{"id":1394,"date":"2017-10-27T16:32:02","date_gmt":"2017-10-27T16:32:02","guid":{"rendered":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/total-internal-reflection\/"},"modified":"2017-11-08T03:27:02","modified_gmt":"2017-11-08T03:27:02","slug":"total-internal-reflection","status":"publish","type":"chapter","link":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/chapter\/total-internal-reflection\/","title":{"raw":"Total Internal Reflection","rendered":"Total Internal Reflection"},"content":{"raw":"\n<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain the phenomenon of total internal reflection.<\/li>\n<li>Describe the workings and uses of fiber optics.<\/li>\n<li>Analyze the reason for the sparkle of diamonds.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1973618\">A good-quality mirror may reflect more than 90% of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce <em data-effect=\"italics\"><em data-effect=\"italics\">total reflection<\/em><\/em> using an aspect of <em data-effect=\"italics\"><em data-effect=\"italics\">refraction<\/em><\/em>.<\/p>\n<p id=\"import-auto-id1460702\">Consider what happens when a ray of light strikes the surface between two materials, such as is shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since <em data-effect=\"italics\">[latex]{n}_{1}&gt;{n}_{2}[\/latex]<\/em>, the angle of refraction is greater than the angle of incidence\u2014that is, <em data-effect=\"italics\">[latex]{\\theta }_{1}&gt;{\\theta }_{2}[\/latex]<\/em>.) Now imagine what happens as the incident angle is increased. This causes <em data-effect=\"italics\">[latex]{\\theta }_{2}[\/latex]<\/em> to increase also. The largest the angle of refraction <em data-effect=\"italics\">[latex]{\\theta }_{2}[\/latex]<\/em>can be is <em data-effect=\"italics\">[latex]\\text{90\u00ba}[\/latex]<\/em>, as shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(b).The <span data-type=\"term\" id=\"import-auto-id1389437\">critical angle<\/span><em data-effect=\"italics\">[latex]{\\theta }_{c}[\/latex]<\/em> for a combination of materials is defined to be the incident angle [latex]{\\theta }_{1}[\/latex]<em data-effect=\"italics\"> that produces an angle of refraction of <em data-effect=\"italics\">[latex]\\text{90\u00ba}[\/latex]<\/em>. That is, <em data-effect=\"italics\">[latex]{\\theta }_{c}[\/latex]<\/em> is the incident angle for which <em data-effect=\"italics\">[latex]{\\theta }_{2}=\\text{90\u00ba}[\/latex]<\/em>. If the incident angle [latex]{\\theta }_{1}[\/latex] is greater than the critical angle, as shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(c), then all of the light is reflected back into medium 1, a condition called <span data-type=\"term\" id=\"import-auto-id1340457\">total internal reflection<\/span>.<\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2736712\" data-label=\"\">\n<div class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Critical Angle<\/h1>\n<p id=\"import-auto-id1319605\">The incident angle [latex]{\\theta }_{1}[\/latex]<em data-effect=\"italics\"> that produces an angle of refraction of <em data-effect=\"italics\">[latex]\\text{90\u00ba}[\/latex]<\/em> is called the critical angle, [latex]{\\theta }_{c}[\/latex]. <\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1435341\">\n<div class=\"bc-figcaption figcaption\">(a) A ray of light crosses a boundary where the speed of light increases and the index of refraction decreases. That is, <em data-effect=\"italics\">[latex]{n}_{2}&lt;{n}_{1}[\/latex]<\/em>. The ray bends away from the perpendicular. (b) The critical angle <em data-effect=\"italics\">[latex]{\\theta }_{c}[\/latex]<\/em> is the one for which the angle of refraction is  . (c) Total internal reflection occurs when the incident angle is greater than the critical angle.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3201448\" data-alt=\"In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1.\"><img src=\"\/resources\/634f5b5a7c4f3dea344af4ef3272c80ca4064ca4\/Figure 26_04_01.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id3038479\">Snell\u2019s law states the relationship between angles and indices of refraction. It is given by <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{n}_{1}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{1}={n}_{2}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1824435\">When the incident angle equals the critical angle ([latex]{\\theta }_{1}={\\theta }_{c}[\/latex]), the angle of refraction is <em data-effect=\"italics\">[latex]\\text{90\u00ba}[\/latex]<\/em> ([latex]{\\theta }_{2}=\\text{90\u00ba}[\/latex]). Noting that [latex]\\text{sin 90\u00ba}\\text{=1}[\/latex], Snell\u2019s law in this case becomes<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{n}_{1}\\phantom{\\rule{0.25em}{0ex}}\\text{sin}\\phantom{\\rule{0.25em}{0ex}}{\\theta }_{1}={n}_{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1447668\">The critical angle [latex]{\\theta }_{c}[\/latex] for a given combination of materials is thus<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{c}={\\text{sin}}^{-1}\\left({n}_{2}\/{n}_{1}\\right)\\phantom{\\rule{0.25em}{0ex}}\\text{for}\\phantom{\\rule{0.25em}{0ex}}{n}_{1}&gt;{n}_{2}\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id1882896\">Total internal reflection occurs for any incident angle greater than the critical angle [latex]{\\theta }_{c}[\/latex], and it can only occur when the second medium has an index of refraction less than the first. Note the above equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in the figure.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2994315\">\n<div data-type=\"title\" class=\"title\">How Big is the Critical Angle Here?<\/div>\n<p id=\"eip-id2771873\">What is the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air?<\/p>\n<p id=\"import-auto-id1449078\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3105089\">The index of refraction for polystyrene is found to be 1.49 in <a href=\"#import-auto-id1244947\" class=\"autogenerated-content\">(Figure)<\/a>, and the index of refraction of air can be taken to be 1.00, as before. Thus, the condition that the second medium (air) has an index of refraction less than the first (plastic) is satisfied, and the equation [latex]{\\theta }_{c}={\\text{sin}}^{-1}\\left({n}_{2}\/{n}_{1}\\right)[\/latex] can be used to find the critical angle <em data-effect=\"italics\">[latex]{\\theta }_{c}[\/latex]<\/em>. Here, then, [latex]{n}_{2}=1\\text{.}\\text{00}[\/latex] and [latex]{n}_{1}=1\\text{.}\\text{49}[\/latex].<\/p>\n<p id=\"import-auto-id1329220\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3137583\">The critical angle is given by <\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]{\\theta }_{c}={\\text{sin}}^{-1}\\left({n}_{2}\/{n}_{1}\\right)\\text{.}[\/latex]<\/div>\n<p id=\"import-auto-id2296664\">Substituting the identified values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">[latex]\\begin{array}{}{\\theta }_{c}={\\text{sin}}^{-1}\\left(1\\text{.}\\text{00}\/1\\text{.}\\text{49}\\right)={\\text{sin}}^{-1}\\left(0\\text{.}\\text{671}\\right)\\\\ \\text{42.2\u00ba.}\\end{array}[\/latex]<\/div>\n<p id=\"import-auto-id2774473\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id852321\">This means that any ray of light inside the plastic that strikes the surface at an angle greater than [latex]\\text{42.2\u00ba}[\/latex] will be totally reflected. This will make the inside surface of the clear plastic a perfect mirror for such rays without any need for the silvering used on common mirrors. Different combinations of materials have different critical angles, but any combination with <em data-effect=\"italics\">[latex]{n}_{1}&gt;{n}_{2}[\/latex]<\/em> can produce total internal reflection. The same calculation as made here shows that the critical angle for a ray going from water to air is [latex]\\text{48}\\text{.}6\u00ba[\/latex], while that from diamond to air is [latex]\\text{24}\\text{.}4\u00ba[\/latex], and that from flint glass to crown glass is [latex]\\text{66}\\text{.}3\u00ba[\/latex]. There is no total reflection for rays going in the other direction\u2014for example, from air to water\u2014since the condition that the second medium must have a smaller index of refraction is not satisfied. A number of interesting applications of total internal reflection follow.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2941186\">\n<h1 data-type=\"title\">Fiber Optics: Endoscopes to Telephones<\/h1>\n<p>Fiber optics is one application of total internal reflection that is in wide use. In communications, it is used to transmit telephone, internet, and cable TV signals. <span data-type=\"term\" id=\"import-auto-id2941186\">Fiber optics<\/span> employs the transmission of light down fibers of plastic or glass. Because the fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the critical angle and, thus, be totally reflected (See <a href=\"#import-auto-id1244947\" class=\"autogenerated-content\">(Figure)<\/a>.) The index of refraction outside the fiber must be smaller than inside, a condition that is easily satisfied by coating the outside of the fiber with a material having an appropriate refractive index. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal refraction. Rays are reflected around corners as shown, making the fibers into tiny light pipes.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1244947\">\n<div class=\"bc-figcaption figcaption\">Light entering a thin fiber may strike the inside surface at large or grazing angles and is completely reflected if these angles exceed the critical angle. Such rays continue down the fiber, even following it around corners, since the angles of reflection and incidence remain large.