{"id":27724,"date":"2021-10-06T13:00:12","date_gmt":"2021-10-06T13:00:12","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27724"},"modified":"2024-08-11T16:43:36","modified_gmt":"2024-08-11T16:43:36","slug":"refraction-at-spherical-interfaces","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/","title":{"rendered":"Refraction at Spherical Interfaces"},"content":{"rendered":"<p><center><\/p>\n<h1>Refraction at Spherical Interfaces<\/h1>\n<p><em><strong>Summary:<\/strong><br \/>\nIn this lesson, we will analyze Refraction at Spherical Interfaces, highlighting how light behaves when passing through spherical surfaces and how images are formed. The key equations for calculating the position and size of images are presented. Practical cases, such as lenses and the estimation of apparent depths, are also explored.<\/em><\/p>\n<p><strong>Learning Objectives:<\/strong><br \/>\nBy the end of this lesson, the student will be able to:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Understand<\/strong> the refraction of light when passing through spherical interfaces.<\/li>\n<li><strong>Derive<\/strong> and use the object-image relationship for spherical interfaces.<\/li>\n<li><strong>Apply<\/strong> Snell&#8217;s Law in the context of spherical interfaces.<\/li>\n<li><strong>Determine<\/strong> the position of the image formed by a spherical interface.<\/li>\n<li><strong>Calculate<\/strong> the magnification of the image through refraction at spherical surfaces.<\/li>\n<li><strong>Understand<\/strong> the sign convention for the position and size of objects and images.<\/li>\n<li><strong>Relate<\/strong> spherical interfaces to flat interfaces as a limiting case.<\/li>\n<li><strong>Analyze<\/strong> the formation of extended images through spherical interfaces.<\/li>\n<\/ol>\n<p><strong>TABLE OF CONTENTS<\/strong><br \/>\n<a href=\"#1\"><strong>Introduction<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>The Object-Image Relationship for Refraction at Spherical Interfaces<\/strong><\/a><br \/>\n<a href=\"#3\">Extracting Relationships Between Angles<\/a><br \/>\n<a href=\"#4\">Introducing Snell&#8217;s Law<\/a><br \/>\n<a href=\"#5\"><strong>Formation of Extended Images by Refraction on the Other Side of Spherical Interfaces<\/strong><\/a><br \/>\n<a href=\"#6\">Synthesis<\/a><br \/>\n<a href=\"#7\"><strong>Flat Interfaces as a Limiting Case of Spherical Ones<\/strong><\/a><br \/>\n<a href=\"#8\"><strong>Exercises<\/strong><\/a><\/p>\n<p><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/13kntUA9n-I\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen=\"allowfullscreen\"><\/iframe><br \/>\n<\/center><\/p>\n<p><a name=\"1\"><\/a><\/p>\n<h2>Introduction<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=146s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">We have already studied how refraction works;<\/span><\/strong><\/a> that is, what happens when light passes from one medium to another. But we have done all this assuming that the interface separating the media is a flat surface. However, both in nature and in practical applications, it is not difficult to find refraction processes at spherical interfaces. Examples of these include the human eye (and almost any animal eye, actually) and most optical devices used in everyday life and industrial applications.<\/p>\n<p style=\"text-align: justify; color: #000000;\">In the following figure, we see how a lens is constructed through two spherical surfaces.<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-xuCmdVLtNcc\/YVuCvXoOfZI\/AAAAAAAAFmc\/mCuYiHaNfxM4I6RcxEJcLln6uLC5MMIKQCLcBGAsYHQ\/s0\/lente%2Bde%2Bvidrio.PNG\" width=\"661\" height=\"378\" alt=\"Glass lens formed with two spherical surfaces\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-xuCmdVLtNcc\/YVuCvXoOfZI\/AAAAAAAAFmc\/mCuYiHaNfxM4I6RcxEJcLln6uLC5MMIKQCLcBGAsYHQ\/s0\/lente%2Bde%2Bvidrio.PNG\" width=\"661\" height=\"378\" alt=\"Glass lens formed with two spherical surfaces\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">For a detailed study of this type of device, it is necessary to review how light behaves when passing from one medium to another through a spherical interface.