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Physics 136-3: General Physics Lab Laboratory Manual - Waves and Modern Physics Northwestern University Version 1.1 December 8, 2016 Contents Foreword 5 1. Introduction to the Laboratory 1.1 Objectives of Introductory Physics Laboratories 1.2 Calculus vs Non-Calculus Based Physics . . . . 1.3 What to Bring to the Laboratory . . . . . . . . 1.4 Lab Reports . . . . . . . . . . . . . . . . . . . . . . . . 7 7 8 8 8 . . . . . . . . . 11 11 12 12 14 15 16 20 21 23 3. Experiment 1: Sound 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 25 36 37 4. Experiment 2: Snell’s Law of Refraction 4.1 Introduction . . . . . . . . . 4.2 The Experiment . . . . . . . 4.3 Part 2: Index of Refraction . 4.4 Analysis . . . . . . . . . . . 4.5 Conclusions . . . . . . . . . . . . . . 39 39 41 42 46 46 5. Experiment 3: Geometric Optics 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 49 64 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Understanding Errors and Uncertainties in the Physics Laboratory 2.1 Introduction: Measurements, Observations, and Progress in Physics . 2.2 Some References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 The Nature of Error and Uncertainty . . . . . . . . . . . . . . . . . . 2.4 Notation of Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Estimating Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Quantifying Uncertainties . . . . . . . . . . . . . . . . . . . . . . . . 2.7 How to Plot Data in the Lab . . . . . . . . . . . . . . . . . . . . . . . 2.8 Fitting Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Strategy for Testing a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CONTENTS 5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Experiment 4: Wave Interference 6.1 Introduction . . . . . . . . . . . . . . 6.2 The Experiment . . . . . . . . . . . . 6.3 Analysis . . . . . . . . . . . . . . . . 6.4 Conclusions . . . . . . . . . . . . . . 6.5 APPENDIX: The Helium-Neon Laser 6.6 The Helium-Neon Laser . . . . . . . 6.7 Laser Safety . . . . . . . . . . . . . . 7. Experiment 5: Diffraction of Light Waves 7.1 Introduction . . . . . . . . 7.2 Analysis . . . . . . . . . . 7.3 Conclusions . . . . . . . . 7.4 APPENDIX: Intensity Distribution from . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 67 71 79 80 81 85 89 91 . . . . . . . . . . . . . . . . . . . . . . . . . . 91 . . . . . . . . . . . . . . . . . . . . . . . . . . 100 . . . . . . . . . . . . . . . . . . . . . . . . . . 102 a Single Slit . . . . . . . . . . . . . . . . . . . 102 8. Experiment 6: Intensity Distributions of Diffraction Patterns 8.1 Introduction . . . . . . . . . . . . . 8.2 The Apparatus . . . . . . . . . . . 8.3 Analysis . . . . . . . . . . . . . . . 8.4 Conclusions . . . . . . . . . . . . . 8.5 APPENDIX: The Semiconductor Diode Laser . . 8.6 The PIN Diode Photodetector . . . 9. Experiment 7: Light Polarization 9.1 Introduction 9.2 Part 1 . . . 9.3 Analysis . . 9.4 Conclusions . . . . . . . 65 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 . . . . . . . . . . . . . . . . . . . . . 121 . . . . . . . . . . . . 10. Experiment 8: The Spectral Nature of Light 10.1 Introduction: The Bohr Hydrogen Atom 10.2 Analysis . . . . . . . . . . . . . . . . . . 10.3 Conclusions . . . . . . . . . . . . . . . . 10.4 APPENDIX: The Vernier Principle . . . A. Physical Units . . . . 109 109 113 116 117 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 123 128 134 134 . . . . 135 135 147 147 148 151 2 CONTENTS B. Using Vernier Graphical Analysis 155 C. Using Microsoft Excel 157 C.1 Creating plots and curve fits . . . . . . . . . . . . . . . . . . . . . . . . . 157 C.2 Performing Calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 D. Using Microsoft Word 161 E. Submitting a Report in Canvas 163 3 CONTENTS 4 Foreward Welcome to the general physics laboratory! This laboratory experience is designed to guide your learning of fundamental concepts of experimentation and data collection, delivered through the medium of hands-on experiments in electricity and magnetism. As a student, you should be aware that you and your colleagues will have a broad set of backgrounds in math, science, and writing and a similarly broad set of career trajectories. Even with the diversity of participants in a course such as this, everyone can share an appreciation of the scientific process. It is our job as instructors (TAs, faculty, and other assistants) to help facilitate this learning independent of your preparation level. Some will find this easier than others, but we will have done our job if you, regardless of background, walk away appreciating a little more deeply what it means for a scientist to claim that “I know something” based on experiments. Some passages of text have been emphasized and color coded to make finding them later more convenient. Checkpoint Checkpoints are intended to cause the student to be sure he is understanding and remembering the material before continuing to waste his time. Helpful Tip Helpful tips offer the student an opportunity to learn a shortcut or otherwise to make better use of his effort. Historical Aside A Historical Aside informs the student of some of the history associated with the discussion topic. History itself is not helpful in performing the experiment or in understanding the physics; however, it sometimes helps the student understand why we do things as we do. 5 CONTENTS WARNING Warnings are exactly what they seem. Defying warnings can result in some personal injury (likely not serious), in some disruption of the apparatus (time-consuming to repair), etc. General Information General Information is usually very helpful with respect to understanding the discussion topic in the broader context of the physical world. We hope these decorations improve the student’s experience and help him/her to learn to experiment more effectively. 6 Chapter 1 Introduction to the Laboratory Physics is an experimental science. As part of basic education in Physics, students learn both physical principles and problem solving (130/135 lecture) and concepts of experimental practice and analysis (136). Physics 136-2 is designed to provide an introduction to experimental techniques in the laboratory, focused on experiments in electricity and magnetism. We will build on the concepts covered in mechanics and use them to explain the indirect observations of electric charges and currents. Since electricity itself is invisible, these studies are considerably more abstract than students are accustomed; however, the process of using a set of tools to yield data and then of analyzing the data to reach conclusions is the same. The primary purpose of the Physics Laboratory is NOT to duplicate the concepts of lecture, although reinforcement is certainly beneficial and intended. This lab is an independent course covering independent concepts. The topics of the lecture serve as examples that we will explore in the lab to learn how to trust and to believe in physical principles. The schedule of topics in each lecture may not correspond directly with the material in the lab, which will be focused on measuring physical phenomena. These two components complement each other, but they seldom track each other. Taken together, 130/135 and 136 should provide the knowledge, problem solving skills, intuition, and practical experience with apparatus and data collection expected of a first year in college-level physics. 1.1 Objectives of Introductory Physics Laboratories In this course, students should expect to advance several learning goals that are broadly relevant in science, technology, and general understanding of human knowledge. These objectives are outlined by the American Association of Physics Teachers at http://www.aapt.org/Resources/policy/goaloflabs.cfm: • Develop experimental and analytical skills for both theoretical problems and data. • Appreciate the “Art of Experimentation” and what is involved in designing and analyzing a data-driven investigation, including inductive and deductive reasoning. • Reinforce the concepts of physics through conceptual and experiential learning. • Understand the role of direct observation as the basis for knowledge in physics. 7 CHAPTER 1: INTRODUCTION • Appreciate scientific inquiry into creatively exploring how the world works. • Facilitate communication skills through informative, succinct written reports. • Develop collaborative learning skills through cooperative work. 1.2 Calculus vs Non-Calculus Based Physics The same set of experiments are given to students in both calculus-based and algebra-based physics courses. The work in this laboratory is designed to be independent of calculus, but it is natural that students with more math background can better appreciate the subtleties of the physics probed in these experiments. Calculus is never required in this course, and your grading will not be affected by your knowledge of calculus (or lack thereof). For completeness the physical laws and principles will be presented in their most general form and that typically does require calculus; however, the student will receive the same grade if he simply ignores these derivations and goes directly to the solutions. These solutions frequently contain algebra and trigonometry but they can always be understood without resorting to calculus. 1.3 What to Bring to the Laboratory You should bring the following items to each lab session, including the first session of the course. There is no additional textbook. 1) A bound quadrille ruled lab notebook. You must have your own, and you cannot share with your lab partner. A suitable version is sold by the Society of Physics Students in Dearborn B6. This lab notebook can be reused for future physics labs. 2) This Physics Laboratory 2nd Quarter lab manual. Accessing the online version on a computer is acceptable in lieu of any printed copy, and likely preferable. 3) A calculator. 4) A pen. 5) You will need to transfer electronic data files and figures from lab to your lab reports. This can be done by email, a cloud storage account, or you may prefer to bring a USB drive to transfer from lab computers. 1.4 Lab Reports You will write lab reports and submit them electronically. The purpose of this exercise is both to demonstrate your work in lab and to guide you to think a bit more deeply about what you are doing. The act of technical writing also helps improve your communication skills, which are broadly relevant far beyond the physics lab. 8 CHAPTER 1: INTRODUCTION The appendices of this lab manual provide some guidance on how best to prepare these reports. You should keep in mind that these are not publishable manuscripts, but concise and clear descriptions of your experiments. They will follow a clear format to communicate your work best. They are not meant to be long. In the past, similar reports were written in class in about 30 minutes. . . these at-home reports are a bit more involved than that, but not by much. Students average one-two hours to complete them if their time in the laboratory is optimized. Each chapter of this lab manual describes what should be contained in your reports. There are also opportunities for bonus points. Be certain to read these before class, since they require additional lab work and cannot be accomplished during the writeup without the necessary data. 9 CHAPTER 1: INTRODUCTION 10 Chapter 2 Understanding Errors and Uncertainties in the Physics Laboratory 2.1 Introduction: Measurements, Observations, and Progress in Physics Physics, like all natural sciences, is a discipline driven by observation. The concepts and methodologies that you learn about in your lectures are not taught because they were first envisioned by famous people, but because they have been observed always to describe the world. For these claims to withstand the test of time (and repeated testing in future scientific work), we must have some idea of how well theory agrees with experiment, or how well measurements agree with each other. Models and theories can be invalidated by conflicting data; making the decision of whether or not to do so requires understanding how strongly data and theory agree or disagree. Measurement, observation, and data analysis are key components of physics, equal with theory and conceptualization. Despite this intimate relationship, the skills and tools for quantifying the quality of observations are distinct from those used in studying the theoretical concepts. This brief introduction to errors and uncertainty represents a summary of key introductory ideas for understanding the quality of measurement. Of course, a deeper study of statistics would enable a more quantitative background, but the outline here represents what everyone who has studied physics at the introductory level should know. Based on this overview of uncertainty, you will perhaps better appreciate how we have come to trust scientific measurement and analysis above other forms of knowledge acquisition, precisely because we can quantify what we know and how well we know it. Scientific ideas themselves are not sacrosanct; contrarily, they can be replaced with improvements and refinements to understanding. This progress is measurable through analysis of uncertainty in data, measurement, and calculation. Scientific progress is based on skepticism and continued critical analysis; uncertainties guide us in our inquiries. 11 CHAPTER 2: UNCERTAINTIES 2.2 Some References The study of errors and uncertainties is part of the academic field of statistics. The discussion here is only an introduction to the full subject. Some classic references on the subject of error analysis in physics are: • Philip R. Bevington and D. Keith Robinson, Data Reduction and Error Analysis for the Physical Sciences, McGraw-Hill, 1992. • John R. Taylor, An Introduction to Error Analysis; The Study of Uncertainties in Physical Measurements, University Science Books, 1982. • Glen Cowan, Statistical Data Analysis, Oxford Science Publications, 1998 2.3 The Nature of Error and Uncertainty Error is the difference between an observation and the true value. Error = observed value - true value The “observation” can be a direct measurement or it can be the result of a calculation that uses measurements; in which case, the “true” value might also be a calculated result. Note that according to this definition, an observation has an error even if we do not know what the true value is. (Estimating this ‘true value’ is often the point of doing the experiment!) Therefore, we will actually be analyzing the uncertainty, the estimate of the expected error in an observation. Example: Someone asks you, what is the temperature? You look at the thermometer and see that it is 71◦ F. But, perhaps, the thermometer is mis-calibrated and the actual temperature is 72◦ F. There is an error of −1◦ F, but you do not know this. What you can figure out is the reliability of measuring using your thermometer, giving you the uncertainty of your observation. Perhaps this is not too important for casual conversation about the temperature, but knowing this uncertainty would make all the difference in deciding if you need to install a more accurate thermometer for tracking the weather at an airport or for repeating a chemical reaction exactly during large-scale manufacturing. 2.3.1 Sources of Error No real physical measurement is exactly the same every time it is performed. The uncertainty tells us how closely a second measurement is expected to agree with the first. Errors can arise in several ways, and the uncertainty should help us quantify these errors. In a way the uncertainty provides a convenient ‘yardstick’ we may use to estimate the error. 12 CHAPTER 2: UNCERTAINTIES • Systematic error: Reproducible deviation of an observation that biases the results, arising from procedures, instruments, or ignorance. Each systematic error biases every measurement in the same direction. • Random error: Uncontrollable differences from one trial to another due environment, equipment, or other issues that reduce the repeatability of an observation. They may not actually be random, but deterministic (if you had perfect information): dust, electrical surge, temperature fluctuations, etc. In an ideal experiment, random errors are minimized for precise results. Random errors are sometimes positive and sometimes negative; they are sometimes large but are more often small. In a sufficiently large sample of the measurement population, random errors will average out. Random errors can be estimated from statistical repetition and systematic errors can be estimated from understanding the techniques and instrumentation used in an observation. Other contributors to uncertainty are not classified as ‘experimental error’ in the same scientific sense, but still represent difference between measured and ‘true’ values. The challenges of estimating these uncertainties are somewhat different. • Mistake, or ‘illegitimate errors’: This is an error introduced when an experimenter does something wrong (measures at the wrong time, notes the wrong value). These should be prevented, identified, and corrected, if possible, and ideally they should be completely eliminated. Lab notebooks can help track down mistakes or find procedures causing mistakes. • Fluctuations: Sometimes, the variability in a measurement from its average is not a random error in the same sense as above, but a physical process. Fluctuations can contain information about underlying processes such as thermal dynamics. In quantum mechanics these fluctuations can be real and fundamental. They can be treated using similar statistical methods as random error, but there is not always the desire or the capacity to minimize them. When a quantity fluctuates due to underlying physical processes, perhaps it is best to redefine the quantity that you want to measure. (For example, suppose you tried to measure the energy of a single molecule in air. Due to collisions this number fluctuates all over the place, even if you could identify a means to measure it. So, we invent a new concept, the temperature, which is related to the average energy of molecules in a gas. Temperature is something that we can measure, and assign meaningful uncertainties to. Because of physical fluctuations caused by molecular collisions, temperature is a more useful quantity in most cases.) 2.3.2 Accuracy vs. Precision • Accuracy: Accuracy is how closely a measurement comes to the ‘true’ value. It describes how well we eliminate systematic error and mistakes. 13 CHAPTER 2: UNCERTAINTIES precise accurate precise not accurate not precise accurate not precise not accurate Figure 2.1: Several independent trials of shooting at a bullseye target illustrate the difference between being accurate and being precise. • Precision: Precision is how exact a result is determined without referring to the ‘true’ value. It describes how well we suppress random errors and thus how well a sequence of measurements of the same physical quantity agree with each other. It is possible to acquire two precise, but inaccurate, measurements using different instruments that do not agree with each other at all. Or, you can have two accurate, but imprecise, measurements that are very different numerically from each other, but statistically cannot be distinguished. 2.4 Notation of Uncertainties There are several ways to write various numbers and uncertainties. • Absolute Uncertainty: The magnitude of the uncertainty of a number in the same units as the result. We use the symbol δx for the uncertainty in x, and express the result as x ± δx. Example: For δx = 6 cm and a length L of 2 meters, we would write L = 2.00±0.06 m. • Relative Uncertainty: The uncertainty as a fraction of the number. Example: 0.06 δx = = 0.03 x 2.00 14 CHAPTER 2: UNCERTAINTIES inches centimeters Figure 2.2: Measuring a string with a ruler. A reasonable measurements from this might be reported as 7.15 ± 0.05 cm. We must be very careful to say when our uncertainty is relative x = 2.00 m ± 0.03(rel) = 2.00 m ± 3% (2.1) • Percent Difference: The difference between measurements can sometimes be represented as a percentage difference. The percentage difference between two values x1 and x2 can be written as the difference divided by the average times 100% or: Percentage difference = 200% × 2.5 2.5.1 x1 − x2 x 1 + x2 (2.2) Estimating Uncertainties Level of Uncertainty How do you actually estimate an uncertainty? First, you must settle on what the quantity δx actually means. If a value is given as x ± δx, what does the range ±δx mean? This is called the level of confidence of the results. Note that the true value of a measurement does not need to fall in the range of the uncertainty. This is a common misconception! In fact, for a 68% confidence interval, the actual value does not fall in the range 32% of the time. In results of final measurements, it is common to use 68%, 95%, and 99% confidence levels for reporting data, but unfortunately, most practicing scientists are not always clear what they mean with an uncertainty. In this course, we will follow the convention of 68% confidence intervals, so that δx represents one standard deviation of the data set. Section 2.6 discusses the standard deviation of a data set. 2.5.2 Reading Instrumentation Measurement accuracy is limited by the tools used to measure. In a car, for example, the speed divisions on a speedometer may be only every 5 mph, or the digital readout of the odometer may only read down to tenths of a mile. To estimate instrumentation accuracy, assume that the uncertainty is one half of the smallest division that can be unambiguously 15 CHAPTER 2: UNCERTAINTIES read from the device. Instrumentation accuracy must be recorded during laboratory measurements. In many cases, instrument manufacturers publish specification sheets that detail their instrument’s errors more thoroughly. In the absence of malfunction, these specifications are reliable; however, ‘one half of the smallest division’ might not be very reliable if the instrument has not been calibrated recently. 2.5.3 Experimental precision Even on perfect instruments, if you measure the same quantity several times, you would obtain several different results. For example, if you measure the length of your bed with a ruler several times, you might find a slightly different number each time. The bed and/or the ruler could have expanded or contracted due to a change in temperature or a slightly different amount of tension. These unavoidable uncertainties are always present to some degree in an observation. Even if you understand their origin, the randomness cannot always be controlled. We can use statistical methods to quantify and to understand these random uncertainties. 2.6 Quantifying Uncertainties Here we note some mathematical considerations of dealing with random data. These results follow from the central limit theorem in statistics and analysis of the normal distribution. This analysis is beyond the scope of this course; however, the distillation of these studies are the point of this discussion. 2.6.1 Mean, Standard Deviation, and Standard Error The mean: Suppose we collect a set of measurements of the same quantity x, and we label them by an integer index i: {xi } = (x1 , x2 , . . . xN ). What value do we report from this set of identical measurements? We want the mean, µ, of the population from which such a data set was randomly drawn. We can approximate µ with the sample mean or average of this particular set of N data points: x̄ = 1 X xi N i (2.3) Of course, this is not the true mean of the population, because we only measured a small subset of the total population. But it is our best guess and, statistically, it is an unbiased predictor of the true mean. The standard deviation: How precisely do we know the value of x? To answer this question of statistical uncertainty based on the data set {xi }, we consider the squared deviations from the sample mean x̄. The sample variance s2x is the sum of the squared 16 CHAPTER 2: UNCERTAINTIES deviations divided by the ‘degrees of freedom’ (DOF). For N measurements the DOF for variance is N − 1. (The origin of the N − 1 is a subtle point in statistics. Ask if you are interested.) The sample standard deviation, sx , is the square root of the sample variance of the measurements of x. sP N 2 i=1 (xi − x̄) sx = (2.4) DOF The sample standard deviation is our best ‘unbiased estimate’ of the true statistical standard deviation σx of the population from which the measurements were randomly drawn; thus it is what we use for a 68% confidence interval for one measurement. The standard error: If we do not care about the standard deviation of a measurement but, rather, how well we can rely on a calculated average value, x̄, then we should use the standard error or standard deviation of the mean sx̄ . This is found by dividing the sample √ standard deviation by N : sx sx̄ = √ (2.5) N 2.6.2 Reporting Data Under normal circumstances, the best estimate of a measured value x predicted from a set of measurements {xi } is given by x = x̄ ± sx̄ . If we perform N more measurements of x, average them, and find their standard error, these two averages will agree within their standard errors 68% of the time. 2.6.3 Error Propagation One of the more important rules to remember is that the measurements we make have a range of uncertainty so that any calculations using those measurements also must have a commensurate range of uncertainty. After all the result of the calculation will be different for each number we should choose within range of our measurement. We need to learn how to propagate uncertainty through a calculation that depends on several uncertain quantities. Final results of a calculation clearly depend on these uncertainties, and it is here where we begin to understand how. Suppose that you have two quantities x and y, each with an uncertainty δx and δy, respectively. What is the uncertainty of the quantity x + y or xy? Practically, this is very common in analyzing experiments and statistical analysis provides the answers disclosed below. For this course we will operate with a set of rules for uncertainty propagation. It is best not to round off uncertainties until the final result to prevent accumulation of rounding errors. Let x and y be measurements with uncertainty δx and δy and let c be a number with negligible uncertainty. We assume that the errors in x and y are uncorrelated; when one value has an error, it is no more likely that the other value’s error has any particular value or trend. We use our measurements as described below to calculate z and the propagated 17 CHAPTER 2: UNCERTAINTIES uncertainty in this result. • Multiplication by an exact number: If z = c x, then δz = c δx (2.6) • Addition or subtraction by an exact number: If z = c + x, then δz = δx (2.7) • Addition or subtraction: If z = x ± y, then δz = q (δx)2 + (δy)2 (2.8) • Multiplication or division: If z = xy or z = xy , then v u δz u δx =t z x !2 δy + y !2 (2.9) • Power: If z = xc , then δz δx =c z x (2.10) The important pattern in these rules is that when you combine multiple uncertainties, you do not add them directly, but rather you square them, add, and then take the square root. The reason for this is intuitive: if one error is randomly positive, the other one is sometimes negative, which reduces the total error. Therefore, it is incorrect to estimate the combination of two uncertainties as their sum since this overestimates the average size of the combined error. 3 As one example consider z = xy . Before we can use the division formula, Equation (2.9), we must use the power formula to get the error in the numerator. Let w = x3 and use Equation (2.10) to find that δw = 3w δx = 3x2 δx. Next, Equation (2.9) gives our result x δz = z v u u t δy y !2 δw + w !2 =z v u u t δy y !2 3 δx + x !2 . The exponent makes the numerator three times more sensitive to uncertainty than the denominator. 18 CHAPTER 2: UNCERTAINTIES 2.6.4 Optional Reading – Advanced Topic: The General Formula for Error Propagation There is a general calculus formula encapsulating the rules for uncertainty propagation for a general function of several variables A = f (x, y, z) . Derivation of this is beyond the scope of this course. If you continue in your study of science, you will become more familiar with equations like this. δA = 2.6.5 v u u t ∂f ∂x !2 ∂f (δx)2 + ∂y !2 ∂f (δy)2 + ∂z !2 (δz)2 (2.11) Significant Figures The significant figures of a number are the digits in its representation that contribute to the precision of the number. In practice, we assume that all digits used to write a number are significant. Therefore, completely uncertain digits should not be used in writing a number and results should be rounded to the appropriate significant figure. For example, you should not express your height as 70.056 inches if your uncertainty is ±0.1 inch. It would more appropriately be written as 70.1 inches. Uncertainties specified using only significant digits are always ±5 times a power of 10; the least significant displayed digit was the result of rounding up or down by as much as 0.5 of that digit. Usually we know our uncertainty to be something close to this but yet different. Further, results of simple calculations should not increase the number of significant digits. Calculations transform our knowledge; they do not increase our knowledge. The rounding should be performed at the final step of a calculation to prevent rounding errors at intermediate steps from propagating through your work but one or two extra digits suffice to prevent this. Zeros are also considered significant figures. If you write a number as 1,200, we assume there are four significant digits. If you only mean to have two or three, then it is best to use scientific notation: 1.2 × 103 or 1.20 × 103 . Leading zeros are not considered significant: 0.55 has just two significant figures. There are some guidelines for tracking significant figures throughout mathematical manipulation. This is useful as a general method to keep track of the precision of a number so as not to carry around extra digits of information, but you should generally be using more formal error estimates from Sections 2.5 and 2.6 for reporting numbers and calculations in the physics lab. • Addition and Subtraction: precise input number. The result is known to the decimal place of the least Example: 45.37 + 10 = 55, not 55.37 or 55.4 √ Why? δ = 0.0052 + 0.52 = 0.5 Where we used the sum formula Equation (2.7). 19 CHAPTER 2: UNCERTAINTIES • Multiplication and Division: The result is known to as many significant figures as are in the least precise input number. Example: 45.4 × 0.25 = 11, not 11.4 r Why? δ = 11 0.05 45 2 + 0.005 0.25 2 = 0.2 > 0.05 Where we used the product formula Equation (2.9). Example: If you measure a value on a two-digit digital meter to be 1.0 and another value to be 3.0, it is incorrect to say that the ratio of these measurements is 0.3333333, even if that is what your calculator screen shows you. The two values are measurements; they are not exact numbers with infinite precision. Since they each have two significant digits, the correct number to write down is 0.33. For this lab, you should use proper significant figures for all reported numbers. We will generally follow a rule for significant figures in reported numbers: calculate your uncertainty to two significant figures, if possible, using the approach in Sections 2.5 and 2.6, and then use the same level of precision in the reported error and measurement. This is a rough guideline, and there are times when it is more appropriate to report more or fewer digits in the uncertainty. However, it is always true that the result must be rounded to the same significant figures as the uncertainty. The uncertainty tells us how well we know our measurement. 2.7 How to Plot Data in the Lab Plotting data correctly in physics lab is somewhat more involved than just drawing points on graph paper. First, you must choose appropriate axes and scales. The axes must be scaled so that the data points are spread out from one side of the page to the other. Axes must always be labeled with physical quantity plotted and the data’s units. Then, plot your data points on the graph. Importantly, you must add your error bars to your data points. Often, we only draw error bars in the vertical direction since these tend to dominate, but there are cases where it is appropriate to have both horizontal and vertical error bars. In this course, we use one standard deviation (standard error if appropriate for the data point) for the error bar. This means that 68% of the time the ‘true’ value should fall within the error bar range. Do not connect your data points by lines. Rather, fit your data points to a model (often a line), and then add the best-fit model curve to the figure. The line, representing your theoretical model, is the best fit to the data collected in the experiment. Because the error bars represent just one standard deviation, it is fairly common for a data point to fall more than an error bar away from the fit line. This is OK! Your error bars are probably too large if the line goes through all of them! The fitting parameters are usually important to our experiment as measured values. These measured parameters and other observations help us determine whether the fitting 20 CHAPTER 2: UNCERTAINTIES model agrees or disagrees with our data. If they agree, then some of the fitting parameters might yield measurements of physical constants. More information can be found in Appendix B on page 155 and in Appendix C on page 157. 2.8 Fitting Data Fully understanding this section is not required for Physics 136. You will use least-squares fitting in the laboratory, but we will not discuss the mathematical justifications of curve fitting data. Potential physics and science majors are suggested to internalize this material; it will become an increasingly important topic in upper division laboratory courses and research. In experiments one must often test whether a theory describes a set of observations. This is a statistical question, and the uncertainties in data must be taken into account to compare theory and data correctly. In addition, the process of ‘curve fitting’ might provide estimates of parameters in the model and the uncertainty in these parameter estimations. These parameters tailor the model to your particular set of data and to the apparatus that produced the data. Curve fitting is intimately tied with error analysis through statistics, although the mathematical basis for the procedure is beyond the scope of this introductory course. This final section outlines the concepts of curve fitting and determining the ‘goodness of fit’. Understanding these concepts will provide deeper insight into experimental science and the testing of theoretical models. We will use curve fitting in the lab, but a full derivation and statistical justification for the process will not be provided in this course. The references in Sec. 2.2, Wikipedia, advanced lab courses, or statistics textbooks will all provide a more detailed explanation of data fitting. 2.8.1 Least-Squares and Chi-Squared Curve Fitting For curve fitting, we make several measurements of a pair in a data set {xi , yi }, labelled by an integer index i. We want to understand how well a theoretical curve predicts y given x. Typically, we also want to extract some parameters {aj } in a model function f , such as y = f (x, {aj }). The method to do this is based on the ‘method of maximum likelihood’, which we do not cover in detail here. In words, we want to find the set of theoretical parameters {aj } for a curve that maximizes the probability of obtaining the observed data set {yi } from {xi }. This will be our best guess of the theory curve, and the result will be an estimate of {aj }. Here, the index j is a different label than i: j labels the fit parameters {aj } and i labels the data points {xi , yi }. Let’s assume that there is no uncertainty in x, and we have N measurements of y, each with some uncertainty. We have a set of N values yexp,i . We also have N predictions from the theory curve ypred,i . These predictions may depend on unknown parameters {aj }. For the least squares method, we ignore any error bars δyi and choose the parameters {aj } to 21 CHAPTER 2: UNCERTAINTIES minimize the function N X (yexp,i − ypred,i )2 (2.12) i=1 This maximizes the probability of obtaining the measured data if the theory were correct. Of course, it is possible that each measurement is not equally well known. Each data point could have its own uncertainty δyi . In this case, we would minimize the variance with each term weighted by how well we know it. This is called Chi-squared fitting and it is the standard data fitting method used in physics laboratories. We seek to minimize some function: N X (yexp,i − ypred,i )2 (2.13) χ2 = δyi2 i=1 We allow the parameters {aj } to vary to give the smallest χ2 . This can be done in some cases exactly from the data, and sometimes it needs an iterative procedure to find the best fit solution. There are also procedures to extract the variance in the parameters {δaj } from the method. Typically, software packages are used for this analysis. 