# PHY208FALL2008. Week2HW. Introduction to Two-Source Interference. Due at 11:59pm on Friday, September 12, View Grading Details [ Print ]

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1 Assignment Display Mode: View Printable Answers PHY208FALL2008 Week2HW Due at 11:59pm on Friday September View Grading Details [ Print ] The following three problems concern interference from two sources or along two paths. Introduction to Two-Source Interference Description: This problem provides students with a basic introduction to the interference of two identical waves. The conditions for constructive and destructive interference are discussed in terms of phase difference and path length difference. Learning Goal: To gain an understanding of constructive and destructive interference. Consider two sinusoidal waves (1 and 2) of identical wavelength period and maximum amplitude. A snapshot of one of these waves taken at a certain time is displayed in the figure below. Let and represent the displacement of each wave at position location ( at time. If these waves were to be in the same ) at the same time they would interfere with one another. This would result in a single wave with a displacement given by. This equation states that at time the displacement of the resulting wave at position is the algebraic sum of the displacements of the waves 1 and 2 at position at time. When the maximum displacement of the resulting wave is less than the amplitude of the original waves that is when the waves are said to interfere destructively because the result is smaller than either of the individual waves. Similarly when the resulting wave is larger than either of the individual waves. Notice that. the waves are said to interfere constructively because To further explore what this equation means consider four sets of identical waves that move in the +x direction. A photo is taken of each wave at time and is displayed in the figures below. Rank these sets of waves on the basis of the maximum amplitude of the wave that results from the interference of the two waves in each set. Rank from largest to smallest To rank items as equivalent overlap them. View When identical waves interfere the amplitude of the resulting wave depends on the relative phase of the two waves. As illustrated by the set of waves labeled A when the peak of one wave aligns with the peak of the second wave the waves are in phase and produce a wave with the largest possible amplitude. When the peak of one wave aligns with the trough of the other wave as illustrated in Set C the waves are out of phase by and produce a wave with the smallest possible amplitude zero! Page 1 of 15

2 Part B Consider sets of identical waves with the following phase differences: A. B. C. D. E. F. G. Identify which sets will interfere constructively and which will interfere destructively. Enter the letters of the sets corresponding to constructive interference in alphabetical order and the letters corresponding to sets that interfere destructively in alphabetical order separated by a comma. For example if sets A and B interfere constructively and sets C and F interfere destructively enter ABCF. BCEGADF Do you notice a pattern? When the phase difference between two identical waves can be written as where the waves will interfere constructively. When the phase difference can be expressed as where the waves will interfere destructively. Consider what water waves look like when you throw a rock into a lake. These waves start at the point where the rock entered the water and travel out in all directions. When viewed from above these waves can be drawn as shown where the solid lines represent wave peaks and troughs are located halfway between adjacent peaks. Part C Now look at the waves emitted from two identical sources (e.g. two identical rocks that fall into a lake at the same time). The sources emit identical waves at the exact same time. Identify whether the waves interfere constructively or destructively at each point A to D. Page 2 of 15

3 Hint C.1 How to approach the problem Recall that constructive interference occurs when the two waves are in phase when they interfere so that the peak (or trough) of one wave aligns with the peak (or trough) of the other wave. Destructive interference occurs when waves are out of phase so that the peak of one wave aligns with the trough of the other wave. Study the picture to find where each type of interference occurs. For points A to D enter either c for constructive or d for destructive interference. For example if constructive interference occurs at points A C and D and destructive interference occurs at B enter cdcc. ccdd Each wave travels a distance or from its source to reach Point B. Since the distance between consecutive peaks is equal to from the picture you can see that Point B is away from Source 1 and away from Source 2. The path-length difference is the difference in the distance each wave travels to reach Point B:. Part D What are the path-length differences at Points A C and D (respectively and )? Enter your answers numerically in terms of separated by commas. For example if the path-length differences at Points A C and D are and respectively enter = Knowing the path-length difference helps to confirm what you found in Part C. When the path-length difference is where the waves interfere constructively. When the path-length difference is where the waves interfere destructively. Part E What are the path-length differences at Points L to P? Enter your answers numerically in terms of separated by commas. For example if the path-length differences at Points L M N O and P are and respectively enter Page 3 of 15

