{"id":27738,"date":"2021-10-06T13:00:07","date_gmt":"2021-10-06T13:00:07","guid":{"rendered":"http:\/\/toposuranos.com\/material\/?p=27738"},"modified":"2024-08-11T17:22:11","modified_gmt":"2024-08-11T17:22:11","slug":"refraction-sur-les-interfaces-spheriques","status":"publish","type":"post","link":"https:\/\/toposuranos.com\/material\/fr\/refraction-sur-les-interfaces-spheriques\/","title":{"rendered":"R\u00e9fraction sur les interfaces sph\u00e9riques"},"content":{"rendered":"<p><center><\/p>\n<h1>R\u00e9fraction sur les interfaces sph\u00e9riques<\/h1>\n<p><em><strong>R\u00e9sum\u00e9:<\/strong><br \/>\nDans cette le\u00e7on, nous analyserons la r\u00e9fraction sur les interfaces sph\u00e9riques, en mettant en avant le comportement de la lumi\u00e8re lorsqu&#8217;elle traverse des surfaces sph\u00e9riques et comment les images se forment. Les \u00e9quations cl\u00e9s pour calculer la position et la taille des images sont pr\u00e9sent\u00e9es. Des cas pratiques, tels que les lentilles et l&#8217;estimation des profondeurs apparentes, sont \u00e9galement explor\u00e9s.<\/em><\/p>\n<p><strong>Objectifs d&#8217;apprentissage:<\/strong><br \/>\n\u00c0 la fin de cette le\u00e7on, l&#8217;\u00e9tudiant sera capable de :<\/p>\n<ol style=\"text-align:left;\">\n<li><strong>Comprendre<\/strong> la r\u00e9fraction de la lumi\u00e8re lorsqu&#8217;elle traverse des interfaces sph\u00e9riques.<\/li>\n<li><strong>D\u00e9river<\/strong> et utiliser la relation objet-image pour les interfaces sph\u00e9riques.<\/li>\n<li><strong>Appliquer<\/strong> la loi de Snell dans le contexte des interfaces sph\u00e9riques.<\/li>\n<li><strong>D\u00e9terminer<\/strong> la position de l&#8217;image form\u00e9e par une interface sph\u00e9rique.<\/li>\n<li><strong>Calculer<\/strong> l&#8217;agrandissement de l&#8217;image par r\u00e9fraction sur des surfaces sph\u00e9riques.<\/li>\n<li><strong>Comprendre<\/strong> la convention des signes pour la position et la taille des objets et des images.<\/li>\n<li><strong>Relier<\/strong> les interfaces sph\u00e9riques aux interfaces planes comme cas limite.<\/li>\n<li><strong>Analyser<\/strong> la formation d&#8217;images \u00e9tendues \u00e0 travers des interfaces sph\u00e9riques.<\/li>\n<\/ol>\n<p><strong>TABLE DES MATI\u00c8RES<\/strong><br \/>\n<a href=\"#1\"><strong>Introduction<\/strong><\/a><br \/>\n<a href=\"#2\"><strong>La relation objet-image pour la r\u00e9fraction sur les interfaces sph\u00e9riques<\/strong><\/a><br \/>\n<a href=\"#3\">Extraction des relations entre les angles<\/a><br \/>\n<a href=\"#4\">Introduction de la loi de Snell<\/a><br \/>\n<a href=\"#5\"><strong>Formation d&#8217;images \u00e9tendues par r\u00e9fraction de l&#8217;autre c\u00f4t\u00e9 des interfaces sph\u00e9riques<\/strong><\/a><br \/>\n<a href=\"#6\">Synth\u00e8se<\/a><br \/>\n<a href=\"#7\"><strong>Les interfaces planes comme cas limite des sph\u00e9riques<\/strong><\/a><br \/>\n<a href=\"#8\"><strong>Exercices<\/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;\">Nous avons d\u00e9j\u00e0 \u00e9tudi\u00e9 comment fonctionne la r\u00e9fraction ;<\/span><\/strong><\/a> c&#8217;est-\u00e0-dire ce qui se passe lorsque la lumi\u00e8re passe d&#8217;un milieu \u00e0 un autre. Cependant, nous avons fait cela dans le cas o\u00f9 l&#8217;interface qui s\u00e9pare les milieux est une surface plane. Toutefois, aussi bien dans la nature que dans les applications pratiques, il n&#8217;est pas rare de trouver des processus de r\u00e9fraction sur des interfaces sph\u00e9riques. Des exemples de cela incluent l&#8217;\u0153il humain (et de presque tous les animaux en r\u00e9alit\u00e9) ainsi que la plupart des dispositifs optiques utilis\u00e9s dans la vie quotidienne et dans les applications industrielles.<\/p>\n<p style=\"text-align: justify; color: #000000;\">Dans l&#8217;illustration suivante, nous voyons comment une lentille est construite \u00e0 travers deux surfaces sph\u00e9riques.<\/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=\"Lentille en verre form\u00e9e par deux surfaces sph\u00e9riques\" 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=\"Lentille en verre form\u00e9e par deux surfaces sph\u00e9riques\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">Pour l&#8217;\u00e9tude d\u00e9taill\u00e9e de ce type de dispositifs, il est n\u00e9cessaire de revoir comment la lumi\u00e8re se comporte lorsqu&#8217;elle passe d&#8217;un milieu \u00e0 un autre \u00e0 travers une interface sph\u00e9rique.