2.8.2 Optional Reading – Advanced Topic: Chi-Square ‘Goodness of Fit’ Suppose that we have a curve with given parameters {aj }. How well does it fit some experimental data? There is a statistical test to determine the goodness of fit. We calculate the χ2 statistic as above, and then divide by the degrees of freedom. If the parameters are obtained from a fit, then the degrees of freedom is N − M where M is the number of fit parameters. Basically, we use our N measurements to calculate the M parameters and the DOF tells us how much ‘information’ we have left in our data set for optimizing the fit. Think of it this way: If we had N independent fit parameters (unknowns) and N exact data points (equations), we could in principle solve for all N unknowns. Instead we have only M fit parameters so the problem is over-specified. On the other hand our data is also uncertain so that we can use this over-specification to average out much of our uncertainties. This is quite similar to making a measurement several times and then to average them to get a better predictor for the ‘true’ value. We calculate the reduced χ2 by dividing by the degrees of freedom. This statistic is then compared to tables of numbers or using software to see how good the model fits the data. If the uncertainties are correct, and the data is truly distributed as expected, and the theory curve truly describes the underlying process, then the reduced χ2 should be about 1. If it is larger, then there is too much variance and either the fit is bad or the uncertainties are underestimated. If it is significantly less than 1, then the uncertainties may be overestimated or the data might be insensitive to the variation predicted by the theory curve. The chisquare value should be near 1 for a properly fit theory curve correctly describing the physical process with reasonable uncertainty estimates. 22 CHAPTER 2: UNCERTAINTIES 2.8.3 Example: Linear Regression The optimization procedure is most often performed for a fit to a linear function y = mx + b. Here, the two parameters that we wish to estimate are m and b, given a data set {xi , yi } with N pairs. The best fit results can be derived using calculus: m= b= xi y i − i xi i y i P 2 P N i xi − ( i xi )2 (2.14) P 2P P P xi i yi − xi xi yi P P 2 2 (2.15) N P P i N i xi − ( P i xi ) Assuming the uncertainties in y dominate and are all equal to δy, the variances can be written: 2 = σm σb2 N N 2 P 2 (δy) 2 i xi − ( i xi ) (2.16) x2i 2 P 2 (δy) 2 x − ( x ) i i i i (2.17) P P = i N P The uncertainties of the parameter estimators are found from the square root of the variances. The goodness of fit can be found from Equation (2.13). A proper statistical determination of parameter uncertainties from real data in linear and nonlinear regression is a subtle topic that is beyond the scope of this course. 2.9 Strategy for Testing a Model Experiments are designed and performed for the purpose of testing whether we can rely upon a particular hypothesis. Suppose we wish to check whether w = f (x, y, z) is a natural truth. Then we must arrange to measure x ± δx, y ± δy, and z ± δz so that we can predict the corresponding value of wp = f (x, y, z). We can also utilize what we have learned in Section 2.6.3 to determine how the uncertainties δx, δy, and δz lead to the uncertainty in our prediction δwp . Together, then we know that wp ± δwp if our model is correct. We cannot quit there, however; just because the model made a prediction does not necessarily mean it is correct. We must also arrange to measure wm ±δwm while our apparatus is ‘set’ to x, y, and z. If the hypothesis is correct and our apparatus faithfully represents f , then this measurement must match f ’s prediction. 2.9.1 A Comparison of Measurements If we make two measurements of (hopefully) the same quantity, both measurements contains the inherent uncertainty of the tool(s) used to make the measurements. Then how can we decide whether the two measurements agree? Statistics allow us to answer this question. 23 CHAPTER 2: UNCERTAINTIES First, we form a null hypothesis by subtracting the two (hopefully equal) measurements ∆ = |wp − wm | . (2.18) Even if these two numbers were taken from the same population, this difference has expected error, σ, due to δwp and δwm . We can find this error using our propagation formula for difference (Equation (2.8)) σ= q (δwp )2 + (δwm )2 . (2.19) Since δwp already contains δx, δy, and δz, σ contains all of our measurement uncertainties and we cannot reasonably expect ∆ < σ. Since our σ has 68% confidence level, we would actually expect ∆ > σ to happen 32% of the time, ∆ > 2 σ to happen 4.6% of the time, and ∆ > 3 σ to happen 0.3% of the time. Larger disagreements are less and less likely. Because of these statistics, we tend to consider when ∆ < 2 σ that the two numbers agree, wp = wm , and that any minor discrepancy is more likely to be due to randomness in the experiment and ‘other sources of error’. After all, claiming inequality would make us wrong about once in twenty times; this is far too often for most scientists. On the other hand, 3 in 1000 times is not too bad; we especially like these odds if they persist as we try to reconcile the difference. Therefore we consider ∆ > 3 σ to indicate that the two numbers disagree. A disagreement could mean that the data contradicts the theory being tested, but it could also just mean that one or more assumptions are not valid for the experiment; perhaps we should revisit these. Disagreement could mean that we have underestimated our errors (or even have overlooked some altogether); closer study of this possibility will be needed. Disagreement could just mean that this one time the improbable happened. These possibilities should specifically be mentioned in your Analysis according to which is most likely, but further investigation will await another publication. One illegitimate source of disagreement that plagues students far too often is simple math mistakes. When your data doesn’t agree and it isn’t pretty obvious why, repeat your calculations to make sure you get the same answer twice. 24 Chapter 3 Experiment 1: Sound 3.1 Introduction Sound is classified under the topic of mechanical waves. A mechanical wave is a term which refers to a displacement of elements in a medium from their equilibrium state; but to be a wave this displacement must then propagate through the medium. The speed at which the wave propagates is inversely related to the mass density of the propagating medium and directly related to the forces attempting to restore the equilibrium condition. A mechanical wave can propagate through any state of matter: solid, liquid, and gas. Mechanical waves can be of two types: transverse or longitudinal. A transverse wave is characterized by a displacement from equilibrium which takes place at right angles to the direction the wave propagates; longitudinal waves have the displacement from equilibrium along the axis of propagation. Since two directions are perpendicular to the direction of propagation, transverse waves have two independent polarization directions. The form of the equations describing these two types of waves is very similar. However, transverse waves can only exist in solid media, where intermolecular bonds prevent molecules from sliding past one another easily. Such sliding motion is called shear. Solids support shear forces and will spring back rather than continue to slide; this intermolecular connection will transmit the transverse wave from molecule to molecule. Longitudinal waves rely only on pressure and can exist in both solids and fluids. They depend on the compressibility of the media. Solids and fluids all show a resistance to compression. Sound waves are longitudinal waves that are transmitted as a result of compression displacement of molecules of the medium. We usually discuss sound in air, but sound travels in everything except empty space. A sound wave can be generated in solids, liquids, or gasses and can continue to propagate in a different medium. The equation used to describe a simple sinusoidal function that propagates in space is given by h i Y(x, t) = A0 sin k(x − vt) p̂ (3.1) 25 CHAPTER 3: EXPERIMENT 1 where Y is the time and position dependent displacement of the media from equilibrium, A0 is the maximum displacement or amplitude of the medium’s motion, v is the velocity of the wave which depends on the characteristics of the media. This particular wave travels along the x-axis. . . x must increase at speed v to keep up with vt. p̂ is the polarization of the wave. The case a longitudinal wave has p̂ = x̂ and a transverse wave has p̂ = py ŷ + pz ẑ some combination of y and/or z polarization. k is a constant that is determined by both the speed of the wave and the frequency of the wave. The constant k is usually expressed as k= 2π , λ (3.2) where λ is the wavelength. The wavelength is related to the wave velocity v and the wave frequency, f , by the expression v = λf. (3.3) A periodic mechanical wave is characterized by a frequency of oscillation, f , which is determined by the source of vibration motion that creates the disturbance. Thus, the frequency and the speed of the wave in the media determines the wavelength. The source can choose to oscillate at any frequency it chooses, but the medium decides the velocity of propagation. Checkpoint What is the difference between a displacement wave and a pressure wave? Checkpoint Is sound a displacement wave, a pressure wave, or may it be considered as both? Equation (3.1) describes the oscillations of particles with equilibrium position x. These equations describe either longitudinal or transverse waves. The difference lies in the interpretation of the displacement which is described in the equation. For a transverse wave, Equation (3.1) describes oscillations of the y and/or z coordinates of the particles at x. For transverse waves the actual wave looks very similar to the plot of the displacement and is easily visualized. For a longitudinal wave, Equation (3.1) describes oscillations of the x coordinate of the particles at x in equilibrium. This results in a sinusoidal variation in the density of media along the axis of propagation. This generally is much harder to visualize, and there are few natural examples that can be easily observed. One such example would be the pulse of compression which can be generated in a slinky spring. A sound wave is a longitudinal wave and since the displacement of the wave causes a variation in the density of air molecules along the direction of the wave, it can be viewed as either a displacement wave or a pressure wave. The above equation may be used to describe either picture. The displacement maximum is usually 90 degrees out of phase with the 26 CHAPTER 3: EXPERIMENT 1 Pressure Wave (a) (b) (c) Displacement Wave Figure 3.1: An illustration of the two representations of a longitudinal wave. The displacement representation is the position of particles with respect to their average positions, but the pressure increases as the particles move toward each other (compression) and decreases as the particles move away from each other (rarefaction). The displacement wave leads the pressure wave by 90◦ . pressure maximum as shown in Figure 3.1. A sound wave is shown with both displacement and pressure representations. The picture represents the density of the medium as the wave passes through it. Checkpoint What determines the pitch of a sound wave? The source, the medium? Which determines the speed of sound, the sound generator, the medium, or both? Which determines the wavelength of sound, the sound generator, the medium, or both? 3.1.1 Superposition When two sound waves happen to propagate into the same region of a medium, the instantaneous displacement of the molecules of the medium is normally the algebraic sum of the displacements of the two waves as they overlap. If at one time and place each individual wave would happen to be at a maximum amplitude, say Y1max and Y2max the net result would be a displacement of the medium at a value equal to the sum of Y1max and Y2max . This is 27 (a) Y1 + Y2 Y2 Y1 CHAPTER 3: EXPERIMENT 1 (c) (b) Figure 3.2: An illustration of constructive interference. Y1 and Y2 are in phase at all times so that their sum has amplitude equal to the sum of Y1 ’s and Y2 ’s amplitudes. (a) Y1 + Y'2 Y'2 Y1 shown in Figure 3.2. If on the other hand, the second wave were at Y20max = −Y2max , the net displacement equal to the sum of Y1max + Y20max , which in effect would be the difference Y1max − Y2max or zero if the amplitudes are equal, as shown in Figure 3.3. (c) (b) Figure 3.3: An illustration of destructive interference. Y1 and Y2 ’ are out of phase by 180◦ or half a wavelength. The sum of the two waves is zero if the two amplitudes are equal. Checkpoint What happens when two sound waves overlap in a region of space? 3.1.2 Reflection We most often think of a reflection as occurring when a wave encounters the border of the medium in which it is traveling. Anytime a wave encounters a sharp change in wave velocity, due to a change in the nature of the medium, a reflection is generated and some or all of the energy of the wave is redirected to the reflected wave. The amplitude and phase of the reflected wave is determined by the boundary conditions at the point of reflection. 28 CHAPTER 3: EXPERIMENT 1 In this lab, we will consider the effects of reflection from a solid boundary, such that the boundary condition requires that the sum of the waves have a displacement of zero at the point of reflection. Air molecules cannot be displaced from equilibrium at the wall. They cannot move into the wall and atmospheric pressure presses them into the wall; they simply have nowhere to go. This condition can only exist if we were to superpose a second wave moving in the opposite direction with exactly the same amplitude, and 180 degrees out of phase with the original wave. Hence, in order to satisfy the boundary condition, a reflection wave is generated with exactly these properties. Pressure 0.97 0.47 (a) -0.03 AN N AN N -0.53 AN N AN N -1.03 (b) (c) Wall The resulting superposition of incident and reflected waves in the region in front of the boundary also sets up a second null area where the amplitudes cancel at a distance of one half of a wavelength from the boundary as shown in Figure 3.4. The null areas are called nodes. If the wave didn’t loose amplitude as it traveled, a null would be present at successive half wavelength intervals over the entire region. As it is, the wave looses amplitude as it propagates, and the cancellation is only partial. 0.97 0.47 (d) -0.03 AN -0.53 -1.03 N AN N AN N AN N Displacement Figure 3.4: Illustrations of pressure waves and displacement waves reflected at a wall. The incident wave and reflected waves interfere to produce a series of nodes (N) and anti-nodes (AN) spaced every half wavelength. (a), (b), and (c) are pressure standing waves and (d) is the displacement. The pressure in (b) becomes the pressure in (c) after the gas moves like the arrows between indicate. The gas ‘sloshes’ back and forth between the dotted lines, but on average does not cross them. 29 The same boundary does not place such restrictions on the pressure wave. The pressure at the boundary may rise and fall, as is required. The wall can easily support whatever pressure results from the superposition of any two waves. A suitable pressure wave which takes advantage of the boundary restriction (which is none) is directed in the opposite direction with an equal amplitude to conserve energy, and is in phase with the incident wave. The resulting superposed waves show a maximum or anti-node at the CHAPTER 3: EXPERIMENT 1 boundary (see Figure 3.4) and a null point or node at a distance of one quarter wavelength away from the boundary. If the amplitude is sustained as the wave travels, a second null appears a half wavelength from the first, or at a point three quarters of a wavelength from the boundary. Between each node, the wave is seen to oscillate between maximum positive and maximum negative amplitudes, where the maximum amplitude is the sum of the maximum amplitudes of the waves considered separately. These areas are called anti-nodes. Such a wave is referred to as a standing wave because it stands still. In either case, successive null points or nodes occur at intervals of half of the wavelength of the traveling waves. By measuring the distance between nodes of the standing wave, we can determine the wavelength of the incident and reflected traveling waves. Since we also know the frequency at which we excited the wave, we can find the speed of the wave. 3.1.3 Part 1: Measuring the Speed of Sound in Air From personal experience one can get a sense that the speed of sound in air is rapid. You notice no delay in hearing a word that is spoken by a person nearby and the movement of the speaker’s mouth. That would be quite a distraction, like watching a movie with the sound track out of synch! And yet, when you sit in the outfield bleachers at a baseball game, you can sense a noticeable delay between the arrival of the light showing the ball being hit and the sound of the crack of the bat on the baseball. The speed of sound is noticeably slower than the speed of light over distances the size of a baseball field. In principle one could measure the speed of sound by timing how long after ones sees the ball hit that the sound arrives if one knew how far away they were from home plate. Instead, we will employ an oscilloscope simulation to observe the very short time delay as sound travels a distance on the order of a meter. In this experiment, a signal generator is used to produce a repeating electrical pulse to drive a speaker. The pulse causes the speaker to emit a ‘click’ or pulse of sound whose speed we will measure. A small microphone is used as a sensor. Its output is connected to one input of the oscilloscope. The wave generator signal is also fed directly into the oscilloscope. This signal will serve as a time reference against which to compare the microphone signal. The delay introduced due to the distance in air that the sound travels from the speaker to the microphone could be used to measure the speed of the click. Are the delays introduced to the process of converting the electrical signal to mechanical sound and back, and in the travel of electrical signals through the wire negligible? They probably are; however, we do not have an exact location for where the sound is produced in the speaker or sensed in the microphone. This could be a problem which we must deal with. A sound wave will be sent down a tube and be reflected off a piston head back to a microphone. We will measure the speed of the sound wave by observing the amount of time delay that is introduced to the arrival of the echo as the distance between the speaker (and microphone) and the piston head is increased. This way we do not need to know exactly where the sound originates or is detected. If the position of the speaker and microphone are unchanged, the only contribution to time is piston position. 30 CHAPTER 3: EXPERIMENT 1 Figure 3.5: A sketch of the apparatus we will use to measure the speed of sound in air. A wave generator and speaker will create a wave that travels down a tube and reflects from a piston. A computer senses the travel time for the wave. Helpful Tip To avoid unnecessary interference with the measurements of other lab students, and to spare the hearing and sanity of your Lab Instructor, leave your speaker on for ONLY those times you are making measurements. Set-up: Familiarize yourself with the equipment as shown in Figure 3.5. Pasco’s 850 Interface will be used to supply signals to the speakers from “Output 1”, to supply power for the microphone from “Output 2”, to digitize the speaker’s signal, and to digitize the microphone’s output in “Voltage D”. Check these connections. A suitable configuration for Pasco’s Capstone program (“Sound 1.cap”) can be found on the lab’s website at http://groups.physics.northwestern.edu/lab/sound.html Click the “Monitor” button at the bottom left to start taking data. Observe the two signals from A and B inputs displayed at the same time. Are the two signals in phase? You may have to readjust the voltage scale to see the complete sine wave of the microphone. Note the relative phase relation by drawing the two signals in your Data. Be sure to label which is which. 31 CHAPTER 3: EXPERIMENT 1 Measure the Speed of Sound To measure the speed of sound we want a square wave output and a frequency of about 5 - 20 Hz. To adjust the signal, click “Hardware” at the left and change only the frequency for “Output 1”. Figure 3.6: A sketch of the oscilloscope display showing the microphone’s response to square wave ‘clicks’. The sound reflects off of the tube ends and travels back and forth down the tube. The microphone measures each time the click passes by. The pulse generated by the speaker travels down the tube and reflects off the moveable piston back to the microphone. Drag the vertical numbers away from zero until the microphone signal occupies most of the Scope. Set the moveable piston to 80 cm from the speaker. As you move the piston note that part of the signal moves to the right; these are the clicks’ echoes passing the microphone as the sound bounces back and forth. Drag the time numbers to the right until the first echo observed by the microphone is near the right side of the Scope and t = 0 is at the left side. You should see something similar to the display shown in Figure 3.6. Move the piston closer to the mike. Does the spike shift on the time scale? Sometimes it is hard initially to identify the reflected pulse. It must move as the distance the clicks must travel changes and it must be the first one to do so. The easiest way to identify the reflected click is to move the piston around and to look for a pulse shifting around on the scope signal. As you move the piston away from the mike you are introducing a delay in the time the microphone picks up the 32 CHAPTER 3: EXPERIMENT 1 sound. You might also see other peaks shifting as you move the piston. These may be second and third echoes of the pulse bouncing off the speaker end of the tube. Set the piston at some minimum distance for which you can readily observe the first echo on the oscilloscope trace (20 cm is a good place). Note the piston position. How accurately can you determine the piston’s position? Use the Smart Tool’s cross-hairs’ icon to locate the leading edge of the pulse and note the time. Right-click the center of the SmartTool, choose Properties, and increase the number of significant digits to 5-6. Be careful to write the correct units and how accurately your time is known. Now move the piston to a new position along the tube far from the speaker and note the position again (70 cm is a nice choice. . . why?). Using the cross-hairs, determine the new time of the shifted echo peak also. Remember that the extra distance you have introduced to the sound travel is twice the change of position of the piston (going toward the piston and coming back). Calculate the speed of sound by dividing the extra distance added to the round trip of the sound pulse by the corresponding increase in travel time. The pulse travels twice as far as the piston moved but the oscilloscope measured the time (not double the time and not half of the time), 2∆x . (3.4) v= ∆t The width of the echo’s leading edge can be an indicator of the uncertainty of the measurement. If the echo moves around, this will increase your measurement error estimate. Measure the latest time and subtract off the earliest time that might reasonably be assigned to the time of the pulse’s echo. Let δ = latest - earliest and δt = 12 δ is a reasonable estimate of the uncertainty in your time measurements. You need measure this uncertainty only once since nothing substantial changes between measurements; each time measurement will have this same uncertainty. 3.1.4 Part 2: Measuring the Wavelength Observing Standing Waves Sound incident on a barrier will interfere with its reflection, setting up a standing wave near the reflector. The distance between nodes in the standing wave is a measure of half the wavelength of the original sound wave when the wave travels at right angles to the reflector. Because of the inefficiency of the reflector and other losses, the nodes may be only partial nodes. Additionally, the microphone has a finite size and will average the sound intensity over a range of positions where only one position is at the intensity minimum. The wavelength of sound should be expected to be on the order of meters for audible sounds. Diffraction effects are commonly observed for sound waves passing through apertures like doors and windows on the order of meters in size. For this part of the experiment adjust the acrylic tube so there is about a centimeter gap between the speaker and the end of the tube. This will release the pressure in the tube and 33 CHAPTER 3: EXPERIMENT 1 force this end of the tube to atmospheric pressure; this end will be a pressure node (and a displacement anti-node). This part of the experiment will use “Sound 2.cap” that can be downloaded from the lab’s website at http://groups.physics.northwestern.edu/lab/sound.html Set the frequency of the generator to 450 Hz. Click “Signal Generator” at the left to access “Output 1” control panel. Keep the microphone near the piston and move the piston and microphone to where the sound resonates in the tube (the microphone output goes to a maximum). Use the mike as a probe to measure the intensity of sound in the region between the speaker and the piston by noting the amplitude of the signal on the scope as you move the mike around back and forth inside the tube. This is the sound pressure level (SPL) as a function of position, P (x), for the standing wave. Place the mike near the piston and note whether the piston head is a node or an anti-node by observing the variation in the intensity of the sound as you move the mike around near the piston. Note your observation in your notebook. What would you expect for a pressure wave or a displacement wave? Is the microphone a pressure sensor or an amplitude sensor? Place the microphone near the speaker end of the tube, and note whether this is a maximum or minimum. Explain this result in your notebook. You would think that near the speaker you would get a large response from the microphone. Is that what you see? Now, move the microphone to locate the first node away from the piston (remember that right next to the piston is a displacement node) where the microphone’s output goes through a maximum. Start with the mike right near the piston and move it away from the piston and toward the speaker. Measure the first position of maximum response away from the piston. Can you detect the next node? A third node? Record your observations. Record the positions of the microphone where its output is minimum. Don’t forget your units and error estimates; how accurately can you position the microphone at the maxima/minima and how accurately can you read the centimeter scale? Move the microphone and remeasure one node and one anti-node several times to check your error estimates. Are nodes and anti-node measurements equally precise? Calculate the wavelength of the sound from the distance between the first node of the standing wave and the second node. If the nodes are close together, you can skip a node, measure the distance between two nodes, and divide by two to get a better accuracy. After all, your measurement errors will be divided as well since the denominator will be twice as large. Calculate the speed of sound using the wavelength just determined and the frequency from the signal generator’s LED display; use Equation (3.3). Note your results in your lab book. 34 CHAPTER 3: EXPERIMENT 1 Checkpoint The distance between successive nodes in a standing wave is a measure of what? 3.1.5 Part 3: Sound Speed at Several Frequencies Set the generator to something low around 450 - 500 Hz. Verify that the gap between the speaker and tube end is still about 1 cm. Keep the microphone near the piston and vary the position of the piston and microphone to maximize the microphone response. Place the microphone at the end of the tube nearest the speaker. Put the piston all the way into the tube so that it is close to the mike. Slowly withdraw the piston and observe the position where the piston produces a maximum sound intensity. Move the microphone to get a maximum response and see if the piston’s location can increase the microphone’s output. Repeat until the piston’s position remains constant. This is where the tube is in resonance with that particular frequency. Note this piston position, its units, and your error. The position of the microphone will not need to change again until the frequency changes. Withdraw the piston further and note a second resonance position. How precisely can you position the piston and measure its position? If you cannot see a second resonance the wavelength of the sound may be too long and you will need to increase the signal frequency. These successive resonance positions are the positions of nodes for the standing wave at this frequency. Since the distance between nodes is half a wavelength, merely doubling this distance and multiplying it by the frequency of the sound as read off the signal generator display will determine the speed of sound. Repeat the measurements of successive resonance positions for four higher frequencies. Make a table of wavelengths, frequencies, and calculated sound speeds for each frequency. Plot your data using Vernier Software’s Graphical Analysis 3.4 (Ga3) program. A suitable configuration for Ga3 can also be downloaded from the lab’s website. Does the speed vary systematically with frequency? Use Ga3 to determine an average speed v̄ and a standard deviation sv by drawing a box around the data points and by using the Analyze/Statistics feature. Compute the deviation of the mean sv̄ using sv sv̄ = √ N What frequency has a speed most similar to the speed measured in Experiment 1? Your measurement’s best predictor is vs = v̄ ± sv̄ 35 CHAPTER 3: EXPERIMENT 1 Checkpoint For a displacement wave reflecting off a solid wall is the boundary a node or an antinode in the resulting standing wave? For a pressure wave reflecting off a solid wall is the boundary a node or an anti-node in the resulting standing wave? 3.1.6 Part 4: Spectral Content The Fourier Transform Most of us are familiar with the link between musical pitch and sound frequency. A soprano voice has high pitch and high frequency whereas a bass voice has low pitch and low frequency. Probably you have already noticed while performing Part 3 above that higher frequencies have higher pitch. You might also have noticed how ‘boring’ a pure sine wave sounds. For contrast, select the square and triangle waveforms for “Output 1”; use a frequency of a few hundred Hz and select the different wave shapes. Which shapes have the most pleasant sound? Record your observations in your notebook. We now want to investigate the frequency content of sounds more closely. The “Sound FFT.cap” from the lab’s website provides a suitable configuration for Pasco’s Capstone program. This will perform a ‘fast Fourier transform’ (FFT) on the microphone’s signal and the speaker’s excitation. In graduate school you will learn more thoroughly that the Fourier transform identifies the frequency or spectral content of functions of time. Study the response vs. frequency for various waveform shapes and note your observations in your data. The square wave should have responses at all integer multiples of the signal generator’s frequency (the fundamental frequency). The triangle wave should have only odd multiples of the fundamental. The amplitudes decrease for higher multiples or harmonics. Try singing a pure note into the microphone and noting the spectral content of your voice. 3.2 Analysis Calculate the Difference between the speed of sound found using the two different methods, ∆ = |v1 − v3 | . Combine the uncertainties in these two measurements using σ= q (δv1 )2 + s2v̄ Discuss the similarities and differences between this difference and the computed errors. What other subtle sources of error can you think of that might have affected your measurements. Communicate with complete sentences. Which measurement method gives the best accuracy and how do you know? 36 CHAPTER 3: EXPERIMENT 1 3.3 Conclusions What physical relations have your measurements supported? Contradicted? Which were not satisfactorily tested? Communicate with complete sentences and define all symbols. What values have you measured that you might want to know in the future? Include your units and errors. 37 CHAPTER 3: EXPERIMENT 1 38 Chapter 4 Experiment 2: Snell’s Law of Refraction 4.1 Introduction In this and the following lab the light is viewed as a ray. A ray is a line that has an origin but does not have an end. Light is an electromagnetic disturbance and, as such, is described using Maxwell’s equations, which expresses the relationship between the electric and magnetic fields in an oscillating wave. Light propagates as a wave; yet, many optical phenomena can be explained by describing light in terms of rays. In the model for light, rays in a homogeneous medium travel in straight lines. This model is referred to as Geometric Optics and is a very elementary theory. In this theory light travels from its origin at a source in a straight line, unless it encounters a boundary to the medium. Beyond this boundary may be another medium which is distinguished by having a speed of light different from the original medium. In addition, light may be reflected at the boundary back into the original medium. A light ray that returns to the original medium is said to be “reflected”. A ray that passes into the other medium is said to be “refracted”. In most interactions between light and a boundary, both reflection and refraction occur. In order to frame laws that govern these phenomena we must define some terms. The boundary between two media is defined as a surface. The orientation of a surface at any specific point is characterized by a line perpendicular to the surface that we call the normal. A ray may encounter a boundary at any arbitrary incidence angle. The angle of incidence is measured with respect to the normal line. A reflected ray will have an angle of reflection that is also measured with respect to the normal. The refracted ray will be oriented by the angle of refraction measured between the ray and the normal to the surface. Checkpoint For geometric Optics what assumption is made about the nature of light? 39 CHAPTER 4: EXPERIMENT 2 What distinguishes the two media is that the speed of light is different from one medium to the other. We define the index of refraction n to be the measure of how much different the speed of light is in a certain medium from that of light through a vacuum. Light travels through a vacuum at 299,792,458 m/s. This speed is thought to be a universal constant and the highest speed allowed in nature as postulated in Einstein’s theory of Special Relativity. We use the symbol c to represent this speed. The index of refraction is a characteristic of the medium. It is the only thing that distinguishes one medium from another in geometric optics. It is defined as the ratio of the speed of light in a vacuum to the speed in a particular medium of interest, n= c c or v = . v n (4.1) Therefore, the value of the index of refraction is always greater than unity. Gasses have an index of refraction close to 1 (nair = 1.00028), while for water the index is about 1.33 and for plastic it is approximately 1.4. Depending on the type of glass the index of refraction of glass can vary from 1.5 to 1.7. Normally we might think that the index of refraction is a constant that is the same for all light. The index of refraction actually depends on the frequency (color) of the light wave to a small degree across the visible part of the spectrum and as such is different for different colors of light. The rules for reflection of light are: 1) The angle of incidence is equal to the angle of reflection, θ1 = θ10 (4.2) where θ1 is the angle of incidence and θ10 is the angle of the reflected ray that propagates in the same medium. (This is the commonly known rule, but this next rule is rarely stated though equally important) 2) The incident ray, the reflected ray, and the normal to the surface, all lie in the same plane. Checkpoint What is the law of reflection? We will not formally investigate these rules in this lab although you will be able to observe the phenomena of reflection as a side issue while performing this lab experiment. The rules for refraction are not so obvious although they where well known to the ancients. 1) The first rule is often cited as Snell’s Law; it is: sin θ1 n2 = or n1 sin θ1 = n2 sin θ2 . sin θ2 n1 40 (4.3) CHAPTER 4: EXPERIMENT 2 where θ2 is the angle of refraction of the ray that is transmitted into the second medium. 2) The incident ray, the refracted ray and the normal to the surface, all lie in the same plane. Checkpoint What is Snell’s Law? What phenomenon does Snell’s Law describe? In general the path of a light ray is reversible in that if a light ray were to be reversed it would follow the same path. A ray traveling from a low index of refraction to a high index of refraction will experience a bending toward the normal. However a ray passing from a high index of refraction to a lower index will experience a bending away from the normal. The angle of refraction will be larger than the angle of incidence. So, what happens when the angle of refraction is greater than 90◦ for a given incidence angle. In this case light cannot be transmitted through the interface and as such it is reflected totally. The efficiency for this reflection is 99.99% (as compared to 95% for a typical silvered surface mirror). The largest angle for which a ray will be transmitted is the critical angle. One can show that the sine of this angle is the inverse of the ratio of the index of refraction of the first medium to the index of the second medium. If the second medium is air (n = 1.00028), the sine of the angle is effectively the reciprocal of the index of refraction of the first material. Checkpoint In optics angles are always measured with respect to what? 4.2 The Experiment The experiment consists of a single thin bundle of light rays exiting a light box. This ray will be incident upon various objects so that we may deduce whether the laws of reflection and of refraction are obeyed by the interaction between the light and the objects. A picture of the ray box is shown in Figure 4.1. 