{"id":27685,"date":"2021-09-18T13:00:28","date_gmt":"2021-09-18T13:00:28","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27685"},"modified":"2024-08-10T23:54:04","modified_gmt":"2024-08-10T23:54:04","slug":"the-refraction-of-light-and-snells-law","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/","title":{"rendered":"The Refraction of Light and Snell&#8217;s Law"},"content":{"rendered":"<p><center><\/p>\n<h1>The Refraction of Light and Snell&#8217;s Law<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nIn this class, we will explore the refraction of light through the analysis of Snell&#8217;s Law. The concept of the refractive index will be explained, Snell&#8217;s Law will be derived using Fermat&#8217;s principle, and the study of how this law allows us to calculate the path of a light ray as it passes through different media will be covered. Additionally, the phenomena of reflection and total reflection will be discussed, applying these concepts to a series of practical exercises. The goal is to understand and apply Snell&#8217;s Law in optical problems.<\/em><\/p>\n<p><strong>Learning Objectives<\/strong><\/p>\n<ol style=\"text-align:left\">\n<li><strong>Understand<\/strong> the concept of the refractive index and its relationship with the speed of light in different media.<\/li>\n<li><strong>Apply<\/strong> Fermat&#8217;s principle to understand how light follows the path that minimizes the travel time between two points.<\/li>\n<li><strong>Demonstrate<\/strong> Snell&#8217;s Law from Fermat&#8217;s principle to determine the path of a light ray passing through different media.<\/li>\n<li><strong>Calculate<\/strong> the angles of incidence and refraction using Snell&#8217;s Law in situations with different refractive indices.<\/li>\n<li><strong>Understand<\/strong> the concept of total internal reflection and how it relates to the critical angle and refractive indices.<\/li>\n<li><strong>Determine<\/strong> the critical angle for total internal reflection at the interface between two media.<\/li>\n<\/ol>\n<p><strong>CONTENT INDEX<\/strong><br \/>\n<a href=\"#1\"><strong>The Refractive Index<\/strong><\/a><br \/>\n<a href=\"#2\">Fermat&#8217;s Principle<\/a><br \/>\n<a href=\"#3\">Snell&#8217;s Law of Light Refraction<\/a><br \/>\n<a href=\"#4\"><strong>Refraction, Reflection, and Total Reflection of Light<\/strong><\/a><br \/>\n<a href=\"#5\"><strong>Exercises<\/strong><\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/LxhWbErujpo\" 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 name=\"1\"><\/a><\/p>\n<h2>The Refractive Index<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=186s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The refractive index is defined<\/span><\/strong><\/a> of a medium as the ratio between the speed of light in a vacuum and the speed of light in that medium. This is a dimensionless quantity and is generally represented by the letter <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_k:<\/span><\/span><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_k=\\displaystyle \\frac{c}{c_k}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> is the speed of light in a vacuum, and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_k<\/span><\/span> is the speed of light in the medium <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Since light always moves slower in any medium than in a vacuum, the refractive index is always greater than or equal to 1.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>Fermat&#8217;s Principle<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=397s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">The speed of light depends<\/span><\/strong><\/a> on the medium in which it travels. The higher the refractive index of the medium, the slower the speed of light when traveling through it; and in relation to this, Fermat&#8217;s principle is stated:<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #80ff80;\">When light travels from one point to another, it does so along the path that minimizes the travel time.<\/p>\n<p style=\"text-align: justify; color: #000000;\">This principle holds even when light passes through different media.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Snell&#8217;s Law of Light Refraction<\/h3>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=608s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Based on what is established by Fermat&#8217;s principle<\/span><\/strong><\/a>, it is possible to formulate an optimization problem that will allow us to determine the path that a light ray will follow as it passes through different media. This is what ultimately leads to Snell&#8217;s Law, whose formulation and demonstration we will see below.