<\/p>\n<p><a name=\"2\"><\/a><\/p>\n<h2>La relation objet-image pour la r\u00e9fraction sur les interfaces sph\u00e9riques<\/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;\">Nous commencerons notre \u00e9tude en enqu\u00eatant<\/span><\/strong><\/a> sur la mani\u00e8re dont la lumi\u00e8re se comporte lorsqu&#8217;elle passe d&#8217;un milieu \u00e0 un autre \u00e0 travers une interface sph\u00e9rique. Pour ce faire, nous consid\u00e9rerons une sph\u00e8re de rayon <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span> faite d&#8217;un mat\u00e9riau avec un indice de r\u00e9fraction <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">n_b<\/span><\/span> immerg\u00e9e dans un milieu avec un indice de r\u00e9fraction <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 sph\u00e9rique s\u00e9parant deux milieux\" 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 sph\u00e9rique s\u00e9parant deux milieux\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p><a name=\"3\"><\/a><\/p>\n<h3>Extraction des relations entre les angles<\/h3>\n<p style=\"text-align: justify; color: #000000;\">Si nous analysons les angles impliqu\u00e9s dans cette figure, nous constaterons 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>D\u00e9monstration<\/h4>\n<p style=\"text-align: justify; color: #000000;\">La premi\u00e8re \u00e9quation est obtenue \u00e0 partir du fait que la somme des angles int\u00e9rieurs d&#8217;un triangle est \u00e9gale \u00e0 deux angles droits :<\/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;\">La deuxi\u00e8me \u00e9quation est obtenue de mani\u00e8re analogue :<\/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>Introduction de la loi de Snell<\/h3>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de la figure, on peut \u00e9galement obtenir les expressions suivantes :<\/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 sph\u00e9rique s\u00e9parant deux milieux\" 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 sph\u00e9rique s\u00e9parant deux milieux\" 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;\">Et \u00e0 partir de la loi de Snell, nous avons<\/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;\">Maintenant, si nous prenons l&#8217;approximation dans laquelle <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_a<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\theta_b<\/span><\/span> sont petits, alors <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\alpha, \\beta<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">\\phi<\/span><\/span> le seront \u00e9galement, et il se produira :<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de la figure, on peut \u00e9galement obtenir les expressions suivantes :<\/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;\">Ensuite, \u00e0 partir de cela et de la loi de Snell, on 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{(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;\">Maintenant, \u00e0 partir de (7), (1) et (2), nous avons<\/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;\">Enfin, \u00e0 partir de (8), des approximations et des \u00e9quations (3), (4) et (5), nous obtenons :<\/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;\">Ce dernier r\u00e9sultat est ce que nous appelons <strong>Relation objet-image pour la r\u00e9fraction sur les interfaces sph\u00e9riques.<\/strong><\/p>\n<p><a name=\"5\"><\/a><\/p>\n<h2>Formation d&#8217;images \u00e9tendues par r\u00e9fraction de l&#8217;autre c\u00f4t\u00e9 des interfaces sph\u00e9riques<\/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;\">Voyons maintenant ce qui se passe lorsque nous changeons<\/span><\/strong><\/a> la source lumineuse ponctuelle par un objet \u00e9tendu. Cela est illustr\u00e9 dans la figure suivante :<\/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=\"objet \u00e9tendu devant une interface sph\u00e9rique\" 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=\"objet \u00e9tendu devant une interface sph\u00e9rique\" class=\"alignnone size-full lazyload\" \/><\/noscript><\/center><\/p>\n<p style=\"text-align: justify; color: #000000;\">L&#8217;analyse pr\u00e9c\u00e9dente nous indique d\u00e9j\u00e0 la relation entre <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime,<\/span><\/span> maintenant nous devons seulement trouver la relation entre les tailles de l&#8217;objet et de l&#8217;image.