4.2.1 Part 1: Reflections In this experiment you will use the Light Ray Box shown in Figure 4.1. It consists of a light source and a Multi-slit Slide. The light source housing is mounted on a colored plastic base in which it can slide back and forth. The utility of this feature will be explained in a future experiment. The Multi-slit Slide is a flat square piece of plastic with notches cut into each 41 CHAPTER 4: EXPERIMENT 2 of the four edges. The Multi-slit Slide slips into a slot on the end of the ray box to create rays of light. Choose the side with just one narrow notch and place that side down as you slip the a Multi-slit Slide into place. A single narrow beam should be observed emerging from the ray box. Place the ray box on the end of a sheet of paper provided, such that the ray is projected onto the paper. Draw a long line for the surface of a mirror using the ruler. Construct the surface normal 90◦ from this surface line using the protractor and the ruler. Position the flat side of the shiny triangle exactly on the surface line. Carefully slide the paper (and shiny triangle) so that the ray from the ray box hits the shiny triangle at Figure 4.1: A photograph of the light ray the position of the surface normal. Carefully box and the multi-slit slide that we will use rotate the paper and mirror about a vertical axis to investigate ray optics. passing through the surface-normal intersection and note the relation between the incident and reflected rays. If the mirror moves on the paper, carefully reposition it before marking the ray positions. Now mark the positions of one incident ray and its reflected ray. One point far from the mirror and the point where the ray hits the mirror (and the surface normal) define each of the two rays. Remove the triangle, draw the incident and reflected rays, and measure the incident and reflected angles. Be sure to record your units and to estimate your experimental errors. How accurately can you construct the surface normal? How accurately can you position the mirror on the surface line? How accurately can you point the ray at the intersection between the surface and the surface normal? How accurately can you mark the incident and reflected rays? How accurately can you construct the rays from the three marks? How accurately can you position the protractor? How accurately can you read the protractor? All of these contribute to your measurement errors and must be considered while you contemplate your error estimates. Is your angle of reflection equal to your angle of incidence? Are your error estimates large enough to cover the difference? If not, repeat the experiment more carefully. 4.3 Part 2: Index of Refraction In this experiment you will also use the Light Ray Box shown in Figure 4.1. Locate the oddly shaped Lucite block. Place the block on the paper with the frosted side down. Aim the ray at one of the two parallel sides of the block. Note that the ray emerges from the opposite side of the block in a direction parallel to the original ray but displaced slightly due to the refraction of the ray inside the block. One option is to print the template in Figure 4.4 and to use it instead of constructing everything on a blank page. You will now draw lines representing the incident and refracted rays so that you can measure the angle of incidence and refraction for several angles of incidence. Arrange the block for a specific angle of incidence. You will repeat the following steps through a series of five angles from a shallow angle to a steep angle of incidence. 42 CHAPTER 4: EXPERIMENT 2 Use a pencil or pen and a ruler to draw a long line to represent the surface of incidence. Use the protractor to construct a surface normal perpendicular to the first line. (Alternately, the worksheet following these instructions may be photocopied or removed and used directly.) Place the Lucite block’s surface exactly on the first line and carefully slide the paper and block (friction Figure 4.2: A sketch of light refraction at the will move the Lucite if the paper is moved surface of a Lucite block. The ray enters the without sudden jerks) so that the light ray block at angle θ1 from the surface normal and enters the Lucite above the intersection of is refracted to angle θ2 from the same normal. the surface and the normal constructed earBy noting the point the ray enters the block lier. It might be possible to improve the and exits the block, we have two points on the length of the rays or their brightness by line of the refracted ray. elevating the front or back of the light ray box with the ruler and/or protractor. Mark the location of the incident ray at another point along the ray, such that its direction can be recreated later. It is advisable that this point be near the ray box. Lastly, draw a two points on the opposite side of the block where the ray emerges. These points should be separated as far as possible to make reproducing the ray’s angle as accurate as possible. These points should be sufficient to recreate the geometry of the refraction. Label these points “1” and carefully drag the paper and Lucite block to set another incident angle with the ray once again entering the Lucite block at the location of the normal line. Mark the incident and refracted rays as before and label them “2”. Continue to mark three more rays for a total of five or more. If the Lucite block should move off of the surface line, carefully reposition it before marking the refracted ray. Remove the block and draw lines representative of the incident and refracted rays. On the refracted side, construct the line between each pair of points and extend it back until it intersects the Lucite block’s back surface. This is the point where the ray emerged from the Lucite block, so inside the block the ray proceeded from O in a straight line to this point of emergence. Connect O to this point of emergence with the ruler and extend the line to a length that will allow you to use the protractor to measure its angle from the normal. Use a protractor to measure the incidence and refraction angles; keep in mind that these angles must be measured from the normal line and not from the surface. Record these measured angles in a properly formatted table. Be sure to include the units and error estimates for these measured angles. How accurately can you position the Lucite block along the surface line? How accurately can you point the ray at the surface normal? How accurately can you mark the incident and refracted rays? How accurately can you construct the incident and refracted rays through the points that mark the rays? How accurately can you position the protractor? How accurately can you read the protractor? All of these questions should be considered while you are considering the sizes of your angle errors. Order the results in a table in your lab book from smallest to largest angle. Plot the index of refraction as a function of increasing incident angle using Vernier Software’s Graphical Analysis 3.4 (Ga3) program. Your lab instructor will show you how to use the program if 43 CHAPTER 4: EXPERIMENT 2 you haven’t already used it in previous labs. GAX can also calculate your index of refraction in a “calculated” column. Observe if there are any systematic changes in the index of refraction. To do this, one must have an idea of how much deviation in the index can be attributed to random variations. One way to assess this is to find an average value for the index, and from that to determine a standard deviation. The average and standard deviation can be obtained from the Analyze/Statistics menu item in Ga3. You will need to draw a box around your data points before the Statistics will be active. Note the result in your Data. Use the std. dev. (sn ) to find the standard error or the deviation of the mean using sn sn̄ = √ . N N is the number of values averaged (probably 5. . . ). refraction as n = n̄ ± sn̄ . 4.3.1 (4.4) Specify your measured index of Part 3: Total Internal Reflection In this experiment you will also use the Light Ray Box shown in Figure 4.1. One option is to print the template in Figure 4.5 and to use it instead of constructing everything on a blank page. Draw a long line for the slanted surface and another at 90◦ for the surface normal. Position the slanted end of the Lucite block along the surface line. Slide the paper to arrange the ray to enter the Lucite block at the squared off end of the block. This is the end opposite the slanted Figure 4.3: A sketch illustrating internal end. Have the ray cross through the block reflection at the surface of a Lucite block. We and exit the slanted side. Observe the ray can see and note where the ray enters the block, that exits this side as you turn the paper and where the ray hits the surface, and where the block to vary the angle that the ray strikes reflected ray exits the block. the inside surface of the slanted side; this is the incident angle θ1 . You should also observe that there is a significant amount of intensity in the ray reflected off the inside surface. Record your observations in your Data. As you increase the incident angle the angle of refraction, θ2 , will also increase. Since θ2 will always be larger than θ1 , there will come an angle of θ1 for which θ2 should equal the maximum of 90◦ . At all angles of incidence larger than this critical angle, total reflection occurs and you should observe a noticeable increase in the intensity of the reflected ray. Total internal reflections are the most efficient optical reflections. Record your observations in your Data. Adjust the block so that the refracted ray just skims the slanted surface of the block and the incident ray hits the surface at the surface normal line. Note any observations of interest as you do this operation. Decide on what position best represents the critical angle 44 CHAPTER 4: EXPERIMENT 2 for the incident ray. Now mark the point where the incident ray enters the block and where the reflected ray emerges. If the block moves on the paper, reposition it on the surface line before marking the rays. Now, remove the block, use the points (and the point of incidence) to recreate the incident and reflected rays as they appeared inside the block. Measure the angle of incidence. Check this with the angle of reflection since they are equal. Be sure to record your units and to estimate your measurement errors. Consider all of the steps needed to construct the rays and to measure the angles. Use this critical angle to calculate the index of refraction for the Lucite block, n= 1 nair ≈ , sin θc sin θc and compare with the value you got in part one. Does it agree within the uncertainty of the first measurement as determined from the standard error? Might the error in your measurement of critical angle be large enough to explain any disagreement? Comment on this in your Analysis. Checkpoint If two adjacent media have the same index of refraction, n, can you observe the phenomena of reflection or refraction? Checkpoint What is a critical angle? What are the two conditions that allow total internal reflection to take place? 4.3.2 Part 4: Optical Fiber One of the more recent and ubiquitous applications of total internal reflection is the optical fiber. Fiber optics has revolutionized the communications industry. A fiber consists of a long thin thread of glass called the core surrounded by an outer shell of a material with a lower index of refraction called the cladding. Light entering the end of the core of an optical fiber is transmitted or piped to the other end with very little loss in intensity even though the fiber may be bent in a circular shape. The thinness of the fiber and the lower index of refraction in the cladding ensures that light will always strike the fiber side at an angle greater than the critical angle for total internal reflection and be totally reflected back into the fiber. Thus the light bounces off the inside as it caroms down the length of the fiber following every bend or twist. Find the optical fiber among the items on your lab table. Be careful in handling the fiber. It is, after all, a piece of glass and will break if bent too sharply. Align one end of the 45 CHAPTER 4: EXPERIMENT 2 fiber with the ray emerging from the ray box. Observe the light emerging from the other end of the fiber. It will look like a bright pinpoint of light. Point this end of the fiber vertically down onto a sheet of paper on the table with the end of the fiber held about five millimeters above the table. You should see a bright spot on the paper where the light rays coming from the fiber hit the paper. The light has been channeled through the fiber even around the bends as the ray bounces off the inside surface of the fiber, possibly many times. The light persists with little loss of intensity because of the great efficiency of these reflections that are at such steep angles to the surface normal as to always be in the realm of total reflection. Now, carefully rotate the incident end of the fiber to introduce and angle between the incident ray and the fiber end’s normal. Observe that the spot on the paper has expanded to form a ring of light. Note these observations and try to explain them in your Data. Checkpoint In the case of fiber optics do you expect the core or the cladding to have a greater value for the index of refraction? Why are optical fibers immune to electrical noise? 4.4 Analysis Which is larger between your ability to construct the light rays on paper and to measure the angles and the difference between your incident and reflected angles? Between the accepted index of refraction (1.495) of Lucite and those measured in Part 2 and/or Part 3? Between the two measurements of Lucite’s index of refraction from Part 2 vs. Part 3? This last question suggests the self-consistency among your measurements. How accurately have you constructed and measured your ray diagrams? What other sources of error have you noticed and how has each affected your data? Might these errors be large enough to explain your differences? 4.5 Conclusions What physical relations do your data support? Contradict? Which are not satisfactorily tested? Communicate with complete sentences and define all symbols. What have you measured that you might need to use in the future? 46 1 2 3 4 5 O 3 3 3 CHAPTER 4: EXPERIMENT 2 Figure 4.4: A template to aid the several measurements of index of refraction. 47 2 2 O 1 1 CHAPTER 4: EXPERIMENT 2 Figure 4.5: A template to aid the observation of total internal reflection and the critical angle. 48 Chapter 5 Experiment 3: Geometric Optics 5.1 Introduction In this and the previous lab the light is viewed as a ray. A ray is a line that has an origin, but does not have an end. Light is an electromagnetic disturbance, and as such is described using Maxwell’s equations, which expresses the relationship between the electric and magnetic fields in an oscillating wave. Light propagates as a wave, yet many optical phenomena can be explained by describing light in terms of rays, which in the model for light travel in straight lines in a homogeneous medium. This model is referred to as Geometric Optics and is a very elementary theory. In this theory, light travels from its origin at a source in a straight line, unless it encounters a boundary to the medium. Beyond this boundary may be another medium which is distinguished by having a speed of light different from the original medium. In addition, light may be reflected at the boundary back into the original medium. A light ray that returns to the original medium is said to be “reflected”. A ray that passes into another medium is said to be “refracted”. In most cases, light passing across a boundary between two media is partly reflected and partly refracted. In this lab we will observe how the laws governing reflection and refraction, which were explored in the previous lab, can be applied to explain how images are formed by mirrors and lenses. Any mirror or lens can be referred to as an optical element. Imaging refers to the process by which a single optical element or a set of optical elements produce an image or reproduction of the light coming from an object. We perceive an object by observing light coming from the object. We are accustom to the fact that light diverges from an object and travels in a straight line from the object. Our two eyes and our sense of perception use these facts to locate objects in our environment and to create a mental picture of our surroundings. An optical element can redirect diverging light rays so that they appear to diverge from a different source. Thus an image is formed apart from the original object. The image can have one of two possible natures. If the light is merely redirected to appear to diverge from a different location, and thus the light in fact was not focused at that point, it is said to be a 49 CHAPTER 5: EXPERIMENT 3 virtual image. The light’s source point is not in a virtual image, but the rays’ new directions make it seem like they came from the virtual image. The most common example would be the reflection in a plane mirror. The image would appear to be behind the mirror surface, and yet behind the mirror may be a brick wall, hardly conducive to propagation of light. On the other hand, light may be redirected to converge at a point, after which it would diverge from that point in a similar manner to the way it originally diverged from the object. The resulting image is said to be a real image when light actually diverges from the location of the image. A screen or white paper placed at a real image plane will show a duplicate of the object. Perhaps it is obvious that a screen behind a mirror will show no such duplicate. The principle objective of geometric optics is to be able to determine the location of an image for certain optical elements arranged in a specific geometry. This may be accomplished in two ways. One can sketch key ray paths in a scale drawing of the geometry or one can calculate the image distance and properties using a set of equations. The Plane Mirror The mirror is the simplest of optical elements to understand. Everyone is familiar with the imaging properties of a plane mirror. If you stand in front of a plane mirror you see your image behind the mirror. The location of the image of an object placed in front of a mirror can be diagrammed knowing that the surface of the mirror reflects light with an angle of reflection equal to the incident angle. Let us look briefly at the properties of the image formed by a plane mirror. It is formed behind the mirror, it is right side up with respect to the object, and it is the same size. seem to come from here ho rays extrapolated rays normal do Object hi 5.1.1 di Mirror Image Figure 5.1: Light rays reflected by a plane mirror are illustrated. It looks as if the rays originated behind the mirror from a virtual image. General Information Note: For the image left is right and right is left. The image seems to be left - right inverted, but this is a separate issue. By convention we measure the object distance do as a positive value when in front of the mirror and image distance di when behind the mirror as negative. Hence for the case of a 50 CHAPTER 5: EXPERIMENT 3 plane mirror where the image is behind the mirror as far as the object is in front then di = −do (5.1) The relative size of the image compared to the object is the magnification, M , defined by the relation hi M= . (5.2) ho For a plane mirror the magnification is M = +1. We emphasize the positive sign because the sign of M has a significance we will only appreciate later. It tells us if the image is right-side-up (upright) or upside-down (inverted) relative to the object. Figure 5.1 shows a diagram locating the image of a plane mirror. In the case of a plane mirror where the image is oriented the same as the object, M is positive. Even more interesting is the fact that the magnification is directly related to the relative distances between the object or image and the mirror. In general M can be calculated from M =− di . do (5.3) Using the relation in Equation (5.1) we get M = +1 again. Such will not always be the case for mirrors in general, but seem to be true for plane mirrors. We note finally that the rays of light form an observable image because they appear to diverge from a point other than where the object is located. Since our eyes observe diverging rays, we can see the image. For a plane mirror this point is behind the mirror and we can see the image. In fact, the rays cannot actually be traced back to that point; if we try, we get stopped by the mirror. They never really pass through the point of apparent divergence. That point may very well lie inside a solid wall or behind the wall in another room. We refer to the image as a virtual. Not all images are virtual. But all virtual images are erect and have negative image distances in the case of a single mirror. Checkpoint What does the relative size of object and image depend on? 5.1.2 The Concave Mirror A mirror surface can be fabricated as part of a spherical surface of radius R. By convention a positive radius is one that is measured to the left of the surface assuming light to be approaching from the left to be reflected off the surface. In fact, all distances are positive on the side where light is bouncing around. The reflecting surface would be termed a concave mirror when R > 0. The principle axis is a line that strikes the mirror in the center at normal incidence. Any light ray that is parallel to this axis when it is reflected off the mirror will arrive at the same point along the principle axis. This is called the focal point and it 51 CHAPTER 5: EXPERIMENT 3 can be shown that this point lies a distance f from the mirror where f = R/2. These acts provide the circumstances for drawing two rays (1 and 2) diverging from the tip of the object that will re-converge at a specific point to locate the image position for a particular object location. We show how these rays can be used to locate an object in Figure 5.2. A more useful interactive diagram called an applet can be found on the internet at http://groups.physics.northwestern.edu/vpl/optics/mirrors.html do 1 ho 4 2 R f 3 hi Object di Image Mirror Figure 5.2: Light rays reflected from a concave mirror are illustrated. Four rays are easy to draw for mirrors: Incident parallel to the axis passes through the focus after reflection, incident through the focus reflects parallel to the axis, incident through the sphere’s center reflects through the center, hits the mirror on the axis reflects at the same angle. The diverging rays are focused at a real image point. The applet makes use of a third ray which, if it strikes the mirror normal to the surface at zero incidence angle, is reflected back on itself. A fourth ray can be obtained where the ray strikes the mirror at the vertex, where the principle axis meets the mirror surface. Such a ray is reflected at an angle equal to the angle between the ray and the principle axis. The most obvious thing about the image formed by this configuration is that the image is inverted. It also appears in front of the mirror. Such an image is so foreign to us that you really have to see it to appreciate it. Take a common metal spoon and hold it at arms length with the concave portion facing you. When you look into the spoon you should see your reflection and everything else behind you in the room inverted. It is harder to discern that the images are in front of the mirror; never-the-less, it is the case. By convention we say that an image located in front of the mirror has a positive image distance. From the definition of magnification in Equation (5.3), M must be negative if both do and di are positive. This agrees with the fact that the image is inverted. 52 CHAPTER 5: EXPERIMENT 3 The image in Figure 5.2 can be located analytically using the expression 2 1 1 1 = = + . R f do di (5.4) The image is reduced in size. This further bears out the relation in Equation (5.3) that ties the magnification to the object and image distances. Here the image lies closer to the mirror than the object and hence is smaller. We also note that the rays of light actually pass through the image location and re-diverge. By convention we say that this image is a real image. Visibly it looks like an object is really located at that point, and the phenomenon is the source of many magic tricks and optical illusions. The fact that the image is inverted and located at a positive image distance goes along with the real nature of the image. A real image will be formed if the object is located anywhere between infinity and the focus of the mirror. Real image can be projected. That is, if you were to place a white card or projection screen at the point where the image is formed, the image will appear in good focus on the screen. -di 1 do hi 3 Object ho 2 4 Image f R Mirror Figure 5.3: Light rays emitted from a very close-up object are reflected by a concave mirror. In this case the rays do not focus; however, they appear to come from a virtual image behind the mirror. Our eyes can then focus these diverging rays and see the virtual image behind the mirror despite its absence. See the caption of Figure 5.2 to learn how to draw these four easy rays. If the object is moved to a point closer than the focus of the mirror the rays will no longer re-converge to form a real image. Instead the rays appear to diverge from a point behind the mirror as shown in Figure 5.3. Such an application would be when you use a makeup mirror. Note that the virtual image is enlarged and right-side-up. Virtual images are not projectable, since the rays never come to a focus at any real point in space. 53 CHAPTER 5: EXPERIMENT 3 5.1.3 Convex Mirror A convex mirror (R < 0) is often called a divergent mirror. Light rays parallel to the principle axis are reflected away such that they appear to diverge from a focus behind the mirror. An image distance marked off behind the mirror is considered negative and locates a virtual image. In a similar manner according to the convention, a focus behind the mirror is negative, and virtual. Indeed, the rays never do pass through this focus just like rays never pass through a virtual image. The images formed by a convex mirror are virtual regardless of where the object is placed in front of the mirror. Figure 5.4 shows the ray diagram locating the image. Note that ray 1 appears to diverge from the virtual focus, and ray 2 appears to be headed for the virtual focus though neither ray reaches this focus. Note also that both the radius and the focus would have numerical values with a negative sign. Checkpoint What is the difference between a real and a virtual focus? Checkpoint What is the difference between a real and a virtual image? 5.1.4 Lenses Not everyone has personal experience using lenses, or so it might seem. Besides the most common application in eyeglasses used to correct imaging problems, campers, for example, have been know to use a focusing lens to start a campfire without matches using the sun’s energy. It gets more personal than that, however. Each of us uses a natural imaging system in our eye to see, which in part relies on a lens within each eye. The surface of a piece of glass can act as an optical element that changes the direction of a light ray as it passes through the surface. We explored Snell’s Law that describes this phenomenon in the last lab. In most applications of an air-glass surface the piece of glass is thin and the rays quickly encounter the opposite side of the glass at a glass-air boundary where the light emerges from the glass. There are many applications where only this one interface is considered such as looking at objects underwater. The cornea of the eye is an example of such a surface. We will not deal with this imaging process, but consider only applications where the light emerges from the optical element into air again. Such problems are referred to as thin lens applications. A plane piece of glass with parallel surfaces usually results in a slight sideways displacement of the original beam but no noticeable angular deflection. A pane of glass is transparent 54 CHAPTER 5: EXPERIMENT 3 do 1 ho 3 4 2 Object hi di f R Image Mirror Figure 5.4: Light rays reflected from a convex mirror appear to come from a virtual image behind the mirror. and transmits rays from an object undistorted. Such a simple case is worth a quick look compared to its reflection analog the case of a plane mirror. When looking at an object through a pane of glass we are in fact looking at the image of the object which is located at nearly exactly the same spot as the object itself. We have two choices. We can rewrite the imaging equations (5.1), (5.2), (5.3) and (5.4) to include the fact that the image is now on the same side of the optical element as the object, or we can change the convention to say that a virtual image is formed on the opposite side of the optical element from the observer. The accepted convention is to choose the latter. The accepted convention is that a real object (+do ) is on the side where light enters the lens, but a real image (+di ), a real radius (+R), and a real focus (+f ) are on the side where light exits the lens. We will see the ramifications of this in the images formed by a thin lens. We should also mention that lenses have two foci; each is a distance |f | from the lens’ center where one is on each side of the lens and both are on the optical axis. Convergent lenses have positive foci and divergent lenses have negative foci. 5.1.5 Convergent Lens A convergent lens redirects parallel rays such that they come to a focus. It is the lens analog to the concave mirror. An example of a convergent lens is one with two convex surfaces although other combinations of surface curvatures can also form a convergent lens. As we did for mirrors, we define the principle axis to be a line that strikes the lens in 55 CHAPTER 5: EXPERIMENT 3 Figure 5.5: Light rays passing through a convergent lens focus at a point behind the lens. Lenses have three rays that are easy to draw: Rays parallel to the axis are focused through the second focal point, rays passing through the first focal point are transmitted parallel to the optical axis, and rays passing through the lens’ center are undeflected. the center at normal incidence. Any light ray that is parallel to this axis before it passes through the lens will be redirected to pass through a single point. This is called the focal point and it can be shown that this point lies a distance f from the lens’ center. Unlike the mirror the radius of the surface of the lens is not easily related to the focus; never-the-less, all rays diverging from a point on the object will re-converge at another specific image point after passing through the lens. One of these rays (1) is emitted parallel to the principle axis and is bent to pass through the positive focus on the far side of the lens. Because a thin lens has two operational sides, (it can be turned around and work just as well), it has a second focus on the same side as the object. Another ray passing through this focus (2) is redirected to be parallel to the optical axis. We show how these rays can be used to locate an object in Figure 5.5. A more useful interactive diagram called an applet can be found on the internet at http://groups.physics.northwestern.edu/vpl/optics/lenses.html The applet makes use of a third ray that strikes the lens at the vertex, where the principle axis meets the lens surface. Such a ray (3) passes through the lens undeflected. The most obvious thing about the image formed by this configuration is that the image is inverted just as with the concave mirror. Contrary to the mirror analog, the image appears behind the lens. Such an image is so foreign to us that we really have to see it to appreciate it. Such a convergent lens is not as readily available as a mirror. If you have access to a magnifying glass you can demonstrate the real image formed by such a lens. By convention we say that an image located after the lens has a positive image distance. 56 CHAPTER 5: EXPERIMENT 3 Checkpoint What is a lens? General Information Note the simple change in convention from mirrors. From the definition of magnification in Equation (5.3), M must be negative if both do and di are positive. This agrees with the fact that the image is inverted. Checkpoint What is the principle axis of a mirror and of a lens? Figure 5.6: Light rays emitted from a very close object do not focus; however, they appear to come from a virtual image instead of the object. The ray directed away from the first focus is transmitted parallel to the axis, the ray incident parallel to the axis is transmitted through the second focus, and rays passing through the lens’ center are undeflected. For do < f , these rays diverge less than before. The image in Figure 5.6 can be located analytically using the same expression we used for mirrors, 1 1 1 = + , (5.5) f do di 57 CHAPTER 5: EXPERIMENT 3 but the focal length is more difficult to compute for a lens. The image is reduced in size. This further bears out the relation in Equation (5.3) that ties the magnification to the object and image distances. Here the image lies closer to the lens than the object and hence is smaller. We also note that the rays of light actually pass through the image location and re-diverge. By convention we say that this image is a real image. Visibly it looks like an object is really located at that point, and a plane card placed at that point will display the image projected on it. Real images are projectable for lenses as well as for mirrors. This phenomenon is applied to camera imaging as well as to how our eyes work. The fact that the image is inverted and located at a positive image distance goes along with the real nature of the image. A real image will be formed from a convergent lens if the object is located anywhere between infinity and the focus of the lens. If the object is moved to a point closer than the focus of the lens, the rays will no longer re-converge to form a real image. Instead the rays appear to diverge from a point in front of the lens as shown in Figure 5.6. Such an application would be when you use a magnifying glass. Note that the virtual image is enlarged and right-side-up. Virtual images are not projectable, since the rays never come to a focus at a real point in space, but the resulting diverging rays can be focused by our eyes. 5.1.6 Divergent lenses Lens do 1 3 hi ho 2 Object -di -f -f Image Figure 5.7: Light rays passing through a divergent lens appear to come from a virtual image instead of the object itself. Three easy rays for a divergent lens are parallel rays are deflected away from the axis and appear to come from the first focus, rays heading toward the second focus are deflected parallel to the optical axis, and rays passing through the lens’ center are undeflected. A divergent lens is the analog of a convex mirror. Light rays parallel to the principle axis get redirected away from the optical axis such that they appear to have originated from the same point. An image distance marked off in front of the lens is considered negative and 58 CHAPTER 5: EXPERIMENT 3 locates a virtual image. In a similar manner according to the convention, a focus in front of the lens is negative, and virtual. Indeed, the rays never do pass through this focus just like rays never pass through a virtual image. This focus is not to be confused with the real focus of a convergent lens. The images formed by a divergent lens are virtual regardless of where the object is placed in front to the lens. Figure 5.7 shows the ray diagram locating the image. Note that ray 1 appears to diverge from the first virtual focus, and ray 2 appears to be headed for the second focus though neither ray reaches its focus. Note also that the virtual focal distances are negative. Checkpoint What is the difference in the convention for mirrors and the convention for lenses? 