The University Of Reading Department Of Physics

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The University of Reading Department of Physics Experimental Physics 1 An instruction manual to accompany module PH1004  Copyright 1996-2005 Department of Physics, The University of Reading All rights Reserved Last Revised August 2005 University of Reading Department of Physics Contents Disclaimer and Safety Assessment .........................................................................................................2 Chapter 1 Introduction to Experimental Physics ..........................................................................3 Chapter 2 Assessment .......................................................................................................................5 Chapter 3 Specimen Experiment - Charge Transport in Materials ..............................................8 Chapter 4 Statistics..........................................................................................................................17 Chapter 5 Introduction to Skills Sessions .....................................................................................19 Skills Session 1 Data Collection ......................................................................................................20 Skills Session 2 Graph Plotting.......................................................................................................23 Skills Session 3 Uncertainties and Errors .....................................................................................31 Skill Sessions 4 Electronic Instrumentation..................................................................................39 Project 1 Electricity ..........................................................................................................................41 Experiment A: - DC Networks.................................................................................................................. 41 Experiment B: - Resonance....................................................................................................................... 44 Project 2 Waves and Interference....................................................................................................47 Experiment A: - Optical Interferometry................................................................................................... 47 Experiment B: - Sound Waves .................................................................................................................. 51 Project 3 Applications of Electronics...............................................................................................56 Experiment A: - The Strain Gauge........................................................................................................... 56 Experiment B: - Electrons and Semiconductors...................................................................................... 61 Project 4 Classical Physics ................................................................................................................63 Experiment A: - The Charge to Mass Ratio of the Electron................................................................... 63 Experiment B: - Angular Momentum ....................................................................................................... 66 Project 5 Spectroscopy ......................................................................................................................68 Experiment A: - The Hubble Redshift ...................................................................................................... 68 Experiment B: - Atomic Spectroscopy...................................................................................................... 72 Electronics 1: - Operational Amplifiers – Theory ..............................................................................76 Electronics 2: - Operational Amplifiers – Practice.............................................................................82 Electronics 3: - Digital Electronics – Theory.......................................................................................84 Electronics 4: - Digital Electronics – Practice .....................................................................................90 The Final Project....................................................................................................................................92 Appendix A – Real Op-Amps ...............................................................................................................93 Appendix B – “Bread boards”..............................................................................................................94 Appendix C – Logic Integrated Circuits..............................................................................................96 Appendix D – “Sensing” Logic Levels .................................................................................................97 Experimental Physics I 1 Module PH1004 University of Reading Department of Physics Disclaimer and Safety Assessment This booklet describing the Experimental Physics Module PH1004 is based on the information available at the time of publication. The University reserves the right at any time to change the contents of this Module. As much notice as possible of any alterations will be given; anyone who is uncertain of the up-to-date position should enquire of the Laboratory Supervisor Student Supervision Health and Safety Aspects for Experimental Physics Laboratory Supervisors: Alternative Supervisor: Dr P.A. Hatherly, Prof. A.C.Wright Dr. D.R Waterman This assessment is valid until 1st July 2006. This assessment was made in August 2005 Nature of Work Laboratory work associated with Experimental Physics as described in this document, Experimental Physics 1 – An instruction manual to accompany Module PH1004. Hazards Electrical equipment Medium strength light sources Cryogenic Liquid Agreed precautions, control measures and personal protective equipment required Precautions as detailed in this instruction manual, Experimental Physics1 - An instruction manual to accompany Module PH1004. In Projects 2A, 3A and 5B in particular, you should avoid staring directly at the light sources. Safety spectacles will be issued for Project 3B; these should be returned to the laboratory supervisor at the end of the session. Risk Category for Supervision Work may proceed because workers are adequately trained and competent in the procedures involved1. Supervisors Dr P.A. Hatherly, Prof. A.C. Wright, Dr D.R. Waterman August 2005 1 This assessment refers only to health and safety aspects of supervision. Students must not work in the laboratory outside the timetabled hours for this module. Experimental Physics I 2 Module PH1004 University of Reading Department of Physics Chapter 1 Introduction to Experimental Physics 1.1 - Introduction This booklet is the manual that accompanies the Part 1 Module Experimental Physics PH1004. The contents of this booklet provide the background for developing skills in experimental physics and specific instructions for the different activities in the Module. 1.2 - Objectives This Part 1 Module is the first of a progressive series of modules offered by the Department of Physics, which is provided to enable you to develop the expertise and experience necessary to conduct practical work in physics. This Module sets out to: • Show how physicists approach the design and execution of experiments in order to test quantitatively theories and models. • Develop the skills in observation, sampling and recording of data and the subsequent analysis that is required to make such quantitative assessments. • Provide practical experience in a range of fundamental physics topics. 1.3 - Requirements In this Module you will first complete four one-week Skill Sessions and then proceed with four Experimental Projects lasting three weeks each and four sessions comprising an introduction to practical electronics. A Final Project will be carried out under semi-examination conditions in the first week of the Summer Term. Each three-week Project contains two related Experiments. All students will keep a detailed experimental logbook throughout the course of the Module. Typically you will work in pairs (except in Electronics, where teams of five to eight are typical) but the assessment will be made on an individual basis. Each Experiment should involve six to eight hours of work, in total. Before each experiment you will be required to conduct some preparatory work, without which, you will not be able to complete the Experiment effectively. Consequently, you will not be allowed into the laboratory to begin an Experiment until your preparatory work has been examined and signed-off by a member of the Experimental Physics Team. At the end of the first two laboratory sessions of each three-week project and each Skills and Electronics session, you should make sure that your logbook is officially stamped and signed by one of the Experimental Physics Team. If the analysis stages of a Project are not completed during the laboratory sessions, you should work on these in your own time. 1.4 - Absence You must sign the register sheet before 10:15 a.m., otherwise you will be considered absent from the session. If you are absent from a laboratory session you will be given an unclassified mark (U) for that session, contributing a mark of zero, unless you are able to show that you were ill or involved in a prior approved activity. You also risk not having notes on which to base any subsequent formal write-up. If you are unable to attend the session you must leave a message either by phone on extension 8541 (0118 378 8541) or by email to [email protected] or [email protected] (copy to [email protected] and [email protected] please) prior to its start. 1.5 - Monitoring of Laboratory Log Books Your logbook must be signed in by one of the Experimental Physics Team at the beginning of each session to ensure that you have undertaken sufficient preparatory work (data analysis) to begin (continue) the Experiment. When you leave the laboratory session, at the end of weeks one and two of a three-week project, you must also make sure that one of the Experimental Physics Team stamps and initials your logbook. Logbooks without initials and a stamp will not be given a mark. Experimental Physics I 3 Module PH1004 University of Reading Department of Physics You must leave your logbook for marking in the appropriate section of the bookshelf in the laboratory as follows:• • • • • • • • Skills Sessions – at the end of session 4 Three week Projects – at the end of the third session Electronics Sessions – at the end of session 4. Final Project – at the end of the session Logbooks not handed in at these times will not be given a mark. If you miss one of the first two weeks of a three-week project, you must complete the work within the remaining sessions. If you miss the third week, for a valid reason, it is your responsibility either to arrange for someone else to hand in your logbook or to make other arrangements with the laboratory teaching staff; prior to the end of that session. Logbooks will be available for collection from 14:00 hours on the following Tuesday. 1.6 Safety All of the Projects and Skill Sessions in this Module take place in the Part 1 Laboratory within the J.J.Thomson Physical Laboratory. This is a laboratory and you should act and think in the manner befitting physicists working in a laboratory environment. Therefore: • All coats and bags must be left at the designated sites outside the experimental physics area. • Food and drinks must not be brought into the laboratory. • Smoking is not allowed in any part of the building. • You must follow the instructions of the Experimental Physics Team in terms of safe working practice. • You are not allowed to work unsupervised and therefore will not normally be allowed into the laboratory outside timetabled hours. • For some Projects there are additional specific safety procedures and these are listed in the appropriate chapter. • Many of the Projects make use of electrically operated equipment. If you suspect that the equipment is faulty you should disconnect it from the main supply if necessary and report the fault immediately to one of the Experimental Physics Team. • You should never attempt to repair of modify the equipment. It is important that you have an appreciation of the capabilities of your apparatus and so avoid overloading any instruments. Experimental Physics I 4 Module PH1004 University of Reading Department of Physics Chapter 2 Assessment 2.1 - Preparatory Work Each Experiment involves some background reading prior to the laboratory session in order to ensure that you are fully conversant with necessary theory. This will involve reading through the chapter associated with the Experiment and the additional reading indicated at the start. Each Experiment is prefaced with some questions and it is essential that you record your answers to these in your logbook. A member of the Experimental Physics Team must sign these to indicate that you are appropriately prepared to commence work. Most prior reading is contained within FLAP modules, "Physics" (International Student Edition) by H.C.O’Hanian, published by W.W.Norton or other texts recommended in your degree handbooks. 