Transverse Waves Lab: Wave Speed, Standing Waves, and Diffraction, Lab Reports of Physics

Download Transverse Waves Lab: Wave Speed, Standing Waves, and Diffraction and more Lab Reports Physics in PDF only on Docsity! Lab #1: Transverse Waves Reading Assignment: Chapter 17 Introduction: The speed, v, of a wave depends upon the properties of the medium through which it propagates. This is often a misunderstood feature of wave phenomena. Instead, it is commonly thought that the frequency of a wave determines the speed with which it travels. In particular, high frequency waves are thought to travel faster than low frequency waves. However, if this were true, then listening to a music concert would not be possible. If higher pitched notes traveled faster than lower pitched notes, then they would reach the back of the auditorium prior to the arrival of the lower pitched note created at the same time. The resulting sound heard by the listener would be a haphazard mix of notes that could hardly be called “music”. If an oscillating source, such as someone’s hand, is put into contact with the end of a rope it will raise the end of the rope by exerting an upward force on the rope. The resulting acceleration of the rope is inversely proportional to the mass of the segment of the rope upon which the force acts. Since the force does not act on the entire rope at the same time, we can approximate the response by dividing the rope into many small segments of rope having length dl and mass dm. The mass per unit length of the rope is just dm/dl = m/l. In the diagram below a pulse encounters a segment of stretched rope. The rope rises in response to the force but the inertia of the segment of the medium prevents it from rising instantaneously. Once the mass reaches the maximum displacement, it falls due to the restoring force of the medium, FT = Tension. Experimentally and theoretically, it can be shown that the speed of propagation of a wave pulse is determined by the following: v = FT m l The source of a periodic oscillation determines the frequency at which pulses are produced in the medium. For example, a person who shakes his/her hand back and forth determines the frequency at which the pulses enter the rope. The frequency of the source is the same as the frequency of the resulting wave pulses. It is a combination of the frequency of the source and the speed of the medium that determines the wavelength (λ ) of the periodic wave that enter the medium. If the source sends a pulse into a relatively fast medium then the first part of the pulse will have the opportunity to travel a great distance before the next pulse is sent behind it, causing a longer wavelength. Whereas, if the medium is slow, the first part of the pulse will not travel as far into the medium before the next one is sent behind it, causing a shorter wavelength. A simple relation can be obtained that summarizes these relationships. v = λf Lab #12 – Transverse Waves This equation should not be misinterpreted. It states mathematically that the product of the frequency and the wavelength determines the speed, v, of the waves. However, as stated above, it is actually the combination of the frequency of the source and the speed of the medium that determines the wavelength, λ, of the wave. Since v = λf and v = FT m l , it must follow that: FT m l = λf In this experiment we investigate this relationship. Another known property of waves is that they reflect off of a boundary. A boundary is defined as the place where a medium changes or ends. The reflected waves interfere with the waves that are still traveling toward the boundary. Under certain conditions, the superposition of the incident and reflected waves produce what is known as standing wave patterns. For an oscillating medium fixed at both ends, the waves continue to bounce back and forth between two boundaries. The condition that must be met to produce standing waves in a fixed-fixed medium is that the total length, L, of the medium must be an integer multiple of a half wavelength of the propagating wave: L = n 2 λ , where n is any integer. λ L For a string of fixed mass per unit length (non-stretchable) that has a fixed frequency source of oscillation attached to it, there should be multiple lengths of that string that permit standing waves for any given tension. That is:       ⋅⋅= ⋅= ⋅= f FnL f n LF fv l m T l m T 2 1 2 λ Other lengths of the string will produce superposition patterns that are haphazard in appearance. Lab #12 – Transverse Waves Template #1: Transverse Waves Name: Section #: Name: Section #: Name: Section #: Activity 1: Standing Waves on a String (24 pts) 1st Tension f (Hz) λ (m) FT (N) v (m/sec) 8. Your best estimate for the wavelength would be the average of these three measurements. 