pro campo electrico.<\/p>\n<p style=\"text-align:justify;\">Similiter omnino fit pro campo magnetico.<\/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 deinde<\/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;\">Ex hoc dicitur campos electromagneticos in vacuo multos modos posse habere, et una familia horum modorum formam undae electromagneticae accipit quae per spatium et tempus propagatur.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Celeritas lucis est constans universalis<\/h2>\n<p style=\"text-align:justify;\">Aliter dictum, perturbationes in campis electromagneticis semper propagantur celeritate <span class=\"katex-eq\" data-katex-display=\"false\">c = 1\/\\sqrt{\\epsilon_0 \\mu_0}\\approx 3\\cdot 10^8[m\/s],<\/span> quae est celeritas lucis in vacuo. Experimentis observatur hanc celeritatem eandem esse omnibus referentialibus inertialibus, quod non congruit cum iis quae obtinerentur si transformationes Galilaeanae adhiberentur, ut ostenditur in <a href=\"http:\/\/toposuranos.com\/material\/es\/las-transformaciones-de-galileo-y-sus-limitaciones\/\" rel=\"noopener\" target=\"_blank\">Transformationes Galilaeanae et earum Limitationes<\/a>; nam secundum has, ipsa quoque structura undae mutatur cum ab uno referentiali inertiali ad alterum transitum fit. Haec resultata sunt pars clavissima ad transformationes Galilaeanas reiciendas, locum facientes transformationibus Lorentzianis relativitatis specialis, quia: transformatio coordinatarum recte formata debebit leges physicae omnibus observatoribus inertialibus servare.<\/p>\n<div style=\"background-color:#F3F3F3; padding:20px;\"><a name=\"4\"><\/a><\/p>\n<h2>Conclusiones<\/h2>\n<p style=\"text-align:justify;\">\n        Hoc studium de undis electromagneticis et celeritate lucis in vacuo aspectus fundamentales physicae modernae patefecit. Aequationes Maxwellianae in vacuo non solum propagationem camporum electromagneticorum ut undarum describunt, sed etiam constantem universalem eminent: celeritatem lucis. Hoc inventum notiones classicas physicae, ut Transformationes Galilaeanas, provocat atque momentum Transformationum Lorentzianarum in relativitate speciali illustrat. Constantia celeritatis lucis in omnibus systematibus inertialibus fundamentum est in nostra intellegentia universi, quod ultra intuitionem classicam procedit et ad altiorem explorationem legum physicae viam pandit.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>Celeritas Lucis et Undae Electromagneticae in Spatio Vacuo Summarium: In hac lectione recognoscemus quomodo, ex comportatione undarum electromagneticarum in vacuo, obtinetur ex resolutione aequationum Maxwellianarum electromagnetismi in spatio vacuo. Quamobrem habetur celeritatem propagationis undarum electromagneticarum in vacuo constantem esse quae nullo systemate inertiali refertur. PROPOSITA DISCENDI Post hanc lectionem discipulus poterit: Demonstrare nexum inter aequationes [&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":4,"footnotes":""},"categories":[1256,1250,1286],"tags":[],"class_list":["post-34810","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electromagnetismus","category-physica","category-relativitas"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v28.0 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Celeritas Lucis et Undae Electromagneticae - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Disce quomodo aequationes Maxwellianae celeritatem lucis tamquam constantem universalem revelent. 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