4 = Every point along the line connecting Points L to P corresponds to a path-length difference every point along this line waves from the two sources interfere constructively.. This means that at The figure below shows two other lines of constructive interference: One corresponds to a path-length difference and the other corresponds to. It should make sense that the line halfway between the two sources corresponds to a path-length difference of zero since any point on this line is equally far from each source. Notice the symmetry about the line of the and the lines. A similar figure can be drawn for the lines of destructive interference. Notice that the pattern of lines is still symmetric about the line halfway between the two sources; however the lines along which destructive interference occurs fall midway between adjacent lines of constructive interference. Constructive Interference Description: Short quantitative problem on a double-slit experiment and interference pattern. Students should know how to find the wavelength of light in different materials. Based on Young/Geller Quantitative Analysis The figure shows the interference pattern obtained in a double-slit experiment with light of wavelength. Page 4 of 15

5 Identify the fringe or fringes that result from the interference of two waves whose phases differ by exactly. Hint A.1 How to approach the problem In a double-slit experiment when two waves reach the screen shifted by an integer number of wavelengths such as at that point the waves are in phase and interfere constructively. Their amplitudes add up and the resulting band or fringe on the screen is a bright region where light intensity is at a maximum. Note that several bright fringes can be seen in the interference pattern shown in the figure. Each of them is produced by waves that are shifted by a different (integer) number of wavelengths. Choose those that correspond to a path difference of. or.2 Find which fringes correspond to constructive interference Which of the fringes shown in the figure correspond to constructive interference? Hint A.2.a Constructive interference fringes Recall that in a double-slit experiment waves that interfere constructively strike the screen in regions where light intensity is maximum creating a series of bright bands or fringes. fringe C only fringes A B and C fringes A B and D fringes A B and E fringes A D and E fringes A B D and E Which of these fringes correspond to a path difference of and which to a path difference of? Hint A.3 Constructive interference pattern In a double-slit experiment fringes of maximum intensity are produced by waves whose path difference is an integer number of wavelengths such as. The bigger this number the farther from the central bright band is the corresponding fringe. For example if the center of the fringe produced by waves that are shifted by is at 1 from the center of the central band the center of the fringe produced by waves shifted by will be at approximately 4 from the central band. Moreover the fringes appear symmetrical with respect to the central band. Thus when the center of the fringe produced by waves that are shifted by is at 1 from the center of the central band the center of the fringe produced by waves shifted by will be located at about the same distance from the center of the central band but on the opposite side. Enter the letter(s) indicating the fringe(s) in alphabetical order. For example if you think that fringes A and C are both correct enter AC. AD Part B The same double-slit experiment is then immersed in water (with an index of refraction of 1.33) and repeated. When in the water what happens to the interference fringes? Hint B.1 How to approach the problem When in water the wavelength of light changes. Will this affect the location of the interference fringes on the screen? Recall that in a double-slit experiment the positions of the centers of the bright fringes on the screen can be expressed in terms of the difference in path length traveled by the waves. Part B.2 Find the wavelength of light in water Consider light with wavelength in air. What is the wavelength of the same light in water? Hint B.2.a Wavelength of light in a material The wavelength of light is different in different materials. In particular the wavelength of light in a material of index of refraction is given by where is the wavelength of the same light in vacuum. This is because in any material the velocity of a wave is given by the product of frequency and wavelength. Since the speed of light in a material is less than it is in vacuum but its frequency does not change its wavelength must be less than the wavelength of the same light in vacuum. Note that the wavelength of light in air is essentially the same as that of light in vacuum. Express your answer in terms of. Page 5 of 15