9. Based upon your average measured wavelength, what is the speed of the waves along the string? (Show your calculation.) Activity 2: Speed and Tension (51 pts) 1. Repeat Activity 1 for at least five different tensions. Be sure to vary the tension significantly. 2nd Tension f (Hz) λ (m) FT (N) v (m/sec) 3rd Tension f (Hz) λ (m) FT (N) v (m/sec) Lab #12 – Transverse Waves 4th Tension f (Hz) λ (m) FT (N) v (m/sec) 5th Tension f (Hz) λ (m) FT (N) v (m/sec) 2. Record the values for the Tensions used and the (average) Speeds obtained in the chart below. Tension (N) Speed (m/sec) 6. Import the Excel graph of Speed vs. Tension. 7. Import the Excel graph of Speed Squared vs. Tension. 8. Using the graphs above as a reference, answer the following questions: • Do your graphs verify the relationship given by the following equation? Explain. v = FT m l • What is the mass per unit length of your string? Explain how this value was determined. Physics 214 Lab #12 – Transverse Waves Lab #2: Inverse Square Law and Polarization of Light NAME: ____________________________________ PARTNERS: ____________________________________ ____________________________________ PHYS 214R SECTION: __________________________ PHYS 214R INSTRUCTOR: __________________________ DATE: __________________________ EMAIL ADDRESS: __________________________ Lab #12 – Transverse Waves Phys 214R Lab #2: Inverse Square Law and Polarization of Light Activity 1: The Inverse Square Law The aim of this experiment is to examine the inverse square law. Set up • Place the Light Source at a convenient place on the Optical Bench. • Place the Light Sensor at a distance of 10 cm away from the Light Source. Allow enough room on the Optical Bench to move the Light Sensor an additional 70 cm further away from the Light Source. • Rotate the aperture bracket and place the perforated screen in front of the Light Sensor. • Turn off the room lights. • Open Science Workshop and install the Light Sensor in port A. Drag a Digits Meter onto the Light Sensor and set it to display two digits. • Spend a few minutes to see what sensitivity setting on the Light Sensor results in readings being available along the entire length of the bench. Be careful not to overwhelm the Sensor at its closest distance. You can adjust the Light Sensor by adjusting the sensitivity button. Collecting data • Open Excel and set up the following data table (make sure to specify the units for all measured quantities): Distance, R Intensity, I • Record the Light Sensor reading at every 5 cm interval, moving away from the Light Source, and enter the data into the above table Q1. Using Excel, plot and analyze your experimental data in a manner that tests whether the inverse square law is obeyed. Q2. Comment on any sources of error or approximations that enter into your analysis of the experiment. Lab #12 – Transverse Waves Activity 2: Study of Malus’ Law The aim of this experiment is to use a polarizer and analyzer to study Malus’ Law. Figure 1: The setup for verification of Malus' law Set-up • Mount the laser light source, the two polarizers, and the Light Sensor on the optical bench, as shown in the figure below. Adjust the Light Sensor to allow light to pass through the narrowest slit. • Open Science Workshop and install the Light Sensor in port A. Drag a Digits Meter onto the Light Sensor. Calibration of The Light Sensor. • Set both polarizers at zero degrees and adjust the sensitivity of the Light Sensor to read less than 40 units. • Rotate one of the polarizers to 90o. Double click on the Light Sensor icon to open the calibration window. • Click the READ button to the right of the Low Value windows. This sets the zero value of the Light Sensor at whatever value it is currently reading due to the background light from the room entering the device. Lab #12 – Transverse Waves Setting the display format of the digits meter • Double click on the display window of the Digits Meter. • The Data Type should have the Current Value radio button darkened. • The Display Format should have 3 digits left of the decimal and 1 digit right of the decimal. Collecting Data • Return both polarizers to the 0o position. • Open the Excel spreadsheet provided. • Click on the MON Button on Science Workshop to monitor the readings of the Light Sensor. • Record the values being displayed by the Digits Meter in the Excel spreadsheet. Rotate the analyzer 10o and record the new value. • Continue to rotate the analyzer by 