5.1.7 Part 1: Mirrors Set up the ray box used in the last experiment but use the 5-slit mask. The ray box may have to be adjusted to make the rays parallel; use at least two meters for this adjustment. The piece of the box holding the slits will slide in the piece holding the light and allow you to cause the rays to emerge parallel to each other. Always be sure the rays are parallel while you are measuring the focus of optical elements. Locate the mirror block. It is roughly a triangular silver plastic piece. One side is flat, one side is concave, and one side is convex. With the flat side placed in the beam of parallel rays observe how the rays are redirected while maintaining the parallel structure. Note too that the deflection angle is twice the incident angle. A small turn of the block results in twice as much change in the direction of the rays. Briefly note your observations in your Data. With the concave side placed in the beam, observe how the rays are redirected to a focal point and from there re-diverge. It might help to elevate the front or the back of the ray box by placing the ruler and/or the protractor under it. With the convex mirror facing the beam the rays diverge and appear to emanate from a focus behind the surface. Place the concave mirror so that the center ray hits the center of the mirror and rotate the mirror until the focused rays are on the optical axis; they intersect the center ray at the same point. Carefully mark the mirror surface with a pen or pencil on the record paper. If the mark is not exactly below the mirror, move the mirror exactly to the mark and verify that the rays still converge on the optical axis. Mark the point where the rays converge and circle the area where one might reasonably choose an alternate focus. It might help to block the center ray at the ray box temporarily. Remove the mirror and paper, turn on your lamp, and carefully measure the mirror’s focal length. The radius of the circle around the focus is a reasonable estimate of the error in this measurement. Record your focal point measurement, units, and error in your Data. 59 CHAPTER 5: EXPERIMENT 3 Place the convex mirror so that the center ray hits the center of the mirror and is retroreflected back into the center slit. Carefully mark the mirror surface with a pen or pencil on the record paper. If the mark is not exactly below the mirror, move the mirror exactly to the mark and verify that the center ray still reflects along the optical axis. Mark two points on each incident ray and each reflected ray; separate each pair of points by as much distance as possible to make reproducing the rays’ angles more reliable. Remove the mirror and paper, turn on your lamp, and carefully reconstruct the five incident and the five reflected rays. Extend all rays to the mirror’s surface. Each incident ray should intersect its reflected ray at the mirror’s surface; if this is not the case, use your records on the paper to reconstruct the experiment with the ray box. This will allow you to adjust your record points to agree better with the rays so that they intersect at the mirror as they should. Extend all five reflected rays through the mirror’s surface and a few cm further than they converge at a virtual focus. Mark the point where the rays appear to originate and circle the area that one might reasonably choose as an alternate focus. Use the ruler to measure the mirror’s (negative) focal length. The radius of the circle around the focus is a reasonable estimate of the error in this measurement. Record your focal point measurement, units, and error in your Data. 5.1.8 Part 2: Lenses Set up the ray box used last week but use the 5-slit mask. The ray box may have to be adjusted to make the rays parallel; use at least two meters for this adjustment. The piece of the box holding the slits will slide in the piece holding the light and allow you to cause the rays to emerge parallel to each other. Always be sure the rays are parallel while you are measuring the focus of optical elements. Locate the two-dimensional convergent lens (convex surfaces). Place this lens in the beam of parallel rays and observe how the rays converge to a focus on the side of the lens away from the source. Move the lens around and note how the emerging rays respond to the various lens positions. Place the lens perpendicular to the rays such that the center ray hits the lens center. This placement should cause the rays to converge on the optical axis (where the center ray would be without the lens). If this is not the case, adjust the lens position to make this true. Mark both lens surfaces (or the center of both sides) so that the lens center can be located later. Mark the point where the rays converge and circle the area one might reasonably choose an alternate focus. Remove the lens and paper and turn on your lamp. Use your ruler to reconstruct the center ray; extend this ray back past the lens location. Use your ruler to locate the center of the lens. Carefully measure the distance between the lens center and the focus. The radius of the circle of alternate foci is a reasonable estimate for the error in this measurement. Record your measured focus, its error, and its units in your Data: fc.l. = (f ± δf ) U. Locate the two-dimensional divergent lens (concave surfaces). 60 CHAPTER 5: EXPERIMENT 3 WARNING Be careful not to drop this lens because its thin center makes it easy to break. Be sure the rays from the ray box are parallel. Place the lens in the beam of parallel rays and observe how the rays appear to diverge from a point in front of the lens. Move the lens around and note how the diffracted rays respond to the various lens positions. Place the lens perpendicular to the rays such that the center ray hits the center of the lens. When this is true, the center ray should remain on the optical axis. Mark both surfaces of the lens (or the center of both edges) so that the lens center can be located later. Mark two points on each diffracted ray so that each ray can be reconstructed later. Remove the lens and paper and turn on your lamp. Use your ruler to reconstruct the center ray; extend this ray back past the apparent point of origin for the five rays. Use your ruler to locate the center of the lens. Use your ruler to reconstruct all five rays and extend each ray past the lens and a few cm past the optical axis. The extended rays should converge to a point; if not, use your record paper to reconstruct your experiment and to determine how to adjust your points for better agreement. Mark the point of convergence and draw a circle around the area one might reasonably choose an alternate focus. Carefully measure the (negative) distance between the lens center and the virtual focus. The radius of the circle of alternate foci is a reasonable estimate for the error in this measurement. Record your measured focus, its error, and its units in your Data: fd.l. = (f ± δf ) U. 5.1.9 Part 3: A Lens System Place the convergent lens in the beam and form a focused beam. Adjust the lens position to optimize the focus. Use the focus as an “object”. Place the divergent lens in the path of the diverging rays at least 6 cm after the focus and observe that the rays are redirected by the lens so that they appear to be diverging from a different point. See position 1 in Figure 5.8. The lens should be perpendicular to the optical axis and the center ray should hit the center of the lens. The center ray will remain on the optical axis when this is true. The virtual image is where the new rays seem to come from. Mark both divergent lens surfaces (or the center of both edges) so that the lens’ center can be located later. Mark the location of the “object” and draw a circle around the area that might alternately be chosen as the focus (i.e. the “object”). Mark two widely separated points on each diffracted ray after the divergent lens. Remove the lens and paper and turn on your lamp. Reconstruct the center ray and extend it back to the “object”. Use the ruler to locate the lens center. Reconstruct the five diffracted rays and extend each back a few cm past where they cross the optical axis. All five should intersect the optical axis at the same point. If this is not the case, use your record paper to reconstruct your experiment and to adjust your points for better agreement. Measure the (negative) virtual image distance between the lens center and the virtual image. The radius of the circle of alternate radii is a reasonable estimate of the measurement error. Record your measurement, its error, and its units in your Data. Repeat for your “object” distance and calculate the divergent lens’ focal length using Equation (5.5). Does this focus 61 CHAPTER 5: EXPERIMENT 3 agree with the focus measured directly? We will check this numerically in our Analysis. Since your object distance and image distance each have error, it should be obvious that your calculated focal length also has error due to both of these sources. Calculate the error in your focal length using v δf = f u u 2t δdo d2o !2 δdi + d2i !2 (5.6) . do -di Object Virtual Image (a) Virtual Object Image -do di (b) Figure 5.8: An illustration of two strategies for measuring the focal length of a divergent lens. In (a) the rays that diverge from the first lens’ focus are spread even further by the concave lens. In (b) the rays that would have converged at one focus are redirected to a different focus. Now place the divergent lens at position 2 in Figure 5.8, between the convergent lens and the focal point and at least 5 cm from the focal point. Observe that the rays converge to a new focal point a little further away from the lenses. This is a real image. But where is the object? The original focus has disappeared when the lens was placed in the beam. It has become a virtual object! Remove the divergent lens and mark the location of the original focus; draw a circle around the area of alternate possible foci. Place the lens in the beam perpendicular to the optical axis such that the center ray hits the lens center. The center ray should remain on the optical axis when this is true. Mark the image location (the new focus point), the area of possible alternate foci, and the divergent lens location. Locate the lens center and measure the object and image distances; be sure to assign the correct signs. Record these measurements, their errors, and their units in your Data. Again use Equation (5.5) to calculate the focus of the lens and use Equation (5.6) to estimate its error. Don’t forget to record this measurement, its units, and its error in your Data. How well does it match the previous two measurements? 62 CHAPTER 5: EXPERIMENT 3 Checkpoint What is a virtual object? 5.1.10 Part 4: Lenses Measure the focus of a real lens. Using a source far away (several meters) find the point where the image of this distant source is brought to a focus on a screen. With the room lights dimmed use the light coming through the lab door or window as an object. How does this compare with the value printed on the lens housing? Keep in mind that your measurement might not be very accurate. Place the optical source at one end of the optical bench and place the white screen on the opposite end of the optical bench. Place Figure 5.9: A good way to measure the focal a convergent lens between the source and the length of a thin lens. There will be two lens screen but nearer to the source and find the positions that focus the object’s image on the spot where the image of the source projected screen. Note these positions so that you can onto the screen is in focus. Observe that the find the distance, d, between them. image is larger than the source because the image distance is larger than the source distance to the lens. How far can you move the lens before the image is noticeably out of focus. Use this distance as an estimate of your measurement error. Record the lens location, its error, and its units: l1 = (x ± δx) U. Calculate the object distance do and the image distance di and use Equation (5.5) to find the focal length of the lens. Is it comparable to the value observed above using the open door? Now we will measure the focus in another way. There are two possible locations where a focused image can be achieved. Move the lens closer to the screen until a second smaller focused image appears on the screen. What has changed? Note the position of the lens for this geometry. How far can you move the lens before the image is noticeably out of focus? What does this distance mean? Record this lens location: l2 = (x ± δx) U. Calculate the object distance do and the image distance di and use Equation (5.5) to find the focal length of the lens. Calculate the distance, d = |l2 − l1 |, between the positions of the lens for which the image was in focus as shown in Figure 5.9. Estimate the error in this distance using the errors in the two lens positions, q δd = (δl1 )2 + (δl2 )2 . 63 CHAPTER 5: EXPERIMENT 3 This distance can be used as another way to measure the focal length of the lens. Measure the total distance D between the object and the screen. Estimate the error in this distance δD. The focal length of the lens is calculated using f= D 2 − d2 . 4D The errors in this calculated measurement is due to the errors in d and D. Use the errors δd and δD v !2 !2 u u d d D 2 + d2 t δf = δd + δd . δD ≈ 2 2D 4D 2D to estimate the error in f . 5.2 Analysis We have measured three different values for the focal length of the diverging cylindrical lens. Label them such that f1 < f2 < f3 and compare f1 to f2 and f2 to f3 using Difference, ∆i = fi − f2 , i = {1, 3}. Note that this “NULL hypothesis” was computed using two numbers that have experimental error. This means that ∆ also has error given by σi = q (δf2 )2 + (δfi )2 , i = {1, 3}. (5.7) If f1 = f2 , then ∆ = 0 or NULL; however, the results of measurement almost never turn out to be the same so the natural question becomes, “How nonzero must ∆ be before the difference is significant?” Use Equation (5.7) to convert the units of ∆ from cm to σ. Is ∆ < σ? Is ∆ < 2σ? Is ∆ > 3σ? We have also measured three different values for the focal length of the real lens on the optical bench. Repeat the analysis above for these three values. For each of these four experiments, determine whether the focal lengths are statistically distinct. One sigma disagreements happen about 1:3 even when the two numbers are the same just because random things happen to measured data. Two sigma disagreements happen about 1:22; even these chances are too large for us to decide that our hypothesis is definitely false. Three sigma disagreements happen only about 1:370, so differences this large lead us to admit that the two numbers that we wound up comparing are probably not the same. Larger differences are yet less likely. Statistics can tell us that the two numbers are probably not equal, but statistics cannot tell us why. Perhaps we have underestimated one or more of our errors. Perhaps the devices in our experiments have properties that contribute to our data and of which we have not properly compensated. Perhaps the environment in which we performed our experiment has contributed to our data. Perhaps our hypothesis is, in fact, false. As experimenters, we must evaluate all possibilities and decide which is the truth. Enumerate 64 CHAPTER 5: EXPERIMENT 3 some of these other sources of error and estimate how each affects your data. In your view is it more likely that we have underestimated how these other effects affected our data or that the theory we tested is false? Keep in mind that others have investigated these theories also and that their results are in the literature. 5.3 Conclusions What physics do your data support? What do they contradict? What did you test and fail to resolve? Do your data support Equation (5.5)? Might Equation (5.3) be true? We did not test Equation (5.3) numerically, but your observations on the optical bench might allow you to address this qualitatively. Always define the symbols used in these equations; it should not be necessary for your readers to read anything else to understand your Conclusions fully. Are there any limitations on the focus equation? What can be used as objects? Did you measure anything whose value might be useful later? The focal lengths of lenses and mirrors might be useful information if we should use these optical elements again. Always include measurement errors and units in these values. 65 CHAPTER 5: EXPERIMENT 3 66 Chapter 6 Experiment 4: Wave Interference WARNING This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 89 before attending this week’s laboratory. Helpful Tip Before beginning this experiment, it is probably a good idea to review the first several pages of the Sound Experiment on page 25. Waves of all kinds are quite similar in most respects so that the same interference between an incident and reflected sound wave that caused standing waves in the sound tube leads to the effects that we will observe today using light waves. 6.1 Introduction In this and the following lab the wavelike nature of light is explored. We have learned that waves usually exhibit some periodic or oscillatory nature such as the water level at a particular point on a pond steadily rising and falling as waves pass. For light (electromagnetic radiation) it is the electric and magnetic fields which oscillate in strength and direction. At visible frequencies these oscillations occur so fast that they are exceedingly difficult to observe directly (1014 Hz.) Another important characteristic of waves is that they may interfere with each other. In simple terms interference is the result of oscillating dips and rises adding together or canceling out. A good example of interference is seen when the circular wave patterns from two stones thrown into a pond meet. High points from the separate waves add to make even 67 CHAPTER 6: EXPERIMENT 4 T T 1 2 t (T) 1 3 2 3 t (T) (b) (a) T Figure 6.1: In each illustration the top two waveforms add point by point to yield the bottom waveform. The resulting effect can add as in the first case, cancel as in the second case, or anywhere in between. The difference between the three cases is the amount of constant phase that has been 3 1 2 t (T) added to the second waveform. This phase is (c) illustrated as a fraction of the total period. Waves have exactly the same dependence on position where the constant phase is in that case a fraction of wavelength instead of period. higher points, dips from the two waves also add making lower dips, and when a dip from one wave passes the same point as a high from the other they cancel. Waves are partly characterized by surfaces of constant phase. If time is frozen, then the argument of the wave’s function varies only in space and the function ωt = ϕ = k · r defines a surface in space. In the specific case where ϕ = k · r = kx, x is constant but y and z are arbitrary. Each x defines the function of a plane parallel to the y-z plane. This is a plane wave. Alternately, ϕ = kr defines surfaces of constant radius or spheres centered at r = 0. This is a spherical wave. Many other surfaces are possible, but these two are the most important ones. Plane waves are by far the easiest to treat mathematically and spherical waves are generated by point sources like atoms, molecules, and surface structures. Checkpoint What is a plane wave? Figure 6.1 shows how two sinusoidal waves add together to produce a new wave. In this lab we will see how interference affects the intensity pattern of light that has passed through two narrow adjacent slits, many parallel slits (diffraction grating), and a special bulls-eye pattern called a Fresnel zone plate. 68 d CHAPTER 6: EXPERIMENT 4 Plane Wave Figure 6.2: A plane wave incident upon two slits is transmitted as an interference pattern. In the center of the pattern (m = 0), the distance L1 from the top slit to the screen equals the distance L2 between the bottom slit and the screen. The time needed to transit this distance is exactly the same for the two paths. The two waves are in phase as in Figure 6.1(a). As we move down off the optical axis, L2 gets shorter and L1 gets longer until their difference is half a wavelength and they cancel as in Figure 6.1(b). Further off axis the path difference is a full wavelength and the peaks and valleys are aligned again (m = −1) as in Figure 6.1(a). 6.1.1 Part 1: Two Slit Interference Young’s Double Slit Experiment Historical Aside In 1801 Thomas Young devised a classic experiment for demonstrating the wave nature of light similar to the one diagrammed in Figure 6.2. By passing a wide plane wave through two slits he effectively created two separate sources that could interfere with each other. (A plane wave is a wave consisting of parallel wavefronts all traveling in the same direction.) For a given direction, θ, the intensity observed at point P on the screen will be a superposition of the light originating from every slit. Whether the two waves add or cancel at a particular point on the screen depends upon the relative phase between the waves at that point. (The phase of a wave is the stage it is at along its periodic cycle; the argument of the sine or cosine function.) The relative phase of the two waves at the screen varies in an oscillatory manner as a function of direction causing periodic intensity minima and maxima called fringes (see Figure 6.2). The spacing of the fringes depends on the distance between the two slits and the wavelength of the light. If the width of the slits is narrow compared to a wavelength, the intensity of the two-slit 69 d CHAPTER 6: EXPERIMENT 4 Figure 6.3: At an angle θ above the optical axis, the distance between the screen and the bottom slit is longer than the distance between the screen and the top slit by δ = d sin θ as shown by the geometry of the right triangle indicated. interference pattern is given by E(θ) = E0 cos β and I(θ) = I0 cos2 β ∝ |E(θ)|2 πd β= sin θ λ for (6.1) where θ is the angle between the initial direction, the location of the slits, and the observation. d is the distance between the two slits, λ is the wavelength of the light, E is the electric field, and I is the light intensity with units of energy per area per second. Equation (6.1) is easily derived. The amplitude of the electromagnetic wave emanating from either slit may be written as E = E0 sin(kx − ωt). After the wave emanating from one of the slits travels a distance l the wave undergoes a phase shift ϕ = kl = 2πl so that λ the electric field is of the form E = E0 sin(ωt + ϕ). The light arrives at the two slits at the 70 CHAPTER 6: EXPERIMENT 4 same time, so the phase difference is due to the difference in distances between the point of observation, P, and the respective slit. This difference in distance is shown in Figure 6.3 as a function of angle. It is seen that the wave front emanating from one slit lags behind the other by ∆l = d sin θ. This gives a relative phase difference of ∆ϕ = 2πd 2π∆l = sin θ. λ λ Since we are interested only in relative differences we can measure from the midpoint between the slits and write the fields due to each slit as 1 E1 = E0 sin ωt + ∆ϕ 2 1 and E2 = E0 sin ωt − ∆ϕ . 2 Using the trigonometric identity, sin A + sin B = 2 sin 12 (A + B) cos 21 (A − B) gives 1 E = E1 + E2 = 2E0 sin ωt cos ∆ϕ = 2E0 cos β sin ωt. 2 (6.2) When squared and averaged over time, this produces Equation (6.1). Maximum intensity peaks occur at angles such that |cos β| = 1 and the difference between the two path lengths is an integral multiple of wavelengths ∆l = d sin θ = mλ (see Figure 6.3). This translates to β = mπ (m = ±1, ±2, . . .) and thus the angles of maximum intensity satisfy, mλ = d sin θ and m = 0, ±1, ±2, . . . (6.3) If the width of the slits is greater than a wavelength the interference fringes are seen to oscillate in brightness as seen in Figure 6.4. This is due to diffraction and will be studied in a later experiment. Checkpoint Sketch two sinusoidal waves out of phase by 45◦ . Checkpoint Two waves in phase and each one with an amplitude A and intensity I meet at the same point. What is the intensity of the signal observed at that point? 6.2 The Experiment A helium-neon (HeNe) laser will be used to illuminate various patterned slides at normal incidence. A two-meter stick and a ruler will be used to quantify the locations of the slides, 71 y CHAPTER 6: EXPERIMENT 4 d x Plane Wave Figure 6.4: The intensity profile of light transmitted through two actual slits. By changing only the slit width a, we can verify that the intensity modulation indicated by the yellow trace is due to the slit width a and NOT to the slit separation d. Presently, we are interested in the closely-spaced, bright spots (the RED trace). The distance x between the slits and the screen and the distance y between the optical axis and each spot can be measured with a meter stick to determine θ using tan θ = xy . the original beam path, and the observation points that have bright spots. This information will be used to determine the angle θ needed to relate the laser’s wavelength to the distance between two slits’ centers. 6.2.1 Part 1: Two-Slit Interference You are provided a slide that has several sets of slits on it with separations and widths as shown in Figure 6.5. You use the HeNe laser as your plane wave light source. The experimental set up is as shown in Figure 6.6. If the laser beam is pointed onto a pair of slits, an interference pattern similar to Figure 6.7 may be observed on a screen placed nearby. The provided screen will hold masking tape where we may record the spot locations. This allows the distance between fringes to be marked and measured easily and conveniently. Observe the interference patterns. Draw what you see for each pattern on the slide in your Data. Be sure also to note the slit dimensions for each pattern. How does the slit separation affect the interference pattern? Can you detect any intensity variations in the different fringes? Why does this occur? For each set of double slits identify the intensity distribution caused by the width, a, of each slit as well as the interference fringes whose spacing is related to the distance, d, between the two slits. Two of the double-slits in Figure 6.5 are particularly interesting to observe because they have the same slit width, a, 72 CHAPTER 6: EXPERIMENT 4 but different slit separations, d. Consequently the minimum of the diffraction pattern must occur at the same θ value but the spacing of the interference fringe pattern varies. Also, two of the double-slits in Figure 6.5 have the same slit separation, d, but different slit widths, a. This combination of four double-slit patterns allows you to isolate the effects of these two independent parameters. You will now make measurements using one double-slit pattern. When the slit mask is perpendicular to the beam, its reflection will re-enter the laser. Orient the screen perpendicular to the beam as well and mark the locations of the interference fringes (the finely spaced bright spots) on a strip of fresh tape. Mark all visible spots; if fewer than five spots are visible on each side of the beam axis, choose a different double-slit pattern. Also measure the distance from the double-slit slide to the screen, x, taking care not to disturb the screen. Remove the double-slits and circle the undeflected laser spot; this is the zero order (m = 0) fringe. Take care not to tear your tape as you remove it from the screen, make a table in your Data, and measure the distances from the beam axis to the Figure 6.5: A photograph of Pasco’s bright spots, ym . It will be more accurate to measure OS-8523 basic optics slit set. We are 2ym (the distance between the pair of dots on opposite interested in the double slits at the sides of the beam axis) and then divide by two. Be bottom left. sure to note your error estimates and units for these measurements. Using λ = 632.8 nm as the wavelength of the laser-light and Equation (6.3), calculate the slit separation. Do the distances, Rm , between the different bright spots and the slits vary significantly more than the total of your measurement errors? If not then we can make an approximation that sin θm = so that d = mλ sin θm ≈ ym ym ≈ = tan θm Rm x (6.4) mλx . ym You can download a suitable setup file for Ga3 from the lab’s website at http://groups.physics.northwestern.edu/lab/interference.html Otherwise, load one column in Vernier Software’s Graphical Analysis 3.4 (Ga3) program with the order numbers (you can Data/Fill to do this quickly), load your measurements of ym (or 2ym if this is what you measured) into another column, and create a calculated column to hold your slit width results. Double-click the column headers to correct the labels and to add the correct units. What will be the units of d? (If x and ym have the same units, then these units will cancel in the ratio; otherwise, it will be more difficult for you to determine the units of d.) Use the “Column” button to choose column “m” and column “ym ” to enter the appropriate formula for d into the “Equation” edit control. (You will see 73 CHAPTER 6: EXPERIMENT 4 Figure 6.6: A sketch of our apparatus showing how the HeNe laser can be directed through the slits and projected onto a view screen. your column names instead of mine.) If you measured 2ym , you will need to divide this column (the denominator) by two or to multiply the numerator by two. You will also need to type in appropriate numbers for λ and x; Ga3 only works with numbers. You will also need explicitly to tell Ga3 to multiply “*”. y3 y-7 Figure 6.7: One example of an interference pattern due to HeNe laser light passing through double slits. Large spots indicate large intensity. If the slit dimensions satisfy d = n a for integer n, there will be missing orders at every nth spot as indicated by the black circles containing no spot. The positions of the spots are measured as indicated for m = 3 on the right and for m = −7 on the left. We MUST count the black circles despite the missing spot. Calculate an average value for each double-slit separation and compare with those listed in Figure 6.5. Plot the slit separation column on the y-axis and the order, m, on the x-axis. Click on the smallest y-axis number and change the range beginning to 0. Do your values of d vary more than you expect your errors to explain? If not, draw a box around your data points with the mouse and Analyze/Statistics. Move the parameters box away from your ¯ is the best guess for your measurement of slit separation and data points. The “mean”, d, the “s. dev.” (or standard deviation, sd ) can be used to estimate the standard error or error in the mean sd (6.5) sd¯ = √ . N You can specify your slit separation using d = (d¯ ± s ¯) U. d 6.2.2 Part 2: Diffraction Gratings A system consisting of a large number of equally spaced, parallel slits is called a “grating”. The diffraction pattern expected from a grating can be calculated in a way similar to the 74 CHAPTER 6: EXPERIMENT 4 Figure 6.8: Two Fresnel Zone Plates. Dark rings block light that would be out of phase and would interfere destructively on the optical axis. Much smaller Fresnel Zone Plates can focus X-rays without a lens. These are suitable for microwaves. two slit system. A simple way to do this calculation is to represent the amplitude and phase of each wave originating from one of the grating slits and impinging on a point on the screen by a vector called a phasor (see textbook). Summing the phasors corresponding to each slit gives the correct amplitude and phase of the total wave. The intensity distribution of the light scattered off a grating with N very narrow slits is sin N β I(θ) = I(0) N sin β !2 (6.6) where β is defined in Equation 6.3. This relation is very similar to the one for the interference of two slits. In fact if N = 2, then substituting the identity sin 2β = 2 sin β cos β in the last term of Equation 6.6) gives sin N β N sin β !2 = sin 2β 2 sin β !2 = 2 sin β cos β 2 sin β !2 = (cos β)2 and it is seen to reduce to the two slit formula Equation (6.2). sin N β The angular dependence of the interference maxima are described by the function N . sin β With a large number of slits, this function has a numerator rapidly varying with θ and a denominator slowly varying. As the number of illuminated slits increases, the intensity peaks will narrow. The position of the fringe maxima generated by a grating is given by Equation 6.3. Gratings are commonly used to measure the wavelength of light, λ. This may be determined using Equation (6.6) and by measuring the angle between the maxima. These measurements are called spectroscopy and will be very important in our last experiment. Place the diffraction grating in the path of the green laser and perpendicular to the beam. 75 CHAPTER 6: EXPERIMENT 4 The number of slits per inch should be written on one side of the slide. Because the spacing between slits is so small the angles of the maxima are quite large. You will need to keep the screen close so that the 1st and 2nd order maxima can be seen. Measure the distance between each of these spots and the central maximum. Don’t forget your units and error. Also measure the distance between the screen and the grating slide. How much does the intensity vary for the central, the 1st , and the 2nd order maxima? For both measurements (or all four if you did not measure 2 ym ) made in step 2a) calculate values of the wavelength of the light. Note that these angles are not small so that sin θ ≈ tan θ does NOT apply to this case. You must use ym d sin tan−1 . λ= m x (6.7) Compare these values to the given wavelength of the laser, 532.0 nm. Since we used ym , x, and d to calculate λ, we must expect that their errors will contribute to an experimental error in λ. Estimate the error in λ using λ δd d q d 3 −1 ym cos tan (xδym )2 + (ym δx)2 δλ2 = m x q δλ1 = δλ = (δλ1 )2 + (δλ2 )2 . Calculate the maximum number of fringes that you are expected to observe with the grating. (Hint sin θ ≤ 1). Do you observe all of the expected fringes? Checkpoint What are gratings useful for? 6.2.3 Part 3: The Fresnel Zone Plate In the Fraunhofer formalism light is represented by plane waves, and the distance between light-source and scattering object or scattering object and observer is assumed to be very large compared to the dimensions of the obstacle in the light path. If you drop these conditions, the light phenomena observed are described by the Fresnel formalism and the light waves are represented by spherical waves rather than plane waves. Figure 6.9 shows the spherical surface corresponding to the primary wave front at some arbitrary time t after it has been emitted from S at t = 0. As illustrated the wave front is divided into a number of annular regions. The boundaries between these regions correspond to the intersections of the wave front with a series of spheres centered at P and having radius r0 + 12 λ, r0 + λ, r0 + 32 λ, etc. 76 CHAPTER 6: EXPERIMENT 4 Figure 6.9: The Fresnel Zones. The surface of the spherical wavefront generated at point S in the figure above has been divided into several ‘Fresnel Zones’. Each area is comprised of points that are close to the same distance from point P and thus all the secondary wavelets emanating from within the same Fresnel Zone will add constructively at P. These are Fresnel zones or half-period zones. The sum of the optical disturbances from all m zones at P is E = E1 + E2 + . . . + Em (6.8) Checkpoint What is the difference between Fraunhofer and Fresnel scattering? The path length for the light passing through each consecutive zone increases by λ/2 so the phase changes by π and the amplitude, Eζ , of zone ζ alternates between positive and negative values depending on whether ζ is odd or even. As a result contributions from adjacent zones are out of phase and tend to cancel. The surface of the spherical wavefront generated at point S in Figure 6.9 has been divided into several ‘Fresnel Zones’. Each area is comprised of points that are close to the same distance from point P and thus all the secondary wavelets emanating from within the same Fresnel Zone will add constructively at P. This suggests that we would observe a tremendous increase in irradiance (intensity) at P if we remove all of either the even or the odd zones. A screen which alters the light, either in amplitude or phase, coming from every other half-period is called a zone plate. Examples are shown in Figure 6.8. Suppose that we construct a zone plate which passes only the first 20 odd zones and obstructs the even ones, E = E1 + E3 + . . . + E39 77 (6.9) CHAPTER 6: EXPERIMENT 4 Zone Plate rm Rm S r0 O P Figure 6.10: A schematic diagram showing how travel distance can be determined for light passing through a zone plate at a distance Rm from the optical axis. If this distance is within ρ0 + r0 + (m ± 14 )λ, the zone plate is transparent; otherwise it is opaque. and that each of these terms is approximately equal. For a wavefront passing through a circular aperture the size of the fortieth zone, the disturbance at P (demonstrated by Fresnel but not obvious) would be E1 /2 with corresponding intensity I ∝ (E1 /2)2 . However, with the zone plate in place E ∼ = 20E1 at P. The intensity I of the light at P has been increased by a factor of 1600 times. The zone plate acts as a lens with the focusing being done by interference rather than by refraction! To calculate the radii of the zones shown in Figure 6.8, refer to Figure 6.10. The outer edge of the mth zone has radius Rm . By definition, a wave which travels the path S-Rm -P must arrive out of phase by m2 λ with a wave which travels the path S-O-P, that is, Path difference = (ρm + rm ) − (ρ0 + r0 ) = According to the Pythagorean theorem ρm = q mλ . 2 2 + ρ2 and r = Rm m 0 q (6.10) 2 + r 2 . Expand both Rm 0 2 these expressions using the binomial series and retain only the first two terms, ρm ≈ ρ0 + 2Rρm0 2 and rm ≈ r0 + 2Rrm0 . Substituting into Equation (6.10) gives the criterion the zone radii must satisfy to maintain the alternating λ/2 phase shift between zones, mλ R2 R2 2 = (ρm + rm ) − (ρ0 + r0 ) ≈ m + m and Rm ≈ 2 2ρ0 2r0 1 ρ0 mλ . + r10 (6.11) The width of zone m is proportional to √1m . Since the circumference (length) of the ring is √ 2πRm ∼ m, the area and transmitted intensity is the same for all m. For m = 1 we may rewrite Equation (6.11) as 1 1 λ 1 + = 2 = (6.12) ρ0 r0 R1 f1 78 CHAPTER 6: EXPERIMENT 4 to put it in a form identical to the thin lens equation ( d1o + d1i = f1 ) with primary focal length R2 f1 = λ1 . Once the zone plate is made, the ring radii are fixed so that the focal length is different for every wavelength. This property might be utilized in a spectrometer. The spacing of the zones on the Fresnel Zone Plate are very fine. If a narrow beam is passed through the plate it will act as a diffraction grating. Put the Fresnel Zone Plate in the red laser path and observe the diffraction pattern. Move the plate back and forth so that you observe the results when light hits the center and the outside of the plate. Is the slit separation smaller on the outside or center of the plate? What have you observed that supports this answer? Can you sketch the zones in your Data? This kind of experiment used with X-rays allows us to quantify the properties of crystalline solids. Measure the primary focal length of your setup’s zone plate. Do this by centering the laser beam on the double concave lens which will diffuse the beam and generate on the screen a uniformly illuminated circle. The divergent beam emanating from the double concave lens now acts as a point source or ‘object’. Now place the zone plate between the lens and the screen, and change the position of the screen along the direction of the beam to observe the refocusing of the laser beam. Measure the two quantities ρ0 and r0 (Figure 6.10) for at least 2 different initial values of r0 and then use Equation (6.12) to calculate f1 . Don’t forget your units. How far can you move the zone plate before the spot is out of focus? You must consider all of these locations when estimating your errors. Often zone plates are made of plastic material or metal with adjacent rings supported by spokes so that the transparent regions are devoid of any material. These will function as lenses in the range from ultraviolet to soft (low energy) X-rays (1 to 1000 Å) where ordinary glass is opaque. Zone plates have also been used to focus low energy neutron beams. Checkpoint What is a Fresnel half-period zone? Sketch a zone plate. What are the zone plates used to do? 6.3 Analysis Calculate the difference between the manufacturer specified double-slit spacing and the mean of your measurements ∆ = Measured - Manufacturer (6.13) Calculate the exptected error in this differences using σ= q (δMeasured)2 + (δManufacturer)2 . Repeat for the measured green laser wavelength versus 532.0 nm. 79 (6.14) CHAPTER 6: EXPERIMENT 4 For each, is ∆ < 2σ? Is ∆ > 3σ? Recall that ∆ > σ about 1:3 and ∆ > 2σ about 1:22 just from random fluctuations; these probabilities are too large for us to assert that the compared two numbers are not equal. But ∆ > 3σ only 1:370 due only to random fluctuations, so differences this large suggest that we conclude that the two numbers are indeed different; statistics cannot tell us why, however. Can you think of anything in the three experiments that might reasonably affect the measurements that we have not already included in our measurement errors? Discuss each of these using complete sentences and paragraphs. In your view is it more likely that the theory that has survived for a hundred years of professional tests is false or that you have made some mistake or overlooked some significant sources of error that might explain your nonzero difference(s)? 