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1500825\" data-alt=\"Light ray enters an S-shaped tube and undergoes multiple reflections, finally emerging through the other end.\"><img src=\"\/resources\/e7180682af9dada10106ba9ed7d93c48251b6cfc\/Figure 26_04_02.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"Light ray enters an S-shaped tube and undergoes multiple reflections, finally emerging through the other end.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1856471\">Bundles of fibers can be used to transmit an image without a lens, as illustrated in <a href=\"#import-auto-id2976033\" class=\"autogenerated-content\">(Figure)<\/a>. The output of a device called an <span data-type=\"term\">endoscope<\/span> is shown in <a href=\"#import-auto-id2976033\" class=\"autogenerated-content\">(Figure)<\/a>(b). Endoscopes are used to explore the body through various orifices or minor incisions. Light is transmitted down one fiber bundle to illuminate internal parts, and the reflected light is transmitted back out through another to be observed. Surgery can be performed, such as arthroscopic surgery on the knee joint, employing cutting tools attached to and observed with the endoscope. Samples can also be obtained, such as by lassoing an intestinal polyp for external examination. <\/p>\n<p id=\"import-auto-id1954488\">Fiber optics has revolutionized surgical techniques and observations within the body. There are a host of medical diagnostic and therapeutic uses. The flexibility of the fiber optic bundle allows it to navigate around difficult and small regions in the body, such as the intestines, the heart, blood vessels, and joints. Transmission of an intense laser beam to burn away obstructing plaques in major arteries as well as delivering light to activate chemotherapy drugs are becoming commonplace. Optical fibers have in fact enabled microsurgery and remote surgery where the incisions are small and the surgeon\u2019s fingers do not need to touch the diseased tissue.<\/p>\n<p id=\"import-auto-id1709277\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id2976033\">\n<div class=\"bc-figcaption figcaption\">(a) An image is transmitted by a bundle of fibers that have fixed neighbors. (b) An endoscope is used to probe the body, both transmitting light to the interior and returning an image such as the one shown. (credit: Med_Chaos, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2889912\" data-alt=\"Picture (a) shows how an image A is transmitted through a bundle of parallel fibers. Picture (b) shows an endoscope image.\"><img src=\"\/resources\/fef311450b954ae2967617571d697888b14385bb\/Figure 26_04_03.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"Picture (a) shows how an image A is transmitted through a bundle of parallel fibers. Picture (b) shows an endoscope image.\" width=\"250\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1500855\">Fibers in bundles are surrounded by a cladding material that has a lower index of refraction than the core. (See <a href=\"#import-auto-id1447642\" class=\"autogenerated-content\">(Figure)<\/a>.) The cladding prevents light from being transmitted between fibers in a bundle. Without cladding, light could pass between fibers in contact, since their indices of refraction are identical. Since no light gets into the cladding (there is total internal reflection back into the core), none can be transmitted between clad fibers that are in contact with one another. The cladding prevents light from escaping out of the fiber; instead most of the light is propagated along the length of the fiber, minimizing the loss of signal and ensuring that a quality image is formed at the other end. The cladding and an additional protective layer make optical fibers flexible and durable.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1447642\">\n<div class=\"bc-figcaption figcaption\">Fibers in bundles are clad by a material that has a lower index of refraction than the core to ensure total internal reflection, even when fibers are in contact with one another. This shows a single fiber with its cladding.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2296155\" data-alt=\"The image shows a bundle fiber with a medium of refractive index n sub 1 inside surrounded by a medium n sub 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core.\"><img src=\"\/resources\/3c00c5635426db8fda047e75ab5bb7aeb8759273\/Figure 26_04_04.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The image shows a bundle fiber with a medium of refractive index n sub 1 inside surrounded by a medium n sub 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core.\" width=\"200\"><\/span><\/p><\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1431124\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Cladding<\/div>\n<p id=\"import-auto-id1495090\">The cladding prevents light from being transmitted between fibers in a bundle.<\/p>\n<\/div>\n<p id=\"import-auto-id1849890\">Special tiny lenses that can be attached to the ends of bundles of fibers are being designed and fabricated. Light emerging from a fiber bundle can be focused and a tiny spot can be imaged. In some cases the spot can be scanned, allowing quality imaging of a region inside the body. Special minute optical filters inserted at the end of the fiber bundle have the capacity to image tens of microns below the surface without cutting the surface\u2014non-intrusive diagnostics. This is particularly useful for determining the extent of cancers in the stomach and bowel. <\/p>\n<p id=\"import-auto-id2833751\">Most telephone conversations and Internet communications are now carried by laser signals along optical fibers. Extensive optical fiber cables have been placed on the ocean floor and underground to enable optical communications. Optical fiber communication systems offer several advantages over electrical (copper) based systems, particularly for long distances. The fibers can be made so transparent that light can travel many kilometers before it becomes dim enough to require amplification\u2014much superior to copper conductors. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">low loss<\/em><\/em>. Lasers emit light with characteristics that allow far more conversations in one fiber than are possible with electric signals on a single conductor. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">high bandwidth<\/em><\/em>. Optical signals in one fiber do not produce undesirable effects in other adjacent fibers. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">reduced crosstalk<\/em><\/em>. We shall explore the unique characteristics of laser radiation in a later chapter.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1451652\">\n<h1 data-type=\"title\">Corner Reflectors and Diamonds<\/h1>\n<p id=\"import-auto-id2998544\">A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object, shown in <a href=\"\/contents\/60b4727b-829e-4ea7-9238-9140b6a1b20c@4#import-auto-id2794931\" class=\"autogenerated-content\">(Figure)<\/a>, is called a <span data-type=\"term\">corner reflector<\/span>, since the light bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. It was more expensive for astronauts to place one on the moon. Laser signals can be bounced from that corner reflector to measure the gradually increasing distance to the moon with great precision.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1857674\">\n<div class=\"bc-figcaption figcaption\">(a) Astronauts placed a corner reflector on the moon to measure its gradually increasing orbital distance. (credit: NASA) (b) The bright spots on these bicycle safety reflectors are reflections of the flash of the camera that took this picture on a dark night. (credit: Julo, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1462279\" data-alt=\"Picture (a) shows the lunar expedition with the astronauts and their space shuttle. Picture (b) shows rectangular and round shaped bicycle reflectors.\"><img src=\"\/resources\/14f25ed4d62c8841eb2317798cba0e0f025cf0e0\/Figure 26_04_05.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"Picture (a) shows the lunar expedition with the astronauts and their space shuttle. Picture (b) shows rectangular and round shaped bicycle reflectors.\" width=\"200\"><\/span><\/p><\/div>\n<p id=\"import-auto-id1474266\">Corner reflectors are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a critical angle that is less than [latex]\\text{45\u00ba}[\/latex]. One use of these perfect mirrors is in binoculars, as shown in <a href=\"#import-auto-id2093405\" class=\"autogenerated-content\">(Figure)<\/a>. Another use is in periscopes found in submarines.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2093405\">\n<div class=\"bc-figcaption figcaption\">These binoculars employ corner reflectors with total internal reflection to get light to the observer\u2019s eyes.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1451652\" data-alt=\"The picture shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and suffers total internal reflection and then falls on the second prism and gets total internally reflected and emerges out through one of the eyepiece lenses.\"><img src=\"\/resources\/cf27ff6b5c481a7ea3ad2cb0fcc58796a1923164\/Figure 26_04_06.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The picture shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and suffers total internal reflection and then falls on the second prism and gets total internally reflected and emerges out through one of the eyepiece lenses.