10o and collect readings for a complete revolution of the analyzer. Make sure to additional readings around the maxima and minima. Q3. In Excel, create a graph of transmitted intensity (I) vs. angle (θ). Explain briefly how you would convert this into a plot showing the variation of the amplitude of the transmitted electric field vs. angle. Q4. Using Excel, plot and analyze your data in a manner that tests whether Malus’ Law is obeyed. Include any relevant plots and analysis with your report. Q5. Comment on sources of error in your measurements. Lab #12 – Transverse Waves Activity 2: Study of Malus’ Law (PASTE YOUR DATA TABLE HERE.) Q3. In Excel, create a graph of transmitted intensity (I) vs. angle (θ). Explain briefly how you would convert this into a plot showing the variation of the amplitude of the transmitted electric field vs. angle. (Paste your plot here and follow with analysis.) Q4. Using Excel, plot and analyze your data in a manner that tests whether Malus’ Law is obeyed. Include any relevant plots and analysis with your report. (Paste your plot here and follow with analysis.) Q5. Comment on sources of error in your measurements. Activity 3: Studying the effect of a λ/2−Wave Plate (PASTE YOUR DATA TABLE HERE.) Q6. In Excel, create a graph of intensity (I) vs. angle (θ). (Paste your plot here and follow with analysis.) Q7. From your plot in Q6, what do you conclude about the effect of the λ/2-wave plate on a linearly polarized wave incident at 450 to the optic axis? Explain your answer. Q8. Someone suggests that perhaps the λ/2-wave plate works by introducing a 1800 phase difference between two orthogonally polarized waves as they travel through the plate. Develop this argument further. Lab #12 – Transverse Waves Physics Post-lab 214 Inverse square law and Polarization Name:__________________________ Section:_____ Date:__________ Q1. If the Sun delivers about 1 kW/m2 of electromagnetic flux to the Earth’s surface, what is the total power output by the Sun? Q2. In the figure below, initially unpolarized light is sent through three polarizing sheets whose polarization directions make angles of θ1 = θ 2 = θ 3 = 300 with the y-axis as shown. Calculate the percentage of the incident intensity that is transmitted by the system of three sheets. Lab #12 – Transverse Waves Pre-Lab #3: Interference and Newton’s Rings Name: ____________________ Section #: _________________ Questions: 1. Halliday, Resnick and Walker, 6th Ed. Problem 36-49P. 2. Halliday, Resnick and Walker, 6th Ed. Problem 36-50P. 3. Halliday, Resnick and Walker, 6th Ed. Problem 36-51P. Lab #12 – Transverse Waves d) Determine h, the difference between the two readings. e) Repeat this measurement several times, and calculate the average value of h. f) Place the spherometer on a sheet of paper, and make an impression of the three legs and the screw point as in Figure 2(b). With a vernier caliper, measure the distance X from the center point to the point of each leg. g) Calculate the average value of X. h) Calculate R using Eq. (7) Activity 3: Determining the Wavelength of the Monochromatic Light 1. In Excel, create a graph of D2 vs. m. Copy the graph and paste it into the Template. 2. From your graph determine the values of the slope and hence the wavelength λ Template #3: Interference and Newton’s Rings Lab #12 – Transverse Waves Date__________________ Names: _____________________________________ Section _______________ Partner _____________________________________ Partner _____________________________________ Activity 1: Observing Newton’s Rings 6. Open Excel and set up the following …. -Insert completed table here- Activity 2: Determining the Radius of the Lens b) Place the lens on the three legs of the spherometer as shown in Figure 2(a), and advance the screw until it just makes contact with the glass. Record the reading. ___________________ c) Replace the lens with a flat glass plate. Advance the screw (in the same direction to avoid backlash) until it touches the surface, and record the reading. ___________________ d) Determine h, the difference between the two readings. h =____________________ e) Repeat this measurement several times, and calculate the average value of h. have =____________________ g) Calculate the average value of X. Xave =____________________ h) Calculate R using R =____________________ Activity 3: Determining the Wavelength of the Monochromatic Light 3. In Excel, create a graph of D2 vs. m. Copy the graph and paste it into the Template. -Insert completed graph here- Lab #12 – Transverse Waves 4. From your graph determine the values of the slope and hence the