Refraction of Light Finding the Index of Refraction and the Critical Angle

Finding the Index of Refraction and the Critical Angle OBJECTIVE Students will verify the law of refraction for light passing from water into air. Measurements of the angle of incidence and the angle of refraction, along with the critical angle will be utilized to determine the index of refraction of water. LEVEL Physics T E A C H E R P A G E S NATIONAL STANDARDS UCP.3, A.1, A.2, B.6 TEKS 2(A), 2(B), 2(C), 2(D), 2(E), 2(F), 3(A), 7(B), 8(A) CONNECTIONS TO AP IV. Waves and optics, C. Geometrical optics, 1. Reflection and refraction TIME FRAME 45 minutes MATERIALS (For a class of 28 working in pairs) 14 laser pointers 14 semicircular plastic dishes water paper 14 protractors or sheets of polar graph paper 14 viewing screens or backing papers 14 metric rulers powdered non-dairy creamer Jello (optional) TEACHER NOTES A nice opening demonstration is to place a meter stick into a small aquarium filled with water. Demonstrate the apparent bending of the meter stick by inserting it at a 45 angle. Point out that the apparent bending is due to the way light interacts with matter. Further reinforcement of light refraction can be done with a demonstration using a penny placed at the bottom of an opaque coffee cup and slowly filling the cup with water allowing the penny to come into view to a student standing far enough away that he or she could not see the coin initially. Laser pens or pointers may be purchased at any discount store or scientific catalogue, and are much less expensive than a decade ago. The semicircular plastic dishes can be purchased from any of several scientific catalogues, such as Sargent Welch www.sargentwelch.com. Jello may be substituted as the medium for water. It is actually much easier to see the angles of reflection and refraction with Jello as 612 Laying the Foundation in Physics

the medium. You may want to have the students perform the experiment with water first, and then a more dense medium such as Jello. Light and matter appear quite different, but there must be an underlying connection at some level because they interact with each other. Interaction implies some fundamental relationship between them. To observe and verify this interaction between light and matter, we will determine the index of refraction of water and the critical angle of water. In any homogeneous material, light travels in straight lines. When light encounters a boundary (a change in optical medium) some of the light reflects back obeying the law of reflection and some of the light is transmitted into the new medium. The transmitted light does not travel in the same direction as the original light. Instead it is bent (refracted) at the boundary and travels in a different direction. This phenomenon is called refraction. Incident light ray Figure 1 Reflected light ray Refracted light ray The refraction of light at the interface between two materials is described mathematically by Snell s Law. In Figure 1 above, the long dashed line represents the normal, a line perpendicular to the surface. The angle θ measures the angle of incidence relative to the normal. The angle φ measures the angle of refraction relative to the normal. Snell s Law states that T E A C H E R P A G E S n sin θ = n sin φ i The quantity n i is the index of refraction for the medium in which the light was incident. The quantity nr is the index of refraction for the medium in which the light was refracted. The index of refraction n of a material is a measure of the speed of light in that medium. It is defined as the ratio of the speed of light in vacuum c to the speed of light v in the medium. r n = c v The index of refraction of the vacuum is 1. The index of refraction of air, which depends somewhat on the temperature and density of the air, is very nearly 1 as well. Laying the Foundation in Physics 613

40 POSSIBLE ANSWERS TO THE CONCLUSION QUESTIONS AND SAMPLE DATA DATA AND OBSERVATIONS Index of Refraction Worksheet T E A C H E R P A G E S 50 60 70 90 80 30 20 0 10 614 Laying the Foundation in Physics

θ water θ air sin θ water sin θ air n water 5 5 0.087 0.087 1.00 10 12 0.174 0.208 1.19 15 18 0.259 0.309 1.19 20 0.342 0.438 1.28 25 33 0.423 0.545 1.29 30 41 0.500 0.656 1.31 35 50 0.574 0.766 1.34 40 60 0.643 0.866 1.35 45 69 0.707 0.934 1.32 50 55 60 65 70 T E A C H E R P A G E S 75 80 85 Critical angle = 49.7 Laying the Foundation in Physics 615

