Year 1 Laboratory Manual (2014–2015)

Department of Physics November 10, 2014

Resistor colour chart Resistors are colour coded, indicating their resistance and tolerance. Most of the resistors you will encounter in first year lab will have four bands as shown below.

The colours are coded as follows:

Band Band Band Band

A corresponds to the first significant digit B corresponds to the second significant digit C represents the decimal multipler D indicates the tolerance

For example: a resistor with red, black, brown, gold bands corresponds to

20 × 101 ± 5% Ω = 200 ± 5% Ω

Colour

Value

Black Brown Red Orange Yellow Green Blue Violet Gray White Gold Silver

0 1 2 3 4 5 6 7 8 9 — —

Tolerance — ±1% ±2% — ±5% ±0.5% ±0.25% ±0.1% ±0.05% — ±5% ±10%

Capacitor values Capacitors vary in physical size and appearance depending on their type and capacitance. In the first year lab you will use two types:

Electrolytic capacitors are packaged in metal cans of varying size. They have a polarity that should be observed, the negative leg is marked and is shorter than the positive. The capacitance is printed on the side.

Ceramic capacitors are small ceramic discs. They have no preferred polarity. Their capacitance is marked using three digits. The first two digits indicate the first two digits of the capacitance, the third is the decimal multiplier and the result expressed in pF. For example 154 corresponds to 15x104 pF which would be written conventionally as 150nF

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Contents 1 Introduction to the First Year Laboratory 1.1 Reasons for Experimental work . . . . . 1.2 The Structure of First Year Lab . . . . . 1.3 Safety (good laboratory practice) . . . . 1.4 General Experimental Strategy . . . . . . 1.5 General Computing Strategy . . . . . . . 1.6 Laboratory Reports . . . . . . . . . . . . 1.7 Laboratory Marks and Assessment . . . .

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4 Introduction to Experiments in Optics 4.1 A. Ray Optics—Lens Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 B. Wave Optics—Single-Slit Diffraction . . . . . . . . . . . . . . . . . . . . . .

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2 Introduction to the Computing Facilities 2.1 Before the End of the First Week . . . . 2.2 Introduction . . . . . . . . . . . . . . . 2.3 Basic Text Editing . . . . . . . . . . . . 2.4 Useful Packages . . . . . . . . . . . . . 2.5 LaTeX . . . . . . . . . . . . . . . . . . 2.6 World Wide Web . . . . . . . . . . . . . 2.7 Email . . . . . . . . . . . . . . . . . . . 2.8 Blackboard e-Learning System . . . . . . 2.9 Using Your Own PC or Laptop . . . . .

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3 Measurements and Uncertainties Exercises 3.1 Introduction . . . . . . . . . . . . . . . . . . . . 3.2 Means, Standard Deviations, and Standard Errors 3.3 Propagation of Errors . . . . . . . . . . . . . . . 3.4 Combinations of Errors . . . . . . . . . . . . . . 3.5 Plotting Graphs and Line-Fitting . . . . . . . . . 3.6 Practical Example . . . . . . . . . . . . . . . . . 3.7 Non-Linear Fits . . . . . . . . . . . . . . . . . . 3.8 Histograms . . . . . . . . . . . . . . . . . . . . . 3.9 Bibliography . . . . . . . . . . . . . . . . . . . . Appendix 3.A Solutions to Exercises . . . . . . . . . .

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Original authors are given at the end of each chapter when possible. The contents of this manual has subsequently been revised by Ned Ekins-Daukes and Yoshi Uchida, and also incorporates improvements that have been suggested by numerous demonstrators (postgraduates, postdocs, and academics) over the years.

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CONTENTS 5 Introduction to Electronics 5.1 Direct Current Equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Alternating Current Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Practical Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6 Experiments in Optics 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . 6.2 Working Practices . . . . . . . . . . . . . . . . . . 6.3 Interference and Diffraction . . . . . . . . . . . . . 6.4 Experiment 2a: Young’s Double Slit Interference and 6.5 Experiment 3: Imaging with Lenses . . . . . . . . . 6.6 Experiment 4: Polarisation of Light . . . . . . . . . 6.7 Experiment 5: Verification of Malus’ Law . . . . . .

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55 55 57 58 62 65 70 72

7 Experiments in Electromagnetism 7.1 Experiment 1: The Parallel Plate Capacitor . . . . . . . . . . . . . . . . . . . . 7.2 Experiment 2: Ampère’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Experiment 3: Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8 The 8.1 8.2 8.3

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Grating Spectrometer Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Setting up the spectrometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spectrometer theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9 The Charge-to-Mass Ratio of 9.1 Background . . . . . . . . 9.2 Experimental procedure . . 9.3 Details . . . . . . . . . . . Appendix 9.A Helmholtz coils .

the Electron . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

10 The (not so) Simple Pendulum 10.1 Introduction . . . . . . . . . . . . . . . 10.2 Experiment 1: Amplitude dependence . . 10.3 Experiment 2: The equivalence principle 10.4 Experiment 3: Coupled pendulums . . . Appendix 10.A Finite Amplitudes . . . . . . . Appendix 10.B Coupled Pendulums . . . . . . 11 Measuring the Speed of Light 11.1 Introduction . . . . . . . . . . . . . . . 11.2 Heterodyne Detection . . . . . . . . . . 11.3 The Apparatus . . . . . . . . . . . . . . 11.4 Experimental Procedure . . . . . . . . . Appendix 11.A Random and Systematic Errors

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12 A Guide to Using Origin® 108 12.1 Performing Calculations on Data . . . . . . . . . . . . . . . . . . . . . . . . . 109 12.2 Adding Error Bars to Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 12.3 Fitting Straight Lines to Data . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3

CONTENTS 12.4 Exporting Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Appendix 12.A Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 13 Keeping a Good Laboratory Record Book 13.1 Why keep a lab book? . . . . . . . . . . . . . . . 13.2 What should a lab book be? . . . . . . . . . . . . 13.3 What should your lab book include? . . . . . . . . 13.4 Some notes on computing-based experiments . . . 13.5 Neatness versus speed: Getting the balance correct

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115 115 116 116 117 117

14 Some Guidelines on Report Writing 14.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14.2 About Report Writing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix 14.A Dos and Don’ts . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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15 Use 15.1 15.2 15.3

of other peoples’ work in assessed essays, The College definition of plagiarism . . . . . . College Regulations . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . .

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summaries . . . . . . . . . . . . . . . . . . . . .

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and lab reports 128 . . . . . . . . . . . 128 . . . . . . . . . . . 129 . . . . . . . . . . . 131

Appendix A Equipment Specifications 133 A.1 Hameg Instruments Oscilloscope HM303-6 . . . . . . . . . . . . . . . . . . . . 133 A.2 Thurlby Thandar Instruments Power Supply EL301 . . . . . . . . . . . . . . . . 137 A.3 Thurlby Thandar Instruments Digital Multimeter 1604 . . . . . . . . . . . . . . 139 Appendix B Physics Department Calculators

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Appendix C Measurement and Uncertainties Summary Sheet

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Chapter 1 Introduction to the First Year Laboratory Contents 1.1 1.2

1.3 1.4 1.5 1.6 1.7

1.1

Reasons for Experimental work . . . . . The Structure of First Year Lab . . . . Experiments in Optics & Electromagnetism . Computing . . . . . . . . . . . . . . . . . . Demonstration Experiments . . . . . . . . . Summer Project . . . . . . . . . . . . . . . Opening Hours . . . . . . . . . . . . . . . . Safety (good laboratory practice) . . . General Experimental Strategy . . . . . General Computing Strategy . . . . . . Laboratory Reports . . . . . . . . . . . Being Assigned and Submitting Your Report Laboratory Marks and Assessment . . .

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Reasons for Experimental work

Experimental science is concerned with the observation of phenomena, the recording of results and the interpretation of those results in a broader logical framework. This process is illustrated in Fig. 1.1. In first year lab you will develop key skills for taking accurate measurements, keeping accurate laboratory records and for interpreting your data and assessing its accuracy. The apparatus you use to observe the natural world may be as basic as a wooden metre stick, but it may also involve using computer-controlled electronic circuitry. When dealing with large quantities of data, you will be glad of the assistance of a computer to help you analyse your data, and also to use the power of a computer as you calculate what you would expect from a theoretical model and compare it to data. As a consequence of this, a substantial part of your training in the first year lab will be focused upon learning techniques in electronics, computer programming and handling uncertainties in measurements. In addition to this, you will have the opportunity to perform some classic 5

Introduction to the First Year Laboratory

Figure 1.1: A possible graphical representation of the nature of physics work. experiments in physics. The Undergraduate Laboratories are designed to build the following laboratory skills: • • • • •

1.2

assembling and operating experimental equipment following and recording experimental procedures determining the errors that arise from experimental and statistical uncertainties taking and interpreting data presenting the results of an experiment in a written report.

The Structure of First Year Lab

You will start in first year lab with a series of introductory sessions designed to give you a basic 1 minimum training in Optics, Electronics and Measurement and Errors. Then in late October you will embark on three cycles of lab experiments lasting four weeks each. At the end of each lab cycle you will write a report on an aspect of the experiment you have been performing, this will be marked and returned to you with constructive feedback. The three lab cycles encompass:

Experiments in Optics & Electromagnetism You will learn, by example, basic principles of geometrical and physical optics, investigating the resolving power of lenses, interference and polarisation. In the electromagnetism experiments you will repeat some of the classic experiments that established the field, “rediscovering” it through the work of Faraday and Ampere. Later in the year you will receive formal lectures on this topic, building a consistent theoretical framework around the findings you make in the laboratory.

Computing The computer lab aims to introduce you to computers as tools for advanced calculations and the analysis of data. You will learn the programming language Python and write programs that simulate physical phenomena, exercising your knowledge of physics and building skills in logical thinking.

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Introduction to the First Year Laboratory

Demonstration Experiments In addition to teaching basic laboratory skills, we offer four experiments that demonstrate some key physics: a) b) c) d)

Measuring the speed of light The Grating Spectrometer: measuring hydrogen emission lines The charge-to-mass ratio of the electron The not-so-simple pendulum

Using the equipment supplied, you will spend two lab sessions on each experiment making careful measurements and drawing from your knowledge of physics to interpret the data. At the end of each lab cycle you submit a report on your work that is marked and returned to you within a fortnight. The lab report and an assessment of your performance in the lab count equally towards your final lab mark.

Summer Project In the third term you will perform a project where you will investigate some aspect of physics together with a lab partner and supervised by a knowledgeable demonstrator. Unlike the work that precedes this, there is no “script” to follow—you and your parter, in consultation with your demonstrator (a research physicist), will guide the project to its conclusion as you see fit, taking issues such as safety into full consideration. At the end of the third term, you will build a presentation display and demonstrate your project over two afternoons when the Physics Department opens its doors to other members of the college and visiting school groups. You will also submit a written report on your project. Dates and deadlines are printed on the rear cover of the lab manual.

Opening Hours The laboratory operates during timetabled sessions: Mondays 2pm–5pm Tuesdays 9am–12pm Thursdays 9am–12pm Fridays 2pm–5pm See your timetable to find out whether you are in the laboratory on Mondays/Tuesdays (Groups A–F) or Thursdays/Fridays (Groups G–L). The laboratory will be closed for lunch promptly at 12.00 on Tuesdays and Thursdays, but is generally available at other times if you want to do additional work (or to catch up!). Demonstrators are only available during the official laboratory sessions: the technical staff may be able to answer your questions, but you should not expect them to demonstrate the experiments.

1.3

Safety (good laboratory practice)

You are only allowed in the laboratory when there is a ‘responsible person’ present such as a demonstrator or the Laboratory Staff. Most aspects of safety in the UG lab are a matter of common sense and good housekeeping (i.e. being tidy): 7

Introduction to the First Year Laboratory • Do not stray significantly from the experimental procedure you have been given, without thinking through the consequences. If you do need to deviate from the manual, discuss with a demonstrator first. • Keep the lab tidy—clear up after yourself and ensure that bags and coats do not impede on walkways. • Keep cables and other trailing leads out of walkways and in a tidy manner. • All electrical equipment is checked annually by the Department to make sure it is electrically safe—however you must check all electrical equipment before you use it in case it has been damaged between the annual checks. If you find any damage, inform a demonstrator and do not use the equipment. • Do not attempt to fix any equipment—if it is not working or is damaged, inform a demonstrator. • Never switch off the mains using the master switches mounted on the walls: a temporary power break can do enormous damage to the equipment and to the computers. • Switching on/off the 13 amp mains should be done using the bench sockets in the conventional way. If in doubt, ASK! • It is important that you make a point of reading the “Risk Assessment” sheet issued with each experiment before you start work on the experiment. This will detail specific control measure you need to take and what to do if there is an emergency situation. • There must be NO eating, drinking or using of personal headphones in the laboratory. This last point is not because we dislike your choice of music but because you must remain aware of all activity around you and be able to hear people trying to warn you of problems. • We ask that you arrive at the laboratory wearing closed shoes (not flip-flops or sandals)— this is to protect your feet from falling equipment and debris that may be on the floor—and wear suitable clothing, as laboratories are hazardous places with many risks. Please ensure you do not become a casualty. • A well thought out and planned experiment will not only yield good results but also ensure you and your classmates are kept safe.

1.4

General Experimental Strategy

Our aim in lab is to give you a firm grounding in experimental physics. This requires thinking about the experimental strategy, collecting, recording, and analysing the data, and then writing an appropriate report. The following guidelines apply: 1. Read the relevant section(s) of the laboratory manual before the lab session starts and think about what the experiment is designed to establish. 2. Set up the apparatus and check that it is working as expected. 3. It is often appropriate to collect a set of ‘quick look’ data. Apply these data to any formula needed to obtain the required quantity. Is the answer ’about right’ or have you blundered somewhere? Check the calculations. Think. If there is a problem, return to Step 1. 4. Before collecting your serious data: a) THINK about systematic errors: e.g. is the timebase on your oscilloscope correctly calibrated? b) THINK about random errors: e.g. is the total error sensitive to one particular measurement? Is it insensitive to some other parameter, which does not need to be measured with (time wasting) accuracy? c) THINK about the overall strategy. For example: “should the data be taken at regular intervals in x? Is it better to concentrate on taking data at high values of x or over 8

Introduction to the First Year Laboratory

5. 6. 7. 8.

1.5

as wide a range as possible, or should you concentrate on a region where there is significant non-linear behaviour in your data?” How do you measure or estimate any uncertainties and the effect that they will have on the final result? This will form part of the procedure that you will be carrying out. Write down the strategy in your lab book. It may be quite clear now, but you may have forgotten it by next term and certainly by the following year. Take your definitive sets of data, including the measurements that you need to estimate any uncertainties. Keep a well documented lab book. You will receive a lecture on this topic and it is discussed further in this manual (see “keeping a good lab book”). Analyse your data for the results and for the random uncertainties. The exercises on measurement and uncertainty are bound in this lab manual and a summary of the most frequently used ideas is printed at the back. Write your laboratory report. This will be done outside of laboratory hours.

General Computing Strategy

Our aim in computing is to give you a firm grounding in computational techniques. This requires thinking about program structure, numerical strategy, coding, debugging, testing your programs, using them to produce your results (which are equivalent to experimental data in this context), and analysing the data, and then writing an appropriate report. The following guidelines apply: 1. Read the first section(s) of the laboratory script, and think about what the computing is designed to achieve. 2. Make sure you are familiar with the computing language to be used and how the operating system and development environment work. This may involve some quick and simple exercises. 3. Before embarking on your serious coding: a) THINK about the available numerical techniques and decide what is appropriate (for example, which integration technique you should use), given the level of accuracy you would like to achieve. b) THINK about errors: what type of data (integer, real, double precision, etc.) do you need to use to avoid unacceptable rounding errors and which parts of the computation are these likely to be most problematic? c) THINK about the overall strategy. If you use a particular technique, is the computation going to take weeks to complete? Do you really want to difference two large numbers or is there a better way to preserve accuracy 4. Write down the strategy in your laboratory notebook. It may be quite clear now, but you may forget it by next term or next year. 5. When coding your programs, be sure to include copious useful comments within it. Next week you will have forgotten what you were doing. In a commercial environment coding is often done sequentially by different authors and each need to add a level of comment sufficient to get the next author going both efficiently and effectively. 6. Design your program so that all parts of it can be tested. Give it a good structure and think about how you can progressively test it. Significant results from tests should be recorded in your notebook. 7. Once you are confident the code is working, take your definitive data, analyse the results and check for numerical accuracy and for errors. 8. Write your laboratory report. This will be done outside of laboratory hours. 9

Introduction to the First Year Laboratory

1.6

Laboratory Reports

The purpose of the report is to convey particular information to the reader. It should therefore be written assuming that the reader is a physicist who is not familiar with the experiment (or the computing problem) and who has not read the lab manual. The report should be self-contained, but may have appendices if necessary. The style to be adopted is that of a scientific journal article. You will only be asked to write a laboratory report on a part of the work you have done in each of the long experiments in the first half-unit module. As you will write it retrospectively, your notebook must have sufficient experimental and/or computational detail. Your demonstrator will assign the precise topic at the end of the four-week lab session (see Laboratory Timetable). Do make sure that you fully understand what is required, and if in doubt, ask your demonstrator.

Being Assigned and Submitting Your Report • Your demonstrator will, on the days marked R1 on the timetable, discuss with you a suitable topic to be written up, and the title of the report. • Write the report by the agreed date (R2 on your Timetable), and attach the Report Form. The deadline on the R2 date is 4pm. You must ensure a paper copy and electronic copy are submitted by this time. You must sign the Report Form to certify that the work is your own and complies with the College rules on plagiarism (see copy on noticeboard, and also Chapter 15). • Hand your report in to the staff in the Level 4 Laboratory Office (not to anyone else) and submit an electronic copy via Turnitin∗ . The lab staff will log the arrival time of the paper copy and pass it on for marking. Note that late reports receive an automatic mark of zero—this is a College Regulation—so plan to submit your report well ahead of the deadline † . • On the agreed R3 date, your report will be available for collection in the First Year Lab office. You should read the written remarks on the report and make an appointment with your demonstrator to discuss the report outside of lab hours. The aim of this final feedback session is to help you understand where you can improve your report writing. • To summarise: R1: you are set the topic and title for your report R2: you hand in the report, on paper and via Turnitin before 4pm. R3: the report is returned to you together with verbal and written feedback. Dates and deadlines are printed on the rear cover of the lab manual. Make sure you submit your report well ahead of the 4pm deadline.

1.7

Laboratory Marks and Assessment

Your practical laboratory marks are derived from two sources. Half of the marks are awarded for General Performance in the laboratory, and the other half for the Laboratory Report. Specific submission instructions are subject to change as we improve the procedures. You will be informed if any changes are made. At any time, ask the heads of experiment or lab, or the laboratory technicians if in doubt. † Any computer can be expected to fail at some point, and we all make silly mistakes; so it is your responsibility to ensure that all your work is backed up so that you do not lose your report draft if your computer is broken, or if you accidentally delete the file you are working on—this is also an important professional skill. ∗

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Introduction to the First Year Laboratory The General Performance Mark is based on factors such as your motivation and independence, your comprehension, your results, the quality of your record keeping, and the ‘thoroughness’ of your work. The ‘thoroughness’ does not refer to whether you have got to the end of the experiment or not, but to whether or not you have tried to cut corners, missed out parts of the experiment, etc. It is not necessary to complete every experiment or exercise to obtain good marks. It is generally much better to have done some of the experiments or exercises well rather than to rush through everything. If your progress is very slow, your demonstrator will advise you on how to speed things up or which sections to skip. Your General Performance Mark equals the assessment made by your demonstrators multiplied by your fractional attendance. Be sure to email the Head of Laboratory if you miss lab because of illness or other sound reason! The Laboratory Report Mark is based on the following criteria: a) b) c) d) e)

Does the Report read well? Does it convey all the essential information? Is it logically set out, following the Guidelines in Chapter 14? Is good use made of diagrams? Have the experimental uncertainties been reasonably measured or estimated and correctly combined? f) Are the results, including the errors, reported correctly and with proper justification? g) Are the graphs complete and annotated? h) Does the Report, and especially the conclusions, show insight? Are areas for improvement and further work identified? Once again: If your report is submitted late, it will automatically receive zero marks. This is in accordance with Imperial College policy for late submission and only in very exceptional circumstances can an extension be made. If you find yourself unable to submit your report by the deadline, you should contact your personal tutor urgently. The process for dealing with mitigating circumstances involves extensive documentation and does not guarantee that an allowance or extension can be made. Please ensure that you meet the report submission deadlines. To guard against accidents, we recommend submitting your work well ahead of the 4pm deadline. The report will be assigned a percentage mark, which relate as follows to degree classifications: more than 70% 60-70% 50-60% 40-50% less than 40%

Equivalent to a First Class degree Equivalent to an Upper Second Class degree Equivalent to a Lower Second Class degree Equivalent to a Third Class degree Not at a sufficient level to pass First Year Laboratory

Your marks will be posted onto “Gradebook”, an online repository that you can access from your “Blackboard” account. In addition to this, your personal Tutor will be sent your grades. You will receive the Laboratory Report marks and both written and verbal feedback from your demonstrator once the report is returned on the R3 date.

