{"id":27727,"date":"2021-10-06T13:00:05","date_gmt":"2021-10-06T13:00:05","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27727"},"modified":"2024-08-11T16:52:25","modified_gmt":"2024-08-11T16:52:25","slug":"refracao-em-interfaces-esfericas","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/refracao-em-interfaces-esfericas\/","title":{"rendered":"Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas"},"content":{"rendered":"<p><center><\/p>\n<h1>Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nNesta aula, analisaremos a Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas, destacando como a luz se comporta ao passar por superf\u00edcies esf\u00e9ricas e como as imagens s\u00e3o formadas. S\u00e3o apresentadas as equa\u00e7\u00f5es chave para calcular a posi\u00e7\u00e3o e o tamanho das imagens. Tamb\u00e9m s\u00e3o explorados casos pr\u00e1ticos, como lentes e estimativas de profundidades aparentes.<\/em><\/p>\n<p><strong>Objetivos de Aprendizagem:<\/strong><br \/>\nAo final desta aula, o estudante ser\u00e1 capaz de:<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Compreender<\/strong> a refra\u00e7\u00e3o da luz ao passar por interfaces esf\u00e9ricas.<\/li>\n<li><strong>Derivar<\/strong> e utilizar a rela\u00e7\u00e3o objeto-imagem para interfaces esf\u00e9ricas.<\/li>\n<li><strong>Aplicar<\/strong> a Lei de Snell no contexto de interfaces esf\u00e9ricas.<\/li>\n<li><strong>Determinar<\/strong> a posi\u00e7\u00e3o da imagem formada por uma interface esf\u00e9rica.<\/li>\n<li><strong>Calcular<\/strong> a amplia\u00e7\u00e3o da imagem atrav\u00e9s da refra\u00e7\u00e3o em superf\u00edcies esf\u00e9ricas.<\/li>\n<li><strong>Compreender<\/strong> a conven\u00e7\u00e3o de sinais para a posi\u00e7\u00e3o e tamanho de objetos e imagens.<\/li>\n<li><strong>Relacionar<\/strong> interfaces esf\u00e9ricas com interfaces planas como um caso limite.<\/li>\n<li><strong>Analisar<\/strong> a forma\u00e7\u00e3o de imagens estendidas atrav\u00e9s de interfaces esf\u00e9ricas.<\/li>\n<\/ol>\n<p><strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\"><strong>Introdu\u00e7\u00e3o<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>A Rela\u00e7\u00e3o Objeto-Imagem para Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas<\/strong><\/a><br \/>\n<a href=\"#3\">Extraindo Rela\u00e7\u00f5es entre os \u00c2ngulos<\/a><br \/>\n<a href=\"#4\">Introduzindo a Lei de Snell<\/a><br \/>\n<a href=\"#5\"><strong>Forma\u00e7\u00e3o de Imagens Estendidas pela Refra\u00e7\u00e3o do Outro Lado das Interfaces Esf\u00e9ricas<\/strong><\/a><br \/>\n<a href=\"#6\">S\u00edntese<\/a><br \/>\n<a href=\"#7\"><strong>Interfaces Planas como Caso Limite das Esf\u00e9ricas<\/strong><\/a><br \/>\n<a href=\"#8\"><strong>Exerc\u00edcios<\/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>Introdu\u00e7\u00e3o<\/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;\">J\u00e1 estudamos como funciona a refra\u00e7\u00e3o;<\/span><\/strong><\/a> ou seja, o que acontece quando a luz passa de um meio para outro. Mas fizemos tudo isso supondo que a interface que separa os meios \u00e9 uma superf\u00edcie plana. No entanto, tanto na natureza quanto nas aplica\u00e7\u00f5es pr\u00e1ticas, n\u00e3o \u00e9 dif\u00edcil encontrar processos de refra\u00e7\u00e3o em interfaces esf\u00e9ricas. Exemplos disso incluem o olho humano (e quase qualquer olho de animal, na verdade) e a maioria dos dispositivos \u00f3pticos utilizados na vida cotidiana e em aplica\u00e7\u00f5es industriais.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Na figura a seguir, vemos como uma lente \u00e9 constru\u00edda atrav\u00e9s de duas superf\u00edcies esf\u00e9ricas.<\/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=\"Lente de vidro formada por duas superf\u00edcies esf\u00e9ricas\" 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=\"Lente de vidro formada por duas superf\u00edcies esf\u00e9ricas\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">Para um estudo detalhado deste tipo de dispositivo, \u00e9 necess\u00e1rio revisar como a luz se comporta ao passar de um meio para outro atrav\u00e9s de uma interface esf\u00e9rica.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>A Rela\u00e7\u00e3o Objeto-Imagem para a Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas<\/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;\">Iniciaremos nosso estudo investigando<\/span><\/strong><\/a> como a luz se comporta ao passar de um meio para outro atrav\u00e9s de uma interface esf\u00e9rica. Para isso, consideraremos uma esfera de raio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span> feita de um material com \u00edndice de refra\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span> imersa em um meio com \u00edndice de refra\u00e7\u00e3o <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=\"Interface esf\u00e9rica separando dois meios\" 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=\"Interface esf\u00e9rica separando dois meios\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Extraindo Rela\u00e7\u00f5es entre os \u00c2ngulos<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Se analisarmos os \u00e2ngulos envolvidos nesta figura, perceberemos que:<\/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>Prova<\/h4>\n<p