ANALYSIS 1. Calculate the index of refraction of water for each incident angle using Snell s Law. Record your values for n water in the data table. Average all of your measurements. Calculate your percent error. The accepted value for the index of refraction for water is 1.33. n sin θ = n sin θ w w a a 1.00 (sin 45 ) n = = 1.32 w sin 69 1.00 + 1.19 + 1.19 + 1.28 + 1.29 + 1.31 + 1.34 + 1.35 + 1.32 n = = 1.25 averagewater 9 % error = 1.33 1.25 100 = 5.85 % 1.33 T E A C H E R P A G E S Average of all your measurements: n water = 1.25 % Error = 5.85% 2. On the axes below, plot a graph of sin θ air (y-axis) vs. sin θ water (x-axis). Be sure to sure to use proper graphing techniques, including a title, scaling, labeling the axes, and drawing the best-fit curve that represents the average of the data. Linear Fit For: Angle of Incidence vs. Angle of Refraction: Angle of Incidence y = mx+b m(slope): 1.39 º/º b(y-intercept): 0.0373º Correlation: 1.00 Angle of Incidence (º) Angle of Refraction (º) 616 Laying the Foundation in Physics

3. Determine the index of refraction from the slope of the graph. Slope = 1.39 Graphical Estimate of n water = 1.39 % Error = 4.5% 4. In the space provided, calculate the index of refraction for water and your percent error. n air sin θ = c n n water water n air 1.00 = = = 1.31 sin θ sin(49.7 ) c Critical angle = 49.7 n water (from the critical angle) = 1.31 % Error = 1.41% 5. Using n water = 1.33, determine the velocity of light in water. 3.0 10 1.33 = s v water 8 m v =2. 10 m/s water 8 6. If a medium has a large index of refraction, what does that say about the speed of light in that medium? What can you say about the way the light ray bends in relation to the perpendicular (or the normal) to the surface to the media? A large index of refraction indicates that light will slow down more than in a medium with a smaller index of refraction. The light ray will bend toward the normal in a medium with a large index of refraction. T E A C H E R P A G E S 7. What happens when light travels to a medium of lower refractive index? The light speeds up in the new medium and bends away from the medium. 8. Will light be refracted more while passing from air into water or while passing from water into glass (n = 1.50)? Explain. Light will be refracted more while passing from air into water since the difference in the indexes of refraction for the two media is greater than for light going from water into glass. 9. Will light traveling from air into water undergo total internal reflection? Explain. No, total internal reflection occurs only when light goes from a more optically dense medium to a less optically dense medium. Laying the Foundation in Physics 617

CONCLUSION QUESTIONS 1. A diligent physics student is given the following equipment: a transparent acrylic cube, a visiblespectrum laser, a metric ruler, a protractor, and a viewing screen. Her instructor asks her to devise a method to measure the index of refraction of the transparent solid. After much reflection, the lights come on and she readily measures the index of refraction of the transparent solid. Describe in detail how she determined the index of refraction. The results of her illumination are shown in the two diagrams below. P 2 is the path of the light from the laser through the air and P 1 represents the path of the light through the transparent cube. T E A C H E R P A G E S s P 1 P 2 The student measures the length s of one side of the cube. She determines the angle of refraction by examining the distance between the undeflected laser beam and the exit point P 1 of the beam in the cube. Taking the arctangent of the ratio of l 1 to side s gives the refraction angle. Likewise, the distance l 2 between the undeflected laser beam and the exit point P 2 yields the angle of incidence. s θ 1 s θ 2 l 1 l 2 P 1 P 2 1 1 1 2 θ =tan θ 1 ls =tan 2 ls Finally by applying Snell s Law the index of refraction of the transparent solid is readily determined. n sin θ = n sin θ 1 1 2 2 618 Laying the Foundation in Physics

2. Use Snell s Law to determine the path of the light through this rectangular sheet of glass ( n = 1.50). Draw a normal, perform the appropriate measurements and calculations for the entry point and draw the refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal, perform the appropriate measurements and calculations for the exit point and draw the refracted ray for the light exiting the glass. Show all your calculations in the space below. n sin θ = n sin θ 1 1 2 2 (1.00) sin 44 = 1.50 sin θ θ = 27.6 (1.50) sin 27.6 = (1.00) sin θ θ =44 T E A C H E R P A G E S 3. Use Snell s Law to determine the path of the light through the triangular glass ( n =1.50). Draw a normal, perform the appropriate measurements and calculations for the entry point and draw the refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal, perform the appropriate measurements and calculations for the exit point and draw the refracted ray for the light exiting the glass. Show all you calculations in the space below. Laying the Foundation in Physics 619

