Experiment 15- Geometric Optics

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Feb 20, 2024

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Report for Experiment #15 Geometric Optics TA: June 23, 2023

Abstract A light source was set up to produce five emerging rays on top of white paper for the initial investigation. Then, a concave mirror was positioned so that it reflected all 5 of the beams. The focal length was determined after tracing each of these rays, even those that crossed the mirror's edge. The convex mirror, concave, and convex lenses (using refracted rays) underwent the same procedure. The focal lengths were determined to be 5.6, 6.8, 12.4, and 12.1 cm, with an error of 0.3 cm for each. Only the concave mirror did not fall within the range of uncertainty of the values provided when compared to their nominal values. For the second experiment, a light source was set up to emit 1 beam that traveled through an acrylic trapezoid and landed on top of a white piece of paper. The refracting ray was located by tracing the incident and departing rays and connecting them inside the trapezoid contour. These were used to determine an acrylic block's index of refraction. The four values of n were used to obtain an average acrylic index of refraction of 1.471 after this was repeated three more times. This was within the range of uncertainty of the known value 1.49. For the third investigation, the focal length and magnification of the lens were determined by positioning an object, a 100 mm lens, and a white metal screen onto an optic rail at precise locations. The lens was moved until the image was sharp, focused, and magnified. The object was positioned at 10 cm and the screen at 110 cm. h o and h i were measured, as well as where the lens was. This was repeated six times, each time bringing the screen in 10 cm closer. These data were used to determine d o and d i , which were then used to produce two graphs: one showing the lens's focal length and the other its magnification. The found focal length of 10.7 cm was outside the range of the previously reported value of 10 cm. The magnification -1.011 fell within the known value -1's range. Introduction In today's world, there are many instances of optical devices that we use on a daily basis, such as contact lenses for vision correction, microscopic lenses to concentrate laser beams, and cameras that can employ both lenses and mirrors. Without understanding how light moves and interacts with mirrors and lenses, none of these would be conceivable. This field of research, known as geometric optics, makes the assumption that most of the time light moves in straight lines, or rays. All angles are calculated according to the surface normal, which is the direction perpendicular to a surface. This holds true for both curved and flat surfaces. The intuitive law of reflection asserts that the angle of incidence 1 and angle of reflection 𝜃 ! must be identical for every surface that reflects some light, including mirrors. This law also applies to light beams that reflect off spherical or paraboloidal mirrors. The focal point (F) is the location through which all parallel rays that reflect off a paraboloidal mirror pass. Spherical mirrors, on the other hand, have many points that combine to generate a curved focus plane rather than a single focal point. By making sure that the spherical mirror's size is tiny relative to its radius, we can approximate the paraboloidal mirror with one. This allows us to roughly infer that a spherical mirror has a focal point. With this, we can use the following equation to get the focal length, which is defined as the separation between the mirror's face (O) and the focal point (F): 𝑟

𝑓 = 2 Concave and convex spherical mirrors are the two varieties. The definition of focal length is unchanged in a convex mirror. In a vacuum, light moves at a constant speed of about 3.00*10 8 m/s. It slows down as it passes through transparent materials. For each unique substance, the index of refraction (n) is the ratio of the two different speeds: 𝑛 = " , where c is the speed of light in a vacuum and v is its speed in the material # A portion of the light that enters a transparent material is reflected off of its surface. As the light passes through the medium, it begins to bend. This is also referred to as refraction and alters the direction of the light ray. It's vital to remember that light always bends in the direction of the normal when it enters a medium with a higher index of refraction, and away from the normal when it enters a media with a lower n. Snell's Law (the law of refraction) can be used to determine the angle of refraction: 𝑛 $ sin 𝜃 $ = 𝑛 ! sin 𝜃 ! where θ 1 is the angle of incidence and θ 2 is the angle of refraction and n 1 and n 2 are the respective index of refraction for the two mediums Unlike mirrors, which are opaque, lenses are clear and let light rays pass through. Converging and diverging lenses will each be the subject of a separate experiment. In convergent lenses (such as double convex lenses), parallel rays are bent in the direction of one another to form a focus point. Parallel rays are diverged using diverging lenses (such as twin concave lenses) so they never contact. To prevent any picture errors (aberrations), it is assumed that the concave and convex lenses are both spherical. The incident rays will be assumed to enter the lens at shallow angles close to its axis (also known as paraxial rays) and the lens thickness will be assumed to be tiny relative to the lens's radius. A lens, as opposed to a mirror, has two focal points: F and F'. This is helpful when locating an image that is presented on opposing sides of a lens using a ray diagram. A single point somewhere on the lens' focus plane is where parallel rays converge when they enter a double convex lens. Instead, if all of the rays come from one object point, they will all be focused to one image point on the other side of the lens. The lens equation can be used to illustrate the relationships between the object and image positions with the focal point: 1 1 1 + = 𝑑 % 𝑑 & 𝑓 The object size and the image size it corresponds to are typically different in situations like these. The ratio of the picture height to the object height is known as the magnification (M): 𝑀 = ℎ & = − 𝑑 & ℎ % 𝑑 %

