Lab Report 15

pdf

School

Harvard University *

*We aren’t endorsed by this school

Course

102

Subject

Physics

Date

Feb 20, 2024

Type

pdf

Pages

Uploaded by PrivateStar23629

Report for Experiment #15 Geometric Optics Sebastian Krauss Lab Partner: Hayley Ha TA: Dave Anderson 10/30/2023

Abstract In this experiment we examine three interactions of light. With each of the three examining a different aspect of geometric optics. The first being reflection and refraction of concave and convex lenses and mirrors. The second being about indexes of refraction, and the third being about magnification. With each of these investigations we had a goal in mind: To investigate parallel rays reflecting off plane and spherical mirrors, to investigate the refraction of parallel rays through a concave and convex lens, to determine the index of refraction of the rhombus prism, to find the focal length of a thin lens using the optics bench. For our study we will be using the PASCO ray optics kit, PASCO optics bench, PASCO 100 mm lens, PASCO optics viewing screen, PASCO light source, stand and power supply. We will calculate these values based on our measurements and compare them to expected values after propagating error, we will develop plots for focal length, and magnification ration for investigation three and compare with expected results.

Introduction The applications of lenses are immense in modern technology, from DVD players to corrective eye surgery. The principles of which are studied in the field of Geometric Optics, providing an understanding of how light travels and interacts with lenses and mirrors. Geometric optics includes the assumption that light travels in a straight line, with the lines being called rays. This leads to the ray model of light, which can be further described with a series of interactions between light and a reflective or transparent surface. The angles involved in describing rays are measured with respect to the normal of the surface. With the normal being the direction perpendicular to the surface, creating a 90 degree angle. This remains true for all surfaces regardless of curvature. This further translates in the intuitive law called the law of reflection, which states, “The angle of incidence θi must equal the angle of reflection θr” . As discussed, the rays of light that bounce off of a spherical or paraboloidal mirror also follow the law of reflection. For paraboloidal mirrors, incoming parallel light rays will hit the surface of the mirror and change angle such that they all focus through a single point, called the focal point. For spherical mirrors, the opposite will happen as the light is focused on a collection of points that form a curved focal plane. We approximate a parabolic mirror by ensuring that the size of the sphere mirror is small compared to its radius. We then approximate that a sphere mirror has a focal point. With the focal length f being equal to half of the radius of the curve r of the mirror 𝑓 = 𝑟 2 (1) Within spherical mirrors we have two subcategories. The concave and convex mirrors. The concaves mirror has an indentation, and is usually thicker on the sider than in the middle. The convex mirror is usually thicker in the middle than in the sides. For concave mirrors, the focal point is in front of the mirror surface, for convex mirrors the focal point is behind the mirror surface. In a vacuum, light travels at approximately 3.00 * 10^8 m/s. As the light passes through other transparent materials or mediums, this speed will change, and the light will slow down. The ratio of the speed of light (vacuum) to the speed of light in a transparent material is called the index of refraction, further defined by 𝑛 = ? 𝑣 (2) With n being the index of refraction, c, the speed of light in a vacuum, and v, the speed of light in a medium. When light enters a medium, two events occur. Firstly, the light is reflected off of the surface of the medium, resulting in the application of the law of reflection. Next the remaining portion of the light bends as it passes through the medium. This directional difference as the light changes mediums is called refraction. With the light bending towards the normal when entering a medium with a higher index of refraction than a vacuum. And with the light bending away from the normal when entering a medium with a lower index of refraction. The angles for each can be found using Snell ’ s law, or the law of refraction: 𝑛 1 sin 𝜃 1 = 𝑛 2 sin 𝜃 2 (3)