n sin θ = n sin θ 1 1 2 2 (1.00) sin 46 = (1.50) sin θ θ = 28.7 (1.50) sin 28.7 = (1.00) sin θ θ =46 T E A C H E R P A G E S 4. In our experiment, the beam of light actually passed through three different media (air, plastic, and water). We assumed that the interaction of the light with the plastic could be ignored. Is that assumption reasonable or is it an additional source of error? We will examine this question by imagining three media in layers as shown in the diagram below. The beam passes from air into medium X and then from medium X into the water. There are four angles to measure. Use your knowledge of geometry and Snell s Law to find the relationship between angles 1 and 4. Given that relationship, how important is it to find the index of refraction of X, assuming we are interested in knowing the index of refraction of the water? Air 1 X 3 2 Water 4 Angles 2 and 3 are congruent since they are alternate interior angles. Hence angle 1 and angle 4 must also be congruent by applying Snell s Law. The angle of incidence from the air into material X is equal to the angle of refraction from material X into the air. Thus, medium X can be ignored, just as we ignored the plastic dish in determining the index of refraction of water. 620 Laying the Foundation in Physics

Finding the Index of Refraction and the Critical Angle Light and matter appear quite different, however, there must be an underlying connection at some level because they interact with each other. Interaction implies some fundamental relationship between them. To observe and verify this interaction between light and matter, you will determine the index of refraction of water and the critical angle of water. In any homogeneous material light travels in straight lines. When light encounters a boundary (a change in optical medium) some of the light reflects back obeying the law of reflection and some of the light is transmitted into the new medium. The transmitted light does not travel in the same direction as the original light. Instead it is bent (refracted) at the boundary and travels in a different direction. This phenomenon is called refraction. Incident light ray Reflected light ray Refracted light ray Figure 1 The refraction of light at the interface between two materials is described mathematically by Snell s Law. In Figure 1 above, the long dashed line represents the normal, a line perpendicular to the surface. The angle θ measures the angle of incidence relative to the normal. The angle φ measures the angle of refraction relative to the normal. Snell s Law avers: n sin θ = n sin φ i The quantity n i is the index of refraction for the medium in which the light was incident. The quantity nr is the index of refraction for the medium in which the light was refracted. The index of refraction n of a material is a measure of the speed of light in that medium. It is defined as the ratio of the speed of light in vacuum c to the speed of light v in the medium. r n = c v The index of refraction of vacuum is 1. The index of refraction of air, which depends somewhat on the temperature and density of the air, is very nearly 1 as well. We will use this approximation for the index of refraction of air. Laying the Foundation in Physics 621

The phenomenon of total internal reflection occurs when the light travels from a medium with a higher index of refraction to a medium with a lower index of refraction. When n i > nr, the refracted ray bends away from the normal. If the angle of incidence is large enough, the angle of refraction will be 90 and the light travels parallel to the interface between the two media. The angle of incidence for which this occurs is called the critical angle. If the angle of incidence is increased further, then the calculated value of the angle of refraction is greater than one which is mathematically impossible! At the critical incident angle, the light does not pass through the surface, it reflects off the surface, such that the surface becomes a mirror and obeys the law of reflection. Therefore, the critical angle for light passing from a more dense medium of n1 to a less dense medium of n 2, where n > n, is 1 2 sin θ = c n n 2 1 The phenomenon of total internal reflection is important in numerous fiber optic technologies, from communication to surgical procedures. Total internal reflection explains how light (and information) can be transmitted great distances with little loss of energy. PURPOSE In this activity you will investigate the refraction of light as it passes from water into air. Measurements of the angle of incidence and the angle of refraction, along with the critical angle will be utilized to determine the index of refraction of water. MATERIALS laser pointer water protractor or polar graph paper metric ruler semicircular plastic dish paper viewing screen or backing paper powdered non-dairy creamer Safety Alert Caution Do NOT look into the laser and do NOT direct the laser at others. 622 Laying the Foundation in Physics

