{"id":27687,"date":"2021-09-18T13:00:17","date_gmt":"2021-09-18T13:00:17","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27687"},"modified":"2024-08-10T23:52:56","modified_gmt":"2024-08-10T23:52:56","slug":"a-refracao-da-luz-e-a-lei-de-snell","status":"publish","type":"post","link":"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/","title":{"rendered":"A Refra\u00e7\u00e3o da Luz e a Lei de Snell"},"content":{"rendered":"<p><center><\/p>\n<h1>A Refra\u00e7\u00e3o da Luz e a Lei de Snell<\/h1>\n<p><em><strong>Resumo:<\/strong><br \/>\nNesta aula, exploraremos a refra\u00e7\u00e3o da luz atrav\u00e9s da an\u00e1lise da Lei de Snell. O conceito de \u00edndice de refra\u00e7\u00e3o ser\u00e1 explicado, a Lei de Snell ser\u00e1 derivada usando o princ\u00edpio de Fermat, e o estudo de como essa lei nos permite calcular o caminho de um raio de luz ao passar por diferentes meios ser\u00e1 abordado. Al\u00e9m disso, os fen\u00f4menos de reflex\u00e3o e reflex\u00e3o total ser\u00e3o discutidos, aplicando esses conceitos a uma s\u00e9rie de exerc\u00edcios pr\u00e1ticos. O objetivo \u00e9 entender e aplicar a Lei de Snell em problemas de \u00f3ptica.<\/em><\/p>\n<p><strong>Objetivos de Aprendizagem<\/strong><\/p>\n<ol style=\"text-align:left\">\n<li><strong>Compreender<\/strong> o conceito de \u00edndice de refra\u00e7\u00e3o e sua rela\u00e7\u00e3o com a velocidade da luz em diferentes meios.<\/li>\n<li><strong>Aplicar<\/strong> o princ\u00edpio de Fermat para entender como a luz segue o caminho que minimiza o tempo de percurso entre dois pontos.<\/li>\n<li><strong>Demonstrar<\/strong> a Lei de Snell a partir do princ\u00edpio de Fermat para determinar o caminho de um raio de luz ao passar por diferentes meios.<\/li>\n<li><strong>Calcular<\/strong> os \u00e2ngulos de incid\u00eancia e refra\u00e7\u00e3o usando a Lei de Snell em situa\u00e7\u00f5es com diferentes \u00edndices de refra\u00e7\u00e3o.<\/li>\n<li><strong>Compreender<\/strong> o conceito de reflex\u00e3o total interna e como ele se relaciona com o \u00e2ngulo cr\u00edtico e os \u00edndices de refra\u00e7\u00e3o.<\/li>\n<li><strong>Determinar<\/strong> o \u00e2ngulo cr\u00edtico para a reflex\u00e3o total interna na interface entre dois meios.<\/li>\n<\/ol>\n<p><strong>\u00cdNDICE DE CONTE\u00daDOS<\/strong><br \/>\n<a href=\"#1\"><strong>O \u00cdndice de Refra\u00e7\u00e3o<\/strong><\/a><br \/>\n<a href=\"#2\">O Princ\u00edpio de Fermat<\/a><br \/>\n<a href=\"#3\">A Lei de Snell da Refra\u00e7\u00e3o da Luz<\/a><br \/>\n<a href=\"#4\"><strong>Refra\u00e7\u00e3o, Reflex\u00e3o e Reflex\u00e3o Total da Luz<\/strong><\/a><br \/>\n<a href=\"#5\"><strong>Exerc\u00edcios<\/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>O \u00cdndice de Refra\u00e7\u00e3o<\/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;\">O \u00edndice de refra\u00e7\u00e3o \u00e9 definido<\/span><\/strong><\/a> de um meio como a raz\u00e3o entre a velocidade da luz no v\u00e1cuo e a velocidade da luz nesse meio. Esta \u00e9 uma quantidade adimensional e \u00e9 geralmente representada pela letra <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;\">Onde <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c<\/span><\/span> \u00e9 a velocidade da luz no v\u00e1cuo, e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">c_k<\/span><\/span> \u00e9 a velocidade da luz no meio <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">k.<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Como a luz sempre se move mais lentamente em qualquer meio do que no v\u00e1cuo, o \u00edndice de refra\u00e7\u00e3o \u00e9 sempre maior ou igual a 1.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h3>O Princ\u00edpio de Fermat<\/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;\">A velocidade da luz depende<\/span><\/strong><\/a> do meio no qual ela viaja. Quanto maior o \u00edndice de refra\u00e7\u00e3o do meio, menor ser\u00e1 a velocidade da luz ao viajar por ele; e em rela\u00e7\u00e3o a isso, enuncia-se o princ\u00edpio de Fermat:<\/p>\n<p style=\"text-align: center; color: #000000; background-color: #80ff80;\">Quando a luz viaja de um ponto a outro, ela o faz pelo caminho que minimiza o tempo de percurso.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Este princ\u00edpio se mant\u00e9m mesmo quando a luz passa por diferentes meios.<\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>A Lei de Snell da Refra\u00e7\u00e3o da Luz<\/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;\">Com base no estabelecido pelo princ\u00edpio de Fermat<\/span><\/strong><\/a>, \u00e9 poss\u00edvel formular um problema de otimiza\u00e7\u00e3o que nos permitir\u00e1 determinar o caminho que um raio de luz seguir\u00e1 ao passar por diferentes meios. Isso \u00e9 o que leva finalmente \u00e0 Lei de Snell, cuja formula\u00e7\u00e3o e demonstra\u00e7\u00e3o veremos a seguir.