\" width=\"200\"><\/span><\/p><\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1388858\">\n<h1 data-type=\"title\">The Sparkle of Diamonds<\/h1>\n<p id=\"import-auto-id2844799\">Total internal reflection, coupled with a large index of refraction, explains why diamonds sparkle more than other materials. The critical angle for a diamond-to-air surface is only [latex]\\text{24}\\text{.}4\u00ba[\/latex], and so when light enters a diamond, it has trouble getting back out. (See <a href=\"#import-auto-id1280991\" class=\"autogenerated-content\">(Figure)<\/a>.) Although light freely enters the diamond, it can exit only if it makes an angle less than [latex]\\text{24}\\text{.}4\u00ba[\/latex]. Facets on diamonds are specifically intended to make this unlikely, so that the light can exit only in certain places. Good diamonds are very clear, so that the light makes many internal reflections and is concentrated at the few places it can exit\u2014hence the sparkle. (Zircon is a natural gemstone that has an exceptionally large index of refraction, but not as large as diamond, so it is not as highly prized. Cubic zirconia is manufactured and has an even higher index of refraction ([latex]\\approx 2.17[\/latex]), but still less than that of diamond.) The colors you see emerging from a sparkling diamond are not due to the diamond\u2019s color, which is usually nearly colorless. Those colors result from dispersion, the topic of <a href=\"\/contents\/c221d1fc-6368-440d-9d75-00f45fc0570d@5\">Dispersion: The Rainbow and Prisms<\/a>. Colored diamonds get their color from structural defects of the crystal lattice and the inclusion of minute quantities of graphite and other materials. The Argyle Mine in Western Australia produces around 90% of the world\u2019s pink, red, champagne, and cognac diamonds, while around 50% of the world\u2019s clear diamonds come from central and southern Africa.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1280991\">\n<div class=\"bc-figcaption figcaption\">Light cannot easily escape a diamond, because its critical angle with air is so small. Most reflections are total, and the facets are placed so that light can exit only in particular ways\u2014thus concentrating the light and making the diamond sparkle.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1234997\" data-alt=\"A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected, and this reflected ray further undergoes multiple reflections when it falls on other faces.\"><img src=\"\/resources\/2732debc83d0ee14fd680aa185e2ec6eac02b9d0\/Figure 26_04_07.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected, and this reflected ray further undergoes multiple reflections when it falls on other faces.\" width=\"225\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Bending Light<\/div>\n<p id=\"eip-id1530089\">Explore bending of light between two media with different indices of refraction. See how changing from air to water to glass changes the bending angle. Play with prisms of different shapes and make rainbows.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2602263\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d3f7e1460b9b084fa90cc0701e04d79ad0cb77bc\/bending-light_en.jar\">Bending Light<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_26.4\" data-alt=\"\"><a href=\"\/resources\/d3f7e1460b9b084fa90cc0701e04d79ad0cb77bc\/bending-light_en.jar\" data-type=\"image\"><img src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\"><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p><\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3096337\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2637388\">\n<li>The incident angle that produces an angle of refraction of [latex]\\text{90\u00ba}[\/latex] is called critical angle.<\/li>\n<li>Total internal reflection is a phenomenon that occurs at the boundary between two mediums, such that if the incident angle in the first medium is greater than the critical angle, then all the light is reflected back into that medium.<\/li>\n<li>Fiber optics involves the transmission of light down fibers of plastic or glass, applying the principle of total internal reflection.<\/li>\n<li>Endoscopes are used to explore the body through various orifices or minor incisions, based on the transmission of light through optical fibers.<\/li>\n<li>Cladding prevents light from being transmitted between fibers in a bundle.<\/li>\n<li>Diamonds sparkle due to total internal reflection coupled with a large index of refraction.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1290850\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1279267\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1485785\">\n<p id=\"import-auto-id1242828\">A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible when submerged. Can it be a diamond? Explain. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1217783\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id895923\">\n<p id=\"import-auto-id3135909\">A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1333608\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1486462\">\n<p id=\"import-auto-id2732165\">Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices of refraction and angles, perhaps referring to <a href=\"#import-auto-id2981932\" class=\"autogenerated-content\">(Figure)<\/a>. Some of us have seen the formation of a double rainbow. Is it physically possible to observe a triple rainbow? <\/p>\n<p id=\"import-auto-id1703371\">\n<\/p><div class=\"bc-figure figure\" id=\"import-auto-id2981932\">\n<div class=\"bc-figcaption figcaption\">Double rainbows are not a very common observance. (credit: InvictusOU812, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2268054\" data-alt=\"A double rainbow with spectacular bands of seven colors.\"><img src=\"\/resources\/27d96cec65a1187c6be048d47d54b2a8a49f5d39\/Figure 26_04_08.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"A double rainbow with spectacular bands of seven colors.\" width=\"325\"><\/span><\/p><\/div>\n<p><strong data-effect=\"bold\"><\/strong><\/p><\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3100229\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1228243\">\n<p id=\"import-auto-id2834278\">The most common type of mirage is an illusion that light from faraway objects is reflected by a pool of water that is not really there. Mirages are generally observed in deserts, when there is a hot layer of air near the ground. Given that the refractive index of air is lower for air at higher temperatures, explain how mirages can be formed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2853826\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1388837\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2870264\">\n<p id=\"import-auto-id1220875\">Verify that the critical angle for light going from water to air is [latex]\\text{48.6\u00ba}[\/latex], as discussed at the end of <a href=\"#fs-id2994315\" class=\"autogenerated-content\">(Figure)<\/a>, regarding the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1753678\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1203648\">\n<p id=\"import-auto-id1169466\">(a) At the end of  <a href=\"#fs-id2994315\" class=\"autogenerated-content\">(Figure)<\/a>, it was stated that the critical angle for light going from diamond to air is [latex]\\text{24}\\text{.}4\u00ba[\/latex]. Verify this. (b) What is the critical angle for light going from zircon to air?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1857932\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1391172\">\n<p id=\"import-auto-id2939231\">An optical fiber uses flint glass clad with crown glass. What is the critical angle?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2822191\">\n<p id=\"import-auto-id1215625\">[latex]\\text{66}\\text{.}3\u00ba[\/latex]<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1486542\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1678854\">\n<p id=\"import-auto-id2015114\">At what minimum angle will you get total internal reflection of light traveling in water and reflected from ice?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2033416\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1062885\">\n<p id=\"import-auto-id1447722\">Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is [latex]\\text{45}\\text{.}0\u00ba[\/latex], what must be the minimum index of refraction of the material from which the reflector is made? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3099790\">\n[latex]&gt;1\\text{.}\\text{414}[\/latex]\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3148372\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3193368\">\n<p id=\"import-auto-id1515789\">You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of [latex]\\text{68}\\text{.}4\u00ba[\/latex] when submerged in water? What is the substance, based on <a href=\"\/contents\/18eef263-8513-4954-bcc8-07aa263f0a50@7#eip-69\" class=\"autogenerated-content\">(Figure)<\/a>? (b) What would the critical angle be for this substance in air? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1826801\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3087862\">\n<p id=\"import-auto-id3041680\">A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown in <a href=\"#import-auto-id1827588\" class=\"autogenerated-content\">(Figure)<\/a>. What is the index of refraction for the liquid and its likely identification?