6 = Thus when the experiment is immersed in water the wavelength of light decreases. How does this affect the location of the interference fringes? Part B.3 Find how the centers of the fringes change with the wavelength of light Let be the position of the center of the mth band (measured from the center of the central band) when the experiment is in air and let be the same position when the experiment is in water. Which of the following expressions is correct? Hint B.3.a Using proportional reasoning To solve this problem use proportional reasoning to find a relation between the position of the mth band when the experiment is in air and the same position when the experiment is in water. Find the simplest equation that contains these variables and other known quantities from the problem. Write this equation twice once to describe and again for. You need to write each equation so that all the constants are on one side and your variables are on the other. Since your variable is in this problem you want to write your equations in the form. To finish the problem you need to compare the two cases presented in the problem. For this question you should find the ratio. Hint B.3.b Position of the centers of the bright fringes in a double-slit experiment In a double-slit experiment with light of wavelength waves whose path difference is (where is an integer) interfere constructively and create the bright band on the screen. The position of the center of the mth band (measured from the center of the central band) on the screen is given by where and are respectively the distance from the screen to the slits and the distance between the slits. As you found out when the experiment is immersed in water the centers of the bright fringes are closer to the central band than when the experiment is in air. But does the location of the center of the central band change when the wavelength of light decreases? Part B.4 Find how the center of the central band changes with the wavelength of light In a double slit-experiment the central fringe of the interference pattern corresponds to waves that traveled the same distance from the source. If the wavelength of the waves changes will the position of the center of this fringe change? Assume the slits are horizontal so that the fringe pattern displayed on a screen is oriented vertically.. Hint B.4.a How to approach the question In a double-slit experiment the central fringe is created by waves that travel the same distance from the slits and its center is always located at the same distance from both slits. Thus its position changes only if the location of the slits changes regardless of the properties of the light that shines through the slits. The position of the center of the central fringe will shift upward. The position of the center of the central fringe will shift downward. The position of the center of the central fringe will not change. They are more closely spaced than in air by a factor of They are more widely spaced than in air by a factor of They are spaced the same as in air. They are shifted upward. They are shifted downward. When the experiment is immersed in water the wavelength of light decreases because the index of refraction of water is higher than that of air. Since the positions of the centers of the bright bands depend on the wavelength of light light with a smaller wavelength will produce interference fringes that are more closely spaced; the higher the index of refraction the more closely spaced are the fringes. Page 6 of 15

8 .3 What is the path length of reflected waves? Consider the lengths of the paths taken by the two reflected waves. Note that the path from the transmitter to the building is larger for one wave while the path from the building to the antenna is larger for the other. What is the difference in length between the path of the reflected wave from the transmitter to A and the path of the reflected wave from the transmitter to B in integer multiples of the wavelength? Express your answer in wavelengths as a fraction. = wavelengths Part B Your FM station has a frequency of megahertz. The speed of light is about meters per second. What is the distance between points A and B? Express your answer in meters to two significant figures. = This problem discusses a diffraction grating. Problem Description: For your science fair project you need to design a diffraction grating that will disperse the visible spectrum ( nm) over 30.0 degree(s) in first order. (a) How many lines per millimeter does your grating need? (b) What is the firstorder... For your science fair project you need to design a diffraction grating that will disperse the visible spectrum ( nm) over in first order. How many lines per millimeter does your grating need? lines/mm Part B What is the first-order diffraction angle of light from a sodium lamp = The following two problems concern thin-film interference. Thin Film (Oil Slick) Description: This problem explores thin film interference for both transmission and reflection. A scientist notices that an oil slick floating on water when viewed from above has many different rainbow colors reflecting off the surface. She aims a spectrometer at a particular spot and measures the wavelength to be 750 (in air). The index of refraction of water is The index of refraction of the oil is What is the minimum thickness of the oil slick at that spot? Page 8 of 15