2.2 - Laboratory Logbook All scientists keep a laboratory logbook in which they record all of the details, results and calculations related to their work. Such logbooks are a working document and scientists record ideas and discussions as well as hard data in their logbook. As part of this Module, you are required to keep an official hardback laboratory logbook; subsequent experimental physics modules will require separate laboratory logbooks. These may be obtained in the Experimental Physics Laboratory. All of your work must be contained in this logbook and, therefore, graphs etc must be securely glued into it, without folding, at appropriate points. Working on sheets of paper within the laboratory is not allowed! Your logbook is not a formal report but more a running commentary and should be sufficiently detailed for another physicist to follow your work. Indeed, you may be required to generate a formal report from these notes later in your course. Your logbook will, at some point, contain mistakes; these should be clearly crossed out, and a note added to explain what went wrong. At the end of each Experiment you will prepare an abstract. The following notes should give you a clear idea of the appropriate format and detail. If you are in any doubt as to what is required, please ask one of the Experimental Physics Team. 2.2.1 - What to Record in your Logbook • Each Experiment should start on a fresh page with a title and the date; all text must be written in ink. • You should start the two experiments of each three-week project at opposite ends of your logbook. • All answers to questions undertaken as part of your preparatory work should be included as an introduction to your written work. • You need not copy information from the manual, but you may need to refer to the diagrams and methods described in it. • However, include any diagrams that show exactly how you set up the experiment, if they are not already in the manual. In a few cases (e.g. circuit diagrams), it may be helpful to reproduce and annotate the diagram already in the manual as part of your notes. • Results should be tabulated where possible; tables must be drawn on a separate page for clarity and must include appropriate units and estimates of uncertainty. • You should aim to make all calculations and plot all graphs during the laboratory session. • When plotting graphs choose sensible scales and include error bars. Make sure that the axes are properly labelled and include a legend to describe the graph. • Quantitative estimates of the errors are extremely important (see chapters 4-6) and must therefore be included at each stage of any calculation. Experimental Physics I 5 Module PH1004 University of Reading Department of Physics 2.2.2 - The Abstract The abstract, which should follow the entry you have made on the particular Experiment, is not just a series of scrappy notes. It should be written with care and should extend to no more than 200 words. The precise format will vary from one project to another but will include: • A brief statement of the purpose of the Experiment in your own words; you should not give detail copied from the manual or a textbook. • A short summary of the experimental techniques used together with a clear statement of key results along with your estimate of the error involved (details of the error calculation should be in your laboratory notes not the abstract). This summary shows what you have learnt from the activity and also gives you experience in preparing concise and informative reports. • Brief comments about anomalous or inconsistent results. Could these be the result of systematic errors etc? Remember at all times to think about the physical significance of what you are doing and of any numbers you calculate. 2.3 - Assessment Procedure This Module is assessed completely by continuous assessment. After each Project, you will hand in your laboratory logbook, which will be examined by one of the Experimental Physics Team. You will be provided with personal feedback for each constituent Experiment on: • • • • • The quality of your preparatory work and work outside the laboratory session. Your approach to and operation of the project. The analysis and conclusions you draw from the results obtained. Your analysis of uncertainties and errors. Your communication skills as demonstrated in your abstract. Feedback will be in the form of specific comments on your work; together with an assessment sheet, which must be glued into your logbook at the end of each experiment. These will enable you to identify those skills in experimental physics that you are developing satisfactorily, and those that require further effort. Your project will be given a mark on the usual A to F scale and these marks will collectively contribute to the overall assessment for this Module. The weightings for the factors are as follows: • • • • • Preparatory/outside work Laboratory notes Error Analysis Abstract Overall performance 20% 20% 20% 20% 20% Students should expect to obtain a 100% assessment if all Skill Sessions and five Projects (or equivalents) are completed in a perfect manner. If you are absent from a laboratory session you will be awarded an unclassified grade (U) for that session, unless you are able to show that you were ill or involved in a prior approved activity. If you are unable to attend the laboratory session you should leave a message with the Physics Office on internal extension 8541 (0118 378 8541 from outside the University) or by email to Experimental Physics I 6 Module PH1004 University of Reading Department of Physics [email protected] or [email protected] (copy [email protected] please) before the start of the session. to [email protected] and Work that has not been initialled and officially stamped will be deemed unclassified (U). Similarly, logbooks that are not made available for marking by the specified deadline will be given an unclassified mark (U). Logbooks will be available for collection from 14.00 hours on the following Tuesday. 2.4 - Calculation of Final Mark The marks for each component of the module are allocated as follows: • Four Skills Sessions 40 marks • Three three-week Projects* 120 marks (40 marks each) • Four Electronics Sessions 40 marks • The Final Project 40 marks • Total 240 marks * In calculating the final mark, the lowest non-U grade for a three-week project is disregarded. The marks for the Skills Sessions, the Final Project and Electronics Sessions cannot be dropped. If a U grade is obtained for weeks one or two of a three-week project, either due to unofficial absence or failure to have a logbook stamped and signed on exit from the laboratory, zero marks will be awarded for that experiment. A U grade for week three will result in any marks for that project being reduced by 50%, assuming the logbook is handed in at the end of that session. In either case, the mark for the project in question cannot be dropped when calculating the final mark. Failure to hand in a log book for marking at, or before, the end of a skills session, the last session of a three-week project, or the final project session will result in an overall U grade (zero mark) for that activity. An overall U grade will also be awarded if U grades are obtained for all of the weeks of a three-week project. The Electronics Sessions are assessed by two theory tests (one in each of the theory sessions) and two practical sessions. Each theory test consists of 10 questions examining the full range of topics discussed. Some questions (which will be indicated) are considered to test advanced ability and carry double marks. Five of the ten questions will be marked for assessment (the same five for all students) and will contain a fair mix of basic and advanced questions. Each practical session is in two parts – the first testing basic competence and the second, more advance ability. The division of marks reflects this. The practicals are assessed from log books in the same manner as the three week projects. Each component of the electronics sessions carries equal marks. Experimental Physics I 7 Module PH1004 University of Reading Department of Physics Chapter 3 Specimen Experiment - Charge Transport in Materials 3.1 - The Manuscript Objectives (i) To study the process of electrical conduction in a metal. (ii) To compare this with electrical conduction in a disordered semiconductor. (iii) To gain experience in handling cryogenic liquids. 3.2 - Prior Reading O’Hanian Chapters 28 and 44. 3.3 - Safety Procedure Liquid nitrogen is very cold (77K) and prolonged contact will result in a severe burn. This liquid must therefore be handled with extreme caution and particular care must be taken to avoid contact with your eyes. Safety goggles must be worn at all times! In addition, observe standard procedures associated with electrical equipment. 3.4 - Introduction In this experiment you will investigate the process of electrical conduction in different types of material, including a metal and a disordered organic semiconductor. In both of these, the conductivity, which is the reciprocal of resistivity, will depend upon the number of charge carriers that are available to transport charge through the bulk of the material and the rate at which these are able to move; i.e. their mobility. In a metal the current is associated with the movement of electrons, in this semiconductor, the charge carriers are more complex. You will investigate the process of electrical conduction by considering the temperature dependence of the resistance. In a metal, the number of charge carriers available for conduction is effectively fixed. Then, the transport process is governed by the way in which the moving electrons interact with each other, or with the atoms of the crystal lattice. These phenomena are known as scattering processes. If the electron mobility is determined by electron-electron scattering: R = K1 T2 where R is the resistance of a given specimen, T the absolute temperature, in Kelvin and K1 a constant. If the electron mobility is determined by thermal interactions with the crystal lattice of the metal (for T>θD, the Debye Temperature): R = K2 T K2 is another constant. If the electron mobility is determined by thermal interactions with the crystal lattice of the metal (for T<θD): R = K3 T Experimental Physics I K3 is another constant. 8 Module PH1004 University of Reading Department of Physics In semiconductors, the process of electrical conduction is very different. In these, the conductivity is generally determined by the number of carriers available to transport the charge through the material. In disordered systems, the conductivity, σ, can often be described by an equation such as:  −T  σ = σ 0 exp 1 / 02  T  where σ0 and T0 are constants. The conductivity for a specimen is given by the equation: 1 l σ= = ρ RA Where l is the length of the specimen and A is its cross-sectional area. Since for a given specimen l and A are constants: 1 σ∝ R 3.5 - The Experiment Set up the circuit shown below, such that you can apply a known fixed voltage to the specimen and measure the current flowing through it. You now need to develop an experimental procedure that will enable you to determine the resistance of a specimen as a function of its temperature. It is important to apply a fixed voltage since, as demonstrated by Project 3B in this manual, not all materials are linear; i.e., the current may not be proportional to the applied voltage. Each specimen contains a different material, to which are attached two leads. In addition, each also contains a K-type thermocouple, which will enable you to measure its temperature. The thermocouple enables you to read off the temperature of the sample directly. 3.6 - Laboratory Notes 15th December CHARGE TRANSPORT IN MATERIALS 3.61 - Objectives In this experiment we will investigate how electricity is conducted in different types of material. The objectives of the work are: (i) To study the process of electrical conduction in a metal. (ii) To compare this with electrical conduction in a disordered semiconductor. (iii) To gain experience of handing cryogenic liquids. 3.6.2 - Setting-up The Equipment To investigate the charge transport through two different samples we will first set up the circuit shown below, which consists of a stabilised voltage source in series with a digital multimeter, set to measure current, and the specimen. An AVOmeter is connected across the power supply to provide a direct measure of its output voltage. The analogue meter is chosen to measure the voltage which, once set, should not change during the course of the experiment. DC Power Supply Experimental Physics I AVO 9 Sample DMM Module PH1004 University of Reading Department of Physics In addition, the thermocouple output from the sample is connected to the digital thermometer to provide a direct measurement of the sample temperature; this meter must be set to the K range to match the characteristics of the thermocouple. The two digital instruments are finally positioned adjacent to one another to enable the current through the sample and the sample temperature to be measured at the same instant. 3.6.3 - Preliminary Measurements Before making any detailed measurements we need to find a suitable voltage value that will enable us to make accurate readings of current as a function of temperature. To do this we will introduce each sample, in turn, into the above circuit and measure its resistance at the two extreme temperatures; i.e. at room temperature and as close as possible to the boiling point of liquid nitrogen (77K). As indicated in the laboratory manual, the chosen voltage should be such that the specimen current never exceeds 10mA. However, to provide the maximum accuracy, we will initially choose the voltage applied to each sample such that the maximum current is just below 2mA, so that all measurements can employ the maximum number of significant figures on the meter. NB (i) (ii) (iii) (iv) Specimen Voltage ±0.2 / V RED BLACK 10.0 10.0 Current at max. temp. ±0.001 / mA 1.783 1.392 Voltage ±0.2 / V 10.0 10.0 Current at Min. temp. ±0.001 / mA 1.820 0.024 The laboratory manual states that the red specimen is metallic. The laboratory manual states that the black specimen is a disordered semiconductor The maximum temp ≡ room temperature - this was measured to be 19.5±0.1oC using the digital thermometer and the thermocouple embedded in each specimen prior to immersion in liquid nitrogen. For each sample, the minimum temperature attainable, as measured using the thermocouple, fell within the range –193 ± 1oC. The errors quoted above were derived as follows. The error in the voltage corresponds to the error associated with the accuracy of the meter - ±2% full-scale deflection (i.e. ±0.2V on the 10V scale). Since the scale can be read to ±0.02V, this reading error was neglected. The error in the current is taken at all times to be ± the final digit on the display - ±0.001mA in this case. No other information is available concerning the accuracy of this meter. The error in the temperature similarly corresponds to the final digit on the display - ±0.1oC in this case. No other information is available concerning the accuracy of this meter. From the results listed in the above table, it is clear the resistance of the red specimen does not change dramatically with temperature and, therefore, the choice of voltage is determined, largely by behaviour of black sample. For the rest of the experiment, the applied voltage will be set at 10.0±0.2V. 3.6.4 - Temperature Dependence of Conductivity In this part of the project, we will need to vary the temperature of the material of interest and record how the current flowing changes with temperature. However, since we are only able to hold the sample at either room temperature or close to 77K, we will have to proceed as follows. (i) (ii) Record the current flowing at room temperature Cool the specimen in liquid nitrogen and measure the current flowing at the minimum temperature we can reach in a reasonable time. Experimental Physics I 10 Module PH1004 University of Reading (iii) Department of Physics Allow the specimen to warm up slowly, measuring the current as a function of temperature. To develop a reasonable procedure for this final step we will first try a number of different approaches in an attempt to control the heating rate of the specimen. (i) We first immersed the red sample and left it for ~1min after the minimum temperature had been reached, so that it was at equilibrium. We then removed it from the liquid nitrogen and left it to warm up on the bench. From low temperatures, its temperature was seen to increase very quickly and, because of the response time of both meters, it was very difficult to make accurate measurements. Nevertheless, as the sample approached room temperature, the heating rate became reasonable. (ii) In an attempt to reduce the heating rate at low temperatures we took a polystyrene cup and poured in some liquid nitrogen. After about a minute we removed the nitrogen, leaving us with a cooled cup full of cold gas. Repeating the above experiment demonstrated that this procedure reduced the heating rate over the complete temperature range. However, it was still rather fast at the lowest temperatures. (iii) Finally, we cooled the specimen and simply removed if from the liquid nitrogen, leaving it just above the surface of the nitrogen; i.e. within the evolving gas. Under these circumstances, the heating rate was acceptable at even the very lowest temperatures. A combination of the approaches described above seems to give reasonable behaviour. Nevertheless, to minimise the errors, we will repeat the cooling/heating cycle for each specimen. 