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Chapter 2 Introduction to the Computing Facilities Contents 2.1 2.2

2.3 2.4

2.5 2.6 2.7

2.8 2.9

Before the End of the First Week . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computing suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logging on and off . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disks and file storage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Printers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic Text Editing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Useful Packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Microsoft Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Styles and formatting . . . . . . . . . . . . . . . . . . . . . . . . . . Equations in Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . Importing graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . Other packages/programs . . . . . . . . . . . . . . . . . . . . . . . . . . . . LaTeX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . World Wide Web . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Email . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From within Imperial using a College computer . . . . . . . . . . . . . . . . . From outside Imperial or using your own computer . . . . . . . . . . . . . . . Blackboard e-Learning System . . . . . . . . . . . . . . . . . . . . . . Using Your Own PC or Laptop . . . . . . . . . . . . . . . . . . . . . . Connecting your laptop to the College network . . . . . . . . . . . . . . . . . Wireless connection . . . . . . . . . . . . . . . . . . . . . . . . . . . Wired (Ethernet) connection . . . . . . . . . . . . . . . . . . . . . . Access when you are away from campus . . . . . . . . . . . . . . . . . . . . Working from your own computer: access to your College files . . . . . . . .

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13 13 13 13 14 15 15 15 16 16 17 17 17 18 18 18 18 19 19 20 20 20 21 21 21 21 22 22

Introduction to the Computing Facilities

2.1

Before the End of the First Week

You should have set up the following to avoid future problems. • You should have set a password on your Imperial College account • You should have checked that the printer is set up correctly on your account • You should have checked that you can send and receive email using your new Imperial College account. • You should have checked that you can access the course material for First Year Computing on Blackboard. The instructions for how to do all these things are found in this document∗ .

2.2

Introduction

This document helps you set yourself up for your new Imperial College account and familiarise yourself with the facilities available. This will not take long for those of you who are familiar with the Microsoft Windows operating system. If you are unfamiliar with this environment, do not hesitate to ask for help from your demonstrators.

Computing suite The computing facilities in the Blackett Laboratory for the use of our undergraduates are located on Level 3. The computing suite is open from 8am to 10pm Monday to Friday and 9am to 10pm on Saturday and Sunday. Access is by use of your College security card. The suite consists of a large room and a smaller annex. The annex is reserved for teaching use during 1st and 2nd year lab hours so that lab sessions are not disrupted by general traffic. Otherwise, you are welcome to use any available PC at any time during opening hours. The machines in the main room of the computing suite have Intel processors with at least 4GB RAM. All PCs are networked and run the same software packages.

Rules College rules must be obeyed when using a College PC. They are found at: http://www3.imperial.ac.uk/secretariat/collegegovernance/provisions/policies/ infosystems/policies/policy2 They are summarised in the Policy statement on misuse of computing facilities at Imperial College, a copy of which is posted in the Computing Suite. Deliberate misuse of College computers is a severe College disciplinary offence and may carry legal penalties. The key points of this policy are: • You can only use a computer when you have permission to do so. • You cannot copy any software whatsoever without permission of the copyright owner. • You must not use the computer system to store, transmit or display offensive material, such as sexual, pornographic, racially offensive or abusive material. If you have any doubt Parts of these instructions date back to the times when many aspects of computing were still novel to many incoming students; it is realised that most students now do not need an introduction to email and the web etc., but sections on such topics have been retained in this manual for completeness. ∗

13

Introduction to the Computing Facilities about any material’s suitability, then do not touch it. Web access is monitored both within and outside our own department. Failure to comply with this rule will result in disciplinary action and could mean removal from Imperial College. (This has happened.) • Under the Data Protection Act, you may not record on a computer system any information concerning living persons without authorisation, unless your use is specifically exempt. In addition to this, we have some local rules concerning the physics cluster: • No games: there is a high demand for the PCs throughout the day • Do NOT share your password with anyone • Do not shut down or switch off PCs: they are updated remotely via the network throughout the day. (When you have finished, simply Log Off.) • No food or drink can be brought into lab. • Do NOT unplug any cables - there is a separate area if you wish to connect a laptop, and Wifi is available throughout the room • Switch off mobile phones. PLEASE!

Getting started A username and temporary password have been set up automatically for you by College Information and Computing Technologies (ICT). If you have not already activated your account, here is how you can do it at the Computing Suite: 1. Enter the username activate and then the password Activate! (Note the Capital A and the exclamation mark in the password.) This will automatically take you to the webpage https://www.imperial.ac.uk/ict/activateaccount 2. Read the Conditions of Use. Click on the box Accept at the bottom of the page to accept that you will abide by these rules when using the Imperial College network. 3. In order to identify you, you will now be asked for your first and last names, your date of birth and your College identifier (CID). Your CID can be found on most official correspondence from College or on your College security card. 4. You will now have to choose a secure password for yourself. Make sure that you choose a password you can remember (even after a long summer holiday!) Resetting a password has to be done manually by authorised ICT personnel. Go to “How to choose a strong password” for advice on how to choose a password that minimises the risk of it being hacked. In particular, the password should • Be much more than 8 characters long • Include at least 3 out of the following 4 categories: lower-case letters, upper-case letters, numbers and special symbols. Weak passwords are rejected by the system. Never tell anyone your password, including personnel from any support desk. 5. When you have decided on your password, enter it twice into the online form. 6. When you have completed the form, click on the Activate My Account button. Your account should be activated automatically immediately. Your new username and your @imperial.ac.uk email address will be displayed on the screen. Write these down somewhere 14

Introduction to the Computing Facilities safe so that you will not lose them. (But do not write down the password in the same note!)

Logging on and off Now that you have activated your account, you can log on to the College network. To log on, hold the Ctrl, Alt and Del keys down at once. [We will use the notation + (e.g. Ctrl + Alt + Del) for simultaneous use of keys.] This will bring up a window which prompts you for your username and password. Enter your username and password and you will be logged into your account. A setup script will run automatically: you will see its progress in a window with a black background. Do not close that window prematurely , otherwise your session will not be set up properly, e.g. you might not have access to any printers. If an error message pops up saying that you will only be using a temporary profile, please bring this to the attention of a demonstrator. We will contact ICT to fix the problem. The screen you see when you log on is called the ‘Desktop’. It contains several useful icons for direct access to frequently used programs. More utilities and software can found if you click on the Start button at the bottom left of the desktop. When you have finished your session, remember that you have to log off from the network. To do so, you also use the Ctrl+Alt+Del keys, but this time choose Log off . You can also use Start followed by Log Off . This will leave it ready for the next user to log on. Do NOT shut down the computer.

Disks and file storage Files are stored on disks (or drives) on each PC and also on the College file server. Click on the My Computer icon located near the top left corner of the desktop. A window will open up showing you the disks that you have access to. The local disk on each PC is the C: disk . You can write files for temporary use to C:\temp\ your_username but these are deleted every evening. You cannot modify files on the C: disk. All files that you want to keep should be put on the H: disk . These are not physically stored on the PC. Instead, they are stored on a central file server in College. This means that you can access your files from any computer in College. When you log on, to any of the PCs in the cluster, you will automatically be linked to your own H: disk. You have a storage allocation of 1GB . You will need to purge files regularly to keep well within this limit. Music and video files take up a lot of disk space, and are generally not needed for your work. If you exceed this quota, you will find that your programs will not run due to lack of disk space.

Software The operating system is Windows 7 Enterprise and you should spend a little time making yourself familiar with it. To find out what programs you can use, click on the Start button at the bottom left corner of the desktop, and then click on All Programs entry. All the packages you need for your coursework can be found here. Word, Excel and Outlook can be found under the Microsoft Office menu.

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Introduction to the Computing Facilities Also accessed through Start are a number of other facilities, including the Search button which you can use to find files. On the left hand side of the “desktop” there are a number of icons including: My Computer: alternate route to disks, files and control panel Internet Explorer: access to the world wide web Outlook: access to email Spend a little time familiarizing yourself with what is available and where to find it.

Printers You should be able to print a file from whatever package you happen to be working on. To do so, you typically choose File | Print (meaning click on File and then Print from the submenu that pops up) from the menu (or click on the printer icon directly). On Microsoft Office 2007 applications, the Print command is found on the drop-down menu that appears when you click on the Microsoft Office logo at the top left corner of the window. The printers you use for computing lab will be the ones in the Level 3 computing suite. You should be able to see the available printers after you have selected Print. • For colour, choose ICTprintservice colour (PCL) • For black and white, ICTprintservice mono (PCL) Then, click OK to the file to the printer. To collect the document, go over to the printer you have selected. Run your Imperial College swipe card through the reader. Press PULL PRINT from the options menu on the printer. Your document should now be listed. Press the Print button to print the file. After the document has printed, make sure that you log out of the printer so that it is ready for the next user. Printing costs 3p per sheet for black & white and 12p for colour. You will receive £5 worth of printing credit free at the start of each year. Subsequently you have to buy credits online or using a machine in the Central Library. You can check your print credit by clicking on the desktop icon “ICTprintservice”. The print quota should be set up by the end of the first week of term. This ICT service in fact allows you to use any printer in College, for example, for high-quality colour printing. For more information on the printing service, see https://www.imperial.ac. uk/ict/printservice/ .

2.3

Basic Text Editing

Much science and computing-related writing involves typing in plain text (ASCII, or more generally, UTF)—this includes programming code, HTML for writing web pages, emails and documention and LaTeX (see below). While more “graphical” editors such as Notepad or TextEdit, which may seem more intuitive at first, can be used to edit plain text, there are many tools that are widely used and incorporate incredibly helpful features that make writing and editing documents much easier.

16

Introduction to the Computing Facilities The most popular examples of such editors are Vim (http://www.vim.org) and Emacs (https: //www.gnu.org/software/emacs/). These are both free and are availble on almost all computing platforms. Because of their popularity, there is an abundance of information, including some excellent tutorials∗ , available online. While it is not imperative that you learn how to use such a tool—and they are a little more difficult to get started with than tools aimed at the layperson—it would be a good investment of time if you were to spend a weekend or two acquainting yourself with a good text editor.

2.4

Useful Packages

Many of the packages you will use are in the integrated suite called Microsoft Office. We are now using the 2010 edition of this software. This section contains very brief introductions to Word for word processing. Use of Origin for presenting results will be discussed during the computing lab session on Measurements and Errors.

Microsoft Word MS Word is a word processing program with which you can create reports, letters and documents. Its functionality is integrated with other MS Office programs. For example, you can take an Excel graph and insert it into a Word document. Word is recommended for lab reports. You can find Microsoft Word under the Start menu. As usual, you double click on the icon to open the program. Word opens a blank document into which you may start typing your report. You can get to the commands to save and print the document and to open new documents by clicking on the Microsoft Office logo at the top left corner of the window. There are other tabs for different menus across the top of the window. Under Home, you have the Clipboard where copy and paste functions are found. You also have the Font, Paragraph and Style menus for formatting text (see below). Under Insert, you find functions to insert tables, figures, cross-references, etc . Under Review, you can find the spell checking facility. Styles and formatting One of the few things you must know right away in Word is the way it handles paragraphs. If you type a carriage return, the program thinks you are starting a new paragraph and will indent and space out the new paragraph according to the present paragraph settings. You should only type returns when you really want to have this sort of formatting; otherwise, just keep on typing and Word will sort out newlines automatically for you. Formatting the paragraph can either be done manually by clicking on Home, choosing the options under Paragraph as appropriate or by clicking on a choice of established styles in the Styles box. This will probably start off with the Normal or Plain Text style but the dropdown menu offers numerous others including different ranks of headings which can make a document much clearer. If you construct a style you like and want to save you can do this using Format then Style then New Style giving it a name you can remember. If you use the automatic numbering of headings or of bullets (see toolbar) then you can add in new items and delete others and the lists will be automatically re-numbered. Using the functions under Home | Font you can change fonts, italicize words, make them bold, make them super- or sub-scripts etc. There are many options. The best way to learn is to try ∗

One highly accessible example, as of this writing, is http://www.openvim.com/tutorial.html.

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Introduction to the Computing Facilities them out and use the Get Started facility. If you see a word underlined in a wiggly red line, it thinks you have misspelt something. You can try its automatic correction facility (click the right mouse button with the cursor over the word) but beware: it sometimes makes embarrassingly wrong choices. You may also want to set the spell checker to use UK instead of US spellings. This is found under Review | Proofing | Spelling & Grammar. Equations in Word If you want to insert an equation that is anything more complicated than E = mc 2 then the Equation Editor comes in useful. To use this facility click on Insert then Object then Microsoft Equation; you can then type into the box, using any symbols required from the drop-down menus. Importing graphics There are many ways to import graphics into your Word document. You can use Insert | Picture. Once inserted you can reshape, resize and reposition your picture by clicking and dragging the handles on the corners/edges of the plot.

Other packages/programs A number of other packages are installed on the PCs including: Origin: for manipulating data and presenting results as graphs or tables. PowerPoint: for presentations. MatLab, Mathematica maths packages and graphics. Adobe Reader: for reading documents written in portable document format (pdf files). Ghostscript: for reading files written in postscript (.ps files). Python: Enthought Python Distribution. GSL: GNU numerical library (routines can be incorporated into C++.) Firefox: alternative open source web browser. Most of these have online help facilities if you wish to teach yourself how to use them. The software provision is updated from time to time, so this list may be incomplete.

2.5

LaTeX

While Microsoft Word is a commonly-used package for writing documents, for professional scientific writing, and especially in physics, LaTeX is often considered a superior typesetting system∗ . LaTeX is free and is available on almost any computing platform, with the basic files being written plain text. Therefore there are no issues of document format compatibility, and editing can make the use of all the features that advanced text editors (see Sec. 2.3 possess. The format allows you to focus on the content of your document, rather than its appearance, as the way it is typeset can be adjusted easily once it is written, unlike with most other word processors. ∗

This laboratory manual is typeset entirely in LaTeX.

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Introduction to the Computing Facilities Most importantly, the typesetting quality of LaTeX is at the level of that from a professional publishing organisation, in particular for mathematical equations. In first year lab, we provide optional LaTeX templates for report writing to help you get started.

2.6

World Wide Web

The World Wide Web has become an essential part of scientific life: not too surprising, given that it was invented by physicists∗ ! We publish scientific reports and share our results on the web. Most scientific journals have online editions and the College library subscribes to them. We do not prohibit non-science usage on the College network. However, we discourage excessive use in the computing suite especially during busy times. Of course, you should not use the network inappropriately in violation of the Conditions of Use [see Section 2.2] to which you have agreed when you activated your account. To browse the web, double click on the Internet Explorer browser icon on the Desktop (you can also use Firefox which can be found inside the programme menu under Mozilla Firefox). The first page you find is your home page, probably the Physics Department’s home page. It has an address (or URL for “universal resource locator”) with the name http://www.imperial. ac.uk/physics/ . You can jump to other pages by clicking on highlighted words (“links”) or on the horizontal menu items near the top of the page. Click on some of the items and learn more about your department. You can add the names of sites you need to access often to your Favorites (see menu bar) to avoid having to remember the names of sites that you visit frequently. If you know the URL of a site you want, you can type it in the Address box and press Return. Some useful sites are: ICT information for new students: http://www.imperial.ac.uk/ict/services/newstudents ICT software for students: http://www.imperial.ac.uk/ict/services/ personalcomputersupportandmobileservices/softwarepurchase Institute of Physics Physics World :http://physicsworld.com/ Google search engine: http://www.google.co.uk/ DuckDuckGo search engine (much better privacy than Google): http://www.duckduckgo.com/ Imperial College website: http://www.imperial.ac.uk/ You can access all areas of the Imperial College website from anywhere. Some pages, such as those containing teaching material and security information, are restricted for College members only. You will be prompted for your Imperial username and password when accessing these pages.

2.7

Email

This is now the principal for communication in College. The teaching staff (e.g. the senior tutor, your academic and personal tutors, demonstrators, heads of lab) will use this to send you important information, e.g. scheduling tutorials, assessment meetings, etc. ∗

See http://home.web.cern.ch/about/topics/birth-web/where-web-was-born.

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Introduction to the Computing Facilities

IMPORTANT: Check your College email account regularly. You can keep 500MB of email messages on the College mail server (separate from the 1GB quota for files stored in the H: disk).

From within Imperial using a College computer You can use Microsoft Outlook for your email. This is located under the Microsoft Office menu under the Start menu. [NB: Do not use Outlook Express]. A list of messages received in your Inbox will be automatically displayed. You can read a message by double-clicking it. To delete or reply to messages or to write new ones, use the icons on the toolbar. Send yourself an email to make sure that your new email address works!

From outside Imperial or using your own computer There are two ways to access your email from outside College. If you are not an experienced user of email applications, use the first method: it is simplest and safest. 1. Use the web-based email interface of the College mail server. Find any PC with a web browser and go to the website https://icex.imperial.ac.uk. Type your College email address without the domain name (i.e. leave out the @imperial.ac.uk) in the box and then, when prompted, your College username and password. • Many email client applications will, by default, download the messages from the mail server onto your PC and then remove them from the server. This means that they will not be there when you next check your messages on a College computer! This default setting should be changed so that the mail application leaves messages on the server . (In technical terms, this issue applies to the POP3 protocol, but not the IMAP protocol.) 2. With your personal computer, you can set up any email client to access the College mail server. You need to have administrative privileges on the PC to set this up. Different email applications have different setup procedures. The key information you need is that the College mail server is located at the address exchange.imperial.ac.uk . It supports the POP3 and IMAP protocols for how email applications communicate with mail servers. To find out more, look up the web page for the computer centre (ICT) at http://www.imperial.ac.uk/ict/ and follow the link to Services | User, Email, File and Directory Services | Email . Be careful about a couple of things. • You should switch on a security feature on your email application that would encrypt your College password so that it cannot be “sniffed” as it travels across the internet. This feature is usually referred to as SSL or TLS, and there should a tick box to enable this security feature in some Preferences or Configuration menu.

2.8

Blackboard e-Learning System

Most of our teaching material is on the Blackboard system (formerly WebCT) at http://learn. imperial.ac.uk/ . This is a virtual learning environment. It contains material for your courses, such as lecture notes, problem sheets, solutions and online quizzes. There is also a facility to communicate with your lecturer and fellow students. Information can be found at: http://www.imperial.ac.uk/ict/services/teachingandresearchservices/elearning/ vle/forstudents/ 20

Introduction to the Computing Facilities IMPORTANT: Make sure that you can log on to Blackboard and that you can see the course material for Year 1 Computing. If not, tell your demonstrator.

2.9

Using Your Own PC or Laptop

It is not necessary to own a PC or laptop to carry out all the work required for the physics degree. It can, of course, give you the freedom to work on programs, reports etc. in your own time. However, for the formal taught computing labs, you are required to spend the appropriate sessions in the computing lab so that your progress can be monitored and assessed by demonstrators. NB: lab marks are weighted by attendance. If you need to buy software for your machine, there are discounted offers for students available through the College ICT shop, see http://www.imperial.ac.uk/ict/services/software/ shop/ . In particular, you can obtain Symantec anti-virus software for free.

Connecting your laptop to the College network When you are on campus, you can connect your computer to the Imperial College network (which includes the halls of residence) using a wireless connection or an ethernet connection. The information can be found at the ICT web pages http://www.imperial.ac.uk/ict/ : follow the links to Services | Security, Network, Data Centre and Telephony Services | College Network | Network Connections . The following is a brief summary. You need to register your laptop/PC with ICT. This is to ensure that only members of College have access to the College network. For further information see http://www3.imperial.ac. uk/ict/services/networks/networkconnections . Wireless connection You will need a wireless network card (802.11b or 802.11g). The /section network name (SSID) will appear as Imperial-WPA . (Do not log on to any wireless networks with SSIDs that you do not recognise: they may be illegal networks that will collect all your passwords!) It will prompt you for your Imperial username and password to authenticate you as an authorised user of the network. Wired (Ethernet) connection Your computer should have a built-in network card. You will need an ethernet cable which can be purchased from the Union Shop. You should set up the machine to obtain its IP address automatically (from a College DHCP server) and to obtain the DNS server address automatically. On Windows 7, you can set this up as follows: Start, Control Panel, Network and Internet, Network and Sharing Centre, Local Area Connection, Properties, select Internet Protocol Version 4 (TCP/IPv4), Properties, then check the boxes marked Obtain IP address automatically and Obtain DNS server address automatically . [NB if you previously had any other IP/DNS settings which you might need to use again, take a note of them before carrying out the above procedure as they will be removed. However, most internet service providers will assign these addresses automatically.] Non-Windows machines can also be connected to the network. Instructions for Linux and MacOS can be found on the ICT webpages.

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Introduction to the Computing Facilities

Access when you are away from campus If you are not on campus and you need to access files stored on the College network, you will need to establish a Virtual Private Network (VPN) connection to College. For details, go to http://www.imperial.ac.uk/ict/, and follow the links to Services | Security, Network, Data Centre and Telephony Services | College Network . This will allow your computer to behave as if it is on the College network (even though you may be half way across the world!) With this connection, you can • access your College files using the “Add a Network Place” method described in Section 2.9 • access Imperial College web pages that are restricted to College members • access services restricted to Imperial College members, such as electronic journals to which the College library has subscribed. Note that, while the VPN connection is up, the Conditions of Use of the College network applies to all the internet traffic of your PC because it will be routed through the College network. Non-Windows machines can also be connected to the network from outside College. Instructions for Linux and MacOS can be found on the ICT webpages.

Working from your own computer: access to your College files You would often want to transfer your reports, programs, results, reports etc. between the College network and your personal computer. There are many ways to do this. For small files, you can simply send them to yourself by email. (You can check College emails from outside College, see Section 2.7 above.) For files larger than 1MB, you can copy them onto a removable disk, such as a USB storage device (“memory stick”). These methods, although inelegant, have the advantage that they are simple, reliable and need no further setup. If you are using a Windows machine, you can use “Add a Network Place” to access the College file servers directly. You can find the name of the file server where your files are stored in the Network Drives listing of My Computer when you log in on a College computer. It should be something like: \\icfs6.cc.ic.ac.uk\your_username . You should be able to make use of this facility if your computer is on the College network. If you are outside College, then you need to establish a VPN connection first (see Section 2.9).

