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Assignment 11- Solutions 1) Assignment 11- Solutions 1) The picture above shows a source of light (the red dot), with the light shining on the origin, which is marked by the position of the green dot. The horizontal axis is the interface separating one medium from another. Some of the light reflects back into the first medium, and some refracts into the second medium. Choose the three correct statements about this situation from the list below. The first medium (where the source is) has a higher index of refraction than the second medium. FALSE- since The angle inside the blue region is smaller than outside ͢'0 Ƙ ͢*0/ The first medium has the same index of refraction as the second medium. FALSE- same. The first medium has a lower index of refraction than the second medium. TRUE-same reason. If the index of refraction of the second medium was increased, the angle of the reflected beam would change. FALSE- for the reflected wave the interface has acted like a plane mirror , it is reflected at the same angle, regardless of the second medium ’s refraction index. Changing the index of refraction of the second medium would have no effect on the angle of the reflected beam. TRUE- as the previous one, interfaces acts like a mirror. If the light source was moved to the position of the purple dot in the second medium, and the light was incident on the origin again, the light would definitely experience total internal reflection. FALSE- Snell’s law doesn’t change if we reverse the direction of the light rays. As long as the angle in the blue region is the same we have the same equation : ͢'0 sin '0 Ɣ ͢*0/ sin *0/ And so the outgoing ray will be at the same angle. The only difference is that we won’t have the reflected ray outside (instead we get one inside.) If the light source was moved to the position of the purple dot in the second medium, and the light was incident on the origin again, some light would refract out into the upper medium, traveling toward the position of the red dot. TRUE- described above. If the light source was moved to the position of the purple dot in the second medium, and the light was incident on the origin again, some light would refract out into the upper medium, but the light would not travel toward the position of the red dot. FALSE 2) The picture above shows a source of light in the lower medium (the purple dot), with the light shining on the origin, which is marked by the position of the green dot. The horizontal axis is the interface separating one medium from another. In this case, all of the light reflects back into the lower medium. Choose all the correct statements about this situation from the list below. The lower medium (where the source is) has a larger index of refraction than the upper medium. TRUE- We are observing total internal reflection while going from the lower medium to the upper one. This means that there exists a critical angle beyond which this happens and critical angles are only for going from larger index of refraction to smaller . So lower medium has larger index. The lower medium has the same index of refraction as the upper medium. FALSE- read above The lower medium has a smaller index of refraction than the upper medium. FALSE- read above The angle of incidence in this case is greater than the critical angle for total internal reflection to occur. TRUE- read above. The angle of incidence in this case is smaller than the critical angle for total internal reflection to occur. FALSE- read above. If the light source was moved to the position of the red dot in the upper medium, and the light was incident on the origin again, the light would definitely experience total internal reflection. FALSE- there won’t be total internal reflection in going from lower index to higher and, as we found before, the lower medium has higher index of refraction. If the light source was moved to the position of the red dot in the upper medium, and the light was incident on the origin again, some light would refract into the lower medium, traveling toward the position of the purple dot. FALSE- Some light would indeed refract into the lower medium, but it won’t be towards the purple dot . This we can see from Snell’s law ( blue for the lower medium): ͥ͢ sin ͥ Ɣ ͢'0 sin '0 But we know that for the in the picture no can be found ( is beyond the critical '0 ͥ '0 angle), and that is why it gets reflected. So if we put some in the equation, the that we ͥ '0 get will never be the angle in the diagram. If the light source was moved to the position of the red dot in the upper medium, and the light was incident on the origin again, some light would refract into the lower medium, but the light would not travel toward the position of the purple dot. TRUE- see above. 