# Interference. Physics 102 Workshop #3. General Instructions

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1 Interference Physics 102 Workshop #3 Name: Lab Partner(s): Instructor: Time of Workshop: General Instructions Workshop exercises are to be carried out in groups of three. One report per group is due by the end of the class. Each week s workshop session would typically contain three sections. The first two sections must be completed in class. The third section should be attempted if there is time. 1. A pre-lab reading and assignment section 2. Experiment section 3. Practice questions and problems This week s workshop session helps in understanding the concept of interference. Part I: Pre-lab reading assignment Light is a form of electromagnetic waves. Waves move differently than particles. Waves diffract (deflect in various directions) when passing through a narrow opening, while particles continue along a straight line. Interference is exhibited by all types of waves - sound waves, waves on water, electromagnetic waves, etc. You will observe and investigate interference patterns using visible light during this workshop session. Essential definitions Monochromatic beam of light: A monochromatic beam of light is a beam of light of a single wavelength (or color). Coherrent sources or waves: Two or more sources are called coherent sources if the waves that leave the two sources bear a definite relationship to each other. If the two waves do not bear a definite relationship to each other then they are said to be incoherent. Figure 1 is a sketch of two coherent sources. The difference between phases of the swings of each wave is constant in time and space. The waves do not need to be continuous. However, if the wave has a discontinuity, the other wave must change in exactly the same wave. Figure 2 is a sketch of two incoherent sources. The waves have discontinuities at different points, thus the phase difference changes. If waves have different wavelengths then they are incoherent since the phase difference changes continuously. 1

2 Figure 1: Coherent waves Figure 2: Incoherent waves Phase: Two coherent waves are said to be exactly in phase if a crest (trough) of one wave exactly coincides with the crest (trough) of the other wave. The two waves shown in Figure 1 are exactly in phase with respect to each other. Constructive and destructive interference: When one wave encounters the other they simply add and create one resultant wave (superposition principle). When two waves having the same amplitude and wavelength and exactly in phase with respect to each other interfere with one another the amplitude of the resulting wave is twice the amplitude of the individual waves and the two waves are said to undergo constructive interference. Similarly, when two waves with the same amplitude and wavelength and exactly out of phase with respect to each other interfere with one another the amplitude of the resulting wave is zero and the two waves are said to undergo destructive interference. Source of waves must be coherent to create a stable interference pattern. Waves passing through a narrow slit: When a plane wave encounters a barrier with a narrow opening, it bends and fans out as a circular wave as it passes through the opening. (Figure 3a shows this). On the other hand a parallel beam of particles while passing through a narrow opening do not spread out (Figure 3b). 2

3 Figure 3: Waves through a narrow slit Double-slit interference pattern: Consider a parallel, monochromatic beam of light (for example light from a laser beam) incident on a barrier that consists of two closely spaced narrow slits S1 and S2. The narrow slits split the incident beam into two coherent beams of light. After passing through the slits the two beams spread out in all directions. They overlap each other, hence interfere (Figure 4a). If this transmitted light is made to fall on a screen some distance away one observes an interference pattern of bright and dark fringes on the screen (depicted in Figure 4b and Figure 4c). The bright fringes correspond to regions where the light intensity is a maximum (constructive interference) and the dark fringes correspond to regions where the intensity is a minimum (destructive interference). Figure 4: Double-slit interference 3

4 How is interference patterns produced? Figure 5: Interference production The bright fringe at the center of the screen between the two slits is considered for analysis. Figure 5 shows the waves reaching the point P from the slits S1 and S2. The waves emerging from slits S1 and S2 are exactly in phase. Since they travel the same distance to reach the point P they are in phase at the point P. The two waves therefore interfere constructively at the point P and hence there is a bright fringe (or a maximum) at the center of the screen. To explain how the other bright fringes seen on either side of the central maximum are produced refer to Figure 6. Let λ be the wavelength of the incident light. Let d be the distance between the two slits and L be the distance from the slits to the screen. Suppose the point P1 corresponds to the position of the first maximum above the central bright fringe. Figure 6: First bright fringe From fig. 6 it is clear that the line from S2 is greater than the line from S1 which means that the wave from S2 travels an extra distance to reach point P1. If we assume that the slit-to-screen length L is much greater than the slit separation d (L >> d) then the lines 4