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>The Object-Image Relationship for Refraction at Spherical Interfaces<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=235s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">We will begin our study by investigating<\/span><\/strong><\/a> how light behaves when passing from one medium to another through a spherical interface. To do this, we will consider a sphere of radius <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span> made of a material with a refractive index <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span> immersed in a medium with a refractive index <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a.<\/span><\/span><\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Spherical interface separating two media\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Spherical interface separating two media\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Extracting Relationships Between Angles<\/h3>\n<p style=\"text-align: justify; color: #000000;\">If we analyze the angles involved in this figure, we will notice that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{(1)}&amp; \\theta_a &amp; =\\alpha + \\phi \\\\ \\\\\n\n{(2)}&amp; \\phi &amp; =\\beta + \\theta_b\n\n\\end{array}<\/span><\/span><\/p>\n<h4>Proof<\/h4>\n<p style=\"text-align: justify; color: #000000;\">The first equation is obtained from the fact that the sum of the interior angles of a triangle is equal to two right angles:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-0ILYFKDJRo0\/YVvJaKwoZxI\/AAAAAAAAFms\/arUYkIwSz18wvNntSE_WHuknjoYolOXSQCLcBGAsYHQ\/s0\/triangulorectangulo-optica.PNG\" width=\"346\" height=\"148\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-0ILYFKDJRo0\/YVvJaKwoZxI\/AAAAAAAAFms\/arUYkIwSz18wvNntSE_WHuknjoYolOXSQCLcBGAsYHQ\/s0\/triangulorectangulo-optica.PNG\" width=\"346\" height=\"148\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; \\alpha + \\phi + (\\pi - \\theta_a) = \\pi\\\\ \\\\\n\n\\equiv &amp; \\alpha + \\phi - \\theta_a = 0 \\\\ \\\\\n\n\\equiv &amp; \\color{blue}{\\theta_a = \\alpha + \\phi}\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">The second is obtained in a similar manner:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-R5siuLq4gJs\/YVvJ22k7m6I\/AAAAAAAAFm0\/D6gmIAHjXFY4cNmUwrnq4yvxQ_2PcBSjgCLcBGAsYHQ\/s0\/triangulorectangulo-optica2.PNG\" width=\"329\" height=\"216\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-R5siuLq4gJs\/YVvJ22k7m6I\/AAAAAAAAFm0\/D6gmIAHjXFY4cNmUwrnq4yvxQ_2PcBSjgCLcBGAsYHQ\/s0\/triangulorectangulo-optica2.PNG\" width=\"329\" height=\"216\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp; \\beta + \\theta_b + (\\pi - \\phi) = \\pi\\\\ \\\\\n\n\\equiv &amp;  \\beta + \\theta_b - \\phi = 0\\\\ \\\\\n\n\\equiv &amp; \\color{blue}{\\phi = \\beta + \\theta_b }\n\n\\end{array}<\/span><\/span><\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h3>Introducing Snell&#8217;s Law<\/h3>\n<p style=\"text-align: justify; color: #000000;\">From the figure, we also have the following expressions:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Spherical interface separating two media\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-x0g44iZ6LDo\/YVuDO2nkD3I\/AAAAAAAAFmk\/8gQHFBnQxvEG-5B6XUAgTTZHtxntx6YogCLcBGAsYHQ\/s0\/inteface%2Besf%25C3%25A9rica.PNG\" width=\"987\" height=\"371\" alt=\"Spherical interface separating two media\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rll}\n\n{(3)}&amp;\\tan(\\alpha) &amp;=\\displaystyle \\frac{h}{s+\\delta}\\\\ \\\\\n\n{(4)}&amp;\\tan(\\beta) &amp;=\\displaystyle \\frac{h}{s^\\prime - \\delta}\\\\ \\\\\n\n{(5)}&amp;\\tan(\\phi) &amp;=\\displaystyle \\frac{h}{R - \\delta}\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And from 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\">\\begin{array}{rl}\n\n{(6)} &amp;  n_a\\sin(\\theta_a) = n_b \\sin(\\theta_b)\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now, if we take the approximation where <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b<\/span><\/span> are small, then <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> will also be small, and it will follow that:<\/p>\n<p style=\"text-align: justify; color: #000000;\">From the figure, we also have the following expressions:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\sin(\\theta_a) &amp;\\approx \\theta_a \\\\ \\\\\n\n\\sin(\\theta_b) &amp;\\approx \\theta_b \\\\ \\\\\n\n\\delta &amp;\\approx 0 \\\\ \\\\\n\n\\tan(\\alpha) &amp;\\approx \\alpha \\\\ \\\\\n\n\\tan(\\beta) &amp;\\approx \\beta \\\\ \\\\\n\n\\tan(\\phi) &amp;\\approx \\phi\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Then, from this and 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\">\\begin{array}{rl}\n\n{(7)} &amp; n_a \\theta_a \\approx n_b \\theta_b \\\\ \\\\\n\n\\equiv &amp;  \\theta_b \\approx \\displaystyle \\frac{n_a}{n_b} \\theta_a\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now, from (7), (1), and (2) 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}\n\n{(8)} &amp; \\phi - \\beta \\approx \\displaystyle \\frac{n_a}{n_b}(\\alpha + \\phi) \\\\ \\\\\n\n\\equiv &amp; \\phi \\approx \\beta + \\displaystyle  \\frac{n_a}{n_b}(\\alpha + \\phi) \\\\ \\\\\n\n{}\\equiv &amp;  n_b\\phi \\approx n_b\\beta + n_a \\alpha + n_a\\phi \\\\ \\\\\n\n\\equiv &amp; \\color{blue}{n_a \\alpha + n_b\\beta \\approx (n_b - n_a) \\phi }\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Finally, from (8), the approximations, and the equations (3), (4), and (5), we arrive at:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n{(9)} &amp; \\displaystyle n_a \\left( \\frac{\\color{red}{h}}{S + \\underbrace{\\delta}_{\\to 0}} \\right) + n_b \\left(\\frac{\\color{red}{h}}{S^\\prime - \\underbrace{\\delta}_{\\to 0} } \\right) \\approx (n_b - n_a) \\left(\\frac{\\color{red}{h}}{R-\\underbrace{\\delta}_{\\to 0}}\\right) \\\\ \\\\\n\n\\equiv &amp; \\displaystyle \\color{blue}{\\frac{n_a}{S } + \\frac{ n_b}{S^\\prime } \\approx \\frac{n_b - n_a}{R} }\n\n\\end{array}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">This last equation is what we call the <strong>Object-Image Relationship for Refraction at Spherical Interfaces.<\/strong><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Formation of Extended Images by Refraction on the Other Side of Spherical Interfaces<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1211s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Now let&#8217;s see what happens when we change<\/span><\/strong><\/a> the point light source to an extended object. This is illustrated in the following figure:<\/p>\n<p><center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-Nnk1qsJET0k\/YVvLLFbEWPI\/AAAAAAAAFm8\/zWNnZMpqHSMT_38PJgKqIhqzLO53CCMSwCLcBGAsYHQ\/s0\/objeto-extendido-frente-interfaz-esferica.PNG\" width=\"1005\" height=\"409\" alt=\"extended object in front of a spherical interface\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-Nnk1qsJET0k\/YVvLLFbEWPI\/AAAAAAAAFm8\/zWNnZMpqHSMT_38PJgKqIhqzLO53CCMSwCLcBGAsYHQ\/s0\/objeto-extendido-frente-interfaz-esferica.PNG\" width=\"1005\" height=\"409\" alt=\"extended object in front of a spherical interface\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">The previous analysis already indicates the relationship between <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span><\/span> now we only need to find the relationship between the sizes of the object and the image.<\/p>\n<p style=\"text-align: justify; color: #000000;\">From the figure, 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}\n\n\\tan(\\theta_a) &amp; =\\displaystyle \\frac{y}{S} \\\\ \\\\\n\n\\tan(\\theta_b) &amp; =\\displaystyle - \\frac{y^\\prime}{S^\\prime}\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">We will combine this with Snell&#8217;s Law<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a\\sin(\\theta_a) = n_b\\sin(\\theta_b). <\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And for this, we will rely on the fact that for small angles the approximation holds<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n\\sin(\\theta_a) &amp; \\approx \\tan(\\theta_a) \\\\ \\\\\n\n\\sin(\\theta_b) &amp; \\approx \\tan(\\theta_b)\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">So that we can write<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\begin{array}{rl}\n\n&amp;\\displaystyle n_a \\frac{y}{S} \\approx- n_b \\dfrac{y^\\prime}{S^\\prime} \\\\ \\\\\n\n\\equiv &amp; \\displaystyle \\dfrac{y^\\prime}{y} \\approx - \\dfrac{n_a S^\\prime}{n_b S} \\\\ \\\\\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Now, recalling what we have seen for spherical mirrors, we have something analogous. At this point, we can (re)define the magnification factor <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span> as:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\nm=\\displaystyle \\frac{y^\\prime}{y}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">so that:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\displaystyle \\color{blue}{m\\approx -\\frac{n_a S^\\prime}{n_b S}}\n\n<\/span><\/span><\/p>\n<p><a name=\"6\"><\/a><\/p>\n<h3>Synthesis<\/h3>\n<p style=\"text-align: justify; color: #000000;\">In summary, so far we have extracted