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Suponha que um raio parte de um ponto A e chega a um ponto B, cruzando uma interface que separa dois meios com \u00edndices de refra\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2<\/span><\/span>, respectivamente. Nosso objetivo ser\u00e1 encontrar uma rela\u00e7\u00e3o que nos permita calcular o caminho do raio de luz seguindo o princ\u00edpio de Fermat do tempo m\u00ednimo de percurso, e para isso, \u00e9 montado o seguinte diagrama:<\/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=\"Lei de Snell\" 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=\"Lei de Snell\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">O racioc\u00ednio come\u00e7a analisando a forma do tempo de percurso do raio de luz. 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}{Tempo\\,de\\,Percurso} &amp; =\\displaystyle \\frac{{Dist\u00e2ncia}}{{Velocidade}} \\\\ \\\\ &amp; \\displaystyle =\\frac{{Dist\u00e2ncia\\,no\\,meio\\,1}}{{Velocidade\\,no\\,meio\\,1}} + \\frac{{Dist\u00e2ncia\\,no\\,meio\\,2}}{{Velocidade\\,no\\,meio\\,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;\">Feito isso, mantendo fixos os pontos A e B, o tempo de percurso \u00e9 determinado pelo ponto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> em que o raio toca a interface entre os meios. Com isso, podemos definir uma fun\u00e7\u00e3o de tempo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x)<\/span><\/span> como<\/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;\">Agora, como o princ\u00edpio de Fermat estabelece que a luz segue o caminho que minimiza o tempo de percurso, \u00e9 poss\u00edvel a partir disso encontrar o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> que minimiza a fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x).<\/span><\/span> Estamos diante de um problema de otimiza\u00e7\u00e3o.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Derivando <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t<\/span><\/span> em rela\u00e7\u00e3o a <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span>, temos:<\/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;\">Agora, observe que:<\/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;\">Assim, substituindo isso na derivada do tempo, temos:<\/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;\">Finalmente, se o ponto <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">x<\/span><\/span> minimiza a fun\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">t(x),<\/span><\/span> ent\u00e3o a derivada deve ser nula, e temos:<\/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;\">Esta \u00e9 a Lei de Snell para a refra\u00e7\u00e3o de um raio de luz que passa entre dois meios, mostrando a rela\u00e7\u00e3o entre o \u00e2ngulo de incid\u00eancia <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1<\/span><\/span> e o \u00e2ngulo refratado <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>Refra\u00e7\u00e3o, Reflex\u00e3o e Reflex\u00e3o Total da Luz<\/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;\">Vimos que quando a luz<\/span> <\/strong><\/a>passa de um meio para outro, ela se refrata, mas, em geral, o que ocorre \u00e9 uma combina\u00e7\u00e3o de refra\u00e7\u00e3o e reflex\u00e3o; e dependendo dos \u00edndices de refra\u00e7\u00e3o e do \u00e2ngulo de incid\u00eancia do raio de luz, a refra\u00e7\u00e3o pode desaparecer, restando apenas a reflex\u00e3o.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Suponha que um raio de luz incide de um material <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">a<\/span><\/span> para outro <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">b<\/span><\/span> com \u00edndices de refra\u00e7\u00e3o <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a<\/span><\/span> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span>, respectivamente. Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a \\gt n_b,<\/span><\/span> pela Lei de Snell, temos:<\/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;\">Como <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_a\/n_b \\gt 1,<\/span><\/span> acontece que <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(\\theta_b) \\gt \\sin(\\theta_a),<\/span><\/span> o que implica que o raio refratado se desvia afastando-se da normal. Isso significa que deve existir algum <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a\\lt 90^o<\/span><\/span> para o qual <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\sin(\\theta_b)=1<\/span><\/span> e, portanto, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b=90^o,<\/span><\/span> como mostrado na figura a seguir.