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: Dr P.A. Hatherly, Prof. A.C.Wright Alternative Supervisor: 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
1This 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 20% • Laboratory notes 20% • Error Analysis 20% • Abstract 20% • Overall performance 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 to [email protected] and [email protected] please) before the start of the session.
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 K3 is another constant.
Experimental Physics I 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: -
− T0 σ = σ 0 exp T 1 / 2 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.
DMM e DC l AVO p
Power m a
Supply S
Experimental Physics I 9 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.
Specimen Voltage ±0.2 / V Current at max. Voltage ±0.2 / V Current at temp. ±0.001 / Min. temp. mA ±0.001 / mA RED 10.0 1.783 10.0 1.820 BLACK 10.0 1.392 10.0 0.024
NB (i) The laboratory manual states that the red specimen is metallic. (ii) The laboratory manual states that the black specimen is a disordered semiconductor (iii) 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. (iv) 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) Record the current flowing at room temperature (ii) 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 Department of Physics
(iii) 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 RUN 2 Temp. ±1/oC Current Temp. ±1/oC Current ±0.001/mA ±0.001/mA -190 1.820 -190 1.820 -180 1.819 -180 1.818 -160 1.815 -160 1.815 -140 1.812 -140 1.811 -120 1.808 -120 1.808 -100 1.804 -100 1.804 -80 1.801 -80 1.801 -60 1.797 -60 1.797 -40 1.793 -40 1.793 -20 1.789 -20 1.789 0 1.786 0 1.786 19 1.783 19 1.784
Table 1 Current flowing through specimen A as a function of temperature.
Temp. ± 1 /oC RUN 1: RUN 2: 1/I ± 0.0003/ mA-1 1/I ± 0.0003/ mA-1 -190 0.5495 0.5495 -180 0.5497 0.5501 -160 0.5510 0.5510 -140 0.5519 0.5522 -120 0.5531 0.5531 -100 0.5543 0.5543 -80 0.5552 0.5552 -60 0.5565 0.5565 -40 0.5577 0.5577 -20 0.5590 0.5590 0 0.5599 0.5599 19 0.5608 0.5605
Table 2 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: -
− T0 σ = σ 0 exp 1/ 2 T
Experimental Physics I 13 Module PH1004 University of Reading Department of Physics
RUN 1 RUN 2 Temp. /oC Current Temp. /oC Current ±0.001 /mA ±0.001 /mA -190 0.027 -190 0.023 -180 -180 0.046 -170 0.080 -170 0.085 -160 0.118 -160 0.129 -150 0.192 -150 -140 0.290 -140 0.278 -120 0.475 -120 0.498 -100 0.753 -100 0.803 -80 0.947 -80 0.977 -50 1.051 -50 1.092 -30 1.208 -30 1.186 0 1.305 0 1.364 19 1.342 19 1.401
Table 3 Current flowing through specimen B as a function of temperature.
1/T1/2 / K-1/2 RUN 1: Current / mA RUN 2: Current / mA 0.1100 0.027 0.023 0.1037 0.046 0.0985 0.080 0.085 0.0941 0.118 0.129 0.0902 0.192 0.0867 0.290 0.278 0.0808 0.475 0.498 0.0760 0.753 0.803 0.0720 0.947 0.977 0.0670 1.051 1.092 0.0642 1.208 1.186 0.0605 1.305 1.364 0.0584 1.342 1.401
Table 4 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
0.56
1 -
A 0.558 m /
t 0.556 n e r r
u 0.554 C / 1
0.552
0.55
-200 -150 -100 -50 0 50
Temp/ oC
Conductivity Data for the Black Specimen
10
1 A m / t n e r r u
C 0.1
0.01
0.06 0.07 0.08 0.09 0.1 0.11 1/ T 0.5 /K -0.5
Experimental Physics I 16 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.)
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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!
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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.
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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 clamp instruments.
Set up the apparatus as shown in figure 1. Record this figure thread Be sure to record the length of the pendulum, Lab stand 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. bob
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 Figure 1. Apparatus for measuring the period of a simple pendulum
Experimental Physics I 20 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 ª /sº 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.
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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?
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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:
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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 b/s as a guide, plot the a parameter on the x axis, and b on the y axis. On the a 10 14.3 axis, a range of 0 – 80 would be suitable – devise a suitable range for b 20 22.1 25 26 Be sure to properly label the graph – see the sketch above. 32 31.46 45 41.6 60 53.3 77 66.56
(b) Draw a line through the data points, and extract the gradient, A and intercept, B
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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 ª best fitº straight line can still be drawn. There are mathematical techniques for determining the best fit line, but for our purposes, ª eyeballingº 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 b/s Judge for yourself now the axes scales. 13 21 36 33 39 40 46 48 57 52 64 61
73 67
(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.
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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: 1 2 1 l l 2 p = 2π = 2π (recall that x = x ) g g
Squaring both sides of the equation, we obtain: 4π 2 p 2 = 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
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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) = log2π g Recalling the basic properties of logarithms, this becomes: l log( p) = log(2π )+ log g which can be further manipulated to: 1 1 log(p) = log(2π )− 2 log(g) + 2 log(l)
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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.
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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 Useful Graphical Form On y axis plot: On x axis plot: y = aebx ln(y) = ln(a) + bx ln(y) x y = ax b ln(y) = ln(a) + b ln(x) ln(y) ln(x) b or simply y = ax b if b is known y x a 1 b x c 1 x c y = c = + b + x y a a y
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 (i) = exp(Ax+B) y
1 A (ii) = 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.
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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.
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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
(i) A = cos (B) and (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 1 i 1
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The Correct Method Take an even number of readings, and pair them as shown below for n = 12.
12 - 6 11 - 5 10 - 4 9 - 3 8 - 2 7 - 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 2 ∂A ∂A ∆A = ∆B + ∆C ∂B ∂C
Experimental Physics I 32 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 ∆A = ∆B2 + ∆C2
(ii) Difference A = B - C ∆A = ∆B2 + ∆C 2
(iii) Product A = B x C 2 2 ∆A ∆B ∆C = + A B C
(iv) Quotient A = B / C
2 2 ∆A ∆B ∆C = + A B C
(v) Constant power 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
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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 ª startº and ª stopº, 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 Error on period / s 1 10 9.64 ± 0.2 0.96 ± 0.02 2 10 9.65 ± 0.2 0.97 ± 0.02 3 10 9.69 ± 0.2 0.97 ± 0.02 4 10 9.52 ± 0.2 0.95 ± 0.02 5 10 9.44 ± 0.2 0.94 ± 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 = 5×0.022 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 number of readings Note the error has been rounded to 2 decimal places
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 p/s ∆p / s p2 / s2 ∆p2 / s2 0.10 0.001 0.67 0.01 0.45 0.01 0.15 0.001 0.83 0.01 0.68 0.02 0.20 0.001 0.89 0.01 0.79 0.02 0.30 0.001 1.13 0.01 1.28 0.02 0.40 0.001 1.28 0.01 1.64 0.03
Note that ∆p2 has been calculated using relation (v), such that: 2 S S 2 = 2 or, S = 2 p∆p p 2 p
Careful when manipulating errors: ∆p2 ≠ (∆p)2
We can now put ª error barsº 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:
1 1 log(p) = log(2π )− 2 log(g) + 2 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 ∆l /m p/s ∆p / s log(l) ∆log(l) log(p) ∆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 ª black boxº. You will also learn how to access the uncertainty in your measurements.