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1827588\">\n<div class=\"bc-figcaption figcaption\">A light ray inside a liquid strikes the surface at the critical angle and undergoes total internal reflection.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2192306\" data-alt=\"A light ray travels from an object placed in a denser medium n1 at 15.0 centimeter from the boundary and on hitting the boundary gets totally internally reflected with theta c as critical angle. The horizontal distance between the object and the point of incidence is 13.4 centimeters.\"><img src=\"\/resources\/92257714264be61fb25bc1cd9a84f0f2f741938d\/Figure 26_04_09.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"A light ray travels from an object placed in a denser medium n1 at 15.0 centimeter from the boundary and on hitting the boundary gets totally internally reflected with theta c as critical angle. The horizontal distance between the object and the point of incidence is 13.4 centimeters.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id769217\">\n<p id=\"import-auto-id1338529\">1.50, benzene<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1234232\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2760474\">\n<p id=\"import-auto-id1449204\">A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown in <a href=\"#import-auto-id1338159\" class=\"autogenerated-content\">(Figure)<\/a>. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1338159\">\n<div class=\"bc-figcaption figcaption\">A light ray enters the end of a fiber, the surface of which is perpendicular to its sides. Examine the conditions under which it may be totally internally reflected.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2031974\" data-alt=\"The figure shows light traveling from n1 to n2 is incident on a rectangular transparent object at an angle of incidence theta 1. The angle of refraction is theta 2. On refraction, the ray falls onto the long side and gets totally internally reflected with theta 3 as the angle of incidence.\"><img src=\"\/resources\/b91ba49c56280d5625a0526efb0c6ab62568f9bd\/Figure 26_04_10.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The figure shows light traveling from n1 to n2 is incident on a rectangular transparent object at an angle of incidence theta 1. The angle of refraction is theta 2. On refraction, the ray falls onto the long side and gets totally internally reflected with theta 3 as the angle of incidence.\" width=\"250\"><\/span><\/p><\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1133862\">\n<dt>critical angle<\/dt>\n<dd id=\"fs-id2986456\">incident angle that produces an angle of refraction of  [latex]\\text{90\u00ba}[\/latex]<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2893618\">\n<dt>fiber optics<\/dt>\n<dd id=\"fs-id2847537\">transmission of light down fibers of plastic or glass, applying the principle of total internal reflection<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2775308\">\n<dt>corner reflector<\/dt>\n<dd id=\"fs-id1426773\"> an object consisting of two mutually perpendicular reflecting surfaces, so that the light that enters is reflected back exactly parallel to the direction from which it came<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>zircon<\/dt>\n<dd id=\"fs-id2780383\"> natural gemstone with a large index of refraction<\/dd>\n<\/dl>\n<\/div>\n\n","rendered":"<div class=\"textbox learning-objectives\">\n<h3 itemprop=\"educationalUse\">Learning Objectives<\/h3>\n<ul>\n<li>Explain the phenomenon of total internal reflection.<\/li>\n<li>Describe the workings and uses of fiber optics.<\/li>\n<li>Analyze the reason for the sparkle of diamonds.<\/li>\n<\/ul>\n<\/div>\n<p id=\"import-auto-id1973618\">A good-quality mirror may reflect more than 90% of the light that falls on it, absorbing the rest. But it would be useful to have a mirror that reflects all of the light that falls on it. Interestingly, we can produce <em data-effect=\"italics\"><em data-effect=\"italics\">total reflection<\/em><\/em> using an aspect of <em data-effect=\"italics\"><em data-effect=\"italics\">refraction<\/em><\/em>.<\/p>\n<p id=\"import-auto-id1460702\">Consider what happens when a ray of light strikes the surface between two materials, such as is shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(a). Part of the light crosses the boundary and is refracted; the rest is reflected. If, as shown in the figure, the index of refraction for the second medium is less than for the first, the ray bends away from the perpendicular. (Since <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-52b3c5132bf81daa4ab4e333416f8b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#62;&#123;&#110;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/em>, the angle of refraction is greater than the angle of incidence\u2014that is, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-415a4f62d5c49c329b1fcbec67c74624_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#62;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"55\" style=\"vertical-align: -4px;\" \/><\/em>.) Now imagine what happens as the incident angle is increased. This causes <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e852402f4009281864616033bb269c45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/em> to increase also. The largest the angle of refraction <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-e852402f4009281864616033bb269c45_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"15\" style=\"vertical-align: -3px;\" \/><\/em>can be is <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/em>, as shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(b).The <span data-type=\"term\" id=\"import-auto-id1389437\">critical angle<\/span><em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/><\/em> for a combination of materials is defined to be the incident angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-59e926cba2867871cf3ac57fce328409_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> that produces an angle of refraction of <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/em>. That is, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/><\/em> is the incident angle for which <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a61eea893c855d7256370e7178b429f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\" \/><\/em>. If the incident angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-59e926cba2867871cf3ac57fce328409_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/> is greater than the critical angle, as shown in <a href=\"#import-auto-id1435341\" class=\"autogenerated-content\">(Figure)<\/a>(c), then all of the light is reflected back into medium 1, a condition called <span data-type=\"term\" id=\"import-auto-id1340457\">total internal reflection<\/span>.<\/em><\/p>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id2736712\" data-label=\"\">\n<div class=\"bc-section section\" data-depth=\"1\">\n<h1 data-type=\"title\">Critical Angle<\/h1>\n<p id=\"import-auto-id1319605\">The incident angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-59e926cba2867871cf3ac57fce328409_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"14\" style=\"vertical-align: -4px;\" \/><em data-effect=\"italics\"> that produces an angle of refraction of <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/em> is called the critical angle, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/>. <\/em><\/p>\n<\/div>\n<\/div>\n<div class=\"bc-figure figure\" id=\"import-auto-id1435341\">\n<div class=\"bc-figcaption figcaption\">(a) A ray of light crosses a boundary where the speed of light increases and the index of refraction decreases. That is, <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-c2a33125ebba0202309d91eff330fdbf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#60;&#123;&#110;&#125;&#95;&#123;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"59\" style=\"vertical-align: -4px;\" \/><\/em>. The ray bends away from the perpendicular. (b) The critical angle <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/><\/em> is the one for which the angle of refraction is  . (c) Total internal reflection occurs when the incident angle is greater than the critical angle.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id3201448\" data-alt=\"In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1.\"><img decoding=\"async\" src=\"\/resources\/634f5b5a7c4f3dea344af4ef3272c80ca4064ca4\/Figure 26_04_01.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"In the first figure, an incident ray at an angle theta 1 with a perpendicular line drawn at the point of incidence travels from n1 to n2. The incident ray suffers both refraction and reflection. The angle of refraction is theta 2. In the second figure, as theta 1 is increased, the angle of refraction theta 2 becomes 90 degrees and the angle of reflection corresponding to 90 degrees is theta c. In the third figure, theta c greater than theta i, total internal reflection takes place and instead of refraction, reflection takes place and the light ray travels back into medium n1.