6.4 Conclusions What physics have your experiments supported? Contradicted? What have you investigated and failed to resolve? Is mλ = d sin θ? What do these symbols mean? Does light exhibit wave properties? If so, which properties? Do Fresnel zone plates focus light? Don’t forget that we communicate using complete sentences. Have you measured anything that might be of value in the future? If so, state what they are, give your measured values, their errors, and their units. 6.4.1 References Sections of this write up were taken from: • Bruno Gobi, Physics Laboratory: Third Quarter, Northwestern University • D. Halliday & R. Resnick, Physics Part 2, John Wiley & Sons. • R. Blum & D. E. Roller, Physics Volume 2, Electricity, Magnetism, and Light, HoldenDay. • E. Hecht/A. Zajac, Optics, Addison-Wesley Publishing 80 CHAPTER 6: EXPERIMENT 4 6.5 APPENDIX: The Helium-Neon Laser The laser (Light Amplification by Stimulated Emission of Radiation) is a source of radiation capable of delivering intense coherent electromagnetic fields. The laser beam coherence is manifest in two ways: 1) it is highly monochromatic, and 2) it is spatially coherent; the emitted light has a nearly constant wavefront and directionality. The special characteristics of laser radiation are directly attributable to the phenomenon of stimulated emission and to the positive feedback mechanism provided by the cavity structure. Figure 6.11, a schematic diagram of a typical gas laser, highlights the major elements of a gas system. It consists of an active medium (lasing material), which in this case is a gas, but may consist of a solid or a liquid. A power supply pumps energy into the active medium, exciting the atoms. To achieve oscillations mirrors are placed perpendicular to the axis of the active medium to form an optical resonant cavity. Stimulated emission in the active medium results in the required amplification, whereas the mirrors supply the feedback required for regenerative action and oscillation. The rest of this appendix explains these concepts in greater detail. (a) (b) Figure 6.11: Two energy diagrams illustrating photo-emission and photo-absorption. In the first case an atom gives its energy of excitation to the creation of a photon. In the second case a photon collides with an electron and promotes it to a more energetic atomic orbit. Checkpoint Which ones are the unique properties of light emitted by a laser? 6.5.1 Atomic Levels The atom (see 8th lab) consists of a small heavy nucleus surrounded by orbiting electrons. Quantum Mechanics describes the orbits of the electrons as standing waves. Consequently, 81 CHAPTER 6: EXPERIMENT 4 Figure 6.12: Two energy diagrams illustrating photo-absorption and stimulated emission. If the atom is initially not excited, the photon excites the atom. If the atom is initially excited, the incident photon stimulates the emission and both photons are identical to the incident photon. the electrons belonging to a specific orbit have a discrete energy value. Of the infinite number of allowed electron orbits (energy levels), only the lowest energy levels are filled when the atom is in the ground state. When the atom is excited, electrons are transferred from the low energy stable orbits to higher states, leaving vacancies in the original orbits. The excited atom will then decay (the electron in the excited state jumps back down to lower levels) with a time constant characteristic of the level. (The time constant of an atomic level has exactly the same meaning as the time constant of an RC circuit.) This time is usually around 10−8 s, but in some cases it is on the order of a second (it is then called metastable). Figure 6.11 in the 8th lab shows, as an example, the energy levels of hydrogen. For the sake of simplicity, we shall consider an atom having only two possible energy states: an upper state E2 and a lower state E1 as shown in Figure 6.12. If the atom is in the upper state (excited state) and makes a transition to the lower state, then ∆E energy can be emitted in the form of radiation with the frequency E2 − E1 (6.15) ν= h (h is the Planck constant: h = 6.634 × 10−34 J·s). On the other hand, if the atom is initially in the lower energy state, E1 , and makes a transition to the higher state, E2 , then energy and hence radiation with a frequency given by Equation 6.15, must be absorbed. Albert Einstein showed that emission (transition from E2 → E1 ) can occur in two ways: 1) by the atom randomly changing to the lower state with lifetime τ . This is called spontaneous emission and is contrasted with absorption in Figure 6.11. 2) by a photon having an energy equal to the energy difference between the two levels interacting with the atom in the upper state and causing it to change to the lower state with the creation of a second photon. This process can be thought of as the converse of absorption and is known as stimulated emission. See Figure 6.12. There are two very important points about stimulated emission upon which the properties 82 CHAPTER 6: EXPERIMENT 4 of laser light depend. First, the photon produced by stimulated emission is of almost equal energy to that which caused stimulated emission and hence the light waves associated with them must be of nearly equal frequency. Second, the light waves associated with the two photons are in phase - they are said to be coherent. In the case of spontaneous emission the random emission of photons results in waves of random phase and the light is said to be incoherent. Checkpoint Sketch an energy level diagram of an atom. Indicate one transition and the frequency of the transition. How many possible energy levels are present in an atom? Checkpoint What does it mean that a level of an atom has a lifetime of 10−8 s? What is a metastable level (or state) of an atom? Checkpoint What is the difference between spontaneous emission and stimulated emission? 6.5.2 Population Inversion Consider two two-level energy systems representing an atom in an upper and lower state. Suppose that a photon of energy equal to the energy difference between the two levels approaches the two atoms, then which event is more likely to take place, absorption or stimulated emission? Einstein showed that under normal circumstances, both processes are equally probable. A larger population (number of atoms) in the upper level will result in stimulated emission dominating, while if there are more atoms in the lower level, there will be more absorption than stimulated emission. Normally, there are more atoms in the lower level, since in equilibrium such a system tends to minimize its energy (remember your chemistry!) Hence, under the normal thermodynamic circumstances absorption will dominate. However, if we can somehow put energy into the system in such a way that there are more atoms in the upper level, then stimulated emission will dominate. When we have such a situation, where we have put more atoms into the upper level than there are in the lower level, we have achieved population inversion. 83 CHAPTER 6: EXPERIMENT 4 Figure 6.13: Two energy diagrams illustrating how an excited He atom collides with an unexcited Ne atom; the collision transfers the excitation from the He to the Ne’s longlived metastable state. Identical collisions occurring throughout the gas mixture causes more excited Ne atoms than ground-state Ne atoms. 6.5.3 Excitation of Atoms Atoms can be excited by electrical discharge, radio frequency signals (RF), heat, chemical reactions, collisions with electrons, or collisions with other atoms. The excitation by collision (two different kinds) is the most relevant process in a gas laser. 1. Collisions of the first kind involve the interaction of an energetic electron with an atom in the ground state. The impact of the electron causes it to exchange some of its energy with the atom and the latter becomes excited. This process can be represented by an equation similar to the one used in chemical reactions, i.e. He + e1 → He∗ + e2 (6.16) the * signifies that the He is in an excited state. The kinetic energy of the electron before and after the collision is designated by e1 and e2 respectively. Since all electrons are identical, we cannot know whether two electrons exchange places in the collision; indeed, many argue that the distinction is meaningless. 2. A collision of the second kind involves the collision of an excited atom in a metastable state with another atom of a different element in an unexcited state, the final result 84 CHAPTER 6: EXPERIMENT 4 being a transfer of energy from the metastable state to the unexcited atom, thus exciting it. The metastable atom therefore loses energy and reverts to a lower state. An example of a collision of the second kind is represented by the following equation, He∗ + Ne → He + Ne∗ . (6.17) The excited Ne atom is then ready to emit a photon and to return to its ground-state. Checkpoint How do you excite an atom? 6.6 The Helium-Neon Laser Figure 6.13 shows juxtaposed the energy levels of He and Ne. Only the levels relevant to the operation of this laser are indicated. The boxes in the figure indicate groups of tightly spaced levels (energy bands) rather than sharp single ones, and the letter M indicates that the level is metastable (long lifetime). The beam of light of the Helium-Neon Laser is produced by the stimulated emission of radiation produced in the decay of excited electrons in the 3S and 2S levels of neon. These levels of the neon atom, however, are not easily excited because the outer orbitals have a much greater probability of being if excited before them. Instead, energy supplied by an electrical current (in the form of energetic electrons) is stored in the metastable levels of helium and later transferred by collisions of the second kind to neon. The two 21 S and 23 S metastable levels of helium are very close in energy to the 2S and 3S levels of neon which we wish to populate with excited electrons. Therefore, collisions between helium and neon (which are ensured by selecting a ratio of He to Ne gas of 10 to 1) leave the 2S and 3S levels of neon well populated. The observed light of the HeNe laser is generated only by the decay of 3S electrons to the 2P state (632.8 nm). Sustained operation of the laser is ensured by the He/Ne ratio and by the fact that the lower level (2P) decays 10 times faster (decay time t = 10−8 s) than the upper (3S) one (t = 10−7 s). (So that basically all electrons that decay to the 2P level undergo further decay). Light generated in any other forms of decay is suppressed (for example by inserting a prism into the cavity, which diverges the unwanted frequencies away from the axis of the laser tube). We have so far described how energy is pumped into the plasma to achieve population inversion. To have an operating laser we still need signal amplification and the proper feedback to sustain the oscillation. This will be discussed next. 85 CHAPTER 6: EXPERIMENT 4 Figure 6.14: A schematic diagram of a resonant cavity filled with a plasma tube. The light gain medium in the cavity replaces cavity losses to sustain the laser oscillator. Checkpoint In a He-Ne laser, which gas emits the observed transition of 632.8 nm? Checkpoint How are the Ne atoms excited in a He-Ne laser? Checkpoint What is the meaning of “population inversion” when explaining the operation of a laser? Checkpoint How do you eliminate radiation of unwanted wave length from a laser beam? Checkpoint Why is the ratio of the He- to Ne- gas, 10 to 1, in the He-Ne laser? 6.6.1 Amplification and feedback We have indicated above that the observed light of the HeNe laser is generated only by the transition of 3S electrons to the 2P state (632.8 nm). In this process of spontaneous emission, a great part of the radiation propagates away from the cavity in arbitrary directions and is lost. (Though some will be absorbed by a 2P state of neon and will repopulate the 3S 86 CHAPTER 6: EXPERIMENT 4 level.) A small portion, however, will strike other neon atoms (excited to the 3S level) and cause stimulated emission. Of the photons emitted here, however, only the small portion emitted in a direction perpendicular to the mirrors may stay in the cavity and contribute to the amplification process. Upon successive reflections by the mirrors, light traveling through the laser medium is further coherently amplified. Since the HeNe laser light amplification is only 1.02 on each pass of the beam from one mirror to the other, all losses must be kept below 2%. Losses in the cavity formed by plane-parallel mirrors will be extremely sensitive to the parallelism of the two mirrors. Slight angular misalignment might result in the reflected beam missing the opposite mirror and lead to excessive loss and difficulties in attaining oscillation. Even under perfect parallelism, it can be shown that a resonator formed by plane-parallel mirrors has higher diffraction losses than a cavity formed by various combinations of plane and curved mirrors. Since the laser tube and the mirrors form an optical resonant cavity, the optical path length between successive reflections of the mirror must be an integral number of wavelengths to produce reinforcement of the wave. This arrangement of mirrors was familiar long before the advent of lasers and is known as the Fabry-Perot interferometer. Checkpoint What physical effect is responsible for the amplification of the light signal of a laser? 6.6.2 The Laser’s Mirrors To produce an external laser beam, the mirror at the front of the laser tube is a partial reflector which reflects 99% of the light and transmits approximately 1%. The mirror at the back end of the laser tube has a higher reflectivity and reflects about 99.9% of the light while transmitting less than 0.1%. Laser mirrors are always made by successive deposition of materials with high and low refractive indices onto substrates of the highest optical quality. By making the optical thickness of each layer a quarter of a wavelength thick, very high reflectivities can be achieved. The mirror is the most expensive part of a laser - often good quality mirrors are made of 20 or more layers of dielectric material. Checkpoint Why is the plasma tube of a gas laser enclosed by two parallel mirrors normal to the axis of the tube? How do you extract the light from a laser? 87 CHAPTER 6: EXPERIMENT 4 Figure 6.15: A sketch of light propagating through an excited active gain medium. Spontaneous emissions with a radial component quickly leave the cavity after very little amplification; contrarily, the axial mode gets amplified continuously and repetitively. 6.6.3 Mode Configuration A beam of light propagating down the optical axis of the laser will be amplified, and providing sufficient gain is available, will emerge from the laser output mirror. Generally speaking, a photon traveling at right angles to the optical axis will not be amplified to form a laser beam. Although it may cause many stimulated emissions, there will not be a large enough number to overcome all losses or, in other words, the gain will be inadequate. However, in the situation where a wave is traveling only slightly off axis, the wave may be able to zig-zag or “walk” between the mirrors a sufficient number of times to produce enough gain to overcome the losses. This produces the laser output, which consists of a variety of complicated patterns. These patterns are the result of the laser operating in what is referred to as transverse modes. (Some light rays are not aligned on the laser’s axis.) Such modes are designated by the notation TEMpq modes (TEM standing for transverse electromagnetic). The most commonly used mode is the TEM00 mode. A laser operating in this mode has a Gaussian flux density pattern over the beam cross section. This pattern - a single disk area - produces a single spot of light. Checkpoint What does the beam spot look like for a laser operating in the TEM00 mode? 88 CHAPTER 6: EXPERIMENT 4 6.6.4 References Sections of this write up were taken from: • M.J. Beesley, Lasers and Their Applications, Taylor Z Francis Ltd., and Barnes & Noble Inc., 1971. • Edited by S.S. Charschan, Lasers in Industry, Western Electric Series, Van Nostrand Reinhold Company. 6.7 Laser Safety WARNING Lasers emit extremely bright light. Laser light can damage your eyes. Lasers exist that cut and weld metals; however, this lab does not contain them. Care must be exercised to avoid exposing your eyes to the laser’s light. Powerful pulsed lasers are used for welding and are also capable of drilling holes in metal. High-power lasers are used in surgery, in resister trimming, in non-destructive testing, and even in static and dynamic art exhibits. The laser used in the lab has a nominal 0.5 mW visible output which places the device within the Class III category, as defined by the Department of Health, Education, and Welfare for laser safety. To insure absolute safety, however, the following precautions and safety procedures MUST be observed. 1) Treat all laser beams with great respect. 2) Never look into the laser window (even when turned off) or stare into the beam (on axis) with either the naked eye or through binoculars or a telescope at a distance. 3) Do not rely on sunglasses as eye-protecting devices. 4) Never point the laser beam near anyone else’s eyes. 5) Cover all windows to protect passers-by. 6) Never leave lasers unattended while activated. If not in use disconnect A.C. power cable. 7) Room illumination in the work area should be as high as is practicable to keep the eye pupil small and reduce the possibility of retinal damage due to accidental exposure. 8) Remove all superfluous and highly reflective objects from the beam’s path. These include rings, watches, metallic watchbands, shiny tools, glassware, etc. 89 CHAPTER 6: EXPERIMENT 4 90 Chapter 7 Experiment 5: Diffraction of Light Waves WARNING This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 89 before attending this week’s laboratory. 7.1 Introduction In this lab the phenomenon of diffraction will be explored. Diffraction is interference of a wave with itself. According to Huygen’s Principle waves propagate such that each point reached by a wavefront acts as a new wave source. The sum of the secondary waves emitted from all points on the wavefront propagate the wave forward. Interference between secondary waves emitted from different parts of the wave front can cause waves to bend around corners and cause intensity fluctuations much like interference patterns from separate sources. Some of these effects were touched in the previous lab on interference. General Information We will observe the diffraction of light waves; however, diffraction occurs when any wave propagates around obstacles. Diffraction and the wave nature of particles leads to Heisenberg’s Uncertainty Principle and to the very reason why atoms are the sizes that they are. In this lab the intensity patterns generated by monochromatic (laser) light passing through a single thin slit, a circular aperture, and around an opaque circle will be predicted and experimentally verified. The intensity distributions of monochromatic light diffracted 91 CHAPTER 7: EXPERIMENT 5 m=4 m=3 m=2 y m=1 m=0 a x m = -1 m = -2 Diffraction Pattern m = -3 m = -4 Plane Wave Figure 7.1: A sketch of a plane wave incident upon a single slit being diffracted and the resulting intensity distribution. The intensity is on a logarithmic scale; since our eyes have logarithmic response, this much more closely matches the apparent brightness of the respectve dots. Keep in mind that this week we are concerned with the dark intensity MINIMA and that m = 0 is NOT a minimum. from the described objects are based on: 1. the Superposition Principle, Intensity vs. Position 2. the wave nature of light Disturbance: 1 (a) Amplitude: A = A0 sin(ωt + ϕ) (b) Intensity: I ∝ ( P 0.8 A)2 , 0.6 3. Huygen’s Principle - Light propagates in such a way that each point reached by the wave acts as a point source of a new light wave. The superposition of all these waves represents the propagation of the light wave. 0.4 0.2 0 -7 -5 -3 -1 1 3 5 7 y (cm) All calculations are based on the assumption that the distance, x, between the slit and the Figure 7.2: A graphical plot of an examviewing screen is much larger than the slit width ple diffraction intensity distribution for a a:, i.e. x >> a. This particular case is called single slit. Fraunhofer scattering. The calculations of this 92 CHAPTER 7: EXPERIMENT 5 type of scattering are much simpler than the Fresnel scattering in which case the x >> a constraint is removed. Checkpoint What is the difference between Fraunhofer and Fresnel scattering? Checkpoint Are the eyes sensitive to the amplitude, to the phase, or to the intensity of light? Are the eyes’ response linear, logarithmic, or something else? 7.1.1 Part 1: Diffraction From a Single Slit A narrow slit of infinite length and width a is illuminated by a plane wave (laser beam) as shown in Figure 7.1. The intensity distribution observed (on a screen) at an angle θ with respect to the incident direction is given by Equation (7.1). This relation is derived in detail in the appendix and every student must make an effort to go through its derivation. The mathematics used to calculate this relation are very simple. The contributions from the field at each small area of the slit to the field at a point on the screen are added together by integration. Squaring this result and disregarding sinusoidal fluctuations in time gives the intensity. The main difficulty in the calculation is determining the relative phase of each small contribution. Figure 7.2 shows the expected shape of this distribution; mathematics can describe Figure 7.3: A photograph of Pasco’s this distribution, OS-8523 optics single slit set. We are sin α 2 πa I(θ) interested in the single slits at the top = with α = sin(θ), (7.1) I(0) α λ right and the circular apertures at the top left. where a is the width of the slit, λ is the wavelength of the light, θ is the angle between the optical axis and the propagation direction of the scattered light, I(θ) is the intensity of light scattered to angle θ, and I(0) is the transmitted light intensity on the optical axis. When θ = 0, α = 0, and the relative intensity is undefined (0/0); however, when θ is very small and yet not zero, the relative intensity is very close to 1. In fact, as θ gets closer to 0, this ratio of intensities 93 CHAPTER 7: EXPERIMENT 5 gets closer to 1. Using calculus we describe these situations using the limit as α approaches 0, sin α lim = 1. α→0 α Although 0/0 might be anything, in this particular case we might think that the ratio is effectively 1; this is the basis for the approximation sin θ ≈ θ when θ << 1 radian. Certainly, the intensity of the light on the axis is defined, is measurable, and is quite close to the intensity near the optical axis. The numerator of this ratio can also be zero when the denominator is nonzero. In these cases the intensity is predicted and observed to vanish. These intensity minima occur when 0 = sin α for each time πm = α and mλ = a sin θ for m = ±1, ±2, . . . (7.2) Figure 7.4: A sketch of our apparatus showing the view screen, the sample slide, the laser, and the laser adjustments. Our beam needs to be horizontal and at the level of the disk’s center. We might note the similarity between this relation and the formula for interference maxima; we must avoid confusing the points that for diffraction these directions are intensity minima and for diffraction the optical axis (m = 0) is an exceptional maximum instead of an expected minimum. This relation is satisfied for integer values of m. Increasing values of m give minima at correspondingly larger angles. The first minimum will be found for m = 1, the second for m = 2 and so forth. If is less than one for all values of θ, there will be no minima, i.e. when the size of the aperture is smaller than a wavelength (a < λ). This indicates that diffraction is most strongly caused by protuberances with sizes that are about the same dimension as a wavelength. Four single slits (along with some double slits) are on a disk shown in Figure 7.3 photograph of Pasco’s OS-8523 optics single slit set. We are interested in the single slits at the top right and the circular apertures at the top left.. This situation is similar to the one diagrammed in Figure 7.1. To observe diffraction from a single slit, align the laser beam 94 CHAPTER 7: EXPERIMENT 5 parallel to the table, at the height of the center of the disk, as shown in Figure 7.4. When the slit is perpendicular to the beam, the reflected light will re-enter the laser. The diffraction pattern you are expected to observe is shown in Figure 7.5. Figure 7.5: Observed Diffraction Pattern. The pattern observed by one’s eyes does not die off as quickly in intensity as one expects when comparing the observed pattern with the calculated intensity profile given by Equation (7.1) and shown in Figure 7.2. This is because the bright laser light saturates the eye. Thus the center and nearby fringes seem to vary slightly in size but all appear to be the same brightness. Observe on the screen the different patterns generated by all of the single slits on this mask. Note the characteristics of the pattern and the slit width for each in your Data. Is it possible from our cursory observations that mλ = a sin θ? If a quick and cheap observation can contradict this equation, then we need not spend more money and time. Calculate the width of one of the four single slits. This quantity can be calculated from Equation (7.2) using measurements of the locations of the intensity minima. The wavelength of the HeNe laser is 6328 Å, (1 Å= 10−10 m). The quantity to be determined experimentally is sin θ. This can be done using trigonometry as shown in Figure 7.6. Measure the slit width using several intensity minima of the diffraction pattern. Place a long strip of fresh tape on the screen and carefully mark all of the dark spots in the fringes. Be sure to record the manufacturer’s specified slit width, a. Avoid disturbing the laser, the slit, and the screen until all of the minima are marked and the distance, x, between the slit and the screen is measured. Is your screen perpendicular to your optical axis? Remove the slit and circle the undeflected laser spot. Ideally, this spot is exactly the center of the diffraction pattern. Make a table in your Data and record the order numbers, m, and the distances, ym , between the optical axis and the dark spots. It is more accurate to measure the distance 2 ym between the two mth minima on opposite sides of the axis; you might wish to avoid this additional complication. Note the units in the column headings and the errors in the table body. Enter your data table into Vernier Software’s Ga3 graphical analysis program. A suitable setup file for Ga3 can be downloaded from the lab’s website at http://groups.physics.northwestern.edu/lab/diffraction.html Otherwise, change the column names and units to represent your data. Add a calculated column to hold your measured slit widths. Enter the column name and appropriate units and type 95 CHAPTER 7: EXPERIMENT 5 R y x Figure 7.6: A sketch of the trigonometric functions sine and tangent. “m” * 632.8 * x / (c * “ym”) into the Equation edit control. Use the “Column” button to obtain the quoted values; choose your column names instead of mine. Instead of “x” type the number of cm, m, etc. between your screen and slit; use the same units for x as for ym so that they cancel in the above ratio. What units remain to be the units of a? If you like, you can use the value “c” to convert these units to nm (c = 1), µm (c = 1000), mm (c = 1000000), or m (c = 109 ). Be sure to use the correct units in the column heading to represent your numbers. Plot your slit widths on the y-axis and the order number on the x-axis. Click the smallest number on the y-axis and change the range to include 0. Does it look like all of your data are the same within your error? If so, draw a box around your data points and Analyze/Statistics. Move the parameters box off of your data points. Interpret the “mean” ā of your data to be the best estimate of the slit’s true width and the “std.dev” to be the best estimate of each data point’s error, sa . Estimate the error in the mean to be the standard error sa sā = √ . N Specify your best estimate of slit width as a = (ā ± sā ) U. Checkpoint The relation of sin2 α/α2 for α = 0 is an undefined expression of 0/0. What is the limit of this relation for the limit α → 0? Checkpoint Which relation gives the position of the diffraction minima for a slit of width a, illuminated with plane wave light of wavelength λ? 96 CHAPTER 7: EXPERIMENT 5 Checkpoint If you want to sharpen up a beam spot by inserting a narrow vertical slit into the beam, will the beam spot get more and more narrow as you close the slit? Explain. 7.1.2 Part 2: Diffraction by a Circular Aperture For a circular hole of diameter, d, the diffraction pattern of light with wavelength λ consists of concentric rings, which are analogous to the bands which we obtained for the single slits. The pattern for this intensity distribution can be calculated in the same way as for the single slit (see Appendix), but because the aperture here is circular, it is more convenient to use cylindrical coordinates (z, ρ, φ). The superposition principle requires us to integrate over a disk, and the result is a Bessel function. The condition for observing a minimum of intensity is found from the zeroes of the Bessel function, λ sin θ = km . d (7.3) For the first order (m = 1) minimum k1 = 1.22. Higher order minima will have larger km coefficients. The disk provided with the apparatus has circular openings of two different diameters. Put the apertures in the laser beam one at a time and observe how the diameter of the dark rings depends on the aperture diameter. Record your observations in your Data and choose the configuration which gives the best diffraction pattern for detailed study. Calculate the diameter of the aperture by measuring the distance between the center of the dot pattern and the first minimum and by using Equation (7.3). To make this measurement accurately, make the distance between the object and the screen large; you may choose to use the wall as your screen. The radius can be measured most accurately by measuring the dark ring’s diameter and dividing by 2. The radius then determines sin θm = rxm as shown in Figure 7.6. If the dark circle is irregular, use a typical value instead of the largest or smallest radius and consider its shape when estimating the error. Measure the distance, x, between the screen and the aperture. Don’t forget to include your units and errors. The roundoff error in k1 is limited to ±0.005 and you have estimated errors for x and r1 . The error in our wavelength is much smaller than these so estimate the error in aperture diameter using v u u δr1 1.22λx d≈ and δd = dt r1 r1 !2 0.005 + 1.22 2 . (7.4) Measure the value of the constant km in front of the ratio λd in Equation (7.3) for the second order minimum, m = 2, and the third order minimum, m = 3. Do this by using as aperture diameter d the value previously obtained above and by measuring sin θ 97 CHAPTER 7: EXPERIMENT 5 corresponding to the second minimum and then the third minimum, v u u δd drm and δkm = km t km ≈ λx d !2 δrm + rm !2 . In measuring the constant km you are determining the zeroes of a function called a Bessel function that seems to describe situations with cylindrical symmetry. Checkpoint Beams of particles act like waves with very short wavelengths when scattered by protuberances such as other particles. If a target of Pb and one of C are bombarded with a beam of protons, which target will show the sharpest diffraction pattern, the large size Pb target or the small C one? (Hint: The pattern due to an aperture is identical to the pattern caused by its photographic negative: a disk in empty space.) Checkpoint What size object will generate an observable diffraction pattern if placed in the path of light with wavelength λ? Checkpoint What is responsible for the factor 1.22 in Equation (7.3)? 7.1.3 Part 3: The Poisson Spot Figure 7.7: A sketch illustrating the optical beam line used to observe the Poisson spot in the center of a sphere’s shadow. 98 CHAPTER 7: EXPERIMENT 5 Historical Aside In 1818 Augustin-Jean Fresnel entered a competition sponsored by the French Academy to try clear up some outstanding questions about light. His paper was on a wave theory of light in which he showed that the refraction and reflection of light and all of their characteristics could be explained if we only allowed light to be treated as a wave. His theory even explained the results of Young’s double-slit experiment. This was about 30 years after Sir Isaac Newton presented a corpuscle theory of light in which light particles moved from a source to a destination but could be bent by material surfaces. Everyone knows to hide behind a rock or a tree if someone is throwing stones or shooting bullets at him since the projectiles will either be stopped by the obstacle or pass harmlessly by. One is not so safe from waves in a swimming pool, however, because waves will bend around obstacles having the size of their wavelengths or smaller. Historical Aside Many scientists at the meeting were having problems with Fresnel’s paper because everyone can see that objects cast shadows. Simeon Denis Poisson, one of the judges, was particularly disturbed by Fresnel’s paper and he used wave theory to show that if light is to be described as a wave phenomenon, then a bright spot would be visible at the center of the shadow of a circular opaque obstacle, a result which he felt proved the absurdity of the wave theory of light. This surprising prediction, fashioned by Poisson as the death blow to wave theory, was almost immediately verified experimentally by Dominique Argo, another member of the judges committee. The spot actually exists! The spot is still called the Poisson spot despite Argo’s having discovered it. I guess this just goes to show that some of our mistakes will never be lived down. . . Historical Aside Fresnel won the competition, but it would require James Clerk Maxwell (sixty years later) to locate the medium in which the waves of light propagate. In less than 60 seconds you can now settle a controversy that has preoccupied the minds of the brightest philosophers and scientists for centuries. Is light described as a stream of particles or as a wave phenomenon? In other words, does the Poisson spot exist? Observe the Poisson Spot. Record your observations in your Data. Can you explain the Poisson spot as constructive interference from the thin ring of light wave-fronts that just barely miss the sphere? What must be true for all of this light to interfere constructively at the location of the spot? 99 CHAPTER 7: EXPERIMENT 5 Use the concave (diverging) lens included with the apparatus to expand the beam slightly. Place the circular obstacle in the beam, (see Figure 7.7), and observe on the screen, at the center of the shadow generated by this obstacle, a bright spot - the Poisson spot! By varying the obstacle’s position between the lens and screen, you can optimize the intensity of the Poisson spot sort of like focusing the Fresnel zone plate. Checkpoint What is the Poisson spot? What does the existence of the Poisson spot demonstrate? 7.1.4 Late 20th Century View of Light The distinction between wave and particle relies on two types of experiments. Observation of interference phenomena demonstrates the presence of waves. Experiments which show discrete as opposed to continuous changes such as the photoelectric effect demonstrate particle phenomena (each photon of light knocks one electron off an atom). Light is a wave (you observe interference) and light is also a particle called the photon (you observe scattering and absorption of single photons). These two descriptions of light are not mutually exclusive. The wave nature of light is observable in certain natural phenomena, whereas the particle nature becomes apparent in other natural phenomena. The surprising conclusion is that light behaves sometimes like a wave, and sometimes like a particle - the nature of light depends upon the particular experimental situation. This is called the “wave - particle duality.” Checkpoint What is our present view of light? Is light a wave in some medium or a stream of particles? 