<\/p>\n<p style=\"text-align: justify; color: #000000;\">\u00c0 partir de la figure, nous avons :<\/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;\">Nous allons combiner cela avec la loi 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;\">Et pour cela, nous nous baserons sur le fait que pour de petits angles, l&#8217;approximation suivante est valable :<\/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 sorte que nous pouvons \u00e9crire :<\/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}<\/span><\/span><\/p>\n<p style=\"text-align: justify; color: #000000;\">Maintenant, en nous rappelant ce que nous avons vu pour les miroirs sph\u00e9riques, nous avons quelque chose d&#8217;analogue. \u00c0 ce stade, nous pouvons (re)d\u00e9finir le facteur de grossissement <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">m<\/span><\/span> de la mani\u00e8re suivante :<\/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 sorte 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>Synth\u00e8se<\/h3>\n<p style=\"text-align: justify; color: #000000;\">En r\u00e9sum\u00e9, jusqu&#8217;\u00e0 pr\u00e9sent, nous avons extrait deux r\u00e9sultats qui nous permettent de d\u00e9duire la formation d&#8217;images lorsque la lumi\u00e8re \u00e9mise par un objet traverse une interface sph\u00e9rique. Ces \u00e9quations sont les suivantes :<\/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;\">Avec ces deux \u00e9quations, vous pouvez calculer \u00e0 la fois la position de l&#8217;image ainsi que son orientation et sa taille, et elles fonctionneront ind\u00e9pendamment de la surface de l&#8217;interface, qu&#8217;elle soit concave ou convexe. \u00c0 ce stade, cependant, il est n\u00e9cessaire de clarifier la convention des signes.<\/p>\n<h4>Convention des signes<\/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;\">Avec ces deux \u00e9quations, vous pouvez calculer \u00e0 la fois la position de l&#8217;image ainsi que son orientation et sa taille,<\/span><\/strong><\/a> et elles fonctionneront ind\u00e9pendamment de la surface de l&#8217;interface, qu&#8217;elle soit concave ou convexe. \u00c0 ce stade, cependant, il est n\u00e9cessaire de clarifier la convention des signes.<\/p>\n<p style=\"text-align: justify; color: #000000;\">L&#8217;interface s\u00e9pare l&#8217;espace en deux r\u00e9gions : l&#8217;une o\u00f9 se trouve l&#8217;objet et l&#8217;autre o\u00f9 se trouve l&#8217;image. En fonction de cela, nous avons :<\/p>\n<ul style=\"text-align: justify; color: #000000;\">\n<li><strong>Position de l&#8217;objet <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S<\/span><\/span> :<\/strong> Positive si elle se trouve du c\u00f4t\u00e9 de l&#8217;objet, n\u00e9gative si elle se trouve du c\u00f4t\u00e9 de l&#8217;image.<\/li>\n<li><strong>Position de l&#8217;image <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">S^\\prime<\/span><\/span> et le rayon de courbure <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R<\/span><\/span> :<\/strong> Positive si elle se trouve du c\u00f4t\u00e9 de l&#8217;image, n\u00e9gative si elle se trouve du c\u00f4t\u00e9 de l&#8217;objet.<\/li>\n<li><strong>Taille de l&#8217;objet et de l&#8217;image, <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y<\/span><\/span> et <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">y^\\prime<\/span><\/span> :<\/strong> Positive si elle est au-dessus de l&#8217;axe optique, n\u00e9gative si elle est en dessous de l&#8217;axe optique.<\/li>\n<\/ul>\n<p><a name=\"7\"><\/a><\/p>\n<h2>Les interfaces planes comme cas limite des sph\u00e9riques<\/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;\">Tout ce que nous avons d\u00e9velopp\u00e9 pour les interfaces sph\u00e9riques<\/span><\/strong><\/a> est \u00e9galement utile pour mieux comprendre les interfaces planes. En fait, nous pouvons comprendre une interface plane comme un morceau d&#8217;interface sph\u00e9rique avec un rayon de courbure tr\u00e8s grand ; en fait, si nous prenons les limites sur la relation objet-image pour les interfaces sph\u00e9riques lorsque le rayon tend \u00e0 l&#8217;infini, nous avons :<\/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;\">Et si nous calculons le facteur de grossissement \u00e0 partir de cela, nous obtenons :<\/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;\">Autrement dit, l&#8217;image conserve sa taille et son orientation, ce qui change, c&#8217;est sa position observ\u00e9e.<\/p>\n<p><a name=\"8\"><\/a><\/p>\n<h2>Exercices<\/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>Devant une tige en verre cylindrique, une particule est plac\u00e9e comme indiqu\u00e9 ci-dessous<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>Si la particule se trouve \u00e0 30[cm] de la tige et que son extr\u00e9mit\u00e9 est approximativement sph\u00e9rique avec un rayon de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">R=1,5[cm],<\/span><\/span> calculez la position de l&#8217;image g\u00e9n\u00e9r\u00e9e \u00e0 l&#8217;int\u00e9rieur de la tige.