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

With θ1 being the angle of incidence, θ2 the angle of refraction and n1 and n2 the indices of refraction for the respective mediums. Lenses are transparent optical devices which allow the transmission of light rather than reflection of light. In this experiment, two types of these lenses will be investigated, the converging lens and diverging lens. A converging lens refracts parallel rays through a single focal point after light passes through. A diverging lens will cause the parallel light rays to diverge and separate in opposite directions. These directions can be traced back to a focal point. Using spherical lenses, we will make two assumptions in order to avoid any image errors (aberrations). The lens thickness will be small compared to the curvature radius, making the lens relatively thin. Secondly, we will assume that the parallel rays will enter the lens at a shallow angle. As previously observed, parallel rays entering a double convex lens converge at a single point located on the lens focal plane. However, if the ray originated ay a single object point, they would be focused to an image point on the opposite side. Relating the positions of the object, the image, and the focal point is the expression called the lens equation. With the object distance do, the image distance di, and the focal length f all being measured from the center line of the lens. 1 ? 𝑜 + 1 ? 𝑖 = 1 𝑓 (4) In this setup, the differing sizes of the object and its resulting image define the magnification, M, of the setup. It is the ratio of the image height hi, to object height ho. 𝑀 = ℎ 𝑖 ℎ 𝑜 = − ? 𝑖 ? 𝑜 (5) When constructing a ray diagram, there are usually three rays used to trace and locate the position and height of the image. This starts with the bottom of the object on the lens axis. Then three rays are drawn from the top of the object. A single ray leaves parallel to the axis, passing through the focal point. The second through the center of the lens. The third though the focal point f ’ and the exits parallel to the axis. These rays will mark the position and height of the image. The real image is formed when the rays pass the lens and an image is formed. A virtual image is formed when rays diverge after passing through the lens. The position of the virtual image can be determined by tracing the rays back to their apparent point of origin. For our study we will be using the PASCO ray optics kit, PASCO optics bench, PASCO 100 mm lens, PASCO optics viewing screen, PASCO light source, stand and power supply. Our objectives are the following: To investigate parallel rays reflecting off of plane and spherical mirrors, to determine the index of refraction of the rhombus prism, to investigate the refraction of parallel rays through a concave and convex lens, To find the focal length of a thin lens using the optics bench. Throughout our study we will be using various forms of measurement, each of which involves some aspect of error. We will study focal points, indexes of refraction, the position of objects, and magnification. In order to preserve the validity of our numbers, we must use propagation of error when computing variables and for the error bars on a graph. The equation for which is the following.

(6)

Investigation 1 — Ray Tracing In this investigation, we will see how five rays reflect off of a concave and convex mirror. We will then directly measure the focal length of the mirrors. Start in a dark room, with the power supply connected to the light source, connected to the wall outlet. Adjust the plastic slider such that five rays emerge from the light source. Over a white piece of paper, take a triangular mirror and place it on the paper with the open cavity facing down, so that the concave side is reflecting the rays. Trace both the incident and reflected rays on the white sheet of paper, along with the outline of the mirror itself. The lines will then converge at the focal point, allowing us to record the focal length. Because the rays have a non-zero width, we will find a region of intersection. The length of which will determine our uncertainty. We found that the region was around .6 cm. We then repeat this experiment with the convex side of the triangle. This time tracing the ray back into the outline of the triangle, such that the focal length is emerging from the other side of the mirror surface. The region of the intersection was recorded to be around .6 cm, the same as with the concave mirror. The resulting focal lengths are shown below. Table 1. Focal lengths for concave and convex mirrors with 5 rays and respective uncertainties Our values for measured focal length align with the nominal value of 6.3 +/- .3cm and is within the experimental uncertainties of the expected value. As before, we take a new sheet of white paper and place the light source at the end. This time, we take the convex lens, rather than the mirror, and place it on the paper such that the flat side is resting on the ground. It will stand upright, and the 5 rays will pass through it across the curve of the lens. Then trace both the incident, the refracted rays, and the outline of the lens on the paper. Similarly, to the previous procedure, there should be an intersection region where the rays converge. This will then allow us to find the focal point, and subsequently the focal length. Determine the uncertainty in the measurement by noting the length of the region and taking half of its length. We found that the region was 1 cm long, making our uncertainty .5cm. Then we repeated this procedure with the concave lens, this time tracing the rays back into the outline of the lens, and from the opposite side of the refraction origin compared to the convex lens. The region was found to be 1cm long, making our uncertainty .5 cm. The values for both are shown below. Table 2. Focal lengths for concave and convex lens with 5 rays and respective uncertainties Focal Length Convex Lens 13.4cm +/- .5cm Concave Lens 11.5cm +/- .5cm Focal Length Concave Mirror 6.1 cm +/- .3cm Convex Mirror 5.9 cm +/- .3cm

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Our value for measured focal length align with the nominal value of 12.9 +/- .5cm for the convex lens only and is within the experimental uncertainties of the expected value. Our value for concave lens however is close to the expected value, we may attribute this discrepancy to human error. Investigation 2- Refraction of light through Acrylic In this investigation, we will see how light refracts as it passes through an acrylic rhombus. We will then directly measure the index of refraction through incident and refracted angles. We start in a dark room, with the light source slider adjusted such that a single ray is emitted. The source will then be pointed towards the Acrylic trapezoid with the frosted side down, with both over a white sheet of paper, as seen in the previous investigation. Rate the trapezoid such that the single ray is going into side 2 of the trapezoid (incident to), at an angle between 35 and 70 degrees normal to the surface. The single ray should then exit through side 4 with the refracted ray passing through the trapezoid. A figure is shown below denoting the sides and ray. Figure 1. Trapezoid with incident, refracted, and exiting rays, as described in the investigation. Then trace the incident ray, the outline of the trapezoid, and the exiting ray. Then connect the incident ray and exit ray to form the refracted ray. Using equation 3 from the introduction, we can the find the index of refraction for the trapezoid. By measuring the angles of refraction, we can solve of n. This is then repeated for three other incident angles within the 35-to-70-degree range. The measurements for each of the trials are shown below. The error for the angle measurements is seen to be half a degree, and the calculated error for each trial for the index of refraction is calculated for each trial Table 3. Incident ray angle, and internal angle with experimental error. Index of refraction (n) with calculated error. Trial 1 Angle n incident ray 55 1.43