10 substantially easier so you should keep both approaches in mind. Why Butterfly Wings Shimmer Description: Basic conceptual and quantitative questions on thin film interference followed by an application to the iridescence of butterfly wings. Learning Goal: To understand the concept of thin-film interference and how to apply it. Thin-film interference is a commonly observed phenomenon. It causes the bright colors in soap bubbles and oil slicks. It also leads to the iridescent colors on many insects and bird feathers. In this problem you will learn how to work with thin-film interference and see how it creates the dazzling display of a tropical butterfly's wings. When light is incident on a thin film some of the light will be reflected at the front surface of the film and the rest will be transmitted into the film. Some of the transmitted light will be reflected from the back surface of the film. The light reflected from the front surface and the light reflected from the back surface will interfere. Depending upon the thickness of the film this interference may be constructive or destructive. We will be studying the interference of light normal to the surface of the film. The figure shows the light entering at a small angle to normal only for the purpose of showing the incident and reflected rays. For this problem you will only be concerned with the geometric aspects of thin-film interference so ignore phase shifts caused by reflection from a medium with higher index of refraction. (Because of the structure of a butterfly's wings such phase shifts do not contribute much to what you actually see when you look at the butterfly.) Assume that light is incident normal to the surface of a film of thickness back surface travel than the light reflected from the front surface? Express your answer in terms of.. How much farther does the light reflected from the Part B For constructive interference to occur the difference between the two paths must be an integer multiple of the wavelength of the light (as is true in any interference problem) i.e. the general criterion for constructive interference is where is a positive integer. This is usually stated in the slightly more explicit form. Given the thickness of the film what is the longest wavelength that can exhibit constructive interference? Express your answer in terms of. = Part C If you have a thin film of thickness 300 what is the third-longest wavelength of light that exhibits constructive interference with the reflected light? Note that this corresponds to. Express your answer in nanometers to three significant figures. = Page 10 of 15

11 Part D The criterion for destructive interference is very similar to the criterion for constructive interference. For destructive interference to occur the difference between the two paths must be some integer number of wavelengths plus half a wavelength: or where is a nonnegative integer. What is the second-longest wavelength that will not be visible (i.e. will have strong destructive interference for the reflected waves) when reflected from a film of thickness 300? Note that the longest wavelength corresponds to for destructive interference. This is why the notation used for the second-longest wavelength is instead of. Express your answer in nanometers to three significant figures. = The blue morpho butterfly lives in tropical rainforests and can have a wingspan greater than 15 cm. The brilliant blue color of its wings is a result of thin-film interference. A pigment would not produce such vibrant pure colors. What cannot be conveyed by a picture is that the colors vary with the viewing angle which causes the shimmering iridescence of the actual butterfly. The scales of the butterfly's wings consist of two thin layers of keratin (a transparent substance with index of refraction greater than one) separated by a 200- gap filled with air. Part E What is the longest wavelength of light that will exhibit constructive interference at normal incidence? The keratin layers are thin enough that you can think of them as representing the surfaces of a 200- "film" of air. Express your answer in nanometers to two significant figures. = This wavelength lies near the cutoff between visible (violet) and ultraviolet light. The shorter wavelengths that are strongly reflected (corresponding to in ) will not be relevant to what humans see when they look at the butterfly. To understand why the color changes with viewing angle try drawing a diagram of light incident on a thin film at a large angle. The distance the light travels within the film will be increased. However at large angles the light reflected from the front surface will actually have to travel farther to an observer outside of the film than the light reflected from the back surface. The increased distance outside of the material for the front surface reflection actually makes the net pathlength difference smaller than it would be for normal incidence. As viewing angle increases the largest wavelength that experiences constructive interference gets shorter. Thus although the butterfly is blue at normal incidence at large angle of incidence no particular wavelength of visible light is short enough to be strongly reflected. Part F The wavelength that appears in the interference equations given in Parts B and D represents the wavelength of light within the medium of the film. So far we have assumed that the medium composing the film is air but many thin-film problems will involve films with an index of refraction different from that of air. Suppose that the butterfly gets wet thus filling the gaps between the keratin sheets with water ( in air will be strongly reflected now? ). What wavelength Page 11 of 15