3.6.5 - Charge Transport in Specimen A The current flowing through specimen A will now be measured as a function of temperature, as described above and recorded in Table 1. To obtain these data, the sample was positioned over the boiling nitrogen until its temperature had risen to between 100oC and -80oC, when it was removed and placed in a cooled polystyrene cup. Once the temperature had risen to between -40oC and -20oC, it was finally placed on the bench to increase its heating rate. Each measurement of current was made at a temperature within the range ±1oC of the quoted value since the display of the digital thermometer did not vary continuously. For each individual current reading, the quoted errors are assumed to be in line with the static equivalents listed above. However, additional errors may be incurred during this dynamic experiment as a consequence of temperature gradients within the specimen, problems associated with attempting to read two changing digital meters simultaneously etc. As a consequence, we chose to repeat the run but, as is evident from the Table 1, any additional errors cannot be large. To compare with theory, where the resistance is expected to increase linearly with temperature, we now need to evaluate the reciprocal of each current, since this is proportional to resistance at a fixed voltage. This is shown in Table 2. Experimental Physics I 11 Module PH1004 University of Reading Department of Physics RUN 1 Temp. ±1/oC Current ±0.001/mA -190 1.820 -180 1.819 -160 1.815 -140 1.812 -120 1.808 -100 1.804 -80 1.801 -60 1.797 -40 1.793 -20 1.789 0 1.786 19 1.783 Table 1 Current flowing through specimen A as a function of temperature. Temp. ± 1 /oC -190 -180 -160 -140 -120 -100 -80 -60 -40 -20 0 19 Table 2 RUN 2 Temp. ±1/oC Current ±0.001/mA -190 1.820 -180 1.818 -160 1.815 -140 1.811 -120 1.808 -100 1.804 -80 1.801 -60 1.797 -40 1.793 -20 1.789 0 1.786 19 1.784 RUN 1: 1/I ± 0.0003/ mA-1 0.5495 0.5497 0.5510 0.5519 0.5531 0.5543 0.5552 0.5565 0.5577 0.5590 0.5599 0.5608 RUN 2: 1/I ± 0.0003/ mA-1 0.5495 0.5501 0.5510 0.5522 0.5531 0.5543 0.5552 0.5565 0.5577 0.5590 0.5599 0.5605 Processed current/temperature data for sample A. Experimental Physics I 12 Module PH1004 University of Reading Department of Physics These data are plotted on the linear graph shown in Figure 1. From this it is clear that the resistance of the red specimen varies linearly with temperature at a fixed voltage of 10.0±0.2V and can be described by the following equation over the temperature range studied: - 1 = mT + c I where I is the current flowing at a temperature T, m is the gradient and c is the intercept. From the graph: m = (5.93 ± 0.13) x 10-5 mA-1 / oC and c = 0.5599 ± 0.003 mA-1 Changing the units of temperature to the more useful Kelvin. m = (1.74 ± 0.15) x 10-4 mA-1 / K and c = 0.5436 ± 0.006 mA-1 Thus, since the resistance is proportional to conductivity, we can conclude that the red specimen behaves like a metal within the above temperature range. 3.6.6 - Charge Transport in Specimen B We will now measure the current flowing through specimen B as a function of temperature, using the same technique. Again, for each reading, the errors are assumed to be in line with the static equivalents listed above. However, additional errors will be incurred as a consequence of the dynamic nature of this experiment. From these data, shown in Table 3, it is clear that the behaviour of this material is highly non-linear with temperature. As a result, we will plot our data using log - linear graph paper. The above data are shown in Table 4 in the required form since, for a disordered semiconductor, the current should increase exponentially with temperature. In view of the scatter in the current data over that which would be expected based upon the accuracy of a static equilibrium measurement, we will neglect the relatively small (generally less than 1%) error in temperature. This large scatter is probably a consequence of small temperature variations within the mass of each sample; since the current is strongly dependent upon temperature, these have significant consequences. As a result, the static error bars are too small to be usefully shown. In the previous case, the conductivity of the red specimen was not strongly dependent upon temperature and, consequently, such variations were not so obvious. Nevertheless, it is clear from this graph that the current flowing through the black specimen increases exponentially with temperature at a fixed voltage of 10±0.2V. However, at higher temperatures (above ~170K), there is a change in the gradient, which might indicate a change in the conduction mechanism. Therefore, we can conclude that our low temperature data is in line with the theory given in the laboratory manual:  −T  σ = σ 0 exp 1 / 02  T  Experimental Physics I 13 Module PH1004 University of Reading Department of Physics RUN 1 Temp. /oC Current ±0.001 /mA -190 0.027 -180 -170 0.080 -160 0.118 -150 0.192 -140 0.290 -120 0.475 -100 0.753 -80 0.947 -50 1.051 -30 1.208 0 1.305 19 1.342 Table 3 Current flowing through specimen B as a function of temperature. 1/T1/2 / K-1/2 0.1100 0.1037 0.0985 0.0941 0.0902 0.0867 0.0808 0.0760 0.0720 0.0670 0.0642 0.0605 0.0584 Table 4 RUN 2 Temp. /oC Current ±0.001 /mA -190 0.023 -180 0.046 -170 0.085 -160 0.129 -150 -140 0.278 -120 0.498 -100 0.803 -80 0.977 -50 1.092 -30 1.186 0 1.364 19 1.401 RUN 1: Current / mA 0.027 0.080 0.118 0.192 0.290 0.475 0.753 0.947 1.051 1.208 1.305 1.342 RUN 2: Current / mA 0.023 0.046 0.085 0.129 0.278 0.498 0.803 0.977 1.092 1.186 1.364 1.401 Processed current/temperature data for sample B. Experimental Physics I 14 Module PH1004 University of Reading Department of Physics From the graph we can evaluate -To, which is the gradient. To = 100 ± 6 K1/2 However, we are unable to evaluate σ0 from the current voltage characteristics without knowing about the geometry of the sample. Nevertheless, from the above analysis we can conclude that that black specimen behaves like a disordered semiconductor at low temperatures, but that some change in conduction mechanism may occur as room temperature is approached. 3.6.7 - Abstract The electrical conductivity characteristics of two different materials have been studied. A voltage of 10±0.2V was found to give a suitable current in each specimen and this value was used throughout the rest of the experiment. Before making any detailed measurements, a method was devised by which the temperature of the specimen could be varied between a value close to the boiling temperature of liquid nitrogen and room temperature. By varying the local environment from adjacent to the liquid nitrogen, thought cold nitrogen gas, to ambient, the heating rate could be maintained at a suitable level for current values to be recorded at different temperatures. The above voltage was then applied to the metallic specimen and the current flowing was measured as a function of temperature from close to the boiling point of liquid nitrogen to room temperature. Over this temperature range, it was found that the resistance increased linearly with temperature at a rate given by (5.93 ± 0.13) x 10-5 mA-1 /K. On applying the same fixed voltage to the semiconductor an exponential increase in current with temperature was observed. Detailed comparison with theory revealed that the characteristics could be divided into two regimes; an apparent change in the conduction process was observed at ~170K. Below this temperature, the linear gradient was quantified to give a value for To of 100 ± 6 K1/2. Experimental Physics I 15 Module PH1004 The University of Reading Department of Physics Conductivity Data for the Red Specimen 1/Current /mA -1 0.56 0.558 0.556 0.554 0.552 0.55 -200 -150 -50 -100 0 50 o Temp/ C Conductivity Data for the Black Specimen Current/mA 10 1 0.1 0.01 0.06 0.07 0.08 0.09 1/ T Experimental Physics I 16 0.5 0.1 /K 0.11 -0.5 Module PH1004 The University of Reading Department of Physics Chapter 4 Statistics 4.1 - Errors and Variability When performing an experiment, you collect data in order to answer a question. However, the answers are not immediately obvious because you always have to contend with variability; if you perform the same experiment twice you don’t get exactly the same results each time. So which answer is right? Furthermore, you always have to contend with less data than you would wish, simply because of the available time, or because of the constraints of the equipment. In statistical terminology, you can only take a sample of all the measurements that are theoretically possible. This combination of variability and sampling creates peculiar difficulties for researchers, but these difficulties can often be overcome by the use of statistical techniques. Our definition of "statistics" should, therefore, be based on variability and sampling. Statistics is a body of knowledge that can be of use to anyone who has taken a sample from a population in which there is variability from item to item. The three words underlined in this definition are at the very heart of data analysis. The presence of variability in the data tends to obscure the truth that the researcher is seeking. However, statistical techniques can help you to penetrate this variability, expose the truth and draw valid conclusions. In addition to the inherent variability associated with experimentation, we have the further difficulty, which arises because our data is based on a sample (small number of measurements) whilst we wish to draw conclusions about the whole population (the universe at large). 4.2 – Definitions Some terms commonly used in discussing experimental errors are defined below: Population A population is simply a large group of items being investigated by the scientist. Sample A sample is a group extracted from the population and, clearly, any conclusions we draw will be invalid if the sample is not representative of the population from which it was taken. Whilst this point is obvious, it is well worth making because, although sampling is often taken for granted, it may be the weak link in any investigation. Variability Every professional scientist or technologist appreciates the need to recognise and measure variability; it is present in all our experiments. Despite our efforts to work as thoughtfully as possible, uncertainties or errors will appear. Random Errors Things fluctuate! Imagine counting the number of ants emerging from an anthill over a 1 minute period. In the first instance, you might count 105. In the second, 97. Over a large number of minutes, you might come to an average figure of 100. This isn’t the whole story though, because looking at your results, you would find that the number for any individual minute would fluctuate between about 90 and 110! This ± 10 fluctuation would be the random error on the measurement, and the number of ants emerging per minute would be written as 100 ± 10 ants s-1. (Incidently, the ± 10 arises from the statistics of counting – for a count of N events, the error is N . This topic will be covered in future aspects of your degree programme.) Experimental Physics I 17 Module PH1004 The University of Reading Department of Physics Systematic Errors A traditional method of measuring the old unit of length of the yard was to use the distance from the tip of your nose to the ends of the fingers of your outstretched arm. If however, your yard measured this way were 10% greater than the “standard” yard, your measurements would always be out by 10%. This is a systematic error. Such errors arise when: • Measuring devices are mis-scaled or mis-calibrated • Measuring devices have offsets Systematic errors can be very subtle and difficult to quantify. As an example of the way errors work, imagine 5 scientists attempt to measure the focal length of a given lens. Inevitably, they will produce 5 different numbers; which is right? None of them will be exactly correct but all are likely to be approximately correct provided there was nothing fundamentally incorrect about the measurement method. Each measurement will be subject to random errors (for example, how exactly do you determine where a focus is?). Of course, there may have been systematic errors associated with the measuring apparatus (e.g. a tape measure might have been stretched). 4.3 - Blob Chart One way to describe the variability in a set of data is to draw a diagram like the simple blob chart shown below. From this, you can see how the data are scattered, or spread, and how each measurement deviates from the mean value of 47.3. We can see how many measurements were above average, and how many were below average. We can also immediately see if any data point appears anomalous and this could then be ignored. A blob chart shows the distribution of data in one dimension (i.e. it only involves data on one axis, (x). In two dimensions, such a plot is termed a graph (i.e. two axes, x and y). Like blob charts, graphs have many advantages when it comes to the presentation of data; it is easy to visualise what is going on and it is easy to spot anomalous results, which can then be checked and re-measured or otherwise accounted for. 0 10 20 30 40 50 60 70 80 90 Blob Chart 4.4 – Final Remarks It should be emphasised that the term error in the context of experimental uncertainty has NOTHING TO DO WITH GETTING THINGS WRONG! Rather, you should view the determination of experimental errors as part of establishing your confidence in the precision and accuracy of the experiment. Don’t be afraid to quote large errors – this is not necessarily a reflection of your ability, rather of the capabilities of the experiment. In many areas of research, “errors” in quoted results as large as 10 –20% are commonplace. Indeed, results with smaller errors may be treated with suspicion! Experimental Physics I 18 Module PH1004 The University of Reading Department of Physics Chapter 5 Introduction to Skills Sessions 5.1 Introduction A vital part of work in the laboratory is having the necessary skills. As with anything of this nature, skills have to be developed and practiced. Hence, the first four weeks of the module will be devoted to instilling some basic skills and techniques. In week 1, we will develop the skill of data collection through a simple, yet elegant experiment. In week 2, you will use this data to plot graphs and analyse the results. In the third week, we will take the analysis further by introducing the concept of experimental uncertainties (“errors”), what they mean and how to handle them. In the final skills session, you will be introduced to a range of electronic equipment and their operation. A brief summary is provided below. Skills Session 1 – Data Collection You are provided with the following apparatus: a lab stand, some thread, a set of weights, a rule and a stopwatch. Using this apparatus, construct a simple pendulum and record the period of the pendulum for a series of lengths of thread. Skills Session 2 – Graph Plotting Graph plotting will be discussed, and you will carry out a number of exercises on plotting graphs and analysis. You will then take the data from Skills Session 1, plot appropriate graphs, and draw some conclusions. Skills Session 3 – Errors The meaning, use and calculation of errors will be discussed. You will carry out a number of exercises on error calculation, and apply your skills to estimating the errors in the data from Skills Session 1. (Note – the apparatus will be provided again, to allow you to do this). Skills Session 4 – Electronic Instrumentation In this skills session, you will be introduced to the use of analogue and digital multimeters and oscilloscopes. You will be provided with “quick start” guides for each instrument, enabling you to use them with the minimum of fuss. You will practice their use by analysing the outputs of a “black box”, providing a set of unknown signals. 