The Black Box You are provided with a ª black boxº 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 ª black boxº 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,
1 τ V = V 2 (t)dt rms ∫ 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 ª CALº (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 ª black boxº, 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 ª black boxº
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 (ª free runº 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 R2 R3 (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 R2 R3 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 V in series with a resistance R . Th Th 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 RL supply
is equivalent, in Thevenin terms, to:
R Th
VTh R L
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 RL Power supply
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,
1 Z = R + jωL + . Total jωC
Therefore the amplitude ip and phase φ are given by,
V I = 2 2 1 R + ωL − ωC
(ωL − 1 ) tanφ = ωC R
Experimental Physics I 44 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,
1 ω 2 = 0 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,
I V . VC = = jωC 2 2 1 jωC R + ωL − ωC
The phase φ is given by, R tanφ = . C (ωL −1/ωC)
In this case VC /V is a maximum when,
2 ω = ω 0 (1 − R / 2LC) .
For most circuits of interest and for the circuit you will study in this project R2 << 2LC. In this case o ωr≈ω0 and φc ≈90 , and the voltage at resonance is given by,
VP L [VC ]max = = Vω 0 . ω 0CR 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,
2 ∆ω=RCω0
ω 0 1 = = Q ∆ω 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.
2 Plot a graph to show that 1/ω0 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,
ω L Q = 0 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, ω Q = 0 ∆ω
Compare this with the value obtained from the voltage magnification at resonance.
Compare your measured results with theory and explain any differences.
A B Inductance
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: - λλ′ λ 2 ∆λ = = 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 / M M 1 & 1 Screen Beam Splitter B
Compensator
Micrometer M 2
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
d M M M 1 M 2 1 2 (a) (b)
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, λλ′ 2 ∆λ = ≈ λ 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) = asin(ωt - kx) Check that, B ω = k is a solution by substitution. ρ 0
(iii) Satisfy yourself that:- U(x,t) = asin(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 1/2 velocity v= ω/k = (B/ρ0) 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 P + dP ∆x dx
F ∆x 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, dP P + ∆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, ∆V ∆P = -B 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 d2U - (∆P)∆x.A = B ∆x.A dx dx2
The equation of motion is,
d 2U d 2U ρoV o = B ∆x.A dt2 dx2
Experimental Physics I 52 Module PH1004 The University of Reading Department of Physics
Such that, d2U B d2U = 2 2 dt ρo dx A solution to this equation is, U(x,t) = asin(ωt - kx)
Where a is an arbitrary amplitude, and, B ω = k ρo If this wave is then perfectly reflected at x = 0 we produce an additional travelling wave propagating in the negative x-direction,
Ur (x,t) =-asin(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
Utotal (x,t) = −2asin(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, dU ∆P = P − P = −B 0 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
dP V γ + Pγ V γ −1 = 0 dV
dP V = γ P dV
∆P B = − ∆V /V0
dP B = -vo 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)
− Rsin(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 computer- generated 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. R 2
R 1 -
+
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 ª matchedº 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.
lU l knife s d s knife N edge edge lL t mg mg t
r r 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,
lU ∝ r + t, lN ∝ r and 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 R R strain gauge U O
VB
lower R strain gauge RL O
-V
Figure 3 When the specimen is undeformed, RU = RL = R.
When the strip is deformed, RU = R + ∆R and 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 +5V
R R 2 R O U
R A 1 -
R Bal R B 1 +
Vout R R Rl null L RO 2 0V
-5V -V
Figure 4
(i) Earth the two inputs to the amplifier at the points A & B (zero input) and adjust the ª offset nullº 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< 3B.4 Safety Procedures Liquid nitrogen is very cold and prolonged contact will result in a severe burn. This liquid must 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! 3B.5 Introduction Many electronic devices contain Semi-conducting materials such as silicon, germanium and gallium arsenide. In particular, junctions between different semiconductors (p.n. junctions, bipolar transistors), semiconductors and insulators (field effect transistors), and semiconductors and metals (Schottky diodes) are often employed to produce electronic devices. In this experiment you will study the behaviour of a simple electronic component, namely a silicon diode. By considering the response of this device to various applied voltages you will then be able to use it to measure the temperature of boiling liquid nitrogen. Most of the electronic components that you have encountered obey Ohm©s Law. That is, the current, i, which flows in the device is proportional to the voltage, V that is applied to it, i ∝ V Examples of so-called linear devices are resistors, capacitors and inductors. A diode is a device that will only conduct electricity in one direction: even in this, the forward direction, the current is not proportional to the voltage. A diode is therefore a non-linear electronic device. Diodes contain a junction between two dissimilar materials, such as a metal and a semiconductor, and electrons move easily in one direction but find it almost impossible to move in the opposite direction. A number of processes may be involved at the junction, but the net result is the injection of electrons from one material into the other when a suitable electric field is applied. A suitable Experimental Physics I 61 Module PH1004 The University of Reading Department of Physics electric field equals a voltage applied in the forward direction: the injection of electrons equals a current, which flows throughout the circuit. The behaviour of such a device can be analysed using the following empirical relationship: - eV i = i [exp( ) −1] s nkT In the above equation, V is the applied voltage, i is the current that flows at a temperature T, and e, k, is, and n are constants. e is the charge on an electron, k is the Boltzmann constant, is is the saturation current and n is an ideality factor which equals one for an ideal diode but which, in practice, is somewhat higher. 3B.6 The Experiment Set up the circuit shown below, which will enable you to investigate the current/voltage relationships of the diode. Because of the non-linear behaviour of this device, before taking any measurements, take a few minutes to explore the way in which the current varies with the applied voltage in the forward direction. Be careful not to exceed the maximum current that the AVOmeter or the diode can handle! The current should always be less than 10mA. Repeat the above with the diode connected in the reverse direction. What is the maximum current that you can observe with the given power supply? What happens if you disconnect the voltmeter? Now, you should be able to estimate the magnitude of is. Discuss your conclusion with a demonstrator before proceeding. Now, returning to the forward direction, take measurements of the current as a function of applied voltage and, by plotting a suitable graph on log-linear graph paper, evaluate the diode ideality factor, n. Also, by re-plotting your data, produce a better estimate of is. How does this value compare with your previous estimate and is the approximation made in Section 3B.3 justified? You are now in a position to use the diode to measure temperature, since you can measure i as a function of V, and you know e, n and k. Get a demonstrator to pour out some liquid nitrogen for you and then immerse the diode, leave it for a few minutes to cool. AVOmeter D.C Digital Diode Volts Voltmeter Take measurements of the current as a function of the applied voltage and plot a graph on log-linear graph paper. From this you can evaluate the temperature of the diode, which will be very close to that of liquid nitrogen. Experimental Physics I 62 Module PH1004 The University of Reading Department of Physics Project 4 Classical Physics Experiment A: - The Charge to Mass Ratio of the Electron 4A.1 Objectives (i) To perform Lenard's classic experiment to determine e/m (ii) To evaluate the ratio e/m and associated errors. 4A.2 Prior Reading O' Hanian Chapter 31: FLAP module P8.1. 