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id3038479\">Snell\u2019s law states the relationship between angles and indices of refraction. It is given by <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-054128432f3870b9e4befeafe7214d23_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"158\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1824435\">When the incident angle equals the critical angle (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f9c28ccb285eac96921ea15eb844cc84_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"54\" style=\"vertical-align: -4px;\" \/>), the angle of refraction is <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/em> (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a61eea893c855d7256370e7178b429f0_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#50;&#125;&#61;&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"58\" style=\"vertical-align: -3px;\" \/>). Noting that <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-6a8d59088e11159afdcf2a1e766f0bec_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#32;&#57;&#48;&ordm;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#61;&#49;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"67\" style=\"vertical-align: -1px;\" \/>, Snell\u2019s law in this case becomes<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-90444b53a2fb1e3ef8b198b6e4268457_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#49;&#125;&#61;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"111\" style=\"vertical-align: -4px;\" \/><\/div>\n<p id=\"import-auto-id1447668\">The critical angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/> for a given combination of materials is thus<\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-47004b196fe631d6cef9ac50b34db756_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#47;&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#102;&#111;&#114;&#125;&#92;&#112;&#104;&#97;&#110;&#116;&#111;&#109;&#123;&#92;&#114;&#117;&#108;&#101;&#123;&#48;&#46;&#50;&#53;&#101;&#109;&#125;&#123;&#48;&#101;&#120;&#125;&#125;&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#62;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"238\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id1882896\">Total internal reflection occurs for any incident angle greater than the critical angle <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/>, and it can only occur when the second medium has an index of refraction less than the first. Note the above equation is written for a light ray that travels in medium 1 and reflects from medium 2, as shown in the figure.<\/p>\n<div data-type=\"example\" class=\"textbox examples\" id=\"fs-id2994315\">\n<div data-type=\"title\" class=\"title\">How Big is the Critical Angle Here?<\/div>\n<p id=\"eip-id2771873\">What is the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air?<\/p>\n<p id=\"import-auto-id1449078\"><strong>Strategy<\/strong><\/p>\n<p id=\"import-auto-id3105089\">The index of refraction for polystyrene is found to be 1.49 in <a href=\"#import-auto-id1244947\" class=\"autogenerated-content\">(Figure)<\/a>, and the index of refraction of air can be taken to be 1.00, as before. Thus, the condition that the second medium (air) has an index of refraction less than the first (plastic) is satisfied, and the equation <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-68fcb9018d9794610bef0fae93ef2084_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#47;&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"140\" style=\"vertical-align: -5px;\" \/> can be used to find the critical angle <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d9fca074a8363712b97af1699b172376_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"14\" style=\"vertical-align: -3px;\" \/><\/em>. Here, then, <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a938ae31effd02de3dbc2427cfcfdf62_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"74\" style=\"vertical-align: -3px;\" \/> and <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-a95f5fa22915464ed97843a1cd74e153_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#61;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"16\" width=\"74\" style=\"vertical-align: -4px;\" \/>.<\/p>\n<p id=\"import-auto-id1329220\"><strong>Solution<\/strong><\/p>\n<p id=\"import-auto-id3137583\">The critical angle is given by <\/p>\n<div data-type=\"equation\" class=\"equation\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-638096fcaa814afaf0e9018bb94d949c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#123;&#110;&#125;&#95;&#123;&#50;&#125;&#47;&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"21\" width=\"148\" style=\"vertical-align: -5px;\" \/><\/div>\n<p id=\"import-auto-id2296664\">Substituting the identified values gives<\/p>\n<div data-type=\"equation\" class=\"equation\">\n<pre class=\"ql-errors\">*** QuickLaTeX cannot compile formula:\n&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;&#95;&#123;&#99;&#125;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#48;&#48;&#125;&#47;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#57;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#61;&#123;&#92;&#116;&#101;&#120;&#116;&#123;&#115;&#105;&#110;&#125;&#125;&#94;&#123;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#50;&#46;&#50;&ordm;&#46;&#125;&#92;&#101;&#110;&#100;&#123;&#97;&#114;&#114;&#97;&#121;&#125;\n\n*** Error message:\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#35;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#32;&#105;&#110;&#32;&#97;&#108;&#105;&#103;&#110;&#109;&#101;&#110;&#116;&#32;&#112;&#114;&#101;&#97;&#109;&#98;&#108;&#101;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#36;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#36;&#92;&#98;&#101;&#103;&#105;&#110;&#123;&#97;&#114;&#114;&#97;&#121;&#125;&#123;&#125;&#123;&#92;&#116;&#104;&#101;&#116;&#97;&#32;&#125;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#69;&#120;&#116;&#114;&#97;&#32;&#125;&#44;&#32;&#111;&#114;&#32;&#102;&#111;&#114;&#103;&#111;&#116;&#116;&#101;&#110;&#32;&#36;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n&#77;&#105;&#115;&#115;&#105;&#110;&#103;&#32;&#125;&#32;&#105;&#110;&#115;&#101;&#114;&#116;&#101;&#100;&#46;\r\n&#108;&#101;&#97;&#100;&#105;&#110;&#103;&#32;&#116;&#101;&#120;&#116;&#58;&#32;&#46;&#46;&#46;&#45;&#49;&#125;&#92;&#108;&#101;&#102;&#116;&#40;&#48;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#55;&#49;&#125;&#92;&#114;&#105;&#103;&#104;&#116;&#41;&#92;&#92;&#32;&#92;&#116;&#101;&#120;&#116;\r\n\n<\/pre>\n<\/div>\n<p id=\"import-auto-id2774473\"><strong>Discussion<\/strong><\/p>\n<p id=\"import-auto-id852321\">This means that any ray of light inside the plastic that strikes the surface at an angle greater than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-d5ed38e3fcb28386ede06cac3c8e165c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#50;&#46;&#50;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"31\" style=\"vertical-align: -1px;\" \/> will be totally reflected. This will make the inside surface of the clear plastic a perfect mirror for such rays without any need for the silvering used on common mirrors. Different combinations of materials have different critical angles, but any combination with <em data-effect=\"italics\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-52b3c5132bf81daa4ab4e333416f8b4c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#123;&#110;&#125;&#95;&#123;&#49;&#125;&#62;&#123;&#110;&#125;&#95;&#123;&#50;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"15\" width=\"60\" style=\"vertical-align: -4px;\" \/><\/em> can produce total internal reflection. The same calculation as made here shows that the critical angle for a ray going from water to air is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-172c64203deb82c61cf79cd1ae021fcf_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#54;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>, while that from diamond to air is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5b70de700831a7e480f53dbf8892e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>, and that from flint glass to crown glass is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee164ea374df43d5c20d2f8b619eecc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/>. There is no total reflection for rays going in the other direction\u2014for example, from air to water\u2014since the condition that the second medium must have a smaller index of refraction is not satisfied. A number of interesting applications of total internal reflection follow.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id2941186\">\n<h1 data-type=\"title\">Fiber Optics: Endoscopes to Telephones<\/h1>\n<p>Fiber optics is one application of total internal reflection that is in wide use. In communications, it is used to transmit telephone, internet, and cable TV signals. <span data-type=\"term\" id=\"import-auto-id2941186\">Fiber optics<\/span> employs the transmission of light down fibers of plastic or glass. Because the fibers are thin, light entering one is likely to strike the inside surface at an angle greater than the critical angle and, thus, be totally reflected (See <a href=\"#import-auto-id1244947\" class=\"autogenerated-content\">(Figure)<\/a>.) The index of refraction outside the fiber must be smaller than inside, a condition that is easily satisfied by coating the outside of the fiber with a material having an appropriate refractive index. In fact, most fibers have a varying refractive index to allow more light to be guided along the fiber through total internal refraction. Rays are reflected around corners as shown, making the fibers into tiny light pipes.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1244947\">\n<div class=\"bc-figcaption figcaption\">Light entering a thin fiber may strike the inside surface at large or grazing angles and is completely reflected if these angles exceed the critical angle. Such rays continue down the fiber, even following it around corners, since the angles of reflection and incidence remain large.