12 Hint F.1 How to approach the problem In Part E you found the wavelength that exhibits constructive interference; represents the wavelength of the light within the medium of the film (the medium being water in this case). Now find the wavelength of light in air that once in water has a wavelength of. Hint F.2 Relating the wavelength in air to the wavelength in water Recall that the wavelength within a medium is related to the wavelength in vacuum (or air to the accuracy we are working with) by the equation where is the index of refraction of the medium. Be careful not to confuse this with the arbitrary positive integer in the equations introduced earlier in this problem. Express your answer in nanometers to two significant figures. = Part G Several thin films are stacked together in each butterfly wing scale. How would these multiple layers of thin films affect the light reflected by the butterfly's wings? Hint G.1 A picture of the situation The figure shows a model of the layers in the scales on the butterfly's wing. Make a similar drawing so that you can draw in the rays of light to help you visualize what is occurring in this situation. Keep in mind that when light strikes a transparent boundary some of the light is reflected and some is transmitted. There would be no change because all of the 400- light would be reflected by the first scale and so the light is unaffected by the remaining scales. More 400- light would be reflected because some of the light transmitted through the first layer could be reflected by the second layer light transmitted by the second layer could be reflected by the third etc. Less 400- light would be reflected because light reflected from the second layer can interfere destructively with light reflected by the first layer. There would be no change because the interference effects would all occur at the first layer. More 400- light would be reflected because the effective gap would be three times as wide. Less 400- light would be reflected because the effective gap would be three times as wide. Layering of thin films is used to make wavelength-specific mirrors used widely in laser applications that are far more reflective than metal mirrors. Layered films are also used to make antireflective coatings for camera lenses. These last three problems are about diffraction. Problem Description: The second minimum in the diffraction pattern of a a-wide slit occurs at theta. (a) What is the wavelength of the light? The second minimum in the diffraction pattern of a wide slit occurs at What is the wavelength of the light? = nm Page 12 of 15

14 Diffraction due to a circular aperture is important in astronomy. Since a telescope has a circular aperture of finite size stars are not imaged as points but rather as diffraction patterns. Two distinct points are said to be just resolved (i.e. have the smallest separation for which you can confidently tell that there are two points instead of just one) when the center of one point's diffraction pattern is found in the first dark ring of the other point's diffraction pattern. This is called Rayleigh's criterion for resolvability. Consider a telescope with an aperture of diameter Part D What is the angular radius of the first dark ring for a point source being imaged by this telescope? Use nanometers for the wavelength since this is near the average for visible light. Express your answer in degrees to three significant figures. = Part E Two stars in a certain binary star system have angular separation of resolved with the telescope described above? degrees when viewed from earth. Can they be yes no Resolving Power of the Eye Description: Use the measured resolving power of a person's eye to calculate the diameter of the pupil of the eye by assuming that the resolving power is diffraction limited. If you can read the bottom row of your doctor's eye chart your eye has a resolving power of one arcminute equal to If this resolving power is diffraction-limited to what effective diameter of your eye's optical system does this correspond? Use Rayleigh's criterion and assume that the wavelength of the light is 570. Hint A.1 Definition of Rayleigh's criterion Recall that Rayleigh's criterion for resolving objects is that two objects are just distinguishable from each other if the center of one object's diffraction pattern lies on the first minimum of the other object's diffraction pattern. Hint A.2 Diffraction equation for a circular aperature The diffraction equation for a circular aperature is where is the wavelength of light used is the diameter of the circular aperture and is the angle at which the first minimum lies (relative to a centerline from the aperature). The factor 1.22 comes from the fact that we are using a circular aperture for the diffraction pattern instead of an infinite slit. Page 14 of 15

15 Express your answer in millimeters to three significant figures. Summary 0 of 9 items complete (0% avg. score) 0 of 45 points Page 15 of 15

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