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When light rays leave an object and travel through a lens to form an image on the other side, as is the case with convex lenses, a true picture is produced. On the other hand, concave lenses produce a virtual picture when the object's rays diverge after passing through the lens and the image can be traced back to the same side as the item. In this experiment, the focal lengths of concave mirrors and lenses, the index of refraction in an acrylic block, and the focal length and magnification of a 100 mm lens will all be determined using concave and convex mirrors and lenses, a transparent acrylic block, and an optical rail. Investigation 1 Ray tracing was employed in the initial experiment to determine the focal lengths of concave and convex mirrors and lenses. All of the room's lights were turned out before we began. The light source was attached to the power supply, which was plugged into an outlet. Five rays of light emanated from the light source after the plastic slide was moved to the left or right. The light source was at one edge of a white piece of paper that was placed on the table. In order to avoid leaving fingerprints on the mirror surface, the triangular mirror from the ray optics kit was carefully put on the paper with the open cavity facing the paper and the hole on top. The placement of the mirror was then marked out on the paper by tracing the incident and reflected rays using a ruler. When a region of intersection is drawn on paper using the reflected rays, it is because they all met at the focal point. The focus point was chosen to be in the center of this area, and the distance in cm from the mirror's center to this point was measured. The intersection's uncertainty was estimated by multiplying the intersection's length by two. The paper was then moved to a fresh section, and the convex mirror procedure was repeated. The corresponding focal lengths were then compared to the nominal values provided by the manufacturer, together with any associated errors. Then, a fresh white sheet of paper was taken and set, once more, near the border of the light source. To avoid getting fingerprints on the lens surface, the convex lens was carefully placed on the paper with the flat side down after being removed from the ray optics kit. The lens was then positioned so that all five rays traveled through it and were parallel to its axis. A ruler was used to sketch the contour of the lens as well as the incident and refracted rays. Following the trace, the focal point's uncertainty was estimated by taking the center point from the area of intersection. After that, the paper was moved to a fresh area, and the identical procedures were carried out while utilizing the concave lens. The corresponding focal lengths were then compared to the nominal values provided by the manufacturer, together with any associated errors. Table 1 Concave: Measurement Uncertainty Mirror Focal length (cm) 5.6 0.3 Nominal value(cm) 6.3 0.3

Lens Focal length (cm) 12.4 0.3 Nominal value(cm) 2.9 0.5 Table 2 Convex: Measurement Uncertainty Mirror Focal length (cm) 6.8 0.3 Nominal value(cm) 6.3 0.3 Lens Focal length (cm) 12.1 0.3 Nominal value(cm) 12.9 0.5 Investigation 2 In the second inquiry, ray tracing was employed to determine an experimental value for an acrylic block's index of refraction (n). The plastic slide on the light source was adjusted to generate a single beam after starting with the same setup as Investigation 1. A fresh piece of white paper was placed with the end towards the light source, and the acrylic block was positioned on top with the frosted side facing the paper. This gave us access to the trapezoid's inside. In order to ensure that the incidence ray angle was between 35° and 70°, the acrylic block was rotated until a single ray incident was upon side II of the trapezoid as seen below:

Fig. 1 In relation to either the surface or the normal, the incident angle was measured. A ruler was used to trace the incident ray, the ray that left the acrylic block, and the acrylic block's contour. The incident ray, exit ray, and two spots where these rays intersect the block were all attached to the block's contour using the ruler once more. The finished illustration resembled Fig. 1 above. Using a protractor, the incidence and refracted angles at surface II of the block were then measured with respect to the normal; the uncertainty was expressed as the smaller of the two measurements (half a degree). The following was then transformed into a radian angle. Following that, the index of refraction within the acrylic block was determined using the formula below, where n 1 (index of refraction in air) was assumed to be 1 (real value: 1.000293). In order to create a total of four indexes of refraction, these operations were then repeated for three other incident angles. Then, using the standard deviation as its uncertainty, the average of the four observations for the index of refraction was determined. The standard value of 1.49 was then used to compare the average index of refraction and its uncertainty. Table 3 Incident (°) Refracted (°) 𝛿 Degree (°) Incident (R) Refracted (R) n

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Trial 1 33 22 0.5 0.576 0.384 1.454 Trial 2 37 23 0.5 0.646 0.401 1.540 Trial 3 47 30 0.5 0.820 0.524 1.463 Trial 4 42 28 0.5 0.733 0.489 1.425 Table 4 n acrylic 1.471 𝛿 n 0.049 n theory 1.490 The computed index of refraction (n) for acrylic, which includes the theoretical index of refraction (n) for acrylic of 1.490, ranges from 1.422 to 1.520 when compared to the theoretical value. The fact that trail 1's incident angle was beyond the permitted range of 35° and 70° (33°) was one potential source of ambiguity in this inquiry. Investigation 3 In the third experiment, a lens, white screen, and item were used to determine the lens's magnification and focal length. Making sure that the object was facing the white metal screen, the light source was fitted in the stand/bracket that was mounted on an optical rail. With an error of half the lowest increment, the object's inner circle's circumference ho was measured and recorded (cm). When the object was situated at 10 cm ± 0.5 cm on the optic rail, the stand/bracket was shifted by loosening the screw at the bottom. The 100 mm lens was then positioned in the optical rail's channel between the subject and the display. Once a sharp, focused, and magnified image was visible on the screen, the lens was adjusted up and down the optical rail. Then, with an error of half the smallest increment (±0.5 cm), the positions of the item, lens, and screen were recorded. The inner circle of the image on the screen's diameter hi was measured with a ruler, with an error of ±0.5 cm. The preceding steps were then repeated after moving the screen 10 cm closer to the object. Up until a total of six data sets were gathered, this was repeated. The screen's distance from the optical rail was at least 60 cm. The object distance d o , the image distance d i , and their uncertainties were computed for each data set that was discovered. After that, the values d o d i and d o +d i were computed with their uncertainty. The focal length of the lens and its uncertainty were then calculated by plotting the graph of d o d i vs. d o +d i . Then, this focal length was contrasted with the standard focus length of 10 cm. In order to account for their uncertainties, the numbers h i /h o and d i /d oi were subsequently determined.

The magnification and associated uncertainty were then determined by plotting a second h i /h o versus d i /d o graph. The theoretical value for magnification of -1 was then compared to this. Table 5 Thin Lens Object Location (cm) Lens Location (cm) Screen Location (cm) h i (cm) 𝛿 distance (cm) d o (cm) 𝛿 d o (cm) d i (cm) 𝛿 d i (cm) Trial 1 10 22.2 110 -7.2 0.05 12.2 0.071 87.8 0.071 Trial 2 10 22.4 100 -6.3 0.05 12.4 0.071 77.6 0.071 Trial 3 10 22.7 90 -5.3 0.05 12.7 0.071 67.3 0.071 Trial 4 10 23.2 80 -4.3 0.05 13.2 0.071 56.8 0.071 Trial 5 10 23.9 70 -3.3 0.05 13.9 0.071 46.1 0.071 Trial 6 10 25.5 60 -2.2 0.05 15.5 0.071 34.5 0.071 Table 6 do*di (cm 2 ) 𝛿 (do*di) (cm 2 ) do + di (cm) 𝛿 (do + di) (cm) 1071.16 6.27 100 0.10 962.24 5.56 90 0.10 854.71 4.84 80 0.10 749.76 4.12 70 0.10 640.79 3.40 60 0.10 534.75 2.67 50 0.10 Table 7