wavelength λ λ =____________________ Show your calculation of the slope Lab #12 – Transverse Waves Reading Assignment: Halliday, Resnick, and Walker: Chapter 37 (especially 37.2, 37.4, 37.6, 37.7) Fundamentals of Optics by Jenkins and White Optics by Hecht and Zajac Geometrical and Physical Optics by Longhurst. Introduction: When a beam of light passes through a narrow aperture it spreads out into the region of the geometrical shadow. Such an effect is known as diffraction. We will be studying diffraction from various types of apertures in the first part of this experiment. Diffraction by a single slit The diffraction pattern obtained from a single slit is shown in figure 1. The intensity as a function of angle θ is given by 2 2 sin sin sinsin      =                   = α α λ θπ λ θπ oo Ia a II Eq. 1 Figure 1: The single slit diffraction pattern where α = πa sinθ λ Eq. 2 a is the slit width and λ is the wavelength of the light used. Io is the intensity at θ = 0, corresponding to the principal maximum. The diffraction pattern is formed in a direction which is perpendicular to the length of the slit. The positions of the diffraction minima in the pattern are found from the condition α = ±π, ±2π, ±3π, … which yields the formula Lab #12 – Transverse Waves sinθ = ±n λ a , n = 1, 2, 3, ... Eq. 3 Diffraction by a double slit Figure 2: The double slit diffraction pattern Consider a diffracting aperture consisting of two long parallel slits each of width a and separated by a distance d. The transmitted intensity is given by β α α λ θπ λ θπ λ θπ 2 22 2 cossin4sincos sin sinsin 4      =                              = oo I d a a II Eq. 4 where α was defined in eq. 2 and λ θπβ sind= Eq. 5 The factor of 4 arises from the fact that the amplitude of the wave is twice what it would be if one slit were covered. The "diffraction'' term in eq. 4 is recognizable as the intensity distribution for a single slit and serves as the envelope for the "interference'' term cos2β which describes the interference between the diffracted beams from each slit. The appearance of the intensity as a function of sinθ for the case d = 3a is shown in figure 2. If a is small, the diffraction pattern from either slit will be essentially uniform over a broad central region and interference fringes will be evident in that region. Diffraction minima are located at sinθ = ±n λ a , n = 1, 2, 3, ... Eq. 6 while the interference minima are located at Lab #12 – Transverse Waves sinθ = ± m + 12( ) λ d , m = 1, 2, 3, ... Eq. 7 If the slit width a is kept constant and the separation d is increased (or decreased) the nature of the interference pattern changes, though the diffraction envelope itself remains unaltered. The number of interference fringes in the central diffraction maximum is (2d/a) -1. (The factor of 2 comes from identical patterns on either side of the central point. The term –1 arises because the m=+0 and m = -0 are the same) Template #4: Single and Double Slit Diffraction Name:_______________________________ Section #:________ Lab #12 – Transverse Waves • Divide the distances between side orders by two to get the distances from the center of the pattern to the first and second order minima. Record those values of y in the table provided in the template. • Using the average wavelength of the laser (670 nm for the Diode Laser), calculate the slit width twice, once using the first order (a1) and once using the second order (a2) minima. Record the results in the table. • Calculate the percent differences between the experimental slit widths and 0.04 mm. Record in the table. Distances Slits to Screen Between m=1 Center to m=1 Between m=2 Center to m=2 Slit Spacing (d) Slit Width (a1) Slit Width (a2) % difference Post Lab #4: Single and Double Slit Diffraction Name:_______________________________ Section #:________ Questions: 1. Consider a double-slit interference experiment. If the index of refraction, n, of the material between the slits and the screen is doubled and the frequency of the incident light is halved, in what way would the spacing between the interference maxima change? Lab #12 – Transverse Waves 2. Light passes through a rectangular slit that is narrow but tall. (a) Will the resulting diffraction pattern be more spread out in the horizontal or vertical direction? (b) If you wanted to construct a loud speaker, would you build a narrow and tall one or one which is wide and short? Explain. 