7.2 Analysis We have measured a slit width and the slit manufacturer measured the width of the same slit. We form a NULL hypothesis by computing the difference between these two measurements, ∆ = Measured - Manufacturer. If our measurement is the same as the manufacturer’s measurement, then this difference is zero or NULL. Unfortunately, all measurements have error, so we cannot reasonably expect this difference to vanish after all. A natural question to ask ourselves, then, is “How big can 100 CHAPTER 7: EXPERIMENT 5 our difference be before it is significant?” If only the experimental errors are responsible for the nonzero result, then the difference must be bounded by our (and the manufacturer’s) errors. We computed ∆ using two numbers containing errors (Measured and specified by a Manufacturer) so the error in ∆ can be computed from these errors in the numbers used to compute ∆, q σ = (δMeasured)2 + (δManufacturer)2 . The manufacturer’s specified slit width has tolerance 1% so δManufacturer = 0.01 a. If all of our errors are random, we should expect ∆ > σ with probability 1:3, ∆ > 2 σ with probability 1:22, ∆ > 3 σ with probability 1:370, etc. Larger disagreements are progressively less likely IF random statistical fluctuations are the only reason for the difference. Certainly one sigma disagreements cannot be used contradict a theory that has served us well. Even two sigma disagreements occur frequently enough that we admit the probability that we were simply unlucky rather than our theory is wrong. This is especially true if we consider that we might have overlooked something in our apparatus or its environment that has contributed to our data and that we have not corrected nor compensated for. This is especially true if one or more of our assumptions was not realized as well as we thought. This might also be true if we were too optimistic about how well we performed the experiment; if our error is too small, then our difference will be too many of them. (We must conclude our Analysis with a paragraph or two that describes suspicious items such as these and how each may have affected our data.) Three sigma and larger disagreements are sufficiently rare that we must admit the likelihood that our measured and specified values are indeed different for some reason. Statistics can tell us that the numbers are probably different; however, statistics cannot tell us why. . . we must figure out why they are different for ourselves. Since you are new to this, you have probably made some actual mistakes in processing your data; try to find these mistakes and/or ask for help from your instructor, but don’t waste too much time or you will not finish. Another common problem for beginners is that they underestimate some of their errors, ask your instructor if your estimates are reasonable. Once we have verified that no other source of error can explain ∆ > 3 σ and the effect is reproduced consistently, we must then suggest that something is wrong with the models that we used to produce our measured value; the theory we are testing might have a hole in it. We have also measured the diameter of an aperture and the manufacturer has specified a value for comparison. Each of these measurements has error also, so perform all of the analysis above for the circular aperture data as well. What properties of our apparatus and its environment (NOT already included in σ) might affect the data that we have gathered? For each estimate how it might have influenced our results. (Communicate with complete sentences.) Consider all of your observations (not just the numerical, statistical analysis above) and proffer a brief argument about whether light is best represented as a wave or a stream of particles. 101 CHAPTER 7: EXPERIMENT 5 7.3 Conclusions We are now ready to state simply and briefly what our data says about light. We have already detailed the reasons why and should not repeat those reasons here; Conclusions are merely a series of results that our data supports. Always communicate with complete sentences to introduce valid equations (equations are equations NOT sentences despite the fact that we tend to read them as though they were sentences) and define the symbols (d, λ, θ, . . .) in the equations. We need not argue why we conclude as we do at this point; however, we must state our conclusions clearly so that our reader can understand them completely without reading anything else in our report. Finally, report any measured quantities that we or other experimenters might find useful in future work. If we expect that we might re-use our slit or aperture, we might report their measured dimensions. The Bessel functions help us describe many phenomena having cylindrical symmetry, so the locations of their zeroes is useful information. In each case communicate with complete sentences and include the units and errors for the measurements. 7.3.1 References Sections of this write up were taken from: • Bruno Gobi, Physics Laboratory: Third Quarter, Northwestern University • D. Halliday & R. Resnick, Physics Part 2, John Wiley & Sons. • R. Blum & D. E. Roller, Physics Volume 2, Electricity, Magnetism, and Light, HoldenDay. • E. Hecht/A. Zajac, Optics, Addison-Wesley Publishing. 7.4 APPENDIX: Intensity Distribution from a Single Slit The calculation of the intensity distribution of diffraction phenomena is based upon the superposition principle and Huygen’s principle. Light is a wave and if we choose to characterize it by the magnitude of its electric field vector, E, then it can be represented as, E = E0 sin(ωt − kx). 102 (7.5) CHAPTER 7: EXPERIMENT 5 If two waves with responses E1 and E2 reach the same point P at the same time, then the superposition principle gives the total displacement of E as P E0 = E1 + E2 . (7.6) y a This total displacement depends upon the relative phase ϕ between the two waves. Looking at Figure 7.8, try to imagine what happens when E2 shifts with respect to E1 (this shift is given by the phase ϕ = −kx). By lining up peaks with peaks and valleys with valleys, one gets a maximum displacement of the wave medium, E0 in this case. In the opposite case, when one lines up peaks Figure 7.8: A schematic illustration of a plane wave incident upon a single slit and with valleys, the response, E0 , becomes zero. the angular coordinate used to locate points of The light intensity, I, is related to the amplitude, E0 , by the important relation observation. I= ε0 c |E0 |2 . 2 (7.7) Note that amplitudes must be summed first before the square is taken for the intensity, |E0 |2 = |E1 + E2 |2 . Figure 7.8 shows the quantities used to calculate the diffraction from a single slit of width a. Each point in the slit along the y-direction will generate its own wavelet (Huygen’s principle) and each of these wavelets will have the same wavelength, λ, as the incident plane wave. A wavelet generated at point y and reaching point Note that amplitudes must be summed first will disturb the electric field at P by the amount Ey (t) = with ω = 2π T E0 sin(ωt + ϕ(y)) a = 2πf . The phase ϕ(y) can be determined from the path difference ϕ(y) = k∆ = 2πy sin θ λ (7.8) so that the electric field response at P due to the wavelet emanating from y in the slit is 2πy E0 sin ωt + sin θ . a λ Ey (t) = (7.9) The superposition principle determines the total amplitude at P to be the sum of the 103 CHAPTER 7: EXPERIMENT 5 contributions from every point in the slit. This is obtained by integrating, E0 2πy E0 Z α sin(ωt + ϕ) dϕ sin ωt + sin θ dy = E(t) = Ey (t) dy = λ 2α −α −a/2 −a/2 a E0 sin α =− (cos(ωt + α) − cos(ωt − α)) = E0 sin ωt. (7.10) 2α α Z a/2 Z a/2 We used Equation (7.8) to change integration variables from y to ϕ and Equation (7.1) to simplify to α. Figure 7.9 shows how this function varies with position on a view screen. We are interested in the light intensity; note that the of amplitude sin α Equation (7.10) is EP = E0 α , and the intensity at P is 0.9 0.7 0.5 0.3 0.1 -10 -8 -6 -4 -2 -0.1 0 2 4 6 8 10 I(θ) = I(0) ε0 c 2 EP 2 ε0 c 2 E0 2 sin α = α 2 , (7.11) where I(0) is the intensity on the optical sin θ, a is the slit width, λ is axis, α = πa λ y (cm) the wavelength, and θ is the direction to P from the slide. I(0) is always the maximum Figure 7.9: The amplitude of the electric intensity of the diffraction pattern. Equafield’s oscillations at each position on the tion (7.11) will have a value of zero each time viewscreen. The negative ‘amplitudes’ reflect a that sin2 α = 0. This occurs when α = ±mπ half-period phase shift, but the intensity is the for integer m 6= 0 and yields Equation (7.2) same in this case as for positive amplitudes. for predicting the directions to points with minimum light intensity from a single slit. The shape of the distribution given by Equation (7.11) is shown in Figure 8.1. -0.3 We might note as one final characteristic of diffraction that the intensity maxima are not midway between the intensity zeroes as we might naively expect. The division by α shifts the maxima toward the optical axis. Toward the optical axis corresponds to division by smaller a and a larger intensity. Since dividing zero by a still yields zero, the zeroes are not shifted. Calculus allows us to determine that the maximum intensities occur when tan αm = αm . 7.4.1 Diffraction from Double-Slit A very similar procedure will allow us to analyze the diffraction expected from two identical slits. Figure 7.10 shows an appropriate coordinate system; one might note the similarity to Figure 7.8 that we used to analyze the single slit. In the double-slit case, we must integrate over the bottom slit where − d2 − a2 < y < − d2 + a2 and then add this to the integral over the top slit where + d2 − a2 < y < + d2 + a2 . We will make the same change of variable from y to ϕ and the same simplification from a to α as we used for the single-slit case. Additionally, we 104 CHAPTER 7: EXPERIMENT 5 y P a d y=0 Figure 7.10: A schematic diagram illustrating a plane wave incident upon two identical slits and the angular coordinate used to locate points of observation. will use β = πd λ sin θ to simplify from d to β in the same fashion. " d+a d−a Z E0 Z − 2 2 E(t) = sin (ωt + ϕ(y)) dy + sin (ωt + ϕ(y)) dy d−a a − d+a 2 2 " # Z (β+α) E0 Z −(β−α) = sin (ωt + ϕ) dϕ + sin (ωt + ϕ) dϕ 2α −(β+α) (β−α) −(β−α) (β+α) E0 E0 = − cos (ωt + ϕ) + − cos (ωt + ϕ) 2α 2α −(β+α) (β−α) 105 # CHAPTER 7: EXPERIMENT 5 E0 E(t) = cos (ωt − (β + α)) − cos (ωt − (β − α)) 2α + cos (ωt + (β − α)) − cos (ωt + (β + α)) = E0 h cos (ωt − α − β) + cos (ωt − α + β) 2α i − cos (ωt + α − β)) − cos (ωt + α + β) i E0 h 2 cos (ωt − α) cos β − 2 cos (ωt + α) cos β 2α i E0 h = cos (ωt − α) − cos (ωt + α) cos β α sin α = 2E0 cos β sin(ωt) α = (7.12) We have used two trigonometric identities cos(A − B) + cos(A + B) = 2 cos A cos B cos(A − B) − cos(A + B) = 2 sin A sin B the first was used once with A = ωt + α and B = β and once with A = ωt − α and B = β. The second was used in the last step before a re-arrangement of multiplicands. An alternate and less rigorous approach is to consider the net amplitude at P after singleslit diffraction from each slit individually to be the emission from each slit. Each of these two waves then travels its individual distance to the screen and they interfere at P. When a plane wave is incident on two (or more) identical slits, the amplitude of the electric field response at P for each slit is given by Equation (7.10). Figure 7.10 shows the quantities used to calculate the double slit diffraction intensity distribution. Since the slit widths a are equal, we expect the intensity of the waves from them arriving at P to have equal amplitudes, E1 = E2 = E0 sin α . α (7.13) If we measure the distances between the slits and P from the midpoint between the slits, we can determine the relative phases for the two waves ∆ ϕ1 = k − 2 ! =− 2π 1 πd · d sin θ = − sin θ. λ 2 λ (7.14) If we let πd sin θ, then ϕ1 = −β and similarly ϕ2 = β. λ The total electric field from the slits, then, is β= 106 (7.15) CHAPTER 7: EXPERIMENT 5 E(t) = E1 (t) + E2 (t) = E0 = 2E0 sin α [sin(ωt − β) + sin(ωt + β)] α sin α cos β sin(ωt) α (7.16) sin A+B . The intensity distribution is found where we have used sin A + sin B = 2 cos A−B 2 2 to be sin α 2 I(θ) = cos2 β, (7.17) I(0) α where we note that the electric field amplitude for two slits on the optical axis is twice as large as for one slit. We might note that the diffraction from two slits is the same as for one slit, as a comparison of Equation (7.10) with Equation (7.17) easily shows. The interference between the two slits yields fringes described by cos2 β and the diffraction merely modulates the fringes’ peak brightness values. Examination of the definitions of β (Equation (7.17)) and α (Equation (7.1)) allows us to note that β must change faster with θ than α; after all, the slit widths cannot exceed the separation; a = b means one big slit twice as wide instead of two discrete slits. Checkpoint 2 The distribution of light from two slits is represented by the product sinα α (cos β)2 . Which one of these two terms is called the diffraction term and which one is the interference term? Which term is responsible for the interference fringes? General Information This product of interference between two point sources and diffraction of one extended source is one example of a more general strategy. When linear superposition applies, the response of several identical extended sources can be determined by dividing the problem into two simpler problems: solve one extended source, solve point sources at each of their centers, and multiply the two solutions. This process is called the convolution of the two solutions. 107 CHAPTER 7: EXPERIMENT 5 108 Chapter 8 Experiment 6: Intensity Distributions of Diffraction Patterns WARNING This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 89 before attending this week’s laboratory. 8.1 Introduction In the electricity and magnetism laboratory, we observed the resonance of a RLC circuit. This is the frequency distribution of the circuit’s current response to an AC voltage stimulus. The Normal Distribution used in Statistics relates the probability of an observation to the value observed. Distributions are quite common in nature so it is fitting that we should study them. In this lab you will observe and study the intensity distribution of light that passes through a single slit and later a pair of identical slits; in this case we will monitor the intensity as the angle between the original beam direction and the location of the observer is varied. In two previous laboratory exercises you have observed these distributions using your eyes as detectors. In this laboratory you will utilize a silicon PIN diode photodetector to get numerical values of intensity that can be plotted and analyzed graphically. Along the way, you will 1) learn about photodiode light intensity detectors, 2) learn to use rotary optical encoders to monitor position, 3) measure the intensity distribution of light diffracted by a single slit, 4) measure the intensity distribution of light diffracted by double slits, and 109 CHAPTER 8: EXPERIMENT 6 Intensity vs. Position 1 0.8 0.6 0.4 0.2 0 -7 -5 -3 -1 1 3 5 7 y (cm) Figure 8.1: A graphical plot of one example intensity distribution for light transmitted through and diffracted by a single slit. 5) evaluate the theory of light as waves using this data. You should find that these quantitative observations have maxima and minima in precisely the same places that we observed in the previous experiments. The detailed theory of light transmitted through slits is described in the appendix following the writeup for this experiment. You will be quizzed on the material in the appendix as well as on the lab write up. 8.1.1 Intensity Distribution of Light Diffracted by Single Slits We have learned in the fifth laboratory that the intensity distribution, I(θ), generated by a narrow slit of width, a, and infinite length, when illuminated with coherent light of wave 110 CHAPTER 8: EXPERIMENT 6 length λ is given by sin α I(θ) = I(0) α 2 for α = πa sin θ, λ (8.1) θ is the angle along which direction the intensity is observed, and I(0) is the intensity of the principal maximum. Figure 8.1 shows the expected ratio as calculated using Equation (8.1). Minima occur when sin2 α = 0 or when α = ±mπ and m = 1, 2, 3, . . . (8.2) Intensity maximum positions are given for values of α at which the derivative of Equation (8.1) becomes zero. This leads to the equation tan α = α. An approximate solution is 1 π, α=± m+ 2 m = 1, 2, 3, . . . (8.3) The ratio of intensity of a secondary maximum (order m) I(θ) to the principal maximum I(0) is h i 1 2 sin m + π I(θ) 1 2 (8.4) ≈ h i2 = h i2 1 I(0) m+ 2 π m + 12 π since sin α = ±1 at the approximate maxima. In the first part of this lab you will compare the intensity ratios calculated with Equation (8.4) to the measured ratio. Additionally, you will see how well Equation (8.1) describes the intensity distribution details by graphing the measured intensity vs. the position of the detector and by fitting the results to the Equation (8.1) model. We will finish the first part by verifying that the slit width predicted by the graphical analysis is the same as that specified by the manufacturer. 8.1.2 Intensity Distribution of Light Transmitted Through Double-Slits Historical Aside In 1801 Thomas Young devised a classic experiment for demonstrating the wave nature of light similar to the one diagrammed in Figure 6.4. By passing a wide plane wave through two slits he effectively created two separate sources that could interfere with each other. (A plane wave is a wave consisting of parallel wave-fronts all traveling in the same direction.) For a given direction, θ, the intensity observed at point P on the screen will be a 111 CHAPTER 8: EXPERIMENT 6 R y x Figure 8.2: A schematic diagram of our experiment showing how we can determine the scattering angle, θ, by measuring the distance from the optical axis and the distance between the slits and the screen. superposition of the light originating from every slit. Whether the two waves add or cancel at a particular point on the screen depends upon the relative phase between the waves at that point. (The phase of a periodic function is the stage it is at along its periodic cycle; the argument of the sine or cosine function is its phase.) The relative phase of the two waves at the screen gradually increases as a function of direction causing periodic intensity minima and maxima called fringes (see Figure 6.2). The spacing of the fringes depends on the distance between the two slits and the wavelength of the light. Equation (8.1) gives the intensity versus direction for each of the two slits; however, interference occurs by the superposition of the two electric fields E = E0 sin α [sin(ωt − kx1 ) + sin(ωt − kx2 )] α (8.5) where the two amplitudes are equal for a uniform plane wave incident upon two equal width slits. As the observer moves around the mask with the two slits, x1 and x2 will vary. We let the width of the two slits be a and the distance between the centers be d so that the notation is consistent with previous developments. We place the origin of coordinates midway between the slits so that x1 = x − d2 sin θ and x2 = x + d2 sin θ. We substitute this above into Equation (8.5) and use a trigonometric identity sin A + sin B = 2 sin 12 (A + B) cos 21 (A − B) to find sin α 1 1 E = E0 sin ωt − kx + kd sin θ + sin ωt − kx − kd sin θ α 2 2 sin α 1 1 = E0 2 sin (2ωt − 2kx) cos (kd sin θ) α 2 2 sin α 1 = 2E0 cos (kd sin θ) sin (ωt − kx) α 2 If we square this and average over a long time, the result becomes the intensity sin α I(θ) = I(0) cos β α 2 112 sin α = I(0) α 2 cos2 β (8.6) CHAPTER 8: EXPERIMENT 6 where 1 1 2π πd kd sin θ = · d · sin θ = sin θ = β. (8.7) 2 2 λ λ The average of sin2 ϕ is 21 and the constants out front get absorbed into I(0). From the result in Equation (8.6) we see that it is the simple product of the single slit result in Equation (8.1) and the very narrow double-slit result in Equation (6.1). This combination is illustrated graphically in Figure 8.3. You will observe this intensity distribution in Part 2. We will take data using the PIN photodetector and treat the intensity versus detector position graphically. We will see if the model of Equation (8.6) fits the data adequately and whether it predicts the same a and d that the mask’s manufacturer specified. Figure 8.3: Graphical plots of light intensity for inIf the model satisfies these terference, diffraction, and the combined effects for light aspects of your data, you will transmitted through matched double slits. have very strong evidence that Huygen’s principle that was used in each individual slit to achieve the observed diffraction is valid and that the superposition principle for electric and magnetic fields is valid. A similar analysis for N equally spaced identical slits yields sin α I(θ) = I(0) α 2 sin N β N sin β !2 . (8.8) This is very useful for predicting the detailed interaction of light with diffraction gratings. 8.2 The Apparatus We will be using Pasco’s optical bench shown in Figure 8.4. Pasco’s laser diode module fits their optical bench and will supply the coherent light for our experiment. The light will illuminate Pasco’s slit set shown in Figure 6.5 and Figure 7.3 at normal incidence. 113 CHAPTER 8: EXPERIMENT 6 The resulting interference and diffraction patterns will then be detected by Pasco’s PIN photodetector. The photodetector can be moved across the optical axis and its position can be monitored by their rotary encoder and rack-and-piñon gear. Pasco’s rotary encoder, CI-6538, reports the angle of its shaft and pulley rotation in radians (rad). We would like to determine the detector’s perpendicular location, y, in meters (m). After examining the rack-and-piñon gears carefully, we have determined that the piñon gear has 24 teeth. The rack is threaded at 3 teeth/cm. As the encoder makes one revolution, the computer will report 2π rad and the carriage will move the distance of 24 teeth. We can then find the distance, y, that the carriage will move to be y(ϕ) = 24 teeth 2π rad 3 teeth 0.01 m ×ϕ= 0.04 m ϕ. π rad (8.9) πy and we Additionally, both the slit width α and the separation β contain z = πλ sin θ ≈ λx can choose to separate our slits from our screen by 1.000 m; so we can let Ga3 calculate a (0.04 m)πϕ πy 0.04 ϕ column for z ≈ λx = (1.000 = 670×10 −6 mm using m)πλ 0.04 * “Phi” / (670 * 10^-6) (You need to define “z” somewhere in your Procedures or Data so your reader will know the meaning of your x-axis: z = πλ sin θ.) The units of z are “1/mm”; we will let this be our horizontal axis and the light intensity be our vertical axis when we plot our data. Figure 8.4: A photograph of the apparatus used to measure the intensity distribution of our light patterns. The x and y axes are indicated as are the diode laser light source, the slit pattern wheel, and the rotary encoder used to move the detector along the y-axis and to measure its motion. 8.2.1 Part 1: Light Intensity Distribution for Single Slit Configure Pasco’s Capstone program to gather your data. A suitable setup file can be downloaded from the lab’s website at http://groups.physics.northwestern.edu/lab/intensity.html 114 CHAPTER 8: EXPERIMENT 6 Measure the intensity distribution of the diffraction pattern generated by the narrowest vertical slit on the detector shape wheel. Set your sample slits ‘exactly’ 1.000 m from your detector shape wheel. Turn on your laser diode and verify that the reflection from the sample pattern has the same horizontal (y-direction) position as the laser aperture. This assures that the laser and slit pattern are very nearly perpendicular. It might be necessary to fold up some scrap paper and to wedge it between the pattern wheel holder and the optical bench track to obtain correct performance. Move the detector along the rack to one side of the diffraction pattern where the pattern is very dim but visible in low light. Click “Monitor” at the bottom left and very, VERY slowly turn the rotary encoder wheel at constant speed until the detector passes through the diffraction pattern. As you turn the wheel, the computer should plot your diffraction pattern on the graph. Click “Stop” at the bottom left. Copy and paste your data into Vernier Software’s Ga3 graphical analysis program. A suitable configuration file for Ga3 is also available for download from the website. Plot the intensity on the y-axis and “z” on the x-axis. Try to fit your data to the theoretical model in Equation (8.1). It would be wise to save your data. Draw a box around your data points, Analyze/Curve Fit..., and select the “single slit” model from the bottom of the list. Be sure the model agrees with bg + I0 * (sin (a * (x-x0)) / (a * (x-x0)))^2 It might be necessary to fit manually until the model is fairly close to the data points. “bg” is the vertical offset due to background room light and detector leakage due to thermal energy, “I0” is the height of the central maximum, “x0” is the location of the central maximum on the x-axis, and “a” is the width a of the single slit. Once the fit is fairly close, ‘Done’ to exit the fitter and to replace the default parameters. Now activate the fitter again (Analyze/Curve Fit...) and try to autofit. Make sure you have uncertainties specified for your fit parameters. If not right-click the parameters box, ‘Properties...’, and enable “Show Uncertainties”. If you need more significant figures, change the parameters properties again to get more. Calculate from your data the ratio I(θ)/I(0) for the first (m = 1), second (m = 2), and possibly third (m = 3) subsidiary maxima. What are your measurement errors in these ratios? Carry out your calculations for both peaks √ corresponding to +m and −m, and give as a result the average value. Divide your error by 2. Make sure you subtract the background “bg” from all of your intensity measurements. Predict these ratios using Equation (8.4). Keep enough significant digits so that your roundoff error is significantly less than your measurement error above; now you can treat these numbers as exact. 8.2.2 Part 2: Light Intensity Distribution for Double Slits Now, observe the various intensity distributions for the available double-slit pattern masks. Choose one for detailed analysis and repeat the procedure in Section 8.2.1 above except now 115 CHAPTER 8: EXPERIMENT 6 the fit function must describe the distribution for the “double-slit” sample pattern, bg + I0 * (sin (a * (x-a0)) * cos (d * (x-d0)) / (a * (x-a0)))^2 In this case “a0” is the location of the envelope’s central maximum, “d0” is the horizontal offset of the narrow fringes, and “d” is the distance d between the slits. The remaining parameters are the same as for the single slit model. 8.3 Analysis Our consideration of light waves interfering and diffracting led us to Equation (8.1) and Equation (8.6). Try to imagine inventing these models without using wave interference; how well do you imagine your model would fit your data? Keep your answers to these questions in mind when you are deciding what your conclusions should be later. Before we do that, however, our beautiful model spat out slit dimensions that we can compare to measurements from the manufacturer obtained via other means. . . probably using a microscope and micrometer scale. So a natural question is whether these two measurements yield the same values for these respective dimensions. Surely by now we are all getting familiar with the statistics needed to determine the answers to these questions. Calculate the difference between the manufacturer specified single-slit width and the measurement delivered by your model fitting ∆ = Fitted - Specified. Calculate the error in this difference using σ= q (δFitted)2 + (δSpecified)2 . For each, is ∆ < σ? Is ∆ < 2 σ? Is ∆ > 3 σ? Recall that ∆ > σ about 1:3 and ∆ > 2 σ about 1:22 just from random fluctuations; these probabilities are too large for us to assert that the compared two numbers are not equal. But ∆ > 3 σ only 1:370 due only to random fluctuations, so differences this large suggest that we conclude that the two numbers are indeed different; statistics cannot tell us why, however. Repeat this quantitative analysis for the single slit width, a1 , the double-slit width, a2 , and the double-slit separation, d2 . Also analyze the measured peak height ratios with those predicted by Equation (8.4). Can you think of anything in the experiments that might reasonably affect the measurements that we have not already included in our measurement errors? Discuss each of these using complete sentences and paragraphs. In your view is it more likely that the theory that has survived for a hundred years of professional tests is false or that you have made some mistake or overlooked some significant sources of error that might explain your nonzero difference(s)? 116 CHAPTER 8: EXPERIMENT 6 8.4 Conclusions We are now ready to state simply and briefly what our data says about light. We have already detailed the reasons why and should not repeat those reasons here; Conclusions are merely a series of results that our data supports or contradicts clearly stated. Always communicate with complete sentences to introduce valid equations (equations are equations NOT sentences despite the fact that we tend to read them as though they were sentences) and define the symbols (d, λ, θ, . . .) in the equations. We need not argue why we conclude as we do at this point; however, we must state our conclusions clearly so that our reader can understand them completely without reading anything else in our report. Finally, report any measured quantities that we or other experimenters might find useful in future work. If we expect that we might re-use our slit patterns, we might report their measured dimensions. In each case communicate with complete sentences and include the units and errors for the measurements. 8.4.1 References Sections of this write up were taken from: 1. D. Halliday & R. Resnick, Physics Part 2, John Wiley & Sons. 2. R. Blum and D.E. Roller, Physics Volume 2, Electricity Magnetism and Light; HoldenDay. 3. E. Hecht/A. Zajac, Optics, Addison-Wesley Publishing Company. 117 CHAPTER 8: EXPERIMENT 6 8.5 APPENDIX: The Semiconductor Diode Laser Semiconductors are materials that can be pretty good electrical conductors in some situations and pretty good insulators in other situations. The most widely used semiconductor material is silicon (Si), but carbon (diamond) and germanium (Ge) have similar properties. Each silicon atom in a solid crystal is bonded to four nearest neighbor silicon atoms. Since silicon is in column IVA of the periodic table of the elements, this means that every atom is very content with its bonding structure; every atom shares one electron covalently with each of its four nearest neighbors and effectively has a full valence shell of eight electrons. All of the electrons in the crystal are already busy holding the crystal together so that no electrons are free to move about and to conduct electrical current. In solid state physics we say that an energy gap has developed between the valence band and the conduction band, the valence band is completely filled, and the conduction band is completely empty. Pure silicon is a very good electrical insulator at low temperature and moderate electric fields. Pure silicon is impossible to obtain, however, and no object has infinite volume so the surface atoms have no nearest neighbors with which to share electrons. If we imagine substituting a column IIIA atom like gallium (Ga) or boron (B) for one of the atoms in the silicon crystal, we see that boron has only three electrons to share with its four nearest neighbors. One of the boron atom’s neighbors wants another electron that boron cannot give. It turns out that this “hole” (h+ ) can be filled by any one of the four shared electrons near this hole moving into this position, but this leaves behind another hole. This hole, in turn, can be filled by any of the four electrons near its position. If we consider that the positively charged atomic nuclei are not moving along with their electrons, we will see that the volume around the boron atom has become negatively charged and that the volume around the hole has become positively charged. The negatively charged boron atom is fixed in place by the crystal lattice, but the positively charged hole is now moving around inside the material and conducting electrical charge. The addition of type IIIA elements to a silicon lattice turns it into a conductor of positive charge. We call this “p-type” semiconductor material. In solid state physics we say that the substitution of boron has added a hole to the valence band. This is illustrated in Figure 8.5(a). If we imagine substituting a type VA atom like phosphorous (P) or arsenic (As) for one of the atoms in the silicon crystal, we see that arsenic has five electrons to share with its four nearest neighbors. This leaves an extra electron (e− ) that is not needed to hold the crystal together. This extra electron, then, is free to move around inside the solid and to take its charge along with it. The arsenic nucleus, however, is fixed in place by the four bonds to the lattice and when we removed one of its electrons the volume around the arsenic core became positively charged. Similarly, the mobile volume around the extra electron is negatively charged. This charge motion is an electrical current. The addition of type VA atoms to a silicon crystal turns it into a conductor of negative charge. We call this “n-type” semiconductor material. Figure 8.5(b) illustrates this situation. In solid state physics we say that the substitution of arsenic has added electrons to the conduction band. 118 CHAPTER 8: EXPERIMENT 6 Figure 8.5: Two illustrations of how doping silicon can generate a conductor. In (a) a boron (B) atom replaces a silicon (Si) atom and creates an electron vacancy or hole in the valence band. In (b) an arsenic (As) atom replaces a Si atom and leaves an extra free electron in the conduction band. It is also possible to form semiconductors using molecules such as GaAs or InP instead of atoms to build up the crystal. These binary III-V semiconductors are generally doped by replacing their atoms with silicon; replacing the column III atoms with silicon yields conduction electrons and replacing the column V atoms with silicon yields valence band holes. Ternary semiconductors like GaAlAs also have properties different than other materials; we can exploit specific properties of the various materials to build up the particular set of properties we need for a particular device to have. Finally, there are organic semiconductors that usually exploit a molecular substrate instead of a crystal lattice to obtain semiconductive properties. This allows organic semiconductors to be very flexible instead of brittle like silicon. If we imagine building a semiconductor device by beginning with a p-type substrate and adding layer after layer of n-type material, the result is a pn-junction. The converse process of beginning with a n-type substrate and adding p-type layers also generates a pn-junction. The second law of thermodynamics would suggest that the extra electrons diffuse from the n-type material to fill the entire volume of the crystal uniformly with electrons. Similarly, the holes should diffuse from the p-type material and fill the crystal uniformly with holes. The crystal does, in fact, try to obey the second law of thermodynamics; however, electrons and holes are each electrically charged so that a very large electric field develops in the vicinity of the junction. As electrons moved into the junction, they left behind a positive charge; similarly, holes left behind a negative charge. This makes a large electric field point from the n-type material toward the p-type material as shown in Figure 8.6. Ordinarily, this electric field would result in a large current but most of the electrons that enter the junction recombine with the holes that enter the junction and become a silicon bond once again. Sometimes a hole and an electron can form a metastable “exciton” wherein they orbit each other like the electron and proton in the hydrogen atom. Once all of the charges are thus immobilized in the junction region, the junction becomes a very good insulator that cannot conduct electric current; the junction region has no free mobile charges so we call it an “intrinsic” semiconductor. In photodetectors we intentionally add pure (intrinsic) 119 CHAPTER 8: EXPERIMENT 6 semiconductor material between the p and n materials to increase the likelihood of absorbing a photon in this volume; this forms a p-type - intrinsic - n-type or pin-junction. Figure 8.6: An illustration of charges flowing across a pn-junction. Most of the time the electrons and holes simply recombine once again to become an immobile, neutral silicon bond. Sometimes an electron and hole form a metastable exciton where they orbit each other like the charges in a hydrogen atom. In (c) these processes are shown from the band theory of solids perspective. If we attach the positive terminal of a battery to the p-type material and the negative terminal to the n-type material, current will flow. In fact, so much current will flow that if we do not somehow limit the current, the device will quickly heat up and burn out. As the battery helps cancel the electric field in the junction region, the electrons and holes enter the junction as a battery current and recombine. Each time an electron-hole pair recombines the energy of attraction between them is released as a photon. The energy of the photon is . A pn-junction is a equal to the band gap energy Eg of the semiconductor, Eg = hf = hc λ light emitting diode (LED) but the light is outside the human visible range (400-700 nm) for most diodes. Gallium arsenide (GaAs) diodes emit the bright red light used in calculator displays in the 1970s. Over the years we have developed ways to trap the photons emitted by some pn-junctions by making the surfaces around the junction region highly reflective. By containing the light in this manner, we greatly increase the rate that these trapped photons stimulate photon emissions and pair recombination. As we may have learned earlier in studying the HeNe laser, these stimulated emissions have exactly the same energy (wavelength), direction (momentum), and phase as the photon that stimulated the emission; the two photons are coherent and have the same quantum state. Since photons satisfy Bose-Einstein statistics instead of Fermi-Dirac statistics, it is not forbidden for many photons to have the same state. These two photons are now twice as likely to stimulate another photon in their state as any of the singlet photons in the device. Once one of them does stimulate another emission, the three photons will be three times more likely to stimulate another like themselves, etc. Very soon a very large fraction of the photons in the laser have the same wavelength, direction, and phase. The light emitted by the semiconductor diode laser is coherent and monochromatic. From this discussion one should not deduce that all semiconductor diode lasers are made from silicon. In fact, only recently have we figured out how to make lasers from silicon at 120 CHAPTER 8: EXPERIMENT 6 all. Additionally, the color of the light is determined by the semiconductor’s energy gap; since there are lasers having many different colors, they must be made from many different semiconductors. Since the light wavelength is determined by the semiconductor’s band gap, it is possible to tune the wavelength over a few dozen nanometers range by adjusting the diode’s temperature. At higher temperature the crystal expands so that the atoms are further apart and the attraction between them (the energy gap) is reduced slightly. 