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Suppose a ray departs from point A and arrives at point B, crossing an interface that separates two media with refractive indices <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2<\/span><\/span> respectively. Our goal will be to find a relationship that allows us to calculate the path of the light ray following Fermat&#8217;s principle of minimum travel time, and for this, the following diagram is set up:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-1CdoEOp5SHc\/YUDNPnQyOxI\/AAAAAAAAFjE\/RG-kYgV4KKwAE3QwiM9nB3cA-OOXesONQCLcBGAsYHQ\/s0\/n1n2leydeSnell.PNG\" width=\"875\" height=\"518\" alt=\"Snell's Law\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-1CdoEOp5SHc\/YUDNPnQyOxI\/AAAAAAAAFjE\/RG-kYgV4KKwAE3QwiM9nB3cA-OOXesONQCLcBGAsYHQ\/s0\/n1n2leydeSnell.PNG\" width=\"875\" height=\"518\" alt=\"Snell's Law\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">The reasoning begins by analyzing the form of the travel time of the light ray. We have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}{Travel\\,Time} &amp; =\\displaystyle \\frac{{Distance}}{{Speed}} \\\\ \\\\ &amp; \\displaystyle =\\frac{{Distance\\,in\\,medium\\,1}}{{Speed\\,in\\,medium\\,1}} + \\frac{{Distance\\,in\\,medium\\,2}}{{Speed\\,in\\,medium\\,2}}\\\\ \\\\&amp; =\\displaystyle \\frac{\\sqrt{a^2 + x^2}}{c_1} + \\frac{\\sqrt{b^2 + (d-x)^2}}{c_2}\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Having done this, keeping points A and B fixed, the travel time is determined by the point <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> at which the ray touches the interface between the media. With this, we can define a time function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x)<\/span><\/span> as<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x) = \\displaystyle \\frac{1}{c_1}\\sqrt{a^2 + x^2} + \\frac{1}{c_2}\\sqrt{b^2 + (d-x)^2}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now, since Fermat&#8217;s principle states that light follows the path that minimizes travel time, it is possible from this to find the <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> that minimizes the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x).<\/span><\/span> We are dealing with an optimization problem.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Differentiating <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t<\/span><\/span> with respect to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span>, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\begin{array}{rl}\\dfrac{dt}{dx} &amp;\\displaystyle = \\frac{1}{c_1}\\frac{d}{dx}\\sqrt{a^2 + x^2} + \\frac{1}{c_2}\\frac{d}{dx}\\sqrt{b^2+(d-x)^2}\\\\ \\\\ &amp;\\displaystyle = \\frac{1}{c_1} \\frac{2x}{2\\sqrt{a^2 + x^2}} + \\frac{1}{c_2}\\frac{2(d-x)(-1)}{2\\sqrt{b^2+(d-x)^2}} \\\\ \\\\ &amp;\\displaystyle = \\frac{1}{c_1} \\frac{x}{\\sqrt{a^2 + x^2}} - \\frac{1}{c_2}\\frac{(d-x)}{\\sqrt{b^2+(d-x)^2}} \\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now notice that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\\sin(\\theta_1) &amp;\\displaystyle =\\frac{x}{\\sqrt{a^2 + x^2}}\\\\ \\\\ \\sin(\\theta_2) &amp;\\displaystyle = \\frac{(d-x)}{\\sqrt{b^2+(d-x)^2}} \\\\ \\\\ c_1 &amp; \\displaystyle = \\frac{c}{n_1} \\\\ \\\\ c_2 &amp; \\displaystyle = \\frac{c}{n_2} \\end{array} <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">So, replacing these in the time derivative, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{dt}{dx} = \\frac{n_1}{c} \\sin(\\theta_1) - \\frac{n_2}{c}\\sin(\\theta_2)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Finally, if point <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> minimizes the function <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x),<\/span><\/span> then the derivative must be zero, and we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\color{blue}{n_1 \\sin(\\theta_1) = n_2 \\sin(\\theta_2)}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">This is Snell&#8217;s Law for the refraction of a light ray passing between two media, showing the relationship between the angle of incidence <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1<\/span><\/span> and the refracted angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_2.<\/span><\/span><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h2>Refraction, Reflection, and Total Reflection of Light<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=1614s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">We have seen that when light<\/span> <\/strong><\/a>passes from one medium to another, it refracts, but generally, what happens is a combination of refraction and reflection; and depending on the refractive indices and the angle of incidence of the light ray, refraction may disappear, leaving only reflection.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Suppose a light ray strikes from a material <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> to another <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> with refractive indices <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span> respectively. If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a \\gt n_b,<\/span><\/span> according to Snell&#8217;s Law, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\sin(\\theta_b) = \\frac{n_a}{n_b}\\sin(\\theta_a)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Since <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a\/n_b \\gt 1,<\/span><\/span> it happens that <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(\\theta_b) \\gt \\sin(\\theta_a),<\/span><\/span> which implies that the refracted ray deviates away from the normal. This means that there must be some <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a\\lt 90^o<\/span><\/span> for which <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(\\theta_b)=1<\/span><\/span> and, therefore, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b=90^o,<\/span><\/span> as shown in the following figure.