5.2 Assessment The skills sessions contribute the equivalent of one three-week experimental project to the module. The first week of data collection is unassessed, but it is in your interest to participate fully as the data collected will be used in sessions 2 and 3, which are assessed. Books should be handed in for marking at the end of Week 4. Experimental Physics I 19 Module PH1004 The University of Reading Department of Physics Skills Session 1 Data Collection Introduction In this session, you will practice the technique of data collection. This is not simply a matter of writing down numbers, but rather carefully considering: • • • • • Am I collecting the correct data? Am I collecting them in the most appropriate way? Am I recording what the numbers mean? Am I collecting enough? Am I recording what I’m doing? In this Skills Session, the notes here will provide you with guidance on the above points. You should be recording items or tasks marked in italics to reflect good laboratory and log-keeping practise. Be sure to carry the experience of this skills session into future sessions and experiments. The Experiment The aim of this experiment is to collect data on the period of a simple pendulum, varying the length of the pendulum and the mass of the bob. Record these aims The objectives of this experiment are to: • Practice recording data • Practice correct log keeping • Collect data useable in subsequent Skills Sessions Record these objectives You are provided with the following equipment: • A lab stand and clamp • A reel of thread • A set of weights • A ruler • A stopwatch Record the equipment used, including any model numbers and serial numbers of instruments. Set up the apparatus as shown in figure 1. Record this figure Be sure to record the length of the pendulum, and record where you measure it to and from. For example, from where the thread is tied to the clamp to where it’s tied to the weight. Think carefully about this. We are now in a position to think about recording data. First, what do we mean by the period of the pendulum? The usual definition would be a complete swing from the starting Experimental Physics I 20 clamp Lab stand thread bob Figure 1. Apparatus for measuring the period of a simple pendulum Module PH1004 The University of Reading Department of Physics point, to the far side of the swing, and back again.. Now, how do we measure the period? Clearly, we are going to use the stopwatch, but are you able to time a single swing? Try it – you’ll probably find it’s not possible to react fast enough, especially if the period is short! In any event, for reasons that we’ll discuss in a future session, a single reading can never be considered reliable. Try recording, say, 10 periods and dividing the total time by 10 (or however many you choose). Is this more reliable? Repeat the recording – do you get the same number? (Note – ensure the pendulum does not swing through too large an angle – the analysis in future sessions will be based on the so-called “ small angle approximation” ) Record your observations. Record what you plan to do, based on these observations. We can now take the data for the first length of pendulum and the first mass of bob You should record your results in an appropriate table, clearly labelled and readable. Your results should look something like the table below: Deliberate mistake number one – something is missing from this table – a title! Always include a descriptive title so you can easily find the table you are looking for. In this case, a title like “Pendulum Period table 1 – l = 23 cm” might be appropriate Notice that the experiment has been repeated a number of times to ensure the results are reproducible. This also enables an average of the data to be taken, again improving the reliability of the result. In this case, the average is simply the sum of all the periods divided by the number of trials. Notice that all experimental parameters have been noted – in this case, the pendulum length and the bob mass. Note that the units have been recorded. Stating that the length of the pendulum is 23 is meaningless – do you mean 23 cm? m? inches? In the table columns, the units of the period have been recorded as seconds by putting “ /s” after the parameter. Note the variability of the total time. This is due to differences in starting and stopping the stopwatch. This effect will be discussed in a later Skills Session. Note the number of decimal places the times have been recorded to. Writing the period as 0.957438 s is meaningless- is your apparatus really capable of measuring microseconds? Only record the number of decimals consistent with the abilities of your apparatus. Experimental Physics I 21 Module PH1004 The University of Reading Department of Physics Repeat the above procedure for a number of lengths of thread – 5 should be enough, spread over as large a range as possible. As a suggestion, try lengths of about 10 cm, 15 cm, 20 cm, 30 cm and 40 cm (it doesn’ t matter exactly what the measurements are – just so long as you record them). We have completed the first part of the experiment. You should note some variability of the period depending on the length of the pendulum. Record your observations and any conclusions you can draw at this stage The second part of the experiment involves examining the effect of bob mass on the period of the pendulum. For this, we need to fix the length of the pendulum, and vary the mass. You should generate a series of tables similar to that for the first section, but now change the mass between each table. Again, a set of 5 masses should be sufficient. As a suggestion, try 10 g, 20 g, 50 g, 100 g and 200 g. Record the data and record any observations regarding the effect of changing the mass and any conclusions we might draw. In particular, take care when measuring the length of the pendulum when you change the mass. Ensure you record what the length really is. Can you devise a way of ensuring the length is identical (or nearly so) for each mass? Experimental Physics I 22 Module PH1004 The University of Reading Department of Physics Skills Session 2 Graph Plotting Introduction Note: Some useful information is available online at: http://www.bbc.co.uk/education/asguru/maths/12methods/02geometry/10straight/index.shtml In Skills Session 1, you recorded sets of data and compiled them in tabular form. As it stands, this data is valid, but difficult to use. We can use the data in its entirety to plot a set of graphs and draw some physically useful results. In particular: • A theoretical analysis of the simple pendulum reveals the period, p in seconds (s) is given by l p = 2π (equation 1) g where l is the length of the pendulum in metres (m) and g is the acceleration due to gravity in ms-2. Does our data support this conclusion? If not, why not? • The above analysis suggests the mass of the bob is irrelevant. Does our data support this conclusion? If not, why not? Preparatory Work This work should be carried out in your log books before the start of the session. Graph paper is available in the Part I laboratory or from Dave Patrick. As stated previously, graphs should be stuck into your log book. Before coming into the laboratory, you should read through this section carefully, to make sure that you understand it and make any necessary notes in your logbook. You should attempt exercises 1, 2 and 6. You should also equip yourself with a sharp pencil, preferably with a 0.5 mm lead, a rubber and a rule. Graphical Representation In many experiments you will investigate two quantities that are linearly related, that is: y = mx + c where x is the independent variable (what you measure), m is the gradient, c is the intercept and y is the dependent variable (what you calculate) The gradient m is the change in y for a given change in x, and the intercept c is the value of y when x is zero. This type of relation is best displayed graphically because, in addition to the ability to measure c and m directly, you are able to see the extent of scatter on individual data points. You can therefore estimate the overall uncertainty, and also observe whether any data are obviously wrong. How these properties appear on a graph is illustrated in the sketch below: Experimental Physics I 23 Module PH1004 The University of Reading Department of Physics Note – the above is a sketch graph and should not be used for serious analysis. Normally, you should plot your graphs on pre-printed graph paper. Saying that, the sketch graph has a range of uses. For example, it is good practice to plot data as you’re collecting it. This helps you see if you are taking appropriate data and if any points are obviously “wrong” for some reason. Don’t be afraid of sketching! When using graph paper, always label the divisions on your graph so that interpolation or extrapolation is as easy as possible. For example, let 1cm equal 2 or 5 units rather than 3 or 7. When the experimental values are far removed from zero it may be appropriate to suppress the origin (x=0, y=0), but if the relation is one of simple proportionality:y = mx the origin should provide the most accurate point on the graph. However, do not force the graph through the origin; your measuring instrument may have a zero error or there may be other factors in play. Exercise 1: (a) Plot the following dataset; the data is believed to follow the relationship b = Aa +B where A and B are constants and a and b are the parameters (or variables) a/mA 10 20 25 32 45 60 77 b/s 14.3 22.1 26 31.46 41.6 53.3 66.56 as a guide, plot the a parameter on the x axis, and b on the y axis. On the a axis, a range of 0 – 80 would be suitable – devise a suitable range for b Be sure to properly label the graph – see the sketch above. (b) Draw a line through the data points, and extract the gradient, A and intercept, B Experimental Physics I 24 Module PH1004 The University of Reading Department of Physics In the above exercise, you will note the data fall perfectly on a straight line. In a real experiment, this is very unlikely to be the case. Referring back to the sketch graph, you will note that the points are scattered, yet a “ best fit” straight line can still be drawn. There are mathematical techniques for determining the best fit line, but for our purposes, “ eyeballing” the line is sufficient. As a guide, judge the best fit line so that roughly equal numbers of points are above and below the line. You may be able to safely ignore points which obviously lie far from the line (if possible, it would be better practice to repeat the experiment for that data point) Exercise 2: (a) The following data is similar to that in Exercise 1, but now represents a real data set. Plot a suitable graph for this data: a/mA 13 36 39 46 57 64 73 b/s 21 33 40 48 52 61 67 Judge for yourself now the axes scales. (b) Assuming again that the data follows a relationship of the form b = Aa +B, draw a best-fit line through the data and determine A and B (note – these will not necessarily be the same as in exercise 1) In the above, we have assumed that the data follow a linear relationship. However, this is not always the case. Let’ s take the example now of the data from the pendulum experiment last week. You should have a data set which looks something like: l /m p/s 0.10 0.67 0.15 0.83 0.20 0.89 0.30 1.13 0.40 1.28 Exercise 3: Plot the data you collected last week, choosing the scales sensibly. At this stage, you need not include the origin. Sketch a best-fit line through your points – you will find that you can’ t quite put a straight line through it! This tells you that the period-length relationship is not linear. Experimental Physics I 25 Module PH1004 The University of Reading Department of Physics You should obtain something like (based on the above dataset): Clearly, for our data, plotting p vs. l isn’ t too helpful, as the physical relationship isn’ t linear. How can we obtain a linear form of the graph, and hence obtain some useful information? There are two ways we can tackle this. Technique 1 rearrangement of the function we wish to test. The relationship we wish to test is: l l p = 2π = 2π   g g 1 2 1 (recall that x = x 2 ) Squaring both sides of the equation, we obtain: 4π 2 p2 = l (equation 2) g In other words, the square of the period is linearly dependent on the length. We can also see that this is a relationship of the form y = mx, with y = p2 and x = l and the gradient, m, as (4π2/g) We should now generate a new dataset, based on our original, which should look like: l /m p/s p2 / s2 0.10 0.67 0.45 0.15 0.83 0.68 0.20 0.89 0.79 0.30 1.13 1.28 0.40 1.28 1.64 Experimental Physics I 26 Module PH1004 The University of Reading Department of Physics Exercise 4: Generate a table similar to the above using your data, and plot an appropriate graph. You should include the origin. Draw a best-fit straight line through your data and obtain the gradient. You should obtain something similar to the graph below: Note that the line hasn’ t been forced through the origin Recalling that the gradient = (4π2/g), find a value for g, the acceleration due to gravity at the Earth’ s surface (remember the units – ms-2) (In the above case, the gradient is 3.99 s2m-1 (note the gradient has units – always quote these!). Hence, the value of g for the sample data is: g = (4π2/gradient) = 9.89 ms-2) We might ask why the graph doesn’ t pass through the origin. This is a question we’ ll address in the next Skills Session. Meantime, try to measure it on your graph (it won’ t be the same as above) – if your intercept appears to give a negative p2 value, don’ t worry – estimate the intercept on the above graph. Technique 