internal angle 35 error 0.5 0.0198 Trial 2 Angle n Incident ray 57 1.63 internal angle 31 error 0.5 0.02538 Trial 3 Angle n Incident ray 74 1.44 internal angle 42 error 0.5 0.01438 Trial 4 Angle n Incident ray 45 1.373 internal angle 31 error 0.5 0.02326 The propagated error resembles the following equation, which in this example involves the angles in trial 1. With R being the index of refraction, V being the incident ray angle, and m being the internal ray angle. Figure 2. Propagated error of trial 1 The average of these four measurements was then calculated, this is shown in the table below. The error of the average is propagated using the error values calculated for the individual indexes of refraction. Table 4. Average of the index of refraction with the propagated error.

n avg 1.46825 n error 0.010559 With the error propagation for the average of the index of refraction resembling the following equation. Figure 3. Propagated error of average of index of refraction. Our value does not align with the expected value for the index of refraction of Acrylic = 1.49 without our estimate for error. Despite this the value is very close, being only .01 off from the expected value. Investigation 3- Focal Length of a thin Lens In this investigation we will examine the various positions and markers of an image to find the focal length of a thin lens using the optics bench. We then further find magnification and create plots examining the relationship between focal lengths and heights of images. We start with the light source in the stand that is on the optical rail, making sure the object is facing the white metal screen. With the light on, measure and record the diameter, ho, of the inner circle of the object, then slide the stand until the object is located at 10cm on the optical rail. Continue and position the screen at 110 cm on the optical rail. Then place the 100mm lens in the channel of the optical rail such that the light passes through it and onto the screen. Adjust the lens up and down the optical rail such that the resulting image is sharp, focused, and magnified on the screen. Record the positions of the object, lens, and screens, along with the errors in the positions. The record the diameter, hi, of the inner circle of the image on the screen. We repeat this procedure 6 times, moving the screen 10cm closer with each data set. For each set of data, we then calculate the object distance, do, image distance, di, and the error for each. This can be seen in the table collection below.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Table 5. Diameter of inner circle, object position, lens position, screen position, and diameter of inner circle of image lens. Along with the object distance and image distance with propagated error. Trial 1 Error Object Distance Prop Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 14 cm 0.0707 86 cm 0.0707 Object Position 10 cm 0.05 Lens Position 24 cm 0.05 Screen Position 110 cm 0.05 Diameter (Image Lens - Inner Circle) 7 cm Trial 2 Error Object Distance Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 14.1 cm 0.0707 75.9 cm 0.0707 Object Position 10 cm 0.05 Lens Position 24.1 cm 0.05 Screen Position 100 cm 0.05 Diameter (Image Lens - Inner Circle) 6 cm Trial 3 Error Object Distance Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 14.4 cm 0.0707 65.6 cm 0.0707 Object Position 10 cm 0.05 Lens Position 24.4 cm 0.05 Screen Position 90 cm 0.05 Diameter (Image Lens - Inner Circle) 5 cm Trial 4 Error Object Distance Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 15 cm 0.0707 55 cm 0.0707 Object Position 10 cm 0.05 Lens Position 25 cm 0.05

Screen Position 80 cm 0.05 Diameter (Image Lens - Inner Circle) 4.2 cm Trial 5 Error Object Distance Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 15.5 cm 0.0707 44.5 cm 0.0707 Object Position 10 cm 0.05 Lens Position 25.5 cm 0.05 Screen Position 70 cm 0.05 Diameter (Image Lens - Inner Circle) 3.2 cm Trial 6 Error Object Distance Error Image Distance Error Diameter (Object Lens - Inner Circle) 1 cm 0.05 17.2 cm 0.0707 32.8 cm 0.0707 Object Position 10 cm 0.05 Lens Position 27.2 cm 0.05 Screen Position 60 cm 0.05 Diameter (Image Lens - Inner Circle) 2.1 cm We will examine the relationship between dodi and do+di by plotting dodi vs do + di including error bars, these values are listed in the table below with their respective errors. From this plot, we can then find the focal length f of the lens as the slope and the propagated error respectively. Table 6: Object and image distance, product and sum with propagated errors. d0di error d0+di error 1204 6.1602 100 0.09998 d0di d0+di 1070.19 5.458 90 0.09998 d0di d0+di 944.64 4.748 80 0.09998