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Chapter 3 Measurements and Uncertainties Exercises Contents 3.1 3.2

Introduction . . . . . . . . . . . . . . . . . Means, Standard Deviations, and Standard Mean . . . . . . . . . . . . . . . . . . . . . . . Sample standard deviation . . . . . . . . . . . . Standard error of the mean . . . . . . . . . . . Use of Origin . . . . . . . . . . . . . . . . . . . 3.3 Propagation of Errors . . . . . . . . . . . . Multiplying factors . . . . . . . . . . . . . . . . Powers . . . . . . . . . . . . . . . . . . . . . . 3.4 Combinations of Errors . . . . . . . . . . . Sums and differences . . . . . . . . . . . . . . Products . . . . . . . . . . . . . . . . . . . . . 3.5 Plotting Graphs and Line-Fitting . . . . . . Plotting . . . . . . . . . . . . . . . . . . . . . Straight line fits . . . . . . . . . . . . . . . . . Multiple datasets . . . . . . . . . . . . . . . . 3.6 Practical Example . . . . . . . . . . . . . . 3.7 Non-Linear Fits . . . . . . . . . . . . . . . 3.8 Histograms . . . . . . . . . . . . . . . . . . 3.9 Bibliography . . . . . . . . . . . . . . . . . Website . . . . . . . . . . . . . . . . . . . . . Textbooks . . . . . . . . . . . . . . . . . . . . Appendix 3.A Solutions to Exercises . . . . . .

3.1

. . . . . . . . . . . . . . . Errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 26 26 27 28 29 29 29 30 30 31 31 32 32 33 35 35 36 36 36 36 38

Introduction

The purpose of this session is to help you to gain experience of the correct treatment of experimental uncertainties (errors) in the laboratory. Some of the exercises in this session will make use of your calculator, as this is what you are likely to use while you are working in the laboratory. Some of the exercises also will make use of Origin® for the calculations. This software package is similar to Excel but its provisions of mathematical analysis and graphing make it a 23

Measurements and Uncertainties Exercises better choice for scientific (rather than business) applications. This three-hour session follows on from the introductory lecture on Measurements and Errors and is designed to reinforce the ideas introduced in the lecture with straightforward practical examples. If you are unsure about anything in this document, discuss it with your fellow students, as it often helps to collaborate to absorb concepts such as these. You should also feel free to ask a demonstrator—they are there to help guide you as you figure out new concepts. Most of the mathematics used for data analysis and error handling is fairly straightforward, but it is important to practise applying these ideas. Data analysis and error handling are practical skills which are used constantly in the laboratory. Practical work is not just about collecting good data but also about the correct analysis of the data and extraction of results from it. This includes the determination of a final error - which tells you how reliable the results are and how they can be used. When you are working in the laboratory, and presenting your results in laboratory reports, you will always be expected to treat experimental uncertainties properly and to show how you have arrived at your final quoted range of values. The exercises you will go through in this session should introduce you to all the ideas that you will need in your first year lab work for the presentation and analysis of data. When working in the laboratory, you will often need to use your calculator for simple calculations. (We recommend Casio fx-83ES.) For example, you will need to check that your measurements make sense and are giving roughly the expected results. Or you may need to estimate the uncertainty of a set of measurements to check how accurately you are able to make the measurements. At a later stage you will analyse your measurements using an Origin® spreadsheet in order to make further calculations (e.g. calculating means and standard deviations of sets of data), and to present the data in tables, plot graphs, and extract slopes and intercepts together with their errors. Important points to remember 1. A measurement without units is meaningless. 2. A measurement without an estimated range of uncertainty (or error ) is also meaningless. 3. Always quote the final result of a measurement in the form a ± b, with a (measurement) and b (error) given to the same number of decimal places. For example, a measurement of the speed of light might give: c = (2.94 ± 0.04) × 108 m s−1 . This expression means that the true value is likely (the probability being approximately 68%, see below) to lie between 2.90 × 108 m s−1 and 2.98 × 108 m s−1 ,. i.e. if you repeat the measurement a large number of times about 68% (i.e. two-thirds) of the results will lie between 2.90 × 108 m s−1 and 2.98 × 108 m s−1 . 4. Errors are usually only known roughly, so quote your errors to no more than two significant figures. One significant figure is usually sufficient. (i.e. 2.94 ± 0.04, or possibly 2.942 ± 0.038, but not 2.94217 ± 0.03794).

3.2

Means, Standard Deviations, and Standard Errors

Often in an experiment you will make several measurements of a quantity. By taking the average of many measurements, the effect of random errors can be reduced because errors of opposite signs tend to cancel each other. Furthermore, the precision of the final result may be estimated by looking at the spread of the data. 24

Measurements and Uncertainties Exercises NB. taking multiple measurements cannot give information on any systematic errors which may be present. When a very large number of measurements are made, the spread of results about the mean value, if determined by random fluctuations, will be represented by a Normal Distribution (also called the Gaussian distribution). This gives the probability P(x) dx

(3.1)

of an individual measurement lying between x and x + dx. The distribution plotted in Fig. 1 has a mean of x = 0, and a standard deviation of σ = 1, although this will not be the case generally. It is always the case, however, that it is normalised so that the total area under the curve is unity (i.e. a measurement is certain to lie somewhere).

Figure 3.1: The Normal (or Gaussian) Distribution P(x) dx. The general form of the Normal Distribution (see Fig. 3.1) is (x − x)2 1 P(x) = √ exp − . 2σ2 σ 2π

(3.2)

In this example, x = 0 and σ = 1. Two important values related to the Normal Distribution are: • about 68% of the measurements are within ±1 standard deviation of the mean • about 95% of the measurements are within ±2 standard deviations of the mean. Example Here is a set of values obtained for the wavelength of a sound wave: 25

Measurements and Uncertainties Exercises 0.76 m, 0.79 m, 0.84 m, 0.75 m, 0.80 m, 0.79 m We write these as x1 , x2 , x3 , x4 , x5 , and x6 , and the whole set can be referred to as xi with the suffix i = 1, 2, · · · , N where the number of measurements made N = 6 in this case. There are three important quantities that we can evaluate from this set of measurements: the mean, the sample standard deviation and the standard error of the mean.

Mean This is the average of the measurements, and is the best estimate of the true value of the quantity that is being measured. It is written as x and is calculated from 1 X xi x= N i=1 N

(3.3)

Exercise 1 Using your calculator you can find this either by adding the numbers together and dividing by 6, or by using the calculator’s built-in statistical functions to calculate the mean directly. You must know how to use both ways reliably. [Find online the manual for your calculator now if you do not know how to use its statistical functions.] Whichever way you calculate it you should find the value 0.788 m. Note that the data do not justify any more precision than is implied by giving 3 decimal places so we do not, for example, write the result as 0.788333 m. [NB. Answers to Exercises are given at the end of this document.]

Sample standard deviation The standard deviation σ is a measure of the spread of the measurements and is defined as the square root of the mean square deviation. For a sample of N measurements the best estimate of σ is given by s (sometimes written as σn−1 ) in the formula  !2  N N X X 1  1 s2 = x2i − xi  (3.4) N − 1 i=1 N i=1 [See lecture for derivation]. Again, you can calculate s manually using your calculator, but it is much easier to calculate it using the built-in statistical functions. Note that unless the number of measurements is rather large, the range of values of s that come out of separate sets of measurements is wider than you would expect, so s is never very accurately determined in real measurements (see Exercise 3 below for examples of this). As we will see, the exact size of the errors are never very well known, so rough calculations with error values are generally acceptable. Use your calculator to find out the value of s for the data above. You should find that s (or σn−1 ) = 0.03 m. 26

Measurements and Uncertainties Exercises

Standard error of the mean We have calculated the mean x which is an estimate of the true value of the quantity we are measuring. How accurate is this estimate? The sample standard deviation s is not a measure of the error in this estimate. We need the Standard Error of the Mean, written as σm , which tells you the accuracy with which the mean of the data points gives the true value of the quantity you are measuring (which we will call x0 ). As you take more and more measurements, the standard deviation of the measurements stays the same (the behaviour of the apparatus is not changing) but the accuracy with which they determine the true value of x0 improves. This is because combining more measurements has the effect of reducing random errors by averaging. The formula for σm is

s σm = √ . N

(3.5)

The quantity σm falls as √1N compared to the sample standard deviation s so, for example, in order to double the accuracy of your estimate for x0 , you have to take four times as many data points ∗ [again remember that this does not affect any systematic errors]. Calculate σm using this formula for the data above. You should find the value 0.013 m. Summarising the results above, for the data set given, you would quote the following for your best estimate of the true value of the wavelength and its error: λ = 0.788 ± 0.013 m.

(3.6)

Quoting the standard error is a more quantitative way to describe the accuracy than simply using an appropriate number of decimal places. It would also be acceptable to write this as: λ = 0.79 ± 0.01 m.

(3.7)

Note that this value is quoted with limits of ±σm . This means that the true value is likely (68%) to lie within the range λ − σm to λ + σm . However, about 33% (one-third) of the time it will lie outside this range. Sometimes errors are quoted with the value ±2σm when the author wants to give a limit on the range of possible values rather than an indication of the likely error. This is sometimes referred to as “95% confidence level” as for large N, 95% of all measurements lie within ±2σ of the mean. If you do this, you should always state clearly that the uncertainty quoted is two standard errors. Exercise 2 The following measurements are obtained in an experiment to measure the speed of light (all in units of 108 ms−1 ): 2.95, 3.04, 2.98, 3.10, 3.01, 2.89, 2.96, 2.47, 3.02, 2.97 Find the values of x, s and σm for this set of data. How would you quote the final result? Is it consistent with the true value of the speed of light? Even if the sample data do not follow a Normal Distribution, the Central Limit Theorem tells us that the statistics of many samples will follow a Normal Distribution with σm as the standard deviation: more on this in the second-year course on Statistics and Probability. ∗

27

Measurements and Uncertainties Exercises Now look at the data more carefully, and you should notice that one of the data points is a long way away from the others. This might be an indication that a mistake was made in taking or writing down this data point. Because this data point is well outside the range of the others (it is about three standard deviations away), you might be justified in discarding this point and redoing the calculation. How does that affect the results?

Use of Origin In laboratory experiments you will often need to plot graphs of your results and for this we use Origin® . This software also provides a multitude of data analysis tools. Here you will need to use a relatively small number of its features but it is advised that you familiarise yourself with its features as many will be required in your lab work during the year. A short introductory guide to Origin® is provided in your lab handbook. Start Origin® and enter the data from Exercise 2 into a column. Put appropriate text into the Long Name and Units boxes. Statistics on the data in a column can be calculated as follows: • • • • •

Select the column Click on Statistics in the menu bar then Descriptive Statistics Statistics on Columns Open Dialog

A window will open with many options. You should see in the Data Range box a reference to the column you have chosen. In the Quantities to Compute/Moments select as many parameters as you find interesting but include Mean, Standard Deviation and SE of Mean. Click OK. If you have used the default Output Settings a new sheet (called DescStatsOnCols 1) will appear. On it you will find a Table containing the values requested. Check that the values are the same as you found using your calculator [NB. Standard Deviation in Origin® gives the sample value s not the population value σ]. Repeat the calculation with different settings; this is easily accomplished by clicking the small padlock icon on the results sheet and selecting Change Parameters. For the rest of the exercises here you will need to load the Origin® files Exercise3 etc which can be found on Blackboard (http://learn.imperial.ac.uk) under the “Year 1 Laboratory and Computing, Measurements and Uncertainty” section. Exercise 3 In the data file you will find five sets of data, each having twenty points. These numbers were generated using a random number generator and have a normal distribution with a fixed mean and standard deviation. Calculate the mean and standard deviation of each of the samples separately, this is easily achieved by selecting all five columns on the data sheet and when choosing parameters for the Statistics on Columns, make sure that Independent Columns is chosen in the Input Data section. Do the means of each set come out close to each other? Are the standard deviations of each set the same? What is the standard error of the mean of each set? Are the differences between the means of all the sets similar to the values of the calculated standard errors? Think hard about the results that you get and whether they correspond to what you expect to see.Finally, find the value for the mean, standard deviation and standard error of the complete data set. To do 28

Measurements and Uncertainties Exercises this select all five columns on the data sheet and select Combined as Single Dataset rather than Independent Columns. How would you quote a final result for this complete set of measurements? You will see that the standard error here is much less than the standard deviation of the data points. Note the benefit of using a relatively large number of data points. However, in real experiments you have to consider two other factors in deciding how many data points to take: 1. Is the effort involved in taking lots of data points worthwhile given that the relative gain gets slower as more points are taken? 2. Are there other sources of error (either random or systematic) in the experiment that will then dominate the final result so that there is no point in further reducing the random error on the measurement of this particular quantity? When you have got to this point, discuss your conclusions with a demonstrator, to check your understanding of the material so far, before moving on.

3.3

Propagation of Errors

In many cases it is necessary to find the value of a function of the quantity actually measured. For example, if you measure the radius r of a circular object but want to know the area (call it A), then you need to calculate the quantity A = πr2 . We therefore need to know the error in A given the error in r. There is a general rule which covers all these cases: the error σz in z when z(x) is a function of x is given, for small errors, by: dz σx (3.8) σz = dx x=x where σx is the error in x and the gradient is evaluated where x = x.

Multiplying factors Exercise 4 We take the function z = 2x as a simple example. Using the column of values in the Origin® file Exercise4, find the mean and standard deviation of the values given for x . Now use Set Column Values (as described in the Performing calculations on data section of Introduction to Origin® document) to set up a column containing the values of 2x and find the mean and standard deviation of these values. Are your results consistent with the general expression above? Note that the same√formula holds for the standard error (σm ) as well as the standard deviation, as the factor of 1/ N is the same in both cases. We write the error in a quantity x simply as σx . Note that in this context the source of the error estimate is irrelevant: it may refer to a standard (random) error derived from a set of measurements, or it may refer to an estimate for the systematic error in x. The mathematics for the propagation and combination of errors is the same in both cases.

Powers Now consider the case of z = x2 :

29

Measurements and Uncertainties Exercises • Will the mean of z be the same as the square of the mean of x? • What do you predict for the relationship between σz and σx ? • Make a column of the values of x2 and then find the mean and standard error. Are the results what you predicted? Using the general expression above we can write (approximately, for small σ) in this case: σ  σ  x z =2 (3.9) z x which shows that the fractional error in z is equal to m times the fractional error in x. We will see later that fractional errors (i.e. percentage errors) are often useful in error calculations. Check that this formula gives the same result as found above. Exercise 5 You are now in a position to find the error in a slightly more complicated case by combining the ideas earlier in this section: Consider how the error in an estimate of the volume V of a sphere depends on the error in measurement of its radius r. [Remember V = 4πr3 /3]. If the measurements of the radius have a mean of 24.7 cm and a standard error of 0.8 cm. What are the best values of V and its error?

3.4

Combinations of Errors

Another important topic to consider is how to evaluate the error in something that depends on the measured values of two different quantities. For example, to get the length of an object you might measure the position of one end of it on a scale and then measure the position of the other end on the same scale. The length is the difference of the two values, but what is the error in the length, given the errors in the position of the two ends? There is a general rule for combining errors. If z is a function of x and y the error σz in z(x, y) is given by: ! 12  2  2 ∂z ∂z (3.10) σ2 + σ2 σz = ∂x x=x x ∂y x=x y y=y

y=y

provided that x and y are independent quantities in the sense that the random errors in measuring them are not correlated with each other. ∂z NB: The curly d in, for example, ∂x implies partial differentiation. This means you differentiate the function z with respect to x while keeping all other variables (y in this case) constant.

Sums and differences Supposing z = x + y; show that the formula above would suggest: σ2z = σ2x + σ2y .

(3.11)

This is called “adding the errors in quadrature”. Note that the same equation for σz applies in the case z = x − y as with z = x + y. [NB. The extension to three or more variables is straightforward.]

30

Measurements and Uncertainties Exercises Exercise 6 An example is given in the Origin® sheet Exercise6 which gives a set of values for a quantity x and a separate set of values for a quantity y. Introduce new columns for x + y and x − y into the table and then find the means, standard deviations and standard errors for x, y, x + y and x − y. Are your results consistent with the theory above? It is sometimes the case that one error is significantly larger than the other, and if this is true we find that the final error is virtually equal to the largest contributing error. For example, if σy = 0.3σx , then for z = x ± y, σz ≈ 1.04 σx . In this case we are justified in ignoring the effect of any error in y contributing to the final error in z. It is important to bear this in mind—do not bother doing calculations that are unnecessary because the errors are dominated by one source. Another important situation where errors are added in quadrature is when a set of measurements have both systematic and random errors.

Products The next case to consider is when z = xy. Show that in this case the general formula gives the approximate result:  σ 2  σ 2  σ 2 x y z = + (3.12) z x y In other words, the fractional errors add in quadrature. [Again the extension to three or more terms is straightforward]. Note that the same formula applies when z = x/y as for z = xy. Find the standard deviations and standard errors for z = xy and z = x/y using the data in the table and see how your results compare with those given by the above formula. Again, if one fractional error is much larger than the other, then the final error is just determined by the larger of the two. It is often useful to express fractional errors as a percentage, and then if one error is more than about three times the other, you can assume that the final fractional error is just equal to the larger of the two contributing errors. Exercise 7 The refractive index of a material determines the ratio of the angle of incidence θi to the angle of refraction θr of light as it crosses the material boundary [Snell’s Law n = sin θi / sin θr ]. Use the General Rule above to find an expression for σ2n /n2 in terms of the means and errors in θi and θr . If measurements of θi and θr have means of π/6 and π/8 respectively and the standard error of each is π/200 what are the best values of n and its error? Discuss your conclusions for Exercises 6 & 7 with a demonstrator before moving on.

3.5

Plotting Graphs and Line-Fitting

The aim of this section is to develop good practice in presenting experimental results as graphs. 31

Measurements and Uncertainties Exercises

Plotting Exercise You need the Origin® file Exercise8 from Blackboard. The first column is temperature (in degrees Kelvin) and the second and third columns the resistivity∗ of copper and aluminium, respectively, measured at each temperature. Select the first two columns of data then use Plot, Symbol, Scatter to produce a graph of the copper values. Following the instructions in the Introduction to Origin® document, now add error bars of 1 K in temperature and 5% in resistivity. Double click on a data point to edit style, size and colour of the points. In the top left hand corner of the plot you will see a small grey box with the number 1. This indicates that this plot only has one layer. Double-clicking on this produces a box indicating the data that have been used; changes to the input data can be made by choosing different parameters in the box. Plots may have more than one layer (up to a maximum of 121); this is useful, for example, in cases where different datasets (with different units/scale) need to be plotted on one graph.

Straight line fits Often with physics experiments we want to test a hypothesis that predicts that the data will behave in a way that conforms to a mathematical model. A simple model might predict that two variables have a linear relationship, so that a plot of one against the other would be a straight line. In all cases we can test the hypothesis by seeing how closely the actual measurements match the predicted line. Now add a straight line fit to your resistivity plot using Analysis, Fitting, and Linear Fit. In the Dialog Box there are lots of options for the fitting and for output. A Table will appear on the plot giving various measures of the fit, as selected in the options. In the top left corner, below the grey layer number box, will appear a small padlock symbol: clicking on this will reveal a menu including the option to Change Parameters in the fitting options which is very helpful for quick adjustments; try some out. As well as the summary statistics box on the plot the fitting procedure can produce a lot more information on a different worksheet. An example, for the Cu resistivity data, is given in Figure 2. Origin® gives results to 5 or more significant figures. It does not recognise that when, for example, we write a value 1.76×−8 Ωm for resistivity we do not measure it to be 1.760000×−8 Ωm. When you use the results you should generally quote the error to not more than 1 or 2 significant figures, and the main result to the same number of decimal places. Thus from Fig. 3.2 we would deduce that the intercept of the resistivity curve is at (−2.31 ± 0.75) × 10−9 Ωm. Some of the most useful information that can be derived/output are: • Confirmation of the Input Data used in the calculation. The resistivity ρ of an electrically-conducting material is defined as the constant of proportionality in the relationship between the resistance R of a wire made of the material, its length L and its cross-sectional area A : R = ρL/A. Its unit is the ohm-metre (Ω m). ∗

32

Measurements and Uncertainties Exercises • The number of degrees of freedom (DF) in the calculation∗ • The values derived for the slope m and intercept c of the line, along with the standard errors of those values and the 95% confidence limits† . • The correlation r between the x and y values. • The Residual Sum of Squares (RSS) which is the sum of squares of the differences between the points and the fit line (see p also the bottom plot in Fig. 3.2). • Root Mean Square Error = RSS/DF (of the fit). • Total Sum of Squares (TSS) which is the sum of squares of the differences between the points and the mean value (determines the standard deviation of the data, not the fit). • The R2 value (coefficient of determination): R2 =

TSS − RSS RSS Explained variation = =1− . Total variation TSS TSS

(3.13)

Its value usually lies between 0 and 1, where 1 indicates a perfect fit and 0 indicates that the data is so far from conforming to a straight line that none can be determined. Note that a high value of R2 may indicate a low value of random error but it says nothing about the presence of systematic error. Take a look at these values in your output and make sure you understand how they relate to the data and fit. There is an option to force the line to go through a particular intercept value (Fix Intercept) but think carefully before using this. Supposing theory says a line should go through the origin: the presence of a systematic error in the measurements will be revealed by the line missing the origin. Not only that but by forcing the line to go through the origin you would make it have the wrong slope. Try this out on the resistivity data and see how the values of m and c respond.