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1500825\" data-alt=\"Light ray enters an S-shaped tube and undergoes multiple reflections, finally emerging through the other end.\"><img decoding=\"async\" src=\"\/resources\/e7180682af9dada10106ba9ed7d93c48251b6cfc\/Figure 26_04_02.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"Light ray enters an S-shaped tube and undergoes multiple reflections, finally emerging through the other end.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1856471\">Bundles of fibers can be used to transmit an image without a lens, as illustrated in <a href=\"#import-auto-id2976033\" class=\"autogenerated-content\">(Figure)<\/a>. The output of a device called an <span data-type=\"term\">endoscope<\/span> is shown in <a href=\"#import-auto-id2976033\" class=\"autogenerated-content\">(Figure)<\/a>(b). Endoscopes are used to explore the body through various orifices or minor incisions. Light is transmitted down one fiber bundle to illuminate internal parts, and the reflected light is transmitted back out through another to be observed. Surgery can be performed, such as arthroscopic surgery on the knee joint, employing cutting tools attached to and observed with the endoscope. Samples can also be obtained, such as by lassoing an intestinal polyp for external examination. <\/p>\n<p id=\"import-auto-id1954488\">Fiber optics has revolutionized surgical techniques and observations within the body. There are a host of medical diagnostic and therapeutic uses. The flexibility of the fiber optic bundle allows it to navigate around difficult and small regions in the body, such as the intestines, the heart, blood vessels, and joints. Transmission of an intense laser beam to burn away obstructing plaques in major arteries as well as delivering light to activate chemotherapy drugs are becoming commonplace. Optical fibers have in fact enabled microsurgery and remote surgery where the incisions are small and the surgeon\u2019s fingers do not need to touch the diseased tissue.<\/p>\n<p id=\"import-auto-id1709277\">\n<div class=\"bc-figure figure\" id=\"import-auto-id2976033\">\n<div class=\"bc-figcaption figcaption\">(a) An image is transmitted by a bundle of fibers that have fixed neighbors. (b) An endoscope is used to probe the body, both transmitting light to the interior and returning an image such as the one shown. (credit: Med_Chaos, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2889912\" data-alt=\"Picture (a) shows how an image A is transmitted through a bundle of parallel fibers. Picture (b) shows an endoscope image.\"><img decoding=\"async\" src=\"\/resources\/fef311450b954ae2967617571d697888b14385bb\/Figure 26_04_03.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"Picture (a) shows how an image A is transmitted through a bundle of parallel fibers. Picture (b) shows an endoscope image.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1500855\">Fibers in bundles are surrounded by a cladding material that has a lower index of refraction than the core. (See <a href=\"#import-auto-id1447642\" class=\"autogenerated-content\">(Figure)<\/a>.) The cladding prevents light from being transmitted between fibers in a bundle. Without cladding, light could pass between fibers in contact, since their indices of refraction are identical. Since no light gets into the cladding (there is total internal reflection back into the core), none can be transmitted between clad fibers that are in contact with one another. The cladding prevents light from escaping out of the fiber; instead most of the light is propagated along the length of the fiber, minimizing the loss of signal and ensuring that a quality image is formed at the other end. The cladding and an additional protective layer make optical fibers flexible and durable.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1447642\">\n<div class=\"bc-figcaption figcaption\">Fibers in bundles are clad by a material that has a lower index of refraction than the core to ensure total internal reflection, even when fibers are in contact with one another. This shows a single fiber with its cladding.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2296155\" data-alt=\"The image shows a bundle fiber with a medium of refractive index n sub 1 inside surrounded by a medium n sub 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core.\"><img decoding=\"async\" src=\"\/resources\/3c00c5635426db8fda047e75ab5bb7aeb8759273\/Figure 26_04_04.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The image shows a bundle fiber with a medium of refractive index n sub 1 inside surrounded by a medium n sub 2. Medium n sub 2 is made up of cladding material and n sub 1 is the core.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" id=\"fs-id1431124\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">Cladding<\/div>\n<p id=\"import-auto-id1495090\">The cladding prevents light from being transmitted between fibers in a bundle.<\/p>\n<\/div>\n<p id=\"import-auto-id1849890\">Special tiny lenses that can be attached to the ends of bundles of fibers are being designed and fabricated. Light emerging from a fiber bundle can be focused and a tiny spot can be imaged. In some cases the spot can be scanned, allowing quality imaging of a region inside the body. Special minute optical filters inserted at the end of the fiber bundle have the capacity to image tens of microns below the surface without cutting the surface\u2014non-intrusive diagnostics. This is particularly useful for determining the extent of cancers in the stomach and bowel. <\/p>\n<p id=\"import-auto-id2833751\">Most telephone conversations and Internet communications are now carried by laser signals along optical fibers. Extensive optical fiber cables have been placed on the ocean floor and underground to enable optical communications. Optical fiber communication systems offer several advantages over electrical (copper) based systems, particularly for long distances. The fibers can be made so transparent that light can travel many kilometers before it becomes dim enough to require amplification\u2014much superior to copper conductors. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">low loss<\/em><\/em>. Lasers emit light with characteristics that allow far more conversations in one fiber than are possible with electric signals on a single conductor. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">high bandwidth<\/em><\/em>. Optical signals in one fiber do not produce undesirable effects in other adjacent fibers. This property of optical fibers is called <em data-effect=\"italics\"><em data-effect=\"italics\">reduced crosstalk<\/em><\/em>. We shall explore the unique characteristics of laser radiation in a later chapter.<\/p>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1451652\">\n<h1 data-type=\"title\">Corner Reflectors and Diamonds<\/h1>\n<p id=\"import-auto-id2998544\">A light ray that strikes an object consisting of two mutually perpendicular reflecting surfaces is reflected back exactly parallel to the direction from which it came. This is true whenever the reflecting surfaces are perpendicular, and it is independent of the angle of incidence. Such an object, shown in <a href=\"\/contents\/60b4727b-829e-4ea7-9238-9140b6a1b20c@4#import-auto-id2794931\" class=\"autogenerated-content\">(Figure)<\/a>, is called a <span data-type=\"term\">corner reflector<\/span>, since the light bounces from its inside corner. Many inexpensive reflector buttons on bicycles, cars, and warning signs have corner reflectors designed to return light in the direction from which it originated. It was more expensive for astronauts to place one on the moon. Laser signals can be bounced from that corner reflector to measure the gradually increasing distance to the moon with great precision.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1857674\">\n<div class=\"bc-figcaption figcaption\">(a) Astronauts placed a corner reflector on the moon to measure its gradually increasing orbital distance. (credit: NASA) (b) The bright spots on these bicycle safety reflectors are reflections of the flash of the camera that took this picture on a dark night. (credit: Julo, Wikimedia Commons)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1462279\" data-alt=\"Picture (a) shows the lunar expedition with the astronauts and their space shuttle. Picture (b) shows rectangular and round shaped bicycle reflectors.\"><img decoding=\"async\" src=\"\/resources\/14f25ed4d62c8841eb2317798cba0e0f025cf0e0\/Figure 26_04_05.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"Picture (a) shows the lunar expedition with the astronauts and their space shuttle. Picture (b) shows rectangular and round shaped bicycle reflectors.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<p id=\"import-auto-id1474266\">Corner reflectors are perfectly efficient when the conditions for total internal reflection are satisfied. With common materials, it is easy to obtain a critical angle that is less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1142d1c44cfaf3459c45a3d6cc399899_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"17\" style=\"vertical-align: -1px;\" \/>. One use of these perfect mirrors is in binoculars, as shown in <a href=\"#import-auto-id2093405\" class=\"autogenerated-content\">(Figure)<\/a>. Another use is in periscopes found in submarines.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id2093405\">\n<div class=\"bc-figcaption figcaption\">These binoculars employ corner reflectors with total internal reflection to get light to the observer\u2019s eyes.