3. In a single-slit diffraction pattern, what is the intensity at a point where the total phase difference between wavelets from the top and bottom of the slit is 66 radians if the central intensity is I0? Lab #5: Atomic Spectra Reading Assignment: Halliday, Resnick, and Walker: Chapters 37 (especially 37.7 and 37.8) and 40 (40.6) Introduction: Lab #12 – Transverse Waves A diffraction grating essentially consists of a number of very narrow parallel slits placed at regular intervals. A grating has a strong dispersive power, i.e., it can separate out light of different wavelengths. Thus, we can examine the components of a given light source by passing it through a diffraction grating. Every element in nature, under certain conditions, emits light with a definite set of wavelengths (its emission spectrum). This set is unique to the element in question, and so we may identify the element by its spectrum. Diffraction by a single slit In Lab #5 we studied the diffraction pattern obtained from a single slit. The intensity as a function of angle θ is given by 2 2 sin sin sinsin      =                   = α α λ θπ λ θπ oo Ia a II Eq. 1 with α = πa sinθ λ Eq. 2 where a is the slit width, λ is the wavelength of the light used and Io is the intensity at θ = 0. Diffraction by a double slit In Lab #5 we also studied the interference pattern produced by a double slit. The transmitted intensity is given by β α α λ θπ λ θπ λ θπ 2 22 2 cossin4sincos sin sinsin 4      =                              = oo I d a a II Eq. 3 with λ θπβ sind= Eq. 4 where d is the separation between slits. Diffraction by multiple slits When a parallel beam of monochromatic light is incident normally on a diffraction grating, there is a Fraunhofer diffraction pattern in the transmitted light. For the case of N slits each of width a and separation d the intensity is given by 22 sin sinsin            = β β α α NII o Eq. 5 with α and β as defined above. For N = 2 this reduces to Eq. 3 for the double slit. The first factor is the single slit diffraction envelope, while the second one represents the interference term Lab #12 – Transverse Waves PreLab #5: Atomic Spectra Name:_______________________________ Section #:________ Questions: 1. What will be the effect on the width of the interference lines, if the number of rulings in a diffraction grating is increased while keeping the distance between the rulings the same? 2. For a given order (e.g. m = 2), is the angle of the diffraction maxima for red light larger or smaller than the angle for blue light? Explain. 3. A grating has 315 rulings/mm. For what wavelengths in the visible spectrum can the 5th order diffraction be observed when this grating is used in a diffraction experiment? 4. The following figure shows red line, a blue line and a violet line of the same order in the pattern produced by a diffraction grating. 1. Is the location of the point corresponding to θ = 0 to the left or right in this figure? 2. If we increased the number of rulings in the grating, eg by removing tape that had covered half of the rulings, would the half width of the lines increase, decrease or remain the same? 3. Would the separation of the lines increase, decrease or remain the same? 4. Would the lines shift to the right, to the left or remain in place? 5. Sketch an energy level diagram of the hydrogen atom with the various levels labeled with the proper value of the quantum number n. Indicate on your diagram which transitions cause the four lines of the Balmer series? Lab #12 – Transverse Waves Lab #5: Atomic Spectra Name:_______________________________ Section #:________ Name:_______________________________ Name:_______________________________ Goals: 1. To study the use of a diffraction grating with sodium light 2. To measure the visible spectrum of hydrogen 3. To determine some of the energy levels in the H atom Equipment Needed: 1. Spectrometer setup: 2. Optics bench (OS- 8518) with cm scale 3. Collimating Slits 4. Collimating lens 5. Focusing Lens 6. Aperture Bracket (CI- 8534) for Light sensor with Post 7. High Sensitivity Light Sensor (CI-6604) with cable 8. Rotary Motion Sensor (CI-6538) 9. Grating Mount 10. Diffraction Grating 11. Sodium Lamp 12. Hydrogen Lamp 13. Science Workshop Figure 1 Lab #12 – Transverse Waves Figure 2 1. 