Multiple datasets You can add a second (or subsequent) curve to an existing plot. Return to the data sheet and add a column giving 5% for the error in the Aluminium values. Select the 4 columns necessary to plot the Al data with error bars. Now return to the Graph sheet and choose Graph, Add Plot to Layer, Scatter. Edit symbol size etc as for copper data. Remove the exponent values (E-08) on the y axis by double-clicking the y axis and choosing Tick Labels, Divide by Factor and choosing an appropriate value. Edit the y axis title correspondingly (double-click on its box and use the Format Toolbar. Another useful facility is the Symbol Map which you can open by right-clicking on the box once opened). You may also need to edit the legend which you can do either by double-clicking its box or using Graph, New Legend. You should end up with a plot looking something like Fig. 3.3. This is the number of data points n minus the number of parameters p to be fitted, which in this case is 2 viz m and c. † For a large number of degrees of freedom (i.e. data points) a range of ±1 standard error is 68% likely to contain the correct value and ±2 standard errors is 95%. These percentages are reduced for a smaller number of degrees of freedom. ∗

33

Measurements and Uncertainties Exercises

Figure 3.2: Results from simple linear regression of Cu resistivity data.

34

Measurements and Uncertainties Exercises

3.5

Equation

y = a + b*x

Weight

Instrumental

Residual Sum of Squares

0.47213

Pearson's r

0.99864 0.99688

Adj. R-Square

Value Resistivity Al Resistivity Al

Slope

Standard Error

-4.03546E-9

5.63119E-10

1.0868E-10

2.14756E-12

Resistivity Al Resistivity Cu Linear fit Al Linear fit Cu

2.5

8

Resistivity (10 ! m)

3.0

Intercept

2.0

1.5

y = a + b*x

Weight

Instrumental

Residual Sum of Squares

2.17754

Pearson's r

0.99343

Adj. R-Square

0.98503 Value

1.0 180

Equation

200

220

240

260

280

Standard Error

Resistivity Cu

Intercept

-2.12656E-9

7.46385E-10

Resistivity Cu

Slope

6.52305E-11

2.84076E-12

300

320

340

360

380

Temperature (K)

Figure 3.3: Origin® plot of resistivity as a function of temperature for copper and aluminium with error bars, linear fits and derived statistics.

3.6

Practical Example

Imagine an experiment where the resistance of a pair of identical resistors is to be found. A measurement is made of the voltage difference across the two resistors and the current running through them is also measured. We expect the resistance of each resistor to be described the equation: 1 V1 − V2 (3.14) R= 2 I where V1 and V2 are the voltages at the two ends of the resistors and I is the current through them. Exercise 9 Consider two approaches to finding the value of R.: 1. We take one measurement of each of V1 , V2 and I and accept the equipment manufacturer’s error estimates giving the following values: V1 = 6.9 ± 0.5 V, V2 = 0.7 ± 0.1 V and I = 0.43 ± 0.03 A. Find a value for R and its error σR using the appropriate methods for combining errors. 2. We take a series of measurements of V1 , V2 and I with results as given in the Origin® file Exercise9. Plot (V1 − V2 ) against I and use a linear fit to find R and σR . Do the two approaches give the same results?

3.7

Non-Linear Fits

So far we have only considered fitting straight lines to a dataset. More complex relationships might require the line to be represented by e.g. a polynomial or an exponential function. These, 35

Measurements and Uncertainties Exercises and other, fits can also be achieved using other options under Analysis and Fitting. Exercise 10 Read in the data ∗ from the Origin® file Exercise10. Try fitting a polynomial of order 2. Add y-errors of magnitude 10% before mid-1993 and 2% after that; how does that affect your fit? Experiment with other functions. Given that the incidence of cosmic rays is modulated by the 11-year solar activity cycle try a plausible sine function (use Parameters to fix the period).

3.8

Histograms

It is often useful to look at spreads in measurements using histograms. Exercise 11 Using the dataset in Exercise11† in Origin® use Plot, Statistics, Histogram to plot a quick result. Now try varying the bin sizes: select data column then use Statistics, Descriptive, Frequency Counts, Computation Control. The results will appear in a new sheet, you can then plot the histograms using these data.

3.9

Bibliography

Website The Measurements and Errors material can be found on the web by going to First Year Lab and Computing on Blackboard.

Textbooks Young and Freedman, sections 1.1-1.6, give an elementary introduction. Hughes, I. F. and T. P. A. Hase Measurements and their Uncertainties (Oxford University Press, 2010). This is a friendly slim volume with nice examples and useful exercises (if you want to test your understanding). It contains more than you need for First Year Lab but will also be useful in later years. Squires, G. L. Practical Physics (Cambridge University Press, 4th edition 2001). This is a good general textbook on experimental measurement and the treatment of errors. Chapters 1 to 5 deal with errors and chapters 6 to 13 deal with experimental technique and record keeping. The first column contains the date and the 2nd and 3rd columns contain indirect measurements of cosmic rays made at ground stations in Colorado, USA and Kiel, Germany. Cosmic rays are energetic particles that are generated in outer space and pass through the Earth’s atmosphere. As they collide with atmospheric particles, they disintegrate into smaller pions, muons and the like, producing a cosmic ray shower. These particles can be measured on the Earth’s surface by neutron monitors. The data are monthly mean neutron counts in units of 100 per hour. The neutron counts at Kiel are consistently higher than those at Colorado because of its more northerly latitude: the charged particles are steered around the Earth’s magnetic field lines so that there is a much higher incidence of cosmic rays at the poles than at the equator. † These are real exam results from one of the 3rd Year Comprehensive Papers! ∗

36

Measurements and Uncertainties Exercises For more details of error theory see: Taylor, J. R. An Introduction to Error Analysis (University Science Books, 2nd edition 1997).

37

Measurements and Uncertainties Exercises

Appendix 3.A

Solutions to Exercises

Exercise 1 Mean = 0.788 m Sample Standard Deviation (s) = 0.032 m Standard Error of the Mean (σm ) = 0.013 m Exercise 2 Mean = 2.939 × 108 ms−1 (or 2.94 × 108 ms−1 ) Sample Standard Deviation (s) = 0.174 × 108 ms−1 (or 0.17 × 108 ms−1 ) Standard Error of the Mean (σm ) = 0.055 × 108 ms−1 (or 0.06 × 108 ms−1 ) The result should be quoted as (2.94 ± 0.06)×108 ms−1 (this is just consistent with the correct value of c of 3.00 × 108 ms−1 ). With the unreliable point taken out: Mean = 2.991 × 108 ms−1 (or 2.99 × 108 ms−1 ) Sample Standard Deviation (s) = 0.060 × 108 ms−1 (or 0.06 × 108 ms−1 ) Standard Error of the Mean (σm ) = 0.020 × 108 ms−1 (or 0.02 × 108 ms−1 ) The result should now be quoted as (2.99 ± 0.02) ×108 ms−1 (note the lower final error). Exercise 3 For the five data sets we obtain: MEAN STD DEV STD ERROR

SET 1

SET 2

SET 3

SET 4

SET 5

6.50 0.88 0.20

6.64 0.85 0.19

6.34 0.65 0.14

6.21 0.82 0.18

6.50 0.69 0.15

The overall mean is 6.44 and its standard error is 0.08. Note that, even with as many as 20 data points per set, the standard deviations of each set can be quite different from each other. These data points were generated at random with a mean of 6.5 and a standard deviation of 0.8. You can see that the final results for the total set of 100 points are consistent with these values. Exercise 4 The results obtained are x values MEAN STD DEV STD ERROR

0.396 0.040 0.009

38

2x values x2 values 0.793 0.080 0.018

0.159 0.031 0.007

Measurements and Uncertainties Exercises The calculated standard error for 2x is 2 × 0.009 = 0.018 (as above). The calculated standard error for x2 is 2 × 0.396 × 0.009 = 0.007 (as above). Exercise 5 V = 63121 ± 6133 cm3 or better (6.3 ± 0.6) ×104 cm3 or even better (6.3 ± 0.6) ×102 m3 . Can do this either by direct calculation using σV = 4πr2 σr . or using fractional errors: σr /r = 0.8/24.7 = 0.032 (i.e. 3.2%) so σV /V = 3 × 3.2% = 9.6%, which gives the same error as calculated above. Exercise 6 Values for

x

y

MEAN 26.9 STD DEV 2.4 STD ERROR 0.8 %Error 3

20.3 5.1 1.6 8

x + y Values x − y Values x × y 47.2 5.1 1.6 3

6.5 6.2 1.9 30

544.1 138.2 43.7 8

x/y 1.4 0.4 0.1 10

Note that the errors here do not agree exactly with predictions because there are only 10 data points. Exercise 7 Differentiate the expression for n with respect to θi and θr and use the results with the general expression for combining errors to show that: σ2n = {cot2 θi + cot2 θr } × σ2θ n2 Insert values to obtain n = 1.31 ± 0.06. Exercise 9 V12 = 6.2 ± 0.5 V (notice that this error is completely determined by the V1 error) fractional error on V12 is 8% fractional error on I is 7% so fractional error on R is

√ 82 + 72 = 11%, giving R = 7.2 ± 0.8 Ω.

Original Author: Joanna Haigh

39

(3.15)

Chapter 4 Introduction to Experiments in Optics Contents 4.1

A. Ray Optics—Lens Imaging . . . . . . . . . . . . . . . . . . . . . . Ray diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lens formulae . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experimental system 1: Imaging with a lens . . . . . . . . . . . . . . . . . . Experiment 1.1: Experimental imaging . . . . . . . . . . . . . . . . . . . . . Experiment 1.2: Measuring the focal length of a lens . . . . . . . . . . . . . 4.2 B. Wave Optics—Single-Slit Diffraction . . . . . . . . . . . . . . . . . Experimental system 2: Diffraction by a single slit . . . . . . . . . . . . . . .

40 40 41 41 41 42 42 42 42

In this introductory session, you will perform two experiments that will develop your experimental skills and illustrate the ray and wave nature of light • A. Thin Lens Imaging (Ray Optics) • B. Single-slit diffraction (Wave Optics)

4.1

A. Ray Optics—Lens Imaging

In this experiment, you will study the imaging properties of a lens using some basic methods and formulae as applied to imaging systems.

Ray diagrams Figure 1 is a ray diagram that illustrates the object distance s, the image distance s’, all measured from the centre of a thin converging lens with focal length f. When drawing ray diagrams, it is convenient, as shown in Fig 1, to use the 3 principal rays constructed as follows: • A ray passing through the centre of the lens is undeviated. • A ray directed parallel to the optical axis passes through a focal point. • A ray passing towards, or away from, a focal point emerges parallel to the axis. It is useful to denote the front and back focal plane (F and F 0 ) of the lens on the diagram. Any two of the principle rays will allow you to locate the image (the third can be used as a cross-check). The image point is where the principle rays cross. 40

Introduction to Experiments in Optics

ho

F’ O

I

hI

F s

s’ f

f

Figure 4.1: Principal rays used to locate the image formed by a converging lens.

Lens formulae For small angles, the thin lens formula and the linear magnification can be derived from the ray diagram. The thin lens formula is: 1 1 1 + 0 = (4.1) s s f The linear magnification m is given by s0 m=− . s

(4.2)

Exercises The thin lens formulae An object is placed 150 mm from a 100 mm focal length lens. Use the thin lens formula to calculate the image distance. What is the magnification? Comment on the significance of the negative sign in the calculated magnification. Locating the image and finding the magnification using ray diagrams Construct a ray diagram of the previous exercise using a scaled drawing in your lab book. Choose a suitable scaling to fit the diagram in your lab book. Draw two principle rays and hence locate image location. (Check with the third principle ray). Measure image distance on your scaled diagram and rescale to determine the image location (s’). Measure the linear magnification by comparing the heights of the image and object hI /hO .

Experimental system 1: Imaging with a lens image, observed on ground-glass screen

f

object Light source

s

s’

view from behind screen

Figure 4.2: 2: Experimental system for imaging with awith positive lens. Figure Experimental system for imaging a positive lens.

41

Introduction to Experiments in Optics

Experiment 1.1: Experimental imaging Fig. 4.2 shows a simple set-up for imaging with a positive focal length lens. Use the slide with a single letter as an object in the slot-holder in front of the light source. Use the plano-convex lens of nominal focal length 100 mm. Place the lens 150 mm from the object (use your meter rule to measure this). Observe the image of the object slide on the ground-glass screen by looking at the back of the screen and adjusting its position for sharpest image. It may be difficult to see the image until you are close to the image plane. Use your meter ruler to measure as precisely as you can the image distances of your system. Also measure the size of one of the dimensions of the object and its corresponding size in the image plane. Estimate an uncertainty for each of the measurements you make. Use your object and image size to calculate the experimental linear magnification with associated uncertainty. Compare your experimental results to your calculations in Exercise 4.1.

Experiment 1.2: Measuring the focal length of a lens Fig. 4.3 shows a simple method for measuring the focal length of a positive lens. mirror

lens

object Light source

f

Figure 3: Simple method for estimating focal length of a positive lens.

Figure 4.3: Simple method for estimating the focal length of a positive lens. Switch on the light source—the voltage on the supply should be at voltage 5V (max.). Use the slide with a pin-hole as an object in the slot-holder in front of the light source. Use the planoconvex lens of nominal focal length 100 mm. Adjust the height of the lens to get the transmitted light to run parallel to the bench and if necessary tilt the lens so it is at right-angles to the optical axis. Place the mirror on the optical rail to reflect the light back onto the pin-hole slide (just to the side of the pin-hole itself). You should observe that the size of the reflected beam varies as you move the lens along the bench, but there is a position where it is a minimum. This position occurs when the lens is at the focal length from the object. (Check to see if the return spot is affected by the position of the mirror). Use your meter ruler to measure this distance as precisely as you can, taking your measurement to the centre of the lens. Make an estimate of the uncertainty. Ensure you record your measurement of the focal length with associated uncertainty in your lab book.

4.2

B. Wave Optics—Single-Slit Diffraction

The aim of this experiment is to observe the far-field diffraction patterns obtained when a laser illuminates an aperture.

Experimental system 2: Diffraction by a single slit 1. Fig. 4.4 shows the basic set-up for visual observations. In this experiment, the light source is a diode laser emitting at a red wavelength of 670 ± 1 nm. This laser is powered by a 42

Introduction to Experiments in Optics bi-convex lens f=1000mm

observing screen

LASER

f diffracting object

Figure 4.4: Basic arrangement for visual observation of diffraction patterns. Visual observations are made on the screen in the focal plane of the lens: DO NOT STARE DIRECTLY INTO THE LASER BEAM! special stable 5 volt power supply. Use the variable width slit as the object. Set the slit width visually to about 1.0 mm and place it immediately after the laser output. [You can centre the laser on the slit by loosening the laser’s post and rotating it slightly.] Observe the diffraction pattern on the screen placed 1 m from the 1000 mm focal length lens. Now slowly close the slit, and observe and note in your lab book the change in the diffraction pattern. Set the slit width such that the width of the central maximum is approximately 10 mm. Draw a labelled sketch (with a scale marked) of this diffraction pattern, noting your visual estimate of the slit width. According to theory, the “far-field” intensity diffraction pattern I(x) of a single slit of width a, as a function of distance x in the observation plane is given by "
#2 sin πax λf
(4.3) I(x) = I0 πax λf

where I0 is the intensity at the centre, λ is the wavelength and f is the focal length of the focussing lens. From this equation, the zeros of intensity occur at x-coordinates xm = m(λf/a), m = ±1, ±2, ±3. . . . Does the single slit diffraction pattern you see with your eye look as though it is described by Eq. 4.1? Are the “zeroes” of intensity correctly spaced, given the slit width that you used? According to the equation, the first secondary maximum has an intensity that is about 4.5% that of the central intensity. Does that agree with your visual impression? Make these observations carefully and critically: keep a clear and accurate record in your lab book. 2. In this last part of the experiment, the aim is to produce quantitative measurements of the intensity. To measure the intensity a photodiode is used together with a narrow slit (=100 µm) in front of it. The photodiode output is a voltage that is linearly proportional to the light intensity. The slit/photodiode are provided as a single unit that attaches to a precision translation slide (the slit/photodiode can be rotated by loosening the screw on the mounting bracket of the translation slide). The translation slide has a total range of 25 mm. Use the set-up shown in Fig. 4.5. For this experiment, use the slide marked “Single Slit”. The exact width of this slit is not known but it is about 60 µm. Compared to Fig. 4.4, the only significant change is that the 1.0 m focal length lens has been replaced by the 0.5 m focal length lens and the observing screen is replaced by the photodiode assembly. (Check that the power supply for the photodiode is turned on!) 43

Introduction to Experiments in Optics bi-convex lens f=500mm

Photodiode/slit on translation stage

LASER

f diffracting object

voltmeter

Figure 4.5: Basic arrangement for measurement of the intensity of diffraction patterns. The object is placed close to the laser output and the f = 0.5 m lens immediately follows. The slit in the photodiode assembly is placed 0.5 m from the lens. • First, find the centre of the diffraction pattern (the largest intensity) and note the voltage reading and the position on the micrometer scale of the translation stage. [You should choose the best scale on the voltmeter (either 2 V FSD or 200 mV FSD).] • Secondly, try scanning the photodiode quickly across the intensity pattern and note the general shape. Find the first secondary maximum and compare its magnitude to the central maximum. • Finally, find the positions of the first “zeroes” (m = ±1). Hence estimate the slit width with a value for the uncertainty.

44

Chapter 5 Introduction to Electronics Contents 5.1

Direct Current Equipment . . . . . . . Power supplies . . . . . . . . . . . . . . . . Digital multimeters . . . . . . . . . . . . . Protoboards . . . . . . . . . . . . . . . . . Direct current circuits: Ohm’s law . . . . . 5.2 Alternating Current Circuits . . . . . . Function generators . . . . . . . . . . . . . Oscilloscopes . . . . . . . . . . . . . . . . . Output impedance of the function generator 5.3 Practical Notes . . . . . . . . . . . . . . Electric current . . . . . . . . . . . . . . . Voltage . . . . . . . . . . . . . . . . . . . . Drift velocity . . . . . . . . . . . . . . . . . Resistance . . . . . . . . . . . . . . . . . . Electrical power . . . . . . . . . . . . . . . Alternating current . . . . . . . . . . . . . Phase . . . . . . . . . . . . . . . . . . . . Electrical power . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 46 46 47 47 48 48 50 51 51 51 52 52 52 53 54 54

Physics is an experimental science and physicists need to have a working knowledge of electronics in order to make measurements. The purpose of this introductory session is to acquaint you with the basic equipment necessary to build, power and analyse electronic circuits.

5.1

Direct Current Equipment

First you should familiarise yourself with the direct current (DC) equipment on the bench. The equipment you will use is the following:

Power supplies This power supply has meters that display the voltage and current supplied to the circuit. It can operate in two modes, constant current (CC) or constant voltage (CV)and supplied power from the + and − terminals. The knobs on the front of the instrument allow you to set an upper 45

Introduction to Electronics limit to the current and voltage. Depending upon the resistance of the load that you connect to the power supply, one of these settings will be limiting. The red LED lights up to indicate that the power supply is in CC mode. To turn the power supply on, switch the power switch on the left hand side and press the black button below the ON LED.

Digital multimeters The multimeter enables three measurements to be made, current, voltage and resistance. These different modes of operation are selected using different buttons and making connections to the appropriate terminals. There is a common terminal, which you use for all measurements, but dedicated inputs for voltage/resistance measurements and current. Often there are several current inputs to suit measurements of different magnitudes of current. Specifications for these instruments can be found in the Appendices.

Protoboards When constructing circuits, you will need some means for securing components and ensuring good electrical connections between them. This is most conveniently achieved using a protoboard where Components are pushed through small holes in the top of the board and connected by internal metal strips.

46

Introduction to Electronics

Figure 5.1: Simple DC circuit with a 5 V power supply and a resistor. The photograph shows a standard protoboard with enlargements of two areas of the board. The internal connections are overlaid as grey shading on the enlarged images. You will notice that some of the connection lines run the entire length of the board, while others are quite short. The long, vertical and horizontal lines are called “bus strips” and are typically used for connecting a power supply to multiple parts of the circuit. Smaller components are connected together using the shorter “terminal strips” allowing up to five wires or component legs to be connected together in a single strip. Some partially disassembled protoboards are located on your bench with the connections exposed so that you can inspect the internal connections for yourself.

Direct current circuits: Ohm’s law To gain some experience using the DC equipment, you will make a measurement of resistance using ohm’s law and compare with the value obtained using the multimeter. Choose a resistor with a value of R1 = 1 kΩ. (See inside front cover for instructions on how to identify the correct component). Check the value using the digital multimeter (DMM) in ohmmeter mode. How does the quoted value compare with your measurement? Is it within the tolerance stated for the resistor? • • • • •

5.2

Set up the simple circuit shown in Fig. 5.1. Using the DMM as an ammeter, measure the current in the circuit. What do you assume about the resistance of the ammeter? Using the DMM measure the voltage across R1 and confirm Ohm’s law (R= V/I). What do you assume about the resistance of the voltmeter?

Alternating Current Circuits

There are relatively few applications where steady, direct current is used. The electricity supplied to our homes varies sinusiodally with time, audio equipment reproduces complex waveforms that we call music. An AC voltage can therefore be written as: v = V0 sin ωt where V0 is the amplitude of the voltage and ω = 2πf its angular frequency. Modern electronic circuits cover a huge frequency range, from telephony (20–20 kHz) through radio (100 kHz) and mobile phones (GHz) to airport security using electromagnetic waves in the THz region. These AC signals are generated using oscillator circuits and you will now learn to measure AC signals using an oscilloscope and investigate their frequency dependent properties.