8.6 The PIN Diode Photodetector When energetic particles irradiate a semiconductor, the resistivity of the semiconductor decreases. The energetic particles break molecular bonds by colliding with the valence electrons. Each broken bond yields an electron free to conduct electricity in the conduction band; each broken bond also yields a free hole in the valence band and these increases in free charge carriers increase the electric conductivity (σ = ρ1 ) and reduce the resistivity. Eventually, these carriers reach the edge of the semiconductor or recombine with another partner to recover the status quo when the irradiance is removed. This makes semiconductors good photoconductors. In order to generate an electron-hole pair, the incident particle must deposit enough energy upon the bound electron to break the chemical bond. This energy is the semiconductor’s band gap energy, Eg . Very energetic particles might break several bonds each before running out of energy; each collision costs the particle Eg so that counting the broken bonds can give information about the energy of the incident particle. This property makes semiconductors very valuable to high energy physicists. Alternatively, if the energies of incident particles are known to be near the gap energy, then counting the broken bonds yields information about the brightness or intensity of the beam. In our discussion of the pn-junction above, we noted that electrons from the n-type material diffuse across the junction as do holes from the p-type material to recombine and to make several microns thickness of intrinsically pure semiconductor at the junction with a high electric field. When light with sufficiently high frequency encounters this junction, it breaks these bonds. This high electric field then pushes the alleviated holes toward the p-type material and the electrons toward the n-type material to generate a small potential difference between the two sides of the pn-junction. Solar cells can generate electric power utilizing this photoelectric effect, but pn-junctions also make sensitive and linear photodetectors. Since each photon creates one electron-hole pair, the generated current is directly proportional to the light’s intensity. The biggest problem with pn-junction photodetectors is that such a thin junction absorbs very few of the incident photons; most pass through without interacting with the junction. To increase the junction’s thickness, we deposit a layer of intrinsically pure (i) material between the p-type and the n-type materials. These pin-junctions (PIN diodes) are more sensitive and have faster response than regular pn-junctions. 121 CHAPTER 8: EXPERIMENT 6 122 Chapter 9 Experiment 7: Light Polarization WARNING This experiment will employ Class III(a) lasers as a coherent, monochromatic light source. The student must read and understand the laser safety instructions on page 89 before attending this week’s laboratory. 9.1 Introduction We might recall that light is “transverse electro-magnetic waves”. James Clerk Maxwell’s electrodynamics equations describe these waves in detail. Transverse waves have the E and B vectors perpendicular to the direction of wave propagation. Out of 3-dimensional space, this still leaves two dimensions (or two degrees of freedom) in which the vectors might point; they can point in any radial direction around the light ‘rays’. These two degrees of freedom are represented mathematically by two orthogonal coordinate axes. These two coordinates are the basis for specifying the polarization of the light. The electric field vector E can be specified completely using only these two coordinates. Although the same is true for the magnetic field vector, the electrons in dielectrics get moved by the electric field; they experience an electric force −eE. Since their velocities are much smaller that light-speed, any magnetic force is much smaller and their random velocities generally average to zero quickly anyway. (The positively charged nuclei are thousands of times heavier than the electrons so that their response to the waves are neglected; their accelerations are thousands of times less.) This leads us to specify the polarization p̂ of the light to be the direction of the electric field, E(t) = E(t) p̂. We also know that sinusoidal waves are equal parts positive and negative so that there is no difference between positive and negative polarization, p̂ ⇔ −p̂. Linear polarization has the polarization vector p̂ a fixed unit vector in space and time. 123 CHAPTER 9: EXPERIMENT 7 Figure 9.1: A polaroid absorbs oscillating electric fields parallel to its grain. This can polarize natural light or it can check whether light is already polarized. Polarized light will be absorbed when its electric field is parallel to the polaroid’s grain. This is the most common form of polarization and the one we will be studying in this lab. If the plane of vibration rotates around the direction of propagation, then the light is said to be elliptically polarized. If the major and minor axes of an ellipse are equal, the ellipse is a circle and the same is true for circularly polarized light. Depending on the direction of rotation the light may be right- or left-circularly polarized. These two rotation directions are also independent of each other and form an alternate basis for specifying a general polarization. An ordinary light source consists of a very large number of randomly oriented atomic emitters. Each excited atom radiates a polarized wave train of roughly 10−8 s duration (in other words imagine a sinusoidal wave about 3 m long). New wave trains are constantly emitted and the overall polarization changes in a completely unpredictable fashion. These wave trains make up natural light. It is also known as unpolarized light, but this is a bit of a misnomer since, in actuality, every wave train is polarized but the wave trains’ polarization planes are randomly oriented in space (i.e. are different polarization states). The set of all wave trains has every possible polarization in equal abundance. In this lab you will 1) study the phenomenon of light polarization, 2) compare our observations to Malus’ law and to Brewster’s law 3) learn about the function and use of a semiconductor diode laser, and 4) learn about the function and use of a PIN photodiode light detector. The laser is described in the appendix following the writeup on page 118. You will be quizzed on the material in the appendix as well as on the lab write up. Light’s polarization can be used to manipulate light. Just as changing its wavelength inside different dielectrics allows us to bend and to focus light, placing optical elements 124 CHAPTER 9: EXPERIMENT 7 Figure 9.2: A photograph of the light detector and shape wheel. We will use one of the circular apertures. between two polarizers allows us to choose whether the light passes the second polarizers. For example, liquid crystals (nematic fluids) rotate the polarization of light so that we can choose with an electric field whether the polarization matches the second polarizer and is transmitted. Many materials exhibit electro-optical or magneto-optical effects that can also be used in this manner. Birefringent (or dual index of refraction) crystals can be cut to exactly the right length to form half-wave plates or quarter-wave plates whose orientation relative to the incident polarization affects the polarization state of the transmitted light. Finally, light reflected from dielectric surfaces is partially polarized. Light reflected at the Brewster’s angle from a dielectric surface is completely polarized. We will observe the Brewster’s angle for Lucite as part of today’s experiment. Checkpoint What property of electro-magnetic radiation determines its polarization? Checkpoint What is a wavetrain? What is the plane of vibration of light? What is the difference between polarized and natural light? 125 CHAPTER 9: EXPERIMENT 7 Checkpoint Is the light emitted by most light sources polarized? Explain. What is circularly polarized light? 9.1.1 Polarizers An optical device whose input is natural light and whose output is some form of polarized light is known as a polarizer. We know a variety of ways to polarize light. All of these techniques have in common some form of asymmetry associated with the way they operate. Three such techniques are discussed next. 1. Wire-Gridiron Polarizer. Imagine that an unpolarized electromagnetic wave impinges on a gridiron of very fine wire from the left. (See Figure 9.1.) The electric field can be resolved into two orthogonal components; in this case, one chosen to be parallel to the wires and the other perpendicular to them. The y-component of the field drives the conduction electrons along the length of each wire, thus generating a current. This current heats the wires so that energy is transferred from the field to the grid. In contrast the electrons are not free to move very far in the x-direction and the corresponding field component of the waves is essentially unaltered as it propagates through the grid. The transmission axis of the grid is accordingly perpendicular to the axes of the wires. The green arrows in Figure 9.1 indicate the polarization axis. 2. A Polaroid is a molecular analog to the wire gridiron. A sheet of clear polyvinyl alcohol is heated and stretched in a single direction, its long hydrocarbon molecules becoming aligned in the process. The sheet is then dipped into an ink solution rich in iodine. The iodine impregnates the plastic and attaches to the straight long-chain polymeric molecules, effectively forming a chain of its own. The conducting electrons associated with the iodine can move along the chain as if they were long thin wires. The component of E in an incident wave which is parallel to the molecules drives the electrons, does work on them, and is strongly absorbed. The transmission axis of the polarizer is therefore perpendicular to the direction in which the film was stretched. 3. A polarizing beam-splitter separates an incoming beam of light into a transmitted, horizontally polarized beam and a reflected, vertically polarized beam. These polarizers make use of Brewster’s law that will be studied today. Checkpoint How can you produce a beam of linearly polarized light? How can you establish if a beam of light is linearly polarized? How does a polaroid film polarize a beam of light? 126 CHAPTER 9: EXPERIMENT 7 9.1.2 Malus’s Law How do you determine experimentally whether or not a device is actually a linear polarizer? Consider Figure 9.1 where natural (unpolarized) light is incident on an ideal linear polarizer. Only light with a specific orientation (polarization state) will be transmitted. That state will have an orientation parallel to a specific direction which we call the transmission axis of the polarizer. In Figure 9.1 the transmission axis is depicted by green arrows. In other words, only the component of the optical field parallel to the transmission axis will pass through the device unaffected. If the polarizer is rotated about the z-axis the amount of transmitted light will be unchanged because of the complete symmetry of unpolarized light. Now suppose that we introduce a second possibly identical ideal polarizer, or analyzer, whose transmission axis is vertical. If the amplitude of the electric field transmitted by the polarizer is E0 , Figure 9.3: The polarization light perpendicular to only its component parallel to the the plane of incidence is unaffected by reflections at transmission axis of the analyzer dielectric surfaces. will be passed on (assuming 100% efficiency). This component will have magnitude E0 cos θ. The observed intensity is the square of the amplitude, consequently I(θ) = I0 cos2 θ. (9.1) This is known as Malus’s law, having first been published in 1809 by Etienne Malus, military engineer and captain of the army of Napoleon. Observe that I(90◦ ) = 0. This arises from the fact that the electric field that has passed through the polarizer is perpendicular to the transmission axis of the analyzer (two devices so arranged are said to be “crossed”). 127 CHAPTER 9: EXPERIMENT 7 Checkpoint What is the relation between polarizer orientation and transmitted light if the incident light is linearly polarized? Checkpoint What happens to the polarization of light that doesn’t pass the polarizer? Checkpoint What is Malus’ law? 9.2 Part 1 Observe the polarization of your laser beam. Rotate a polaroid film placed in your laser beam, and then look for changes in the beam intensity on the screen (Figure 9.2). The polaroid film has the same function as the analyzer shown in Figure 9.1. If the intensity of the transmitted light does not change when rotating the analyzer, the beam is not linearly polarized. If it varies according to Malus’s law (Equation (9.1)) the light is linearly polarized. Enter your observations in your Data. Observe the polarization states of the light traversing a polaroid film. As described before, light transmitted through a polaroid film is linearly polarized. Place one polaroid film in the laser beam. This will transmit linearly polarized light whether or not your laser generates a polarized beam. This Figure 9.4: A sketch of a green light ray reflected by film is the polarizer. Rotate the a dielectric surface. The polarization in the plane of polarizer to get maximum trans- incidence is greatly reduced by the reflection. 128 CHAPTER 9: EXPERIMENT 7 mission (this will be 90◦ from minimum transmission). Now insert a second polaroid film (the analyzer) downstream of the first one. For the analyzer choose the polaroid with the optical encoder attached. Rotate the analyzer keeping the polarizer fixed. The intensity you observe on the screen will then go through maxima and minima, as predicted by Equation (9.1). When the transmission axis of polarizer and analyzer make an angle of 90◦ , no light will be transmitted (analyzer and polarizer are crossed). Describe your observations in your Data. Execute the Pasco Capstone configuration file from the lab’s website at http://groups.physics.northwestern.edu/lab/polarization.html Be sure the transmitted light enters the detector. Click “Record” at the bottom left and slowly rotate the analyzer at least one revolution. Be careful or the analyzer will fall out of its housing. Click “Stop” at the bottom left. Be careful not to disturb the laser or the detector until experiment 1d) is complete. Click one entry of the table, press ctrl+a to select all of the table, and press ctrl+c to copy the table contents to Window’s clipboard. Now execute Vernier Software’s Ga3 graphical analysis program, click row 1 under the angle header, and press ctrl+v to paste the data into Ga3. Draw a box around the data points on the graph and Analyze/Automatic Fit. Fit the transmitted intensity to Malus’ law near the bottom of the list. This model will fit your data using Figure 9.5: A graphical plot illusb + I0 * (cos(x - x0))^2 trating how reflected light is polarized differently for different angles of in- Verify that the formula is correct and “Try Fit”. If the cidence. model line passes through your data, click “Done” and make sure the uncertainties in the fit parameters are shown. Print your graph for your notebook and copy and paste it into your MS Word report. Place a third polaroid film (use the polaroid with the encoder attached for the middle one) between crossed polarizer and analyzer. The polarizer and analyzer are crossed when (without the middle polarizer) no light is transmitted. When rotating this third (middle) filter the transmitted intensity goes through a maximum which appears each time the filter in the middle is rotated by 90◦ . Explain (briefly in your Data) what you are observing in terms of the fact that an analyzing filter placed in a linearly polarized beam of light effectively rotates the plane of polarization of the light to the axis of the polarizer and reduces the intensity to the projected component. For which orientation of the transmission axis of the filter in the middle do you observe a maximum or a minimum of transmitted light intensity? Copy and paste this data into Gax as before. Fit this data to the “Malus 2” model near the bottom of the list; this model should agree with b + I0 * (sin(2 * x - x0))^2 129 CHAPTER 9: EXPERIMENT 7 Show analytically that the intensity of the transmitted light, in the configuration of experiment 1c), will give maxima at intervals of 90◦ of the middle filter (Apply Malus’ law each time linearly polarized light goes through a polarizing sheet and don’t forget your trigonometry). If the detector or laser was inadvertently disturbed despite the warning, repeat 1b) without disturbing the laser or detector. If only the middle polarizer is the difference between 1c) and 1d), then the peak intensity of 1c) should be four times the peak intensity of 1d). Unfortunately, even when the polarization is aligned with the polarization axis of the polaroid only 76% ± 1% of the light passes. Polaroids are lossy polarizers so we must compensate for the fact that in 1d) the light passed three polaroids and in 1c) the light passed only two by dividing the intensity in 1d) by 0.76. Predict the intensity in 1c) using the intensity in 1d), 4 I0 . (9.2) Ipredicted = 0.76 Combine the errors in the measured intensity with the error in the polarizer efficiency to estimate the error in this prediction, δIpredicted = Ipredicted v u u t δI0 I0 !2 1 + 76 2 . (9.3) Checkpoint What is the efficiency of our polarizers? 9.2.1 Polarization By Reflection One of the most common ways to polarize light is to reflect the light off of some dielectric medium. The glare spread across a window pane, a sheet of paper, or the sheen on the surface of a telephone or a book jacket are all partially polarized. One simple, although incomplete explanation of this polarization is supplied by the electron-oscillator model. Consider an incoming plane wave linearly polarized so that its field is perpendicular to the plane of incidence (see Figure 9.3). The wave is refracted at the interface entering the medium. Its electric field drives the bound electrons, in this case normal to the plane of incidence, and they in turn re-radiate a reflected component of the incident light. The radiation emitted by an electron oscillating along a straight line is called dipole radiation. Its electric field is largely parallel to the direction of motion of the charge and no radiation is emitted along the axis of the dipole (the direction of oscillation). It is consequently clear from the geometry of the dipole radiation pattern that both the reflected and refracted waves must also be in the same polarization state as the incident radiation. On the other hand, if the incoming field is in the incident plane as shown in Figure 9.4 the electron-oscillators near the surface will vibrate under the influence of the refracted wave. 130 CHAPTER 9: EXPERIMENT 7 The flux density of the reflected wave is relatively low because the reflected ray direction makes a small angle ϕ with the dipole axis. If we could arrange things so that ϕ = 0, or equivalently θrefl + θrefr = 90◦ , the reflected wave would vanish entirely. Consider an incoming unpolarized wave represented by two components, one polarized normal to the incident plane and one in the incident plane. Under these circumstances only the component normal to the incident plane, and therefore parallel to the surface, will be reflected. The particular angle of incidence for which this situation occurs is designated by θp and is referred to as the polarization angle or Brewster’s angle, so that θp + θrefr = 90◦ . Hence from Snell’s law n1 sin θp = n2 sin θrefr (9.4) Figure 9.6: A photograph of the Lucite block on the and from the fact that θrefr = 90◦ − θp rotary table and the optical bench. it follows that n1 sin θp = n2 cos θp and tan θp = n2 n1 (9.5) (9.6) This is known as Brewster’s law after the man who discovered it empirically about forty years before light phenomena were described as electromagnetic waves. The degree of polarization of unpolarized light reflected from Lucite (n2 /n1 = 1.5) is shown in Figure 9.5. The reflected beam is 100% polarized at 56.2◦ . Checkpoint What is Brewster’s law? Checkpoint A certain reflected light beam satisfies Brewster’s law. In which plane is the light polarized? 131 CHAPTER 9: EXPERIMENT 7 9.2.2 Part 2: Confirm Brewster’s Law We want to check Brewster’s Law by observing the polarization of the light reflected off the surface of a block of Lucite when the angle of incidence is set at the Brewster’s angle (for Lucite θp = 56.2◦ ). Set the Lucite block on the rotary table (Figure 9.6). Hold the rotary table at 0◦ while orienting the Lucite block perpendicular to the beam. The reflection will return to the laser when this is the case. (See Figure 9.7.) The surface off which you are reflecting the laser beam must be vertical and situated along a diameter of the rotary table. We need some horizontal polarization, but the lasers are oriented to give vertically polarized light. Place a polarizer between the laser and the Lucite and orient its polarization axis at 45◦ . Rotate the rotary table by θ1 = 56.2◦ and measure the polarization of the reflected beam using a polaroid filter as an analyzer. Is the reflected light polarized? Record your observations in your Data. Place a second polarizer oriented horizontally between the first polarizer and the Lucite. Vary the angle θ1 and determine for which value of θ1 the reflected beam has least intensity. The observed degree of Figure 9.7: A photograph of the Lucite block polarization should vary as shown in Fig- aligned to θ1 = 0.0◦ incidence. The reflection ure 9.5. Since the incident light is oppositely has the same horizontal location as the laser polarized, the observed intensity should vary exit port. If the angle is not zero on the scale, as 100% minus the degree of polarization. In subtract its value from the Brewster’s angle this configuration we are using the reflection later. as an analyzer instead of a polarizer. Once the Brewster’s angle is found, vary the second polaroid (closest to the Lucite) to try to get the reflected intensity to be exactly zero. Slowly increase the angle until the reflection vanishes (see Figure 9.8.) and read the angle. Continue increasing the angle until the reflection is visible, slowly decrease the angle until the reflection vanishes again and read the maximum reasonable Brewster’s angle. Now carefully rotate the rotary table back to zero and verify that the surface is still perpendicular to the beam; if it is not, measure the angle that does 132 CHAPTER 9: EXPERIMENT 7 Figure 9.8: Two photographs illustrating how to bracket Brewster’s angle. In the first the reflection is to the right of Brewster’s angle; rotate to the left until the reflection vanishes and measure the angle. In the second the reflection is too far left; rotate right until the reflection vanishes and measure the angle again. This gives estimates of Brewster’s angle and its error. make it perpendicular and subtract this adjusted zero from your measurement of Brewster’s angle. (Remember to keep the correct sign.) Remember to include the units and error with your measurement. Enter in your Data a brief description of what you have observed. Make a drawing. Using your measured Brewster’s angle (corrected for zero) calculate the index of refraction of your Lucite block. Remember that the index of refraction for air is very nearly 1. 133 CHAPTER 9: EXPERIMENT 7 Checkpoint Polarization by reflection determines the plane of polarization. Is it the vertical or the horizontal plane in which the vector of the reflected light oscillates? Explain. 9.3 Analysis Is the peak intensity of the first graph four times the peak intensity of the second graph? Is your measurement of Lucite’s index of refraction consistent with previous measurements (1.495)? Investigate these questions by computing the difference and then comparing the difference to your net error ∆ = n − 1.495 What other sources of error did you notice while performing your experiments? How might each of these have affected your data? Do your models fit your data? Don’t forget that the period of the sine (cosine) function was predicted explicitly by the model and was not varied by the fitter. 9.4 Conclusions What has your data demonstrated? Is Malus’ law supported? Contradicted? Is Brewster’s law supported? Contradicted? Define all symbols used in equations and communicate with complete sentences. Have you measured anything that might be of value in the future? If so state your measured values, units, and errors. 9.4.1 References Sections of this write up were taken from: • D. Halliday & R. Resnick, Physics Part 2, John Wiley & Sons. • R. Blum and D.E. Roller, Physics Volume 2, Electricity Magnetism and Light, Holden- Day. • E. Hecht/A. Zajac; Optics, Addison-Wesley Publishing Company. 134 Chapter 10 Experiment 8: The Spectral Nature of Light 10.1 Introduction: The Bohr Hydrogen Atom Historical Aside The 19th century was a golden age of science. We could explain everything we observed. Near the turn of the century that began to change. The arrival of the microscope allowed us to see random motion of wheat grains. Light emitted from hot objects had spectrum different than expected; indeed the expected spectrum was unreasonable. Light emitted or absorbed by atomic gases had sharp lines instead of continuity. In this lab we will resolve some of the reasons for the confusion surrounding the state of science in nineteen hundred. These explanations involve accepting that light has a particle or quantum nature as well as the wave nature that our interference experiments have verified. These explanations involve an atomic nature for materials instead of the continuous, two charged fluid model suggested by Benjamin Franklin. These explanations even involve stable quantum states in atoms where the accelerating electron does not radiate EM waves as expected. An atom has a diameter of about 10−10 m and is made up of a small (about 10−14 m), heavy, and positively charged nucleus, surrounded by orbiting electrons. The nucleus consists of protons (positive electric charge) and neutrons (no electric charge) bound together by nuclear forces. In a neutral atom there are as many negatively charged electrons as there are protons. Each electron occupies one of the many possible well defined atomic orbit states. Each particular orbit state has an associated total energy (kinetic + potential). Electrons “jumping” from one orbit with less energy to one with more (binding) energy will release discrete quantities of energy in the form of light. The spectrum of quantized frequencies observed in the light emitted by excited atoms is a direct measurement of the allowed transitions of 135 CHAPTER 10: EXPERIMENT 8 -e me e M r Figure 10.1: Schematic force diagram for the nucleus and electron in a hydrogen atom. electrons among different orbits in the atom. The emission spectrum of a given atom is a unique signature of this atom. Even different isotopes in an element family have different spectra. Although the existence of discrete line spectra emitted by atoms had been known for a long time, it was Niels Bohr who first calculated the spectrum of the hydrogen atom. This is the simplest atomic system and consists of one proton as nucleus and one orbiting electron. The electron describes a circular orbit in the electric field of the proton; the force acting on the electron is v2 e2 = Fe = −me ar = −me . (10.1) − 4πε0 r2 r From Equation (10.1) the kinetic energy of the electron is 1 e2 2 KE = me v = . 2 8πε0 r (10.2) The electrostatic potential due to the proton’s positive charge at a distance r is V = e 4πε0 r (10.3) and the potential energy, Ue = (−e)V of the electron is Ue = − e2 = −2KE 4πε0 r 136 (10.4) CHAPTER 10: EXPERIMENT 8 Series Limit Hydrogen Energy Levels 0 Series Limit -1 -2 Series Limit The total energy of the electron-proton system (of the atom) is E = KE + Ue = KE − 2KE e2 (10.5) =− 8πε0 r Brackett Series Paschen Series Bohr quantized the value of r by postulating that the electronic orbit must be a standing wave with wavelength λ, consequently the length of the circle describing the electron’s trajectory must be the product of an integer and λ, 2πrn = nλ (10.6) -3 Balmer Series -4 -6 -7 -8 Series Limit Energy (eV) -5 -9 where n = 1, 2, 3, . . . -10 -11 -12 -13 Lyman Series -14 Figure 10.2: This figure shows an energy level diagram for hydrogen along with some of the possible transitions. An infinite number of energy levels are crowded between n = 6 and the escape energy n = ∞. The French physicist Louis de Broglie postulated that one can assign a wavelength λ to any object having a momentum so that for the electron p = me v; λ = hp = mhe v , where h is Planck’s constant. Combining this with Equation‘(10.6) we see that 2π rn = nλ = nh . me v (10.7) We eliminate the electron’s velocity from Equation (10.2) and Equation (10.7) to find e2 1 1 nh = me v 2 = me 8πε0 r 2 2 2π me r and 1 e = r nh 2 π me . ε0 !2 (nh)2 = 2 8π me r2 (10.8) (10.9) Checkpoint What did Bohr postulate in calculating the energy spectrum of the hydrogen atom? 137 CHAPTER 10: EXPERIMENT 8 Table 10.1: Details about the photon interactions with hydrogen atoms for several ‘lines’. Most humans can see the first 3-4 Balmer lines. Checkpoint What does quantization mean? Name five quantities that are quantized in nature. Now we substitute this into Equation (10.5) to determine the atom’s energy to be e2 e2 =− En = − 8πε0 rn 8πε0 e nh 2 π me me =− ε0 2 e2 2 hε0 !2 1 1 = −R y n2 n2 (10.10) where the Rydberg constant is defined to be me Ry = 2 e2 2 hε0 !2 = 13.605 eV. (10.11) The radius rn of the orbiting electron and the binding energy En are quantized; they can assume only discrete values! The quantity n is called the principal quantum number. The 138 CHAPTER 10: EXPERIMENT 8 Figure 10.3: The spectrum of white light as viewed using a grating instrument like the one in this experiment. The different orders identified by the order number m, are shown separated vertically for clarity. In actuality they would overlap. radius rn increases as n2 . The energy values are negative because we are calculating a binding energy; (one must supply energy in order to achieve an isolated electron and an isolated proton at rest). Figure 10.2 shows the energy of stationary state and their associated quantum numbers. The upper level marked n = ∞, corresponds to a state in which the electron is completely removed from the atom; the atom is ionized. Checkpoint Why does the binding energy of an atomic electron have a negative value? An atom radiates when, after being excited, it makes a transition from one state with the principle quantum number n, and consequently with energy En , to a state m with lower energy Em . The quantum energy carried by the photon is hν = Eγ = En − Em = −Ry 1 1 − . n2 m2 (10.12) The wave speed, the wavelength, and the frequency of oscillation are related by c = λν, so we can find the wavelength of the emitted photon to be λ= c hc = . ν Eγ (10.13) In our experiment we will wish to measure values for the Rydberg constant using our 139 CHAPTER 10: EXPERIMENT 8 Figure 10.4: A sketch of our grating spectrometer. Our gas is ionized to provide the source of light. Lens A collimates the light into a beam the size of the grating. Lens B focuses the light onto the cross-hair (see inset) and the eyepiece allows us to see the spectral lines. The Vernier angular scale allows us to determine the wavelength of the line at the cross-hair. measured wavelengths, Ry = − hν 1 n2 − 1 m2 hc =− 1 λ n2 − 1 m2 =− 1239.84 eV · nm λ 1 n2 − 1 m2 . (10.14) We must keep in mind that Bohr’s model above applies only to hydrogen and to hydrogen-like atoms. To understand more complex atoms we must incorporate the revolutionary theory of quantum mechanics. Even quantum mechanics applied to these complex systems cannot be solved exactly, but we have developed approximate solutions that agree quite well with experiment. Additionally, quantum mechanics can explain the subtle variations in hydrogen’s Figure 10.5: A sketch of the apparatus (side spectrum that are visible in the data you view) illustrating how to view the spectrum of a hot tungsten filament. will gather today. 140 CHAPTER 10: EXPERIMENT 8 The Grating Spectrometer As already discussed in the second laboratory, gratings are used to analyze the frequency spectrum of light. Light of a given wavelength, λ, will be deflected by a grating with a line spacing constant d, by an angle θ, according to the relation mλ = d sin θ with m = 0, 1, 2, . . . (10.15) where m is the order number. Figure 10.3 shows how a grating spectrometer diffracts visible light in groups of intensity distributions as a function of θ, and each of these groups corresponds to a different order m. Previously, our experience with diffraction gratings was limited to illumination by monochromatic lasers and we saw the spots into which the laser was diffracted. Previously we saw spots because we were using only one color (or wavelength, λ) of light. Today, we are illuminating the gratings with many colors simultaneously and each color will excite every order of the gratings. Because of this it is important that we not allow ourselves to become overwhelmed by what the spectrometer does with our light. Adjacent distributions do overlap at the larger orders; third order violet occurs before second order red. Figure 10.4 shows the spectrometer used for this experiment. It consists of a circular table with a diffraction grating at its center and an angular Vernier scale along its circumference. Two pipes are mounted on the table to keep out room light and these pipes intersect in the region of the grating. The mounting of the first pipe is permanent. It is designed to carry parallel light from an external source to shine upon the grating. The second pipe is mounted in such a way as to be able to pivot about the center of the circular table. This rotation makes it possible to separate the light, collecting only what has been diffracted through some chosen angel, θ. If this pipe is set to an angle such as θ = 10◦ , then whatever light the grating diffracts through an angle of 10◦ will travel down the length of the pipe. It will be focused by the adjustable lens (B) and the eyepiece. A pair of cross-hairs identify the center of the pipe and, together with the vernier scale located on the equipment, allow the accurate measurement of the angle, θ. The grating constant d is engraved on the frame holding the grating. Slit S, in Figure 10.4 defines the light source; the slit is parallel to the lines of the grating so the diffracted light will form vertical lines instead of spots. Lens A is separated from the slit by a distance equal to its focal length, f , and consequently it projects a beam of parallel light on to the grating. Such a collimated beam utilizes most of the grating so that the spectra are quite sharp. Lens B focuses the light diffracted by the grating into a focal plane which contains the cross-hair. Both positions of lens B and the cross-hair can be adjusted in order to focus the image better. 