PROCEDURE 1. Place the Index of Refraction Worksheet on a flat surface or table. Fill a semicircular dish with water and center the semicircular dish on the outline of the dish. 2. Sprinkle a small amount of non-dairy creamer on the water. This will make the laser beam visible in the water. 3. The dish has an etch mark on its flat side at the center of the semicircle. Shine the laser into the dish through the curved wall of the dish. Aim the beam so that it hits the etch mark on the flat wall of the semicircle. Vary the angle beginning with an incidence angle of 5 and approaching 90, by moving the laser pen around the curve of the dish. Always shine the laser perpendicular to the curved wall so that the beam strikes the etch mark (midpoint) of the flat wall. Place a viewing screen or some backing paper opposite the flat wall of the dish and perpendicular to the flat surface or table. The purpose of the viewing screen or backing paper is to help you locate the exit point of the laser beam and measure the angle of refraction for each angle of incidence. 4. Draw a ray on the worksheet from the center of the semicircle (the etch mark on the flat surface) through each of the angles of refraction and extend it to the margin of the paper. Draw an arrowhead on each incident ray and all the refracted rays you used showing the path of the light. Measure the angle that the refracted rays make with the normal and record them in the data table. Fill in as much of the table as possible. Use the data and Snell s Law to determine the index of refraction. 5. Move the laser pen around the curved surface of the semi-circular dish and observe the phenomenon of total internal reflection. At some position, the light ray exiting the flat side will reach an angle of 90 and then reflect back out the curved side of the dish. When the refracted angle reaches 90 draw a line along the flat side of the dish (parallel to the interface between the air and the water). Draw a line perpendicular to and through the center of the flat side of the dish. Measure the incident and reflected angles. The angle of incidence (which should also be the angle of reflection) is the critical angle. Determine the index of refraction for the water using the relationship: n sin θ = c n 2 1 Laying the Foundation in Physics 623

40 Name Period Finding the Index of Refraction and the Critical Angle DATA AND OBSERVATIONS Index of Refraction Worksheet 90 80 70 60 50 30 20 0 10 624 Laying the Foundation in Physics

θ water θ air sin θ water sin θ air n water 5 10 15 20 25 30 35 40 45 50 55 60 65 70 75 80 85 Critical angle = Laying the Foundation in Physics 625

3. Determine the index of refraction from the slope of the graph. Graphical Estimate of n water = % Error = 4. In the space provided calculate the index of refraction for water and your percent error. Critical angle = n water (from the critical angle) = % Error = % 5. Using n water = 1.33, determine the velocity of light in water. 6. If a medium has a large index of refraction, what does that say about the speed of light in that medium? What can you say about the way the light ray bends in relation to the perpendicular (or the normal) to the surface to the media? Laying the Foundation in Physics 627

7. What happens when light travels to a medium of lower refractive index? 8. Will light be refracted more while passing from air into water or while passing from water into glass (n = 1.50)? Explain. 9. Will light traveling from air into water undergo total internal reflection? Explain. CONCLUSION QUESTIONS 1. A diligent physics student is given the following equipment: a transparent acrylic cube, a visiblespectrum laser, a metric ruler, a protractor, and a viewing screen. Her instructor asks her to devise a method to measure the index of refraction of the transparent solid. After much reflection, the lights come on and she readily measures the index of refraction of the transparent solid. Describe in detail how she determined the index of refraction. The results of her illumination are shown in the two diagrams below. P 2 is the path of the light from the laser through the air and P 1 represents the path of the light through the transparent cube. s P 1 P 2 628 Laying the Foundation in Physics

3. Use Snell s Law to determine the path of the light through the triangular glass ( n =1. 50 ). Draw a normal, perform the appropriate measurements and calculations for the entry point and draw the refracted ray for the light entering the glass. Continue the ray to the opposite side, draw a normal, perform the appropriate measurements and calculations for the exit point and draw the refracted ray for the light exiting the glass. Show all your calculations in the space below. 4. In our experiment, the beam of light actually passed through three different media (air, plastic, and water). We assumed that the interaction of the light with the plastic could be ignored. Is that assumption reasonable or is it an additional source of error? We will examine this question by imagining three media in layers as shown in the diagram below. The beam passes from air into medium X and then from medium X into the water. There are four angles to measure. Use your knowledge of geometry and Snell s Law to find the relationship between angles 1 and 4. Given that relationship, how important is it to find the index of refraction of X, assuming we are interested in knowing the index of refraction of the water? Air 1 X 3 2 Water 4 630 Laying the Foundation in Physics

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