47

Introduction to Electronics

Function generators The function generator produces sinusoidal, square or triangular oscillating voltages which you can use as an input to any circuit you build to test its response. You can control the amplitude of the waveform and its frequency. These are set using a combination of a knob or dial and a range button. Normally you would expect the voltage to oscillate either side of ground (earth, 0 volts). However, you can also add a DC offset to the oscillating voltage. For example, you may wish to set up a 1 V peak-to-peak oscillation superimposed on a +3 V DC potential. The DC offset knob allows you to achieve this. • Set the function generator to give a 4 V peak-to-peak (p–p)1 kHz sine wave. Think carefully about how the range buttons work to make sure you are really at 1 kHz. • On the bench, you will find some headphones which can be connected to the function generator via the 3.5 mm jack. What does the output sound like? Over what range of frequencies can you hear? How does the volume change as you change the amplitude? Close your eyes and set an amplitude that is half or twice as loud. Is your hearing response linear or logarithmic? Listen to square and triangular waves as well. Musically, an octave corresponds to halving or doubling the frequency. A perfect fifth has a ratio of 3:2, a fourth 4:3, a third 5:4, etc. If you know what these intervals sound like, is it more accurate to set the frequency by ear or by reading the dial? Concert A is 440 Hz.

Oscilloscopes The oscilloscope allows you to measure voltage as a function of time and always measures the voltage between ground and a test point. The co-axial cables used to connect to the oscilloscope are actually composed of two wires and one of these is always connected to ground (just like a TV aerial)∗ . In contrast, the inputs of the DMM can float, i.e. neither terminal needs to be connected to ground. A DMM can be used to measure the voltage across a circuit component even if both sides of the component are at non-zero voltages, something that is impossible with an oscilloscope. If you feed in a constant DC signal the oscilloscope simply acts as a voltmeter, but if you have a time-varying voltage from say an oscillator, you can use it to visualise the signal as it changes in amplitude or frequency. The oscilloscope that you will be using have two input channels, each of which is fed to a separate amplifier via a switch which selects “DC coupling”, “AC coupling” or a “Ground” signal (the importance of this switch will become apparent as you work through the following sections). The amplified signal is then used to drive the vertical part of the display. A separate time-base unit feeds a repetitive sweep to the display which drives the horizontal component (the time axis).

∗

See p. 135 for reference information on the oscilloscope controls.

48

Introduction to Electronics

Figure 5.2: The dual channel oscilloscope used in first year lab. • First use your oscilloscope to look at a signal from the function generator. Set the generator to give a 4 V p–p, 1 kHz sine wave. Connect it to the Channel 1 input of the scope using a BNC to banana cable. Set the scope to 1 V/div vertically, 0.2ms/div horizontally. Set the trigger controls to Auto and then check what happens when you switch the trigger mode between AC and Line. Don’t panic if you can’t get anything to display! • Learning to operate an oscilloscope takes time. Locate the relevant controls on your oscilloscope and try to obtain a meaningful trace on the screen (ask a demonstrator if you get stuck). Once you have some sort of sensible trace, work back through the following sections to explore the controls more fully. If you want to practice from home, check out the virtual oscilloscope webpage: http://www.virtual-oscilloscope.com/. • The trigger unit is the section most new users of scopes have problems with. Unless we explicitly tell the time base when to start a sweep, it will do so at random times relative to our input signal. The result can be a blank display, or one with the trace zipping across the screen in a rather messy and uncontrolled way. The trigger unit tells the time base when to start a sweep, and is used to synchronise the input signal and the timebase. You need to select various options: 1. Where to take a synchronisation (sync) signal from (Channel 1, 2 the AC power line, or an External source). 2. AC or DC coupling of the signal. 3. Whether to trigger from a rising or falling edge. 4. The trigger level at which point the timebase will begin a sweep. Check you understand the effect of these controls. • Now disconnect the oscilloscope input from the oscillator, leave the ground connector free and take hold of the input end of the cable. You should be able to pick up an oscillating signal from your finger on the order of 0.1–0.5 V peak-peak! Sketch this signal accurately in your lab book and measure its frequency and amplitude. Where do you think this 49

Introduction to Electronics signal comes from? Discuss this with your partner and fellow students, and then your demonstrator. This last point is actually a very important result for all sorts of measurements. Any piece of electronic equipment is both a source of electrical noise and a receiver for noise from other nearby equipment. This noise can be transmitted by radio waves (RF) or through the earth line and mains power rails. In general you need to take care to keep electrical noise out of your experiments. Further notes on the oscilloscope: The vertical controls set the gain and offset of the input amplifier. They are usually calibrated to give a set number of volts per division on screen (a ‘division’ is 1cm on the screen), which you can change in discrete jumps. There is also a variable setting which lets you change the volts per division smoothly. This can be useful if you want to expand some feature so that it is easier to see, but in this mode you can not make a meaningful voltage measurement. If you want to accurately measure the size of a signal, make sure that the variable control is set in the CAL or calibrate position. There may also be a X5 or X10 control to let you expand the display about the centre of the screen. You also need to take this into account when measuring a voltage. The vertical position control is an offset which allows you to put 0 V at the centre of the screen, or at any other position on the screen you wish to use. The horizontal (time-base) controls set the speed at which the scope sweeps the signal across the screen in the horizontal direction. Again you can change it in calibrated jumps, or smoothly in an uncalibrated mode. Remember that to make a meaningful measurement the variable control must again be in the CAL position. There will quite often be a 10X button which will let you expand the display about the centre of the screen by a factor of 10. You must take this into account when measuring a frequency or time interval. In DC mode any input voltage goes straight to the amplifier, and the output is used to drive the vertical component of the display. In AC mode, the signal is connected to the amplifier via a capacitor. A capacitor used in series in this way will block any constant (DC) voltage but let through a time-varying (AC) signal. This can be very useful if you want to look at a small oscillating signal superimposed on a large constant background. You can see the effect of swapping from AC to DC coupling as the signal from your oscillator should have both an AC and DC component. In ground mode the amplifier is simply fed a zero voltage signal (the input to the channel has no effect) and you can use this as a reference to find the zero voltage position on screen. In general DC coupling will tell you the most about your signal, and you should not use AC coupling unless you want to remove a constant background level from your measurements.

Output impedance of the function generator All pieces of electronic equipment have an effective internal resistance and sometimes care must be taken to ensure that they operate in the manner that you expect. Take the function generator as an example. If you look carefully at the output terminals you will see that 600 Ω is printed next to the terminals. This means that the output impedance of the function generator is 600 Ω. Impedance is a generalised form of resistance used when treating AC signals. During the electronics course in the second term, you will study AC impedance formally, but for now you can understand it as equivalent to resistance but often with a frequency dependence.

50

Introduction to Electronics We can illustrate the effect of the impedance of the function generator with the following experiment. Set the function generator to 4 V p–p and connect it to the oscilloscope. The circuit diagram shown below the part in the left hand dotted box represents the function generator—in fact it is the equivalent circuit of the generator (if you looked inside the box—NOT RECOMMENDED— you would see a much more complicated circuit but the equivalent circuit reproduces all the behaviour of the real generator with V0 = 4 V p–p and R0 = 600 Ω. The right hand dotted box shows the equivalent circuit of the oscilloscope. This is much simpler and consists of a single input resistor Ri . When connected to the oscilloscope, the value for Ri is extremely large. You can see 1M Ω printed below the input terminal. Therefore if we consider how the voltage will divide, we find that virtually all V0 will be dropped across Ri (the oscilloscope terminals) and virtually none dropped across R0 . The oscilloscope therefore is making an accurate measurement of V0 . Now connect your headphones and measure the output from the function generator. The signal will have dropped, since the impedance of the headphones will be low, typically a few tens of ohms. V0 remains unchanged, but the potential divider now has most of the voltage dropped across R0 , rather than Ri . The consequence of this is that the function generator should only be used to drive high-impedance circuits, or put differently, circuits that draw relatively little current. The definition of ‘little current’ means a voltage/current ratio much greater than 600 Ω.

5.3

Practical Notes

These notes provide some background to the electronics practicals you will encounter during the next two terms. Further information can be found in the recommended textbook Young & Freedman chapters 21 to 31.

Electric current Current is the flow of electric charge, dQ/dt. The charge, Q, might be ions in an electrolyte (like a battery) or a gas discharge (like a fluorescent strip-light) but for metals (the most common material found in electric circuits) it is the movement of electrons that generates current. An electron carries a negative charge of 1.602 × 10−19 coulombs and the unit of current, the ampere, is defined as the passage of 1 coulomb of charge (about 6 × 1018 electrons) per second along a metal wire. Such large numbers are often difficult to comprehend but to give you an idea—this number is greater than the estimated age of the universe in seconds!

Voltage Voltage sources pull electrons into the positive terminal and push them out of the negative terminal. They therefore supply the driving force for current. A voltage is sometimes called a potential difference or an emf (electromotive force) although there are subtle differences in these quantities that you will discover later in the Electricity and Magnetism course. All voltage sources have some internal resistance (or impedance) which means they may not always supply the voltage you think (see output impedance of the function generator). 51

Introduction to Electronics

Drift velocity

Figure 5.3: Cartoon of an electron’s path as it travels in a conductor. Metals conduct electricity because the outer electrons of the metal atoms are free to move. When a voltage (potential difference) is applied across the ends of the wire of length L, it produces an electric field E = V/L and the electrons move in response to the force, F = eE. Since an electron has a negative charge the force acts to move the electron in the opposite direction to E (which, by convention points from the positive to the negative potential). According to Newton’s law the electron should accelerate and the electron speed increase with time. In fact the moving electrons are subject to collisions with the metal ions and other electrons which limit the speed to the drift velocity, vdrift (Fig. 5.1.

Resistance Electrons in a metal wire rapidly achieve a drift velocity on application of a voltage. Thus flowing electrons lose momentum through collisions—this is the origin of resistance. Ohm showed that voltage and current in a wire are proportional and that the constant of proportionality is R leading to Ohm’s law R=V/I, where R is measured in Ω. Using the analogy of water pumped through a pipe, the rate of pumping is dependent on the dimensions of the pipe and the resistance of a wire is given by R= l/A where l is the length of wire, A its cross section and ρ its resistivity, which is a property of the metal. Real metals have low values of ρ and wires consequently have tiny resistances. From Ohm’s law we see that there will be a negligible voltage drop for even sizable currents in a wire. The function of resistors is to limit the current in different parts of a circuit. There are many types but simple thin film types are most common (graphite is regularly used since it has a relatively high ρ). Different film thickness can be used to produce a range of resistors from a few Ωto many MΩ.

Electrical power Power is defined as energy per unit time. In an electrical circuit an electron acquires an energy e × V from the voltage source and loses it when it moves around a complete circuit. If there are N electrons moving in the circuit in time t, then the energy loss is NeV/t and remembering that current I = charge/time, this gives electrical power as V × I measured in Joules. How is this energy lost? We have already seen that collisions with metal ions leads to resistance and 52

Introduction to Electronics these collisions generate heat. So in a simple circuit consisting of a battery and a resistor the energy loss (or dissipation) results in heating of the resistor and its value is given by P = V × I, or invoking Ohm’s law: V 2 /R or I2 R.

Alternating current Although we use batteries in mobile devices and as back-up power supplies, most circuits run on alternating current (AC). This is due to the way electricity is generated; rotating an electrical coil within a magnetic field. The full line in Fig. 5.4 represents the current i = Im sin ωt.



T (= 2π/ω)

-

time [arbitrary units] Figure 5.4: Alternating currents (AC), represented as current versus time. Solid line: i = Im sin ωt, where Im = 1 A here. Dotted line: i = Im sin(ωt + φ), for φ = 55 deg or 0.97 rad. The period of the oscillation T is indicated as the duration between two peaks

time [arbitrary units] Figure 5.5: The same signals as the previous figure, but with the horizontal time axis plotted from time t = −φ/ω, or in this case shifted forward by T × 55/360, where T is the period of the oscillation. The important parameters are: Im , the amplitude of the wave (here equal to 1) and ω its angular frequency. The latter is derived from the period of the wave (repeat time) T measured in s. The frequency f = 1/T (measured in s−1 or Hertz (Hz)) and ω = 2πf, measured in radians per second, or rad/s. Note that time-varying quantities like voltage and current are assigned lower case letters while time invariant quantities like a battery voltage or amplitude are assigned upper case letters. 53

Introduction to Electronics

Phase The voltage and current in purely resistive circuits are always in phase but this is not true of circuits containing capacitors and inductors. The full line in Fig. 5.4 corresponds to a current i = sin ωt. The dotted line is given by i 0 = sin(ωt + φ) where φ is the positive phase shift relative to i (in this case about 55 deg or 0.97 rad). Note that a positive phase shift is equivalent to a shift forward in time by 55/360 × T and i 0 reaches its peak value before i. Alternatively we could have taken a snapshot of these waves at a time 55/360 ×T later. This is shown in Fig. 5.5. With these new axes we might now assign i 0 = sin ωt and i = sin(ωt − φ). This shows that phase is a relative parameter and in dealing with circuits we can decide which component will dictate the zero of phase. Determining the phase relations between components is crucial to understanding the behaviour of an AC circuit.

Electrical power We saw earlier how electrical power was given by V × I for a DC current, but what about energy dissipation in a resistor carrying AC current? Heating within the resistor is unaffected by the direction of current flow, but the instantaneous value of the current (or voltage) varies with time and we have to find the time-averaged value of an AC waveform. If i = I sin ωt and 2 v = V sin ωt, then P = IV sin2 t. The time-averaged 1/2 (can you prove √ value of√sin ωt is √ √ this?) so the average power is P = IV/2 or I/ 2V/ 2 where I/ 2 and V/ 2 are known as the root mean square or RMS values of current and voltage respectively. The domestic AC supply in the UK is 240 V—can you say what the peak to peak value is?

54

Chapter 6 Experiments in Optics Contents 6.1 6.2

6.3

6.4

6.5

6.6

6.7

6.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working Practices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Safety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Your laboratory notebook . . . . . . . . . . . . . . . . . . . . . . . . . . . . Working practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Keep optical components clean! . . . . . . . . . . . . . . . . . . . . . . . . . Interference and Diffraction . . . . . . . . . . . . . . . . . . . . . . . . Experiment 1: Diffraction by a single slit and by a strand of hair . . . . . . . List of equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 1a: single-slit diffraction . . . . . . . . . . . . . . . . . . . . . . Experiment 1b: Determining the diameter of a hair . . . . . . . . . . . . . . Experiment 2a: Young’s Double Slit Interference and Diffraction Gratings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 2b: N-slit diffraction and diffraction gratings . . . . . . . . . . . Experiment 3: Imaging with Lenses . . . . . . . . . . . . . . . . . . . Experiment 3a: Testing the lens formula . . . . . . . . . . . . . . . . . . . . Experiment 3b: Image quality and resolution of lenses . . . . . . . . . . . . . An aside about lens aberrations . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 4: Polarisation of Light . . . . . . . . . . . . . . . . . . . Law of Malus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . List of equipment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Experiment 5: Verification of Malus’ Law . . . . . . . . . . . . . . . .

55 57 57 57 57 57 58 58 58 59 61 62 64 65 66 67 69 70 71 72 72

Introduction

Welcome to Experiments in Optics. For the next two weeks, you will do a variety of experiments that illustrate the ray and wave nature of light. Recent years have seen a revolution in optical technology (e.g. lasers in CD- and DVD-players, telecommunications and medicine), and optics also remains one of the main diagnostic measurement techniques in a wide range of industrial and research fields. However, the aims of this part of the lab go beyond learning some optics— in particular, we want you to appreciate that measurements are never exact and therefore that experiment and “theory” will never agree precisely. The best we can hope for is that experiment 55

Experiments in Optics and theory are consistent, bearing in mind the errors of the experiment and the approximations of the theory. Everyone knows that measurements have errors but it is easy to forget that theories have their “error”—i.e. approximations—and this is illustrated in the present topic. In optics, we consider light to be described by rays or waves or photons, and sometimes by all three in the same experiment! Physical Model of Light

Name Given to Model

Rays Waves Photons

Geometrical Optics Physical Optics Quantum Optics

One important thing to learn in any branch of science is what level of approximation is required to solve a particular problem. For example, quantum optics is meant to provide a complete description of light but if you tried to use it to design a simple lens you would (probably) never get an answer. It turns out that an extremely high quality lens can be designed using the concept of rays (geometrical optics) and some knowledge of the wave model (physical optics). On the other hand, the basic principle of operation of a laser requires the quantum model—no amount of clever playing with rays or waves can fully explain the light emission in a laser. In Experiments in Optics, we shall do experiments that use the models of light that are called Geometrical (Ray) Optics and Physical (Wave) Optics. Ray and wave pictures of light are fairly straightforward, and sufficiently comprehensive, to account for most optical phenomena and devices In Experiments in Optics you will work with a partner. There are four sessions (two per week) and the following table summarises the schedule of experiments: Lab Session

Experiment

1 2 3 4

Single Slit Diffraction Two-slit Interference and Gratings Lenses Polarisation

56

Experiments in Optics

6.2

Working Practices

Safety Laser safety During the course of this series of experiments, you will use a compact diode laser of wavelength 670 nm (red). The power of this laser is approximately 1 mW. Lasers produce a highly collimated (parallel) beam of light; the eye could focus this to a very small spot as small as 10 µm in diameter, giving a power density of 1 kW per square centimetre (but of course only over a tiny area). Retinal damage may occur before the “blink reflex” closes the eye. Therefore, • NEVER LOOK DIRECTLY INTO THE LASER • NEVER POINT THE LASER AT ANY OTHER PERSON • DO NOT USE MIRRORS TO REFLECT LASER BEAMS AROUND THE LAB Electrical safety In this experiment you will also use various items of mains-powered electrical equipment; this should not be tampered with in any way. If you are in any doubt regarding the safety of any piece of equipment or any experimental procedure, you should consult a demonstrator before proceeding. Trip hazards You are often working in darkened conditions, so it is especially important that bags and coats are stowed thoughtfully and passageways around benches are kept clear.

Your laboratory notebook Start by writing the day and time, and title of the experiment. As you do each part of the lab, it is essential to keep a clear written record in your lab notebook. However, do not spend a long time engrossed in your lab book—never more than 10 minutes at a time—remember this is a practical laboratory, not an exercise in writing. Every 10 or 15 minutes, record what you have done in your book. Write clearly, draw lots of clearly labelled sketches, write down any conclusions you have drawn or decisions you have made. It is vital that you describe or draw what you actually see. If a trace on a screen has lots of bumps, wiggles and spots then draw them. Do not draw what you think you might get from a perfect experiment; you might be throwing away important details.

Working practice We expect you to work hard in the laboratory. Concentrate on the experiments and work methodically and creatively—this will then make the work more enjoyable. If you get stuck, first talk the problem through with your partner and other fellow students; if you cannot resolve the issue after about five minutes, ask a demonstrator to help you think it through.

Keep optical components clean! Do not touch the surface of an optical component with your bare hands. If lenses, gratings, polarisers etc. get covered with fingerprints and dirt, they will scatter light and, when quantitative measurements are being made, may produce quite spurious effects.

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Experiments in Optics

6.3

Interference and Diffraction

When two or more waves are superimposed, the resultant amplitude (and hence intensity) depends on the amplitudes and phases of the component waves. This process is known either as interference or diffraction, depending upon the geometry of the particular experiment. When there are a finite number of waves (e.g. light from two very narrow slits or two distinct optical paths) we talk of interference but if there is a continuum of waves (e.g. from light passing through a single slit of non-negligible width) then we talk of diffraction. The aim of this set of experiments is to observe the interference and diffraction patterns obtained when a laser illuminates various apertures and to compare them with those predicted according to the wave theory of light. Analysis of diffraction close to the diffracting aperture, known as near-field diffraction, can become quite mathematically complex. A much simpler case is the socalled far-field diffraction regime in which the plane of observation is very far from the aperture (ideally at infinity)—as seen in Fig. 6.1a. Due to the physical size of optical bench this is not always practical and an equivalent effect is achieved by making all observations at the focal plane of a lens—as seen in Fig. 6.1b. From Fig. 6.1b it is noted that rays diffracted at angle θ are observed on the viewing screen at a distance x from the lens axis where θ = x/f (for small angles).

a) input light

diffracted rays at angle  meet at infinity

b) x

~

 L diffracting object

parallel rays meet at a point in focal plane





very distant observing screen

x f

Lens, f

=x/f (small angles)

observing screen

Figure 6.1: Input light beam (e.g. from laser) is diffracted into rays at new angles. The far-field diffraction pattern can be observed a) on a very distant screen (ideally at infinity) or b) in the focal plane of a lens.

Experiment 1: Diffraction by a single slit and by a strand of hair There are basically two parts to this three hour session. First, you will investigate the diffraction pattern of a single slit. Secondly you will examine the diffraction pattern of a strand of hair and based on single slit diffraction and Babinet’s principle determine its diameter.

List of equipment • • • • • • • •

1.5 m triangular profile optical bench, with five saddles Diode laser, post-mounted, with stabilised 5 V power supply bi-convex lens f = 500 mm, bi-convex lens f = 1000 mm holder for slides set of slides (diffracting slits and gratings) holder for observing screen, post-mounted, with white card precision translation stage with photodiode (and a power supply and portable voltmeter) metre rule 58

Experiments in Optics Single slit diffraction theory The theoretical derivation of the intensity of light in the “far-field” diffraction pattern of a single slit is found by summing the amplitudes of light from each elementary point on the slit. This procedure follows from Huygens’ Principle and the amplitude in this case is the electric field of light and has both magnitude and phase. The intensity pattern is found by squaring the amplitude pattern. For a slit of width a, the intensity I(x) of the diffraction pattern as a function of distance x in the observation plane is given, for small angles θ = x/f, by " I(x) = I0

sin

πax λf
πax λf


#2 ,

(6.1)

where I0 is the intensity at the centre, λ is the wavelength and f is the focal length of the focussing lens. Note that the quantities a, x, λ and f in Eqn. 6.1 all have dimensions of length and should all be expressed in the same units, e.g. millimetres. Fig. 6.2 shows the single slit diffraction pattern.