two results that allow us to infer the formation of images when light emitted from an object passes through a spherical interface. These are the following equations:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\n\\begin{array}{rl}\n\n\\displaystyle \\dfrac{n_a}{S} + \\dfrac{n_b}{S^\\prime} &amp; \\approx \\dfrac{n_b - n_a}{R} \\\\ \\\\\n\nm &amp; \\displaystyle \\approx - \\dfrac{n_a S^\\prime}{n_b S}\n\n\\end{array}\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">With these two equations, you can calculate both the position of the image and the orientation and size of the image, and they will work regardless of whether the interface surface is concave or convex. At this point, however, it is necessary to clarify the sign convention.<\/p>\n<h4>Sign Convention<\/h4>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1682s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">With these two equations, you can calculate<\/span><\/strong><\/a> both the position of the image and the orientation and size of the image, and they will work regardless of whether the interface surface is concave or convex. At this point, however, it is necessary to clarify the sign convention.<\/p>\n<p style=\"text-align: justify; color: #000000;\">The interface divides the space into two regions, one where the object can be found and the other where the image is located. Based on this, we have:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><strong>Object position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>:<\/strong> Positive if it is on the object&#8217;s side, negative if it is on the image&#8217;s side.<\/li>\n<li><strong>Image position <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/span> and radius of curvature <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span>:<\/strong> Positive if it is on the image&#8217;s side, negative if it is on the object&#8217;s side.<\/li>\n<li><strong>Object and image size, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/span> and <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime<\/span><\/span>:<\/strong> Positive if it is above the optical axis, negative if it is below the optical axis.<\/li>\n<\/ul>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Flat Interfaces as a Limiting Case of Spherical Ones<\/h2>\n<p style=\"text-align: justify; color: #000000;\"><a href=\"https:\/\/www.youtube.com\/watch?v=13kntUA9n-I&amp;t=1897s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">Everything we have developed for spherical interfaces<\/span><\/strong><\/a> also helps to better understand flat interfaces. In fact, we can understand a flat interface as a piece of a spherical interface with a very large radius of curvature; indeed, if we take limits on the object-image relationship for spherical interfaces as the radius tends to infinity, we have:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\frac{n_a}{S } + \\frac{ n_b}{S^\\prime} = \\lim_{R\\to \\infty} \\frac{n_a}{S } + \\frac{ n_b}{S^\\prime } \\approx \\lim_{R\\to \\infty} \\frac{n_b - n_a}{R} = 0\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">And if from this we calculate the magnification factor, we get:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m=1\n\n<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">That is, the image retains its size and orientation, what does vary is its observed position.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>Exercises<\/h2>\n<p><center><iframe class=\"lazyload\" width=\"560\" height=\"315\" data-src=\"https:\/\/www.youtube.com\/embed\/bwGJYhN3Gv8\" title=\"YouTube video player\" frameborder=\"0\" allow=\"accelerometer; autoplay; clipboard-write; encrypted-media; gyroscope; picture-in-picture\" allowfullscreen><\/iframe><\/center><\/p>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>In front of a cylindrical glass rod, a particle is placed as shown below<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-ayoBx1gP-cM\/YVvMd0Dv6rI\/AAAAAAAAFnE\/TBz3MtCOMvw5SxoNk0XlR70M11UCopfbACLcBGAsYHQ\/s0\/varilla-de-vidrio1.PNG\" width=\"945\" height=\"327\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-ayoBx1gP-cM\/YVvMd0Dv6rI\/AAAAAAAAFnE\/TBz3MtCOMvw5SxoNk0XlR70M11UCopfbACLcBGAsYHQ\/s0\/varilla-de-vidrio1.PNG\" width=\"945\" height=\"327\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>If the particle is 30[cm] from the rod and the tip of the rod is approximately spherical with a radius <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=1.5[cm],<\/span><\/span> calculate the position of the image generated inside the rod.