Name (printed) LAB TWO DIMENSIONAL WAVE INTERFERENCE INTRODUCTION Tap a stick repeatedly in the water of a pond and you get what you ve always come to expect a succession of circular waves. It makes sense, because if the energy from the stick moves in all directions at the same rate, then the shape of the wave front would geometrically have to be a circle. But try tapping two sticks simultaneously and close together and you would see something very different. You would (if you looked closely enough) see an interference pattern (Figure 5.16). The pairs of outward moving circular waves from the two sources combine (or interfere) in a pattern that is absolutely motionless and stable. The pattern consists of lines (actually pairs of hyperbola) that represent regions of either constructive interference (crests from one source meeting crests from the other source) or destructive interference (crests from one source meeting troughs from the other source). The lines where constructive interference occurs are called antinodes or maxima. The lines where destructive interference occurs are called nodes or minima. To understand the physics of antinodes and nodes, first consider that the two sources of waves are separated by a bit. This means that when each of them is producing waves, there will be points out beyond the two sources where a wave from one source has traveled a different distance than a wave from the other. If the sources are in phase (in step with each other, producing pulses at the same time), then there will be points where the waves will no longer be in step. If at one point they are out of step by a full wavelength, then they are really, back in step. Crests are still aligned with crests and there would be constructive interference at that point Constructive Interference Destructive Interference Figure 5.16: Two sources of waves placed close together, and having the same frequency, will produce a stable interference pattern with regions of constructive and destructive interference. (Figure 5.17). This would also be true if the waves were out of step by two wavelengths (or any integer number of whole wavelengths). However, if the waves at one point were out of step by a half wavelength, then the crest of one would be in line with the trough of the other and they would cancel each other out. This would be a point of destructive interference (Figure 5.17). This would also be the case if the waves were out of step by one-and-a-half wavelengths (or any number of odd half wavelengths). Constructive Interference Destructive Interference Figure 5.17: If two sources of waves are separated by a bit when each of them is producing waves, there will be points out beyond the two sources where a wave from one source has traveled a different distance than a wave from the other. Depending on the difference in distance traveled, the waves can interfere constructively or destructively. 1
PURPOSE To recognize wave interference patterns. To learn to use wave interference patterns to calculate the wavelength of the waves causing the pattern. PROCEDURE (PART 1) In the space below make a neat and careful drawing of the water wave interference pattern you observe. Label the central antinode, the second antinode to the right of the central antinode, and the third node to the left. Figure 5.18: Ripple tank viewed from above. Note the paper towel wave absorbers. 2. Explain the physics of what is causing the second antinode to the right of the central antinode. Figure 5.19: Bobbers producing an interference pattern. Note the flat nodes in between the rippled antinodes. 3. Explain the physics of what is causing the third node to the left of the central antinode. 2
BACKGROUND FOR PART 2 m = 1 m = 0 m = 2 m = 1 m = 2 m = 3 x m = 3 m = 4 m = 4 L L θ Figure 5.20: Interference pattern measurements. S 1 d S 2 You ve seen that when two wave sources have the same frequency and are reasonably close together, they form a stable interference pattern. It s possible to calculate the wavelength of the waves producing this pattern with three measurements from the interference pattern. In order to make these measurements, a point must first be labeled on one of the antinodes. The location of this point is somewhat arbitrary, but should be in the center of the antinode and as far from the sources as possible. Any antinode can be chosen. After the point on the antinode is chosen, three measurements must be made (see figure 5.20): Then, to calculate the wavelength, λ, of the wave, use one the following equations: dsinθ or d x L = mλ The photograph on the following page is of an interference pattern. You can choose points on antinodes and make measurements to calculate the wavelength of the wave producing this interference pattern. L the distance from a point midway between the sources to the point on the antinode. x the distance perpendicular from the middle of the central antinode to the point on the antinode. d the distance between the centers of the sources. You also need to know the number of the antinode, m, that the point is on. Antinodes are numbered from the central antinode (m = 0), starting with m = 1. Antinodes are counted as positive when going either to the left or to the right of the central antinode. (The point in Figure 5.20 is on m = 2.) You can use x and L to calculate the angle, θ, shown in Figure 5.20. Figure 5.21: Marking data on the projected image of a stable interference pattern. 3
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PROCEDURE (PART 2) 1. Locate four distant points on four different antinodes on the photo of the interference pattern on the previous page. It s helpful to have two on each side of the pattern so the drawings don t get too crowded. Use pencils and a ruler to draw lines that show all variables d, x, L. 2. Measure and record d, x, L, and m for each of the points in the data table below. 3. Call me over to make a measurement of your actual wavelength. DATA AND ANALYSIS 1. Calculate the wavelength for each trial as well as the average wavelength. 2. Calculate the experimental error: trial # m d (cm) x (cm) L (cm) λ (cm) 1 2 3 4 Known wavelength: Average exp. wavelength: Percent error: SHOW CALCULATIONS BELOW TRIAL 1 TRIAL 3 TRIAL 2 TRIAL 4 5 GET CHECKED BEFORE MOVING ON