Intensity

1.0

0.8

0.6

0.4

0.2

0.0 -10

-5

0

5

10

Distance Figure 6.2: The single slit diffraction pattern of Eqn 6.1, with the scaling constant I0 = 1. The horizontal axis is in units of the quantity ax/λf. From Eqn. 6.1, the zeros of intensity occur at x-coordinates xm given by  πax 

= mπ, λf where m = ±1, ±2, ±3, . . .. This can be rearranged as

(6.2)

xm = m(λf/a).

(6.3)

m

Experiment 1a: single-slit diffraction Fig. 6.3 shows the basic set-up for the experiments on interference and diffraction. In these experiments, the light source is a red diode laser emitting at a mean wavelength of λ = 670 ± 1 nm. This laser is powered by a special stable 5 volt power supply. [This is the type of laser device used in bar-code readers at Supermarket check-out counters and within DVD players. The basic diode laser inside the housing is an ultra-compact device and it is easy to forget that it may 59

Experiments in Optics

lens, focal length f

Photodiode/slit on translation stage

LASER

f diffracting object

voltmeter

Figure 6.3: Basic arrangement for measurement of the intensity of interference and diffraction patterns. The object is placed close to the laser output. Use the f = 0.5 m lens for measurements on the photodiode assembly. Use the f = 1 m lens for visual observations on a white screen. cause permanent damage if pointed at the eye. Therefore—NEVER LOOK DIRECTLY INTO THE LASER—NEVER POINT THE LASER AT ANY OTHER PERSON.] For this experiment, use the slide marked “Single Slit”. The exact width of this slit is not known but it is about 60 µm. To measure the intensity a photodiode is used together with a narrow slit (=100 µm) in front of it. The amplification circuity of the photodiode output results in a voltage that is linearly proportional to the light intensity. The slit/photodiode are provided as a single unit that attaches to a precision translation slide (the slit/photodiode can be rotated by loosening the screw on the mounting bracket of the translation slide). The translation slide has a total range of 25 mm. Make sure that the photodiode slit is vertical (loosen the screw and rotate the slit/photodiode if necessary). Use the bi-convex lens with f = 500 mm and ensure the distance between the centre of the lens and the photodiode slit is accurately located at the focal length of the lens to observe the far-field diffraction pattern. Looking by eye, check that the centre of the diffraction pattern is approximately coincident with the photodiode slit when the translation stage is set at its mid-point (about 12.5 mm) (make minor adjustments by tilting the laser and re-centring the “single slit” in its holder, if necessary). You should then measure the intensity over the full range of the micrometer movement. • First, find the approximate centre of the diffraction pattern (the largest intensity) and choose the scale on the voltmeter (either 2 V FSD or 200 mV FSD). • Second, get a feel for how often you will have to sample the diffraction pattern to get a good representation of it; every 1 mm, every 0.5 mm, every 0.25 mm...? A little preliminary work here will save you a lot of effort later. [Make sure you know how to read the micrometer; discuss this with your fellow students you have any doubt, and ask a demonstrator if you are unsure]. • Finally, make your set of measurements. You will have probably already noticed that background illumination affects your measurements so try to ensure that this remains constant and/or is very much reduced during the complete run. You may notice that the meter reading fluctuates more-or-less randomly. This is called “noise”; what are the causes of this noise? How might you reduce the effects of noise physically or mathematically in your results? Plot your measured intensity profile (by hand, on graph paper) as a function of x for x = 0 to 25 mm. Find the position of the first zeroes (m = ±1) and hence estimate the slit width.

60

Experiments in Optics

Experiment 1b: Determining the diameter of a hair Babinet’s principle Suppose that the amplitude (electric field) of the light wave at a certain point x in a diffraction pattern of a certain object is A1 (x). Now, replace the object by its complementary form; the complementary form can be pictured as the (photographic) “negative”, i.e., the complementary object is transmitting where the origina1 is opaque, and is opaque where the original is transmitting. For example, the complementary object corresponding to a slit is a thin opaque strip or wire. (i.e. a slit has transmission function T1 (x 0 ) = 1 for x 0 < a/2 and T1 = 0 otherwise, the complementary wire has T2 (x 0 ) = 0 for x 0 < a/2 and T2 = 1 otherwise). Let the complementary screen produce an amplitude A2 (x) at point x in the diffraction pattern. Babinet’s principle states that A1 (x) + A2 (x) = A(x), (6.4) where A(x) is the amplitude of the wave at point x in the diffraction pattern when there is no diffraction screen present at all. This theorem results from the reasoning that no screen at all (T = 1 everywhere) = sum of screen (T1 (x 0 )) + complement of screen(T2 (x 0 )) and hence the sum of the two complementary screens’ diffraction patterns = the diffraction pattern with no screen. Hence, according to Babinet, in regions where A(x) = 0, A1 (x) = −A2 (x). Since intensity is proportional to the square of the amplitude (i.e. I(x) = A(x)2 ), the intensity diffraction pattern of the object is (surprisingly!?) the same as the complementary object in these regions. In the case of the set-up shown in Fig. 6.3, observe the distribution of light at the observing screen when there is no diffracting object (i.e. I(x) = A(x)2 ) with the 1m focal length lens in place and record this in your labbook. Now tape a hair over one of the blank slides provided and observe its diffraction pattern. Use Eqs. 6.1, 6.3 and 6.4 to explain the form of the pattern you see and also to find the diameter of the hair. Quote an error value on the diameter. Try using a micrometer to measure the diameter of the hair and compare the answer with your diffraction-based estimates. Have you made any assumptions to produce your estimate of the diameter of the hair using the diffraction method? If so, make sure these are stated clearly in your lab book.

61

Experiments in Optics

6.4

Experiment 2a: Young’s Double Slit Interference and Diffraction Gratings

In this three-hour laboratory session, you will explore more complex features of interference and diffraction. In the first part you will investigate a two-slit diffraction pattern. In the later part you will explore how gratings, made from multiple slits, diffract light. Use the double slit slide, labelled “Slit 2” as the object and place it immediately after the laser output as in Fig. 6.3. The two slits are each supposedly of the same width (nominally that of the single slit in Experiment 1), separated by roughly four times the width. Observe the pattern on the screen using the 1.0 m focal length lens and observing on the screen placed 1 m from the lens. Draw a labelled sketch (with a scale marked) of this diffraction pattern. Using the scale on a ruler, or otherwise, make a crude estimate of the spacing between successive minima of intensity.

P r1 x

S1

d S2

r2 !

! I dsin! L

Figure 4: Geometry for Young’s double slit interference phenomenon: for large L, the

Figure 6.4: Geometry for Young’s double slit interference phenomenon: for large L, the path difference to point P is approximately d sin θ, where d is the slit separation. (ii) Fig. 6.4 shows the geometry used to derive the form of the interference pattern produced by two infinitely narrow (i.e. idealised) slits. If the slits are infinitely narrow, then the difference between the paths S2 P and S1 P is (for a distant screen) given by δ = r2 − r1 ≈ d sin θ

(6.5)

Since the total light pattern at point P is found by summing the contributions from the two ± ± of± the phases) the positions of the bright and dark fringes are given by slits (with due account constructive or destructive interference conditions: MAXIMA : d sin θ =fmλ   1 λ MINIMA : d sin θ = m + 2

(6.6) (6.7)

where m= 0, ± 1, ± 2, ± 3, . . . . 59

In the experiment, we observe in the focal plane of the focussing lens f, and for small angles, sin θ ≈ θ = x/f, so we can write the position of the bright and dark fringes as: 62

Experiments in Optics

λf MAXIMA : xm = m  d  1 λf MINIMA : xm = m + 2 d

(6.8) (6.9)

where m= 0, ± 1, ± 2, ± 3, . . . . The spacing between fringes (between successive maxima or successive minima) is therefore λf/d. For slits of negligible width, the form of the intensity distribution of the interference pattern is, for small angles:   πdx 2 (6.10) I(x) = 4I0 cos λf where I0 is the intensity due to a single source and is assumed to be uniform across the screen. • Does Eq. 6.10 give a good description of the pattern you observed in part (i)? Describe any differences in your lab book. In fact, when the slits have a finite width a, the intensity is equal to the product of the distribution of the single-slit Eq. 6.1 with Eq. refeq:OpticsExperiments:OriginalEq5:InterferenceIntensity and may be written as: "
#2   sin πax πdx 2 λf
I(x) = 4I0 cos (6.11) πax λf λf where a is the slit width and d is the separation of the slits. The pattern is sketched in Fig. 6.5 for the case where d = 4a.

1.0

Intensity

0.8

0.6

0.4

0.2

0.0 -10

-5

0

5

10

Distance

Figure 6.5: Young’s double slit fringes for the case where the separation of the slits equals four times their widths (assumed to be equal). (iii) In this section you are asked to verify 6.9 for the positions of the minima. You should use the slit/photodiode assembly to locate the minima of intensity. Use the arrangement of Fig. 6.3 with the double slit as the diffracting object and the f = 0.5 m focussing lens. Make sure that the distance from the lens to the measuring slit on the photodiode stage is as close as possible to 500 mm, to within a tolerance of ±2 mm. Start near the centre of the pattern and work your way out, searching for each minimum; note the micrometer reading x for as many orders as you can (at least five). You will therefore end up with a table with the following headings: 63

Experiments in Optics Order m

Value of xm

Use the computer program Origin to draw a graph of the values of xm against order m. The graph should be a straight line whose slope is the separation between orders and according to the theory it should equal λf/d. Compute the slope and standard error of the slope of the least squares best-fit straight line through the experimental data. Using λ = 670 ± 1 nm, d = 230±20 µm and f = 500 ± 1 mm, calculate the theoretical value of the gradient and its standard error. Compare the experimentally determined and calculated values and their standard errors. Have you verified the theory to within the limits of random experimental error?

Experiment 2b: N-slit diffraction and diffraction gratings You are provided with slides marked “multiple slits 3”, “multiple slits 4”, etc. up to 6 slits. Take the three-slit slide and observe the diffraction pattern formed by it carefully. You will probably find it a lot easier to see the pattern if you use the eyepiece provided to magnify the patterns as shown in Fig. 6.6. Make a clear record in your lab book. Do the same for the slide marked four slits. bi-convex lens f=1000mm

eyepiece

observing screen

LASER

f diffracting object

Figure 6.6: Arrangement for observation of magnified interference and diffraction patterns using an eyepiece: observations are made on the screen. 18 16

N=4

Intensity (II0 /)

14 12

N=3

10 8 6

N=2

4 2 0 -4

-2

0

2

4

Distance

Figure 6.7: The intensity interference patterns for 2, 3 and 4 infinitely narrow slits of equal separation. Fig. 6.7 shows the interference patterns for two, three and four infinitely narrow slits, all on the same scale. Is this what you see (be honest!)? 64

Experiments in Optics Now quickly look at the interference patterns for five slits and then six slits. Can you guess what the pattern will look like as the number of slits N → ∞? A diffraction grating consists of a very large number of slits. You are provided with three coarse gratings, labelled “1”, “2” and “3”. Observe the diffraction pattern from each one carefully. You might find it useful to compare the patterns with that from the two slit object. For each grating find 1. The width a of each slit (they are nominally all the same). 2. The centre-to-centre separation d of adjacent slits.

6.5

Experiment 3: Imaging with Lenses

The vast majority of instruments that incorporate some optics have simple items such as lenses and mirrors. The function of these elements is usually to provide even, or controlled, illumination of an area (the reflector behind a torch bulb must be one of the “simplest” examples of this) or to relay an image of an object to a specified plane or surface (e.g. a camera lens). In this series of experiments, you will study the imaging properties of lenses using the concepts of geometrical optics. First we start by imaging with a single lens and test the lens formulae that predict perfect images, free of any kind of defect or aberration. Finally we will examine the quality of images using a resolution test chart.

F’ O

I

F s

s’ f

Figure 6.8: A lens forms an image (I) of an object (O) where s is the object distance and s 0 is the image distance. Simple theory of thin lens imaging Fig. 6.8 is a ray diagram illustrating the imaging action of a lens. The object distance s, the image distance s 0 are all measured from the centre of a thin converging lens. (A similar diagram can be made for a diverging lens but we will not be using one in this experiment so will not consider it here). The diagram is an oversimplification and we will later consider some of the features of real lenses, including the quality of the image formation. For small angles, the “thin lens formula” can be derived from this diagram: 1 1 1 + 0 = s s f

(6.12)

with the following sign convention: • Light travels from left to right; s is positive (real) on the left, negative (virtual) on the right: s 0 is positive on the right, negative on the left: f is positive for a converging lens, negative for a diverging lens. (You should be aware that there are other sign conventions). 65

Experiments in Optics The linear magnification m is defined, in this sign convention, as s0 (6.13) s where the negative sign simply indicates that when a real image is formed it is inverted. m=−

List of equipment • • • • • • • • •

Lathe-type optical bench with 4 standard saddles and 1 adjustable saddle Incandescent light source, post-mounted, with power supply. Maximum voltage is 5 V! Plano-convex lens, nominal focal length 100 mm Plano-convex lens, nominal focal length 160 mm Achromatic doublet lens, nominal focal length 160 mm Ground glass viewing screen, 40 mm diameter Set of object slides including a resolution test chart slide and green filter slide CCD camera Plane mirror, 40 mm diameter

Experiment 3a: Testing the lens formula image, observed on ground-glass screen

f

object Light source

s

s’

Figure 6.9: Experimental system for imaging with a positive lens. The lens formula, Eq. 6.12, can be tested and used to find the focal length, by plotting the reciprocal of the image distance, 1/s 0 , against the reciprocal of the object distance, 1/s. The intercept of this graph on the 1/s 0 axis is the reciprocal of the focal length (1/f). Fig. 6.9 shows a simple set-up for imaging with a positive focal length lens. Switch on the light source—the voltage on the supply should be at voltage 5V (max.)—and use the slide with a pin-hole as an object placed in the slot-holder in front of the light source. Use the plano-convex lens of nominal focal length 100 mm. Observe the image of the object slide on the ground-glass screen by looking at the back of the screen and adjusting its position for sharpest image. It may be difficult to see the image until you are close to the image plane and you may need to centre the image on the screen by adjusting the height of the lens. Use your ruler to measure the s and s 0 values over as wide a range as the length of the bench allows. Start with the smallest possible separation (s + s 0 ) between the object and image planes (how is this distance related to f?) and then use 5 or 6 more object-to-image distances, using the whole length of the optical bench. For each object-to-image distance, there will be two positions of the lens that yield a sharp image, so your graph will consist of 10-12 points. Now use computer program Origin to plot a graph of 1/s 0 versus 1/s and also to determine the focal length and its standard error. Instructions on how to use this program are given elsewhere in this manual. Print out a plot of the graph and paste it in your lab book. 66

Experiments in Optics

Experiment 3b: Image quality and resolution of lenses Previously, we considered imaging with “thin” lenses and within a small-angle geometrical optics (ray) approximation. For real lenses a fuller treatment needs to consider also the physical optics (wave) picture. There are two key reasons for loss of quality by imaging with a real lens: i) the finite diameter D of the lens and ii) lens aberrations due to details of a real (“thick”) lens. Consider light from a point object (e.g. point object A in Fig. 6.10) that expands as a spherical wave (only the centre ray is illustrated in Fig. 6.10). Only the part of this wave, that is within the lens diameter, is imaged. The lens, therefore, acts as a limiting (circular) aperture that will diffract light as well as imaging it. The result is that the point object A is not imaged to a point A

B’ 

x

 A’

B

L

L

Figure 6.10: Resolution limit of a lens due to its finite diameter D (and aberrations) but to an image spread function A 0 . For a lens with no aberrations the characteristic angular size of this image is: λ θ = 1.22 (6.14) D for light of wavelength λ. (The 1.22 is a geometrical factor due to the circular shape of the diffracting object). As a consequence, if two point objects (e.g. A and B) subtend an angle of α n(red) then f(blue) < f(red).

Figure 6.13: Chromatic aberration. The doublet lens is made of two glass materials with refractive index properties that partially compensate for this aberration.

Figure 6.14: A doublet lens which can partially compensate for chromatic aberration.

6.6

Experiment 4: Polarisation of Light

The purpose of this session is to give you an introduction to the concept of the polarisation of light. Light from a normal incandescent or fluorescent lamp is usually “unpolarised” whereas laser light is usually linearly polarised—to understand what these terms mean, we have to consider in some detail the nature of light waves. Light is a transverse electro-magnetic vector wave. For light travelling in the +z direction (say) there is a time- and space-varying electric field E(x, t) = E0 cos(ωt − kz) and magnetic field B(x, t) = B0 cos(ωt − kz) , where E and B are vector quantities, transverse to the propagation direction (i.e. they lie in the x–y plane). In free space, there is a simple relation between E and B, so that specifying one of them—usually we choose E—is still a complete description of the field. We can think of E as a two-dimensional vector, with components Ex (z, t) and Ey (z, t) along mutually perpendicular x and y directions at each point in space z and time t. These components oscillate sinusoidally in space and time, with amplitudes E0,x and E0,y and, in the general case with a phase difference δ. Different states of polarisation are the result of different values of E0,x , E0,y and particularly δ, the phase difference. A simple mathematical representation of the field (at z = 0, say) is: ! E0,x E= cos(ωt) (6.15) E0,y Since the field components oscillate sinusoidally, the resultant field is simply the sum of two simple harmonic motions of the same frequency applied in perpendicular directions. Consider the 70

Experiments in Optics case when Ex and Ey oscillate in phase (δ=0) and we get linearly polarised light. Fig. 6.15 shows the case of light linearly polarised (a) along the x-axis, and (b) when it is polarised at angle θ to the x-axis. Fig. 6.15 also shows the case in which E0,x = E0,y and (c) δ = 45 deg (ellipticallypolarised light), and (d) δ = 90 deg (circularly-polarised light). [In the last case, the circle can be seen to be circumscribed by summing the perpendicular components Ex = E0 cos(ωt) and Ey = E0 cos(ωt − π/2) = E0 sin(ωt)].

a

b

1 E  E0   cos(t ) 0

c



 cos( )  E  E0   cos(t )  sin( ) 

E

d

E 0  cos(t )    2  cos(t   / 4) 

E

E0  cos(t )    2  sin(t ) 

Figure 6.15: Representation of polarised light states. (a) linearly-polarised along x-axis, (b) linearly-polarised angle θ to x-axis (c) elliptical polarisation, δ = 45 deg, (d) circular polarisation, δ = 90 deg. In natural light (e.g. light from a blackbody), the amplitudes E0,x and E0,y and the phase difference δ all vary randomly in time and therefore no fixed state of polarisation is observed; we say it is “unpolarised”. Laser light is produced in a cavity whose geometry usually forces a single polarisation state, typically linear polarisation. Light that is initially unpolarised can be made linearly polarised by a variety of means. In the laboratory experiment, we use sheet “Polaroid” in which long-chain molecules are aligned in such a way as to transmit only the component of the electric field along one direction. We refer to this device as a polariser.

Law of Malus If linearly polarised light of unit amplitude (and hence unit intensity) is incident on a polariser whose transmission axis is the same as that of the incident light, then the amplitude and intensity of the transmitted wave is also unity. In practice, there are bound to be losses from true absorption or from surface reflections and the transmitted intensity will be less than unity. A polariser used to study already polarised light is usually called an analyser (obviously, the same device can be viewed as a polariser or analyser depending on its use). On the other hand, if the linearly polarised light has its electric field component perpendicular to the transmission axis of the analyser, then in principle the amplitude and intensity of the transmitted light is zero. Again, in practice, it will have some small, non-zero value because of imperfections in the analyser. If the angle between the direction of the electric field of the linearly polarised light and the transmission axis of the analyser is θ, then the transmitted amplitude (for a unit amplitude incident wave) is cosθ (consider for example the input field has polarisation as shown in Fig. 6.15b and that the analyser polariser is aligned to x-axis). The transmitted intensity is cos2 θ (since intensity is proportional to square of electric field). This is usually written in the form I = I0 cos2 θ,

(6.16)

where I0 is the incident intensity of the linearly polarised incident beam. Eq. 6.16 is known as the Law of Malus. 71

Experiments in Optics

List of equipment • • • • •

Lathe-type optical bench and fittings used for lenses section Red LED light source, with power supply 1 linear polariser (“Polaroid”) 40 × 40 mm 1 analyser (i.e. another linear polariser) in rotation post mount Photodiode (as used in diffraction experiments with slit removed) on translation stage and power supply.

6.7

Experiment 5: Verification of Malus’ Law

Figure 6.16: Set-up for verification of Malus’ Law. Check that the power supply to the photodiode is switched on before taking any measurements. [This is essential as the photodiode will give a reading even without power supply on but it will not be linear with intensity]. Before placing the polariser and analyser in position, use the f = +160 mm focal length singlet lens to focus the central bright spot of the red LED light source onto the photodiode at a magnification of approximately −0.25 (i.e. use Eqs. 6.12 and 6.13. Place the 40 × 40 mm linear polariser and the analyser in its rotation mount immediately in front of the detector, as shown in Fig 6.16. Be careful with the alignment; make sure that the light from the LED is passing centrally through both the polariser and analyser. If you rotate the analyser, you will find that the signal on the photodiode varies, and in principle should be varying as cos2 θ. In order to reduce any background illumination you can make sensible use of black card and switch off your “reading” light if this is practical (but do not keep switching it on and off as this is going to affect your neighbours’ results). Once you have minimised the background illumination, record the intensity I as a function of the angle ν marked on the rotation mount, for increments of ν of say 10 deg between 0 and 360 deg. The angle ν = θ + , where  is an unknown offset which depends on angle of the transmission axis of the linear polariser (this depends on how you mounted the polariser). Plot a graph of I versus ν. Your graph should be cos2 in form (but not necessarily equal to a maximum at ν = 0 deg) on top of a background level. Determine the offset angle  and the mean background level Iback . Plot a second graph of I − Iback versus cos2 (ν − ) using the computer and determine the least squares straight line fit. Have you verified Malus law?