2. It m Figu lens sens 1 2 3 4 5 The sodium lamp must be switched on at least 10 minutes before you use it in order to let it warm up properly. The hydrogen lamp has a short life-time, so do not keep it on when not required. ight be helpful to review any important points in the description of the setup here (see res 1 and 2). In this experiment, we send light through (1) a collimator slit and collimator , (2) a fine diffraction grating, (3) a focusing lens, (4) an aperture disk, and finally (5) a light or. . The collimators serve to change the original beam shape to a rectangular shape with sharp edges, i.e. collimated. The slit should be set to the smallest size, slit 1 for this experiment. (If it is not already set, you can loosen the black screw and slide the slit plate so that light passes through slit 1.) The collimating lens helps the beam keep its size and shape over the distance it must travel to the light sensor. . The diffraction grating diffracts the beam of light. You have seen what happens when light is diffracted in a previous lab. In particular, the light waves spread out and interfere with each other so that if we allow the resulting light to fall on a surface, we see there are locations where there are maxima and minima. In fact, the angle at which the light maxima occur is given by θ in Eq. 6 above. d is the characteristic separation between grating lines. . The focusing lens serves, not surprisingly, to focus the components of the highly diffracted beam that is spreading out over the distance from the light source to the light sensor. We would like them to be focused on the aperture in front of the light sensor. . The aperture disk is the disk with various slits located just in front of the light sensor. We would also like this set to slit 1, the smallest size. . The high sensitivity light sensor detects the beam hitting its aperture and sends a signal to Science Workshop, which then returns a measurement of the intensity of that light. The sensor has a switch for variable gains that you will have to play with to obtain the best plots. Lab #12 – Transverse Waves Activity 2: Study of the Hydrogen Spectrum. 17. Replace the sodium lamp with a hydrogen lamp (and use slit 3) and repeat the previous procedure (steps 11-15) to obtain the hydrogen dispersion spectrum. If you do not observe any spectral lines, consult your laboratory instructor who will check and replace the bulb if necessary. Color m θ 18. Using your value for d, complete the following table for the m = 1 order of your hydrogen spectrum and compare the measured wavelengths to those expected: Line# Color λ (expected) θ Sin θ λ (meas) Eγ (meas) 1 Red 656.28 nm 2 Cyan 486.13 nm 3 Blue 434.05 nm 4 Violet 410.17 nm 19. The colors of the four visible lines of the hydrogen Balmer Series spectrum are: red, blue- green (cyan), blue and violet. Which initial states ni and which final states nf correspond to each of these colors? Enter your values in the following table: Line# Color ni nf Ei Εf Ei-Ef Eγ (meas) 1 Red 2 Cyan 3 Blue 4 Violet 20. Compare (in the above table) the measured photon energies with those you expect from Eq. 9 and Eq. 10. What do you conclude? Lab #12 – Transverse Waves Template #5: Atomic Spectra Name:_______________________________ Section #:________ Name:_______________________________ Name:_______________________________ Activity 1: Measurement of the Diffraction Grating Spacing (d) using Sodium Light. 2. Record your Grating number 12. Display your data of Intensity vs. Angle (in radians) with the points connected. 13. Verify that the plot exhibits the behavior you might expect. 14. Copy the graph to the template provided. Before deleting data from Science Workshop, record the values of the angles at which the peaks corresponding to the m = 0,1,2,… lines lie: m θ 15. Combine your measurements appropriately (eg, subtract the two values for the same order and divide by 2) so as to obtain measurements of the grating line separation, d, from the following table: m θm Sin θm d 16. Report your best determination of the grating spacing, d. Lab #12 – Transverse Waves Activity 2: Study of the Hydrogen Spectrum. 17. Replace the sodium lamp with a hydrogen lamp and repeat the previous procedure (steps 11-15) to obtain the hydrogen dispersion spectrum. Display your data of Intensity vs. Angle (in radians) with the points connected Color m θ 18. Using your value for d, complete the following table for the m = 1 order of your hydrogen spectrum and compare the measured wavelengths to those expected: Line# Color λ (expected) θ Sin θ λ (meas) Eγ (meas) 1 Red 656.28 nm 2 Cyan 486.13 nm 3 Blue 434.05 nm 4 Violet 410.17 nm 19. The colors of the four visible lines of the hydrogen Balmer Series spectrum are: red, blue- green (cyan), blue and violet. Which initial states ni and which final states nf correspond to each of these colors? Enter your values in the following table: Line# Color ni nf Ei Εf Ei-Ef Eγ (meas) 1 Red 2 Cyan 3 Blue 4 Violet 20. Compare (in the above table) the measured photon energies with those you expect from Eq. 9 and Eq. 10. What do you conclude? Lab #12 – Transverse Waves