2 take logarithms of both sides of the equation. For equation 1, our relationship becomes:  l   log( p ) = log 2π  g   Recalling the basic properties of logarithms, this becomes:  l   log( p ) = log(2π )+ log  g   which can be further manipulated to: log( p) = log(2π ) − 12 log( g ) + 12 log(l ) Experimental Physics I 27 Module PH1004 The University of Reading Department of Physics Once again, we have a function in a linear form, with y = log(p), x = log(l), gradient, m, = ½ and intercept, c, = log(2π) − ½ log(g) We can now generate a table of the form: l /m p/s log(l) log(p) 0.10 0.67 -1.00 -0.16 0.15 0.83 -0.82 -0.10 0.20 0.89 -0.70 -0.04 0.30 1.13 -0.52 0.03 0.40 1.28 -0.40 0.12 Note: be careful! In the above, logs to base 10 (log10) have been used throughout. You may also use natural (log to base e) logarithms with equal validity, but ensure you note what you do and are consistent throughout an analysis. For reasons we’ll see in the next Skill Session, it is preferable to use natural logs. Exercise 5: Generate a table similar to that above from your data, and plot an appropriate graph. You should include the origin. Draw a best-fit straight line through your data and obtain the gradient. You should obtain something similar to the graph below: Measure the gradient and intercept of your graph. The gradient should come out as something close to ½. Now measure the intercept, c, on the log(p) axis. Recall that, for this graph, c = log(2π) − ½ log(g). For the sample graph above, c = 0.28. To obtain g from this, we can rearrange the expression for the intercept to obtain: log(g) = 2(log(2π) − c) hence, assuming we’ ve taken logs to base 10, g = 102(log(2π ) − c ) = 10.87 ms-2. Experimental Physics I 28 Module PH1004 The University of Reading Department of Physics Obtain a value of g from your data. Does it agree with that obtained in exercise 4? (Important note! Notice that the expression in a log function technically has no units. We therefore have to implicitly divide a parameter like g by 1 ms-2. In other words, we force the total expression to be unitless. We then have to remember to add back the units again when we complete the analysis) Manipulating Other Expressions The techniques described above can be applied to most situations to assist in the analysis of data. A few examples of relationships, and an appropriate form of graph are given below. Basic Expression y = ae bx Useful Graphical Form ln( y ) = ln(a ) + bx On y axis plot: ln(y) On x axis plot: x y = ax b ln( y ) = ln(a ) + b ln( x) or simply y = ax if b is known ln(y) y ln(x) xb 1 b xc = + y a a 1 y xc b y= a b + xc Other functions may be put into the correct form either by simple algebraic manipulation, by application of one or more of the above forms or both techniques As an optional exercise, students may wish to verify the above relationships. Exercise 6: Put the following expressions into linear forms (y = mx+c), identifying the gradient, m, and the intercept c: 1 = exp(Ax + B) (i) y (ii) 1 A = y - C B+ x Some General Hints and Tips The general hints and tips below should help you plot good graphs and get the best out of your data. The list isn’ t exhaustive, and there may well be exceptions, but do try to follow them in your regular laboratory work. Experimental data points should be spread as uniformly as possible along the graph. Note that if the relation is of the form: 1 = mx + c z and you are plotting 1/z against x; then you should not take uniformly spaced values of z, rather you should ensure the values of 1/z are evenly spaced. Experimental Physics I 29 Module PH1004 The University of Reading Department of Physics Plot your data as you are collecting it, even if only on a sketch graph. This will enable you to see if, for example, the spacing of the data points is sufficiently uniform or a data point lies off the trend of the rest and maybe should be repeated or investigated. In any event, try, as far as possible, to analyse your data before an experimental session ends. This ensures that, if there is a problem with the data, something can be checked or repeated. The apparatus might not be in the same state next week! For highest accuracy the gradient should be measured over as wide a range as possible. For example, where measuring the change in x you may not be able to read each end of your ruler to better than 0.5mm, so it is better to measure a line 100mm long than one that is only 10mm long. When drawing a best-fit line, ensure you can do so in one go (i.e., is your ruler long enough?). This is especially important if your data points lie far from the y axis, and you need to obtain an intercept. Experimental Physics I 30 Module PH1004 The University of Reading Department of Physics Skills Session 3 Uncertainties and Errors Introduction There will always be some uncertainty in the readings you take and therefore data without an assessment of the associated uncertainty are almost as useless as data presented without units, since other scientists will have no knowledge of your limits on the accuracy and precision of the data, and will hence be unable to compare their work with yours. Where instruments specify accuracy, take this as the error in your readings; for digital instruments, assume an error of plus or minus one digit, unless you know better. In subsequent Modules you will learn the detailed background behind the statistical techniques for the quantification of random errors through repeated or related readings. However, here, only the basic formulae are required. Preparatory Work Before coming into the laboratory, you should read through this chapter, to make sure that you understand it, and make any necessary notes in your log book. To test your understanding, you should derive the expression for the error on A, ∆A if and (i) A = cos (B) (ii) A = ln (B) and the error on B is ∆B. Note that the error is always taken as a positive quantity. You should also replot the graphs prepared in exercises 4 and 5 in the previous Skills Session, as these will be required in exercises in this Skills Session. NB You should bring a calculator to the laboratory session. Averaging a Set of Differences Suppose that you have to measure the wavelength of a standing wave (λ), and also the uncertainty in that value, by measuring the spacing of n successive pressure maxima, which are spaced half a wavelength apart. An average could be obtained by taking the difference between the final and initial scale readings and dividing by (n/2), but this procedure neglects all intermediate readings. How NOT to Proceed Do not average successive differences as shown above. In this case the average is independent of all but the extreme readings. ∆Y1 = Y2 - Y1 ∆Y2 = Y3 - Y2 ∆Yn-1 = Yn - Yn-1 n −1 ∑ ∆Y = Yn − Y i 1 1 Experimental Physics I 31 Module PH1004 The University of Reading Department of Physics The Correct Method Take an even number of readings, and pair them as shown below for n = 12. 12 11 10 9 8 7 - 6 5 4 3 2 1 Note that in this method each reading is used once only. The data yields 6 independent values of 3λ. The variation between the 6 differences enables the random error to be estimated. Accuracy of the Mean Value You will frequently repeat individual readings, each of which is subject to a random error, and then calculate a mean value from 3 or 4 repeats. This is then your best estimate of the parameter, A, of interest. Although there are statistical techniques which enable you to calculate the likely error using such an approach, for the activities in this module, simply repeat a measurement 3 or 4 times and then generally take the error, ∆A, as plus or minus half the spread of readings. You can easily visualize the spread if you choose to plot a blob chart, as in Section 4.3. Systematic Errors A spread of results about an average value is obtained because of random errors; these arise from factors such as the impossibility of reading with complete accuracy a scale marked with finitely wide divisions. A different source of error arises if the divisions on the scale are not what they claim to be. For example, a diffraction grating could have only 590 lines per mm rather than the 600 lines per mm quoted, so that wavelengths measured with this grating will all be in error by more than 1%. Errors of this type are called systematic, and may sometimes be difficult to deduce. In some experiments, it is possible to eradicate systematic errors by calibrating the equipment against apparatus of known high precision. Exercise 1: On the bench where you are working, you have a ruler and a metal block; both are numbered. Measure the longest dimension of the block and record it (include a random uncertainty of the measurement – if the ruler has1 mm divisions, you can probably estimate to ± 0.5 mm at worst, but probably no better than ± 0.25 mm. Ensure you record the number of the ruler and of the block. The data will be collected during the session and the results discussed later… Combining Errors So far we have discussed how to estimate the random error ∆A associated with a quantity A. Usually, however, the end result depends on intermediate values B, C, etc, all of which have their own uncertainties. Statistical theory states that: 2  ∂A   ∂A  ∆A =  ∆B  +  ∆C   ∂B   ∂C  Experimental Physics I 32 2 Module PH1004 The University of Reading Department of Physics where A = f (B,C), ∆B = the error in B and ∆C = the error in C. So, if: A=B+C ∆A = (∆B )2 + (∆C )2 This gives the likely error in A, and a similar approach yields the formulae below. In other cases, it is necessary to derive an expression for the error starting with the general equation given above. Whilst it is generally pessimistic to assume that all the intermediates will exhibit their maximum errors simultaneously, this assumption is simple to use. Such an approach yields a value of the largest possible error, and you should draw attention to this fact. Then: (i) Sum A=B+C 2 ∆A = ∆B + ∆C (ii) Difference A=B-C 2 ∆A = (iii) Product 2 ∆B + ∆C 2 A=BxC 2 ∆A  ∆B   ∆C  =   +  A  B   C  (iv) Quotient A=B/C 2 ∆A  ∆B   ∆C  =   +  A  B   C  (v) 2 Constant power 2 A = Bn ∆A ∆B =n A B How Many Significant Figures? You may be able to measure the wavelength of a spectral line to 1 part in 10,000 but the value of a capacitance to no better than 10%. In the first case you can quote a value to 5 significant figures (e.g. λ = 589.53nm), but in the second case no more than 2 significant figures are justified (e.g. 12mF). Your calculator will probably deliver 8 significant figures, but when you quote a result do not use any more figures than the experiment allows. Remember that the overall accuracy is largely determined by the least accurate part of the experiment. An extreme example of this feature occurs in the measurement of some torsional properties of a long wire. In this case, there is no point in measuring the length with high accuracy, because the small radius (which can vary by as much as 5%), which appears as the fourth power, dominates the overall accuracy. Errors and Uncertainties in Real Experiments Experimental Physics I 33 Module PH1004 The University of Reading Department of Physics In the previous two Skills Sessions, we have collected and analysed data with no regard for the errors and uncertainties there may have been. Based on the above discussions, we will now rectify this situation. (note: The apparatus will again be provided for you to examine, however, you should really estimate the errors as you are doing the experiment; here, we are doing so after the event which is not best practise.) Firstly, what are the errors on the basic parameters of the experiment? (remember, these are: period, length, mass of bob) Example 1: In the pendulum experiment, I estimate I can read the length of the pendulum to ± 1 mm. With my partner shouting “ start” and “ stop” , I estimate the error on my reactions starting and stopping a stop watch is about ± 0.2 s. On weighing the bob masses, I discover they may differ from their nominal mass by ± 5% I can now generate a table based on the data in Skills Session 1, but now including errors: Length of pendulum = 23 cm ± 0.1 cm mass of bob = 10 g ± 0.5g Trial no. of periods Total time /s Error on total / s Period / s 1 10 9.64 0.96 ± 0.2 2 10 9.65 0.97 ± 0.2 3 10 9.69 0.97 ± 0.2 4 10 9.52 0.95 ± 0.2 5 10 9.44 0.94 ± 0.2 Error on period / s ± 0.02 ± 0.02 ± 0.02 ± 0.02 ± 0.02 Average period = 0.96 s ± 0.01 s The error on the average has been calculated using relation (i) for combining errors in summation (ie, 5 readings hence total error on 5 readings = 5× 0.02 2 then divide by 5 for the average. Notice this means that the error on the average can also be written as: 0.02 5 or, more generally: error on a reading places number of readings Note the error has been rounded to 2 decimal Exercise 2: Estimate the random errors on the various parameters for your experiment and write an appropriate table summarising your data. Hints: Length of pendulum – how precisely can you estimate where the top and the bottom of the pendulum are? How precisely can you read the ruler? Period – You base this on your reactions. How fast are they? How well can you define where a swing starts and ends? Remember, if you’ ve timed 10 periods, your error on a single period is (total error/10) Experimental Physics I 34 Module PH1004 The University of Reading Department of Physics You should end up with a table similar to: l /m ∆l /m p/s ∆p / s 0.10 0.001 0.67 0.01 0.15 0.001 0.83 0.01 0.20 0.001 0.89 0.01 0.30 0.001 1.13 0.01 0.40 0.001 1.28 0.01 How do these errors help us in the analysis of the data? We can indicate these errors on the graphs we plotted in the previous Skills Session, taking account, of course, of the fact that we have applied some functions to our data. Example 2: Taking the data in the above table, and applying Technique 1 in the previous Skills Session to plotting it, we obtain the following table l /m ∆l /m 0.10 0.001 0.15 0.001 0.20 0.001 0.30 0.001 0.40 0.001 p/s ∆p / s p2 / s2 ∆p2 / s2 0.67 0.01 0.45 0.01 0.83 0.01 0.68 0.02 0.89 0.01 0.79 0.02 1.13 0.01 1.28 0.02 1.28 0.01 1.64 0.03 Note that ∆p2 has been calculated using relation (v), such that: S2 S =2 or, S 2 = 2 p∆p 2 p p Careful when manipulating errors: ∆p2 ≠ (∆p)2 We can now put “ error bars” on our replotted graph of p2 vs. l, obtaining something similar to that below: Experimental Physics I 35 Module PH1004 The University of Reading Department of Physics Notice that the error bars are, in this case, quite small, and that the best-fit line doesn’ t cross all the error bars. This may indicate some systematic error(s) which haven’ t been accounted for. Normally, you might expect a best-fit line to cross all the error bars, but don’ t force it! We can use the error bars to generate errors on the gradient. This is illustrated below – note the error bars have been enlarged to ± 10% to illustrate the point. ' ' From this graph, we can estimate the maximum gradient to be 4.53 s2m-1 and the minimum.3.43 s2m-1. For clarity, only the working for the minimum gradient is shown on the graph – normally, you should show everything! Recalling that the best-fit gradient was 3.99 