4A.3 Preparatory Work (i) Describe how the current flowing in a solenoid is related to the consequent magnetic field. (ii) For the three orthogonal vectors: - a = b∧ c , draw a diagram showing their relative directions (iii) Write the magnitude of the force experienced by an electron as it travels perpendicular to a magnetic field as a centripetal force and hence show that:- e 2V = m B2 R2 (Hint: Think about the kinetic energy of the electrons: see text for notation). (iv) From the geometry of the electron path show that the radius of curvature, R, is related to the deflection b of the electron beam across the CRT screen by the a2 + b2 equation: - R = where a is the distance from the electron gun to the 2b screen. You can assume that for small deflections, the screen is flat. 4A.4 Safety Procedure This apparatus uses a relatively high voltage and you should ensure that no liquid or other moisture is in close contact with the apparatus. 4A.5 Introduction This experiment is concerned with measuring the charge to mass ratio of the electron (e/m), by monitoring the effect of a magnetic field on an electron beam. In this adaptation of Lenard©s method (first reported in 1900) a small cathode ray tube (CRT) is used to provide the electron beam whilst a large coil of wire, through which a current flows (solenoid), generates the necessary magnetic field. When charged particles move in a magnetic field they experience a force, which deflects them from their otherwise straight path (think about an electric motor). By measuring this deflection as a function of the magnetic field the charge to mass ratio of the moving particles may be calculated. 4A.6 Background The apparatus is shown below. First, you need to think about the coil. How is the direction of the magnetic field, B, within the solenoid related to the current flowing through the wire? The magnitude of the magnetic field within the solenoid may be found by assuming it to be infinitely long: - B=µ0NI Experimental Physics I 63 Module PH1004 The University of Reading Department of Physics Where I is the current flowing through the coil, N is the number of turns per metre and µ0 is the permeability of free space. The direction of the force F acting on a charge q moving with a velocity v in a magnetic field B is given in vector notation by the equation below, and the deflection of the electrons will be in the direction of -F. Lenard's Method CRT Control Amps CRT Coil F = qv∧B Electron Path in Cathode Ray Tube a b R R At what position and orientation should the CRT be placed in the solenoid such that the electrons experience the maximum force and are therefore deflected to the greatest extent? From the above Experimental Physics I 64 Module PH1004 The University of Reading Department of Physics vector equation, the force F is perpendicular to the velocity vector v. Therefore the path of the electrons within the CRT will be circular, of radius R. To estimate the value of a, inspect a CRT. Since it is difficult to see all the electron gun assembly, take the value of this dimension to be 10.7±0.3 cm; assume V=470±30V. 4A.7 The Experiment You can adjust the current flowing through the solenoid, using the integrated power supply unit. Then, in calculating the magnitude of the magnetic field, B, you will need to assume that the solenoid is infinitely long, i.e. you assume that the ends are unimportant. By using the CRT as a probe, investigate the uniformity of the magnetic field along the solenoid and express your results graphically. Based on these measurements, is the above assumption reasonable and within what limits? Measure the deflection of the electron beam across the CRT screen as a function of the current flowing through the solenoid. Determine the ratio, e/m. by a suitable graphical method. Place a piece of graph paper on the CRT and using this as a graduated scale to measure the deflection. Compare your answer with the textbook value. Do the two values agree within the limits of experimental uncertainty? Experimental Physics I 65 Module PH1004 The University of Reading Department of Physics Experiment B: - Angular Momentum 4B.1 Objectives (i) To perform experiments with a rotating system. (ii) To observe the transformation of potential energy to kinetic energy. (iii) To determine whether angular momentum is conserved. 4B.2 Prior Reading O' Hanian Chapter 13: FLAP modules P2.7 and P2.8. 4B.3 Preparatory Work (i) Write down an expression relating linear velocity and angular velocity. (ii) Explain the term, ª moment of inertiaº. (iii) Derive an expression for the moment of inertia, Id, of a uniform disc of mass M and radius R. (iv) Derive an expression for moment of inertia, Ia, of a uniform annulus of mass M, inner radius Ri and outer radius Ro. 4B.4 Safety Procedures You should be careful with the rotating plate, since its edges may cause injury. 4B.5 Introduction In this experiment you will investigate the conservation of angular momentum and the energy losses in a mechanical system that includes friction. A disc is provided which can rotate about a vertical axis. In the first part of the experiment a string is wound around the hub of the disc. It then passes over a pulley and is attached to a mass. If the mass is allowed to fall it will make the disc rotate. You are asked to compare the loss of potential energy of the mass with the gain in kinetic energy of the disc. In the second part of the experiment a second disc is dropped on top of the spinning disc. By measuring the change in angular velocity the conservation of angular momentum can be investigated. The change in energy of the system is also found. 4B.6 Calculation of the Moment of Inertia of the Discs The kinetic energy of a rotating disc is Iω2/2 where I is its moment of inertia and ω its angular velocity. Use the mass values stamped on the components to calculate the moment of inertia I1 of the disc that can rotate about a vertical axis and I2 of the flat disc with masses added near its outer edge. I1 and I2 are the values about an axis through the centre of each disc and perpendicular to the plane of the disc. Will significant errors occur if you neglect the hub of the first disc and the hole in the centre of the second? 4B.7 Comparison of Potential and Kinetic Energies Tie a small loop at one end of a cord and fasten a mass to the other end. Slip the loop over the peg on the hub of the rotatable disc and pass the cord over the pulley at the edge of the bench; then rotate the disc so that the cord is coiled around the hub. Let the mass m drop through a height h and then measure the successive times for a marker on the disc to pass a fixed point. Plot a graph to show how the period of rotation varies with number of revolutions. Extrapolate back to find the effective period and hence ω at the instant the mass was released. Note: - that in this plot the points Experimental Physics I 66 Module PH1004 The University of Reading Department of Physics will be half way between the integer positions. Carry out some preliminary trials to find two suitable values of h for each of two values of m. 1 2 Compare the loss of potential energy mgh with the kinetic energy gained 2 Iω in each case. 1 2 Evaluate the kinetic energy of the falling mass 2 mv , where v is its final speed. Is this important? Are the results consistent with the suggestion that the major loss process is the work done against friction? 4B.8 Conservation of Angular Momentum Set the disc spinning at the highest rate that will enable you to measure the period of rotation, and hold the other disc horizontal with its central hole above the rotating hub. Let the second disc fall as centrally as possible and continue to record the period of rotation. By extrapolating the data points you should be able to determine the instantaneous change in angular velocity when the second disc was added. Estimate the uncertainty in the values of ω before and after the impact. Hence calculate the kinetic energy and the angular momentum before and after impact. Repeat the experiment for a significantly different value of ω.. Comment on the result. Experimental Physics I 67 Module PH1004 The University of Reading Department of Physics Project 5 Spectroscopy Experiment A: - The Hubble Redshift 5A.1 Objectives (i) To use a simulated optical telescope equipped with a camera and an electronic spectrometer to acquire astronomical data. (ii) To calculate a value for the Hubble constant, H0. (iii) To calculate the age of the universe 5A.2 Prior Reading O' Hanian Chapter 17 and Interlude II: Flap module P5.7. Chapter 3 in Hawking' s A Brief History of Time is also worth reading. 5A.3 Preparatory Work (i) Explain the meaning of the Hubble constant. (ii) In calculating the age of the universe for the Hubble constant, what must be assumed? (iii) What does the term observable universe mean? (iv) Why is the Hubble constant so influential in our models of the future universe? 5A.4 Safety Procedures The standard procedures associated with electrical equipment should be observed. 