<\/li>\n<li>Let&#8217;s consider the same rod from the previous exercise, but now it is underwater. If a needle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1[cm]<\/span><\/span> tall is placed in front of it at the same distance of <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">30[cm],<\/span><\/span> calculate the location and height of the image.<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-2Zr1OfLvpAU\/YVvM1MQr1UI\/AAAAAAAAFnM\/9HCljhcB8Ss97L3tLzyXM2E3JzqioJNHgCLcBGAsYHQ\/s0\/varilla-de-vidrio2.PNG\" width=\"948\" height=\"446\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-2Zr1OfLvpAU\/YVvM1MQr1UI\/AAAAAAAAFnM\/9HCljhcB8Ss97L3tLzyXM2E3JzqioJNHgCLcBGAsYHQ\/s0\/varilla-de-vidrio2.PNG\" width=\"948\" height=\"446\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/li>\n<li>A person looks into the bottom of a pool to estimate its depth. As a guide, they use an arrow painted on the bottom. What is the relationship between the real and apparent depth?<center><img decoding=\"async\" src=\"data:image\/gif;base64,R0lGODlhAQABAIAAAAAAAP\/\/\/yH5BAEAAAAALAAAAAABAAEAAAIBRAA7\" data-src=\"https:\/\/1.bp.blogspot.com\/-G8Lr7rWgyCw\/YVvNQqC6rLI\/AAAAAAAAFnU\/mF2xOr5TPPgG62N8FZqQYijhYQg8_co0ACLcBGAsYHQ\/s0\/profundidad-real-y-aparente.PNG\" width=\"957\" height=\"658\" class=\"alignnone size-full lazyload\" \/><noscript><img decoding=\"async\" src=\"https:\/\/1.bp.blogspot.com\/-G8Lr7rWgyCw\/YVvNQqC6rLI\/AAAAAAAAFnU\/mF2xOr5TPPgG62N8FZqQYijhYQg8_co0ACLcBGAsYHQ\/s0\/profundidad-real-y-aparente.PNG\" width=\"957\" height=\"658\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>Refraction at Spherical Interfaces Summary: In this lesson, we will analyze Refraction at Spherical Interfaces, highlighting how light behaves when passing through spherical surfaces and how images are formed. The key equations for calculating the position and size of images are presented. Practical cases, such as lenses and the estimation of apparent depths, are also [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":27723,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"iawp_total_views":37,"footnotes":""},"categories":[835,635],"tags":[],"class_list":["post-27724","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 v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Refraction at Spherical Interfaces - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.\" \/>\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\/refraction-at-spherical-interfaces\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Refraction at Spherical Interfaces\" \/>\n<meta property=\"og:description\" content=\"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/\" \/>\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=\"2021-10-06T13:00:12+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-11T16:43:36+00:00\" \/>\n<meta property=\"og:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg\" \/>\n<meta name=\"author\" content=\"giorgio.reveco\" \/>\n<meta name=\"twitter:card\" content=\"summary_large_image\" \/>\n<meta name=\"twitter:title\" content=\"Refraction at Spherical Interfaces\" \/>\n<meta name=\"twitter:description\" content=\"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.\" \/>\n<meta name=\"twitter:image\" content=\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.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=\"1 minuto\" \/>\n<script type=\"application\/ld+json\" class=\"yoast-schema-graph\">{\"@context\":\"https:\\\/\\\/schema.org\",\"@graph\":[{\"@type\":\"Article\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Refraction at Spherical Interfaces\",\"datePublished\":\"2021-10-06T13:00:12+00:00\",\"dateModified\":\"2024-08-11T16:43:36+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/\"},\"wordCount\":1649,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"articleSection\":[\"Geometrical Optics\",\"Physics\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/\",\"name\":\"Refraction at Spherical Interfaces - toposuranos.com\\\/material\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\"},\"primaryImageOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#primaryimage\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"datePublished\":\"2021-10-06T13:00:12+00:00\",\"dateModified\":\"2024-08-11T16:43:36+00:00\",\"description\":\"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.