s2m-1, we can quickly see that we can write the gradient as: Gradient = 3.99 s2m-1 ±0.55 s2m-1 Experimental Physics I 36 Module PH1004 The University of Reading Department of Physics (note the + and – error on the gradient is about the same in this case, differing by only 0.01 s2m-1. This may not always be the case, so be prepared to quote “asymmetric” errors – eg, 3.99 s2m-1 +0.50 s2m-1 – 0.45 s2m-1) Calculating the values of g from these gradients, we obtain: g = 9.89 ms-2 + 1.62 ms-2 – 1.18 ms-2 note that the “ textbook” value of 9.81 ms-2 lies within the error range. Exercise 3: Repeat the above procedure using your data. Example 3: Recall we could also obtain a value of g from technique 2 in the previous Skills Session. This involved taking logarithms of both sides of the theoretical relationship between p and l, giving us: log( p) = log(2π ) − 12 log( g ) + 12 log(l ) How do we determine the errors on the log function? The answer is differentiation. Supposing we have a function of x, f(x). If we vary x by an amount ∆x, what is ∆ f(x)? To find this, we need to find f(x+∆x) − f(x). You could just calculate this by putting in the numbers, but that’ s not very satisfactory in general, especially if you have a large data set. We can solve this by differentiating f(x), giving us a measure of how fast f(x) is varying with x, then simply multiply by ∆x. Hence: df ( x) ∆f ( x) = ∆x (note: this is called a First Order Approximation, since we are assuming dx that the differential is a smooth, slowly varying function/ Often this is the case, but take care!) Let’ s take the example of our pendulum. If the error on the period is ∆p, then the error on log(p) is: d log( p ) ∆ log( p ) = ∆p dp Assuming we are taking logs to base 10, then we can use the relationship: d log( x) log(e) = where e = 2.718… is the base of the natural logarithm function, ln(x) dx x Then, we have: log(e) ∆ log( p ) = ∆p and similarly for log(l) p We can then generate a table with errors, as shown below: l /m log(p) ∆l /m p/s ∆p / s log(l) ∆log(l) ∆log(p) 0.10 0.001 0.67 0.01 -1.00 0.004 -0.16 0.006 0.15 0.001 0.83 0.01 -0.82 0.003 -0.10 0.005 0.20 0.001 0.89 0.01 -0.70 0.002 -0.04 0.005 0.30 0.001 1.13 0.01 -0.52 0.001 0.03 0.004 0.40 0.001 1.28 0.01 -0.40 0.001 0.12 0.003 Experimental Physics I 37 Module PH1004 The University of Reading Department of Physics Notice the errors become quite small in this case. Yours might well be larger, but that’ s not a problem. The graph can then be plotted as before, maximum and minimum gradients determined, and, more importantly in this case, the error on the intercept can be found – this gives an error on the measure of g again. Exercise 4: Carry out the above analysis for your data, put appropriate error bars on your graph, determine the error on the intercept and hence obtain a value of g with appropriate errors. A Few Hints and Tips on Errors Remember that Experimental Errors are NOT MISTAKES! They are essential to the usefulness of data. Other scientists cannot compare their data with yours unless they know the level of uncertainty on both your data and their own. Estimate errors realistically. For example, a ruler may typically be marked with 1 mm divisions. You can probably therefore estimate to ± 0.5 mm at worst, but probably no better than ± 0.25 mm. Watch out for systematic errors. These can be very hard to spot. They may be an offset or a mis-calibration of an instrument or measuring device, or something more subtle… can you think of a systematic error in the pendulum experiment? Remember that errors have units too. Experimental Physics I 38 Module PH1004 The University of Reading Department of Physics Skill Sessions 4 Electronic Instrumentation Introduction For many of the experimental projects, in this and subsequent laboratory modules, you will need to be familiar with various electronic instruments. In this session, you will use an AVOmeter, a digital multimeter and an oscilloscope to investigate a series of unknown voltages at the output sockets of a “ black box” . You will also learn how to access the uncertainty in your measurements. The Black Box You are provided with a “ black box” having 8 sockets with unknown voltages which may be either DC or AC with various waveforms. Make sure that you record the number of your “ black box” in your log book, since none of the boxes are the same. When using the AVOmeter and digital multimeter, ENSURE THAT YOU ONLY MAKE MEASUREMENTS OF VOLTAGE, since their internal resistance for the current ranges is very small and damage will result. AVOmeter (Amp Volt Ohm meter) Refer to your “QuickStart” sheet - AVOmeters An AVOmeter is a moving coil instrument and must always be used on its back, since in an upright position the needle will be affected by gravity. For AC voltages, an AVOmeter records the RMS (Root Mean Square) voltage, Vrms = 1 τ ∫V 2 ( t )dt 0 where t is time and τ is the period of the oscillatory wave. (Note – the expression under the square root is merely a formalised way of expressing the time averaged value of the square of the voltage over one period). For a sinusoidal signal (of the form V(t) = V0sin(2πft) where f is the frequency) the RMS voltage is 0.707…. (1 2 ) of the peak voltage. The accuracy of an AVOmeter is a percentage of the full scale (FS) and can be found under the bottom of the meter window. As with all meters, to obtain the maximum accuracy, you should always try to use the most sensitive range possible unless this perturbs the circuit being investigated due to too low an input impedance. Your estimate of the uncertainty in your measurements should include both the instrumental accuracy and that to which you can read the scale. NB Note that the input impedance of an AVOmeter on the 2.5V range is only 100Ω V-1 compared to 1000Ω V-1 for the 10V and higher ranges. This means that the AVOmeter can significantly perturb the circuit being investigated. Digital Multimeter Refer to your “Quick Start” Sheet – Digital Multimeters A digital voltmeter may record either average or RMS AC voltages and has an accuracy that is usually a percentage of the reading plus 1 or more digits in the last displayed place. Depending on the model used, the accuracy may either be found on the underside of the instrument or in the associated manual. Whenever using an instrument for the first time, you should photocopy or record the specifications and put it into your logbook, for future reference. Experimental Physics I 39 Module PH1004 The University of Reading Department of Physics Example: Supposing a digital multimeter has a known precision of 1% and has 4 digits. A reading gives: 1.253 V. What is the error on this reading? Firstly, we only have four digits, so we have an uncertainty of ± 1 in the 4th digit. Hence, quoting just the random error, we would write the reading as: 1.253 ± 0.001 V. Including the 1% precision (a systematic error), we would write: 1.253 ± 0.002 V (remember how errors add, and round them appropriately) Oscilloscope Refer to your “Quick Start” Sheet - Oscilloscopes An oscilloscope can be used to display a voltage waveform and hence to distinguish between different types of AC wave. It can also be used to make measurements of both frequency/time and voltage, providing the variable time and voltage knobs are in the “ CAL” (calibrated) position. The Experiment At the beginning of the session, the use of the three instruments will be explained to you and you should make clear notes as you use each one, so that in future you will be able to do so without further help. You should also clearly record what you are doing as you do it. First, investigate the voltage at each of the 8 sockets of your “ black box” , using the AVOmeter and the appropriate DC voltage range. Repeat these measurements using the appropriate AC voltage range. You should record your results in the form of a table which includes the uncertainty on each measurement: Be sure to record the number of the “ black box” Socket DC Voltage Error AC Voltage Error Next repeat these measurements with the digital multimeter. Do the results agree within their combined errors? Finally, display the output from each socket on the oscilloscope. Make any appropriate sketches and notes. Hence determine the magnitude of the DC voltages (“ free run” mode) and the waveform, peak voltage and frequency of the AC signals. Use the AC peak voltages to calculate the RMS value to compare with the results you obtained with the AVOmeter and digital multimeter. Construct a final table, summarising your results for each socket, including the waveform (or DC), frequency and RMS (or DC) voltage. Do the results obtained with the three instruments agree? Can you explain any discrepancies? Which instrument is the most accurate? Is it the same one for each type of signal? To conclude the session, write a short abstract of not more than 100 words. Experimental Physics I 40 Module PH1004 The University of Reading Department of Physics Project 1 Electricity Experiment A: - DC Networks 1A.1 - Objectives (i) To understand the concept of the Thevenin Equivalence and its implications for electronic instrumentation and circuitry. (ii) To perform experiments with DC networks 1A.2 - Prior Reading O’ Hanian Chapters 28 and 29: FLAP module P4.1. 1A.3 - Preparatory Work (i) From Kirchoff’ s Laws, show that the equivalent resistance (R) of three resistors in series (R1, R2, R3), can be written: R = R1 + R2 + R3 (ii) From Kirchoff’ s Laws, show that the equivalent resistance (R) of three resistors in parallel 1 1 1 1 (R1, R2, R3), can be written:= + + R R1 R 2 R 3 (iii) Explain the term voltage divider circuit. (iv) If a voltage generator with internal resistance R0 produces an open circuit voltage V, derive an expression for the output Vout, from a simple two-resistor (R1,R2) voltage divider. 1A.4 - Safety Procedures Please observe the standard precautions associated with electrical equipment. 1A.5 - Introduction For a circuit containing only voltage generators and resistors, Thevenin' s Theorem states that any combination of voltage generators and resistors considered at the terminals A and B is equivalent at those terminals to a single voltage generator, VTh, in series with a single resistor, RTh. VTh is equal to the open circuit voltage between A and B; RTh is the resistance that would be measured between A and B if all the voltage generators were replaced by short circuits. The objective of this project is to test this theorem. 1A.6 - Background There are a number of basic rules that may be used in all network analysis. All resistors (R1, R2, etc), that are connected in series may be replaced by an equivalent resistance R: R = R1 + R2 + R3 + R4 ….. Similarly, for resistors connected in parallel: 1 1 1 1 = + + ….…. R R1 R 2 R 3 This leads to the voltage divider rule; the voltage across two resistances connected in series divides between them in the ratio of their resistances. Resistances connected in parallel act as a current divider. These rules are effectively specific cases of Kirchhoff’s laws. The first states that at any node in a network, at every instant of time, the algebraic sum of the currents at the node is zero. (For this law, currents entering a node are considered positive; those directed out of the node are negative). The second law states that the algebraic sum of voltages across all the components Experimental Physics I 41 Module PH1004 The University of Reading Department of Physics around any loop of a circuit is zero. These rules may be used to analyse any specific circuit but is often useful to exploit Thevenin’s Theorem to simplify the circuit and hence the analysis. Thevenin’s Theorem states:As far as any load connected across its output terminals is concerned, a linear circuit consisting of voltage sources, current sources and resistances is equivalent to an ideal voltage source VTh in series with a resistance RTh. The value of the voltage source is equal to the open circuit voltage of the linear circuit. The resistance is equal to the resistance that would be measured between the output terminals if the load was removed and all sources were replaced by their internal resistances. 1A.7 - The Experiment 1A.7.1 - The Internal Resistance of the Power Supply In this project you will first find the Thevenin Equivalent quantities for a "black box" that is in the form of a mains powered variable DC voltage supply. We do not ask about the circuit inside the box; VTh and RTh must be determined experimentally. You should set the output of the voltage supply to approximately 6V. Note:- If you change the output voltage before the very last step you will need to start again unless you know the setting precisely. Changing voltage will, in effect, give a new black box, because the internal dynamic circuitry of the power supply changes the effective Thevenin values with voltage. This circuit: DC Power supply RL is equivalent, in Thevenin terms, to: R Th VTh RL The open circuit voltage (zero current flow) V between the output terminals can be measured directly using a multimeter, since the meter has a very high resistance and draws negligible current when in the voltage mode. RTh cannot be measured directly using a multimeter. However, it may be obtained indirectly by measuring the current through various resistive loads connected across the Experimental Physics I 42 Module PH1004 The University of Reading Department of Physics voltage generator. You will need to derive an expression that relates the current to RTh. and you will need to decide how to plot the data usefully. Use the resistance box as a variable load. Measure the current for various loads (RL ) carry out your analysis and hence find the value of RTh. N.B. Ensure that you never exceed the current carrying capabilities of the equipment! This factor sets a minimum level for RL. 1A.7.2 - Thevenin’s Theorem and the Voltage Divider Circuit The analysis of complicated electrical networks may often be simplified by the use of Thevenin’s theorem. In this experiment, you will now evaluate experimentally the behaviour of a specific circuit and then set up the Thevenin equivalent circuit to see if it behaves in the same way. The test circuit shown below is the voltage supply with a potential divider across its output. DC Power supply RL Set up the potential divider circuit as shown in this figure. Determine the Thevenin’ s equivalents for this circuit at the output terminals using the same procedure as above. When you have completed these measurements you will able to check the results by calculation. If you measure the resistances in the potential divider, you should be able to use the rules described in Section 1A.6 and the VTh and RTh for the voltage generator, to determine the equivalents for the generator plus divider. How do the values compare? Now, before you dismantle your circuit, you will need to decide how you will determine whether that circuit and the equivalent circuit constructed using VTh and RTh have the same properties. RTh DC Power supply RL When you have a plan, discuss it with a demonstrator. Carry out the approved plan and construct the equivalent Thevenin circuit shown in the above figure. Remember that the voltage generator has an internal resistance and so you will need to think carefully what value you should set RTh to in your equivalent circuit. Use the precision variable resistance as RTh. Were the circuits equivalent? Experimental Physics I 43 Module PH1004 The University of Reading Department of Physics Experiment B: - Resonance 1B.1 Objectives (i) To use an electrical circuit to observe the effect of resonance (ii) To evaluate the properties of a resonant system and their relationship to the components in that system (iii) To determine the Q factor for the circuit. 