5A.5 Introduction In the early 20th century, the astronomer Vesto Slipher observed that absorption lines within the spectra of most spiral galaxies had longer wavelengths than those observed on earth from stationary objects. Assuming that the redshift was caused by the Doppler shift, Slipher concluded that the red- shifted galaxies are all moving away from us. Virtually all galaxies, with the exception of a few nearby ones, are moving away from the Milky Way. In the 1920' s, Edwin Hubble measured the distances of the galaxies for the first time and, when he plotted these distances against the velocities for each galaxy, he noted that the further a galaxy is from the Milky Way, the faster it is moving away. Since there is nothing special about our place in the universe that makes us a centre of cosmic repulsion, astrophysicists readily interpreted Hubble' s relation as evidence of a universal expansion. The distance between all galaxies in the universe is getting bigger with time, such that an observer on any galaxy, not just our own, would see all the other galaxies travelling away, with the furthest galaxies travelling the fastest. This was a remarkable discovery. The expansion is believed today to be a result of the Big Bang, which we believe occurred between 10 and 20 billion years ago, a date that we can calculate by making measurements like those of Hubble. The rate of expansion of the universe tells us how long it has been expanding. We determine the rate by plotting the velocities of galaxies against their distances, and determining the slope of the graph, a number called the Hubble constant, (H0 = 1.6⇒3.2x10-18s-1) which tells us how fast a galaxy at a given distance is receding from us. So, Hubble' s discovery of the correlation between velocity and distance is fundamental in reckoning the history of the universe. Experimental Physics I 68 Module PH1004 The University of Reading Department of Physics 5A.6 Background To determine, distance and velocity for a given galaxy spectral data is commonly used. So, for any galaxy, it is necessary to record its apparent magnitude and the wavelengths of known spectral lines; here the H and K lines of calcium are used. From these data, you can calculate the distance to the galaxy, by comparing its measured apparent magnitude to its absolute magnitude (assumed to be -22.0±0.1 for a typical galaxy), and its speed from the Doppler-shift formula. The galaxy clusters you will observe have been chosen to be at different distances from the Milky Way, giving a suitable range to see the straight line relationship Hubble first determined. The slope of the straight line will give you the value of H0, the Hubble constant, which is a measure of the rate of expansion of the universe. Once you have H0, you can then find the age of the universe. The following relationships enable you to convert your spectral data into distances and velocities: - M = m + 5 − 5log10 D Where m is the measured magnitude of the galaxy and M is the assumed absolute magnitude. D is then the distance of the galaxy in parsecs. ∆λ H vH = c. λ H Where, c is the speed of light, vH is the velocity of the galaxy based upon the wavelength shift ∆λH of the H line of calcium is (λH = 3968.847Å when there is no relative movement). Similarly, for the K line of calcium is (λK = 3933.67 Å): - ∆λ K vK = c. λ K 5A.7 The Experiment The software upon which this project is based was developed at Gettysburg College, PA, as part of their CLEA (Contemporary Laboratory Experiences in Astronomy) programme. In it, you will repeat Hubble’ s experiment using modern techniques of digital astronomy. The software effectively puts you in control of a large optical telescope equipped with a TV camera and an electronic spectrometer. Using this equipment, you will determine the distance and velocities of several galaxies located in selected clusters around the sky and, hence, evaluate H0. How does your value compare with that given in Section 5A.5 (ensure that you use the same units!)? How does the equipment work? The TV camera attached to the telescope allows you to see the galaxies, and “ steer” the telescope so that light from a galaxy is focused into the slit of the spectrometer. You can then turn on the spectrometer, which will begin to collect photons from the galaxy. The screen will show the spectrum plotted as intensity versus wavelength. When a sufficient number of photons are collected, you will be able to see distinct spectral lines from the galaxy, the H and K lines of calcium. The spectrometer also measures the apparent magnitude of the galaxy from the rate at which it receives photons from the galaxy. Using the simulated telescope, initially measure and record the wavelengths of the calcium H and K lines for one galaxy in each of the selected fields. Also, be sure to record the object name, apparent magnitude, and photon count. Use these data, together the assumed absolute magnitude for each galaxy given above to derive the distance and velocity of each galaxy. Experimental Physics I 69 Module PH1004 The University of Reading Department of Physics If you have time, after completing the necessary data analysis, you can return to each field to study more galaxies. 5A.7.1 Using the Hubble Redshift Program The Hubble Redshift program is stored in the c:\hubble directory in your computer; double-click on the Hubble icon to begin. When the CLEA logo appears, click on Log in the MENU BAR and enter your names. Click OK when ready; the title screen of exercise will then appear. In the MENU BAR, the only available (bold-faced) choices are to Start or to Quit, click on Start. Initially, the screen shows the control panel and view window as found in the ª warm roomº at the observatory. Notice that the dome is closed and tracking status is off. To begin to observe the sky, first open the dome by clicking on the dome button. The dome opens and the view you see is from the finder scope, which is mounted on the side of the main telescope and points in the same direction. Because the field of view of the finder scope is much larger than the field of view of the main instrument, it is used to locate objects for analysis. Locate the Monitor button on the control panel and note its status, i.e. finder scope. Also note that the stars are drifting in the view window. This is due to the rotation of the earth and is very noticeable under the high magnification of both the finder and main instruments. In order to have the telescope keep an object centred over the spectrometer opening (slit) to collect data, turn on the drive control motors on the telescope by clicking on the tracking button. The telescope will now track in sync. with the stars. Before you can collect data, you will need to select a field of view and then select an object to study. To see the fields of study, click once on the change field in the MENU BAR at the top of the control panels. To see how the telescope works, change the field of view to Ursa Major II at right ascension (RA) 11hour 0min and declination (DEC) 56degrees 48min. Notice the telescope slew' s (moves rapidly) to the RA and DEC co-ordinates you have selected. The view window shows a portion of the sky and has two magnifications: - Finder View is the view through the finder scope that gives a wide field of view and has a cross hair and outline of the instrument field of view. Spectrometer View is the view from the main telescope with red lines that show the position of the slit of the spectrometer. As in any image of the night sky, stars and galaxies are visible in the view window. It is easy to recognise bright galaxies in this simulation, since the shapes of the brighter galaxies are clearly different from the dot-like images of stars. However, faint, distant galaxies can look similar to like stars when their shape is not well resolved. Now, change fields by clicking on Ursa Major I in the lists of selected galaxies, to highlight the field. Then click the OK button. Locate the Monitor button in the lower left-hand portion of the screen. Click on this button to change the view from the Finder Scope to the Spectrometer. Using the Spectrometer view, carefully position the slit directly over the object you intend to use to collect data. Do this by slewing, or moving, the telescope with the mouse and the N, S, E or W buttons. Place the arrow on the N button and click on the left mouse button. To move continuously, press and hold down the left mouse button. Notice the red light comes on to indicate the telescope is slewing in that direction. You can adjust the speed or Experimental Physics I 70 Module PH1004 The University of Reading Department of Physics slew rate of the telescope by using the mouse to press the slew rate button. (1 is the slowest and 16 is the fastest). When you have positioned the galaxy accurately over the slit, click on the take reading button to the right of the view screen. The more light you get into your spectrometer, the stronger the signal it will detect, and the shorter will be the time required to get a useable spectrum. Galaxies have internal structure and the light intensity varies from place to place. If you position the slit completely off the galaxy, you will just get a spectrum of the sky, which will be mostly random noise. 