<\/p>\n<p style=\"text-align: justify; color: #000000;\">O \u00e2ngulo de incid\u00eancia que faz com que o raio se refrate ao longo da interface \u00e9 conhecido como \u00e2ngulo cr\u00edtico e satisfaz a rela\u00e7\u00e3o:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\sin(\\theta_{cr\u00edtico}) = \\frac{n_b}{n_a}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">O que equivale a dizer:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_{cr\u00edtico} = \\arcsin\\left( \\frac{n_b}{n_a} \\right)<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Se <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a \\gt \\theta_{cr\u00edtico},<\/span><\/span> ent\u00e3o h\u00e1 reflex\u00e3o total.<\/p>\n<p><a name=\"4\"><\/a><\/p>\n<h2>Exerc\u00edcios:<\/h2>\n<ol style=\"text-align: justify; color: #000000;\">\n<li>Considere um raio de luz que passa da \u00e1gua para o vidro, como mostrado na figura a seguir:<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=\"raio de luz passando da \u00e1gua para o vidro\" 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=\"raio de luz passando da \u00e1gua para o vidro\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\nO \u00edndice de refra\u00e7\u00e3o da \u00e1gua \u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1 = 1,33,<\/span><\/span> e o do vidro \u00e9 <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2=1,52.<\/span><\/span> Se um raio de luz que passa da \u00e1gua para o vidro incide na interface que separa os dois meios com um \u00e2ngulo de inclina\u00e7\u00e3o de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1 = 60^o<\/span><\/span> em rela\u00e7\u00e3o \u00e0 normal, com que \u00e2ngulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_2<\/span><\/span> o raio refratado sai? <span class=\"collapseomatic \" id=\"id69e0398e2c90d\"  tabindex=\"0\" title=\"SOLU\u00c7\u00c3O\"    >SOLU\u00c7\u00c3O<\/span><div id=\"target-id69e0398e2c90d\" class=\"collapseomatic_content \">\nUsando a Lei de Snell, temos:<\/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>; Lei de Snell<\/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>; \u00cdndice de refra\u00e7\u00e3o da \u00e1gua<\/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>; \u00cdndice de refra\u00e7\u00e3o do vidro<\/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>; \u00c2ngulo de incid\u00eancia na interface do raio de luz<\/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>; De (1,2,3,4), \u00c2ngulo de refra\u00e7\u00e3o<\/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;\">[v\u00eddeo]<\/span><\/strong><\/a><\/li>\n<li>Tr\u00eas l\u00edquidos separados por duas interfaces t\u00eam os seguintes \u00edndices de refra\u00e7\u00e3o: <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> e <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_3=1,68,<\/span><\/span> e est\u00e3o dispostos como mostrado na figura a seguir:<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=\"Lei de Snell aplicada a tr\u00eas meios\" 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=\"Lei de Snell aplicada a tr\u00eas meios\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center>Se o raio que vai do meio com \u00edndice <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_1<\/span><\/span> para o de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_2<\/span><\/span> incidir na interface com um \u00e2ngulo <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_1=70^o<\/span><\/span>, com que \u00e2ngulo se refratar\u00e1 ao passar para o meio com \u00edndice <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_3<\/span><\/span>? <span class=\"collapseomatic \" id=\"id69e0398e2cb15\"  tabindex=\"0\" title=\"SOLU\u00c7\u00c3O\"    >SOLU\u00c7\u00c3O<\/span><div id=\"target-id69e0398e2cb15\" class=\"collapseomatic_content \">\nDe forma an\u00e1loga ao exerc\u00edcio anterior, temos o seguinte racioc\u00ednio:<\/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>; Lei de Snell para a transi\u00e7\u00e3o do meio n1 para 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>; Lei de Snell para a transi\u00e7\u00e3o do meio n2 para 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>; De(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>Finalmente, substituindo os dados, temos:<\/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 \/>\nObserve que esse racioc\u00ednio nos mostra que podemos fazer os c\u00e1lculos considerando apenas os meios de entrada e sa\u00edda do raio, ignorando completamente o que est\u00e1 no meio.