1B.2 Prior Reading O’ Hanian Chapter 15 and Chapter 34: FLAP modules P5.4 and P5.5. 1B.3 Preparatory Work (i) Explain what you understand by resonance; how is electrical resonance characterised in term of the amplitude and phase of the output wave relative to the input wave? (ii) Briefly discuss the behaviour of inductors, capacitors and resistors with respect to their ability to store and/or dissipate electrical energy. (iii) What is the commonly used SI derived unit of inductance, and how is this expressed in terms of SI base units? (iv) Explain the term angular frequency and show how this relates to a sinusoidal waveform. 1B.4 Safety Procedures The standard procedures for the use of electrical equipment apply. 1B.5 Introduction In any LCR circuit connected to a sinusoidal voltage source of variable frequency, the current through the components and the voltages across them pass through maxima as the frequency is varied. Taking the quantities to be complex, the relation between the current I and the drive voltage V is, V I= , Z Total where ZTotal is the complex impedance of the circuit. For a series LCR circuit, the total impedance at angular frequency ω is given by, ZTotal = R + jωL + 1 . jωC Therefore the amplitude ip and phase φ are given by, I= V 1   R 2 +  ωL −  ωC   tan φ = Experimental Physics I 44 2 (ωL − 1ωC ) R Module PH1004 The University of Reading Department of Physics The amplitude of the current will be a maximum at an angular frequency ω0 such that, ω 20 = 1 LC This is known as the characteristic frequency. At a frequency ω0 the tanφ is zero and the current is in phase with the driving voltage. In this project we shall study the voltage developed across one of the components. Since an inductance coil has an associated resistance, it is difficult to separate L and R. As a consequence we shall study the voltage across the capacitor. The complex voltage V across the capacitor is, VC = I = jωC V 1   jωC R 2 +  ωL −  ωC   2 . The phase φ is given by, tan φ C = R . (ωL − 1 / ωC ) In this case VC /V is a maximum when, ω = ω 0 (1 − R 2 / 2 LC ) . For most circuits of interest and for the circuit you will study in this project R2 << 2LC. In this case ωr≈ω0 and φc ≈90o, and the voltage at resonance is given by, [VC ] max = VP L . = Vω 0 ω 0 CR R The voltage transfer ratio [VC] max/Vp is known as the Q of the circuit. The other characteristic of the resonance, which is of interest, is the width ∆ω of the peak. This is defined as the difference in frequency between the points where the voltage has fallen to [VC] max/√2. Using the approximation that R2 <<2LC it can be shown that, ∆ω=RCω02 ω0 ∆ω = 1 =Q RCω 0 This gives an alternative way of finding the Q of the circuit. Experimental Physics I 45 Module PH1004 The University of Reading Department of Physics 1B.6 The Experiment You are provided with a unit that contains an inductor and a range of capacitors. The capacitors are connected to a switch that will link any of them such that they will be in series with the inductor. The inductor may be considered as an inductance L in series with a resistance R. Connect one beam of the oscilloscope across EA and set the output of the voltage generator to give 1V peak to peak (VP~0.5V). Connect the other beam of the oscilloscope across EB and measure VC as a function of ω, plot your data and find the resonant frequency. Note, It may be necessary to adjust the output of the voltage generator to maintain the test voltage at a set value as the frequency varies. Also record the value of [VC] max relative to Vp. Calculate a value for Q. Repeat for the other capacitors. Plot a graph to show that 1/ω0 2 is proportional to C and use it to find L. Use the calculated values of Q and the value of L to find R at each frequency from the expression, Q= ω0L R Plot R as a function of frequency. Can you account for the variation? Measure the resistance of the coil with an AVO meter. Why is this value different? Find Q from the expression, ω0 Q= ∆ω Compare this with the value obtained from the voltage magnification at resonance. Compare your measured results with theory and explain any differences. A Inductance B E Experimental Physics I 46 Module PH1004 The University of Reading Department of Physics Project 2 Waves and Interference Experiment A: - Optical Interferometry 2A.1 – Objectives (i) To investigate the ways in which waves interact with one another. (ii) To use optical interference to explore the wave-like nature of light. 2A.2 - Prior Reading O’ Hanian Chapter 39: FLAP modules P5.6 and P6.1. 2A.3 - Preparatory Work (i) Explain the term interference. (ii) Describe briefly three situations where optical interference is observed/exploited. (iii) What is the Michelson-Morley experiment and why was it so significant? (iv) Show that, for two closely spaced wavelengths: - ∆λ = λλ ′ λ 2 = 2t 2t where the notation is as defined in the following text. 2A.4 - Safety Procedures The mercury and sodium light sources use relatively high voltages and may become hot. The mercury lamp produces light in the UV region of the spectrum and you should therefore avoid staring directly at the lamp. 2A.5 - Introduction In this experiment you will study optical interference effects using a Michelson Interferometer (see Figure 1). The basic elements of this instrument are an illumination source, a beam splitter and two adjustable mirrors. Light Scource Aperture & Screen M 1/ B M1 Beam Splitter Compensator M2 Micrometer Figure 1 Experimental Physics I 47 Module PH1004 The University of Reading Department of Physics Light from the entrance aperture is directed along two paths (BM1, and BM2) by the beam splitter B, reflected from the two mirrors M1 and M2 and, finally, recombined. So, on looking through the beam splitter what you see is a reflection of the fixed mirror superimposed onto the movable mirror M1. These mirrors are adjustable in a number of ways. Firstly, M1 can be moved along the path BM. However, its inclination to this direction cannot be changed (M1 is always perpendicular to the incident light beam). Unlike M1, the distance between the beam splitter and M2 is more or less fixed. However, the inclination of mirror M2 can be altered with respect to the incident light beam by means of the adjustment screws. Although you see two superimposed reflections (one in M1 and one in M2), these are both reflections of the same source and are therefore spatially coherent. So, if the difference in the path lengths BM1 and BM2 is not too large, then the light from the two images will interfere and some pattern of fringes will be seen. In the figure shown above, the path difference is represented by the distance between M2 and M1, the optical paths BM2 and BM2 being equivalent. In this experiment you will study two situations, one in which the fringes are circular and one where they are straight. 2A.5.1 - Circular Fringes The conditions for circular fringes are that M1 and M2 are parallel, but separated by a small distance. When combining two light beams the condition for constructive interference is that the two beams are in phase, that is, the path difference corresponds to an even number of half wavelengths (i.e. an integral number of wavelengths). Similarly, the condition for destructive interference is that the two beams are 1800 out of phase, such that the path difference is an odd number of half wavelengths. When M1 and M2 are parallel but separated, these two conditions will each be satisfied at certain angles of observation. d M1 M1 M2 d M2 (b) (a) This is shown schematically in figure2 (a), for simplicity, in two dimensions. In three dimensions these conditions are fulfilled anywhere on a circle of radius r; thus circular fringes are seen. Such fringes are called Fringes of Constant Inclination or Haidinger Fringes and are seen at infinity because, as shown above, the light beams that cause them are parallel. 2A.5.2 - Straight Fringes The conditions for straight fringes are the converse of those for circular fringes, that is, M1 and M2 must be coincident but tilted relative to one another so as to form an optical wedge. This is shown schematically in figure 2(b). The general condition for constructive interference will now be met for certain values of d, to give, in three dimensions, a series of straight fringes running along directions where the wedge spacing is constant. These fringes are called Fringes of Constant Optical Thickness or Fizeau Fringes, and are seen in the mirror surface since the two light beams, which cause them, are diverging. Experimental Physics I 48 Module PH1004 The University of Reading Department of Physics 2A.6 - The Experiment 2A.6.1 - Setting up the Interferometer Initially you should use a mercury lamp, together with a green filter, to provide monochromatic illumination. Now look at M1 through the beam splitter. Can you see any fringes? If not, place a marker on the ground glass screen (an ink mark or the point of a pencil are suitable) and observe the image of the marker in M1. Three images will generally be seen. What is the origin of each of the images? Now, adjust the screw at M2 so that two of the markers coincide. If fringes are still not seen, try the other possible combination. Once you have obtained a fringe pattern investigate the effect of each of the mirror adjustments in turn and, finally, adjust M2 carefully until you can see circular fringes that disappear into or appear at the centre of the fringe pattern as M1 is adjusted. (For this, use the low-geared micrometer adjuster.) 2A.6.2 - Calibration Before making any other measurements it is necessary to calibrate the micrometer in terms of the actual displacement of the mirror M1, since the micrometer moves the mirror via a lever. This you will do optically by monitoring changes in the pattern of fringes. Count the number of fringes that disappear into or appear at the centre of the fringe pattern as the micrometer is turned and record the micrometer reading periodically. Alternatively, you may find it more convenient to count the fringes as they pass a pin set up in the field of view. Each fringe corresponds to a movement of M1 of 1/2 wavelength of mercury green light (λ= 546.1nm). Now, before proceeding, plot the micrometer readings against the number of fringes passed. Do your data give a satisfactory straight line? If not, then repeat this part of the experiment but include more fringes. 2A.6.3 - Spectroscopy The Michelson Interferometer can be used for the accurate determination of small wavelength differences. You will now investigate this using a sodium lamp, the yellow light from which contains two lines that are of very nearly equal wavelength (the sodium D lines). As the interferometer spacing is changed the circular fringes for the two wavelengths fall into and out of step giving periodic disappearances of the fringes. Take micrometer readings for a series of positions of maximum contrast and find the average movement, t, of the mirror M1 between successive maxima using your earlier calibration. Now, the condition for constructive interference, for each of the lines in the doublet is: - 2d cosθ = Nλ 2d cosθ = N ′λ ′ Writing cos θ. ≈ 1, because we are always near to normal incidence then, when the fringe pattern is most clear: - 2d = Nλ = N ′λ ′ Adjusting the position of the mirror, such that the fringes pattern fades and subsequently reappears, then: - 2(d + t) = (N + m)λ = (N ′ + m + 1)λ ′ Experimental Physics I 49 Module PH1004 The University of Reading Department of Physics In this, m is the number of fringes of wavelength λ that have passed as a result of moving the mirror a distance t. For the fringe systems to have passed out of and back into step once, one more fringe of wavelength λ′ will have passed. Using the resultant equation, ∆λ = λλ ′ λ 2 ≈ 2t 2t Calculate ∆λ from your measurements of t. (You may take λ to be 589nm). 2A.6.4 - Straight Fringes and White Light Fringes This part of the experiment requires very careful adjustment of the instrument if white light fringes are to be seen. Firstly, replace the sodium lamp with the mercury lamp and again mount the green filter in front of the ground glass screen. Now, with circular fringes in view adjust the micrometer and note what happens to the size of the fringes as you vary d, the effective distance between the mirrors. Adjust the micrometer so that the circular fringes increase in size until the whole field of view appears almost uniformly light or dark (M1 and M2′ approximately coincident). Now adjust one of the screws until straight fringes are visible (inclining M2′ with respect to M1). Replace the monochromatic filtered mercury source with a tungsten lamp and slowly adjust the position of M1 using the micrometer and low gear attachment. When the optical path lengths BM1 and BM2 are exactly the same (M1 and M2 exactly coincident) coloured white light fringes will be seen at the centre of the field of view. This effect is only visible under these very particular conditions because the fringe patterns from the range of wavelength present in the white light spectrum soon get out of step with one another to give a uniformly illuminated field of view. The reason for this is, of course, the same as for the fringe disappearances you previously saw when using sodium light. Experimental Physics I 50 Module PH1004 The University of Reading Department of Physics Experiment B: - Sound Waves 2B.1 – Objectives (i) To explore the physics of longitudinal waves. (ii) To perform experiments with sound and to determine the velocity of sound in air. 2B.2 - Prior Reading O’ Hanian Chapters 17 and 16: FLAP modules P5.6 and P5.7. 2B.3 - Preparatory Work (i) The density of air, at standard temperature and pressure is 1.293kgm-3. Check that this is reasonable by estimating the height of atmosphere required to produce a standard atmospheric pressure (ii) As discussed below, U(x,t) = a sin (ωt - kx) Check that, ω= B k is a solution by substitution. ρ0 (iii) Satisfy yourself that:- U(x,t) = a sin (ωt - kx) represents a travelling sine wave of frequency f=ω/2π and a wavelength λ=2π/k propagating in the positive x-direction at velocity v= ω/k = (B/ρ0)1/2 by sketching the waveform as a function of x for successive fixed values of t. (iv) Resonance occurs when a standing wave is set up. Show that the condition for resonance is, λ 1 l = (n + ) 2 2 where λ is the wavelength of the sound wave, l is the length of the tube and n is an integer. 