style=\"text-align: justify; color: #000000;\">A primeira equa\u00e7\u00e3o \u00e9 obtida a partir do fato de que a soma dos \u00e2ngulos internos de um tri\u00e2ngulo \u00e9 igual a dois \u00e2ngulos retos:<\/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;\">A segunda \u00e9 obtida de forma an\u00e1loga:<\/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>Introduzindo a Lei de Snell<\/h3>\n<p style=\"text-align: justify; color: #000000;\">A partir da figura, tamb\u00e9m temos as seguintes express\u00f5es:<\/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=\"Interface esf\u00e9rica separando dois meios\" 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=\"Interface esf\u00e9rica separando dois meios\" 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;\">E a partir da Lei de Snell temos<\/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;\">Agora, se tomarmos a aproxima\u00e7\u00e3o onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b<\/span><\/span> s\u00e3o pequenos, ent\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> tamb\u00e9m ser\u00e3o pequenos, e ser\u00e1 que:<\/p>\n<p style=\"text-align: justify; color: #000000;\">A partir da figura, tamb\u00e9m temos as seguintes express\u00f5es:<\/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;\">Ent\u00e3o, a partir disso e da Lei de Snell, temos:<\/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;\">Agora, a partir de (7), (1) e (2) temos<\/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;\">Finalmente, de (8), as aproxima\u00e7\u00f5es e as equa\u00e7\u00f5es (3), (4) e (5), chegamos a:<\/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;\">Esta \u00faltima \u00e9 o que chamamos de <strong>Rela\u00e7\u00e3o Objeto-Imagem para a Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas.<\/strong><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Forma\u00e7\u00e3o de Imagens Estendidas pela Refra\u00e7\u00e3o do Outro Lado das Interfaces Esf\u00e9ricas<\/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;\">Agora vamos ver o que acontece quando mudamos<\/span><\/strong><\/a> a fonte de luz pontual para um objeto estendido. Isto \u00e9 ilustrado na figura seguinte:<\/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=\"objeto estendido em frente a uma interface esf\u00e9rica\" 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=\"objeto estendido em frente a uma interface esf\u00e9rica\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">A an\u00e1lise anterior j\u00e1 indica a rela\u00e7\u00e3o entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span><\/span> agora s\u00f3 precisamos encontrar a rela\u00e7\u00e3o entre os tamanhos do objeto e da imagem.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Da figura, temos que:<\/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;\">Vamos combinar isso com a Lei de Snell<\/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;\">E para isso, nos basearemos no fato de que para \u00e2ngulos pequenos a aproxima\u00e7\u00e3o se aplica<\/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;\">De modo que podemos escrever<\/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;\">Agora, lembrando o que vimos para espelhos esf\u00e9ricos, temos algo an\u00e1logo. Neste ponto, podemos (re)definir o fator de amplia\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span> atrav\u00e9s de:<\/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;\">de modo que:<\/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>S\u00edntese<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Resumindo, at\u00e9 agora extra\u00edmos dois resultados que nos permitem inferir a forma\u00e7\u00e3o de imagens quando a luz emitida de um objeto passa por uma interface esf\u00e9rica. Estas s\u00e3o as seguintes equa\u00e7\u00f5es:<\/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;\">Com essas duas equa\u00e7\u00f5es, voc\u00ea pode calcular tanto a posi\u00e7\u00e3o da imagem quanto a orienta\u00e7\u00e3o e o tamanho da imagem, e elas funcionar\u00e3o independentemente de a superf\u00edcie de interface ser c\u00f4ncava ou convexa. Neste ponto, entretanto, \u00e9 necess\u00e1rio esclarecer a conven\u00e7\u00e3o de sinais.<\/p>\n<h4>Conven\u00e7\u00e3o de Sinais<\/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;\">Com essas duas equa\u00e7\u00f5es, voc\u00ea pode calcular<\/span><\/strong><\/a> tanto a posi\u00e7\u00e3o da imagem quanto a orienta\u00e7\u00e3o e o tamanho da imagem, e elas funcionar\u00e3o independentemente de a superf\u00edcie de interface ser c\u00f4ncava ou convexa. Neste ponto, entretanto, \u00e9 necess\u00e1rio esclarecer a conven\u00e7\u00e3o de sinais.<\/p>\n<p style=\"text-align: justify; color: #000000;\">A interface divide o espa\u00e7o em duas regi\u00f5es, uma onde o objeto pode ser encontrado e a outra onde a imagem est\u00e1 localizada. Com base nisso, temos:<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><strong>Posi\u00e7\u00e3o do objeto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span>:<\/strong> Positivo se estiver do lado do objeto, negativo se estiver do lado da imagem.