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1451652\" data-alt=\"The picture shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and suffers total internal reflection and then falls on the second prism and gets total internally reflected and emerges out through one of the eyepiece lenses.\"><img decoding=\"async\" src=\"\/resources\/cf27ff6b5c481a7ea3ad2cb0fcc58796a1923164\/Figure 26_04_06.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The picture shows binoculars with prisms inside. The light through one of the object lenses enters through the first prism and suffers total internal reflection and then falls on the second prism and gets total internally reflected and emerges out through one of the eyepiece lenses.\" width=\"200\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"bc-section section\" data-depth=\"1\" id=\"fs-id1388858\">\n<h1 data-type=\"title\">The Sparkle of Diamonds<\/h1>\n<p id=\"import-auto-id2844799\">Total internal reflection, coupled with a large index of refraction, explains why diamonds sparkle more than other materials. The critical angle for a diamond-to-air surface is only <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5b70de700831a7e480f53dbf8892e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>, and so when light enters a diamond, it has trouble getting back out. (See <a href=\"#import-auto-id1280991\" class=\"autogenerated-content\">(Figure)<\/a>.) Although light freely enters the diamond, it can exit only if it makes an angle less than <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5b70de700831a7e480f53dbf8892e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>. Facets on diamonds are specifically intended to make this unlikely, so that the light can exit only in certain places. Good diamonds are very clear, so that the light makes many internal reflections and is concentrated at the few places it can exit\u2014hence the sparkle. (Zircon is a natural gemstone that has an exceptionally large index of refraction, but not as large as diamond, so it is not as highly prized. Cubic zirconia is manufactured and has an even higher index of refraction (<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1b029e48955934d722db395c270b797c_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#97;&#112;&#112;&#114;&#111;&#120;&#32;&#50;&#46;&#49;&#55;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"50\" style=\"vertical-align: -1px;\" \/>), but still less than that of diamond.) The colors you see emerging from a sparkling diamond are not due to the diamond\u2019s color, which is usually nearly colorless. Those colors result from dispersion, the topic of <a href=\"\/contents\/c221d1fc-6368-440d-9d75-00f45fc0570d@5\">Dispersion: The Rainbow and Prisms<\/a>. Colored diamonds get their color from structural defects of the crystal lattice and the inclusion of minute quantities of graphite and other materials. The Argyle Mine in Western Australia produces around 90% of the world\u2019s pink, red, champagne, and cognac diamonds, while around 50% of the world\u2019s clear diamonds come from central and southern Africa.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1280991\">\n<div class=\"bc-figcaption figcaption\">Light cannot easily escape a diamond, because its critical angle with air is so small. Most reflections are total, and the facets are placed so that light can exit only in particular ways\u2014thus concentrating the light and making the diamond sparkle.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id1234997\" data-alt=\"A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected, and this reflected ray further undergoes multiple reflections when it falls on other faces.\"><img decoding=\"async\" src=\"\/resources\/2732debc83d0ee14fd680aa185e2ec6eac02b9d0\/Figure 26_04_07.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"A light ray falls onto one of the faces of a diamond, gets refracted, falls on another face and gets totally internally reflected, and this reflected ray further undergoes multiple reflections when it falls on other faces.\" width=\"225\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"note\" class=\"note\" data-has-label=\"true\" data-label=\"\">\n<div data-type=\"title\" class=\"title\">PhET Explorations: Bending Light<\/div>\n<p id=\"eip-id1530089\">Explore bending of light between two media with different indices of refraction. See how changing from air to water to glass changes the bending angle. Play with prisms of different shapes and make rainbows.<\/p>\n<div class=\"bc-figure figure\" id=\"eip-id2602263\">\n<div class=\"bc-figcaption figcaption\"><a href=\"\/resources\/d3f7e1460b9b084fa90cc0701e04d79ad0cb77bc\/bending-light_en.jar\">Bending Light<\/a><\/div>\n<p><span data-type=\"media\" id=\"Phet_module_26.4\" data-alt=\"\"><a href=\"\/resources\/d3f7e1460b9b084fa90cc0701e04d79ad0cb77bc\/bending-light_en.jar\" data-type=\"image\"><img decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/clalonde\/wp-content\/uploads\/sites\/280\/2017\/10\/PhET_Icon.png\" data-media-type=\"image\/png\" alt=\"\" data-print=\"false\" width=\"450\" \/><\/a><span data-media-type=\"image\/png\" data-print=\"true\" data-src=\"\/resources\/075500ad9f71890a85fe3f7a4137ac08e2b7907c\/PhET_Icon.png\" data-type=\"image\"><\/span><\/span><\/p>\n<\/div>\n<\/div>\n<div class=\"section-summary\" data-depth=\"1\" id=\"fs-id3096337\">\n<h1 data-type=\"title\">Section Summary<\/h1>\n<ul id=\"import-auto-id2637388\">\n<li>The incident angle that produces an angle of refraction of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/> is called critical angle.<\/li>\n<li>Total internal reflection is a phenomenon that occurs at the boundary between two mediums, such that if the incident angle in the first medium is greater than the critical angle, then all the light is reflected back into that medium.<\/li>\n<li>Fiber optics involves the transmission of light down fibers of plastic or glass, applying the principle of total internal reflection.<\/li>\n<li>Endoscopes are used to explore the body through various orifices or minor incisions, based on the transmission of light through optical fibers.<\/li>\n<li>Cladding prevents light from being transmitted between fibers in a bundle.<\/li>\n<li>Diamonds sparkle due to total internal reflection coupled with a large index of refraction.<\/li>\n<\/ul>\n<\/div>\n<div class=\"conceptual-questions\" data-depth=\"1\" id=\"fs-id1290850\" data-element-type=\"conceptual-questions\">\n<h1 data-type=\"title\">Conceptual Questions<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1279267\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1485785\">\n<p id=\"import-auto-id1242828\">A ring with a colorless gemstone is dropped into water. The gemstone becomes invisible when submerged. Can it be a diamond? Explain. <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1217783\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id895923\">\n<p id=\"import-auto-id3135909\">A high-quality diamond may be quite clear and colorless, transmitting all visible wavelengths with little absorption. Explain how it can sparkle with flashes of brilliant color when illuminated by white light.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1333608\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1486462\">\n<p id=\"import-auto-id2732165\">Is it possible that total internal reflection plays a role in rainbows? Explain in terms of indices of refraction and angles, perhaps referring to <a href=\"#import-auto-id2981932\" class=\"autogenerated-content\">(Figure)<\/a>. Some of us have seen the formation of a double rainbow. Is it physically possible to observe a triple rainbow? <\/p>\n<p id=\"import-auto-id1703371\">\n<div class=\"bc-figure figure\" id=\"import-auto-id2981932\">\n<div class=\"bc-figcaption figcaption\">Double rainbows are not a very common observance. (credit: InvictusOU812, Flickr)<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2268054\" data-alt=\"A double rainbow with spectacular bands of seven colors.\"><img decoding=\"async\" src=\"\/resources\/27d96cec65a1187c6be048d47d54b2a8a49f5d39\/Figure 26_04_08.jpg#fixme#fixme\" data-media-type=\"image\/png\" alt=\"A double rainbow with spectacular bands of seven colors.\" width=\"325\" \/><\/span><\/p>\n<\/div>\n<p><strong data-effect=\"bold\"><\/strong><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3100229\" data-element-type=\"conceptual-questions\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1228243\">\n<p id=\"import-auto-id2834278\">The most common type of mirage is an illusion that light from faraway objects is reflected by a pool of water that is not really there. Mirages are generally observed in deserts, when there is a hot layer of air near the ground. Given that the refractive index of air is lower for air at higher temperatures, explain how mirages can be formed.<\/p>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"problems-exercises\" data-depth=\"1\" id=\"fs-id2853826\" data-element-type=\"problems-exercises\">\n<h1 data-type=\"title\">Problems &amp; Exercises<\/h1>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1388837\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2870264\">\n<p id=\"import-auto-id1220875\">Verify that the critical angle for light going from water to air is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-58debb1c4c6a5ec753e98141b313a532_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#56;&#46;&#54;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>, as discussed at the end of <a href=\"#fs-id2994315\" class=\"autogenerated-content\">(Figure)<\/a>, regarding the critical angle for light traveling in a polystyrene (a type of plastic) pipe surrounded by air.