\",\"breadcrumb\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#breadcrumb\"},\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"ReadAction\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/\"]}]},{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#primaryimage\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"width\":1024,\"height\":369},{\"@type\":\"BreadcrumbList\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/en\\\/refraction-at-spherical-interfaces\\\/#breadcrumb\",\"itemListElement\":[{\"@type\":\"ListItem\",\"position\":1,\"name\":\"Portada\",\"item\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/es\\\/cursos-de-matematica-y-fisica\\\/\"},{\"@type\":\"ListItem\",\"position\":2,\"name\":\"Refraction at Spherical Interfaces\"}]},{\"@type\":\"WebSite\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#website\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"name\":\"toposuranos.com\\\/material\",\"description\":\"\",\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"potentialAction\":[{\"@type\":\"SearchAction\",\"target\":{\"@type\":\"EntryPoint\",\"urlTemplate\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/?s={search_term_string}\"},\"query-input\":{\"@type\":\"PropertyValueSpecification\",\"valueRequired\":true,\"valueName\":\"search_term_string\"}}],\"inLanguage\":\"es\"},{\"@type\":\"Organization\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\",\"name\":\"toposuranos.com\\\/material\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/\",\"logo\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/logo.png\",\"width\":2400,\"height\":2059,\"caption\":\"toposuranos.com\\\/material\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/logo\\\/image\\\/\"},\"sameAs\":[\"https:\\\/\\\/www.facebook.com\\\/groups\\\/toposuranos\",\"https:\\\/\\\/x.com\\\/topuranos\",\"https:\\\/\\\/www.youtube.com\\\/channel\\\/UC16yDm12cPcrwsE0fAM7X1g\",\"https:\\\/\\\/www.linkedin.com\\\/company\\\/69429190\"]},{\"@type\":\"Person\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\",\"name\":\"giorgio.reveco\",\"image\":{\"@type\":\"ImageObject\",\"inLanguage\":\"es\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"contentUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2023\\\/10\\\/1694478625378-96x96.jpeg\",\"caption\":\"giorgio.reveco\"},\"description\":\"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.\",\"sameAs\":[\"http:\\\/\\\/toposuranos.com\\\/material\"],\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/author\\\/giorgio-reveco\\\/\"}]}<\/script>\n<!-- \/ Yoast SEO plugin. -->","yoast_head_json":{"title":"Refraction at Spherical Interfaces - toposuranos.com\/material","description":"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.","robots":{"index":"index","follow":"follow","max-snippet":"max-snippet:-1","max-image-preview":"max-image-preview:large","max-video-preview":"max-video-preview:-1"},"canonical":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/","og_locale":"es_ES","og_type":"article","og_title":"Refraction at Spherical Interfaces","og_description":"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.","og_url":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/","og_site_name":"toposuranos.com\/material","article_publisher":"https:\/\/www.facebook.com\/groups\/toposuranos","article_published_time":"2021-10-06T13:00:12+00:00","article_modified_time":"2024-08-11T16:43:36+00:00","og_image":[{"url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","type":"","width":"","height":""}],"author":"giorgio.reveco","twitter_card":"summary_large_image","twitter_title":"Refraction at Spherical Interfaces","twitter_description":"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.","twitter_image":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","twitter_creator":"@topuranos","twitter_site":"@topuranos","twitter_misc":{"Escrito por":"giorgio.reveco","Tiempo de lectura":"1 minuto"},"schema":{"@context":"https:\/\/schema.org","@graph":[{"@type":"Article","@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#article","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/"},"author":{"name":"giorgio.reveco","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1"},"headline":"Refraction at Spherical Interfaces","datePublished":"2021-10-06T13:00:12+00:00","dateModified":"2024-08-11T16:43:36+00:00","mainEntityOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/"},"wordCount":1649,"commentCount":0,"publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","articleSection":["Geometrical