5A.7.2 Spectral Data Collection To initiate data collection, press start/resume count and, to check progress, click the stop count button. The computer will then plot the spectrum using the available data. Clicking the stop count button also places the cursor in the measurement mode; using the mouse, place the arrow anywhere on the spectrum, press and hold the left mouse button. The arrow changes to a cross hair and all the necessary data will appear in the lower right area of the window. Object: - the name of the object being studied. Apparent magnitude: - the visual magnitude of the object. Photon count: - the total number of photons collected so far, and the average number per pixel. Integration (seconds): - the number of seconds it took to collect data. Wavelength in angstroms: - as read by the cursor in the measurement mode. Intensity: - relative intensity of light from the galaxy at the position marked by the cursor in the measurement mode. Signal-to-noise Ratio: a measure of the quality of the data taken to distinguish the H and K lines of calcium from the noise. Collect enough photons (usually around 100,000) to determine the wavelength of the line accurately and try to get a signal-to-noise ratio in excess of 10 to 1. How does it affect your results if these criteria are either not met or, alternatively, greatly exceeded? To collect data for additional galaxies, press Return and change the monitor to display. Experimental Physics I 71 Module PH1004 The University of Reading Department of Physics Experiment B: - Atomic Spectroscopy 5B.1 Objectives (i) To align a high resolution grating spectrometer and to calibrate it. (ii) To examine the emission spectrum of hydrogen. (iii) To determine the quantised energy levels of hydrogen and to evaluate the Rydberg constant 5B.2 Prior Reading O' Hanian Chapter 43: FLAP modules P8.2 and P8.3. 5B.3 Preparatory Work (i) Derive the grating equation: - nλ = d sinθ (ii) One useful characteristic of a spectrometer is its resolving power. Explain the meaning of this term (iii) Describe briefly the Bohr Model of the hydrogen atom. (iv) Show for the hydrogen system that the frequency of an emitted photon can be 1 1 written: - = Rc - ν { 2 2} n f ni Where the nomenclature is as specified in the following text. 5B.4 Safety Procedures You should be careful with light sources since these operate at high voltages, have hot surfaces and in certain cases emit light in the UV. You should avoid staring directly at the lamps. 5B.5 Introduction In this experiment you will study the emission spectrum of hydrogen and hence determine the electronic energy levels. You will use a spectrometer based around a diffraction grating which you will calibrate using the known wavelengths of light from a mercury vapour lamp. You will then use the same spectrometer to measure the optical wavelengths of the lines in the hydrogen spectrum, and go on to interpret these wavelengths in terms of the energy levels of the hydrogen atom. It is essential to adjust the spectrometer very carefully, because all the measurements must be made with high precision. 5B.6 Background 5B.6.1 Atomic Structure The light emitted by atoms may be characterised by a set of spectral lines, which are specific to each kind of atom. The early theories of atomic structure were based on classical mechanics and floundered on their inability to explain these characteristic spectral lines. However, in 1913 Bohr proposed a semi-classical model in which, for hydrogen, electrons could only occupy discrete energy levels, which are given by: - hRc E = - n2 In this, n is the so-called angular momentum quantum number, which takes values of 1, 2, 3..., h is Planck©s constant, c is the velocity of light and R is the Rydberg constant. You will find physicists Experimental Physics I 72 Module PH1004 The University of Reading Department of Physics often use units in seemingly strange manner, for example common units for energy are J, eV, cm-1; here the units of energy are m-1. The Rydberg constant itself is given by: - 4 me e R = 2 3 8ε0 h c Where me is the mass of an electron and εo is the electric constant. Here the units of the Rydberg constant are J. When an electron changes its orbital state, the difference between the energy levels associated with those orbits is released as a quantum of light with a frequency ν given by: - ν = ∆E / h The frequency ν, and hence the wavelength λ, of each spectral line is given by: - [Ei - E f ] ν = h Where Ei and Ef are the energies of the initial and final states involved in the transition. Thus for the hydrogen system: - 1 1 = Rc - ν { 2 2} n f ni Where ni and nf are the quantum numbers of the initial and final states. The units of frequency here are Hz. If we are able to observe these spectral lines experimentally then interpretation requires assignment of values for ni and nf. 5B.6.2 Spectroscopy The diffraction grating is characterised by the spacing, d, of the lines in the grating. Typically for a visible light spectrometer there will be ~600 lines per millimetre. If light is incident normally on the diffraction grating then maxima in the diffraction pattern will be observed when: - nλ = d sinθ Where n is the diffraction order, λ is the diffracted wavelength and θ is the diffraction angle (i.e. between the incident beam and the diffracted beam). Useful constants are: - h=6.626x10-34 Js; 1eV = 1.602x10-19J 5B.7 The Experiment The experiment involves three stages; checking the alignment of the spectrometer, calibrating the spectrometer and using the spectrometer to study the emission spectrum of hydrogen. You should find the spectrometer aligned reasonably correctly and therefore you should aim to spend the majority of the time on the second and third stages of the project. 5B.7.1 Alignment of the Spectrometer The spectrometer used in these experiments enables you to measure the angles between the incident light beam and the diffracted beam. It is vital that the spectrometer is aligned correctly; otherwise Experimental Physics I 73 Module PH1004 The University of Reading Department of Physics the angles measured will be incorrect. Before starting the alignment checks, you should familiarise yourself with the locks on the rotation of the grating table and the telescope arm. All adjustments have to be referred to the axis of rotation of the instrument. The collimator must provide a parallel beam of light travelling perpendicular to the axis of rotation and perpendicular to the grating surface. The telescope must also be aligned perpendicular to the axis of rotation and that the telescope lies in a plane which is perpendicular to that axis. If you are in doubt ask one the Laboratory Team. Mercury Emission Lines Colour Intensity Wavelength / nm purple medium 404.66 purple weak 407.78 blue strong 435.83 green weak 491.60 green strong 546.07 yellow strong 576.96 yellow strong 579.01 You should check by visual inspection that the plane of the grating is parallel to the axis of rotation Do not make any adjustments to the table or the telescope. You should ensure that the telescope is adjusted for parallel light. Adjust the telescope eyepiece so that the cross-wires are sharply in focus. Take the spectrometer to a window, look through the telescope at a very distant object and adjust the telescope lens until the object is focused on the cross-wires. If you or your partner differs in short or long-sightedness you may independently adjust the eyepiece to focus the cross-wires. You now need to align the collimator and set the slit system. Position the telescope opposite the collimator and obtain an image of the collimator slit. Adjust the collimator lens until the slit in sharp focus at the cross-wires. The collimator is now set to produce a parallel beam of light at the grating when the slits are illuminated. If possible with your spectrometer, illuminate the collimator slit and reduce its height to produce a point source of light as viewed in the telescope. If your instrument does not have this facility, obtain a point source by masking the slit with tape. Now adjust the inclination of the collimator until the image of the point source lies at the centre of the field of view in the telescope eyepiece. At this stage the collimator will be parallel to the telescope. Illuminate the slit of the collimator with the mercury vapour lamp. Ensure that all the spectral lines on either side of the central image are at the same height in the field of view of the telescope. If necessary, make a small rotation of the grating about a horizontal axis. The final step is to ensure that the grating (at zero angle) is perpendicular to the incident beam. Set the telescope to be at right angles to the collimator. Using the mounted grating as a mirror, obtain an image of the collimator slit; the grating will then be at 450 to the directions of the telescope and Experimental Physics I 74 Module PH1004 The University of Reading Department of Physics the collimator. Hence set the diffraction grating so that it is exactly perpendicular to the light emerging from the collimator. 