<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1753678\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1203648\">\n<p id=\"import-auto-id1169466\">(a) At the end of  <a href=\"#fs-id2994315\" class=\"autogenerated-content\">(Figure)<\/a>, it was stated that the critical angle for light going from diamond to air is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-f5b70de700831a7e480f53dbf8892e60_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#50;&#52;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/>. Verify this. (b) What is the critical angle for light going from zircon to air?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1857932\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1391172\">\n<p id=\"import-auto-id2939231\">An optical fiber uses flint glass clad with crown glass. What is the critical angle?<\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id2822191\">\n<p id=\"import-auto-id1215625\"><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-ee164ea374df43d5c20d2f8b619eecc7_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#54;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#51;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"32\" style=\"vertical-align: 0px;\" \/><\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1486542\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1678854\">\n<p id=\"import-auto-id2015114\">At what minimum angle will you get total internal reflection of light traveling in water and reflected from ice?<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id2033416\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id1062885\">\n<p id=\"import-auto-id1447722\">Suppose you are using total internal reflection to make an efficient corner reflector. If there is air outside and the incident angle is <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-1f34d419342e71f9399f081a37a735bd_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#53;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#48;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"14\" width=\"32\" style=\"vertical-align: -1px;\" \/>, what must be the minimum index of refraction of the material from which the reflector is made? <\/p>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id3099790\">\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-8af27444a33dce0d0f4b59fc4f347a6e_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#62;&#49;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#52;&#49;&#52;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"59\" style=\"vertical-align: -1px;\" \/>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id3148372\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3193368\">\n<p id=\"import-auto-id1515789\">You can determine the index of refraction of a substance by determining its critical angle. (a) What is the index of refraction of a substance that has a critical angle of <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-64f674d8d99b9135d032cb3aa9f2be9b_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#54;&#56;&#125;&#92;&#116;&#101;&#120;&#116;&#123;&#46;&#125;&#52;&ordm;\" title=\"Rendered by QuickLaTeX.com\" height=\"13\" width=\"32\" style=\"vertical-align: -1px;\" \/> when submerged in water? What is the substance, based on <a href=\"\/contents\/18eef263-8513-4954-bcc8-07aa263f0a50@7#eip-69\" class=\"autogenerated-content\">(Figure)<\/a>? (b) What would the critical angle be for this substance in air? <\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1826801\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id3087862\">\n<p id=\"import-auto-id3041680\">A ray of light, emitted beneath the surface of an unknown liquid with air above it, undergoes total internal reflection as shown in <a href=\"#import-auto-id1827588\" class=\"autogenerated-content\">(Figure)<\/a>. What is the index of refraction for the liquid and its likely identification?<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1827588\">\n<div class=\"bc-figcaption figcaption\">A light ray inside a liquid strikes the surface at the critical angle and undergoes total internal reflection.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2192306\" data-alt=\"A light ray travels from an object placed in a denser medium n1 at 15.0 centimeter from the boundary and on hitting the boundary gets totally internally reflected with theta c as critical angle. The horizontal distance between the object and the point of incidence is 13.4 centimeters.\"><img decoding=\"async\" src=\"\/resources\/92257714264be61fb25bc1cd9a84f0f2f741938d\/Figure 26_04_09.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"A light ray travels from an object placed in a denser medium n1 at 15.0 centimeter from the boundary and on hitting the boundary gets totally internally reflected with theta c as critical angle. The horizontal distance between the object and the point of incidence is 13.4 centimeters.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<div data-type=\"solution\" class=\"solution\" id=\"fs-id769217\">\n<p id=\"import-auto-id1338529\">1.50, benzene<\/p>\n<\/div>\n<\/div>\n<div data-type=\"exercise\" class=\"exercise\" id=\"fs-id1234232\" data-element-type=\"problem-exercises\">\n<div data-type=\"problem\" class=\"problem\" id=\"fs-id2760474\">\n<p id=\"import-auto-id1449204\">A light ray entering an optical fiber surrounded by air is first refracted and then reflected as shown in <a href=\"#import-auto-id1338159\" class=\"autogenerated-content\">(Figure)<\/a>. Show that if the fiber is made from crown glass, any incident ray will be totally internally reflected.<\/p>\n<div class=\"bc-figure figure\" id=\"import-auto-id1338159\">\n<div class=\"bc-figcaption figcaption\">A light ray enters the end of a fiber, the surface of which is perpendicular to its sides. Examine the conditions under which it may be totally internally reflected.<\/div>\n<p><span data-type=\"media\" id=\"import-auto-id2031974\" data-alt=\"The figure shows light traveling from n1 to n2 is incident on a rectangular transparent object at an angle of incidence theta 1. The angle of refraction is theta 2. On refraction, the ray falls onto the long side and gets totally internally reflected with theta 3 as the angle of incidence.\"><img decoding=\"async\" src=\"\/resources\/b91ba49c56280d5625a0526efb0c6ab62568f9bd\/Figure 26_04_10.jpg#fixme#fixme\" data-media-type=\"image\/jpg\" alt=\"The figure shows light traveling from n1 to n2 is incident on a rectangular transparent object at an angle of incidence theta 1. The angle of refraction is theta 2. On refraction, the ray falls onto the long side and gets totally internally reflected with theta 3 as the angle of incidence.\" width=\"250\" \/><\/span><\/p>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div data-type=\"glossary\" class=\"textbox shaded\">\n<h2 data-type=\"glossary-title\">Glossary<\/h2>\n<dl class=\"definition\" id=\"import-auto-id1133862\">\n<dt>critical angle<\/dt>\n<dd id=\"fs-id2986456\">incident angle that produces an angle of refraction of  <img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-content\/ql-cache\/quicklatex.com-0ef094c705f55f76b4993ff72af9e73f_l3.png\" class=\"ql-img-inline-formula quicklatex-auto-format\" alt=\"&#92;&#116;&#101;&#120;&#116;&#123;&#57;&#48;&ordm;&#125;\" title=\"Rendered by QuickLaTeX.com\" height=\"12\" width=\"18\" style=\"vertical-align: 0px;\" \/><\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2893618\">\n<dt>fiber optics<\/dt>\n<dd id=\"fs-id2847537\">transmission of light down fibers of plastic or glass, applying the principle of total internal reflection<\/dd>\n<\/dl>\n<dl class=\"definition\" id=\"import-auto-id2775308\">\n<dt>corner reflector<\/dt>\n<dd id=\"fs-id1426773\"> an object consisting of two mutually perpendicular reflecting surfaces, so that the light that enters is reflected back exactly parallel to the direction from which it came<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>zircon<\/dt>\n<dd id=\"fs-id2780383\"> natural gemstone with a large index of refraction<\/dd>\n<\/dl>\n<\/div>\n","protected":false},"author":211,"menu_order":1,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":[],"pb_section_license":"all-rights-reserved"},"chapter-type":[],"contributor":[],"license":[56],"class_list":["post-1394","chapter","type-chapter","status-publish","hentry","license-all-rights-reserved"],"part":1385,"_links":{"self":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1394","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/users\/211"}],"version-history":[{"count":1,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1394\/revisions"}],"predecessor-version":[{"id":1395,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1394\/revisions\/1395"}],"part":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/parts\/1385"}],"metadata":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapters\/1394\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/media?parent=1394"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/pressbooks\/v2\/chapter-type?post=1394"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/contributor?post=1394"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/pressbooks.bccampus.ca\/ubcbatessandbox\/wp-json\/wp\/v2\/license?post=1394"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}