di/do 𝛿 (di/do) hi/ho 𝛿 (hi/ho) 7.20 0.04 -7.2 0.364 6.26 0.04 -6.3 0.319 5.30 0.04 -5.3 0.270 4.30 0.04 -4.3 0.221 3.32 0.04 -3.3 0.172 2.2 0.04 -2.2 0.050 Figure 1: the focal length (cm) of the lens using results from Table 6 The slope of Graph 1 is 10.710 cm, and the uncertainty is 0.106. Although they are included, the error bars for do+di (cm) and do*di (cm) uncertainty are too small to be seen. The slope, which does not contain the theoretical value slope of 10 cm, has a range of 10.604 cm to 10.816 cm when compared to the theoretical values. This could be because some unknowns were left out of the computation. For instance, rounding in calculations involving smaller numbers may have piled up to produce a final figure that is the sum of all the tiny rounding errors, leading to a bigger final error. Additionally, because the lab is virtual in nature, it is possible that the uncertainties measured in calculations were underestimated, leading to a final number that differs from the theoretical one. Graph 2 shows magnification of the image through the lens onto the screen using results from Table 7 y = 10.718x - 1.6257 0 200 400 600 800 1000 1200 0 20 40 60 80 100 120 do+di (cm) Focal Length of Lens (cm)

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Graph 2 has a -1.011 slope and a 0.0441 uncertainty. The slope, when measured against the theoretical values, ranges from -0.967 to -1.055, including the theoretical value slope of -1.00. This supports the theory. Conclusion In conclusion, Investigation 1 revealed that the convex and concave mirrors and lenses had respective focal lengths of 5.6, 6.8, 12.4, and 12.1 cm, each with a 0.3 cm uncertainty. Only the concave mirror did not fall within the range of uncertainty of the values provided when compared to their nominal values. This can be partially the result of a manufacturing error (incorrect nominal values). This might also be the result of any handling-related fingerprints that landed on the mirror's surface. The determined index of refraction (n) for acrylic in Investigation 2 ranges from 1.422 to 1.520 when compared to the theoretical value, which includes the theoretical index of refraction (n) for acrylic of 1.490. The fact that trail 1's incident angle was beyond the permitted range of 35° and 70° (33°) was one potential source of ambiguity in this inquiry. Investigation 3's graph 15.1, which excludes the theoretical value slope of 10 cm, has a slope of 10.710 cm, an uncertainty of 0.106, and a range of 10.604 cm to 10.816 cm. The range of Graph 15.2 is from 0.967 to -1.055, which includes the theoretical value slope of -1.00, and the graph has a slope of -1.011 with an uncertainty of 0.0441. The fact that all of the investigations were conducted virtually was one of the main possible sources of uncertainty throughout the investigations that was not taken into consideration during this experiment. Because of this, certain measurement uncertainties were unclear, which could have led to an underestimating of uncertainties. The calculations themselves could be another source of error. Further calculations may have compounded rounding of some calculations, increasing the rounding error. y = - 1.0051x + 0.0211 -8 -7 -6 -5 -4 -3 -2 -1 0 0 1 2 3 4 5 6 7 8 di/do Magnification of Image

Questions 1. What does the white screen on the optical rail represent? The retina of the eye is represented by the white screen on the optical rail. The image is projected from this area of the eye. 2. If the screen was placed 15 cm from the lens in this experiment, where would the object have to be located to get a sharp image on the screen? Would the image be magnified, minified, or the same size? $ $ $ + = m = 0.5, image is minified by half ’ ! ’ " ( 3. A ray is deflected by 2.37 cm by a piece of Acrylic. Find the thickness t of the Acrylic if the incident angle is 50.5º. See Fig. 15.12. By using snell’s law, 𝑛 $ sin 𝜃 $ = 𝑛 ! sin 𝜃 ! 0.77=1.49sin (r) R = 31 degrees i-r= 19.5 degrees t = 6.1 cm 4. Using your value of the index of refraction n for the Acrylic trapezoid, calculate the speed of light in Acrylic. Refractive index of acrylic = 1.49 V= c/n = 2x10 8 m/s 5. The human eye utilizes a convex lens to focus an inverted image on the retina (like Fig. 15.8). Taking this into consideration, why is it that our vision is not inverted? Acknowledgments I’d like to thank the TA, Tianyi, for taking time out of her own day to help me.

Experiment 15- Geometric Optics

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