5B.7.2 Calibration of the Spectrometer In order to make use of the spectrometer you need to know the distance between the lines in the grating, d. This is achieved by determining the diffraction condition for light of known wavelength. Using the mercury lamp measure the diffraction angles, θ, for more than one order of diffraction, n, for each of the strong lines in the mercury spectrum. You should find that particular spectral lines lay symmetrically either side of the straight through or zero order position. The wavelengths of the lines in the mercury spectrum are given in the table. Present your results graphically and hence obtain an accurate value for the grating spacing. Make an estimate of the resolving power of the spectrometer by observing the separation of the doublet in the emission spectrum of a sodium lamp in which the wavelengths are 589.0nm and 589.6nm. 5B.7.3 Emission Spectrum of Hydrogen Set the spectrometer up with a hydrogen lamp and determine the wavelengths of at least three or, if possible, four lines of the hydrogen spectrum. The lines at the blue end of the spectrum become less distinct as the lamp gets hot so measure these first. Do not leave the lamp running when it is not needed. You may assume that all the spectral lines you see have a common value for nf and that the red line involves: - (ni - nf) = 1 The blue-green line: - (ni - nf) = 2 etc Try to fit your observed frequencies to the equation given in section 5B.6.1, and deduce the values for nf for the lower state levels. You can do this graphically with guessed values of nf=1, 2, 3 etc. If you have been careful with your measurements you should be able to deduce the correct nf unambiguously. Hence evaluate the Rydberg constant. Draw an energy level diagram for hydrogen, putting on it a scale of energy in electron volts. Also show the transitions that you have observed. Calculate the ionisation energy of atomic hydrogen. Experimental Physics I 75 Module PH1004 The University of Reading Department of Physics Electronics 1: - Operational Amplifiers – Theory Basic Principles Amplification ± Take a small signal and increase it' s amplitude ª Black Boxº diagram: Vin Amplifier Vout Suppose now we ª feed backº some of the output signal, and subtract it from the input. In effect, as we will see shortly, the gain is reduced. Isn' t this a rather odd thing to do? Advantages of ª negative feedbackº: • Reduces distortion and non-linearities • Controllable frequency response • Predictability • Properties become dependent on the feedback circuit, not the amplifier A basic negative feedback system: Define: ª Naturalº (or ª open-loopº) gain of the amplifier = A Fraction of output fed back to input = B Vin V1 A Vout + − B The signal going into the amplifer is: V1 = Vin − BVout The output of the amplifier section is simply: V = AV out 1 Hence Vout = A(Vin − BVout ) We can therefore write the output of the amplifier plus feedback as: A V = V out 1+ AB in and the resultant (or ª closed-loopº) gain, G, is: A G = 1+ AB To emphasise the terminology: A = open-loop gain; G = closed-loop gain; AB = loop gain; (1+AB) = return difference or desensitivity Experimental Physics I 76 Module PH1004 The University of Reading Department of Physics We can now see the effect of the negative feedback on, for example, the frequency response of the amplifier. Suppose at 1 kHz, the amplifier has an open loop gain, A1k = 10,000 but at 10 kHz, it only has and open-loop gain A10k = 1,000. Clearly, a factor of ten variation is generally very undesirable. Suppose we insert a feedback loop with B = 0.1 At 1 kHz: G = 10,000/(1+1,000) = 9.99 At 10 kHz: G = 1,000/(1+100) = 9.90 Although we have reduced the gain substantially, the variation in gain is now only 1% (Note ± We could recover the gain of 1000 simply by putting three such amplifiers in series) Practical Realisation – The Operational Amplifier (op-amp) An operational amplifier is a device with two inputs, a single output and typical open-loop gains of 105 - 106 Symbol: + Output í + denotes ª non-inverting input − denotes "inverting" input Action of output as a function of input: Non-inverting input goes more positive (negative), output goes more positive (negative) Inverting input goes more positive (negative), output goes more negative (positive) The output is proportional to the difference between the two inputs The “ Golden Rules” of Op-Amp Behaviour Since the open-loop gain is so high, a difference of less than a millivolt between the inputs can swing the output through it' s full range. Hence, if we have any negative feedback, the output will tend to make the difference between the inputs vanishingly small. We can therefore write the first Golden Rule as: 1. The output attempts to do whatever is necessary to make the difference between the inputs zero The input resistance (or impedance) of typical op-amps is very high, hence very little current is drawn (typically nA, or even pA) Hence, the second Golden Rule is: 2. The inputs draw no current Using these two rules, we can understand an enormous amount of op-amp behaviour, and design many useful circuit elements (Important note ± Rule 1 does not imply that the op-amp changes the input voltages itself ± rather it is through the external feedback network that this is achieved) Another consideration is that the open-loop gain is so high, that we may regard it as infinite. Then, the expression we derived for the closed-loop gain, G becomes: 1 G = B Experimental Physics I 77 Module PH1004 The University of Reading Department of Physics Basic Op-Amp Circuits 1, Non-Inverting Amplifier A basic non-inverting amplifier is shown below. What is the gain of this circuit, and how can we calculate it? Vin + Vout V- − R2 R1 We can do this in two ways: Calculate the B factor for the feedback network, or apply the Golden Rules. B factor approach ± remember this is the fraction of the output fed back to the input: We can see that the feedback network is a simple potential divider, and: R1 V− = Vout R1 + R 2 R Hence, B = 1 R1 + R 2 R + R R and the closed loop gain is therefore; G = 1 2 =1+ 2 R1 R1 Golden Rules approach: Applying Rule 1: The difference between the inputs is zero: Vin = V- But, V- is derived from a potential divider, and is given by R1 V− = Vout R1 + R 2 Hence, R1 Vin = Vout R1 + R 2 and the gain is therefore: V R + R R G = out = 1 2 =1+ 2 Vin R1 R1 as was obtained before. It is frequently advantageous to apply the Golden Rules, as this gives more insight into the operation of the circuit. Experimental Physics I 78 Module PH1004 The University of Reading Department of Physics Inverting Amplifier This application illustrates nicely the use of both Golden Rules. R2 R1 Vin − Vout + 1, Since the non-inverting input is grounded (at zero volts), then, by Rule 1, the inverting input must also be at zero volts (this is the ª virtual ground approximationº) This means the voltage across R1 is Vin, and the voltage across R2 is Vout 2, Since there is no current flow at the inputs (Rule 2), the current through R1 must be equal and opposite to that through R2. V in V out i.e., by Ohm' s Law: = − R 1 R 2 Hence, the gain, G is given by: V R G = out = − 2 Vin R1 Examples: 1, A non-inverting amplifier with gain = 100 Vin + Vout V- − R2 R1 Gain = 1+R2/R1 = 100 Hence, R2 = 99R1 With R1 = 2k, R2 = 198k (note ± there are standard resistor values, and for most purposes, the closest to that required for R2 is 200k. The final gain will therefore be 101) Experimental Physics I 79 Module PH1004 The University of Reading Department of Physics 2, Voltage follower (buffer) Vin + Vout V- − This is a non-inverting amplifier with R1 = ∞ and R2 = 0 Then, gain = 1. This circuit is used where it is necessary to isolate Vin from any load that there might be on Vout. Other Feedback Networks We are not restricted to using just resistors in the feedback loop. We can put components whose ª resistanceº (strictly speaking, impedance) is a function of frequency or time, allowing us to shape the frequency response, or do analogue computing! Example 1: In the inverting amplifier seen previously, suppose we replace the feedback resistor, R2, with a capacitor, C C R Vin − Vout + Recall that the current flowing in a capacitor is time dependent, and is given by: dV i = C c dt Recall also that, since we will apply the virtual ground approximation, Vin dVout = −i = −C (since the voltage across the capacitor is always Vout) R c dt Hence, Vout is a function of time, and is given by: 1 V = − V dt + constant out RC ∫ in In other words, this circuit is an integrator, and the output waveform is simply proportional to the integral (or cumulative sum) of the input waveform. Experimental Physics I 80 Module PH1004 The University of Reading Department of Physics Example 2: Suppose we now replace R1 with a capacitor, leaving the feedback resistor in place. R Vin − C Vout + Applying the virtual ground approximation again: dV V C in = − out dt R dV So, V = −RC in out dt In other words, the output is proportional to the differential (or gradient) of the input. The circuit is a differentiator. Experimental Physics I 81 Module PH1004 The University of Reading Department of Physics Electronics 2: - Operational Amplifiers – Practice In this practical session, you will be constructing op-amp circuits using 741 op-amps (Appendix A) and standard resistors and capacitors. You will be building the circuits on breadboards (Appendix B). The first circuit you will build will demonstrate the validity of the calculations discussed in the theory session and exercises, as well as showing some of the limitations of the ideal approximations. This