<br \/>\n<\/div> <a href=\"https:\/\/www.youtube.com\/watch?v=LxhWbErujpo&amp;t=1417s\" target=\"_blank\" rel=\"noopener\"><strong><span style=\"color: #ff0000;\">[v\u00eddeo]<\/span><\/strong><\/a><\/li>\n<li>Do fundo de uma piscina, um raio de luz \u00e9 dirigido para a interface entre o ar e a \u00e1gua. Determine o \u00e2ngulo de incid\u00eancia para que ocorra uma reflex\u00e3o total.<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=\"raio de luz incidindo em \u00e2ngulo cr\u00edtico\" 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=\"raio de luz incidindo em \u00e2ngulo cr\u00edtico\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><br \/>\n<span class=\"collapseomatic \" id=\"id69e0398e2ccf8\"  tabindex=\"0\" title=\"SOLU\u00c7\u00c3O\"    >SOLU\u00c7\u00c3O<\/span><div id=\"target-id69e0398e2ccf8\" class=\"collapseomatic_content \">\nO \u00e2ngulo cr\u00edtico ser\u00e1 dado por:<\/p>\n<p style=\"text-align: center; color: #000000;\"><span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\displaystyle \\theta_{cr\u00edtico}= \\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;\">[v\u00eddeo]<\/span><\/strong><\/a><\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>A Refra\u00e7\u00e3o da Luz e a Lei de Snell Resumo: Nesta aula, exploraremos a refra\u00e7\u00e3o da luz atrav\u00e9s da an\u00e1lise da Lei de Snell. O conceito de \u00edndice de refra\u00e7\u00e3o ser\u00e1 explicado, a Lei de Snell ser\u00e1 derivada usando o princ\u00edpio de Fermat, e o estudo de como essa lei nos permite calcular o caminho [&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":8,"footnotes":""},"categories":[637,837],"tags":[],"class_list":["post-27687","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 v26.7 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>A Refra\u00e7\u00e3o da Luz e a Lei de Snell - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Refra\u00e7\u00e3o e a Lei de Snell Explicadas: Aprenda como a luz muda de dire\u00e7\u00e3o ao passar entre meios, calcule \u00e2ngulos de incid\u00eancia e refra\u00e7\u00e3o, e resolva exerc\u00edcios pr\u00e1ticos.\" \/>\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\/a-refracao-da-luz-e-a-lei-de-snell\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"A Refra\u00e7\u00e3o da Luz e a Lei de Snell\" \/>\n<meta property=\"og:description\" content=\"Refra\u00e7\u00e3o e a Lei de Snell Explicadas: Aprenda como a luz muda de dire\u00e7\u00e3o ao passar entre meios, calcule \u00e2ngulos de incid\u00eancia e refra\u00e7\u00e3o, e resolva exerc\u00edcios pr\u00e1ticos.\" \/>\n<meta property=\"og:url\" content=\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/\" \/>\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=\"2021-09-18T13:00:17+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-10T23:52:56+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=\"A Refra\u00e7\u00e3o da Luz e a Lei de Snell\" \/>\n<meta name=\"twitter:description\" content=\"Refra\u00e7\u00e3o e a Lei de Snell Explicadas: Aprenda como a luz muda de dire\u00e7\u00e3o ao passar entre meios, calcule \u00e2ngulos de incid\u00eancia e refra\u00e7\u00e3o, e resolva exerc\u00edcios pr\u00e1ticos.\" \/>\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\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/#article\",\"isPartOf\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"http:\/\/toposuranos.com\/material\/#\/schema\/person\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"A Refra\u00e7\u00e3o da Luz e a Lei de Snell\",\"datePublished\":\"2021-09-18T13:00:17+00:00\",\"dateModified\":\"2024-08-10T23:52:56+00:00\",\"mainEntityOfPage\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/\"},\"wordCount\":1728,\"commentCount\":0,\"publisher\":{\"@id\":\"http:\/\/toposuranos.com\/material\/#organization\"},\"image\":{\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/#primaryimage\"},\"thumbnailUrl\":\"http:\/\/toposuranos.com\/material\/wp-content\/uploads\/2024\/08\/leydesnell.jpg\",\"articleSection\":[\"F\u00edsica\",\"\u00d3ptica Geom\u00e9trica\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/\",\"url\":\"http:\/\/toposuranos.com\/material\/pt\/a-refracao-da-luz-e-a-lei-de-snell\/\",\"name\":\"A Refra\u00e7\u00e3o da Luz e a Lei de Snell - 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