<\/li>\n<li><strong>Posi\u00e7\u00e3o da imagem <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/span> e o raio de curvatura <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span>:<\/strong> Positivo se estiver do lado da imagem, negativo se estiver do lado do objeto.<\/li>\n<li><strong>Tamanho do objeto e da imagem, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime<\/span><\/span>:<\/strong> Positivo se estiver acima do eixo \u00f3ptico, negativo se estiver abaixo do eixo \u00f3ptico.<\/li>\n<\/ul>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Interfaces Planas como Caso Limite das Esf\u00e9ricas<\/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;\">Tudo o que desenvolvemos para interfaces<\/span><\/strong><\/a> esf\u00e9ricas tamb\u00e9m serve para entender melhor as interfaces planas. De fato, podemos entender uma interface plana como um peda\u00e7o de uma interface esf\u00e9rica com um raio de curvatura muito grande; de fato, se tomarmos limites sobre a rela\u00e7\u00e3o objeto-imagem para interfaces esf\u00e9ricas quando o raio tende ao infinito, temos:<\/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;\">E se a partir disso calculamos o fator de amplia\u00e7\u00e3o, obtemos:<\/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;\">Ou seja, a imagem mant\u00e9m seu tamanho e orienta\u00e7\u00e3o, o que varia \u00e9 sua posi\u00e7\u00e3o observada.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>Exerc\u00edcios<\/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>Em frente a uma haste cil\u00edndrica de vidro, uma part\u00edcula \u00e9 colocada como mostrado abaixo<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>Se a part\u00edcula estiver a 30[cm] da haste e a ponta desta for aproximadamente esf\u00e9rica com um raio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=1,5[cm],<\/span><\/span> calcule a posi\u00e7\u00e3o da imagem gerada dentro da haste.<\/li>\n<li>Consideremos a mesma haste do exerc\u00edcio anterior, mas agora ela est\u00e1 submersa na \u00e1gua. Se em frente a ela for colocada uma agulha de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1[cm]<\/span><\/span> de altura na mesma dist\u00e2ncia de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">30[cm],<\/span><\/span> calcule o local e a altura da imagem.<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>Uma pessoa olha para o fundo de uma piscina com o objetivo de estimar sua profundidade. Como guia, ela usa uma flecha pintada no fundo. Qual \u00e9 a rela\u00e7\u00e3o entre a profundidade real e a aparente?<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>Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas Resumo: Nesta aula, analisaremos a Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas, destacando como a luz se comporta ao passar por superf\u00edcies esf\u00e9ricas e como as imagens s\u00e3o formadas. S\u00e3o apresentadas as equa\u00e7\u00f5es chave para calcular a posi\u00e7\u00e3o e o tamanho das imagens. Tamb\u00e9m s\u00e3o explorados casos pr\u00e1ticos, como lentes e estimativas de profundidades [&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":7,"footnotes":""},"categories":[637,837],"tags":[],"class_list":["post-27727","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-fisica-pt","category-optica-geometrica-pt"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Aprenda sobre a refra\u00e7\u00e3o em interfaces esf\u00e9ricas, como as imagens s\u00e3o formadas e as equa\u00e7\u00f5es chave para calcular sua posi\u00e7\u00e3o e tamanho.\" \/>\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\/pt\/refracao-em-interfaces-esfericas\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas\" \/>\n<meta property=\"og:description\" content=\"Aprenda sobre a refra\u00e7\u00e3o em interfaces esf\u00e9ricas, como as imagens s\u00e3o formadas e as equa\u00e7\u00f5es chave para calcular sua posi\u00e7\u00e3o e tamanho.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/refracao-em-interfaces-esfericas\/\" \/>\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=\"2021-10-06T13:00:05+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-11T16:52:25+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=\"Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas\" \/>\n<meta name=\"twitter:description\" content=\"Aprenda sobre a refra\u00e7\u00e3o em interfaces esf\u00e9ricas, como as imagens s\u00e3o formadas e as equa\u00e7\u00f5es chave para calcular sua posi\u00e7\u00e3o e tamanho.\" \/>\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\\\/pt\\\/refracao-em-interfaces-esfericas\\\/#article\",\"isPartOf\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas\",\"datePublished\":\"2021-10-06T13:00:05+00:00\",\"dateModified\":\"2024-08-11T16:52:25+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/\"},\"wordCount\":1751,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/#primaryimage\"},\"thumbnailUrl\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"articleSection\":[\"F\u00edsica\",\"\u00d3ptica Geom\u00e9trica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/\",\"url\":\"http:\\\/\\\/toposuranos.com\\\/material\\\/pt\\\/refracao-em-interfaces-esfericas\\\/\",\"name\":\"Refra\u00e7\u00e3o em Interfaces Esf\u00e9ricas - 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