Optics","Physics"],"inLanguage":"es","potentialAction":[{"@type":"CommentAction","name":"Comment","target":["http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#respond"]}]},{"@type":"WebPage","@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/","url":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/","name":"Refraction at Spherical Interfaces - toposuranos.com\/material","isPartOf":{"@id":"http:\/\/toposuranos.com\/material\/#website"},"primaryImageOfPage":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#primaryimage"},"image":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#primaryimage"},"thumbnailUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","datePublished":"2021-10-06T13:00:12+00:00","dateModified":"2024-08-11T16:43:36+00:00","description":"Learn about refraction at spherical interfaces, how images are formed, and the key equations to calculate their position and size.","breadcrumb":{"@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#breadcrumb"},"inLanguage":"es","potentialAction":[{"@type":"ReadAction","target":["http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/"]}]},{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#primaryimage","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2021\/10\/interfaces-esfericas.jpg","width":1024,"height":369},{"@type":"BreadcrumbList","@id":"http:\/\/toposuranos.com\/material\/en\/refraction-at-spherical-interfaces\/#breadcrumb","itemListElement":[{"@type":"ListItem","position":1,"name":"Portada","item":"http:\/\/toposuranos.com\/material\/es\/cursos-de-matematica-y-fisica\/"},{"@type":"ListItem","position":2,"name":"Refraction at Spherical Interfaces"}]},{"@type":"WebSite","@id":"http:\/\/toposuranos.com\/material\/#website","url":"http:\/\/toposuranos.com\/material\/","name":"toposuranos.com\/material","description":"","publisher":{"@id":"http:\/\/toposuranos.com\/material\/#organization"},"potentialAction":[{"@type":"SearchAction","target":{"@type":"EntryPoint","urlTemplate":"http:\/\/toposuranos.com\/material\/?s={search_term_string}"},"query-input":{"@type":"PropertyValueSpecification","valueRequired":true,"valueName":"search_term_string"}}],"inLanguage":"es"},{"@type":"Organization","@id":"http:\/\/toposuranos.com\/material\/#organization","name":"toposuranos.com\/material","url":"http:\/\/toposuranos.com\/material\/","logo":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/logo.png","width":2400,"height":2059,"caption":"toposuranos.com\/material"},"image":{"@id":"http:\/\/toposuranos.com\/material\/#\/schema\/logo\/image\/"},"sameAs":["https:\/\/www.facebook.com\/groups\/toposuranos","https:\/\/x.com\/topuranos","https:\/\/www.youtube.com\/channel\/UC16yDm12cPcrwsE0fAM7X1g","https:\/\/www.linkedin.com\/company\/69429190"]},{"@type":"Person","@id":"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1","name":"giorgio.reveco","image":{"@type":"ImageObject","inLanguage":"es","@id":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","url":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","contentUrl":"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2023\/10\/1694478625378-96x96.jpeg","caption":"giorgio.reveco"},"description":"Soy Licenciado en F\u00edsica, Magister en Ingenier\u00eda Industrial y Docente Universitario. Me dedico a desmitificar la f\u00edsica y las matem\u00e1ticas. Mi objetivo es hacer que estos campos sean f\u00e1cilmente comprensibles para todos, proporcionando las herramientas para explorar no solo el mundo que nos rodea, sino tambi\u00e9n las profundidades de nuestra propia existencia y el orden natural que nos conecta con el cosmos.","sameAs":["http:\/\/toposuranos.com\/material"],"url":"http:\/\/toposuranos.com\/material\/author\/giorgio-reveco\/"}]}},"_links":{"self":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27724","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/comments?post=27724"}],"version-history":[{"count":0,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/posts\/27724\/revisions"}],"wp:featuredmedia":[{"embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media\/27723"}],"wp:attachment":[{"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/media?parent=27724"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/categories?post=27724"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/toposuranos.com\/material\/wp-json\/wp\/v2\/tags?post=27724"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}