5 from the two slits to the point P1 are approximately parallel and the excess distance traveled by the wave from S2 or the path difference is approximately d sin θ. Since the two waves interfere constructively at point P1, the two waves reaching point P1 must be exactly in phase and hence the path difference must be equal to one wavelength (λ). d sin θ = λ (1) Equation (1) is also responsible for the presence of the first bright fringe below the central maximum. In general, for all points on the screen where the path difference is some integer multiple of the wavelength the two waves from the slits S1 and S2 arrive in phase and bright fringes are observed. Thus the condition for producing bright fringes is d sin θ = λ, 2λ, 3λ, 4λ, (2) Similarly, dark fringes are produced on the screen if the two waves arriving on the screen from slits S1 and S2 are exactly out of phase. This happens if the path difference between the two waves is an odd integer multiple of half-wavelengths. d sin θ = λ/2, 3λ/2, 5λ/2, 7λ/2, (3) If y is the distance of a bright or dark fringe from central maximum then, tan θ = y / L (4) Our assumption L >> d implies that the angle θ is very small. Hence Sin θ tan θ = y / L (5) Equations (2), (3) and (5) imply that interference maxima (bright fringes) occur at y = 0, λl/d, 2λL/d, 3λL/d, (6) and interference minima (dark fringes) occur at y = 0, λl/2d, 3λL/2d, 5λL/2d, (7) Thus, the distance between adjacent maxima or adjacent minima (also reffered to as fringe separation is y = λl/d (8) 5

6 Review Questions: Does an ordinary light bulb emit monochromatic light? Explain. A laser beam of wavelength 650 nm (650 * 10-9 m) meter is incident on a double-slit with a slit separation of mm. An interference pattern is produced a distance 1.5 m away from the slits. What is the distance between adjacent dark fringes? Show your calculation. 6

7 Part II: Observing Interference This part of the lab involves the use of a LASER. Please be extremely careful. Please DO NOT look into the laser or aim the laser at other students (especially eyes). Doing so would DEFINITELY result in blindness. You will be using a red diode laser, an optical bench, a slit accessory and a screen. You will also need a meter stick, a small plastic ruler and a graph paper to do the experiments. Getting familiar with the setup: Figure 7: the experiment setup 1. Each table is provided with a laser mounted on an optical bench, a slit accessory and a screen (either a wooden board or a piece of paper). 2. Align the optical bench in such a way that it is perpendicular to the screen. 3. Each table is provided with a slit accessory a plastic circular disk with apertures mounted on a metal frame. The figure below (fig. 8) shows the slit set up. All measurements printed on the slits are in millimeters. 4. In the slit accessory, identify the following I. a = 0.04 mm and d = mm as slit #1 II. a = 0.04 mm and d = 0.25 mm as slit #2 III. a = 0.04 mm and d = 0.5 mm as slit #3 IV. Slit #4 is actually the end of the VARIABLE DOUBLE SLIT with the largest slit separation. This corresponds to a double-slit with dimensions a = 0.04mm and d = 0.75 mm. 5. The laser beam safety: AVOID LOOKING DIRECTLY INTO THE LASER BEAM OR THE REFLECTION OF THE BEAM FROM A MIRROR OR METAL SURFACE. Place the laser at the end of the optics bench furthest from the wood board. 6. Switch the laser ON and you should see a bright red spot on the wooden board. 7. Place the slit accessory on the optics bench about 10 cm to 15 cm (100mm to 150 mm) in front of the laser beam with the dark plastic surface facing the laser beam. 7

8 8. Arrange the equipment so that the distance L between the slit accessory and the wood board is exactly 1m. You must maintain this distance unchanged for the duration of the entire lab. 9. Rotate the circular plastic disk so that the laser beam passes through the slit # Adjust the two alignment screws on the back of the laser to direct the laser beam through the slit and maximize brightness of the image on the screen. 11. If your arrangement is correct, you will observe an interference pattern on the screen. The interference pattern consists of closely spaced bright and dark fringes. You may have to get close to the screen to see this pattern. Consult your instructor if you are not sure you have set things up correctly. Interference using double-slits: Figure 8: Double-slit accessory In this part of the lab, you would be determining the wavelength of light emitted by the laser by observing the interference pattern. 1. Follow instructions given in the previous part to observe the interference pattern caused by slit #1. 2. Attach a sheet of paper or graph sheet to the screen in such a way that the entire interference pattern is captured on the paper. You will observe that the dark and bright fringes are evenly spaced. 3. You will have to accurately measure the distance between the fringes 8

9 a. Use a pencil to mark out the position of several (8 to 10 if possible) dark fringes on the sheet of paper by making a thin vertical line at the center of each dark spot. b. Remove the sheet of paper and measure the distance from the first mark to the last mark with a ruler, or use markings on the graph paper. It you use graph paper make sure it is metric or make conversion from inches to meters (1 in = 25.4 mm). c. Count the number of bright fringes enclosed by the first and last marks on the sheet of paper. d. Divide the total distance measured between the first and the last mark by the total number of bright fringes counted. This is the average distance y between adjacent dark fringes. Record your measurements in the table below. 4. Repeat step 3 for slits #2, #3 and #4. 5. From the equation (8) from part I, λ = y d/l 6. Determine the wavelengths from the measurements made on each of the four slits. Express them in m and in nm. 1nm = 10-9 m. 7. Calculate the average value of wavelength. Compare it with the wavelength written on the laser. DOUBLE- SLIT DIMENSIONS (mm) DISTANCE BETWEEN FIRST AND LAST MARK (mm) NUMBER OF FRINGES COUNTED AVERAGE Y (mm) WAVELENGTH (λ) (mm) (m) What is the accuracy of your measurement? (Calculate it) 9

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