{"id":25626,"date":"2022-07-12T00:00:06","date_gmt":"2022-07-12T00:00:06","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=25626"},"modified":"2024-05-21T09:42:29","modified_gmt":"2024-05-21T09:42:29","slug":"the-speed-of-light-and-electromagnetic-waves","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/en\/the-speed-of-light-and-electromagnetic-waves\/","title":{"rendered":"The Speed of Light and Electromagnetic Waves"},"content":{"rendered":"<div style=\"background-color:#F3F3F3; padding:20px;\">\n<center><\/p>\n<h1>The Speed of Light and Electromagnetic Waves in a Vacuum<\/h1>\n<p class=\"eq\"><em><strong>Summary:<\/strong><br \/>\nIn this class, we will review how, from the behavior of electromagnetic waves in a vacuum, it is derived from solving Maxwell&#8217;s equations of electromagnetism in a vacuum. As a consequence, it is found that the speed of propagation of electromagnetic waves in a vacuum is a constant that does not depend on any inertial frame of reference.<\/br><\/em><\/p>\n<p><strong>LEARNING OBJECTIVES<\/strong><br \/>\nAt the end of this class, students will be able to:<\/center><\/p>\n<ol>\n<li><strong>Demonstrate<\/strong> the connection between Maxwell&#8217;s equations in a vacuum and the propagation of electromagnetic waves.<\/li>\n<li><strong>Argue<\/strong> why the speed of light is constant in a vacuum and how this contradicts Galilean transformations.<\/li>\n<\/ol>\n<p><center><\/p>\n<p class=\"indx\"><strong>INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>Maxwell&#8217;s Equations in a Vacuum<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>The Propagation of Electromagnetic Waves<\/strong><\/a><br \/>\n<a href=\"#3\"><strong>The Speed of Light as a Universal Constant<\/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>Maxwell&#8217;s Equations in a Vacuum<\/h2>\n<p style=\"text-align:justify;\">The electromagnetism of vacuum space has some properties worth mentioning. It turns out that Maxwell&#8217;s equations, which describe electric and magnetic fields, take the following form in a vacuum:<\/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;\">From this, we can verify that any disturbance in the electric and magnetic fields propagates as a wave in vacuum space. How do we know this? Because analyzing these expressions yields a wave equation for both fields.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Propagation of Electromagnetic Waves<\/h2>\n<p style=\"text-align:justify;\">From [4] and [5], it follows that the electric field satisfies the following relation:<\/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;\">Then, as any vector field satisfies the 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;\">From [2, 6] and [7], we can write:<\/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;\">This, marked in blue, is just a wave propagation equation for the electric field.<\/p>\n<p style=\"text-align:justify;\">Analogously, the same occurs for the magnetic field<\/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;\">and then<\/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;\">It is from this that electromagnetic fields in vacuum are said to have many possible modes of being, and one family of these modes takes the form of an electromagnetic wave that propagates through space and time.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>The Speed of Light is a Universal Constant<\/h2>\n<p style=\"text-align:justify;\">In other words, disturbances in the electromagnetic field always propagate with the speed <span class=\"katex-eq\" data-katex-display=\"false\">c = 1\/\\sqrt{\\epsilon_0 \\mu_0}\\approx 3\\cdot 10^8[m\/s],<\/span> which is the speed of light in a vacuum. Experimentally, it is observed that this speed is the same for all inertial frames of reference, which does not align with what would be obtained if Galileo&#8217;s transformations were applied, as shown in <a href=\"http:\/\/toposuranos.com\/material\/en\/galileos-transformations-and-their-limitations\/\" rel=\"noopener\" target=\"_blank\">Galileo&#8217;s Transformations and Their Limitations<\/a>; because according to these, even the wave&#8217;s own structure is altered when moving from one inertial frame to another. These results are the key piece to set aside Galileo&#8217;s transformations, giving way to Lorentz&#8217;s transformations of special relativity because: a properly formulated coordinate transformation must preserve the laws of physics for all inertial observers.<\/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        This study on electromagnetic waves and the speed of light in a vacuum has revealed fundamental aspects of modern physics. Maxwell&#8217;s equations in a vacuum not only describe the propagation of electromagnetic fields as waves but also highlight a universal constant: the speed of light. This discovery challenges the classical notions of physics, such as Galileo&#8217;s Transformations, and underscores the importance of Lorentz Transformations in special relativity. The constancy of the speed of light in all inertial frames is a fundamental pillar in our understanding of the universe, which goes beyond classical intuition and opens the door to a deeper exploration of the laws of physics.\n    <\/p>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>The Speed of Light and Electromagnetic Waves in a Vacuum Summary: In this class, we will review how, from the behavior of electromagnetic waves in a vacuum, it is derived from solving Maxwell&#8217;s equations of electromagnetism in a vacuum. As a consequence, it is found that the speed of propagation of electromagnetic waves in a [&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":6,"footnotes":""},"categories":[710,635,691],"tags":[],"class_list":["post-25626","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-electromagnetism","category-physics","category-relativity"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The Speed of Light and Electromagnetic Waves - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Discover how Maxwell&#039;s equations reveal the speed of light as a universal constant. 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