<\/li>\n<li>Consid\u00e9rons la m\u00eame tige de l&#8217;exercice pr\u00e9c\u00e9dent, mais maintenant elle est sous l&#8217;eau. Si une aiguille de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">1[cm]<\/span><\/span> de hauteur est plac\u00e9e \u00e0 la m\u00eame distance de <span dir=\"ltr\"><span class=\"katex-eq\" data-katex-display=\"false\">30[cm],<\/span><\/span> devant elle, calculez la position et la hauteur de l&#8217;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>Une personne regarde vers le fond d&#8217;une piscine dans le but d&#8217;estimer sa profondeur. Comme guide, elle utilise une fl\u00e8che peinte au fond. Quelle est la relation entre la profondeur r\u00e9elle et apparente?<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>R\u00e9fraction sur les interfaces sph\u00e9riques R\u00e9sum\u00e9: Dans cette le\u00e7on, nous analyserons la r\u00e9fraction sur les interfaces sph\u00e9riques, en mettant en avant le comportement de la lumi\u00e8re lorsqu&#8217;elle traverse des surfaces sph\u00e9riques et comment les images se forment. Les \u00e9quations cl\u00e9s pour calculer la position et la taille des images sont pr\u00e9sent\u00e9es. Des cas pratiques, tels [&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":15,"footnotes":""},"categories":[847,647],"tags":[],"class_list":["post-27738","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-optique-geometrique","category-physique"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>R\u00e9fraction sur les interfaces sph\u00e9riques - toposuranos.com\/material<\/title>\n<meta name=\"description\" content=\"Apprenez la r\u00e9fraction sur les interfaces sph\u00e9riques, comment se forment les images et les \u00e9quations cl\u00e9s pour calculer leur position et leur taille.\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/toposuranos.com\/material\/fr\/refraction-sur-les-interfaces-spheriques\/\" \/>\n<meta property=\"og:locale\" content=\"es_ES\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"R\u00e9fraction sur les interfaces sph\u00e9riques\" \/>\n<meta property=\"og:description\" content=\"Apprenez la r\u00e9fraction sur les interfaces sph\u00e9riques, comment se forment les images et les \u00e9quations cl\u00e9s pour calculer leur position et leur taille.\" \/>\n<meta property=\"og:url\" content=\"https:\/\/toposuranos.com\/material\/fr\/refraction-sur-les-interfaces-spheriques\/\" \/>\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=\"2021-10-06T13:00:07+00:00\" \/>\n<meta property=\"article:modified_time\" content=\"2024-08-11T17:22:11+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=\"R\u00e9fraction sur les interfaces sph\u00e9riques\" \/>\n<meta name=\"twitter:description\" content=\"Apprenez la r\u00e9fraction sur les interfaces sph\u00e9riques, comment se forment les images et les \u00e9quations cl\u00e9s pour calculer leur position et leur taille.\" \/>\n<meta name=\"twitter:image\" content=\"https:\/\/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\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/#article\",\"isPartOf\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/\"},\"author\":{\"name\":\"giorgio.reveco\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#\\\/schema\\\/person\\\/e15164361c3f9a2a02cf6c234cf7fdc1\"},\"headline\":\"R\u00e9fraction sur les interfaces sph\u00e9riques\",\"datePublished\":\"2021-10-06T13:00:07+00:00\",\"dateModified\":\"2024-08-11T17:22:11+00:00\",\"mainEntityOfPage\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/\"},\"wordCount\":1894,\"commentCount\":0,\"publisher\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/#organization\"},\"image\":{\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/#primaryimage\"},\"thumbnailUrl\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/wp-content\\\/uploads\\\/2021\\\/10\\\/interfaces-esfericas.jpg\",\"articleSection\":[\"Optique G\u00e9om\u00e9trique\",\"Physique\"],\"inLanguage\":\"es\",\"potentialAction\":[{\"@type\":\"CommentAction\",\"name\":\"Comment\",\"target\":[\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/#respond\"]}]},{\"@type\":\"WebPage\",\"@id\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/\",\"url\":\"https:\\\/\\\/toposuranos.com\\\/material\\\/fr\\\/refraction-sur-les-interfaces-spheriques\\\/\",\"name\":\"R\u00e9fraction sur les interfaces sph\u00e9riques - 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