Using 1. and 2. get can get rid of the angle $\\gamma$ to find $\\theta_r=\\beta-\\delta$, and simplifying 3. we have $\\theta_i = \\alpha+\\beta$.<\/p>\n

Conversely, we know that
\n$$\\tan \\alpha = \\frac{PQ}{oQ}\\,, \\quad \\quad \\tan \\delta =\u00a0\\frac{PQ}{Qi} \\quad \\quad \\mathrm{and} \\quad \\quad \\tan \\beta =\u00a0\\frac{PQ}{QR}\\,,$$
\nwhere, for instance, $PQ$ is the distance between the points $P$ and $Q$. Furthermore, in the paraxial approximation, in which we assume that the rays are barely separated from the optical axis, we have very smalls angles an therefore: $\\tan \\alpha \\approx \\alpha$, \u00a0$\\tan \\delta \\approx \\delta$,\u00a0$\\tan \\beta \\approx \\beta$, $\\sin \\theta_i \\approx \\theta_i$, $\\sin \\theta_r \\approx \\theta_r$ and notably $oQ \\approx s$, \u00a0$Qi \\approx s’$ and $QR \\approx R$\u00a0. Using these approximations we have the following equations:<\/p>\n

\n
1. From Snell’s law we have: $n_1 \\theta_i = n_2 \\theta_r$.<\/li>\n
2. The relation $\\theta_r=\\beta-\\delta$ becomes $\\theta_r =\u00a0\\frac{PQ}{R} –\u00a0\\frac{PQ}{s’}$.<\/li>\n
3. The relation $\\theta_i = \\alpha+\\beta$ becomes\u00a0$\\theta_i = \\frac{PQ}{s}+\\frac{PQ}{R}$.<\/li>\n<\/ol>\n

   Joining these three equations we obtain
   \n$$n_1 \\left( \\frac{PQ}{s}+\\frac{PQ}{R} \\right) =n_2 \\left(\\frac{PQ}{R} –\u00a0\\frac{PQ}{s’} \\right)\\,.$$
   \nDividing the two sides by the distance $PQ$ and rearranging the terms, we get
   \n$$ \\frac{n_1}{s} + \\frac{n_2}{s’} = \\frac{n_2-n_1}{R}\\,,$$
   \nwhich is the final form of our relation.<\/p>\n","protected":false},"excerpt":{"rendered":"

   Let us consider the following scenario: on a sunny day, some people decide to go to the beach and have some fun with the waves. A coastal guard is keeping an eye on the swimmers, and she notices that one of them appears to be drowning. To help the person in distress…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"om_disable_all_campaigns":false,"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"footnotes":""},"categories":[4,6],"tags":[],"class_list":["post-150","post","type-post","status-publish","format-standard","hentry","category-level-4","category-optics","comments-off"],"_links":{"self":[{"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/posts\/150","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/comments?post=150"}],"version-history":[{"count":42,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/posts\/150\/revisions"}],"predecessor-version":[{"id":192,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/posts\/150\/revisions\/192"}],"wp:attachment":[{"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/media?parent=150"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/categories?post=150"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/camilopez.org\/wlmblog\/wp-json\/wp\/v2\/tags?post=150"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}