experiment tests your basic competence and successful completion will carry a maximum of 40% towards this practical session. The second experiment is more advanced and will involve you in designing a circuit for a specific application. This experiment tests your advanced ability and successful completion will carry a maximum of 60 % towards this practical session. The apparatus you have is: • A breadboard with integral power supplies • A signal generator • An oscilloscope • All the components you need (in drawers on the side bench) When you are building the circuits, take care in ensuring the op-amps are connected correctly before applying power (you should use ±15 V). If in doubt, get a colleague, Dr Hatherly or Dr Waterman to check. Experimental Physics I 82 Module PH1004 The University of Reading Department of Physics 1, (Basic Experiment) Signal amplification and departures from the ideal op-amp Design and build a non-inverting amplifier with gain = 20, using standard components. Apply a sine wave of frequency 100 Hz and amplitude 0.1 V to the input, and observe the output on the oscilloscope. Does your observation confirm your calculation? Now increase the amplitude of the sine wave by 0.1 V steps up to 1 V. In each case, sketch the output waveform. What do you observe? Can you account for this behaviour? Now , resetting the amplitude back to 0.1 V, increase the frequency of the sine wave to 300 Hz, 1 kHz, 3 kHz, 10 kHz, 30 kHz, 100 kHz, 300 kHz and 1 MHz (the values have been chosen to be ½ an order of magnitude intervals). In each case, measure the gain. Do your observations agree with the ideal calculation? If not, why not? 2, (Advanced Experiment) Waveform Manipulation A particular application requires a triangular wave of amplitude 5 V and frequency 200 Hz. Design, build and test a circuit to generate such a wave from a square wave, making appropriate choices of input frequency, amplitude and components. What happens to the output amplitude if you vary the input frequency from the design value of 200 Hz? Experimental Physics I 83 Module PH1004 The University of Reading Department of Physics Electronics 3: - Digital Electronics – Theory An Introduction to Logic Unlike analogue electronics, where the inputs and outputs of a circuit can have a continuum of values, digital electronics has, in general, only two states - often referred to as 1 (high) or 0 (low). How these digital values are handled is rooted in the manipulation of logic, or Boolean algebra. Like normal algebra, we have a number of binary operations, such as addition (eg, x+y) and multiplication (xy), although the Boolean operations are rather different and have very different outcomes. Any Boolean operation involving combining values (combinational logic) can be implemented in digital electronics by the use of gates. The basic Boolean operations are summarised below, together with the standard circuit symbol for the relevant gate, and a truth table, which summarises the outcome of combining different input values OR gate: Truth Table: Circuit Symbol A Boolean symbol: A + B = Q Q A B Q B 0 0 0 Summary: Output is high if any of the inputs are high 1 0 1 0 1 1 1 1 1 AND gate Truth Table Circuit symbol: A Boolean symbol: AB = Q A B Q Q B 0 0 0 (note A*B = Q may also be used) 1 0 0 0 1 0 Summary: The output is high only if both inputs are high 1 1 1 Inverter (NOT gate) Truth Table Circuit symbol: %RROHDQV\PERO $¶ 4 RU 4 A Q A Q 0 1 1 0 Summary: The output is the logical complement (or opposite) of the input Experimental Physics I 84 Module PH1004 The University of Reading Department of Physics The NOT function is frequently incorporated into the AND and OR gates, giving NAND and NOR gates: NOR: NAND: A A Q Q B B A B Q A B Q 0 0 1 0 0 1 1 0 0 1 0 1 0 1 0 0 1 1 1 1 0 1 1 0 One other gate is frequently used, but is not as fundamental as the above: Exclusive Or (XOR) Truth Table: A Circuit Symbol: Q A B Q B 0 0 0 Boolean Symbol: A ⊕ B = Q (note – you may see different symbols used for 1 0 1 this gate and operation) 0 1 1 1 1 0 Summary: The output is high only if either of the inputs is high, but not both Boolean Algebra Like conventional algebra, Boolean algebra has many rules, theorems and identities which make life simpler. Many of the more obvious and useful ones are summarised below: Two operands More than two operands 1’ = 0 and 0’ = 1 ABC = A (BC) = (AB)C AB = BA A + B + C = A + (B + C) = (A + B) + C A + B = B + A A(B + C) = AB + AC AA = A A + BC = (A + B)(A + C) A1 = 1 and A0 = 0 A + A = A De Morgan’ s Theorem: A + 1 = 1 and A + 0 = A (A + B)’ = A’ B’ A + A’ = 1 (AB)’ = A’ + B’ AA’ = 0 Examples: Show that: A + BC = (A + B)(A + C) First, expand out the brackets: (A + B)(A + C) = AA + AC + AB + BC Now use AA = A: = A + AC + AB + BC Using AB + AC = A(B + C): = A + A(B + C) + BC And again… = A(1 + (B + C)) + BC Use A + 1 = 1 on (1 + (B + C)): = A + BC QED Experimental Physics I 85 Module PH1004 The University of Reading Department of Physics Deduce that a Boolean expression for the XOR function is: A ⊕ B = A’B + AB’ From the truth table shown earlier, the XOR function is true only when A = 1 and B = 0 or vice versa. In other words when (A = 1 AND B' = 1) OR (A' = 1 AND B = 1). Combining these statements together gives the desired result. Now, draw a reasonable implementation, using standard gates, of an XOR gate. The set of gates below will satisfy the above expression A Q B Note that this implementation is not unique – algebraic manipulation can yield many different forms. Implementing Logic Functions: Karnaugh Maps We will frequently encounter situations where we know the input states of a system and the outputs we desire. The question now is; how do we produce a Boolean expression (and hence implement a logic circuit)? There are many (mainly computational) techniques, but for fairly simple expressions and circuits, with up to four inputs, the Karnaugh Map method is very convenient. It is best to illustrate this with some examples: 1, A NAND gate (a very simple example) Step 1 – Write out a truth table (this has been done earlier) Step 2 – Make the Karnaugh map. This is similar to the truth table, except the inputs are along two axes, with the outputs arranged on a grid: B\A 0 1 0 1 1 1 1 0 Now identify groups of 1, 2, 4, 8… 1’ s: B\A 0 1 0 1 1 This group is 1 if B’ (NOT B) 1 1 0 This group is 1 if A’ (NOT A) Experimental Physics I 86 Module PH1004 The University of Reading Department of Physics The groups you have identified are combined using the OR function: Hence, Q = A' + B' (Notice that we have demonstrated DeMorgan' s Theorem) 2, A Vote Validator Suppose we have an election in which there are three candidates, of which voters should vote for at most two, but not zero. Generate a Boolean expression and design a logic circuit to carry out this check automatically, and reject votes with zero or three votes cast. 1, The Truth Table A B C Q 0 0 0 0 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1 0 1 1 1 1 0 1 1 1 1 0 2, Write the Karnaugh Map ± Arrange the axes so that only one input bit changes going from one square to the next C \ AB 00 01 11 10 0 0 1 1 1 1 1 1 0 1 3, Now identify groups of 1' s (remember, groups of 1, 2, 4,8 etc): C \ AB 00 01 11 10 0 0 1 1 1 1 1 1 0 1 We have identified groups where: C = 1 and A = 0 (B doesn' t matter), which is therefore CA' , C = 0 and B = 1 (A doesn' t matter), which is BC' , and A = 1 and B = 0 (C doesn' t matter) which is AB' . Therefore, our final function is: Q = CA' + BC' + AB' Experimental Physics I 87 Module PH1004 The University of Reading Department of Physics Notice that this isn't a unique solution. There are a number of ways of grouping the 1' s. In particular, we note that there are only a couple of zeros, hence we can create a large group of 1' s with a few squares excluded (ie, we can group zeros as well as 1' s): C \ AB 00 01 11 10 0 0 1 1 1 1 1 1 0 1 This arrangement represents: All squares = 1 (hence Q = 1), and we exclude ABC (ie, all 1' s) and A' B' C' (ie, all zeros). Hence our function is: 1(ABC + A' B' C' )' = (ABC + A' B' C' )' We have therefore reduced our expression to only two terms, at the expense of each term being a multiple rather than binary operation. Another point to note with Karnaugh maps is that their edges connect (or wrap around). Eg, consider the map below: AB \CD 00 01 11 10 00 0 0 0 0 01 1 0 0 1 11 1 0 0 1 10 0 0 0 0 We can group the 1's as two groups of two (representing BC' D' and BCD' ), or as one group with the left hand edge wrapping round to the right hand edge: BD' Experimental Physics I 88 Module PH1004 The University of Reading Department of Physics Binary Numbers In all of the logic we' ve discussed, we' ve taken the inputs to be independent variables. However, they may also be ª digitsº in a binary number. Counting in binary is quite simple ± it is merely base 2, so the only digits available are 1 and 0. In base 10, we are used to the right-hand most digit representing 1' s (100) the next left digit 10' s (101), the next 100' s (102) etc. In binary, the right-hand most digit again represents 1' s (now 20), but the next left is 2' s the next 4' s (22) etc. Hence, the binary number 110 is 1*22 + 1*21 + 