d0di d0+di 825 4.0305 70 0.09998 d0di d0+di 689.75 3.3315 60 0.09998 d0di d0+di 564.16 2.6185 50 0.09998 Figure 4. Plot of dodi against do + di with error bars. The resulting slope is equal to the focal length. The values which and the expected result are also shown in the figure Actual Focal Length 10 cm Calculated Focal Length 12.743 cm Compared to the known value of f, our results do not agree within our estimate for the error of the focal length, the error which can be almost entirely attributed to the error of DoDi, which then translates to an error ranging from +/- .6 to +/- .2. This discrepancy can be attributed to human error. To find magnification M, we must compare hi/ho to di/do. The values for which can be seen below, including calculated error. Table 7. di/do and hi/ho values for each trial including calculated error Trial di/d0 error of di/do hi/h0 error of hi/h0 1 6.142857143 .61602 7 0.3536 2 5.382978723 .5458 6 0.30414 3 4.555555556 .4748 5 0.255 4 3.666666667 .40305 4.2 0.2159

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

5 2.870967742 .33315 3.2 0.1676 6 1.906976744 .26185 2.1 0.1163 Plot the magnification M, which is equal to the slop of( hi/ho) vs. di/do. Find the slope and its error. Figure 5. plot of hi/ho, against di/do, resulting in the slope M, equal to magnification. This slope does not agree with the theory, as the object height against image height ratio does not align with the image and object distance. The resulting slope should equal 1 if the images followed the rules of thin lens. Our calculated values for error throughout this investigation were found using the error propagation table referenced in the IPL. Conclusion This experiment consisted of three investigations. With each of the three examining a different aspect of geometric optics. The first being reflection and refraction of concave and convex lenses and mirrors. The second being about indexes of refraction, and the third being about magnification. With each of these investigations we had a goal in mind: To investigate parallel rays reflecting off plane and spherical mirrors, to investigate the refraction of parallel rays through a concave and convex lens, to determine the index of refraction of the rhombus prism, to find the focal length of a thin lens using the optics bench. We found that our values for focal length mostly aligned with the expected value for focal length for both the mirrors and lens with it being equal to 6.1 and 5.9 cm +/- .3cm for concave and convex mirrors respectively. And 13.4 and 11.5 cm +/-.5cm for lens respectively. We found that our index of refraction was very close to our expected value after error propagation, with an average value of 1.47, with an error

of .01. In our final investigation we found that the focal length, and magnification ratio between height and distance did not resemble our 1 to 1 ratio, with a value of12.743 for focal length and 1.137 for magnification, however it remained close to the value, and as such the relationship is relatively visible. Our error was calculated using the error propagation table found in the IPL and includes propagation for multiplying and adding values with uncertainties attached to them. In order to improve our experiment, the removal of the uncertainties would help immensely. Namely the slits of light, if the resembled actual lines with a zero width, much of the uncertainty that further propagates down the experiment would be removed. The addition of lasers instead of light slits would improve this. Additionally, more precise measurements would reduce the amount of error, and make our measurements more accurate. Lastly, the human aspect of error remains, as much of our measurements are based off of drawn lines, which may or may not be completely accurate depending on the investigator.

Questions 1. What does the white screen on the optical rail represent? The white screen on an optical rail in an experimental setup, such as one involving lenses, is typically used to observe and capture the image formed by the lens. When light passes through a lens, it can focus or disperse to form an image at a certain distance, depending on the properties of the lens and the object distance. The white screen acts as a projection surface where this image can be seen and measured. 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? 30cm (in front of the lens). minified and inverted. M = -.5 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. 3.94 cm 4. Using your value of the index of refraction n for the Acrylic trapezoid, calculate the speed of light in Acrylic. N = 1.46825 c = V=c/n the speed of light in Acrylic is approximately. 2.042×10^8 meters per second. 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? Despite the eye projecting an inverted image onto the retina, our vision is right-side-up due to the processing in the brain. The visual cortex of the brain interprets the signals it receives from the eyes and corrects accordingly so we perceive the world in its actual orientation.

Your preview ends here

Eager to read complete document? Join bartleby learn and gain access to the full version

Access to all documents
Unlimited textbook solutions
24/7 expert homework help

Acknowledgments I would like to thank my Lab Partner Hayley Ha. I would like to thank our TA David Anderson

References [1] H. Young and R. Freedman, University Physics, p 111, Pearson Education, 14th edition. [2] Hyde, Batishchev, and Altunkaynak, Introductory Physics Laboratory, pp 189-201, Hayden-McNeil, 2017. [3] https://nicoco007.github.io/Propagation-of-Uncertainty-Calculator/

Lab Report 15

Related Documents