141 CHAPTER 10: EXPERIMENT 8 Checkpoint Draw the path of two parallel light rays diverging from the spectrometer slit and entering the observer’s eye. Checkpoint What is the physical significance of the order m of the spectrum produced by a grating? The optics of the human eye is shown schematically in Figure 10.6. The optimum focal distance of our eyes is about 25 cm. In order to observe an image at a closer distance, a lens called an eyepiece may be used as indicated in Figure 10.7. The spectrometer has an eyepiece with focal length of 2.5 cm. Each student will have to adjust the position of the of lens B to see a sharp image the spectral fringe. The placement of the cross-hairs should then be adjusted so that the two are in focus. A small light bulb controlled by a pushbutton switch allows the cross-hairs to be illuminated. To determine the axis (θ = 0) Figure 10.6: Focusing system of the eye. of the spectrometer, place the table light Most of the refraction occurs at the first sur- in front of the spectrometer slit, S, remove face, where the cornea C separates a liquid the diffraction grating and look through the with index of refraction n1 from the air. The pivoting arm of the spectrometer directly lens L, which has a refractive index varying at the light source. Adjust the position of from 1.42 in the center to 1.37 at the edge, the slit, of lens B, and of the cross-hair in is focused by tension at the edges. The main order to see a sharp, white image (order volume contains a jelly like ‘vitreous humour’ m = 0). When a sharp vertical image of (n = 1.34). The image is formed on the the slit is centered on the cross-hair, you can retina. Illumination levels are controlled by read the angle corresponding to the axis of the aperture of the iris. the spectrometer; use the Vernier scale as shown in the appendix. All measurements of θ that you will make must be with respect to the spectrometer axis so subtract this θ0 from all remaining angle measurements. Ga3 can do this for you if you simply incorporate the subtraction into the column formula. Don’t forget to record the units and experimental error for your measurements. 142 CHAPTER 10: EXPERIMENT 8 The Light Sources In this laboratory you will observe the spectrum of four gases: H, He, Ne, and Hg. The light source consists of a glass tube which contains one of these gases at a pressure of about 20-30 cm Hg. A 60 Hz voltage of a few kilovolts is applied across the gas in the tube via electrodes at each end. The non-uniform electric field between the two electrodes will Figure 10.7: Simple lens used as a magnifier. An object at O close to the eye can be focused excite the gas in the tube. by the eye to appear to be much larger and at An electron will remain in an excited a more distant point O’. state for a time on the order of 10−8 s before jumping down to a state with higher binding energy. The excited state, with a lifetime τ ∼ 10−8 s, has then decayed into a lower energy state. In order to prolong the usefulness of these tubes do not keep the light source on any longer than necessary. Checkpoint How might you excite an atom and how long does it stay excited? 10.1.1 Part 1: The Spectrum of White Light The electromagnetic spectrum emitted by the hot filament of the table light is a continuous spectrum whose relative intensity depends upon its temperature. The shape of the spectrum of a black body is shown in Figure 10.9. It was Planck who first calculated the shape of this spectrum. Planck introduced a constant h, now known as Planck’s constant, to reproduce mathematically the observed distribution of this spectrum. Observe the visible part of the electromagnetic spectrum emitted by your incandescent lamp. Look at the spectrum Figure 10.8: A sketch of the apparatus (side associated with the order numbers m = view) illustrating how to view the spectrum of 0, 1, and 2. Observe the different colors as- an ionized gas. sociated with different frequencies or wavelengths. 143 CHAPTER 10: EXPERIMENT 8 14000 12000 Watts/(m2 nm) 10000 Planck Law T=5000 K 8000 Rayleigh-Jeans Law T=3000 K 6000 4000 Planck Law T=4000 K 2000 Planck Law, T=3000 K 0 0 0.5 1 1.5 2 2.5 3 Wavelength (microns) Figure 10.9: Blackbody Radiation Spectrum. Planck’s energy density of blackbody radiation at various temperatures as a function of wavelength. Note that the wavelength at which the curve is a maximum decreases with temperature. Measure the spectral sensitivity of your eye by observing the highest frequency your eye can discern. Use the order m = 1 to make these measurements. Measure the angle θ with respect to the spectrometer axis (don’t forget to subtract off θ0 ) and use Equation (10.15) to calculate the corresponding λ. Don’t forget your units and errors. Now measure the lowest frequency your eye can see using the same method. Note the characteristics of the filament’s spectrum in your Data. Are there holes in the spectrum or is it continuous? Humans can see 400 nm < λ < 700 nm so be sure your measured wavelengths are reasonable before continuing. Once you understand how to use the spectrometer, double-check your knowledge by repeating the exercise on the other side of zero (for m = −1). 10.1.2 Part 2: The Hydrogen Spectrum Start Vernier Software’s Ga3 graphical analysis program. A suitable Ga3 configuration file can be downloaded from the lab’s website at http://groups.physics.northwestern.edu/lab/spectral.html The wavelength is calculated as you enter the order m and the diffracted angle θ using Equation (10.15) λ = md sin θ; since the grating specifies lines/inch, this becomes 144 CHAPTER 10: EXPERIMENT 8 2.54 * 10^7 * sin(“Angle” - θ0 ) / (lines * “Order”) in Ga3’s calculation equation. Another calculated column “Rydberg” with units “eV” and formula 1239.84 / (“Wavelength” * (1/4 - 1/“Upper”^2)) is the Rydberg constant and is plotted on the y-axis. Equation (10.14) has been used as this column’s equation. Use the controls at the bottom left to tell Ga3 about your θ0 and lines/inch. Observe and record as many spectral lines as you can see in the spectrum of hydrogen, using a hydrogen gas tube. Make sure that the light source is properly aligned with the spectrometer slit. At 0◦ the red and blue will merge into a bright sort of tan at proper alignment (the same color as the directly viewed hydrogen tube). Hold the table as you move the eyepiece or you will mess up your careful alignment and need to repeat it before continuing. Enter the angle into Ga3 (θ), the diffraction order (m), and the upper electron shell’s principal quantum number (red⇒ n = 3, cyan⇒ n = 4, violet⇒ n = 5). Compare the wavelength of the lines you observe with the value shown in Table 10.1. Which series of lines do you observe? Table 10.1 also gives the principal quantum numbers involved in the observed transitions. Hints for measuring the wavelengths of spectral lines: 1. Align and center the light source to make the lines as bright as possible. 2. Optimize the focus for each measurement using lens B. Make sure the diffraction grating is aligned perpendicular to the incident, collimated beam (a peg on the slide’s bottom should fit into a groove on the mount.) 3. Measure each color using all visible orders on both sides of the optical axis. Each order yields a measured value of the Rydberg constant. Note correlations between Rydberg constants gathered using one color and contrast these with constants gathered using other colors. 4. Use some intuition in matching the measured wavelengths to the chart of known lines. Even with careful measurement procedures your measured wavelengths may easily vary as much as a couple dozen nanometers from those in the chart. Also the chart only lists the brighter lines; other dim lines may also be present. Checkpoint Which one of the hydrogen series describes light in the visible spectrum? 145 CHAPTER 10: EXPERIMENT 8 Table 10.2: Several known lines for helium (He), neon (Ne), and mercury (Hg). These may be used to identify these substances in chemicals when they are burned in a flame. Checkpoint What is the difference between the electromagnetic spectrum produced by an incandescent object and the spectrum generated by excited atoms? 10.1.3 Part 3: The Spectrum of He, Ne, and Hg The He gas tube is labeled while the tubes with Ne and Hg are only marked with colored tape. Record the wavelength of 5 to 10 spectral lines of each one of these elements. Measure these wavelengths only once each rather than once for each visible order and note their units and errors. Notice that the spectrum of these gases is much more complicated, with many more lines than the one of hydrogen. This is due to the larger number of electrons orbiting the nuclei of these elements, the large number of electron-electron interactions, and consequently the larger number of possible transitions corresponding to spectral lines. Table 10.2 gives the wavelengths and the relative intensity of the line spectra of these elements. By comparing your observed values with those on the table you will be able to tell which gas is enclosed in a tube marked with a given color. Chemists find the spectroscopic uniqueness of atoms and molecules quite useful in identifying unknown substances. 146 CHAPTER 10: EXPERIMENT 8 10.2 Analysis Discuss your graph of measured Rydberg energies. It might help to rescale your y-axis from 0 to 15 eV. What trends do you see in the graph? Bohr’s model and quantum mechanics predicts that all of the Rydberg constants are the same; is it conceivable that your data agrees with this within your errors? If so, it makes sense to compute the average Ry and standard deviation sRy using Analyze/Statistics. First, though, you will need to delete any unused rows at the bottom of your table and to draw a box around your data points on your graph. Move the parameters box away from your data so your readers can see every data point. Print the whole screen for each member of your group and find the standard error in your Rydberg constant sRy σ = sRy = √ . N Now, find the difference between your mean and the best measurement of others ∆ = Ry − 13.605 eV. The standard error is the best estimate of the error in the mean, Ry = (Ry ± sRy ). Does your Rydberg constant agree with everyone else’s? Disagree? How well do you know? What do you observe about the environment of the hydrogen atoms that might affect the electrons’ energies? What does this all mean versus Bohr’s model? Now discuss the three more complex spectra. Take a few minutes and try to fit helium’s spectrum to Bohr’s model (Equation (10.10)). Do all of the lines fit or do we need a better theory? Does it seem like we might have better luck with neon or mercury? 10.3 Conclusions Is blackbody radiation characteristically the same or different than gas discharge light? Is Bohr’s model valid? Under what circumstances? Did you measure anything that you and/or others might find of value in the future? Remember to communicate using complete sentences and to define all symbols used in equations. 10.3.1 Reference Sections of this write up were taken from: Bruno Gobi, Physics Laboratory: Third Quarter, Northwestern University 147 CHAPTER 10: EXPERIMENT 8 10.4 APPENDIX: The Vernier Principle A vernier consists of a movable scale next to a stationary scale. The movable scale is marked off in divisions that are slightly smaller than the divisions on the main scale. If you line up the zero mark of the vernier with the zero mark of the fixed scale (see Figure 10.10), you will notice that not all of the other divisions are aligned. However, if you examine the two scales closely, you should observe that the nth division of the movable scale exactly coincides with the n − 1 division of the fixed scale. This implies that n − 1 divisions of the main scale has or 1 − n1 the same length as n divisions on the vernier scale; so each vernier division is n−1 n times the length of a main scale division. Determining the quantity n1 is an important first step in making any measurement with a vernier; n1 is called the “least count”. The least count gives an indication of the precision of the particular measuring device. For example, if we can read the main scale of our device to the nearest degree, and the device has a least count of 1/4 (i.e. 4 vernier divisions is equal in length to 3 main scale divisions) then we can make meaFigure 10.10: An illustration of the Vernier scale. surements with an instrumentally The position of the movable 0 indicates the coarse limited error of ± 1/4 of a degree. reading on the fixed scale. One additional digit of Examine Figure 10.10. What is precision can be obtained accurately from the Movable n? What is the least count of this Scale digit that aligns best with a Fixed Scale number. Vernier scale? If each division of the main scale corresponds to the one degree, what is the precision of the measurements we can make with this device? A well-designed instrument will have accuracy approximately equal to precision; the ability to provide faithful measurements (accuracy) should be about the same as the amount of information conveyed in a reading (precision). An instrument with too few displayed digits defaults to having round-off be its greatest experimental error. Contrarily, having too many displayed digits wastes money on useless precision and implies an accuracy which it is incapable of providing. Keeping this in mind, we should expect that the experimental error associated with an instrument is usually about the same as one or two least significant digits of its carefully read display. In the case of our spectrometers, we should expect 0.1◦ errors in our angle measurements with reasonable care and attention to detail. To read a measurement from a vernier such as the one in Figure 10.11, follow these steps: 1. Determine the least count of your device. 2. Look opposite the zero mark of the movable scale and read off what division of the fixed scale is at the zero mark. If the zero mark does not exactly align and is situated between two divisions, then read off the value of the lower of these two divisions. In Figure 10.11, we read 46 degrees. 148 CHAPTER 10: EXPERIMENT 8 3. Examine each mark along the movable scale and determine which division (from zero to n) is aligned exactly with a division of the main scale. Read off this number from the movable scale. In Figure 10.11, we can see that 3 on the movable scale is aligned with a division on the main scale. So we must add 3 least counts of 3/10 of a degree to the number read from the main scale earlier, 46. Thus our final answer is 46.3 degrees. If the zero mark (and simultaneously the n mark) of the movable scale had exactly coincided with a division of the main scale, as in Figure 10.10, then we would add zero least counts and arrive at an answer of 0.0 degrees. Figure 10.11: An example tutorial in reading the Vernier scale. The movable 0 is between the fixed 46 − 47, so the reading is 46.x. The x is determined by noting that the movable 3 matches the Fixed Scale numbers best (49 in this case); this makes x = 3 and our reading is 46.3. We might also note that the movable 0 is indeed quite close to 46.3 to double-check our result. Incidentally, all of the above is brought to you courtesy of Pierre Vernier (1585-1638), a French engineer who spent most of his life building fortifications, but who designed the Vernier scale in about 1630. 149 CHAPTER 10: EXPERIMENT 8 150 Appendix A Physical Units In science, we describe processes in Nature using mathematics. Math is very concise, structured, and adaptable so that once we have mathematical models that mimic Nature, we can make very specific and accurate predictions about what will happen in other situations. Engineers use this fact to very good effect to obtain useful results in society. Physical laws are usually expressed in terms of physical quantities that have dimensions. The definition of units that are used to describe these dimensions is the raison d’être of the National Institute of Standards and Technology (NIST). An example that we all should have seen by now is Newton’s second law, F(a) = ma (A.1) Given a mass whose “value” is m, to get motion whose acceleration is a we must apply a push or a pull whose “value” is F. Since a push or a pull is an entirely different entity than a mass, we include in F a different multiplicative variable (N, dyne, pounds, etc.) than we include in m’s value (kg, g, slugs, etc.). The specific point that we need to address now is “What about the parameter a?” What intrinsic multiplicative variable should we include in a’s “value” so that the model is mathematically self-consistent? We can use algebra to figure this out. Let the unknown units be x and consider the specific case of F = 1 N and m = 1 kg . To solve our problem we substitute these values into the model and solve for our unknown, x, 1 N = F = ma = (1 kg)(1 x) so x = N kg (A.2) If we multiply all of a’s pure numbers by the units N/kg , then the unknowns (and unknowables) that we call units will always cancel in such a way that forces will always have units N, masses will always have units kg, and accelerations will always have units N/kg. Because all of these are always true, our model is mathematically self-consistent. Of course, kinematics has taught us that N/kg = m/s2 and that we can also determine acceleration by measuring the distances that objects move, the rates that objects move, and the times needed for these motions and rates to occur. One of the most beautiful and compelling aspects of physics is the frequency that a single physical entity (acceleration in this case) can be determined in 151 APPENDIX A: Physical Units more than one way (sometimes many, many more than one). Let us now move on to a more general case. Let us pretend that we have developed a model represented by f (x) = ax + bx3 (A.3) Without more information the units of f , x, a, and b are ambiguous; so, let us stipulate that f is force (N) and that x is position (m). Is this enough information for us to determine the units of a and b? Our first instinct might be to answer that “No, one equation cannot yield two unknowns”; however, the rules of algebra must be obeyed as well. When we add the two quantities ax and bx3 , they must have the same units. Furthermore, these common units will be assigned to f , so the results of these multiplications had better yield units of N! Now our problem has reduced to exactly the same problem as we solved above; we just have to solve it twice in the present case. Let A be the units of a, let B be the units of b, and substitute into the model: kg N = 2 m s N kg so B = 3 = 2 2 m s m 1 N = f = ax = (1 A)(1 m) so A = 1 N = f = bx3 = (1 B)(1 m)3 (A.4) (A.5) Models having any number of terms can be handled in exactly this same way; we just have to solve the problem once for each term as above. Most students will also soon find short-cuts that will make this much more expedient than it may seem just now. Converting units Before looking at more complex situations, we will summarize how to convert between different physical units. Over the years many, many systems of units have been conceived and used for specifying distance. Historically, it has been important for economic, political, legal, personal, and scientific reasons that we have the ability to specify distance concisely and accurately. From a need to know our height or shoe size to a need to know the area of a land tract or to know how far away is the center of government, we need the ability to measure and to communicate distances. The problem is that different purposes benefit from somewhat different units of measure. It does not make sense to measure the distance to Los Angeles in human hair-widths, for example, even though you often hear small things being referred to “the thickness of a human hair” (∼ 100 micrometers, by the way). To compare ‘apples’ to ‘apples’, we invariably have to convert between different units of measure. The simplest strategy to convert between systems of units is to place the unit for distance in one system beside the unit for distance in the other system and simply to see how they compare. We use one of the “yardsticks” to measure the length of the other “yardstick”. As an example, we might use a meter stick to measure the length of a foot long ruler; when we do so we will find that 1 ft = 0.3048 m. We could also use the foot ruler to measure the meter stick and to find that 1 m = 3.2808 ft. Of course, these numbers are inverses of each 152 APPENDIX A: Physical Units other, which is required for the measurements to be consistent. This implies that 0.3048 ft m = 1 = 3.2808 ft m (A.6) Let us suppose that we are given some distance d = 33.24 ft and that we need to communicate this distance to Paris, France, where no one has heard of “ft”; over there everyone measures distance in “m”. We can multiply anything by 1 without changing its value. This applies to d as well. We see that 1 = 0.3048 m/ft so that multiplying by 0.3048 m/ft is just multiplying by 1. m d = 33.24 ft = 33.24 ft × 1 = 33.24 ft × 0.3048 = 10.13 m (A.7) ft Dimensionless Functions Frequently we find expressions like f (x) = f0 eax or g(x) = g0 ln x x0 (A.8) These functions belong to the class of exponential functions (recall that y = ex if and only if x = ln y). What must the units of f0 , a, g0 , and x0 be in order that f and g be force (N) and that x be distance (m)? First, realize that e ≈ 2.7183 is just a number, no different than 3, 8, or 1 as far as units are concerned. Multiplying pure numbers together does not (and cannot) cause units to appear; therefore, raising e to a power (multiplying e by itself over and over) also results in a pure number without units. Then f0 and g0 must have the same units as f and g, respectively. An exponent necessarily has no units. The only way to convert between units is to multiply by ratios of units and only raising to higher exponents (xa )b = xab yields products of exponents. Units are algebraic unknowns so that having units in an exponent is equivalent to having the exponent to be unknown. . . not to know how many times to multiply the base. We might invent a rule or a convention simply to ignore the units – simply pretend that they are not there – just raise the base to whatever number is there. But that would mean that 1.6332 = 50.3048 m = 51.000 ft = 5 (A.9) which is clearly not true. On the other hand, requiring that exponents have no units works out a little better. To show this we re-arrange our english-metric conversions a little 1 = 3.2808 ft ft and = 0.3048 m m (A.10) and manipulate them similar to the way we did above. Let b be some arbitrary number and note that ft b = b1 = b3.2808 m = b(3.2808)(0.3048) = b1 (A.11) 153 APPENDIX A: Physical Units which is self-consistent. Exponents must be pure numbers. For similar reasons the arguments of logarithms must also be unitless, pure numbers. To see this recall that x ln x0 = ln(x) − ln(x0 ) (A.12) If x and x0 have the same units, the units cancel on the left and leave a pure number to be the logarithm’s argument. If they have different units, then any unit conversion applied to x0 on the right would change the value subtracted from ln(x) thus implying that the quantity on the left can simultaneously be equal to both values. So, how do we deal with the right side of Equation (A.12) if x and x0 have units? We understand that we actually mean something different: ln(x) − ln(x0 ) = ln(x) − ln(x0 ) + ln(a) − ln(a) = ln(x/a) − ln(x0 /a) = ln x x0 (A.13) The constant a must have the same dimensions as x and x0 , and although it cancels entirely from the expression, it must be there to make the logarithm’s dimensionless argument make sense. Incidentally, this is also why logarithmic scales like the Richter scale for earthquakes and the dB scale for sound always have a reference intensity (a definite value with units), equivalent to x0 or a in the discussion above. Sine, cosine, and tangent are trigonometric functions whose arguments are usually derived from angles. Trigonometric functions arguments are dimensionless, but they are not unitless; these arguments must have the units of their angular measure. What this means is that if you scale physical dimensions, the angles do not change. . . if you change your angular measure, then the angle’s numerical value (degrees or radians) does change. Frequently, we deal with expressions like x(t) = x0 cos(ωt + φ) (A.14) If x has units of m and t has units of s, what are the units of x0 , ω, and φ? The cosine function itself has no units so x0 has the units of x. The argument of the cosine is a polynomial, so each term must have the angular units that the cosine function needs. The natural units of angle are radians so that a circle has 2π radians. The units of ω must be rad/s. You might notice a similarity between logarithms, exponentials, and trigonometric functions: their arguments are all dimensionless. This is not an accident. Mathematics typically concerns functions applied to numbers. Thus, mathematical expressions tend to be defined in a way such that they have no dimensions. It is only in physics, where we have to compare real measurements of different quantities, where the reference units appear. Therefore, in these mathematical functions, we must convert the argument to pure numbers for the expressions to make sense. 154 Appendix B Using Vernier Graphical Analysis Additional guidance is generally provided in the lab experiment chapters. Helpful Tip There are two version of this software on the lab computers. The older one, ‘Gax’ is not well-liked and has caused many TA and student headaches throughout the years. The newer version, ‘Graphical Analysis 3.4’ (Ga3) is more functional and reliable but requires a few more steps to get it to do what you need it to do. Please use Graphical Analysis 3.4 in this course. Excel and Gax are always available if you prefer, however. When we perform a least squares fit to a mathematical model, we are effectively compiling all of the knowledge gained by our data into a much smaller set of fitting parameters. To the extent that our data applies to this particular model, the experimental errors that are inevitable in measurements are averaged out so that we might utilize the fitting parameters to improve upon a prediction of our model beyond the data points themselves. This does not mean that the fitting parameters have no uncertainties associated with them; indeed they do have and it is essential that we specify each uncertainty when we discuss the parameter. Many times Graphical Analysis 3.4 does not show the uncertainties in fitting parameters by default. We can always ask it to do so by right-clicking the parameters box, “Properties...”, and select “Show Uncertainties”. We can also specify other properties in the fitting parameters such as the number of significant digits. Not only is the older Gax somewhat unstable, but it also offers meaningful estimates of fitting parameter uncertainties only for the separate ‘linear regression’; custom models have no convenient way to get uncertainty estimates. Since fitting parameters are effectively measurements that the computer has deduced from the data, it is essential that we be able to specify the uncertainties and the units that must accompany them. These are the reasons why we prefer Ga3 that has a less obvious user interface. 155 APPENDIX B: Graphical Analysis Software 156 Appendix C Using Microsoft Excel C.1 Creating plots and curve fits Helpful Tip The primary advantage of Excel is that students have access to it at home. However, it is more challenging to perform statistical-based fits in Excel unless you write your own spreadsheet or macros. It is recommended that you perform your data analysis with Graphical Analysis 3.4 in the lab. However, you always have access to Excel in many locations in case you want to plot outside the lab. This short list of resources will help guide you in creating figures in Excel, if you decide to use it. • A good introduction to basic graphing in Excel is provided by the LabWrite project: https://www.ncsu.edu/labwrite/res/gt/graphtut-home.html • Curve fitting in Excel can be performed, but sometimes it is much more painful than using a dedicated data analysis program. For least-squares curve fitting, this link details how to use Excel: http://www.jkp-ads.com/articles/leastsquares.asp Neither do Excel’s least-squares fitting parameters come specified with uncertainty estimates even after you go through these acrobatics. • We can use Excel’s “LINEST” function to provide complete statistics on straight lines. See ‘linest’ in Excel’s Help files to learn more about its usage. 157 APPENDIX C: Using Excel C.2 Performing Calculations All calculations in spreadsheets begin with “=”. First, select the cell that will hold the result of the calculation, type “=”, and enter the formula to be evaluated. Typing “=5+9” and ‘Enter’ will yield “14” in the cell; the cell beneath will then be selected. If you need a cell to contain ‘=’ instead the results of a calculation, precede it with a single quote ‘0’. . . anything you type after that will be displayed verbatim. Even numeric characters won’t be numbers after ‘0’. If this was the extent of their capabilities, spreadsheets would not be very popular. Each spreadsheet cell is at the intersection of a column having alphabetic label at the top and a row having numeric label at the left. ‘A1’ is the cell at the top left corner of the sheet, ‘C8’ is at the eighth row and third column, and ‘Z26’ is at the 26th row and the 26th column. After column ‘Z’ comes column ‘AA’ to ‘AZ’, ‘BA’ to ‘BZ’,. . . , ‘ZA’ to ‘ZZ’. If even more columns are needed, three, four, and five characters can be used. Column ‘numbers’ count up just like decimal digits, but column designations have base 26 instead of base 10. As an exercise we could put the numbers 1-10 in cells A1:A10. Then in, say, B7 we could type ‘=A1+A2+A3+A4+A5+A6+A7+A8+A9+A10’ to add the numbers (55). After we type ‘=’, we could click on cell A1 to get the computer to place the ‘A1’ in the formula. This comes in handy in a large project when you don’t remember the cell designation and it is not visible on the screen. This is pretty nice and the total (cell B7) will automatically update anytime one or more of the numbers in A1:A10 changes. However, this would mean a lot of typing if the sum of, say, A1:A1000 were needed instead. It turns out that spreadsheets have a ‘sum’ function for adding the contents of cells. “=sum(A1:A10)” would also add the contents of these cells. ANY block of cells can be specified using “range” designators. “=sum(a1:z6)” will add the first six rows of the first 26 columns. The function names and column designators are not sensitive to case. The cell into which the result is to be placed MUST NOT be part of the range or an error will result. The ‘A1:A10’ or ‘A1:Z6’ can be generated by the computer by dragging the mouse across the desired range of cells. In addition to ‘sum’, we could ‘average’, ‘stdev’, ‘count’, etc. to perform statistics on a range. Many other functions are part of the standard spreadsheet application. A partitioned list and usage instructions can be generated using the ‘function’ toolbar button; usually the icon has ‘f ’, ‘fx ’, ‘f (x)’, or such. Frequently, we want to generate graphs. To do this we first need to generate the independent (x-axis) values. Enter ‘10’ into say C3 and ‘=0.1+C3’ into C4. Now, go back to C4 and ctrl+c to copy this formula. Next, drag the mouse from C5 to C103 to select this range (the cells turn black) and ctrl+v to paste the formula into all of the selected cells. Now, C3=10, C4=10.1,. . . , C103=20. Click on one of these cells and look at the formula in the formula bar. The computer has automatically replaced the ‘3’ in ‘C3’ with the correct number to add 0.1 to the cell above this one! This always happens by default when you copy and paste the formulas from one cell to another unless you “Edit/Paste Special. . . ” and specify the details you want pasted. 158 APPENDIX C: Using Excel For relatively smooth functions, these 101 values will yield a very nice graph. Smaller increments can also be generated at need for more quickly changing functions. Uniformly spaced numbers are needed so often that a shortcut exists. Type 10 into C3 and 10.1 into C4. Next, drag from C3 to C4 to select both and release the mouse button. The bottom right corner of the selected cells is a drag handle that can be used to extend the pattern 10.0, 10.1, 10.2,. . . by dragging this handle downward with the left mouse button. Rows of incrementing numbers can be generated similarly. Now that we have our abscissa, we need to generate our ordinate. Click on D3 and enter ‘=3*sin(C3)’. Go back to D3 and drag the drag handle at the bottom right downward to D103. Release the mouse button and note that the result is a sinusoid with amplitude 3. This is OK! but we can generate families of curves with varying amplitudes, frequencies, or phases quite easily. Enter ‘2’ into D2, ‘2.2’ into E2,. . . , ‘3’ into I2. Now enter “=D$2*cos($C3)” into D3. Go back to D3, drag the drag handle across to I3, and release the mouse button. Now, drag the drag handle down to I103 and note that we have generated six sinusoids with amplitudes given by row 2. The ‘$’ prevented the ‘2’ in ‘D2’ and the ‘C’ in ‘C3’ from changing. Using ‘$A$1’ or ‘$B$3’ would insert the number in A1 or B3 into all of the formulas; this would allow you to tweak these two numbers by hand to optimize all of the curves at once. Practice generating various abscissas and ordinates so that you can remember these basics. Also practice plotting the results and pasting the plots into a Word document. You will want ‘scatter plots’ so that you can choose the (x, y) points. 159 APPENDIX C: Using Excel 160 Appendix D Using Microsoft Word Helpful Tip A nice template containing a report outline and hints for performing many of the functions science reports, journal publications, and books often employ can be downloaded from each lab’s website. All word processing programs have similar capabilities and many can import the template. . . although perhaps not seamlessly. Hopefully, this will get you started using word processing software. For your electronic write-ups, you can embed figures directly in the Word files, or you can upload them separately. If you want to embed in Microsoft Word while preparing your document, here are some useful sources of assistance. • To embed an Excel figures directly into Word, you can follow the instructions here: https://support.office.com/en-us/article/Insert-a-chart-from-an-Excel-spreadsheetinto-Word-0b4d40a5-3544-4dcd-b28f-ba82a9b9f1e1 • If you want to embed a pdf figure into a Word document, you can follow the instructions here: https://support.office.com/en-za/article/Add-a-PDF-to-a-document9a293b43-45de-4ad2-a0b7-55a734cf6458 • Another set of instructions for embedding pdfs in Word, with a focus on preserving quality, can be found here: https://pagehalffull.wordpress.com/2012/11/20/quick-and-easy-way-to-insert-a-pdfgraph-or-diagram-into-a-microsoft-word-document-without-losing-too-much-quality/ • For embedding more general images in Word, and using word wrapping, see here: https://support.microsoft.com/en-us/kb/312799 161 APPENDIX D: Using Word With Microscoft Word, there are usually many ways to accomplish the same task. Some produce better results than others. This is why you always have the option of uploading figures and images separately from the text of your write-up. Helpful Tip Your goal for writing in this lab is clarity in communication, not professional quality documents. Beginning students have much difficulty including enough background information about their experiment and apparatus and they tend to include too much mundane detail that applies only to their specific apparatus instead. Eventually our work will be read by scientists all over the world, so it is essential that we practice including enough information about our apparatus and procedure that everyone in the world can understand our data while simultaneously avoiding these mundane details that will tend to confuse readers without our apparatus in front of them. 162 Appendix E Submitting a Report in Canvas Preparation Your lab report should be written in either Microsoft Word or be converted to a text-readable pdf. Every student at Northwestern has access to Microsoft Word and Excel for lab report preparation. JPEG format is not allowed for the main text. Both Word and pdf formats allow your TA to provide helpful feedback using Canvas’ SpeedGrader system. Experience has shown that this online feedback is popular to students in the 136 labs. This formatting requirement is therefore beneficial to everyone involved. If possible, it is easiest if all of your files are merged into a single document. In Word, images and figures can be embedded directly into the text file. If you are submitting a pdf prepared in another method, it is possible to attach images and figures saved in the lab as pdf documents into a single pdf file. Adobe Acrobat Professional can accomplish this. Alternatively, you can merge pdf files online. One possible provider is PDFMerge (http://www.pdfmerge.com/). If you cannot put all of your content into a single file, it is accessible to load figures as separate files (such as jpeg files). If you submit as separate files, you should refer to each uploaded figure in some other part of your Lab Report. Your TA is not required to view or grade any document that is not referred to in other graded sections of your report. It is recommended that you name your files “Figure X” where X is a number. Then you can refer to each figure by name. Alternatively, you can upload figures as a single document such as a pdf, and have the appropriate labels on the figures in each document. Remember: a lab report is practice in communication of ideas and activities. Failure to make references to uploaded figures clear to your TA can result in lost points. Submission To submit your lab report, open the assignment in the Module on Canvas. Click the “Submit Assignment” link. 163 APPENDIX E: Submitting to Canvas Under the File Upload tab, select the “Choose File” button and select the appropriate file. You can press “Add Another File” to create another upload link. When all files are ready and you agree to the TurnItIn pledge, you can press “Submit Assignment”. You are able to re-submit your assignment to Canvas, although if it is late it will be flagged. You should also be able to view your TurnItIn similarity score a few minutes after submission. Your TA will be able to see all submitted files in the Canvas grading system. It is probably easiest on your TA if the first file that you upload is your main lab report text, followed by figure files in order. Do not make life harder for your TAs as they grade your work! 164