Original Author: Mike Damzen

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Chapter 7 Experiments in Electromagnetism Contents 7.1

Experiment 1: The Parallel Plate Capacitor . . . . . . . . . . . . . . . Experiment 1a: Measuring the capacitance of a known capacitor . . . . . . . Experiment 1b: Properties of the parallel plate capacitor . . . . . . . . . . . Experiment 1c (extension): Properties of dielectrics . . . . . . . . . . . . . . 7.2 Experiment 2: Ampère’s Law . . . . . . . . . . . . . . . . . . . . . . . Experiment 2a: Measuring the magnetic field associated with current flow . . Experiment 2b: Measuring the magnetic pattern from the wire . . . . . . . . Experiment 2c (extension): Magnetic field of coils . . . . . . . . . . . . . . . 7.3 Experiment 3: Faraday’s Law . . . . . . . . . . . . . . . . . . . . . . . Experiment 3a: Repeating Faraday’s experiment . . . . . . . . . . . . . . . . Experiment 3b: Electromagnet and coil . . . . . . . . . . . . . . . . . . . . . Experiment 3c: (extension): Investigating different magnetic cores . . . . . .

73 75 76 77 77 78 78 79 79 79 80 81

These experiments reproduce some of the classic discoveries in electromagnetism, specifically those performed by Ampère and Faraday. These fundamental experiments were later generalised and famously summarised by Maxwell yielding the four equations that are commonly known by his name. In his derivation, Maxwell discovered that one additional term was necessary to add to Ampère’s law in order to ensure full consistency between these equations. Later, when establishing the theory of Special Relativity, Einstein showed that the equivalence of apparently different electromagnetic phenomena were actually manifestations of the same physical process. Therefore the experiments you will perform have not only established electromagnetic theory, but also led to some surprising advances in modern physics. You may wish to review the relevant chapters of your textbook before attempting the experiments.

7.1

Experiment 1: The Parallel Plate Capacitor

A capacitor is simply a pair of conductors that stores charge. The simplest configuration for a capacitor consists of two metal plates separated by a thin insulating layer. Fig. 7.1 shows a capacitor with its plates connected by a wire, effectively short-circuited ensuring that the potential of both plates is the same. In this configuration free (electrons) and fixed (static positively charged metal atoms) charges are uniformly distributed throughout the circuit. 73

Experiments in Electromagnetism

Figure 7.1: Charge distributions on an uncharged and charged capacitor. Now if we add a battery supplying a voltage V, electrons are pulled from the positive terminal and pushed to the negative terminal. The charge distribution in the circuit now looks like the lower part of the diagram with an excess positive charge on one plate and excess negative charge on the other. This separation of charge across the insulating layer sets up an electric field in a direction opposite to that imposed by the battery trying to drive current around the circuit. When the two are equal there is no further flow of electrons. However the separation of charge represents stored electrical energy. If we removed the battery and replaced it with a short circuit electrons would flow around the circuit again (albeit in the opposite direction). This method of storing electrical energy is used in many applications, for example in a camera flash. The quantity of charge delivered to the capacitor plates is given by Q = CV where C is the value of the capacitor measured in Farads (F). The farad is a very large unit, so typically a capacitor would have values of pF or nF. The ability of capacitors to hold charge was discovered in the 18th century when early pioneers in electromagnetism inadvertently gave themselves electric shocks by touching a conductor that had been previously charged. In this lab experiment we will develop a quantitative and painless method for measuring capacitance and then investigate the properties of a capacitor. If a charged capacitor is suddenly connected to a resistor, current will flow from one plate to another through resistor R. Apparatus for performing this is shown in Fig. 7.2.

Figure 7.2: Circuit schematic for an apparaptus which charges and discharges a capacitor. The current that flows through resistor R, will depend upon the voltage across the capacitor V. 74

Experiments in Electromagnetism Recalling Ohm’s law (V=IR) and the definition of capacitance above, we obtain: I=

Q . RC

(7.1)

Recalling that current is the flow of charge in time we obtain the differential equation dQ Q =− , dt RC which if the capacitor has an initial charge Q0 , has the solution   −t Q = Q0 exp RC

(7.2)

(7.3)

Therefore, we can expect an exponential discharge from the capacitor with a time constant of 1/RC. Since Q = CV and we can measure voltage against time accurately with the oscilloscope, this will form the basis for measuring capacitance. Ultimately, the aim in this experiment is to determine the capacitance of the large parallel plate capacitor sitting on your desk, at different plate separations. By plotting a graph of capacitance against separation, it will be possible to estimate the universal constant ε0 .

Experiment 1a: Measuring the capacitance of a known capacitor To develop your method for measuring capacitance, you will first measure the capacitance of a known ceramic capacitor, commonly used in electronic circuits. Rather than switching: between a battery and resistor, as suggested in Fig. 7.2, you will use the function generator to perform this task. Use a 10 V peak-peak square wave, and set the Offset dial so that the waveform starts at 0V and rises to +10 V (rather than the default +5V to −5V). Set the frequency to approximately 5 kHz and check the waveform using the oscilloscope:. Your waveform should resemble that shown in Fig. 7.3a. If you have trouble with the oscilloscope, review the section

Figure 7.3: (a) Pulse shape for charging and discharging the capacitor. (b) Circuit diagram showing the measurement of voltage across the capacitor. on oscilloscopes in the “Introduction to Electronics” earlier in this lab manual. Ensure the trigger is set to Internal, the CAL dial is turned fully clockwise and DC coupling is selected. 75

Experiments in Electromagnetism Now: construct the circuit shown in Fig. 7.3b using the 470 pF capacitor provided, and a resistor of approximately 100 kΩ. Take care to ensure that the oscilloscope is connected with the correct polarity. The negative terminals of the signal generator and oscilloscope are grounded, so must be connected with a common ground, as shown. On the oscilloscope you should observe an exponential rise and decay of the voltage across the capacitor. You may wish to use the second channel on the oscilloscope to superimpose the function generator signal over that measured across the capacitor∗ . What effect does the frequency have on the trace that you observe? Try switching the resistor and capacitor, so that the oscilloscope now measures the voltage across the resistor. Why is this trace different? There are two methods that you can attempt for measuring capacitance. The first involves measuring the voltage across the capacitor and then using Eqn. 1, which re-arranged and with appropriate substitution yields:   −t V (7.4) = ln V0 RC Taking measurements of V at various times t and measuring the peak voltage V 0 , will enable you to plot a straight line graph of ln(V/V 0 ) vs t, which should have gradient −1/RC. Since you know the value of R, you can determine the value of C. It is advisable to take at least ten datapoints to ensure you can show a linear correlation in your data. Alternatively, you can measure the voltage across the resistor and use this to determine the total charge that flowed onto the capacitor. Since the peak charge Q0 will correspond to the peak voltage V0 across the capacitor, the capacitance can be determined from: Q0 = CV0

(7.5)

where to determine Q0 it is necessary to take a series of current readings at various time intervals and integrate to give the maximum charge. ZT Q0 = I dt (7.6) 0

You will need to take a number of datapoints to obtain an accurate integral and you can choose the integration method, either trapezium rule or the simple graphical method of counting squares on standard graph paper. The accuracy of your value for Q0 and hence C will depend upon the number of datapoints you take and the integration method you use.

Experiment 1b: Properties of the parallel plate capacitor Having established a method for measuring capacitance, you can now replace the ceramic capacitor with the parallel plate capacitor on the bench. From theory, we expect the capacitance to scale inversely with plate separation, it being related by the expression below: C=

κε0 A , d

(7.7)

where κ is the dielectric constant, d is the plate separation, A is the plate area and ε0 is the permittivity of free space. While not essential, you may find it convenient to trigger the oscilloscope externally from the TTL output from the function generator. ∗

76

Experiments in Electromagnetism Make a couple of measurements of capacitance at different plate separations (up to 2 cm). Take care to keep the plates parallel and do not let them touch. Plot a graph of capacitance against 1/d to determine the extent to which this expression holds. The dielectric constant of air is unity, so by making an estimate for the plate area, obtain a value for ε0 and compare it with the accepted value of 8.854 × 10−12 F m−1 .

Experiment 1c (extension): Properties of dielectrics On the lab bench, you will find some sheets of material. These have been chosen since they have different: dielectric constants. Referring back to Fig. 7.1, we learned that charge will accumulate on the plates of a capacitor when an external voltage is applied. Placing a material between these plates can polarise the charge on this material which in turn attracts further charge to accumulate on the capacitor plates. The degree to which this occurs depends upon the charge density of the material and leads to a value for the dielectric constant > 1. Determine the dielectric constant of some of the materials on the bench and see if they correlate with known values. You should be aware that the dielectric behaviour of most materials only remains constant over specific frequency ranges. When driven by a continuously varying AC field, the charge will be continuously fluctuating and there will be certain frequencies where this frequency resonates with electronic transitions in the material. At these points the dielectric “constant” will vary abruptly, but away from these resonant frequencies, the dielectric constant will be largely frequency independent. You should bear in mind however that dielectric constants may be quoted at optical frequencies (1014 Hz) and potentially quite different to those that you measure in your low frequency experiment.

7.2

Experiment 2: Ampère’s Law

The magnetic field associated with a flow of charge is described very generally by Ampère’s law. In your electromagnetic lectures, you will consider the general form of Ampère’s law and apply it to specific geometries. In this experiment we are only concerned with the magnetic field associated with electrical current travelling along a straight wire. In this case Ampère’s law can be stated as: µ0 I (7.8) B= 2πr where B is the magnetic field strength, I is the current, r is the radial distance from the wire and µ0 = 1.257 × 10−6 H m−1 is the permeability of free space, as in Fig. 7.4. The field pattern and geometry is illustrated above.

Figure 7.4: Magnetic field B at a radius r from a current I, as in Equation 7.8.

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Experiments in Electromagnetism

Experiment 2a: Measuring the magnetic field associated with current flow You will start by repeating work performed in 1820 by Ørsted who was among the first to notice that a compass needle was deflected by the passage of current along an adjacent wire. You will use the 2 A power supply as a current source. Since the power supply can control either voltage or current, you should set the voltage at open-circuit to a large value, say 10 V. This will ensure that the power supply limits the current flow, without restriction on the voltage necessary to achieve the specified current. Next, suspend a 30 cm length of wire between the two retort stands provided. Place the compass on a stage below the wire roughly equidistant between the retort stands. The B field produced by the wire will deflect the compass needle, but can be disturbed if the compass is too close to the retort stands (which may be weakly magnetised) and power supplies where the transformer can cause distortion of the B field. You now want to position the compass so that it sits just below the wire (a couple of millimetres at most). This can be done using the plastic spacers to raise the compass above the stage. To achieve maximum deflection, you will want to arrange the experiment so that the magnetic field due to the wire (BF ) is perpendicular to the Earth’s magnetic field (BEarth .); the deflection angle can then be related to the relative magnetic field strength of the wire to the Earth’s magnetic field. Once you’ve decided on the direction, pull the wire taut, then wrap it round the clamp stand and tape it down. Ensure the wire is as straight and with as few kinks as possible. Having excess wire at each end is not a problem, so long as it is kept away from the point of measurement. Connect the wire through the multimeter via the 10 A socket to the power supply. Turn on the power supply and set the current to maximum. You should observe a deflection of the compass needle. It may be necessary to gently tap the compass to encourage the needle to move. Experiment with your technique to find a method that yields consistent results. Turn the power supply off when you are satisfied that you can observe the effect. You now want to make some careful measurements of the deflection of the compass needle. Make a note of the initial angle of the compass needle and estimate the separation between the needle and the wire. Vary the current passing through the wire, ensuring to use the maximum range available and record the deflected compass needle angle. For each measurement, tap the compass first to force the needle to move, as sometimes it gets stuck, and then measure the new needle position. Plot a graph of Tan(displacement angle) against current. If Ampère’s law is obeyed you should obtain a straight line, the gradient of which will enable you to determine the Earth’s magnetic field BEarth (with associated uncertainty).

Experiment 2b: Measuring the magnetic pattern from the wire Ampère’s law predicts that the magnetic field strength will scale with the inverse of the radial distance from the wire. You can measure this by using the same apparatus as above, but setting the current to maximum (2 A) and varying the separation between the compass and the wire by removing the plastic spacers. If Ampère’s law holds, a graph of magnetic field strength vs 1/r will yield a straight line. Consider carefully the measurement intervals for r since you will want roughly equal spacing of datapoints on a graph of field strength against 1/r. You can again estimate the value of BEarth and compare with your earlier value.

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Experiments in Electromagnetism

Experiment 2c (extension): Magnetic field of coils The magnetic field associated with a single straight wire is relatively weak, but can be enhanced by winding the wire into a coil; each coil effectively adding to the field strength of the last. Make a simple coil by winding wire around a pen or pencil and investigate the field strength and pattern by placing the compass at different locations around the coil. From your observations, try to establish a relationship between the number of coils and the magnetic field strength.

7.3

Experiment 3: Faraday’s Law

In 1831 Faraday performed what looks like a very simple: experiment with a coil and bar magnet shown in Fig. 7.5. Faraday observed that a changing magnetic field induces an electric field, or in the case of this experiment, the motion of a permanent bar magnet results in a voltage appearing across the ends of a coil. We can express Faraday’s findings in the simple expression: ε=

dΦ dt

(7.9)

where ε is the induced electromotive force (voltage) and dΦ/dt is the rate of change of magnetic flux through a single loop. The consequences of Faraday’s experiments are profound; in particular the equivalence of moving either the magnet and coil which ultimately led Einstein to the foundation of Special Relativity some 74 years later∗ . Einstein’s work goes beyond what we can reasonably cover in a lab experiment but the experiment you are about to perform puzzled some of the finest minds of the 19th century and ultimately led to one of the landmark achievements of modern physics.

Experiment 3a: Repeating Faraday’s experiment You: will first reproduce Faraday’s experiment shown in Fig. 7.5 using a coil, your oscilloscope

Figure 7.5: Schematic diagram of Faraday’s classic experiment. and a permanent magnet. On the bench you will find two coils, a large 100 turn coil and a smaller 50 turn coil. Connect the smaller coil to the oscilloscope and try moving a permanent magnet through the coil. The time base setting is unimportant since you are simply observing the effect of the non-periodic motion of the magnet on the current in the coil. Observe the effect of moving the magnet faster or slower through the coil and does the orientation of the magnet matter? Spend 10 minutes experimenting and recording your findings in your lab book. Comment on whether your observations are consistent with ε = dΦ . dt ∗

Introduction to Electrodynamics, D.J.Griffiths, 3rd Ed, Addison Wesley (1999).

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Experiments in Electromagnetism

Experiment 3b: Electromagnet and coil To make a more controlled experiment it is convenient to apply your findings of Ampère’s law to this experiment. We will replace the permanent magnet with an electromagnet and check that the relative motion of these two coils also obeys Faraday:’s law. Connect the larger coil to the power supply as shown in Fig. 7.6 and pass 2 A of current though it. Try moving the two coils relative to each other and check that Faraday’s law still applies. Next:, with the coils arranged

Figure 7.6: Concentric coil arrangement with the primary coil connected to the PSU, the secondary coil connected to the oscilloscope. as shown in Fig. 7.6, observe what happens when you turn the power supply off. The current in one of the coils will drop abruptly and induce a large voltage momentarily in the second. You may be able to record a peak voltage on the oscilloscope but to obtain quantitative results, a more controlled means of controlling dΦ/dt is required. The function generation provides a convenient means for controlling dΦ/dt. Since we need to pass an appreciable current through the large, primary coil, the output of the function generator is fed into an amplifier and then passed into the coil via a 18 Ω resistor∗ . Set: up the circuit

Figure 7.7: Concentric coil arrangement with the primary coil driven using a periodic signal supplied by the function generator. shown in Fig. 7.7 and configure the function generator to about 5 kHz, square wave, zero offset and about 1 V pk–pk amplitude. The amplifier is connected to the signal generator using a BNC to Phono cable. You can use the oscilloscope to measure the output from the amplifier, the magnetic field produced by the primary coil and the induced voltage in the secondary coil. Start with the dial on the amplifier low and increase until you can just detect a signal; you can burn out the resistor if it is turned up too high. With the square wave, you should observe something similar to switching the power supply on and off, since dΦ/dt is zero except when there are abrupt changes in primary coil current. The The amplifier and resistor are required since the output impedance of the function generator is too high (600 Ω) to drive any appreciable current through the coil. For reference, the output impedance of the amplifier is about 4 Ω, so can drive a load of roughly similar impedance efficiently. ∗

80

Experiments in Electromagnetism principle advantage of this arrangement is that the signal is now periodic and more easily measured on the oscilloscope screen. Switch to the triangle waveform and observe the waveform appearing across the secondary coil. You can use both channels on the oscilloscope to monitor the waveform across both coils. Does the induced voltage that you measure correspond to what you would expect from Faraday’s law? What is the effect of using a sine wave, does the shape of the waveform change?

Experiment 3c: (extension): Investigating different magnetic cores When measuring capacitance with the parallel plate capacitor, you observed that different dielectric materials can increase the ability of the capacitor to store charge. Similarly inserting a core into our coils, we can concentrate the magnetic field in the core. This arises since the magnetic field produced by the primary coil, serves to align the internal magnetic moment of electrons in the core material with the applied field. When the electrons align parallel to the field, the effect is called paramagnetism, when in opposition to the applied field the effect is called diamagnetism. Some materials exhibit persistent magnetisation parallel to the applied field and are called ferromagnetic. Inserting a core into the coil with a different magnetic material has the result that the magnetic field is concentrated in the core of the material. Inside the core (and therefore in its vicinity) the permeability in Ampère’s law is no longer that of free-space but modified by the material to µ = κm µ0 . The effect is small for paramagnetic and diamagnetic materials∗ , but extremely large for some ferromagnetic materials where κm can exceed 3000. It is convenient to characterise materials using the magnetic susceptibility χm = 1 − κm ; some typical values are given in Table. 7.1. On the bench you will find several different cores that you can insert into the smaller coil. Investigate the effect of inserting these different cores into your two coils. Do they behave as you would expect? If not, can you find an explanation for their behaviour? Returning to the method used to investigate Ampère’s law, you can use the deflection of the compass needle as a means of assessing the magnetic field strength due to a coil wound around an iron core. A large nail is provided for this purpose. Try to estimate the magnetic field strength of your electromagnet with and without the core. Material

Type

Uranium Paramagnetic Aluminium Paramagnetic Carbon Diamagnetic Copper Diamagnetic Iron Ferromagnetic Nickel-Zinc Ferrite Ferromagnetic

Susceptibility 4.0 × 10−4 2.1 × 10−5 −2.1 × 10−5 −9.7 × 10−6 3, 000 20–15, 000

Table 7.1: Selected magnetic materials and their properties.

The diamagnetic behaviour of carbon and water is responsible for a remarkable demonstration of magnetic levitation of a frog reported in “Everyone’s Magnetism” by A.Geim, Physics Today, Sep.1998, page 36-39 http: //scitation.aip.org/content/aip/magazine/physicstoday/article/51/9/10.1063/1.882437 ∗

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Chapter 8 The Grating Spectrometer Contents 8.1 8.2 8.3

8.1

Introduction . . . . . . . . . . . . Setting up the spectrometer . . . Spectrometer theory . . . . . . . . Experiment 1: The Rydberg constant . Experiment 2: Spectrometer resolution

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82 83 84 86 87

Introduction

A spectrometer is an instrument used to study the spectrum of light. The light source can be anything from a cryogenically cooled crystal to a superhot plasma or a star, and the “light” might be anything from the longest infrared all the way down to γ-rays. In the case of light the spectrum generally takes one of two forms: hot solids, such as a tungsten lamp filament, emit spectra where the intensity is a slowly varying function of wavelength- a “continuous spectra”. Hot or electrically excited gasses emit spectra in which the emission is concentrated at a few well defined wavelengths- called “emission line spectra”, from the appearance of the slit image of the spectrometer. In such a spectrum the ratio of intensities of spectral lines can tell you the temperature of the gas, while the line widths can give you the density of the emitting gas or, in an electrical discharge, the density of the free electrons in the discharge. Spectroscopy is a “non-invasive” technique and is one of the most powerful tools available to physicists studying plasmas, stars, flames, semiconductors, etc. In its simplest form, a spectrometer uses a lens or mirror to produces a collimated beam of light and disperses it with a grating or a prism. The dispersed spectrum is then focused and viewed with an eyepiece (or recorded with an electronic detector). This experiment has two parts. In the first you will measure the spectrum of hydrogen and use your data to deduce the Rydberg constant. In the second, you will investigate the resolving power of a grating spectrometer. this is a measure of the smallest wavelength difference that can be resolved in a spectrum. These experimental tasks are open ended—you might want to put more effort into exploring systematic effects in the Rydberg measurement at the expense of determining the resolving power, or vice versa.

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The Grating Spectrometer

8.2

Setting up the spectrometer

Fig. 8.1 shows the important parts of the spectrometer. The sequence of operations below should get the instrument properly aligned and focused.