PURPOSE To create a light interference pattern, measure its features and then calculate the wavelength of the light. LAB LIGHT WAVE INTERFERENCE PROCEDURE 1. Direct either red or green laser light through the 5-slit opening in the Multiple Slits section of the wheel. These are slits with slit separation of d = 0.125 mm. You will see a series of bright antinodes (Figure 5.25). You can place the screen farther away in order to increase the size of the pattern. 2. Fold the top inch of a half sheet of paper so that it hangs over the screen facing the laser. Mark the positions of the m = 1-6 maximum pairs of the interference pattern on the paper. Figure 5.22: Thomas Young s visualization of two sources of light waves interfering and causing characteristic antinodes (bright regions) and nodes (dark regions). Figure 5.23: Light Wave Interference equipment. The diode laser is clamped behind the Multiple Slits Wheel. Light from the laser is directed through the desired pattern, producing an interference pattern on the screen in the background. Figure 5.24: High Precision Multiple Slits Wheel. There are a number of double and multiple slit patterns to direct laser light through. L Figure 5.25: Measure L from the wheel to the screen. You will see an interference pattern like the one to the right (this one shows the m = 0, m = 1, m = 2 antinodes. 6 2 1 0 1 2
DATA AND ANALYSIS 1. Make measurements similar to those made in the Two Dimensional Wave Interference Lab to calculate the wavelength of red and green laser light being used to create this interference pattern. Measure and record the distance between the first order maximum (m = 1) marks, the second order (m = 2) marks through to the sixth order (m = 6) marks. 2. Divide the distance between the marks by two and record this distance (from the center of the pattern to the maximums) in the table. Note this distance, x. 3. Since the angles are small, and sinθ tanθ for small angles, the experimental wavelength can be calculated by using the equation λ = dx, ml max. order, m 1 2 3 4 5 6 slit-to-screen distance, L distance btw. marks (m) maximum distance x (m) average experimental wavelength, λ (m) DATA Color d (m) x (m) L (m) m Red Green CALCULATIONS (SHOW ALL WORK) 1. Calculate the wavelength for red light. The conventional unit for wavelength is the nanometer (nm). 1 nm = 1 x 10-9 m. Express your calculated wavelength in nanometers. 2. Calculate the wavelength for green light in nanometers. 3. Obtain a slide with multiple slit patterns from me and determine the spacing between the slits in the center pattern. Show experimental setup, data table, and all calculations below. 7 GET CHECKED BEFORE MOVING ON
QUESTIONS AND PROBLEMS WAVE INTERFERENCE 1. Specify the necessary condition for the path-length difference between two waves that interfere: a. constructively b. destructively? 2. Sources S1 and S2 are in phase. Point P is on the m = 6 antinode. Calculate the wavelength of the wave. S 1 S 2 P 3. What exactly causes the first node to the left (or right) of the central antinode to occur where it does? 4. What exactly causes the third antinode to the left (or right) of the central antinode to occur where it does? 5. Find the wavelength for an interference pattern from point sources 2.3 cm apart where a point on the first maximum is 2.4 cm from the central maximum, and 5.4 cm from the midpoint of the sources. 6. Two radio antennas simultaneously transmit signals of the same wavelength and in phase. A radio in a car receives the signals. If the car is at the second maximum of the interference pattern produced, what is the wavelength of the signals? Refer to the drawing on the right. 300 m 1400 m 27 m 8
7. A sound interference pattern is created using two audio speakers located 1.25 m away from each other. The first node is located at 10 from the central antinode. What is the pitch of the sound wave? GET CHECKED BEFORE MOVING ON 8. Two slits are 0.010 cm apart. Monochromatic light is directed through the slits. The 3 rd order (m = 3) bright lines are 12.5 mm apart and are seen 0.40 m from the slits. What is the wavelength and color of the light? 9. Red light, having a wavelength of 650 nm is directed through two slits separated by 0.0330 mm. The distance from the slits to a wall where the interference pattern is formed is 2.35 m. What is the distance between the two 2 nd order bright lines? 10. Monochromatic light passing through two slits separated by 0.012 cm falls on a screen 2.0 m away, producing the pattern to the right. What is the wavelength of the light? 11. If the light used in the problem above is directed through a different pair of slits, the pattern to the right forms on the same screen 2.0 m away. What is the distance between this pair of slits? 9
The last eight problems are for Honors Physics only 12. A soapy water film (n = 1.33) is in a plastic loop out in the sun. A portion of the film reflects all but the green light (λ = 550 nm). What is the minimum thickness of the film in that portion? 13. A plastic thin film (n = 1.37) with a thickness of 212 nm is placed on the lenses of a pair of sunglasses. If you looked at someone wearing these sunglasses in full sunlight, what color would the lenses be? 14. If you take the wand of a soap bubble maker and hold it so that the plane of the soapy water is vertical, there are colored bands containing each color of the spectrum from violet through red. Then they repeat themselves several times until the bottom of the soapy water in the wand. Explain the occurrence of the colored bands as well as why they repeat themselves. 15. If the wand in the previous question is held vertically for a long enough period of time, the color disappears from the bands at the top of the wand and then finally the film breaks. Explain the disappearance of the colors at the top of the wand. 10 GET CHECKED BEFORE MOVING ON
16. A soap film is deposited on the surface of some glass (n = 1.50). A portion of it appears red (λ = 6.00 x 10-5 cm in a vacuum) when viewed from a right angle in white light. Determine the smallest non-zero thickness for that portion of the soap film if its index of refraction is 1.40. 17. How thick should the coating of a material be with an index of refraction of 1.35 on a piece of flat glass (n = 1.55) if you want it to be an anti-reflection coating for blue light (λ = 450 nm)? 18. A transparent film (n = 1.40) with a thickness of 1.10 x 10-7 m is deposited on a glass lens (n = 1.55) to form a non-reflecting coating. What is the wavelength of light (in vacuum) for which this film has been designed? 19. A certain transparent substance has an index of refraction of 1.35. What is the thinnest film coating of this substance on glass (n = 1.50) for which destructive interference of green light (500 nm) can take place by reflection? The glass and film are in air. 11