0*20 = 6. Notice that the maximum number that can be represented by a 3-digit binary number is 7 (111), which is equal to 8-1, or 23-1. In general, the maximum number that can be represented is 2n-1 where n is the number of binary digits. The correspondence between 4-bit binary numbers and their unsigned decimal equivalents is shown on the left: Some terminology: Right-hand most bit is the Least Significant Bit (LSB), the left-hand most bit is the Most Significant Bit (MSB). B inary Decimal 0000 0 0001 1 0010 2 0011 3 0100 4 0101 5 0110 6 0111 7 1000 8 1001 9 1010 10 1011 11 1100 12 1101 13 1110 14 1111 15 Experimental Physics I 89 Module PH1004 The University of Reading Department of Physics Electronics 4: - Digital Electronics – Practice In this practical session, you will be constructing combinational logic circuits using 74xx series TTL ICs (Appendix C). As before, you will be building the circuits on breadboards. The first experiment will demonstrate the principles of simple logic circuits and their operation. This experiment tests your basic competence and successful completion will carry a maximum of 40% towards this practical session. The second experiment is more advanced and will involve you in designing a circuit for a specific application. This experiment tests your advanced ability and successful completion will carry a maximum of 60 % towards this practical session. The apparatus you have is: • A breadboard with integral power supplies • All the components you need (in drawers on the side bench) When you are building the circuits, take care in ensuring the ICs are connected correctly before applying power (you should use +5 V and ground). If in doubt, get a colleague, Dr Hatherly or Dr Waterman to check. Experimental Physics I 90 Module PH1004 The University of Reading Department of Physics 1, (Basic Experiment) Simple Logic Gates Confirm the truth tables for AND, OR and NOT gates, using the 7408, 7432 and 7404 ICs respectively. Construct a circuit to produce an XOR gate, and confirm you have achieved this by obtaining its truth table. Construct the circuit shown below. From inspecting its truth table, what is its function? A B Q C D 2, (Advanced Experiment) Guilty or Not Guilty? In Logicland, trials are held in front of juries of 4. A guilty (logic 1) verdict is returned by either a majority or unanimous vote. If less than three jurors record a guilty vote, a not guilty (logic 0) verdict is returned. Jurors record their votes by secretly and simultaneously pressing buttons for guilty, but not doing so for not guilty. The judge should merely see the final verdict. Devise, build and test a logic circuit to indicate to the judge the verdict arrived at. Experimental Physics I 91 Module PH1004 The University of Reading Department of Physics The Final Project Objectives (i) To provide an opportunity for you to put into practise the skills developed in this module. (ii) To plan an experimental project and to execute it. Prior Reading None - but you should be prepared for anything. Safety Procedure Any specific safety procedures associated with this project will be detailed on the project sheet given to you at the start of the final laboratory session. Introduction You will be given the details of this project at the start of the laboratory session. There will be a defined objective for the project. You and your partner will need to consider carefully the various possible approaches and prepare a plan of your experiments. You will have 30 minutes to formulate this plan and enter it into your laboratory logbook. The remaining 140 minutes will be available for you to complete your planned project. At the end of the laboratory session you must hand in your notebook; i.e., all of the work must be completed during the session. Note. In terms of the assessment of this module, the final project carries the same mark as one of the normal three-week projects and cannot be dropped in the calculation of the final grade. Experimental Physics I 92 Module PH1004 The University of Reading Department of Physics Appendix A – Real Op-Amps In the Undergraduate Laboratory, two op-amps are used, interchangeably. They are devices known as the 741 and TL071, and are contained in 8-pin DIP (Dual In-line Package) integrated circuits. Each pin has a specific purpose, and the pin connections are identical for both devices, as shown below (note ± view from above). Notice that pin 1 is designated either by a dot (either painted on the device or as a small pit) or by a notch. (frequently, both!). Notice also that the pins are numbered anticlockwise from the top left. offset 1 8 no connection inverting 2 - 7 supply V+ + non-inverting 3 6 output supply V- (-15V) 4 5 offset null Some typical parameters Supply Input Input offset Open loop Bandwidth Slew rate (V) Resistance (Ω) (mV) gain (MHz) (V/µs) 741 ±5-18 2M 6 2x105 1 0.5 TL071 ±2-15 1012 2 105 3 13 The pins labelled ª offset nullº are used to correct the small offset which occurs between the inputs and which arises from manufacturing tolerances. For the purposes of the circuits we are dealing with, these are not used. You may leave them ª floatingº (ie, unconnected) when you wire a circuit. Experimental Physics I 93 Module PH1004 The University of Reading Department of Physics Appendix B – “ Bread boards” In the practical component of the electronics laboratory, you will be required to construct and test circuits. Clearly, it would be time-consuming and wasteful to construct permanent soldered circuits, so we will use re-useable prototyping boards. (the common name ª breadboardsº comes from their appearance ± they are typically white, and full of holes!) A typical breadboard is illustrated below: The ª holesº are designed to take standard electronic component leads, which are simply pressed in, and are on a 0.1 inch matrix, which is the industry standard for integrated circuit pin spacings. The holes are wired together underneath in the manner shown below: (note only a small section of the breadboard is shown) Important points: 1. The rails at the top and bottom of the boards are connected horizontally and are usually used for power supply lines 2. These horizontal rails typically have a gap half way along the board 3. The rails in the centre of the board are connected vertically in groups of 5 4. The central rails have a gap in the middle Experimental Physics I 94 Module PH1004 The University of Reading Department of Physics The manner in which an op-amp (or indeed any other ic in a DIP) is mounted is shown on the left. Note that the ic straddles the central insulated gap (so that pins 1 and 8, 2 and 7 etc are not shorted) and that up to 4 connections can be made to each pin (further connections can obviously be made by taking a wire to another, free, rail. A few general hints and tips for successful “ breadboarding” • It is strongly recommended that you familiarise yourself with the way the breadboard is wired before you start assembling your circuits. • Plan carefully the layout of your components. A few minutes thought and planning at the beginning is worth an hour of frustration while you try to solve a problem. • Keep interconnecting wires short. Not only will this give your circuit a neater appearance and help in tracing faults, but also you will reduce noise and pickup (especially true in high- gain circuits) Experimental Physics I 95 Module PH1004 The University of Reading Department of Physics Appendix C – Logic Integrated Circuits As with the op-amps discussed previously, many basic (and complex!) logic functions are available as integrated circuits. In the Undergraduate Laboratory, three types of IC are available, performing the AND, OR and NOT logic functions. These belong to the 7400 family of TTL (Transistor-transistor logic) circuits and are contained in 14-pin DIP integrated circuits. The pin designations for each IC are shown below: Important notes on operation: 7408 ± Quad 2-input 7432 ± Quad 2-input OR AND • Operate at Vcc= 5V and GND =0V (VCC Max 5.25V Min 4.75V) • Input pins should not be set to rise above Vcc • Unused inputs should not be left 'floating', they should be held either 'High' '1' or 'Low' '0'. 7404 ± Hex inverter Experimental Physics I 96 Module PH1004 The University of Reading Department of Physics Appendix D – “ Sensing” Logic Levels In the circuits you will be constructing, it will be of great value to ª senseº the logic state of the outputs. Of course, you could look on a multimeter or on an oscilloscope, but far more convenient is to connect light emitting diodes (LEDs) to the output, to give a direct visual indication of the logic state (on = 1, off = 0). The output from a TTL IC will typically be about 5V (Vcc ± see Appendix A). Since LEDs cannot take a large current (~20 mA max), it is necessary to limit the current by, for example, putting a 220Ω resistor in series. LED Symbol: LED Appearance: Side: Base: Note that the current flows from anode to cathode (left to right in all the above diagrams). Note that the cathode is usually denoted by a short lead and/or a flat on the LED body. The LEDs should be wired in the following way: 220 Ω +5V (Logic Output) Experimental Physics I 97 Module PH1004