2B.4 - Safety Procedures Please observe the standard safety precautions for electrical equipment. This project generates noise! Please try to minimise the level of noise so as to minimise the inconvenience to others as well as yourselves. 2B.5 - Introduction A sound wave in a gas is a longitudinal wave. Superimposed on the random motion of the gas molecules, are small oscillating displacements in the direction of the wave, which correspond to variations in pressure along its path. In this experiment, a small loudspeaker is placed near one end of a long circular tube, the other end of which is closed. The experiment investigates the system of standing waves established in the tube through the superposition of incident plane waves travelling down the tube and reflected waves travelling in the opposite direction. The maxima (or minima) of amplitude, or of pressure in the sound wave, are spaced at half-wavelength intervals. Hence, if the frequency of the wave is known, the speed of sound in the tube can be obtained from the spacing of the maxima. Experimental Physics I 51 Module PH1004 The University of Reading Department of Physics 2B.6 - Background Consider a mass m of gas at a pressure Po, which occupies a volume Vo between the planes F and G which are a distance ∆x apart. P + dP ∆x dx P ∆x F G If A is the area of cross section of the tube and ρ0 the density of the gas then Vo=A∆x and m=ρ0Vo. As a sound wave passes down the tube this element of mass is displaced along the x-axis (i.e. the axis of the tube) by a distance U and hence experiences an acceleration d2U/dt2. The displacement is caused by a difference in pressure on the faces of the element. If P is the pressure on the face F, then the pressure on G is given by, P+ dP ∆x dx Hence the net force in the direction x is, - dP ∆xA dx The pressure in air is related to the fractional change in volume (at the point w where the pressure acts) by, ∆P = -B ∆V V Where ∆P = (P-Po) and B is the bulk modulus of elasticity for air. In the tube ∆V the change in the volume originally between F and G, will be given by the difference in U between the two ends,  dU  ∆V =   A∆x  dx  And ∆V dU = V dx The net force on the element can therefore be written as, d d 2U - ( ∆P)∆x. A = B 2 ∆x. A dx dx The equation of motion is, ρo V o Experimental Physics I 2 d U = B d 2U ∆x.A 2 2 dt dx 52 Module PH1004 The University of Reading Department of Physics Such that, d 2U B d 2U = dt 2 ρ o dx 2 A solution to this equation is, U(x,t) = a sin (ωt - kx) Where a is an arbitrary amplitude, and, ω= B k ρo If this wave is then perfectly reflected at x = 0 we produce an additional travelling wave propagating in the negative x-direction, U r (x,t) = -a sin(ωt + kx) The minus sign is required by the condition that the total displacement U + Ur must be zero at x=0 at all times, since the reflector is rigid. The combined displacement of the two travelling waves is, Utotal ( x, t ) = a[sin(ωt − kx ) − sin(ωt + kx )] Or U total ( x, t ) = −2a sin( kx) cos(ωt ) Which describes a standing wave. Sketch this for successive values of t. 2B.7 - The Experiment The sound wave is excited by the diaphragm of a small loudspeaker placed at one end of the tube. A removable plug closes the other end of the tube. The displacement U is a maximum at the diaphragm and zero at the plug. Since the excess pressure is given by, ∆P = P − P0 = − B dU dx It follows that the excess pressure is zero at the diaphragm and a maximum at the plug. The pressure variations in the gas can be explored with a probe microphone. Connect the loudspeaker to the function generator, and the microphone to the oscilloscope. Insert the microphone, ensuring that it is located at a point of maximum pressure variation at all times. Ensure that the plug has a bare metal face. Find two well-defined resonances (in the range 200Hz to 4000Hz) and plot the shape of the resonance peaks. Deduce a value of Q in each case, where 1/Q = ∆f/fR and fR is the resonant frequency and ∆f is the width of the resonance curve at 1/√2 of the maximum amplitude. Determine the positions of minima in pressure along the tube axis by moving the probe microphone within the tube. Use these to determine the wavelength of the sound wave and hence the speed of sound. Repeat using several different resonant conditions. Experimental Physics I 53 Module PH1004 The University of Reading Department of Physics Use your value of the speed of sound to deduce a value for B. It can be assumed that, since the oscillations in the gas are rapid, they take place without heat exchange to the surroundings; such processes are termed adiabatic. For a gas PVγ = constant, where γ is characteristic of the gas, with a value of 7/5 for air or any other diatomic gas. The bulk modulus B is related to γ by B=γPo which can be demonstrated as follows. Since PVγ = constant then, d ( PV γ ) = 0 dV Vγ dP + Pγ V γ −1 = 0 dV V dP =γ P dV B=− ∆P ∆V /V0 B = - vo dP dV B = γ Po Obtain a value for γ from your measurements. If the reflector is imperfect then the amplitude of the reflected wave will be reduced. It may also change sign. The more general combined displacement can be written as, Utotal ( x, t ) = a[sin(ωt − kx ) − R sin(ωt + kx )] where 1 ≥. |R| Use the computer package provided (type sound at the prompt if not running) to explore how this total displacement function varies with the reflection factor R. Enter different values for R and sketch the results. You will see that envelopes are generated which have the form shown in the figure below. Experimental Physics I 54 Module PH1004 The University of Reading Department of Physics a1 V(t) a2 t R is related to the amplitudes a1 and a2 by |R| = (a1-a2)/(a1+a2). Check this from the computergenerated data. Now check out these predictions by attaching the felt disc to the reflecting plug and measuring the maxima and minima in the amplitude of the standing wave system. Evaluate R for the felt disc. Experimental Physics I 55 Module PH1004 The University of Reading Department of Physics Project 3 Applications of Electronics Experiment A: - The Strain Gauge 3A.1 Objectives:(i) To use strain gauges to investigate the strain induced into a bent metal strip. (ii) To become familiar with the use of a bridge circuit to measure small changes in resistance. (iii) To gain experience in the use of Vernier scales. 3A.2 Prior Reading (i) O’ Hanian Sections 14.4 and 29.6. (ii) Introductory material on operational amplifiers, to enable you to answer 3A.3(ii). 3A.3 Preparatory Work To perform this experiment, it is first necessary to derive the relationship between the output voltage from the operational amplifier, Vout, and the displacement, d, of the metal strip. You should do this in stages: (i) Derive an expression for the radius of curvature, r, of the arc followed by the metal strip, in terms of d and the distance between the two knife edges, 2s (Section 3A.6). Simplify the result by neglecting terms in d2, since in practice d is small. Hence, following Section 3A.6.1, obtain an expression for the strain, e, in terms of d, s and the thickness of the metal strip, 2t. From the definition of the gauge factor, G, convert this into an expression for the fractional change in the resistance of the gauges, ∆R/R. (ii) Derive an expression for the voltage gain, Vout/Vin for the simple operational amplifier circuit in Figure 1, in terms of the feedback resistances R1 and R2, explaining the basis of your derivation. R2 R1 - + Figure 1 (iii)Derive an expression for the output voltage of the unbalanced Wheatstone bridge, VB, (and hence the input voltage for the operational amplifier, Vin) in terms of the supply voltage, V, and ∆R/R. Combine this expression with those from (i) and (ii) above to yield a final expression for Vout in terms of d. (iv) Describe a Vernier scale and explain how it is used. Experimental Physics I 56 Module PH1004 The University of Reading Department of Physics 3A.4 Safety Procedures Please observe the standard safety precautions for electrical equipment. 3A.5 Introduction A strain gauge is a device whose resistance changes in response to a deforming load. Most strain gauges consist of either a long thin wire wrapped around a very thin flat former, or a foil of similar shape, which is attached very firmly to a specimen in which the strain is to be measured. If the specimen is then strained along the axis of the strain gauge then it is assumed that the gauge will experience the same strain and, as a consequence of this change in its dimensions its electrical resistance will change proportionally. If the gauge experiences a strain, e, e = ∆1/1 and a fractional change of resistance ∆R/R occurs. The gauge factor G is defined by ∆5  5 G= = the fractional change in resistance per unit strain. H 3A.5.1 Precautions when using Strain Gauges As typical maximum strains to which specimens are subjected are of the order of a few percent, and since G for most gauges is between 1 and 5, then the fractional change of resistance is always small (ie seldom more than 10% at most). Hence accurate bridge circuitry must be used to measure such changes and other physical processes leading to resistance change must be excluded or their effects compensated. The output from the bridge is amplified before being measured by a DVM. The principal such process is the change of resistance with temperature and this has such a large effect that temperature changes of a few degrees may easily produce resistance changes much larger than those produced by the strain. This effect is usually compensated by having two gauges attached to the specimen in close proximity so that both experience the same temperature change but orientated in different directions so that they experience different strains. 3A.5.2 Orientation of Strain Gauges (i) General case – simple strain If the specimen experiences simple axial strain then the only possible positions for a strain gauge are variations of orientation with respect to the axial direction. The usual arrangement is to have one gauge parallel to this axis and the other perpendicular to it. The axial gauge experiences the full axial strain so that (∆R/R) = Ge whereas the perpendicular gauge experiences a smaller (Poisson) compressive strain (∆R/R)⊥ = Ge⊥ = σGe where σ is Poisson’ s ratio for the specimen. (ii) Special case – bending beam Because the axial strains on the inner and outer surfaces of a bent beam of simple symmetric cross-section are equal and opposite (for small strains), both gauges may Experimental Physics I 57 Module PH1004 The University of Reading Department of Physics be orientated axially, provided that the temperature difference between the sites is negligible. 3A.5.3 Thermal Expansion Effects If a strain gauge and a specimen have different coefficients of thermal expansion then any gauge in contact with the specimen will experience strain when the temperature is changed, no matter what its orientation. Two types of gauge having expansion coefficients “ matched” for use with aluminium and steel are readily available. 3A.6 Theory Before you start the experiment it is necessary that you first analyse the situation you are studying. Initially you need to obtain an expression for the strain in the upper and lower strain gauges in terms of the deflection, d, o the centre of the strip, and other geometrical factors. This can be done with reference to the following diagrams which show (I) the geometry of the loaded rod and (ii) a schematic section through the deformed metal strip. In these circumstances the upper surface, U, will be in tension, the lower surface, L, will be in compression and the central neutral plain, N, of the strip will be unstrained. These three planes are shown in the figures. In what follows, the subscripts U, L and N are appended to parameters to similarly specify a particular plane within the sample. s knife edge d s lU lN knife edge lL mg t mg r r t r Figure 2 3A.6.1 Strain Gauges Assume that the mild steel specimen when loaded symmetrically about the two knife edges will bend into a circular arc. In this case, the following expressions apply for the length, l, of each of the indicated planes, and lU ∝ r + t, lN ∝ r lL ∝ r – t, Where 2t is the thickness of the metal strip. Hence obtain an expression for the strain and the associated fractional change in the resistance of the two gauges. 3A.6.2 Wheatstone Bridge Now consider the Wheatstone bridge element. Experimental Physics I 58 Module PH1004 The University of Reading Department of Physics +V upper strain gauge RU RO VB lower strain gauge RO RL -V When the specimen is undeformed, When the strip is deformed, and Figure 3 RU = RL = R. RU = R + ∆R RL = R - ∆R. Hence derive an expression for the bridge output voltage VB, in terms of R and ∆R, when the supply voltage rails are at potentials of ∆V. Combine this result with that from Section 3A.6.1, 1and the gain of the operational amplifier with feedback resistors R1 and R2, to obtain a final expression for the output voltage, Vout, from the operational amplifier. Experimental Physics I 59 Module PH1004 The University of Reading Department of Physics 3A.7 Experiment The complete bridge and operational amplifier circuit is shown in Figure 4. The bridge resistors, Ro and the feedback resistors R1 = 499Ω and R2 = 100kΩ are of the high stability type, ±0.1 % with a very low thermal coefficient of resistance, ±15ppm/0C. +V R2 RO RU +5V A R1 B R1 - R Bal RL + R2 RO Rl Vout null 0V -5V -V Figure 4 (i) Earth the two inputs to the amplifier at the points A & B (zero input) and adjust the “ offset null” potentiometer RΝ to give zero amplifier output as measured by the DVM. You may find it beneficial to use an oscilloscope trace for setting the null point. (ii) With the strain gauges unloaded, remove the two earthing connections made in part (i) and balance the bridge using the potentiometer RBal. (iii) Before starting to take measurements, test the response of the system by gently flexing the strip in both directions. You should see similar output voltages of opposite sign. (iv) Set up the strip on the knife edges so that it may be bent symmetrically. Zero the bridge with the scale pans attached, but with no weights applied. By applying equal weights to both ends of the strip, measure the amplifier output voltage as a function of the deflection of the strip as observed using the travelling microscope. Finally, find the gauge factor G for the strain gauges by plotting a suitable graph. Experimental Physics I 60 Module PH1004 The University of Reading Department of Physics Experiment B: - Electrons and Semiconductors 3B.1 Objectives (i) To investigate the current/voltage characteristics of a semiconductor device. (ii) To analyse quantitatively it’ s exponential characteristics. (iii) To use the response to measure temperature. 3B.2 Prior Reading O’ Hanian Chapter 44.3-44.5: FLAP module P11.4. 3B.3 Preparatory Work (i) Sketch the current voltage characteristics of a diode. (ii) Consider the characteristics of the diode, as represented by the equation given in Section 3B.5. Explain what you understand by the saturation current is and indicate this on the above sketch. (iii) Rearrange the equation given in Section 3B.5 such that it can be plotted as a straight-line graph, assuming that is<