<\/p>\n<p style=\"text-align: justify; color: #000000;\">The angle of incidence that causes the ray to refract along the interface is known as the critical angle and satisfies the relation:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\sin(\\theta_{critical}) = \\frac{n_b}{n_a}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Which is equivalent to saying:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_{critical} = \\arcsin\\left( \\frac{n_b}{n_a} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">If <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a \\gt \\theta_{critical},<\/span><\/span> then there is total reflection.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Exercises:<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Consider a light ray passing from water to glass as shown in the following figure:<br \/>\n<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-98FbTK-_FZo\/YT-61nYxRnI\/AAAAAAAAFiM\/JbBsuAnS6IA8aB-4hvroeZ1qDF2ebxQUwCLcBGAsYHQ\/s0\/n1n2snell.PNG\" width=\"442\" height=\"321\" alt=\"light ray passing from water to glass\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-98FbTK-_FZo\/YT-61nYxRnI\/AAAAAAAAFiM\/JbBsuAnS6IA8aB-4hvroeZ1qDF2ebxQUwCLcBGAsYHQ\/s0\/n1n2snell.PNG\" width=\"442\" height=\"321\" alt=\"light ray passing from water to glass\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\nThe refractive index of water is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1 = 1.33,<\/span><\/span> and that of glass is <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2=1.52.<\/span><\/span> If a light ray passing from water to glass strikes the interface separating the two media with an inclination angle of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1 = 60^o<\/span><\/span> relative to the normal, at what angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_2<\/span><\/span> does the refracted ray exit? <span class=\"collapseomatic \" id=\"id69e0c9c87fd38\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id69e0c9c87fd38\" class=\"collapseomatic_content \">\nUsing Snell&#8217;s Law, we have:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1 \\sin(\\theta_1) = n_2 \\sin(\\theta_2)<\/span><\/span><\/td>\n<td>; Snell&#8217;s Law<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv <\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\sin(\\theta_2) = \\frac{n_1}{n_2}\\sin(\\theta_1)<\/span><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv <\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_2 = \\arcsin\\left(\\frac{n_1}{n_2}\\sin(\\theta_1)\\right)<\/span><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1=1.33<\/span><\/span><\/td>\n<td>; Refractive index of water<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2=1.52<\/span><\/span><\/td>\n<td>; Refractive index of glass<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(4)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1=60^o<\/span><\/span><\/td>\n<td>; Angle of incidence at the interface of the light ray<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(5)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_2 = \\arcsin\\left(\\frac{1.33}{1.52}\\sin(60^o)\\right) \\approx 49.268^o<\/span><\/span><\/td>\n<td>; From (1,2,3,4), Refraction angle<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=1363s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\"><\/span><\/strong><\/a><\/li>\n<li>Three liquids separated by two interfaces have the following refractive indices: <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1=1.33,<\/span><\/span> <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2=1.41<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_3=1.68,<\/span><\/span> and are arranged as shown in the following figure:<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-AAZlxjqC4s4\/YUAdiGNMSQI\/AAAAAAAAFiU\/mOE-xMfybOoxenNH2O8sufjpTuzH6-WIwCLcBGAsYHQ\/s0\/n1n2n3snell.PNG\" width=\"443\" height=\"430\" alt=\"Snell's Law applied to three media\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-AAZlxjqC4s4\/YUAdiGNMSQI\/AAAAAAAAFiU\/mOE-xMfybOoxenNH2O8sufjpTuzH6-WIwCLcBGAsYHQ\/s0\/n1n2n3snell.PNG\" width=\"443\" height=\"430\" alt=\"Snell's Law applied to three media\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>If the ray passing from the medium with index <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1<\/span><\/span> to <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2<\/span><\/span> strikes the interface with an angle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1=70^o<\/span><\/span>, at what angle will it refract when it passes to the medium with index <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_3<\/span><\/span>? <span class=\"collapseomatic \" id=\"id69e0c9c880684\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id69e0c9c880684\" class=\"collapseomatic_content \">\nSimilarly to the previous exercise, the following reasoning is used:<\/p>\n<table>\n<tbody>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(1)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1 \\sin(\\theta_1) = n_2 \\sin(\\theta_2) <\/span><\/span><\/td>\n<td>; Snell&#8217;s Law for the transition from medium n1 to n2<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(2)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2 \\sin(\\theta_2) = n_3 \\sin(\\theta_3) <\/span><\/span><\/td>\n<td>; Snell&#8217;s Law for the transition from medium n2 to n3<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">(3)<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1 \\sin(\\theta_1) = n_3 \\sin(\\theta_3) <\/span><\/span><\/td>\n<td>; From (1,2)<\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\sin(\\theta_3) = \\frac{n_1}{n_3}\\sin(\\theta_1) <\/span><\/span><\/td>\n<td><\/td>\n<\/tr>\n<tr>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\equiv<\/span><\/span><\/td>\n<td><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_3 = \\arcsin\\left(\\frac{n_1}{n_3}\\sin(\\theta_1)\\right) <\/span><\/span><\/td>\n<td><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>Finally, replacing the data, we have:<\/p>\n<p><center><br \/>\n<span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_3= \\arcsin\\left(\\frac{1.33}{1.68}\\sin(70^o)\\right) \\approx 48.0667^o <\/span><\/span><\/center><br \/>\nNote that this reasoning shows that we can do the calculations by considering only the input and output media of the ray, completely ignoring the one in the middle.<br \/>\n<\/div> <a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=1417s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\"><\/span><\/strong><\/a><\/li>\n<li>From the bottom of a pool, a light ray is directed towards the interface between air and water. Determine the angle of incidence for total reflection to occur.<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-GiDk_G3uybI\/YUDhONr0MaI\/AAAAAAAAFjM\/aSGUOspZeCsm7Cz7DG4r-JrCr03QhYyBgCLcBGAsYHQ\/s0\/%25C3%25A1ngulocr%25C3%25ADtico.PNG\" width=\"588\" height=\"358\" alt=\"light ray incident at critical angle\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-GiDk_G3uybI\/YUDhONr0MaI\/AAAAAAAAFjM\/aSGUOspZeCsm7Cz7DG4r-JrCr03QhYyBgCLcBGAsYHQ\/s0\/%25C3%25A1ngulocr%25C3%25ADtico.PNG\" width=\"588\" height=\"358\" alt=\"light ray incident at critical angle\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<span class=\"collapseomatic \" id=\"id69e0c9c880a41\"  tabindex=\"0\" title=\"SOLUTION\"    >SOLUTION<\/span><div id=\"target-id69e0c9c880a41\" class=\"collapseomatic_content \">\nThe critical angle will be given by:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_{critical}= \\arcsin\\left(\\frac{1.00}{1.33}\\right) \\approx 48.7535^o<\/span><\/span><\/p>\n<\/div><a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=1869s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\"><\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>The Refraction of Light and Snell&#8217;s Law Summary: In this class, we will explore the refraction of light through the analysis of Snell&#8217;s Law. The concept of the refractive index will be explained, Snell&#8217;s Law will be derived using Fermat&#8217;s principle, and the study of how this law allows us to calculate the path of [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27682,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":4,"footnotes":""},"categories":[835,635],"tags":[],"class_list":["post-27685","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-geometrical-optics","category-physics"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>The Refraction of Light and Snell&#039;s Law - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Refraction and Snell&#039;s Law Explained: Learn How Light Changes Direction When Passing Between Media, Calculate Angles of Incidence and Refraction, and Solve Practical Exercises.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"The Refraction of Light and Snell&#039;s Law\" \/>\n<meta property=\"og:description\" content=\"Refraction and Snell&#039;s Law Explained: Learn How Light Changes Direction When Passing Between Media, Calculate Angles of Incidence and Refraction, and Solve Practical Exercises.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\" \/>\n<meta property=\"og:site_name\" content=\"toposuranos.com\/material\" \/>\n<meta property=\"article:publisher\" content=\"https:\/\/www.facebook.com\/groups\/toposuranos\" \/>\n<meta property=\"article:published_time\" content=\"2021-09-18T13:00:28+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-10T23:54:04+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/leydesnell.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"The Refraction of Light and Snell&#039;s Law\" \/>\n<meta name=\"twitter:description\" content=\"Refraction and Snell&#039;s Law Explained: Learn How Light Changes Direction When Passing Between Media, Calculate Angles of Incidence and Refraction, and Solve Practical Exercises.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/leydesnell.jpg\" \/>\n<meta name=\"twitter:creator\" content=\"@topuranos\" \/>\n<meta name=\"twitter:site\" content=\"@topuranos\" \/>\n<meta name=\"twitter:label1\" content=\"Escrito por\" \/>\n\t<meta name=\"twitter:data1\" content=\"giorgio.reveco\" \/>\n\t<meta name=\"twitter:label2\" content=\"Tiempo de lectura\" \/>\n\t<meta name=\"twitter:data2\" content=\"7 minutos\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\/\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"The Refraction of Light and Snell&#8217;s Law\",\"datePublished\":\"2021-09-18T13:00:28+00:00\",\"dateModified\":\"2024-08-10T23:54:04+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\"},\"wordCount\":1583,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/leydesnell.jpg\",\"articleSection\":[\"Geometrical Optics\",\"Physics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\",\"url\":\"http:\/\/toposuranos.com\/material\/en\/the-refraction-of-light-and-snells-law\/\",\"name\":\"The Refraction of Light and Snell's Law - 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