Figure 8.1: The spectrometer. Light from the source passes through a narrow slit and is made parallel by the collimating lens. It strikes the diffraction grating at normal incidence and the diffracted light is focused by the camera lens onto the crosshair “×”. A magnified image is viewed using the eyepiece. The inset shows the path length difference d sin θ. 1. Focus the eyepiece on the cross wire. Ideally this should be done so that the image is at a convenient viewing distance, say about 30 cm. The aim is to find a setting that is free of eyestrain. It is better done with both eyes open. 2. Focus the camera lens so that a distant object is focused on the cross wire. Distant in this context means 5 to 10 metres; focused means no parallax between the image and the cross wire. 3. Set the telescope (camera lens and eyepiece) so that it is aligned with the axis of the collimator. Position the hydrogen lamp close to the slit, so that it is illuminated. Without changing the camera focus adjust the focus of the collimator until a sharp image of the slit is focused at the cross wire. You might need to adjust the slit to be near its minimum width while carrying out this step. The cross wires are easier to use if set as an “×” rather than as a “+”. 4. Check out how the clamp screws and the slow motion drives work, then align the camera and collimator so the cross wires are set on the slit image. Note the reading of the two scales. The main scale is divided into thirds of a degree—the vernier thus reads in arc minutes, or sixtieths of a degree. Check with your partner and fellow students to make sure you are reading the scales correctly, and discuss with your demonstrator if you are uncertain. 5. Move the telescope through precisely 90 deg and clamp it in position. Put a grating in the holder, being careful not to touch the (fragile!) surface and to get the rulings approximately vertical. 6. Turn the grating table until the grating surface is at 45 deg to the incident beam from the collimator. You will see a reflected image of the entrance slit in the eyepiece. Adjust the height of this image using the adjustment screws under the grating table. You may need to 83

The Grating Spectrometer use a shortened slit. Use the grating drive to set the grating so the slit image is precisely at the cross wire. 7. Note the reading. Rotate the grating through precisely 45 deg, using the scales, so that it is accurately set at 0 deg (normal) incidence. 8. Move the telescope out to the way and look into the grating. This should give you an approximate idea of the positions of the spectral lines. Move the telescope around to find the diffracted image of the slit- choosing one at a large angle of diffraction. If the grating rulings are not parallel to the rotation axis of the telescope, this image will be above or below the plane defined by the cross wire. Use the grating adjustment screws (think carefully about which one or ones to use) to rotate the grating in the plane of its surface until the image is at the right height. Check by going to large angle on the other side of the grating normal. The spectrometer is now set up to make accurate, repeatable measurements. It is probably the most accurate instrument you will meet in the lab. In careful hands and using the average of a dozen or so measurements, it is capable of a precision of under three parts in ten thousand, or about ±0.2 nm for yellow-orange light.

8.3

Spectrometer theory

The grating can be considered as a large number N of equally spaced very narrow slits, with slit separation d. For light of wavelength λ a principal maximum occurs when the phase difference between neighboring rays is an integer multiple m of 2π: phase difference = path difference ×

2π = 2πm. λ

(8.1)

The path difference is d sin θ, thus 2π d sin θ = 2πm λ which means

d sin θ = mλ

(8.2)

m is called the order number. This path difference is shown in the inset in Fig. 8.1. Fig. 8.2 shows a fringe pattern for the case N = 4. With N slits illuminated, the first minimum occurs at a phase angle φ = 2πm +

2π N

We find the resolving power by considering the simple criteria that the minimum of one diffraction pattern occurs at the same position as the maximum of another pattern, as shown in Fig. 8.3 Light of a second wavelength λ2 = λ − ∆λ will produce its mth principal maximum at a slightly different angle θ2 given by d sin θ2 = mλ2 . It will have a minimum at angle θ if 2π 2π d sin θ = 2πm + , λ2 N λ2 that is, d sin θ = mλ2 + . N 84

(8.3) (8.4)

The Grating Spectrometer

Figure 8.2: The diffraction pattern for N = 4 slits, showing the principal maxima and minima.

Figure 8.3: Diffraction patterns and resolution. The principal maximum of one pattern occurs at the same position as the first minimum of the other.

85

The Grating Spectrometer Taking Eqs. 8.2 and 8.4, we find

λ2 . N We have defined λ − λ2 = ∆λ, where ∆λ  λ. This means mλ = mλ2 +

λ = mN. ∆λ

(8.5)

At this separation the intensity at a point half way between two equal lines is about 20% less than the peak and the two lines can easily be seen to be separate.

Experiment 1: The Rydberg constant Transition wavelengths in hydrogen are given by the remarkably simple formula   1 1 1 = −R∞ − λ n2 p2

(8.6)

where n and p are integers which represent the energy levels of the initial and final states of the atom, respectively. R∞ is the Rydberg constant, which has the measured value 10 973 731.6 m−1 . It can be expressed in terms of more fundamental constants as R∞ =

α2 me c , 2h

where me is the mass of the electron, c the speed of light, h Plank’s constant and α the fine structure constant: e2 . α= 4π0 hc ∼ 137.036. It is dimensionless.∗ Note that 1/α = The Balmer series in hydrogen is given by Eq. 8.6 with p = 2 and n = 3, 4, 5 . . .. This is to say that the atom is dropping down from the nth energy level to the second (p = 2) energy level, giving off a photon of a specific wavelength that depends on n. Balmer transition wavelengths are in the visible part of the spectrum. From Eq. 8.6, you can calcuate the values of the wavelengths, and using a standard colour spectrum chart, determine the colours that you can expect the light to be visible as. The aim of this part of the experiment is to use the spectrometer to measure the wavelengths via the grating equation (Eqn. 8.2) and to thereby deduce R∞ , and its uncertainty. For each transition wavelength, measure θ as a function of order m, considering both positive and negative values. Before recording data determine the optimal slit size for resolution and intensity. Do not change the slit size once you start taking data. You and your partner should alternate readings of the same line to get a value for the uncertainty in θ. Eq. 1 implies a plot of sin θ vs. m will yield a straight line. You will probably want to use a computer to determine the slope and its uncertainty. You might also want to consider other ways to analyze your data. Recall that d (sin θ) = cos θ dθ . With practice you can make wavelength measurements quite quickly and thus build up a large database. (You will probably want to computerize your analysis.) This means your statistical ∗

see http://physics.nist.gov/constants

86

The Grating Spectrometer

Figure 8.4: Representation of two Balmer transitions in a hydrogen atom: initial state n = 3, final state p = 2, and similarly for n = 4, p = 2. uncertainty should be good, but there are numerous possibilities for systematic error in this experiment.∗ Some things to consider are the effects of spectrometer misalignment, defocus, lamp position, grating period, air pressure and humidity, background light etc. Some of these you can test experimentally— all must be estimated using combinations of measurement, logic and well-justified assumptions. As always, these estimations (of the uncertainties) are just as important as the measurement of the central values. Discuss your strategy with your partner, taking on board any ideas that you or your other fellow students may have, and clearly indicate your approach in your lab book.

Experiment 2: Spectrometer resolution Use the spectrometer to test Eq. 8.5. With the sodium source, find an order where you can just resolve the two lines of the yellow doublet. The yellow (or orange, as color can be subjective) sodium wavelengths are 589.0nm and 589.6nm. Gradually reduce the effective number of grating lines by sliding pieces of card just in front of the grating, parallel to the lines. Do not touch the grating with your fingers! Try to determine the number of lines N at which you just fail to resolve the doublet. Repeat this measurement several times and calculate the mean and standard error. Compare your result for ∆λ derived from Eq. 8.5 to the known value. If you have time measure the separation of the pair of green lines at 569 nm or red lines at 616 nm. Is the separation equal to that of the orange lines in wavelength units and/or in frequency units? What does this tell you about common levels for these transitions? At 589 nm, ∆ν = 515.5 GHz.

∗

You might want to reread the discussion at the end of the speed of light script.

87

Chapter 9 The Charge-to-Mass Ratio of the Electron Contents 9.1 Background . . . . . . . . 9.2 Experimental procedure . 9.3 Details . . . . . . . . . . Appendix 9.A Helmholtz coils

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88 89 91 92

In the Standard Model of Particle Physics, the fundamental properties of the electron are its charge and mass, which are fundamental constants, and its spin and lepton number, which are quantum numbers. It is not easy to measure the electron’s mass directly∗ . It is usually easiest to apply an electric or magnetic force to the electron to measure its charge-to-mass ratio and deduce its mass from an independent measurement of its charge. In this experiment you will determine this fundamental ratio. The apparatus that we have to measure the charge-to-mass ratio of the electron consists of an electron source in an evacuated tube and a set of Helmholtz coils † that produce a uniform magnetic field. The tube contains a trace amount of hydrogen gas that is ionized through electron-molecule collisions and the resulting emission allows you to see the trajectory of the electron beam. By varying electrical potentials applied to the source you can vary the energy and focus of the electron beam.

9.1

Background

A particle of mass m with a charge q moving with velocity v in a magnetic field B experiences a force given by the Lorentz force law: F = q (v × B) .

(9.1)

Because of the cross product, the force is always at right angles to the motion, thus the magnitude of the velocity never changes.‡ For some radiusr, the magnitude of the Lorentz force will just The most accurate measurement is to measure the ratio of the electron’s mass to that of an atomic nucleus, and then converting to absolute mass. The conversion factor has a larger uncertainty than the measurement itself † The magnetic field produced by the Helmholtz coils is discussed in the Appendix. ‡ See Young and Freedman, §27.4. ∗

88

The Charge-to-Mass Ratio of the Electron equal the centripetal force needed to keep the particle in circular motion, FC = m

v2 . r

(9.2)

Equating the magnitudes of these forces, we can solve for the charge-to-mass ratio, q v = . m rB

(9.3)

Our apparatus produces a beam of electrons, so we make the substitutions q → −e , m → me . The electron beam is accelerated by an electrical potential U, so an individual electron has kinetic energy eU. This allows us to determine the electron’s velocity through the equality 1 eU = me v2 . 2

(9.4)

Substituting this into Eq.9.3, it is easy to show (you should verify this!) 2U e = . me (rB)2

(9.5)

The experimental strategy should be clear: you can control U through the voltage applied to the electron source and B through the current in the Helmholtz coils. These both affect the radius of the electron beam. By finding the different combinations of B and U that result in the same radius, or, more genearlly, by measuring the radius r for different values of these parameters you can determine the charge-to-mass ratio of the electron.

9.2

Experimental procedure Safety warning: The electron tube operates at high voltage, up to 500 V. All connections should be made using the special shielded banana cables, not ordinary cables. The tube itself is made of very thin glass. Take care not to bump it or drop things on it, as it will implode if broken.

The electrical connections to the electron tube are shown in Fig. 9.1. It should already be connected to its power supply, but it is worth understanding how the tube works. The cathode is heated by the current from a 6.3 VAC supply. The hot cathode emits electrons via thermionic emission. The cathode is held at ground potential, 0 V. It is surrounded by a metal cylinder with a small hole in its end, called a Wehnelt cylinder. The Wehnelt cylinder is held at a negative voltage with respect to the cathode and helps to focus the electron beam. You will probably find a bias voltage of about −20 V to be optimal. The anode is the conical metal structure you can see in the tube. It is held at a large positive voltage, typically about 250 V. The electrons born at the cathode are accelerated towards the anode and emerge from the hole in its tip. There are two small deflector plates near the anode; they should be electrically tied to the anode using the terminals on the tube base. One meter should be set up as a voltmeter to monitor the anode voltage. The Helmholtz coils are powered by the separate DC supply, you can use its built in ammeter to monitor the coil current. The B field at the centre of the coils is proportional to the coil current I. It has the theoretical value  3/2 n 4 I ≡ β I, (9.6) B = µ0 5 R 89

The Charge-to-Mass Ratio of the Electron where µ0 = 4π · 10−7 N A−2 , n = 130 is the number of turns in each coil and R is the coil radius, nominally 150 mm. For convenience we have defined β as the proportionality between the B field and the current.

Figure 9.1: Electrical connections to the electron beam tube. We can rearrange Eqs. 9.5 and 9.6 to obtain U=

e r2 β2 2 I, me 2

(9.7)

in other words, a plot of the anode voltage U vs. I2 for an electron beam with a given radius r should give a straight line whose slope is proportional to the charge-to-mass ratio. 90

The Charge-to-Mass Ratio of the Electron Starting with U = 250 V, adjust I to given an electron orbit with diameter about 8 cm. The set-up incorporates a mirror reflector, and some sliding plastic markers two parallel rails on each side of the tube. Think about how these can be used to help you measure the radius of the electron orbit and the errors on this measurement, in a stable way such that the measurements can be compared over the several hours that you will be using the apparatus. Make five or so additional measurements with U between 200 V and 300 V, adjusting the coil current as necessary. The electron beam loses energy due to collisions with the background hydrogen gas. According to Eq. 9.5, lower energy electrons follow orbits with a smaller radius. This implies that you should measure the outside diameter of the beam to measure the highest energy electrons. On the other hand, the electrons in the beam repel each other, causing the beam to expand. At the same time the ionized background gas tends to cancel this space charge and therefore focus the beam. You might think this would mean you should measure from the centre of the beam. The manufacturer of the equipment does suggest using the outside edge, but you should at least consider this as a possible source of systematic error. Analyse your (U, I) data to extract a value of e/me . Remember the diameter of the orbit is 2r! You should also calculate the uncertainty in your measurement. There are a few different approaches to this, so discuss your strategy thorougly with your partner at this point, and consult your fellow students for ideas. If uncertain, you may want to discuss your analysis strategy further with your demonstrator, who can help you think through the process.

9.3

Details

The expression for the B field given by Eq. 9.6 is only exact for the single point in the centre of the coils. You should take additional sets of measurements for smaller beam diameters to see if your e/me ratio varies systematically with the orbital diameter. The earth’s magnetic field adds an offset to the field generated by your coils. If you rotate the apparatus by 180 deg and repeat your measurement for a particular diameter, you can subtract off this background. In fact, you can also use your results to get the value of the component of the earth’s field parallel to the coil axis. Assuming the spreading in the beam is due entirely to scattering energy loss, can you determine how much energy the beam loses? Hint: how much voltage is required to move the beam by its width? It probably makes sense to use the centre of the windings on the coils to define R in Eq. 9.6. How much difference would it make to your results if you took the inner or outer edge instead? Can you think of any other factors which could systematically alter your results? The accepted value of e/me is 1.7588 × 1011 C kg−1 .

91

The Charge-to-Mass Ratio of the Electron

Appendix 9.A

Helmholtz coils

It is well known∗ that the magnetic field at a distance z along the axis of a current loop of radius R lying in the x–y plane is R2 . (9.8) Bz = µ0 I 2 (R2 + z2 )3 /2 We can simplify the notation by measuring all distances in units of R. The field from two coils separated by their radius, i.e. located at ±R/2, is then   1 µ0 I . (9.9) Bz = (1 + (˜ 2 z)) At z˜ = 0, this reduces to

 3 4 /2, Bz = µ0 I 5

(9.10)

which is exactly Eq. 9.6 when we take account of the number of turns and put back in the radius R. The magic of Helmholtz coils is that both dBz /dz and d2 Bz /dz2 vanish at the centre.† This means the magnetic field the coils produce is very uniform. In fact, it is possible‡ to derive a general expression for all of the field components. It turns out the field is also very uniform as you move in the x and y directions around the origin, though the mathematics becomes a bit messier.

∗ † ‡

See Young and Freedman, §28.5. See also problem 28.67. The first non-constant term in a series expansion of Eq. 9.9 around z = 0 goes as z4 . See §5.5 in J. D. Jackson, Classical Electrodynamics, 2nd Ed., (Wiley, New York, 1975). Original Author: Ben Sauer

92

Chapter 10 The (not so) Simple Pendulum Contents 10.1 Introduction . . . . . . . . . . . . . . . . 10.2 Experiment 1: Amplitude dependence . . 10.3 Experiment 2: The equivalence principle 10.4 Experiment 3: Coupled pendulums . . . . Appendix 10.A Finite Amplitudes . . . . . . . Appendix 10.B Coupled Pendulums . . . . . .

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The aim of this experiment is to explore some relatively complicated physics using a simple measurement device, the pendulum. There are three main topics: measuring the amplitude dependence of the pendulum’s period, measuring the mass dependence, and exploring the physics of coupled oscillators using two pendulums. You should start with experiment 1 to familiarize yourself with the measurement technique, but feel free to do either or both of the other two experiments. The apparatus for this experiment is simple and data collection is very easy. It is very important that you take the time to analyse your data as you go along—in particular it is vital that you calculate the uncertainties in your measurements.

10.1

Introduction

A simple pendulum is an ideal way to demonstrate simple harmonic motion. Figure 1 shows the force on a simple pendulum of length lwhich makes an angle θ with the vertical. For a bob of mass m Newton’s second law is − mlθ¨ = mg sin θ. (10.1) We make the approximation sin θ ≈ θ , which leads to simple harmonic motion with angular p frequency ω = g/l . It is easiest to measure the period T of the pendulum, which is given by s l T = 2π . (10.2) g

93

The (not so) Simple Pendulum

Figure 10.1: The simple pendulum.

10.2

Experiment 1: Amplitude dependence

The expression for the period given by Eq. 10.1 has no dependence on the amplitude of the pendulum. Appendix 1 presents the mathematics of the finite amplitude pendulum, but you might want to consider this an experimental question: Is there any change in period with amplitude? Set up a simple pendulum using one of the brass bobs. Use a thread around 30 cm to 40 cm long clamped to one of the bars attached to a retort stand. The threaded rod the bob attaches to allows you to make fine adjustments to the length, but for this experiment the length should stay fixed. Using the stopwatch, measure the period of your pendulum after starting it with a very small amplitude. Use another retort stand to set up a reference point to measure the initial amplitude. You will have to decide how many swings to time to produce an acceptable uncertainty on the period. Aim for at least 1%, but 0.1% is not out of the question. Is the optimal point to time the swing at the centre of the swing or at the end? The answer is either blindingly obvious or completely counterintuitive. Discuss this with your partner and fellow students before you start. If uncertain, you can ask your demonstrator to help you figure this out. You now want to determine if the period changes as the amplitude increases. Measure the period for several larger amplitudes. Make a plot of period vs. amplitude as you go along (graph paper is available from the lab office). The change in period is likely to be small, so make the vertical scale about five times the uncertainty in your first measurement. Does what you find make any sense with regard to the theory in Appendix 1? Is it more accurate to measure l using a meter stick or to calculate it from the period? To answer this fully you might need to consider local value of g as compared to the global average.

10.3

Experiment 2: The equivalence principle

The mass m in Eq. 10.1 is the same on both sides of the equation, but on the left it is really the inertial mass and on the right the gravitational. There are good theoretical reasons to suppose 94

The (not so) Simple Pendulum they are the same, but their equivalence is certainly open to experimental test. If we call these mi and mg , the period of a simple pendulum is given by the formula s mi l . (10.3) T = 2π mg g Identically sized pendulum bobs made of tungsten, brass, aluminium and nylon are available. By measuring the period using these different materials you can put a limit on the difference between gravitational and inertial mass. You will have to think carefully about systematics. In particular, you should try to keep the length constant by swapping the different bobs onto the same holder. How well can you say the period is independent of the mass? A simple way to interpret your data is to assume that mg /mi = kρ + a0 , where k and a0 are constants and ρ is the density.∗ You might expect a0 to be 1, but are interested in determining a limit on the size of k, which of course depends on your measurement uncertainties and systematics. These sort of experiments date all the way back to Newton. A modern interpretation is that you are measuring the equivalence principle for bodies with different ratios of protons to neutrons. These are usually called “Eötvös experiments”, eg. S. Schlamminger et. al., Phys. Rev. Lett. 100, 041101 (2008). This work used a torsion balance to eliminate some of the systematic effects you undoubtedly have discovered.

10.4

Experiment 3: Coupled pendulums

Figure 10.2: The coupled pendulums. The bobs oscillate in and out of the plane of the paper. Set up the coupled pendulums as shown in figure 2. Fix one end of the support thread to a retort stand and run the other over the pulley to the mass hanger. Use the two brass bobs and start with about 50g on the hanger to provide tension in the support thread. Make d about 10 cm and L about 20 cm. (You can make L longer to increase the coupling.) Each pendulum should be about 30 cm long. It does not matter too much what their length is, but if they are very long your measurements will take more time to complete. It does matter that their periods are the same. Measure each period with the other fixed (by stopping it against a bar attached to a retort stand) and use the threaded holders to make fine adjustments. A 60dB @ 50/60Hz CMR: >90dB @ DC/50Hz/60Hz

45Hz - 400Hz 400mV 4V 40V 400V 750V

0.5% ± 4 dig.

400Hz - 5kHz 1% ± 4 dig.

5kHz - 20kHz 3% ± 4 dig.

2% ± 4 dig. 3% ± 4 dig.

5% ± 4 dig.

1% ± 4 dig.

100μV 1mV 10mV 100mV 1V

Accuracies apply for readings between 10% and 100% of full scale. Additional error at crest factor = 3 is typically 1%. Input impedance = 10MW nominal. Max. input = 750V rms, 1kV pk. (265V rms on 400mV range). 1kW unbalanced CMR = >60dB at DC or 50Hz (60Hz rejection available as factory option).

RESISTANCE 0.15% ± 6 dig.* 0.1% ± 4 dig. 0.1% ± 4 dig. 0.15% ± 4 dig. 0.3% ± 6 dig. 1.0% ± 10 dig.

10mW 100mW 1W 10W 100W 1kW

4kHz 40kHz

* 400W specification applies after null Max. input 265V DC or AC rms on any range. Max. open circuit voltage 4V 40MW accuracy applies up to 20MW thereafter add 1%

Sensitivity selected by AC range setting

FURTHER FUNCTIONS Continuity Selects 4kW range and sounds audible tone for impedance