3) This picture shows a light source at point A, at the top left of the picture. The light which refracts into the lower medium eventually reaches point B, at the bottom right of the picture. Point A is located at x = 0 m, y = +5 m, and point B is located at x = +10 m, y = -5 m. The upper medium is air, with an index of refraction of n = 1.00. The interface separating the two media lies along the line y = 0. Take the speed of light in vacuum to be 3.00 x 10 8 m/s. An interesting fact about light is that the path it takes from A to B always minimizes the total light travel time. ̻ In other words, any other path from A to B, including the shortest-distance path, would take the light more ͥ time. One can show that this is equivalent to Snell's law. In the case shown in the picture, the time it takes the light to travel from point A to the green point, on the ͥ ͉ ̼ interface, is 26.0 ns. By using Snell's law and considering the geometry, answer the following. (a) What is the x-coordinate of the green point? ͦ We can find the distance between A and the green spot ͦ using : ͘ Ɣ ͪͨ ͬ ͡ ̻͉ Ɣ ͪͨ Ɣ 3 Ɛ 10 Ɛ 26.0 ns Ɣ 7.8 m ͧ Now this is the hypotenuse of the upper triangle. For we thus have: ͥ 5 ͦ cos ͥ Ɣ ƴ sin ͥ Ɣ ǭ1 Ǝ cos ͥ Ɣ 0.768 7.8 The horizontal position of the green dot O is therefore: ƴ ͬ Ɣ 7.8 m Ɛ sin ͥ Ɣ 5.99 m x = 5.99 m. (b) What is the index of refraction of the second medium? To figure out the index we need to use Snell’s law and for that we need the sine of the angles which is the ratio of opposite over hypotenuse and so: ͬ ͦ͢ sin ͥ Ɣ Ɣ ̻͉ ͥ͢ sin ͦ 10 Ǝ ͬ ͉̼ We first need to find OB, which is the hypotenuse of the lower triangle: ͦ ͦ ͉̼ Ɣ ǭ5 ƍ ʚ10 Ǝ ͬʛ Ɣ 6.41 m ͦ͢ Ɣ 1.23 (c) What is the total time, in nanoseconds, it takes the light to travel from point A to point B? ͉̼ ͉̼ 6.41 m Ɛ 1.23 ͨͦ Ɣ Ɣ Ɣ Ɣ 26.0 ͧ͢ ͪͦ ͗/ͦ͢ ͬ ͡ 3 Ɛ 10 ͧ Which is the same as the time in the first region (this is not a coincidence, if the vertical distance is the same the times are the same.) So the total time will be . 2 Ɛ 26 ͧ͢ Ɣ 52.2 ͧ͢ 4) A lamp emitting red light is placed at the bottom of a tank of liquid with depth H = 18.0 cm and index of refraction n = 1.56 (for red light). Above the liquid is air, which has an index of refraction of 1.00. On the surface of the liquid above the light, an observer sees a bright circle with radius R. (a) Why does the observer see a bright circle on the surface? ͌ same as the Since the lamp is the region of higher index of refraction, not all of its light-rays will be able to escape into air. The ones that hit the interface at an angle larger ͂ than the critical angle will undergo total internal reflection and beyond that point the observer outside ͆ doesn’t see light rays coming out. The angle depends on the horizontal distance on the interface from the point right above the lamp. Because the lamp is circular. False. Because the pupils in the observer's eyes are circular. False. Because light striking the surface outside of the circle experiences total internal reflection. True. (b) Calculate the radius of the bright circle seen by the observer. The outermost ray that can come out is at the critical angle, since beyond that the rays will reflect. First, the ray going at makes the hypotenuse : ͆ ͆ Ɣ ǭ͌ͦ ƍ ͂ͦ The radius R of the bright circle on the interface is related to by: ͌ ͌ sin Ɣ Ɣ ͆ √͌ͦ ƍ ͂ͦ But also: 1 1 sin Ɣ Ɣ Ɣ 0.641 ͢ 1.56 1 ͌ Ɣ ƴ ǭ͌ͦ ƍ ͂ͦ Ɣ ͌͢ ͢ √͌ͦ ƍ ͂ͦ ͌ͦ ƍ ͂ͦ Ɣ ͦ͌ͦ͢ ͂ͦ Ɣ ʚͦ͢ Ǝ 1ʛ͌ͦ ͂ 18.0 cm ƴ ͌ Ɣ Ɣ Ɣ 15.0 cm √ͦ͢ Ǝ 1 √1.56 ͦ Ǝ 1 (c) From the list below, choose the two correct statements.
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